Growth and applications of photorefractive potassium lithium tantalate niobate (KLTN) - CaltechTHESIS

Growth and applications of photorefractive potassium lithium tantalate niobate (KLTN) - CaltechTHESIS
CaltechTHESIS
A Caltech Library Service
About
Browse
Deposit an Item
Instructions for Students
Growth and applications of photorefractive potassium lithium tantalate niobate (KLTN)
Citation
Hofmeister, Rudolf
(1993)
Growth and applications of photorefractive potassium lithium tantalate niobate (KLTN).
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/zm1c-8w92.
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
This thesis describes the growth of photorefractive potassium lithium tantalate niobate (KLTN) single crystal material and characterization of its physical and photorefractive properties. The band transport model is used to discuss the conventional photorefractive effect. The coupled mode formalism is introduced to determine the interaction of interfering light beams in a photorefractive material. Solutions for intensity coupling and phase coupling between two beams, as well as diffraction off a dynamic index grating, are presented for both the copropagating and counterpropagating experimental geometries. These solutions are obtained for arbitrary photorefractive phase, [...]. The linear- and quadratic electro-optic effects are discussed. The influence of electric field application on the electro-optic tensor is described.
The top seeded solution growth method is reviewed. The design and construction of a crystal growth system is described. The growth procedures of KLTN are enumerated for several compositions and dopant types. Phase diagrams of the KLTN system are determined. Structural properties of the grown crystals are presented. Certain material characteristics of KLTN are discussed. These include the phase transition temperatures, dielectric properties, and the optical absorption properties.
Electric field control of the photorefractive effect, beam coupling and diffraction, is demonstrated for paraelectric KLTN. A theory is developed to describe the diffraction of beams off photorefractive index gratings in paraelectric KLTN. The solutions of the coupled mode equations are used to develop methods of determining the photorefractive phase [...] in a photorefractive material. These methods are experimentally demonstrated for several types of photorefractive material. In addition, they are used to corroborate a theory describing the magnitude and phase of the net holographic grating in paraelectric KLTN under applied electric field.
A new effect, the Zero External Field Photorefractive (ZEFPR) effect is studied, as well as the application of its unique zero phase ([...] = 0) photorefractive gratings. The ZEFPR effect is forbidden by the conventional photorefractive theory; its origin is shown to be due to the creation of strain gratings under spatially periodic illumination. A theory of coordination of microscopic strains by a macroscopic (growth induced) strain is presented. The ZEFPR gratings are shown to possess identically zero phase when no external electric field is applied. This property is employed in the implementation of various new linear phase-to-intensity transduction devices. In particular, an all-optical phase modulation/vibration sensor (microphone) is described. This device is expected to have numerous applications in environments where electric fields cannot be permitted. The possible implementation of ZEFPR gratings in high speed self aligning interferometric data links is discussed, as well as implementation of a novel self aligning holographic image subtraction device.
The final chapter is devoted to the solution of beam coupling and diffraction off of a "fixed" photorefractively written holographic plane grating. The solutions and mathematical tools developed in this chapter are used extensively throughout the thesis: in chapters two and five to describe diffraction off a photorefractive grating, in chapters seven and eight to solve for the beam coupling off a grating when one beam is phase modulated, and in chapter nine to study the spectral response of fixed holographic interference filters. The techniques are presented with sufficient generality to allow application to numerous other problems, not limited to the ones described here.
Item Type:
Thesis (Dissertation (Ph.D.))
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Yariv, Amnon
Thesis Committee:
Unknown, Unknown
Defense Date:
12 May 1993
Record Number:
CaltechETD:etd-08272007-134313
Persistent URL:
DOI:
10.7907/zm1c-8w92
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
3246
Collection:
CaltechTHESIS
Deposited By:
Imported from ETD-db
Deposited On:
06 Sep 2007
Last Modified:
16 Apr 2021 23:03
Thesis Files
Preview
PDF (Hofmeister_rj_1993.pdf)
- Final Version
See Usage Policy.
11MB
Repository Staff Only:
item control page
CaltechTHESIS is powered by
EPrints 3.3
which is developed by the
School of Electronics and Computer Science
at the University of Southampton.
More information and software credits
Growth and Applications of Photorefractive

Potassium Lithium Tantalate Niobate (KLTN)

Thesis by

Rudolf Hofmeister

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, CA
1993
May 12, 1993

Rudolf Hofmeister

-iii-

Acknowledgements

First, I would like to thank Dr. Amnon Yariv for the support and in-
struction he has given me over the last four years as a graduate student, and
earlier as an undergraduate. I am also especially grateful to Dr. Aharon
(Ronnie) Agranat for introducing me to the field of crystal growth, and for the
enthusiasm he shared with me. Also, I am indebted to Drs. Mordechai (Moti)
Segev and Yaakov Shevy for numerous enlightening discussions on physics

and experimental techniques.

Other members, past and present, of the Quantum Electronics Group
and greater Caltech community deserve recognition as well for their assis-
tance and support: Dr. Victor Leyva for introducing me to the study of pho-
torefractive KTN; Drs. George Rakuljic and Koichi Sayano for many helpful
suggestions On experiments; Jana Mercado for her clerical support; Ali
Ghaffari, who assisted in organization in the lab and repair of equipment;
Charles Neugebauer for instructing me at length in the art of electronics and
computers; and Dr. John Armstrong, Suzanne Lin, and Paul Carpenter for as-
sistance with chemical and structural analyses. I had the pleasure of collabo-
rating with Tony Kewitsch, Doruk Engin, Jim Werner, Korhan Gurkan, and

especially Shogo Yagi.

Dr. Peter Fay was my first contact with Caltech; he reinforced my deci-
sion to pursue undergraduate studies at Caltech. During my undergraduate
years, I was slowly lured away from astronomy by physics, and for a brief peri-

od of time, pure mathematics held some attraction for me. Thankfully, noth-

-iv-

ing came of that. My bent towards the study of optics must have been in-
cited by the instruction of Drs. Demetri Psaltis and William Bridges. In partic-
ular, Dr. Psaltis’s in-class demonstrations of lasers and acousto-optics and Dr.
Bridges’s enthusiasm in optics lab classes were instrumental in sparking my

interest in the field.

Finally, I would like to thank my parents and Lori Taylor for their un-

faltering support during my years of study at Caltech.

Abstract

This thesis describes the growth of photorefractive potassium lithium
tantalate niobate (KLTN) single crystal material and characterization of its
physical and photorefractive properties. The band transport model is used to
discuss the conventional photorefractive effect. The coupled mode formalism
is introduced to determine the interaction of interfering light beams in a pho-
torefractive material. Solutions for intensity coupling and phase coupling be-
tween two beams, as well as diffraction off a dynamic index grating, are pre-

sented for both the copropagating and counterpropagating experimental ge-
ometries. These solutions are obtained for arbitrary photorefractive phase, 9.

The linear- and quadratic electro-optic effects are discussed. The influence of

electric field application on the electro-optic tensor is described.

The top seeded solution growth method is reviewed. The design and
construction of a crystal growth system is described. The growth procedures
of KLTN are enumerated for several compositions and dopant types. Phase
diagrams of the KLTN system are determined. Structural properties of the
grown crystals are presented. Certain material characteristics of KLTN are dis-
cussed. These include the phase transition temperatures, dielectric properties,

and the optical absorption properties.

Electric field control of the photorefractive effect, beam coupling and
diffraction, is demonstrated for paraelectric KLTN. A theory is developed to

describe the diffraction of beams off photorefractive index gratings in para-

-vi-

electric KLTN. The solutions of the coupled mode equations are used to de-
velop methods of determining the photorefractive phase 6 in a photorefrac-

tive material. These methods are experimentally demonstrated for several
types of photorefractive material. In addition, they are used to corroborate a
theory describing the magnitude and phase of the net holographic grating in

paraelectric KLTN under applied electric field.

A new effect, the Zero External Field Photorefractive (ZEFPR) effect is
studied, as well as the application of its unique zero phase ( 6 = 0 ) photore-

fractive gratings. The ZEFPR effect is forbidden by the conventional photore-
fractive theory; its origin is shown to be due to the creation of strain gratings
under spatially periodic illumination. A theory of coordination of microscop-
ic strains by a macroscopic (growth induced) strain is presented. The ZEFPR
gratings are shown to possess identically zero phase when no external electric
field is applied. This property is employed in the implementation of various
new linear phase-to-intensity transduction devices. In particular, an all-opti-
cal phase modulation/vibration sensor (microphone) is described. This de-
vice is expected to have numerous applications in environments where elec-
tric fields cannot be permitted. The possible implementation of ZEFPR grat-
ings in high speed self aligning interferometric data links is discussed, as well
as implementation of a novel self aligning holographic image subtraction de-

vice.

The final chapter is devoted to the solution of beam coupling and
diffraction off of a “fixed” photorefractively written holographic plane grat-
ing. The solutions and mathematical tools developed in this chapter are used

extensively throughout the thesis: in chapters two and five to describe diffrac-

-vii-

tion off a photorefractive grating, in chapters seven and eight to solve for the
beam coupling off a grating when one beam is phase modulated, and in chap-
ter nine to study the spectral response of fixed holographic interference filters.
The techniques are presented with sufficient generality to allow application to

numerous other problems, not limited to the ones described here.

-Vill-

Table of Contents

Acknowledgements 0... ccccccecsessescesscsescsesesecsessaeseresseecscsesecsseesaeseressesanacieeeeasersecseeey iii
ADSEIACE oo ..ccceceeccsccscccessessscsscceccsessssececcvsnsuceesecesssecaccsecssensusuccaceeessensecacseceesececsensnsnacaseesenenee V
Table of Contents ........c.ee cece cccccseessecsccccccecesecsensenscccscceseseeeensesscsaaucecceceeseesesessesesensns viii

1. Introduction to the Photorefractive Effect,

Materials, and Applications 00.0... ccc escerececsessseeecsesssscacaseeeseneessaeeeeeseees 1
LT. IMtroductiOn oo... tee eceeseneescesescsensenenecenecsenessenecsesieseseessseeseeeeeey 1
1.2 Photorefractive materials 00.0.0... ccc cece ene re ce caeeecneceeeeseeseesecseneesens 5

1.2.1 Ferroelectric oxygen-octahedra photorefractive materials ........ 6
1.2.2 Sillenites occ es cseessesescseneeessssesesesevessesessssceceeetesseesanecssaeeees 7
1.2.3 Semiconductor- and other photorefractives ........ccceseeeees 7
1.3 Outline of the thesis ........ccccccsceeeseessseeesssceseceescscsescseseseesscseensnseeeaeaes 8
References for chapter ON€ ...c.cccccssseessseeesseessnseeeesseserssesesssseeescessnseseneeesenenes 13
2. The Photorefractive Effect .....ccccecsssssesessseseesessssssssssssssssesscsesesesaesssssssssssseeseesegs 20
2.1 INtrOGUCTION 0.0... eecse ce ceenecseseceesessesesasecseecsesecsesscecssneesssecnscesneceeseeees 20
2.2 Band transport MOdel] oo. css ceesescsesenesseeeseneseraesensssessssasenseneneoregs 21
2.2.1 The rate Equations ........cccceccceeecssssesetesssecseseseescsesesesscecsesesseeeseeees 28
2.2.2 Single-charge carrier SOUHON 2... cece cece cs eteneeeneeeesenesseneeeeas 31
2.2.3 Two-charge Carrier SO]UtION .....cceecceseessestsesseetseseeeeseeceeseeeeeneneney 32
2.2.4 Photorefractive index CHAN ge... ccccccsecccsseseescesescseneceesenseeeseees 33

2.3 Photorefractive beam COUP]NG oo... cece cscs ceeeseeseeeseescresesstessenees 35

-1x-

2.3.1 Coupled mode equations, fixed Grating occa 37
2.3.2 Copropagating geometry, dynamic grating... sees 41
2.3.3 Counterpropagating geometry, dynamic grating «0. 47
2.4 SUMMATY escececccesseseeseeseesesseseeseesssesenseseessenessseesseseesseressereaseesseeseeeeees 51
References for chapter twO ....ccsccsssssesessessscecsescsetesessssnscscsessseeeseseseceseeseneess 52
3. Electro-optic Effect oo. ccccseeeeseeeeseesessesseessensssssessssescsessesseneseseessessesesesees 56
B.1 INtrOCUctiONn ooo. ec cee ee ceecsecesesecnensseseecsseneeeeeeseseseesesseseeeeseeeeneenees 56
3.2 The index ellipSOid oo. cece cece cscsenserseseceeeessenesseeeeseeseseneesas 58
3.3 The linear electro-optic Effect (Pockels effect) ....ccccccsecssteeeeeeseeeee 64
3.3.1 Symmetry properties; third rank tenSOrs .......ceeeteeeteee eens 64
3.3.2 True and indirect linear electro-optic effect occ 70
3.3.3. Application to photorefractive effect 0... eseteeeeteeeesesens 72
3.3.3.1 Electric field applied along Z ....cccecceseeseeteretetsereneneesesens 72

3.3.3.2 Effective electro-optic coefficient for arbitrary

Clectric FILA oe eee ee eseeesneceeecseneeeeessensestecseseastenensensnenens 74

3.4 The quadratic electro-optic effect (Kerr effect) ....ccccccseeeseeseens 77
3.4.1 Symmetry considerations; fourth rank tenSOTrs ........cceeeees 79
3.4.2. Application to photorefractive effect ........ccccecceecseeeeteseeseneenens 81
3.4.2.1 Electric field applied along Z oe cccseccsesesesserseecseeneeseeees 82
3.4.2.2 Electric field applied along arbitrary direction .............. 84

B.5 SUMIMATY oo cecceececeecec cee eensesescseneseeesasesnenesessesenseesseassessneesensnecesnenesenines 86
References for chapter three ........ccccececceccssseessescscseseseeessneseceneceeseaenaney 88

4, Crystal Growth and Material Properties of Potassium Lithium

Tantalate Niobate (KLTN) oon... ceccsssssscececsesssccsceecescesscesaeseessssereeecesecersesenaa 91

A.V INtrOductiOn ..ccccccecscseeeesseteessesesesesseecseseessesesenseseseseeseeeenseseseeeeserenes 91
4.2 Top seeded solution growth method o.......ccccccceccsseeeteeseeeseeseesesanees 93
4.3 Crystal growth SYSteMm o....ccec cece cseeecseneeenecseeeeecsessesseessenenessseeessnaneees 94
4.4 Growth Of KLIN wueccccccsccessesscsesssesseneesseesesseessessesssesseecsesecssseeesseeeseses 97
4.4.1 Sample QrOWths o..cccceccssceecesenseeeseseeceeeesenseecsesesseeeesseneseeseseeeneeans 97
4.4.2 Composition of Zrown crystals ....ccccceeseeesteteresetesseeeeeetsesenens 103
4.5 Discussion of growth characteristics ......cccccsesceseeeseeseneseeesensseeennees 106
4.5.1 Composition COnSIderations 0... cee eeeeseseeeeneeseeeeees 106
4.5.2 Growth paraMeters ....ccccccccesseseessseeseeseseeecsseecsseesestessensesseesseneeees 117
4.6 Influence of lithium/niobium on transition properties .............. 123
4.7 SUIMIMATLY oeeescccesssssceesseeesceseeeseeseseeeeseeesseeeceesecseeeseesessesesseetesseesseeeeseeessseees 135
References for chapter fOUur .....cccccccsseseseseeseneeeesesesseeessesessseceessseeeeeeneess 136

5. Electric Field Control of the Photorefractive Effect in Paraelectric KLTN 139

5.1 IMtrOductiOn oo... cece cee ceececseeescseseeeeseessesesecsesseseeeeseseeeseeseneey 139
5.2 Bea COUPLING ceccccccccsseessceecseesseeneseseesssesessesessesssenstsssseseessseessesesstnenes 141
5.3 Electric field control Of diffraction ......cccceseseessesseseseeeseneretsesesenens 147
5.3.1 Solution of coupled equations oo... seeeeeseeeneneeeneneneenesens 147
5.3.2 Solution of coupled equation with E, = 0 eee 154
5.3.3 Diffraction experiment resultS .....ccccccceeeeeseeenersneeeneeeeseneseeee 156
5.4 Discussion of experimental results ......ccccccsesesseeseesesesesesnesssereeees 161
5.5 SUIMIMATY woeecesssccsessescscsesensssnsesessescessesesssenscscsenesesessesssenssssesessesassesesasecseees 169
References for chapter five ......cccccccsescsscsssessesssessssceesecsescresesaesessneesensees 170

6. Zero External Field Photorefractive (ZEFPR) Effect in

Paraelectric Materials o....... cece cceeseccccccsssecccnccececcccsececscecscecceaaussseceaceceseccanseceseas 172

-xi-

6.1 INtrOGUctiONn ..... cece ec cc cee es cere csenecsesesessesavssssceessceessasaeeeseeseseeeneeees 172
6.2 The ZEFPR effect ...ccccccccccccseecscsenessssesesescsessssescscsecessessesecnsneeeeesaeeeteas 174
6.2.1 The Jahn-Teller relaxation 2.0... ccccecccseesssseceeeteeeeeseeesnessecseesees 175
6.2.2 Strain dependence of the ZEFPR effect oo... cece eeeeeeeeene 176
6.2.3 E.. dependence of the ZEFPR effect 0... 184
6.3 Theory of distortion Coordination 0... ceccsssetseeeceeeeeeeeeeeseeeeeeeens 191
6.4 The photoelastic photorefractive effect oo... esseseseeteneeteteeenes 195
6.5 SUMMATY ooeecsscsessssssesesescsessssscnescsesencacesaesencecaeaescaeseeeusecssseeeseseseseeeeseess 196
References for Chapter Six. ......ccscccssssssessecesssseseereseseeesensaeeuesesensensneneaeeees 198
7. Applications of the ZEFPR Effect ......cccceescssssssseseeeseeeeeseeesseeseseeeseseseecesenseanes 202
ZV IMtrOdUctiOn o..eeccceecccsssssssscscscsesessssessecsssessseesescnesssesseesatseeseeeaeneeeseaees 202
7.2 Self-aligning vibration sensor/ microphone ........c.c ee eeeeeeeees 203
7.21 UNtrOductiONn oo... cee cnseeesenceceneeecseeesseeecseeeesenseeeeaeeeesensereaees 203
7.2.2 EXPOTiMent woccccccccsccseessssrenececssreseseeesesesensnecesseneeeeesneneeesstaceeeaes 206
7.2.3 Coupled wave analySis oo... sseceeeeteseseeereseseeeeteneeeneeearenes 209
7.2.4 DISCUSSION Of TESUIES Lo... cceeeessetetsesesesetessasetecseneteeaeeeeenaees 215
7.3 Other applications of the ZEFPR effect 0... csccesssseeeeseeeneeeeneneeees 216
7.3.1 Interferometric data link oo... cece ee cseeteneeeeneeeeeneeees 216

7.3.2 Phase image subtraction and phase to

INGENSILY CONVETSION oo..ecscseeceresesesesesseeesetesesesetesssesesenssessseseseeeneeenes 219

TA SUMIMALY ooeeescecsseeesessssessesescesessessesssssseesessscsesesssaveneceessaeaeecasaeeeseeeeeees 222
References for chapter SCVEN .....cccccscessseseeseetesenesenesesesesesereseeeaeaeeeateeeeesens 223

8. Determination of Photorefractive Phase and Coupling Constant ............. 225

8.1 IMtrOductiOn ooo... ieee ceseeesscccccecssectcccecesececenssceeccescestscsttaueceeececessnscerses 225

-xli-

8.2 Formulation of the problem... ccc ccessceeseeeeseesseneenesseneenes 227
8.3 Solution of beam COUPLING 0... cseeees esse sees esesseeeseseseneneneeneney 229

8.3.1 Harmonics method .....ccccccccseescsesseeescesseeenscsesessessnesessesenanes 231

8.3.2 Beam coupling/ diffraction Method oo... cece cseeeeeereeeeenees 236

8.3.3 Phase ramping method o.......cc ccc css csesceceeessesensescseeeeseneneeees 237
8.4 Photorefractive coupling and phase in paraelectric materials ..... 238
8.5 Experimental results occ ccc cseeeceecsecsecscseceeeeeceeceaeaseseesaeeaeeeseeseeeneees 240
8.6 Discussion Of TeSUI]ES oo... cece eeeesseteesceeseseesesesaeetsecseaecseesaecesseeees 243
8.7 SUMMATY oo.eeeecceecceccscsesssstsescetsceeeeescarscseecscsenecscsesessesesseeescseseeecaeseneeaenees 247
References for chapter Cight ........c.ccccccccscsesesseseesenscscseseseessssnecseseseneesaneees 248

9. General Solutions of Coupled Mode Equations with Applications

to Fixed Holographic Gratings .......cccccesssssssssseesssseeesssscsescsenessssessscevessenseenseees 250
G21 IMtrOGUCtiON oo. ecseseseseseeceesesesesesescseseeeeesssesesessssssscscsesesesesesesesssansesesesegess 250
9.2 Solution of coupled differential equations 0.0... cece eeeeeeeees 251

9.3 Spectral response of fixed holographic grating

Interference filLETS oo ees ceeeseeeceecsecserseesesseeeeecesesseeeeeeeseeseeseeeneens 255
9.3.1 IMtrOCUCTION ooo. eee cneesseeeceeceessseceeesscaeeeaeeaecaeesaeeaecseseaeseseneeeee 255
9.3.2 Theoretical investigation, lossless CaS€ .........cceeseeeeeesseeseeeeees 256
9.3.3 Investigation, LOSSY CASO .....eceeeeeteteesseeesesseseseeeeeseneeceeseesesseneets 271

9.3.4 Summary of spectral response of fixed holographic
grating interference filters ........ cece cece cseeseeseseeeseeecesseeeseeaes 279
9.4 Response of fixed holographic gratings written in the
COPFOPAagatiNgG QCOMELTY oo... cccsecsccsceecseseeeeseteeseeessessseeseeecseesseesecsereeenes 282
9.4.1 IMtrOductiOn o.c.ccc ce ccccecseseseseseeececessetesessesenenesesscsensesesesessescssasegeees 282

-xiil-

9.4.2 Formulation of the problem oo... cece csessescssesctseseesesnenees 284

9.4.3 Solution of beam COUPLING oo. ces esseeeescecsereeseeseeeseeneees 288

9.4.4 Determination of coefficients oo... cteeecseeeeseeeeseeeeeseeteees 293

D5 SUMMALY oeeeessscsesssseseseeesceeesescesssesssesseseseeseseecssesceeessscessesesesscseeseneaeeseneaes 297
References for chapter nine oo... cscsccsssesesessesessesesseseesenceeseeneensncateneaeaneneass 299

10. Summary and future directions oo... cece eeeseeceeeteceeseeescsecseseetseseneseeees 301
10.1 SUMMATY eee ceeeececeeteesssessessseneesesssnanecseeesuacsseeecseseseeesenssesseaeeeeesensaeseasees 301

10.2 Future Girections occ cece eceesecsscccccceceecesceecsnscsceceececeeevestesstacseeeescerens 303

Chapter One

Introduction to the Photorefractive Effect,

Materials, and Applications

1.1 Introduction

The photorefractive effect refers to a light induced change in the index

of refraction of a material. The effect usually allows large refractive index
changes ( An ~ 104) with relatively low intensities of incident light. The pho-

torefractive effect arises through the photoexcitation, transport, and subse-
quent retrapping of charge carriers. When a photorefractive medium is illu-
minated with a spatially nonuniform beam, for example a pattern of dark and
bright fringes caused by the interference of two intersecting laser beams,
charges are photoexcited in the bright areas, and tend to be retrapped in the
dark areas. If a photoexcited charge is retrapped in a bright area of the grating,
it is likely that it will be repeatedly photoexcited until it becomes trapped in a
dark region where photoexcitation ceases. This process eventually leads to a
spatial pattern of trapped charges, and thus an electric space charge field, both
of which mimic the spatial intensity pattern. In the conventional photorefrac-
tive effect it is this space charge field which leads to a change in the index of

refraction via the electro-optic effect.

-2-

The effect was first noticed in frequency doubling experiments in lithi-

um niobate (LiNbO ) by Ashkin! and independently by Chen2. Since the ef-
fect was deleterious to the nonlinear optical experiment, the effect was termed
“optical damage.” Not until Chen? realized how the “damage” could be har-
nessed to provide optical data storage did the effect receive its present name
and attention as anything but a nuisance. Numerous applications were real-
ized for the new effect. One of the first was holographic data storage, where it
was predicted that the ultimate storage capacity would be ~ 10! bits cm3.4°
The holographic storage of multiple pages of data was another early applica-
tion, culminating in the simultaneous storage of 500 fixed holograms in one
crystal by Staebler®. Since the photorefractive effect leads to an index grating
which, in general, is not in-phase with the intensity pattern, the coupling of
two beams leads to power transfer between them’. Beam coupling was used to
demonstrate amplification of weak signal beams by factors of several thou-

sand8,

In 1978 Yariv? illustrated that the wave formalism used to describe
photorefractive four-wave mixing was identical to the formalism employed
to discuss nonlinear optical degenerate four-wave mixing. Optical phase con-
jugation of a signal beam is the generation of a light beam with an identical
phase front to the signal, but propagating in the opposite direction!2", The
advantage of using the photorefractive effect would be the possibility of per-

forming optical phase conjugation with low-light intensities. Soon after-

wards, this prediction was verified by Huignard!*. This new application led

-3-
to a resurgence of interest in the photorefractive field. Pattern recognition and

various types of image processing were demonstrated!3-!®, as well as optical

distortion correction!”18,

However, various factors inhibited the commercial implementation of

photorefractive based technologies. To the author’s knowledge it was not
until recently!? that a single photorefractive device had ever left the laborato-

ry and successfully entered the market. Severe material limitations are re-

sponsible for most devices remaining on the lab bench.

One of the most serious problems for most photorefractives is the ten-
dency for transmitted beams to “fanout.” Fanout?”-24 is the process of diffuse

scattering of light from an incident beam into a continuum of directions to-
wards the optic axis of the crystal. The scattering tends to build up over a long
period of time (many times longer than the characteristic grating write time).
The name derives from the broad fan of light which forms pointing toward
the optic axis (c-axis ). This process is not fully understood but is believed to
result from amplification of scattered light beams partially generated by opti-
cal inhomogeneities on the surface or in the bulk of a material. Potassium
tantalate niobate (KTN) and potassium lithium tantalate niobate (KLTN) do
not display fanout except with application of large electric fields. Fanout di-
verts optical power from the signal beams, reducing the amount of light
transmitted and usually the efficiency of the process being performed.

Nevertheless, several applications for the phenomenon have been demon-

strated. These include passive phase conjugation*??® and optical limiters2”.

-4-

For most applications, however, the effect is considered deleterious to pho-

torefractive performance.

The second major problem in implementing photorefractive materials
is the erasure of holograms which occurs on readout. The process of writing
and erasure are symmetric. Thus when a stored hologram is read out with

plane wave illumination, the plane wave redistributes the trapped charge,

erasing the grating. Several mechanisms for fixing holograms are known.2&
34 The most well known method is the thermal fixing method wherein a

hologram is stored and the crystal is heated to 80-150°C. At these tempera-

tures ionic defects become mobile and drift under the influence of the space
charge field. This charge drift compensates the ionic charge pattern. The crys-
tal is then cooled and uniformly illuminated to redistribute the electronic
charge, revealing the now fixed ionic charge pattern. This procedure has been
demonstrated in LINbO3 and KNbO3. Other fixing processes including elec-

tric field controlled domain reversal in SBN (strontium barium niobate) have

been documented 9-3’, and recent results at Caltech indicate that other fixing

mechanisms are possible in SBN without an applied field3® . Fixing in KTN

was observed by writing a grating in the paraelectric phase, cooling the crystal

through its three phase transitions, and then revealing the grating with uni-
form illumination at low temperature??. Recently, a proprietary technique

has been reported for efficient hologram fixation in LiNbO,*9. With the possi-

ble exception of the recently reported proprietary technique, these methods

have proved problematic and have not seen widespread implementation.

-5-

Beyond these fundamental problems are the pure material deficiencies.
Most materials lack either large photorefractive coupling constants or fast
time response. Many materials lose photorefractive sensitivity at wave-
lengths longer than approximately 600nm (this cutoff depends somewhat on
the photorefractive dopant used). Some crystals are excessively fragile, espe-
cially when operated near their phase transition temperature. Finally, most
materials, excepting LiNbO3, are still considered difficult to grow. As a conse-
quence, they are expensive and difficult to obtain. The lead time for a high

quality BaTiO, sample, as an example, can be over a year.
1.2 Photorefractive Materials

A material must meet several requirements to become photorefractive.
First, it must possess a mid-gap, partially filled, photoionizable impurity
level. Second, it must be linearly electro-optic to exhibit the conventional
photorefractive effect, or quadratically electro-optic to display the electric field
controlled photorefractive effect. A few photorefractive effects exist which do

not rely on the electro-optic effect. These include the photorefractive scatter-
ing by absorption gratings4!47 and the zero external field photorefractive
(ZEFPR) effect#®. The nonlinear response of materials to electric fields at tem-
peratures near a structural phase transition has also been shown to produce a

photorefractive effect. The dielectric photorefractive effect4?>! is one exam-

ple.

The materials>* which exhibit photorefractive effects include the oxy-

gen-octahedra photorefractives, the sillenites, and certain semiconductors.

-6-

Some characteristics which are used to distinguish the various materials from
each other are the magnitude of the induced index change, the photorefrac-
tive phase (phase between the index grating and the intensity grating), the re-
sponse time, the dark current and possibility of fixing the grating, and the sen-
sitivity of the material. This last quantity is usually defined as the change in

index per unit of absorbed incident intensity.
1.2.1 Ferroelectric Oxygen-Octahedra Photorefractive Materials

The ferroelectric oxygen-octahedra photorefractives are the most wide-
ly known and studied photorefractive materials. They include the ilmenite
structures, lithium niobate (LiNbO 3) and lithium tantalate (LiTaO3), the per-
ovskites potassium niobate (KNbO3), potassium tantalate niobate (KTa,_
,Nb,O3 or KTN), potassium lithium tantalate niobate (Ky_yLi,Ta,_,.Nb,O3 or
KLTN), and barium titanate (BaTiO), and finally the tungsten bronzes stron-
tium barium niobate (Sr,_,Ba,NbO; or SBN), Bay_,St,Ky_yNa,NbsO}5
(BSKNN), and barium sodium niobate (Ba,NaNb-O)s).

The perovskites, KTN, KLTN, and BaTiO, are characterized by a high
temperature centrosymmetric (cubic) phase and undergo successive transi-
tions to tetragonal, orthorhombic, and finally rhombohedral phases as the
temperature is lowered. Lithium niobate and lithium tantalate have a point

group symmetry of 3m at room temperature and only assume a high symme-
try phase at temperatures above 1200°C (lithium niobate decomposes before

reaching this temperature). SBN is the archetype of the tungsten bronze struc-

ture,and becomes tetragonal in its high temperature paraelectric phase at tem-

-7-
peratures from ~0°C - 120°C depending on composition.

All the ferroelectric oxides are readily amenable to doping by photore-
fractively active species (usually first row transition metals or lanthanides).
They tend to have slow response times and relatively low sensitivities be-
cause of their low carrier mobilities. These materials are the only ones in

which hologram fixing has been observed.
1.2.2 Sillenites

The photorefractive sillenites include the materials Bi,,SiOj,) (BSO),
BijyGeO39 (BGO), and Bi,,TiOz, (BTO), and recently, BiyTeO;>4. They are
noncentrosymmetric cubic materials. They tend to have much smaller di-
electric constants and higher photoconductivities than the ferroelectric ox-
ides, thus the photorefractive sensitivities are higher. The mobilities are
about the same as for the ferroelectric oxides, but the electro-optic coefficients

tend to be smaller.
1.2.3 Semiconductor- and Other Photorefractives

The photorefractive effect has been demonstrated in several semicon-
ductor materials°>. These include GaAs, InP°®, and CdFe. The mobilities are
much higher in these materials than in the previous two classes, with similar
electro-optic coefficients, and the sensitivities are the highest for any type of
photorefractive. Semiconductor photorefractives usually respond best in the
near infrared, while the other classes of material discussed respond well in

the visible.

1.3 Outline of the Thesis

The field of photorefractive materials shows great promise for numer-
ous applications in optical processing and data storage. Unfortunately a lack of
high quality photorefractive materials has restricted the development of sal-
able products and devices. It has been the aim of this thesis to develop and
characterize a new type of photorefractive material, potassium lithium tanta-
late niobate (KLTN), to help overcome the material limitations. In addition,
the unique properties of KLTN, such as its composition controlled phase tran-
sition temperature and electric field controlled photorefractive response, were
studied in order to unveil new applications for which such photorefractives

are well suited.

In chapter two a band transport model of the photorefractive effect is
described. Solutions for both a single charge carrier and two charge carriers are
presented. Photorefractive two beam coupling and diffraction are discussed
for holograms written with beams propagating in the same direction (coprop-
agating or transmission geometry) and in opposite directions (counterpropa-
gating or reflection geometry). The electro-optic effect is described in chapter
three. The linear electro-optic effect and its contribution to the photorefrac-
tive effect is discussed first. The distinction between the clamped and un-
clamped electro-optic coefficients is derived. Effects of an applied electric field
on the index ellipsoid are discussed. A study of the quadratic electro-optic ef-
fect follows, including the effective linear electro-optic coefficients induced by
an electric field in a quadratic medium. Symmetry properties of the linear and

quadratic electro-optic tensors are discussed. Lastly, the rotation of the c-axis

-9-
under application of an electric field in a Kerr material is derived.

Chapter four illustrates the design and construction of a crystal growth
system and the growth of paraelectric KLTN with the top seeded solution
growth (TSSG) method. Sample growths are described and a range of feasible
compositions elaborated. Determination of the compositions of the grown
KLTNs along with the seeding temperatures and the flux compositions allows
the construction of phase diagrams for the KLTN system. These phase dia-
grams are presented. Structural characteristics and material properties of the
TSSG grown KLTNs are enumerated. The influence of composition on the
type and number of phase transitions is discussed, as well as the influence of
composition on the phase transition temperature; dielectric properties of sev-
eral compositions of KLTN are presented. The optical absorption properties

are also examined as functions of crystal composition.

Chapter five describes results of experiments demonstrating the electric
field control of the photorefractive response in the paraelectric phase. Crystals
in the paraelectric phase exhibit no linear electro-optic effect so the conven-
tional photorefractive effect can be modulated by an external field. In diffrac-
tion experiments, voltage controlled diffraction efficiencies of 75% are report-
ed. This is the highest reported value for a photorefractive known to the au-
thor. A theory is developed to describe the diffraction off a dynamically writ-
ten grating when the coupling constant g changes between the writing and
the reading phase. The theory is shown to agree well qualitatively with ex-

perimental data.

Chapters six and seven describe the zero external field photorefractive

-10-
(ZEFPR) effect and its device applications. The ZEFPR effect was discovered

in paraelectric KTN and KLTN°”°8 but was not identified as a new photore-

fractive mechanism until recently4®. It is expected to exist in many photore-
fractives but was observed in KTN/KLTN first because the conventional pho-
torefractive effect which would otherwise dominate the ZEFPR effect, is for-
bidden in these materials. Experiments described in chapter six allow the con-
clusion that the ZEFPR effect is due to a valence state dependent Jahn-Teller
relaxation of the oxygen octahedra surrounding the photorefractive centers.
This relaxation yields strain gratings in-phase with the intensity pattern.
Finally, the strain gratings result in a refractive index change via the photoe-
lastic effect. The ZEFPR effect is unique in the respect that the index grating is
identically in-phase with the intensity pattern. This property is shown, in
chapter seven, to be the basis of numerous novel devices. The development
of a ZEFPR based vibration sensor/ microphone is described. This sensor is
all-optical and self aligning, no electric signals are required for operation. It
has potential applications for sound/vibration sensing in environments
where electrical signals cannot be tolerated or are impractical. These include
corrosive or explosive environments. Two other devices are theoretically de-
scribed in chapter seven: a high speed self aligning interferometric data link,
and a self aligning image subtraction device. Both devices would operate by

implementation of the ZEFPR effect.

Chapter eight discusses several methods of determining the photore-

fractive phase and coupling constant of a material. The photorefractive phase

o is the material parameter which determines the nature of the coupling be-

-11-

tween two interfering beams in a photorefractive crystal, i.e., the relative pro-
portions of phase coupling and intensity coupling. The coupling constant de-
termines the overall magnitude of the coupling. The methods are used to de-
termine the photorefractive phase of crystals of LINDO , BaTiO3, and KLTN.
A theory is derived to describe the interaction of a ZEFPR grating with a con-
ventional electro-optic grating as a function of applied field. The predicted
coupling constants and photorefractive phases are shown to agree with exper-

imentally determined values.

Chapter nine essentially develops a new and exact mathematical for-
malism for solving certain first-order coupled equations. In this chapter, re-
sults are derived that have been implemented in chapters two, five, seven,
and eight. The results in all of these prior chapters hinge on the development
of the mathematical tools in chapter nine. In addition, the mathematical
method is applied in chapter nine to the solution of frequency response of the
reflectivity from fixed photorefractive gratings written in the counterpropa-

gating (reflection) geometry.

In chapter nine, the treatment describes the beam coupling and diffrac-
tion of beams off a fixed dynamically written holographic grating. The grating
can be written in either the copropagating or the counterpropagating geome-
try (transmission or reflection mode). The analysis for counterpropagating ge-
ometry is applied to solving the spectral response of fixed holographic grating
interference filters. A numerical study is also reported. The copropagating
analysis is applied to the vibration response of beam coupling to a dynamical-

ly written grating. The diffraction off a grating is also derived. These last two

-12-

results are the ones referred to in chapters two, seven, and eight. The coprop-
agating analysis is also the basis of the theoretical treatment of chapter five de-
scribing the diffraction off a fixed photorefractive grating with a non-constant

coupling value g.

-13-
References for chapter one

[1] A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, H. J.
Levinstein, K. Nassau, “Optically-induced refractive index inhomogeneities

in LiNbO; and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).

[2] FS. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,”

J. Appl. Phys. 38, 3418 (1967).

[3] F.S. Chen, J. T. LaMacchia, D. B. Fraser, “Holographic storage in lithium
niobate,” Appl. Phys. Lett. 13, 223 (1968).

[4] P J. van Heerden, “Theory of optical information storage in solids,” Appl.

Opt. 2, 393 (1963).

[5] T. Jannson, “Structural information in volume holography,” Opt. Appl.

IX, 169 (1979).

[6] D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and
erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182
(1975).

[7] N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L.
Vinetskii, “Holographic storage in electro-optic crystals. I. steady state,”

Ferroelec. 22, 949-960 (1979).

[8] F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator

using two wave interaction in a BaTiO, crystal,” Opt. Comm. 47, 387 (1983).

[9] A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J.
Quant. Elect. QE 14, 650 (1978).

-14-

[10] B. Ya. Zel’dovich, N. F Pilipetsky, and V. V. Shkunov, Principles of

Phase Conjugation, Springer-Verlag (1985).

[11] R.A. Fisher Ed. , Optical Phase Conjugation, Academic Press, Orlando
(1983).

{12] J. P Huignard, J. P. Herriau, P. Aubourg, and E. Spitz, “Phase conjugate
wavefront generation via real time holography in BizySiO4Q crystals,” Opt.

Lett. 4, 21 (1979).

[13] S. K. Kwong, Y. Tomita, and A. Yariv, “Optical tracking filter using tran-
sient energy coupling effect,” J. Opt. Soc. Am. B 5, 1788-1791 (1988).

[14] J. O. White and A. Yariv, “Real-time image processing via four-wave

mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5-7 (1980).

[15] S. K. Kwong, G. A. Rakuljic, and A. Yariv, “Real-time image subtraction
and ‘exclusive or’ operation using a self-pumped phase conjugate mirror,”

Appl. Phys. Lett. 48, 201-203 (1986).

[16] S. K. Kwong, G. A. Rakuljic, V. Leyva, and A. Yariv, “Real-time image
processing using a self-pumped phase conjugate mirror,” Proc. SPIE 613, 36-42
(1986).

[17] M. Cronin-Golomb, B. Fischer, J. Nilsen, J. O. White, and A. Yariv, “Laser
with dynamic holographic intracavity distortion correction capability,” Appl.

Phys. Lett. 41, 219-220 (1982).

[18] V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimu-

lated Brillouin scattering,” Opt. Lett. 2, 4-6 (1978).

-15-

[19] Accuwave corporation has recently introduced a holographic photore-
fractive interference filter tuned to the H, line for solar observation. Its spec-
tral bandpass of 1/8 A is apparently superior to that of conventional Lyot fil-

ters.

[20] M. Segev, Y. Ophir, and B. Fischer, “Nonlinear multi 2-wave mixing, the
fanning process and its bleaching in photorefractive media,” Opt. Comm. 77,

265- 274 (1990).

[21] M. Segev, Y. Ophir, and B. Fischer, “Photorefractive self-defocusing,”
Appl. Phys. Lett. 56, 1086-1088 (1990).

[22] C.L. Adler, W. S. Rabinovich, A. E. Clement, G. C. Gilbreath, and B. J.
Feldman, “Comparison of photorefractive beam fanning using monochro-
matic and achromatic 2-wave mixing in SBN,” Opt. Comm. 94, 609 -618
(1992).

[23] R.S. Hathcock, D. A. Temple, C. Warde, “Beam fanning in BaTiO3-Cr,” J.
Opt. Soc. Am. A 2, 63 -73 (1985).

[24] M.S. Tobin and M. R. Stead, “Measurements of beam fanning in barium-

titanate,” J. Opt. Soc. Am. A 2, 37 -47 (1985).

[25] V. A. Dyakov, S. A. Korolkov, A. V. Mamaev, V. V. Shkunov, and A. A.
Zozulya, “Reflection-grating photorefractive self-pumped ring mirror,” Opt.

Lett. 16, 1614 -1616 (1991).

[26] M. Cronin-Golomb, K. Y. Lau, A. Yariv, “Infrared photorefractive passive

phase conjugation with BaTiO, - demonstrations with GaAlAs and 1.0914m

-16-

Ar & lasers” Appl. Phys. Lett. 47, 567 -569 (1985).

[27] S. E. Bialkowski, “Application of the BaTiO, beam-fanning optical lim-
iter as an adaptive spatial filter for signal enhancement in pulsed infrared

laser-excited photothermal spectroscopy,” Opt. Lett. 14, 1020 - 1022 (1989).

[28] L. Arizmendi, P. D. Townsend, M. Carrascosa, J. Baquedano, and J. M.
Cabrera, “Photorefractive fixing and related thermal effects in LiNbO3,” J.

Phys: Condens. Matter 3, 5399 -5406 (1991).

[29] G. Montemezzani and P. Gunter, “Thermal hologram fixing in pure and

doped KNbO3,” J. Opt. Soc. Am. B 7, 2323 - 2328 (1990).

[30] S. W. McCahon, D. Rytz, G. C. Valley, M. B. Klein, and B. A. Wechsler,
“Hologram fixing in Bi,,TiO,9 using heating and an AC electric field,” Appl.
Opt. 28, 1967 - 1969 (1989).

[31] P Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal holo-
gram fixing and application to LINbO3-Cu,” Phys. St. S. A 104, 855 -862 (1987).

[32] J.P. Herriau and J. P. Huignard, “Hologram fixing process at room-tem-
perature in photorefractive Bi,»SiO.9 crystals,” Appl. Phys. Lett. 49, 1140 -1142
(1986).

[33] F. Elguibaly, “Information storage in ferroelectrics - thermal fixing,” Can.

J. Phys. 66, 655 -658 (1988).

[34] R. Sommerfeldt, R.A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of
volume phase holograms in LINbO3-Cu,” Phys. St. S. A 99, 15 -18 (1987).

[35] F. Micheron and G. Bismuth, “Electrical control of fixation and erasure of

-|7-

holographic patterns in ferroelectric materials,” Appl. Phys. Lett. , 20, 79-81,
(1972).

[36] F. Micheron and J. C. Trotier, “Photoinduced phase transitions in

(Sr,Ba)Nb5O¢ crystals and applications,” Ferroelectrics, 8, 441-442, (1974).

[37] F. Micheron and G. Bismuth, “Field and time thresholds for the electrical
fixation of holograms recorded in (Srg 75Bag 95) Nb Og crystals,” Appl. Phys.
Lett. , 23, 71-72, (1973).

[38] A. Kewitsch, M. Segev, and A. Yariv, unpublished.

[39] V. Leyva, A. Agranat, and A. Yariv, “Fixing of a photorefractive grating in
KTa,_,Nb,O3 by cooling through the ferroelectric phase transition,” Opt. Lett.
16, 554-556 (1991).

[40] G. Rakuljic, A. Yariv, and V. Leyva, “High resolution volume hologra-
phy using orthogonal data storage,” Photorefractive Materials, Effects, and
Devices Conference of OSA, July 29-31, 1991, Beverley, MA, paper MD-3; G.

Rakuljic, private communication.

[41] A. A. Kamshilin, “Simultaneous recording of absorption and photore-

fractive gratings in photorefractive crystals,” Opt. Comm. 93, 350 -358 (1992).

[42] M. H. Garrett, P Tayebati, J. Y. Chang, H. P. Jenssen, and C. Warde,
“Shallow-trap-induced positive absorptive 2-beam coupling gain and light-in-
duced transparency in nominally undoped barium-titanate,” J. Appl. Phys.

72, 1965 -1969 (1992).

[43] R. S. Cudney, R. M. Pierce, G. D. Bacher, and J. Feinberg, “Absorption

-18-

gratings in photorefractive crystals with multiple levels,” J. Opt. Soc. B 8,

1326 -1332 (1991).

[44] T. Jaaskelainen and S. Toyooka, “Analysis of absorption reflection grat-

ings,” Opt. Comm. 71, 133 - 137 (1989).

[45] S. A. Boothroyd, J. Chrostowski and M. S. OSullivan, ”2-wave mixing by
phase and absorption gratings in saturable absorbers,” J. Opt. Soc. Am. B 6, 766
- 771 (1989).

[46] K. Walsh, T. J. Hall, and R. E. Burge, “Influence of polarization state and
absorption gratings on photorefractive 2-wave mixing in GaAs,” Opt. Lett. 12,

1026 -1028 (1987).

[47] M. Gehrtz, J. Pinsl, C. Brauchle, “Sensitive detection of phase and absorp-
tion gratings - phase modulated homodyne detected holography,” Appl. Phys.
B 43, 61 -77 (1987).

[48] R. Hofmeister, A. Yariv, S. Yagi, and A. Agranat, “A new photorefractive
mechanism in paraelectric crystals : A strain coordinated Jahn-Teller relax-

ation,” Phys. Rev. Lett. , 69, 1459-1462, (1992).

[49] A. Agranat and Y. Yacoby, “Dielectric photorefractive crystals as the stor-
age medium in holographic memory systems,” J. Opt. Soc. Am. B 5, 1792 -
1799 (1988).

[50] A. Agranat, Y. Yacoby, “The dielectric photorefractive effect - a new pho-
torefractive mechanism,” IEEE Ultras. 33, 797 -799 (1986).

[51] A. Agranat and Y. Yacoby, “Temperature dependence of the dielectric in-

-19-
duced photorefractive effect,” Ferroelec. Lett. 4, 19 -25 (1985).

[52] G.C. Valley, M. B. Klein, R. A. Mullen, D. Rytz, and B. Wechsler,
“Photorefractive materials,” Ann. Rev. Mater. Sci. , 18, 165-188, (1988).

[53] V. N. Astratov and A. V. Ilinskii, “The evolution of the photoinduced
space-charge and electric-field distribution in photorefractive sillenite

(Bi,yGeOyp, BI,SiOp9) crystals,” Ferroelec. 75, 251 -269 (1987).

[54] I. Foldvari, H. M. Liu, R. C. Powell, and A. Peter, “Investigation of the
photorefractive effect in BiyTeOs,” J. Appl. Phys. 71, 5465 -5473 (1992).

[55] D.F Eaton, “Nonlinear optical materials - the great and near great,” ACS

Symp. S. 455, 128 -156 (1991).

[56] D. D. Nolte, D. H. Olsen, E. M. Monberg, P. M. Bridenbaugh, and A. M.
Glass, “Optical and photorefractive properties of InP-Ti - a new photorefrac-

tive semiconductor,” Opt. Lett. 14, 1278 - 1280 (1989).

[57] A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new pho-

torefractive material: Ky yLyT1 Ny,” Opt. Lett. , 17, 713-715 (1992).

[58] Changxi Yang, Dadi Wang, Peixian Ye, Qincai Guan, and Jiyang Wang,
“Photorefractive diffraction dynamic during writing in paraelectric KTN crys-

tals,” Opt. Letters 17, 106-108 (1992).

-20-

Chapter Two

The Photorefractive Effect

2.1 Introduction

The photorefractive effect can be defined as a change in the refractive
index of a material caused by illumination with light. The incident light
photo-excites free carriers preferentially in regions of high intensity where
they undergo transport, are repeatedly trapped and re-excited, and are finally
trapped in a region of low-light intensity (see Fig. 2-1). The result is a space
charge field spatially correlated with the intensity pattern creating an index of
refraction pattern (grating) via the electro-optic effect. Two beams propagating
in a photorefractive medium can interact with each other by coherent scatter-
ing from the grating formed by their interference. The phases and the intensi-
ties of the two beams can be coupled. This chapter describes the band transport
mechanism of the photorefractive effect, and the formation of the space
charge field. The coupled mode equations are used to calculate the influence
of index gratings on propagation of beams through the material. Since the
index gratings are written dynamically, the solution for the interaction is per-

formed self-consistently.

The photorefractive effect!2> was first noticed in frequency doubling

-21-
experiments in LiNbO, by Ashkin4. Soon thereafter Chen>~” reported the ef-

fect in KTN and proposed that a space charge field could be formed by pho-
toexcitation of electrons, subsequent drift in an electric field, and retrapping in

regions of low-light intensity. This work is the basis of modern theories on
the effect. Since then, the role of defects and transition metal dopants®-!0 has

been recognized as the source of the photoionizable charge carriers. Three

transport processes have been identified: thermal diffusion", drift!2,13 (when

an external field is applied), and the photovoltaic effect!4!°.

A hopping model for charge transport in BaTiO3 has been suggested by
Feinberg!®. In this model, the charges hop from filled to vacant sites when ex-
posed to optical radiation. Although the model is statistical, for short hopping
lengths the results are similar to those obtained with the band transport
model. For large hopping distances the results depend strongly on the statis-
tics of the hopping. This model has not seen a great deal of attention since it is
difficult to ascertain a plausible physical justification for the statistical behav-

ior required to obtain good correlation with experiments.
2.2 Band Transport Model

The band transport model requires the photogeneration of charge carri-
ers. The atomic species which generates the charge carriers (the donor species)

must be stable in the crystal in at least two valence configurations. We write

Np = Nyp* + Ny? where Np is the total donor ion concentration. The more

negative of these two states, designated as Ny? can act as an electron donor al-

-22-

A(z) B(z)

I --- ---7f —— Photo-excited carriers

™ Light intensity
interference pattern

Charge transport due
to drift and diffusion

Trapped charge

0 21 Ant Kg Z

Space charge field
and index change

-23-

Figure 2-1 Diffusion Limited Photorefractive Mechanism (previous page)

The photorefractive effect is a change in the refractive index of a mate-
rial caused by illumination with light. Two incident laser beams A(z) and B(z)
interfere in a photorefractive material. Free carriers are preferentially photo-
excited in regions of high intensity where they undergo transport, are repeat-

edly trapped and re-excited, and are finally trapped in a region of low-light in-
tensity. The steady-state charge distribution p creates a space charge field. In

the diffusion limited case the charge modulation is in phase with the intensi-

ty pattern. The field is shifted exactly one-quarter grating wavelength because

divE = 4xp/e. Thus the phase between the intensity pattern and the space

charge field is 6 = 1/2.

Figure 2-2 Drift Limited Photorefractive Mechanism (next page)

The light intensity pattern forms a space charge field as in Fig. 2-1, but
the large applied field (drift limited case) prevents the charge from accumulat-

ing exactly in phase with the intensity minima; instead it is shifted by a phase
n/2, so that the space charge field is shifted by @ = x from the intensity pattern.
This can be shown by considering the conductivity which must be in phase

with the intensity. Since the current J = GE must be uniform in the steady

state, we conclude that E.. is exactly out of phase with the conductivity o.

-24-

A(z) Biz)
To-4- ---f/ — Photo-excited carriers
fd ™ Light intensity
dale + interference pattern
He t+
l l J
0 2n An Kez
16)
4 Conductivity proportional
to intensity
l |
0 2m An Kgz
Esc o
Space charge field
and index change
0 2n Ant Kez
J The current J = oE is
uniform
| | {
0 21 An Kez

-25-

Conduction Band

N @ODMWOO0O8G8G@O@O@G@O0@
Np Np

Valence Band

Figure 2-3 The Band Transport Model

The energy diagram for the single photorefractive species, two charge

carrier band transport model. A mid-band dopant level is occupied with Np

=N*p + N°, dopant ions. An electron (hole) is photoexcited from an N°,
(N*p)ion by incident radiation to the conduction (valence) band where it dif-
fuses and/or drifts under an applied field before being retrapped at an N*p

(N°) site.

-26-
lowing photoexcitation or thermal excitation of charge to the conduction
band leaving an ionized Nj* site behind. The mobile charge is retrapped at

one of the more positive sites, called Np» turning it into an unionized Ny?

in the process. At the same time, hole mobility can occur when a hole is pho-
toexcited from an Np* site to the valence band and is retrapped at an Np?
site. These processes are illustrated in Fig. 2-3. Charge neutrality is preserved
by postulating a number of non photoactive acceptor sites Na =Np* whose

only purpose is to provide the two species of ions Np. If no Nj” sites existed,

the excited charge would always be forced to recombine with the Np* site

from which it was photoexcited. This would eliminate any transport.

The designations Nj* and Ny? are meant to convey the relative ion-
ization state. In practice, the Np site is always positively ionized. In the mate-
rials discussed in this thesis, the two stable states of ions are given by Cul+

and Cu2+ or Fe?+ and Fe+. In both cases, the less positively charged of the

two ionic species acts as the donor while the more positive acts as the trap.

The earliest recognized form of transport was thermal diffusion.

Amodei realized! that in the diffusion dominated case a space charge is set
up with magnitude E,, = kTK/e which is shifted by one quarter of a wave-
length relative to the intensity grating. K is the grating wavevector, T is the
temperature, and k is Boltzman’s constant. Fig. 2-1 illustrates the diffusion
dominated formation of space charge. The crystal is illuminated with a peri-

odic (sinusoidal) intensity pattern. Charges tend to accumulate in the dark re-

-27-
gions. By Poisson’s equation E,. ~ J p dz, the space charge field is exactly x/2

out of phase with the intensity pattern.

The transport caused by the drift in an electric field leads to a different
result. In Fig. 2-2 the case for the drift dominated case is illustrated. The inten-
sity pattern is again taken to be sinusoidal, and it is known that in the steady
state the current must be constant to prevent the build up of charge within
the crystal (continuity equation). Since the local conductivity is taken to be
proportional to the excitation rate, i.e., to the intensity, the space charge field
must be inversely proportional to the intensity. Only in this way can the

product of the conductivity and the field (the current) remain constant.

Glass et al. were responsible for equations describing the photovoltaic
current!4. All non centrosymmetric crystals can display a photovoltaic effect
in which a photocurrent is generated without the application of an external
field. When charges are photoexcited they are generated with a preferred di-
rection of motion. The retrapping of electrons can also proceed anisotropical-
ly, thus contributing to the current. The effect was explained as an asymmetric
charge transfer process. A directional photocurrent can result if the orbitals of
the defect ions overlap asymmetrically with the host lattice ion orbitals along
the polar axis. This condition is only forbidden in centrosymmetric materials.
Since the photovoltaic effect yields a current in the same way as does applica-
tion of an electric field, it also holds that, in the steady state, the current must
become uniform by the continuity conditions. Thus the space charge field
must be exactly out of phase with the intensity pattern by the same argument

as in the drift dominated case above.

-28-

In photorefractive materials it can happen that any one of these trans-
port processes is dominant, in which case the space charge field behaves as in-
dicated in the preceding paragraphs. In general, however, two or three of the
mechanisms play a significant role and the solution of the space charge field
is determined by a set of rate equations. The development and solution of the

photorefractive rate equations follows.
2.2.1 The Rate Equations

In the following section the rate equations for the space charge field are

solved neglecting the photovoltaic contribution. It is noted that electrons are

excited into the conduction band at a rate [B, + $,I/(hv)] (Np - Np”), and they
combine at a rate y,.nNp*. Similarly, the excitation rate for holes is given by

[B,, + S,1/(hv)] Np* and the recombination rate by y,p (Np- Np*). The vari-

ables are defined on the next page. The rate equations for electrons and holes

are written as

on Je _ I + +
s vie = [st + B.| (Np-Né) - yen Nb (2.1a)
Op . gdh = {cL + | _Nt

The continuity equation is given by

VI = Viet Vein = e 2 (n- NB - ph (2.2)

The current equations for electrons and holes are

je = N€ Ue E+e Dy Vint Ke Se (Np - NO) I (2.3a)

-29-

jh = pe Un E+e Dp Vp + Kh Sh Nb I.

(2.3b)

Equations (2.3a,b) are often expressed using the Einstein relation eD = kTu,

where T is the temperature. Finally, the Poisson equation gives

BaP =. & _Nd -
V-E 285 ae, B+ Na No - p}.

The variables used are defined (in order of appearance)

I = incident light intensity

hv = optical energy of incident photon

Be (B), ) = thermal generation rate for electrons (holes)

S, (Sj, ) = photoexcitation cross section for electrons (holes)
Np = density of defect ions = Np? + Not

Npy°

density of ionized defect ions

Np? = density of unionized defect ions

Ye (Vy ) = recombination rate of electrons (holes)

It

free electron density

li

p = free hole density
Je G,) = total electron (hole) current density

e = electronic charge

(2.4)

-30-

E = electric field
H, (Hy, ) = electron (hole) mobility

De (Dy, ) = diffusion coefficient for electrons (holes)

Ke (kK, ) = photovoltaic coefficient for electrons (holes)
€ = relative dielectric constant
Ey = permittivity of free space

p = total charge density.

The equations given above are used to solve for the space charge field

when the material is illuminated with the intensity pattern given by

Ix) =1d+mei®*4+ce.c.) (2.5)
where m is the modulation index. If the intensity pattern is caused by coher-
ent interfering beams then m is given by m = (I, Ip)1/2/ I where I, and I,
are the intensities of the individual interfering beams and I = 1, +I,. K is the

grating wavevector K = 21/4, which is also directed along x. When m<

spatial dependence of n, p, Np*, and E can be approximated by linearized
Fourier expansions. In the following analysis only terms at the fundamental
spatial frequency are included. In the case of large modulation depth, higher
order harmonics of these quantities should be considered!”!8._ It should be

noted that if one of the interfering beams is an image containing numerous

spatial frequencies, the modulation depth of the image as a whole might be

-31-
quite high while the modulation depth of any individual spatial frequency re-

mains small. In this case, the model for m<<1 would still apply.
2.2.2 Single-Charge Carrier Solution

The simplest case of the band transport model is the case of only a sin-
gle type and species of charge carrier, usually considered to be the electron.

Most materials can be approximated by this case. All parameters describing
holes in equations (2.1) through (2.4) are set to zero: p = y, =S}, = By, = 0. The

photovoltaic contribution is neglected. The following identities hold in this

case:

OND _ On _ yde (2.6)

ot at e

The set of equations (2.1), (2.3), and (2.4) given above reduce to

on -N§) = w{E- Vn + nV-E +kTv’n (2.7)
OND = (s.1_+ Be}(Np-Ni) - Ye n Nb (2.8)
ot hv

. = .--& -N#
V-E ae (n + Na - No}. (2.9)

Linearized Fourier expansions.are postulated as the solutions for Np*,n, and

E. They are of the form
Nb = Nbo + 5 Nou eiKx + ce. (2.10a)

n = no + ym ei Kx 4 Ce, (2.10b)

-32-
E = E+ 5 Ei ei Kx 4 Cc. (2.10c)

Since the crystals are charge balanced N*p g- Ng- Na = 0. In addition the ap-

proximations Np >> Na >> ng and N*p, = Na >> n, are used. After some

work a solution for the space charge field is obtained!929
Ego = -im En{Eo + i Eq) (1 -e“) (2.11)
Eo + i (En + Eq)
Ey = @Na{y Na Eq = Kel (2.12a,b)
eK Np e€

where Ex, is the charge limited space charge field, Ey is the thermal field, and

Eg is the spatially uniform applied field. The time constant?! tis given by

t= ty Hott (Eu. + Ea) (2.13)

Eo +1 (Ex + Eq)

= hv Na 2.14
“0 Se Io (1 + Be hv/(selo)} Np

where

E,, = Na, (2.15)
Ue K

The time constant for approaching the steady-state value of the space charge

field is inversely proportional to the incident intensity. When an electric

field is applied the response time t becomes imaginary so that the space

charge field oscillates before reaching the steady state.

2.2.3 Two-Charge Carrier Solution

-33-
Here the case of a single photorefractive species with two charge carri-
ers - both the electron and the hole- is considered. Equations (2.1) - (2.5) apply

in this case. The solution for the space charge field has been obtained in the

steady state with no applied field2*,23.24. It is

Ea (Gh - Ge)
(Eq + En) (On + Oe) (2.16)

where En and Eg are the same as above. The electron and hole conductivities

are given by?°

— _ © Me Se Io [Np |
Oc = Ne€Ue= - 1 2.17
e He hv ye [Ne ( )
_ _ © Uh Sh Io ie j
Oh = peln= -1|. (2.18)
h = Pen hv yw \Né

When the electron and hole conductivities are equal, no space charge field
can be written in the steady state. Whenever the hole mobility is nonzero,
the space charge field given by (2.17) will be smaller than that predicted by the

single-charge carrier solution (2.11).
2.2.4 Photorefractive Index Change

The space charge field modifies the index of refraction of the material
via the electro-optic effect. As described in further detail in chapter three, the
index ellipsoid is modified by an amount

nij

ABij = A = Tijk Ex + Sijk! €5 (ex-1) (e1-1) E,E} (2.19)

where AB are the electric field induced changes in the “axis lengths” of the

-34-
index ellipsoid, and nj; are the components of the index of refraction tensor.

ijk and Bijkl are, respectively, components of the linear and quadratic electro-

optic tensors, while e, and E, are the dielectric constant and electric field along

the axis i. The approximation e, - 1+ ¢, is usually made.

In noncentrosymmetric materials, Tijk is nonzero, in general. The
quadratic coefficients are often neglected in these materials. If the space charge
field is directed along the c-axis, as it is in the symmetric geometry (Fig. 2-4),

equation (2.19) reduces to

4 = Tij3 Ege (2.20a)
njj

Anj = Onis Esc (2.20b)

where no is the nominal index of refraction with no applied field, and we
have used the relation A(1/n?) =-(2/n%) An. Also, the convention of designat-

ing the caxis the “3” axis has been followed.

For centrosymmetric materials, i.e., ones with a center of inversion, the
linear electro-optic coefficient is required to be zero (see chapter three). Thus
the quadratic electro-optic effect is the lowest order effect allowed. If the space
charge field is directed along axis 3 and an external field Ep is applied along
the same axis, then E3;=E,.+ Ep). The change in the refractive index due to

E,, alone is given by

Anj(Eo+Esc) - Ani(Eo) = Oo gis (e3€0)" ( (Ese + Eo - EB). (2.21)

-35-
Since we are interested in an index grating with the same spatial dependence
as the intensity pattern, that is, eiKZ the E? term is ignored. The useful,

Bragg matched part of the index grating reduces to

Ani; = n3 $ij33 (€3€0)° Esc Eo. (2.21b)

Thus the Bragg matched contribution to the index grating is present only

with, and is proportional to, an applied electric field.

In paraelectric materials the relative dielectric constant obeys the Curie-
Weiss law and near the Curie temperature can reach values as high as 10° 26,
This can lead to large index gratings when a field is applied. When the exter-

nal applied field is time varying the photorefractive response is modulated.

This has been demonstrated in KTN*° and KLTN at frequencies up to 20kHz.

2.3 Photorefractive Beam Coupling

When two beams interfere in a photorefractive medium they create an

index grating as described above. This index grating is shifted in phase with
respect to the intensity grating by an amount ?”. In chapter one it was men-

tioned that this nonzero photorefractive phase leads to energy transfer be-
tween the two beams. The intensity coupling allows several interesting appli-
cations. These include the amplification of a weak signal beam by a pump
beam as well as optical signal processing. But most importantly, the possibility
of gain affords the possibility of oscillation, just as with a laser.

Photorefractive resonators and phase conjugate reflectors are two of a host of

devices28 which utilize this characteristic.

-36-

The magnitude of the intensity coupling, known as the two beam cou-
pling coefficient, is often the basis of characterization for photorefractive ma-
terials. Yet the two-beam intensity coupling is only half of the story, and this
fact must be stressed: The dynamic coupling of the beams affects the phases of

the beams also. The true variables which describe the interaction are the pho-

torefractive phase @ and the coupling constant g (see below); using these fun-
damental variables, the two-beam intensity coupling coefficient is given by I
=2¢ sino (the amplitude coupling coefficient is given by g sing). Meanwhile
the phase coupling is described by a coefficient g cos. Unfortunately, most
experimental results conspire to yield only the value gsind as a parameter, so

it is difficult to experimentally separate g and 6. This may explain the tenden-

cy to conceptually lump the two variables into one. Also, many characteris-
tics of beam coupling and diffraction are calculated ignoring the change of
amplitude of the index grating throughout the material. Instead, the diffrac-

tion, for example, is usually taken to be given by the Kogelnik solution for a
fixed amplitude grating*?. These problems are addressed below and again

more fully in chapter nine.

In this section the coupled mode equations are developed from the
wave equation. Then the equations are used to describe coupling from a thick
constant amplitude index grating. This analysis was first performed by
Kogelnik. Subsequently, we consider scattering off the dynamic gratings writ-
ten with the photorefractive effect. Both copropagating and counterpropagat-

ing (transmission and reflection) geometries are examined.

-37-

2.3.1 Coupled Mode Equations, Fixed Grating

We consider two coherent optical beams?

E(r) = 5 A(r) e “ikir e; + 7 B(r) e -ik2r @) +c. ¢. (2.22)

where e;,@, and k,,k, are the polarization and propagation directions of
the two beams. For simplicity, the polarizations are taken to be parallel; in the

general case, the coupling between the beams would be multiplied by the fac-

tor ey &5 .

The beams propagate in a medium with a spatially modulated index

grating
n(r) = no + i. cos[K-r + 0] (2.23)

where @ is the net shift in phase between the index grating and the intensity

grating, i.e., d includes the individual phases of beams A and B. The beams

obey the scalar wave equation

V-E+o2ue(r) = 0 (2.24)

where the high frequency dielectric constant is given by

e(r) = €gn2(r) = €9[ not Fro ne ilK-r +9) +c. Cc.) ]. (2.25)

Equations (2.22) and (2.23) are used in (2.24) using

PA cc, dA (2.26)
dr? dr

-38-

to obtain

1 [-2ik, dA -kj Ale thin +c. +
2 dr;

1 |-2ik, 4B 3 Ble kor 4 cc. +
2 dr

i(K-r +o)

w* UW Eo [ ng + F(rome - +c.c.)] xX

[A e-ikyr4 8 eitkoar+cic} = 0. (2.27)

It is clear by inspection that cumulative power exchange takes place only

when

k,-k2- K = 0. (2.28)

Note that the power exchange reverses sign with a phase matching length AL
= ™/(|k, - ky - K1). Thus only synchronous terms need to be considered in

(2.27). Using k; = @ €g Ng (2.27) simplifies to

A(z) cos = ea e# Biz) - AW) (2.29a)
B(z) cosB = eo e® A(z) - & BQ). (2.29b)

The loss terms are added phenomenologically to account for optical absorp-
tion, A is the wavelength in the medium, f is the half angle of beam inci-
dence inside the material (see Fig. 2-4), and z is the distance along the bisector

of the beam angles, z = r, 5 cosB. For convenience, z is redefined to be the

scalar distance along the beam propagation direction, z -> z cos. This en-

-39-
ables one to avoid carrying the factor cosB through all the formulas. As a con-

sequence one must take the effective thickness of the crystal as L = d/cosf,

where d is the true thickness of the crystal.

To solve (2.29) the optical absorption term is eliminated by the change
of independent variable A(z) = A(z) exp[ az/2] and B(z) = B(z) exp[ az/2].

The resultant equations are differentiated and are substituted into the result
of the differentiation to obtain simple second-order equations. The boundary
conditions are A(z=0) = A(0) and B(z=0) = B(0). A(0) and B(0) are real because
the phases of the two beams have already been incorporated into the index
grating; this simply means that the only relevant quantity is the relative

phase between the intensity interference pattern and the index grating. The

solution for A and B is easily shown to be?!

A(z) e%/2 = A(0) cos(g z/2) + ie!% B(O) sin(g z/2) (2.30a)
B(z) e%/? = B(O) cos(g z/2) + ie + AQ) sin(g z/2) (2.30b)

from which the intensities are given by

|A(z)° e& = A*(0) cos*(g z/2) + B?(0) sin?(g z/2)
- A(O) B(O) sin(g z) sind (2.31a)
IB(z)P e% = B2(0) cos*(g z/2) + A7(0) sin*(g z/2)
+ A(O) B(O) sin(g z) sind. (2.31b)
We note that the beam coupling is nonzero for all values of 6, except for the

special case @ = 0 and A(0) = B(0). This is at first surprising, because it will be

-40-
shown that in the case of dynamic holography the intensity coupling must be

zero when 6 = 0 for all values of A and B.

When only one beam is incident on the grating, i.e., B(0) = 0, equations
(2.31) give the diffraction from the grating. The transmitted and diffracted in-

tensities are

|A(z)* e% = A*(0) cos?(g z/2) (2.32a)
IB(z)P e% = A?(0) sin?(g z/2). (2.32b)

Comparison with (2.31) yields the interesting observation that when 6 =0

the output intensities of (2.31) are precisely the sum of the diffracted beams
which result from incident beams A(0) and B(0) separately. In other words,
when the index grating and the intensity grating are in phase, the two inci-
dent beams do not affect each others’ intensity diffraction; each beam diffracts
as it would if the other beam were not present. Nevertheless, intensity cou-
pling does occur ( A(0) # A(L) ) unless A(0) = B(0) because each beam diffracts
an intensity proportional to its own incident intensity. This point will be re-

ferred to in the next section.

In the preceding paragraphs we have described the coherent scattering
of two beams incident on a constant amplitude index grating at the Bragg
angle. The scattering associated with photorefractive gratings is more complex
because the index grating is created by the interfering beams themselves. Thus
when the grating causes power or phase transfer between the beams, the
index grating is affected. In the succeeding paragraphs, this problem of scatter-

ing from a dynamically written grating is treated self consistently.

-4]-

2.3.2 Copropagating Geometry, Dynamic Grating

When two beams as in (2.18) interfere within a photorefractive materi-

al (Fig. 2-4), an index grating is written
n(z) = ng + 4 (An(z)eisei K + cc.) (2.33)

where K is the nominal grating wavevector created by the incident beams,
with K =k, - ky, so that K=2k sinB. The magnitude of the index modulation
is independent of the total incident intensity; it only depends on the beam in-

tensity ratios of A(z) and B(z) and is given by?

An(z) = ny A(z) B’(z)/ I) (2.34)
where nj is a material parameter equal to the peak to peak index modulation

when | A(z)! = | B(z)! (see Fig. 2-5). I(z) is the total intensity I(z) = | A(z)|? +

|B(z)|2.. When An(z) is defined in this way, @ becomes the phase between the

index grating and the intensity grating.

After a similar analysis to that leading to (2.29), the coupled equations

are obtained:
A(z) cosB = imp el B(z) - e A(z) (2.35a)
B(z) cosB = im An” ec A(z) -& B@), (2.35b)

Using the value for the dynamic grating in (2.34), (2.35) is rewritten as

-42-

A(z) cosB =i g ei® Ba" A(z) - & A(z) (2.36a)
I(z) 2
B(z) cosB =i g e# AG)" B(z) - & B(z) (2.36b)
I(z) 2

where the coupling constant is defined as g = mn,/A. Note that since g is de-

fined in terms of n, rather than An, it becomes a pure material parameter, the

dependence on the relative beam intensities being factored out. Also (2.36) is
different from (2.29) in that the dynamic grating has canceled the phase de-
pendence of B(z) in (2.36a) and of A(z) in (2.36b). This property leads to a dif-

ferent method of solution.

The method of solution of equation (2.36a) and (2.36b) is straightfor-
ward. The cosf term is eliminated as before by defining z to be the distance
along the propagation directions. The optical absorption term is eliminated by

the change of independent variable A(z) = A(z) exp[ az/2] and B(z) = B(z)

exp[ az/2 ]. Then solutions of the form

A(z) = a(zjeie B(z) = b(z)ei& (2.37)
are postulated where a(z) and b(z) are real. Equations (2.36a, b) can be separat-
ed into two equations each describing the evolution of the amplitude and

phase of the two beams:

a(z) = - sind g b(z)* / I(z) a(z) (2.38a)

b(z) = + sind g a(z)’ /I(z) b(z) (2.38b)

-43-

K Az Symmetric copropagating
A (transmission) geometry

Az Refractive index grating

26

\ /
Nf
k |

aa

Antisymmetric counterpropagating
(reflection) geometry

Figure 2-4. Beam Coupling Geometries

The beam incidence angles for copropagating and counterpropagating

geometries. The grating wavevector is K, = 2k sinf in the copropagating case,

and K, = 2k cos in the counterpropagating case.

-44-

n(z)
A Index change when
- L=I,
ny An
Ny +¥
20 An f4n Zz
Index change when
L< I,

Figure 2-5 Parameters of the Index Grating

The parameters used to describe the index grating in the coupled mode

equations. An is half the peak to peak modulation of the grating and is a

function of the relative beam intensities with An = n, (1,1)! /\. nq is a ma-
terial parameter independent of the incident beams. The coupling constant

g=nyn/n.

-45-
Ci(z) = cos g b(z)° / I(2) (2.39a)
Co(z) = cos g a(z)* / I(z). (2.39b)
Equations (2.38a,b) are solved by converting them to equations for intensities
using I= (a2) =2aa’,and similarly for I,. Note that these “intensities” are
not the true optical intensities, but are related to them by the multiplication

of exp[ az ]. Use of the identity b(z)? = I(z) - a(z)? in (2.38a) and b(z)? = I(z) -

a(z)* in (2.38b) yields simple Bernoulli equations*? which are readily solved

to yield true intensities

I(z) = eal Ti(h +h) (2.40a)
ly +h etl

L(z) = et bh(h+h) (2.40b)
lel#+I

where A(z) = [I,(z)]!/exp[i¢,] and B(z) = [1(2z)]'/2explity], andI = 2g sing is
the power coupling coefficient. As above, L is the effective thickness of the

crystal: L = d/cosB. We have defined 1, =1,(0) and I, =1,(0). Inspection of

(2.40) reveals that when = 0, there is no intensity coupling and, in the ab-
sence of absorption, I;(z) =1,(0). Since the intensities are constant the dynamic
index grating will have a constant amplitude so one might naively expect
(2.40) (with = 0) to correlate with the constant amplitude grating case (2.31).
The analysis above shows that the two equations disagree except when I, =I).
Since the grating leading to (2.31) was Bragg matched to the incident beams,

one must conclude that the grating of (2.40) is not Bragg matched except in the

-46-
case I, = 15. The amount of Bragg mismatch in (2.40) is such that I;(z) = 1,(0). It
follows also that the diffraction off a dynamically written grating will be maxi-

mum for beams not of the same frequency as the writing beams. Maximum
diffraction for = 0 occurs for a frequency mismatch of AB = -g (Iy - 1,)/(@D

(see below and chapter 9). This fundamental result seems to be ignored in the

literature.

The phases of the two beams are readily determined from equations

(2.40a,b) and (2.39a,b) to be
Ci(z) = 5 coto In[I, + Ib et! 7] (2.41a)

C2(z) = -g cosd z - , coto Inf, + lb e*! 2). (2.41b)

Equations (2.40) and (2.41) are used in (2.34) to determine the index

grating written in the material. It is given by

An(z) = n, Vly bb (ly eT #24 ly ett 22 forte (2.42a)
When _ = 0, special care must be taken when using (2.42a). It is easier in this

case to backtrack to equations (2.41) to determine An. The index grating in this

case reduces to

An(z) = ny we exp[ igz(Ib-h)/T. (2.42b)

The influence of the phase coupling has modified the index grating by a pha-
sor exp[ ig z (I, - 1,)/I] in the zero phase case. Thus as predicted above, the

index grating is no longer Bragg matched to the (uncoupled) beams which

-47-
wrote the grating. This Bragg mismatch, in the 6 = 0 case, is just enough to en-

sure that the transmitted intensities are equal to the incident intensities, un-
like (2.31) where the more intense beam diffracted more strongly and ampli-

fied the weaker beam.

The diffraction off the gratings (2.42) is derived in chapter 9; it is ob-
tained by solving the differential equations (2.35) with (2.42) inserted. The re-

sults for the transmitted and diffracted intensities are given by

Bet+R+2h I, e!2/2 cos[ g coso z]|

. I
I,(z) = e% | (2.43a)
+h t+he™
2 Tz 2 el2/2 o¢
biz) = e-% ib le +1-2e'4* cos[ g cosd z)) (2.43b)
+h ILt+hel
Two special cases deserve attention. When I, =I, = I, (2.43) reduces to
hil = ¢ 7 (1+ cos[g cos z]/cosh{g sind z]) (2.44a)
I,(z)/l = © 7 (1- cos[g cos z]/cosh{g sing z}). (2.44b)

When 6 = 0 and the index grating amplitude is constant (2.43) reduces to

I(z/l; = eo [1 - Leng ar sin’[g z/2]] (2.45a)
1+ 12
In(z)/ly = e ae I; x sin’[g z/2]]. (2.45b)
1+ 12

Comparison with (2.32) shows exactly how the Bragg mismatch reduces

diffraction. When I, =I, , (2.45) reduces to (2.32).

2.3.3 Counterpropagating Geometry, Dynamic Grating

-48-
We perform a similar analysis to that done in the previous section for

beams coupled by an index grating written in the counterpropagating geome-

try (Fig. 2-4). Here the beams write an index grating with wavevector _K=k, -
ky. and K =2 k cosf. In exactly counterpropagating geometry K = 2k. In anal-

ogy to work leading up to (2.29) we obtain for this case
A(z) cosB = Lm An ei? B(z) - ° A(z) (2.46a)

B(z) cosB = “im An” e® A(z) + 2 BG). (2.46b)

Using the dynamic grating as before we write

A(z) cosB =i g ei? Ba" A(z) - & A(z) (2.47a)
I(z) 2
B(z) cosB = -i geld A)" B(z) + & B(z). (2.47b)
I(z) 2

It is difficult to obtain solutions of (2.47) when optical absorption is considered

except for certain values of o (see chapter 9); in this section the equations are

solved for the case @ = 0. The solution proceeds similarly as before with ex-

pressions for the intensities and phases of the two beams given by

I,(z) = c2+v2eTz + c} (2.48a)

L(z) = 5 (Vor+v2eTz - c} (2.48b)

-49-

Wilz) = Lg cosd z - coto coth"[V 14+(v/c)eT]} (2.49a)

wo(z) = - L(g cos z + coto coth[V 14+(w/cye™]] (2.49b)

oR

where the definitions c = I,(z)-1,(z) and v* = 41,(0)1,(0) = 41,@) Le!

are used. Also I,(z) = | A(@) \2 and L(z) = | Bi) 12. From (2.48) and (2.49) we

readily calculate the index grating in the material

An(z) = ny Vv. eT2/2 +1 g cosh z (2.50)

where all parameters are as in the previous section.

From (2.50) it is seen that the Bragg mismatch caused by the phase cou-
pling is given by the phasor expli g cos z] regardless of the relative intensities
of the two beams. Thus the reflection (diffraction) maximum off the grating
occurs at a frequency shift AB = (g/2) cos = 2zn Ad/X? for any combination of
beam intensities. This property would be of importance in the manufacture
of holographic interference filters (chapter 9). As a numerical example, if a
grating is written in the counterpropagating geometry with 656nm laser
beams in a material with g cosd = 10/cm and an index of refraction n= 2.0, the
diffraction maximum will be shifted approximately 0.15A. Since the full
width at half maximum (FWHM) response of such a grating scales inversely
with the length, a crystal with a length L 25mm may show little or no reflec-
tivity at the frequency with which the grating was written. These results as-

sume that the temperature of the crystal is kept constant.

The diffraction observed from the grating (2.46) is relatively easy to de-

-50-
termine when the Bragg mismatch term is absent or the grating is constant
amplitude. This occurs under three conditions: 1) when 6 =27/2, 2) when
the frequency of the reading beam is shifted to compensate the g cosd term, or
3) when the grating is constant amplitude (either @ = 0, or c = 0). In other cases
the solution is quite difficult. The general solution for the diffraction is given

in (9.39,40,41) with the modifications n =-cot[o]/2 and B = 1/4. It is too long to

reproduce here.

When = 2/2, the solution, as stated above, simplifies considerably.

One obtains

A@) = Ciy/l+(ePe™ +@MYer (2.51a)
B@) = Caaf +(¥Pe™ +Ca (Yet (2.51b)
where the coefficients are given by
4/ 1+(%Pe™
C; = A(O) c (2.52a)
sf +(¥P aft +(Per [ve Tu
(Yet?
Cy = -A(0) € (2.52b)
C3 = 2G Cy = 2C, (2.52c,d)

The cases when the index grating is constant amplitude are also inter-
esting in that they simplify to conventional looking results#4 and are of peda-

gogic value. This condition occurs either when c = 0, i-e., I;(z) =I,(z), or

-51-
when 6 = 0 with arbitrary intensity. These cases are illustrated and described

in detail in chapter 9.
2.4 Summary

The band transport model of the photorefractive effect was discussed.
Photorefractive rate equations were introduced and solved to yield the pho-
torefractive space charge field magnitude and phase. Limiting cases dominat-
ed by the effects of diffusion, drift, and the photovoltaic effect were men-
tioned. The coupled mode equations were derived and solved in both the co-
propagating and counterpropagating geometries. The solutions for the two
beam coupling as well as the phase coupling allowed the determination of the
index gratings. Formulas for the diffraction off the index gratings were pre-
sented. They are thrashed out in detail in chapter nine. Several important
differences between coupling off a fixed grating and dynamic coupling were

discussed.

-52-

References for chapter two
[1] A. M. Glass, “The photorefractive effect,” Opt. Eng. 17, 470-479 (1978).

[2] T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive ef-
fect - a review,” Prog. Quantum Electr. 10, 77-146 (1985).

[3] G. C. Valley and J. F Lam, in Photorefractive Materials and their
Applications I. , P. Guenther and J. P. Huignard eds. , chapter 3, Springer-
Verlag, Berlin (1987).

[4] A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, H. J.
Levinstein, K. Nassau, “Optically-induced refractive index inhomogeneities

in LiNbO; and LiTaO3, “ Appl. Phys. Lett. 9, 72-74 (1966).

[5] F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,”

J. Appl. Phys. 38,3418 (1967).
[6] F.S. Chen, J. Appl. Phys. 40,3389 (1969).

[7] FS. Chen, J. T. LaMacchia, D. B. Fraser, “Holographic storage in lithium
niobate,” Appl. Phys. Lett. 13, 223-225 (1968).

[8] H. Kurz, E. Kratzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, and
A. Rauber, “Photorefractive centers in LINbO3 studied by optical-, Mossbauer-,
and epr-methods,” Appl. Phys. 12,355 (1977).

[9] W. Phillips, J. J. Amodei, and D. L. Staebler, “Optical and holographic

properties of transition metal doped lithium niobate,” RCA Rev. 33, 94 (1972).

[10] M. G. Clark, F. J. DiSalvo, A. M. Glass and G. E. Peterson, “Electronic

-53-
structure and optical index damage of iron-doped lithium niobate,” J. Chem.

Phys. 59, 6209 (1973).

[11] J. J. Amodei, “Electronic diffusion effects during hologram recording in

crystals,” Appl. Phys. Lett. 18, 22-24 (1971).

{12] J. B. Thaxter, “Electrical control of holographic storage in strontium-bari-

um- niobate,” Appl. Phys. Lett. 15,210 (1969).

[13] J.J. Amodei, “Analysis of transport processes during holographic record-

ing in insulators,” RCA Rev. 32,185 (1971).

[14] A. M. Glass, D. von der Linde, and T. J. Negran, “High voltage bulk pho-
tovoltaic effect and photorefractive process in LINbO3,” Appl. Phys. Lett. 25,
233-235 (1974).

[15] A.M. Glass, D. von der Linde, D. H. Auston, and T. J. Negran, “Excited
state polarization, bulk photovoltaic effect, and the photorefractive effect in

electrically polarized media,” J. Elect. Matls. 4,915 (1975).

[16] J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth,
“Photorefractive effects and light induced charge migration in barium ti-

tanate,” J. Appl. Phys. 51, 1297 (1980).

[17] E. Ochoa, F. Vachss, and L. Hesselink, “Higher-order analysis of the pho-
torefractive effect for large modulation depths,” J. Opt. Soc. Am. B, 3, 181-187
(1986).

[18] A. Bledowski, J. Otten, and K. H. Ringhofer, “Photorefractive hologram
writing with modulation 1,” Opt. Lett. 16, 672-674 (1991).

-54-
[19] N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L.

Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,”

Ferroelectrics 22, 949-960 (1979).

[20] K. Sayano, Ph. D. Dissertation, California Institute of Technology,

Pasadena, California, unpublished, (1990).

[21] A. Yariv, Optical Electronics , chapter 18, Saunders College Publishing,
Philadelphia, (1991).

[22] G.C. Valley, “Simultaneous electron hole transport in photorefractive

materials,” J. Appl. Phys. 59,3363 (1986).
[23] M. B. Klein and G. C. Valley, J. Appl. Phys. 57,4901 (1985).

[24] M. C. Bashaw, T. -P. Ma, R. C. Barker, S. Mroczkowski, and R. R. Dube,
“Theory of complementary holograms arising from electron-hole transport in

photorefractive media,” J. Opt. Soc. Am. B 7, 2329-2338 (1990).

[25] G. Rakuljic, Ph. D. Dissertation, California Institute of Technology,
Pasadena, California, unpublished, (1987).

[26] A. Agranat, V. Leyva, and A. Yariv, “Voltage-controlled photorefractive
effect in paraelectric KTa_.Nb,O3:Cu,V,” Opt. Lett. 14, 1017-1019 (1989).

[27] V. Kondilenko, V. Markov, S. Odulov, and M. Soskin, “Diffraction of cou-
pled waves and determination of phase mismatch between holographic grat-

ing and fringe pattern,” Optica Acta 26, 239-251 (1979).

[28] V. A. Dyakov, S. A. Korolkov, A. V. Mamaev, V. V. Shkunov, and A. A.

Zozulya, “Reflection-grating photorefractive self-pumped ring mirror,” Opt.

-55-

Lett. 16, 1614-1616 (1991).

[29] H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell
Syst. Tech. J. 48, 2909-2947 (1969).

[30] A. Yariv, “Coupled Mode Theory for Guided-Wave Optics,” IEEE J. Q.
Elec. 9, 919-933 (1973).

[31] A. Yariv, private communication (1992).

[32] J.P. Huignard, J. P. Herriau, P. Aubourg, E. Spitz, “Phase conjugate wave-
front generation via real time holography in Bi;SiOvq crystals,” Opt. Lett. 4,
21-23 (1979).

[33] J. Mathews and R. L. Walker, Mathematical Methods of Physics , chapter
1, Addison-Wesley, Redwood City, California, (1970).

[34] A. Yariv, Optical Electronics , p. 498, Saunders College Publishing,
Philadelphia, (1991).

-56-

Chapter Three

The Electro-optic Effect

3.1 Introduction

The electro-optic effect generally refers to a change in the index of re-
fraction of a material under the influence of an applied electric field. This ef-
fect, in conjunction with the photorefractive space charge field, leads to the
index grating formation in the conventional photorefractive effect. Thus the
character of the photorefractive response of a material varies with direction of
light propagation or with direction of an applied field in a way dictated by the
transformation properties of the electro-optic tensor. Furthermore, the re-
sponse of all photorefractive materials hinges on the form of the electro-optic
tensor which, in turn, is governed by the crystallographic space group of the
material. For these reasons, a separate chapter is devoted to an introduction

of the electro-optic effect.

In 1815 Sir David Brewster discovered that transparent jellies devel-
oped artificial birefringence when subjected to mechanical stress.! Further in-

vestigations soon revealed that all materials displayed this photoelastic effect.

About 60 years later Kerr discovered the quadratic electro-optic effect (Kerr ef-

fect)? in which the index change of the material is proportional to the square

-57-
of the applied electric field. The Kerr effect also occurs in all materials. In 1883

Réntgen*4 and Kundt? independently discovered the linear electro-optic ef-

fect (also known as the Pockels effect) in crystals of quartz and tourmaline.

Pockels soon extended their investigations to numerous other materials®7

and developed the first phenomenological theory of the effect.

The Pockels effect has generally replaced the Kerr effect in device appli-
cations because the linear electro-optic coefficients are usually larger than the
quadratic ones. Also the linear electro-optic response can be increased sub-
stantially by applying an alternating electric field at a frequency near a me-
chanical resonance of the crystal. This latter effect is caused by the indirect (or
converse piezoelectric) contribution to the Pockels effect, which is discussed
later. Under certain conditions, however, the quadratic effect remains the pre-
ferred means of index modulation. In particular, it will be shown that the
quadratic effect with an applied electric field can lead to an effective linear
electro-optic coefficient which is useful for voltage control of the photorefrac-
tive response. The strength of the quadratic effect increases when the materi-

al is operated at temperatures near a structural phase transition.

The development of ferroelectric materials beginning with ADP and
KDP®? in the mid 1940s which exhibit large electro-optic coefficients led to the

first applications of the electro-optic effect for light modulation and optical
shutters. Since then, the effect has seen renewed interest as an integral part of
the photorefractive effect. The electro-optic effect is treated theoretically in
the following sections with an emphasis on its role in the photorefractive ef-

fect. The treatment begins with a description of the index ellipsoid.

-58-
3.2 The Index Ellipsoid

Electric fields are known to influence several optical properties of ma-
terials. These include the optical absorption and the spectral lines which can
exhibit splitting in the Stark effect. The electro-optic effect, however, usually
refers to the influence of an electric field on the index of refraction of a mate-
rial, i.e., the modification of the index ellipsoid, also known as the optical in-
dicatrix. We present a brief review of beam propagation in anisotropic mate-

rials which leads to the concept of the index ellipsoid.

The energy density of the stored electric field in a material is given by

U=+AED= EjeyjE; (3.1)

Nr

where the displacement field is defined as D; =e; Fj and summation over in-

dices is implied. ij is the second rank symmetric dielectric tensor. The medi-

um is considered to be nonabsorbing so that the dielectric tensor is real.

Equation (3.1) can be reexpressed as
D2 D2 D2
K+ 24-2 =2U (3.2)
& & &
where “x,” “y,” and “z” are defined as the principal axes of the material, that
is, they describe the coordinate system in which the dielectric tensor has only
on-axis components given by &,, ey and e,. If we define the principal indices

of refraction n.2

j = &/€p G =x, y, 2), and let a vector r be given by D/(2U)!/2 we

obtain

-59-
eye Ve a] (3.3)
nm ny my
which is the equation of an ellipsoid with major axes parallel to x, y, and z

with lengths 2n,,2n,, and 2n,. This construction is known as the index el-

lipsoid (Fig. 1). It is used to determine the two indices of refraction and direc-
tions of polarization for light propagating in an arbitrary direction in an
anisotropic medium!0._ The recipe for determining these parameters is il-

lustrated in figure 3-1 for a positive uniaxial crystal. Here the direction of

propagation is designated s. It is clear that D’s=0 (by Maxwell’s equation

V'D = 0), so that the indices of refraction are given by the major and minor

axes of the ellipse intersecting the optical indicatrix normal to s. For a rigor-

ous derivation justifying this procedure see Refs. 10 and 11.

A crystal may have zero, one, or two optic axes, in which case it is des-
ignated anaxial, uniaxial, and biaxial. We only consider the anaxial and uni-
axial cases here. For a uniaxial material the index ellipsoid is simply an ellip-
soid of revolution as in figure 3-1. Here two axes have the same index of re-
fraction n, (“o” for ordinary) and the remaining (optic) axis has n = n, (ex-

traordinary). Thus the index ellipsoid is given by
ey Voy a] (3.4)
ng ng ng
If no>n, the crystal is positive uniaxial, and if n, < nj it is negative uniaxial.
From the graphical derivation of n(6)!? in figure 3-1, where 0 is the propaga-

tion angle in the material measured from the optic axis, trigonometry yields

-60-

Z optic
axis

n(@)

Figure 3-1. The index ellipsoid method

The index ellipsoid and the coordinate axes defined by equation (3.4)
are illustrated. For a given direction of propagation s, the indices of refraction
and the eigenaxes of polarization can be read off from the semimajor and
semiminor axes of the ellipse formed by the intersection of the index ellipsoid

with the plane normal to s.

-61-

1 ~ cos2@ 4 sin’@ (3.5a)
ng(@) ng ng
no(O) = No. (3.5b)

The designations “ordinary” and “extraordinary” follow from this result.
Quartz, rutile, and water ice are examples of positive uniaxial materials,
while tourmaline, emerald, and barium titanate are negative uniaxial.

Often the index ellipsoid is defined in terms of the optical imperme-
ability tensor which has components Bi = Inj. In analogy to (3.3) it is then

given by

BY ix? + BSoy? + B33z2 + 2BYoxy +

2B33yz + 2B93xz = ] (3.6a)
Oxix; = | (3.6b)
where Bij = B; i is used. Equation (3.6b) is simply (3.6a) using implied summa-

tion notation. In the principal coordinate system (3.6a) reduces to a form sim-

ilar to (3.3):

0 Q 0)
i1X? + Booy? + B33z2 = 1. (3.6c)
As a final note on beam propagation in anisotropic materials the phase

and group velocities are discussed. The phase and group velocities are given
by}8
vp= 2s (3.7a)

Va = Vx@(k). (3.7b)

-62-
They are often graphically calculated with the normal surface, a construction

related to the index ellipsoid. This is the surface defined by constant w(k),

where k = ks!9. Here instead, the coordinate diagram of figure 3-2 is used.
From equation (3.7a) the phase velocity is in the direction of propagation, i.e.,
normal to both Hand D. The group velocity is perpendicular to the tangent
to the normal surface by definition, but it also points in the direction of power
flow!4. Power flow is in the direction of the Poynting vector S=ExH, which
is normal to both E and H. Since D and E are in general not parallel in
anisotropic media, one concludes that the direction of propagation (perpen-
dicular to lines of equal phase) and the direction of power flow are not paral-
lel. Thus two beams of different polarization, propagating in the same direc-
tion, may nevertheless diverge from each other. This phenomenon is known
as “walkoff” and is often a limiting factor of the interaction length of certain
nonlinear optical processes. In nonlinear optics experiments, angles of propa-

gation and geometries are often chosen to minimize walkoff.

As an example, consider two beams propagating at an angle @ from the
z axis in the z-y plane in a medium with a principal dielectric tensor given by
(10 0'
010 (3.8)
10 0 2

Note that this is the dielectric tensor which describes the optical indicatrix of

figure 3-1. One beam is taken to be ordinary polarized so that E=E,=D,. In

this case the phase velocity and the group velocity are parallel. The other

-63-

Figure 3-2. Phase velocity and group velocity

In an anisotropic medium, Eand Dare not generally parallel; in the il-
lustration above they are related by (3.8). The phase velocity is in the direc-
tion of propagation s, normal to both H and D. The group velocity (direction
of power flow) is in the direction of the Poynting vector S = Ex H, which is
normal to both Eand H. For ordinary polarized beams, Dand E are parallel,
For extraordinary beams, E,, = D,,, but 2E z= D z SO

8,0" y y
Thus two beams of different polarization, propagat-

so that sis parallel to v
that s is at angle to Vee
ing in the same direction, may nevertheless diverge from each other. This

phenomenon is known as “walkoff.”

-64-

beam is extraordinarily polarized so that E, =Dy and E, = 2D,. The directions
of phase and group velocity are illustrated graphically in figure 3-2. The two

beams will walkoff from each other as they propagate through the crystal.
3.3 The Linear Electro-optic Effect (Pockels Effect)

3.3.1 Symmetry Properties; Third Rank Tensors

According to Pockels’ theory! the optical indicatrix is modified by ap-

plication of an electric field so that

nij

Bi - By = ABy = 7 1 | = S rijx Ex (3.9)

where E; is the k th Cartesian component of an applied field E. Tijk are com-
ponents of the linear electro-optic tensor. The tensor is of the third rank and
satisfies Tijk = "ik because it relates a vector E with a second rank symmetric
tensor, i.e., the optical indicatrix or optical polarizability tensor. The symme-
try properties of the Pockels tensor are identical to that of the piezoelectric
tensor d, which relates a vector E to a strain field. Thus all piezoelectric crys-
tals can display a linear electro-optic effect and the form of the two tensors is
identical for all crystal classes. The maximum number of independent
Pockels (or piezoelectric) coefficients is 18, reduced from a maximum of 27 for
an arbitrary third rank tensor. This reduction is a result of the symmetry of
the relevant second rank tensor (ijk = Tiik and dijx = jik” i.e., the optic in-
dicatrix for the Pockels effect and the strain field in the piezoelectric effect.

The linear electro-optic effect was first discovered in tourmaline which had

already been noted for its piezoelectric and pyroelectric properties. Its pyro-

-65-

electric effect is particularly strong and as a result tourmaline is known to the

jeweler as the gemstone which attracts dust more rapidly than any other.

Only crystals which lack a center of inversion can display the Pockels

effect. This can readily be shown by the transformation properties of tensors.

One writes!

Vijk = Qi1 &jm Okn Tmn (3.10)

where Oi are 3x3 rotation matrices. The transformation matrix for inversion

is given by Oj = 95 5; i.e., the negative of the identity matrix. Application of

the inversion transformation on (3.10) yields T ii = (-1)(-1)(-1) Tk = Tijk
But for a crystal which has a center of inversion, the inversion transforma-
tion is a symmetry operation and must leave all tensor properties unchanged.
Thus T ik = Tijk: These two requirements can only be met when Tijk = 0.
However not all acentric crystals can possess the linear electro-optic effect.
The octahedral class (O) with symmetry designation 432 is noncentrosymmet-

ric but is forbidden by symmetry from displaying a linear electro-optic effect:

Crystals of space group 432 have three four fold axes (Cy), four triad
axes (C3), and six diad axes (C,). It is the combination of the tetrahedral sym-
metry C, and the characteristic cubic symmetry C3 which conspire to disallow
the linear electro-optic effect. Figure 3-3 shows a hypothetical unit cell of a
crystal with 432 symmetry with the symmetry axes illustrated. It is clear by in-
spection that the structure is noncentrosymmetric; the operation I (inversion)
carries the structure into its mirror image. In fact crystals of 432 symmetry, as

well as many other noncentrosymmetric materials, exist as right- and left

-66-
handed enantiomers.

It can be shown!>.!” that crystals of class 4, i-e., crystals obeying the tetra-

hedral symmetry Cy, have their Pockels tensor components restricted to the

following nonzero values

O 13
O 113
r= ]| 9 O fsx | (3.11)

T4, 15; O

Ts1 -T41 OO
0 0 0
Thus only the above nonzero coefficients need be further considered. If the
above coefficients are subjected to the C3; symmetry operation it can be shown
that r391 =1439 =1493, that is rg, = T59 =I¢3 (see following pages for the contract-
ed index notation definition). Thus comparison with (3.11) yields r4, = 0.
Also it is directly required that rs, = 0 and that ry) = ro =133 = 0 andrj3=1r,)=
0. The nonzero coefficients for a material with three diad axes and a C3 axis

are as illustrated in (3.12)

0 0 0
0 0 0
r=-{ 9 0 0 (3.12)
Tq] 0 0
0 T4] 0

-67-

-68-

Figure 3-3. 432 symmetry unit cell (previous page)

An example of a unit cell displaying 432 symmetry is illustrated. The
chiral “bars” in the (110) positions of the octahedron make the crystal noncen-
trosymmetric. The inversion operation I, when applied to the unit cell, gives
the mirror image of the unit cell. Nevertheless, the cell has Cy symmetry axes
along the (100) directions, C, axes along (110) (and (100)), and C3 axes along
(111). The symmetry axes are shown separately from the unit cell for clarity.
The combination of C, and C3 symmetries conspire to eliminate the possibili-

ty of a linear electro-optic effect.

-69-

Equations (3.11) and (3.12) can only be simultaneously met when all Tijk = 0.
The octahedral 432 class is the only noncentrosymmetric system which is not

piezoelectric and not linearly electro-optic.

The application of an electric field on a centrosymmetric material along
a crystal axis or along a cube diagonal will make the material uniaxial. All
other directions of applied field result in a biaxial material. Likewise, a uniax-
ial material with a field applied along the optic axis remains uniaxial, where-
as a field applied along any other crystallographic axis or along an arbitrary di-
rection yields a biaxial material. In general, a biaxial material remains biaxial
when subjected to a field. Thus the application of an electric (or any other)
field usually lowers the symmetry of a material. Therefore one might expect
that the material, under the influence of that field, might display nonzero
electro-optic coefficients which were previously forbidden. This does occur,
and the magnitude of these previously forbidden coefficients is proportional
to the applied field, so that the field acting on these coefficients yields a
quadratic response. This process is known as a morphic lowering of the sym-
metry and was first observed by Mueller!8. In the case of the electro-optic ef-
fect the change in the refractive index is expected to scale roughly with the ap-
plied electric field relative to the mean interatomic fields. For this reason,
quadratic effects such as morphic symmetry changes and the Kerr effect are
usually ignored when the linear effect is present. These two quadratic effects
are distinct, however, and should not be confused. The Kerr effect is only a

particular subset of the effects which lower crystal symmetry.

Since the maximum number of independent coefficients is 18, it is pos-

-70-

sible to compress the index notation of the r;j,._ In contracted notation the co-
efficients Tijk are compressed to r,,, where (ij) ->m and k=n_ under the fol-

lowing scheme:

(ij) m (3.13)
(11) -> 1
(22) -> 2
(33) -> 3
(32),(23) -> 4
(13),(31) -> 5
(12),(21) -> 6.

This notation often simplifies equations considerably and will be used where
ever convenient. However, it must be used with care because tensors con-
tracted using this notation no longer obey tensor transformation and multi-
plication properties. Additionally, for many tensors encountered, the contract-
ed coefficient does not have the same numerical value as the original, i.e.,
for a tensor T, Tijk1 May not be equal to T,,,. For example in the compliance
tensor, $4441 = $41, but $4449 = $16/2 and $)71 =S¢¢/4. However, the elastic tensor,

which is the inverse of the compliance tensor, has Cijkl = Crm for all i, j, k, 1,

and m, n!?.

3.3.2 True and Indirect Linear Electro-optic Effect

In equation (3.9) we have not specified the mechanical state of the crys-
tal, i.e., whether stress free or strain free (clamped). These two cases will lead

to different electro-optic responses because the applied field causes a strain in

-71-

the unclamped case, and thus to a contribution to the index change from the
photoelastic effect. Considering both electro-optic and photoelastic effects the

changes in the index ellipsoid under application of fields can be written

3 3

ABij = » Tijk Ex + > dijmn Omn (3.14a)
k=1 m,n =1
z 3

AB = >) tie Ex + )) Pijmn mn (3.14b)
k=1 m,n =1

where 6; j and € j are components of the stress and strain tensors respectively.

Gijk1 and Pijk1 are the components of the stress-optical and strain-optical ten-
sors, and r; jk and Tijk are components of the free and clamped electro-optic
tensors. The strain tensor components in (3.14b) are resolved as

3 3
Emn = » Smnop Oop + »y dmnk Ex (3.15)
op=1 k=1

where s is the compliance tensor and d the converse piezoelectric tensor.

Insertion of (3.15) into (3.14a) yields

3 f. 3 3 3
ABi = »y Tijk + »y Pijmn Amnk | Ex + > »y Pijmn Smnop Oop. (3.16)

k=] m,n = 1 mn=l op= 1

Comparison with (3.14a) reveals that

qijkl = Ss Pijmn Smnkl (3.17)
mn= 1
Tijk = Tijk + »y Pijmn Gmnk. (3.18)

m= 1

-72-

Therefore the electro-optic coefficient r’;,, measured with Zero strain

(clamped) differs from the free coefficient Tijk by the product of the converse
piezoelectric tensor with the strain-optical tensor. The clamped value ijk is
usually referred to as the true electro-optic effect while the free value Tijk is
the sum of the true and the indirect electro-optic effects. The indirect contri-
bution can be either negative or positive. Substantial increases in the electro-
optic response can be achieved by operating an electro-optic material with a
time varying field applied at frequencies near a mechanical resonance. As a

historical note, both Rontgen and Kundt believed that the linear electro-optic

effect was due entirely to the indirect effect. It remained for Pockels to prove

the existence of the true linear electro-optic effect®.
3.3.3 Application to the Photorefractive Effect

3.3.3.1 Electric field applied along z

As described in chapter two, when a photorefractive material is illumi-
nated with a spatially periodic intensity pattern a spatially periodic electric
space charge field is formed. If the geometry used is as illustrated in figure 2-
4, then the space charge field will be nominally directed along the z axis: E,. =
Eco cos(K,2)z. Referring to equation (3.9) the index ellipsoid is modified ac-
cording to

ABi =A

Nj

| | = T7j3 Ege. (3.19)

In a crystal of barium titanate with crystal symmetry 4mm the relevant non

zero coefficients are 1413 =1593 and r333 (see equation (3.23) and Refs. 10, 11 for

-73-

a list of material electro-optic coefficients). Thus the index ellipsoid in the

presence of the field is given by

[Boi + risEsck? + (B%. + risBscly? + (B93 + r33Bsclz? = 1 3.20)
where as stated above, Bj, = By. = 1/ Ny? and B33 =1/ No. Since there are no
cross terms in expression (3.20) it is of the form (3.4) and we may read the new

indices of refraction directly. Using A(1/n2) = -2 n-3An we can write

Nx = Ny = No - 2 113 Ese (3.21a)

Ny, = Ne - > 133 Eg. (3.21b)

where it is still to be understood that E, is a function of z. From the above we
can conclude that a plane wave propagating in the x direction and polarized
in the y direction, i.e., ordinary polarization, will see an index grating given

by (3.21a)

Any(z) = - “ons Esccos[K gz] (3.22a)

while the same beam extraordinarily polarized (along z) sees a grating

An,(z) = - 5 133 Esccos[Kgz]. (3.22b)

If the space charge field had been applied along another direction cross

terms would have appeared in an equation analogous to (3.20) signaling a ro-
tation of the index ellipsoid. This effect is covered in the literature!9 and will

be illustrated later in the chapter.

-74-

3.3.3.2 Effective electro-optic coefficient for arbitrary electric field

The effect of a space charge field directed along the z axis on the index
ellipsoid was calculated in (3.22a,b). When two beams interfere in a photore-
fractive material at an arbitrary angle the effective electro-optic coefficient
must be calculated as a function of the beam geometry. In this section we cal-

culate the effective electro-optic coefficient for propagation in crystals such as
barium titanate with 4mm symmetry292!_ The nonzero electro-optic coeffi-

cients of the 4mm class are given by

0 0 143
0 T}3
r = 0 0 133 (3.23)
O rm O
t™yo O O
0 0 0

The index change is then defined in the literature as An = ro¢E,./2n 22 so that
the effective electro-optic coefficient is related to the conventional ones by a
factor of n4. In figure 3-4 two beams are shown propagating at angles o and B
with respect to the optic axis (z axis). Propagation vectors of the two beams
are given by k, = k(0, sina, cosa) and k» = k(0, sinB, cosB) so that the index

grating wavevector is given by

-75-

Figure 3-4. Geometry for arbitrary angle two-beam propagation

Two beams propagate at angles a and B from the optic axis, and in the

zx plane, of a uniaxial photorefractive material. The space charge field has

grating wavevector K,, =k, - kj.

-76-

ky = kz -k; =k(0, cosB-cosa, sinB-sina) =

2k sin(B] 0, cog 8), sin{*P) (3.24)

2 2
= 2k sinf B] (0, ko, ks}

Therefore the unit vector specifying the direction of the grating wavevector

and the space charge field is e,, = (0, ky, k3). Ifa quantity A is defined with A

=e*r'e,.°e€ where ris the electro-optic tensor and € is the dielectric tensor
with nonzero components €; = €) = Ng’ and £4, = n,”, then the index change
is given by2324 An(x) = 1/(2n) e*s ‘A‘e, where e is the polarization vector
of wave a or B. For 4mm materials r*e,. = ijk esck is given by

T13 k3 0 0

P-esc= 0 113 k3 42 ko (3.25)

0 142 kz 33 k3

where, as an example, the first term in the matrix of (3.25) is given by 214, k;

=143k3. For ordinary polarized beams we have e =(1,0,0), and application of

the above relations yields

a+Bl (3.26)

_ 4 -,— nf4 .
Tet = Ng 713 k3 = ng 3 sin 5

* *
For extraordinary polarized beams we have e , = (0, cosa, -sina) ande g = (0,

cosB, -sinB) by inspection of figure 3-4. In analogy to (3.26) we obtain”?

-77-
4 2 2 2(a +B
ng 113 cosa cosB + 2 ng ng r42 cos 5

ret = sin a+) B27)
+né 133 sina sinB

In BaTiO3, for example, r45 >> 143,133 So the best photorefractive response oc-

curs not in the symmetric geometry of Fig. 2-4 because in that case

cos[(a+B)/2] = 0.

The index change as defined after equation (3.23) is proportional to the
space charge field which is also a function of the propagation angles a and B.

When no external field is applied, the magnitude of E,. for extraordinary po-

larized beams is given by (see chapter two)

Ee] = KD —*t_—- cosfa. -B (3.28)
© 1 + (kg/ko)

where kT is the thermal energy, e the electronic charge, k, the grating

wavevector, and kg = [N ,e*/(eegkT)]1/2. When qa and B£ are approximately

equal, E.. will be proportional to k, and thus by (3.23) to sin[(a — B)/2].

In the following section we describe the electro-optic effect in materials
which are forbidden by symmetry from displaying the Pockels effect. The ten-
sor transformation properties become slightly more involved because the rel-
evant tensor is of higher rank, but many problems are simplified by the

isotropy of the index ellipsoid in centrosymmetric materials, i.e., ny = ny =n,.

3.4 The Quadratic Electro-optic Effect (Kerr Effect)

The quadratic electro-optic effect (Kerr effect) was discovered by Kerr

-78-
several years before the Pockels effect was known. It is a higher order effect

and follows from Pockels theory2> when higher order susceptibilities of the

optical indicatrix are considered. If we consider higher order terms in equa-

tion (3.9) we obtain

ABij = qe = Sri Ex + © Kiga Ex Ei
ny k k,l
=Tijk Ex + Kijar Ex Ei (3.29)

where Kix are the quadratic electro-optic coefficients. The quadratic electro-

optic effect is more commonly expressed in terms of polarizations so that

Kijmn ExE) = gijmn PxPi (3.30)

Kj
Siikl = kl (3.31)
€6 (ex - 1) (er - 1)

The quadratic coefficients are usually ignored when the linear effect is pre-

sent.

The Kerr effect can be resolved into a true and an indirect quadratic ef-

fect in an analysis similar to that leading up to equation (3.18). The result is

gta: Sha = YY pijmn Qmnki (3.32)

mn=1
where the subscripts “u” and “c” refer to unclamped and clamped conditions.

Here p is the strain-optic tensor and Qis the electrostrictive tensor.

Contracted notation is often used to reduce the number of indices of

fourth rank tensors. A tensor with components Tijkl is contracted to Ty. y

-79-

where (ij) -> m and (kl) -> n in the same way as performed above for the lin-
ear electro-optic coefficients. The Kerr effect is most often studied for cen-
trosymmetric materials for which only three independent g;;,; exist. They are

211 = 820 = 833, B12 = B13 = B03, and B44 = Yss = Bes, which in the original nota-

tion are

811 = 81111/ 822227 83333
812 = 81122 81133” 82233" 83322 82211 83311
844 = 81212 81313" 82323 83232 82121" 83131

= &1221/ 81331 82332 83223 82112" 83113-

Unless the electric fields are applied only along one axis, numerous terms

must be considered in the calculation of the index ellipsoid.

Of the perovskites used as photorefractive materials, most have com-

parable values of the quadratic electro-optic coefficients. For potassium tanta-

late niobate (KT ¢

and 24, = 0.147 m4/C? 2627. For KLTN the values are comparable. In chapter

six the Kerr coefficients of a KLTN are determined by measuring the

birefringence as a quadratic function of the applied electric field.
3.4.1 Symmetry Considerations, Fourth Rank Tensors

As proved in the previous section the Pockels effect must vanish in
centrosymmetric materials. Another way to see this is to consider the effect of
an applied electric field E on a material. Let the material be centrosymmetric.

If the crystal is reversed with respect to the field it is equivalent to reversing

-80-

the direction of the electric field. By the symmetry of the material, a reversal
of the crystal orientation must not affect the physical situation and the refrac-

tive index change, but the sign of E, and E; is reversed. Thus the index ellip-
soid is modified by AB = TijK Ey) + Ki CECE). Comparison with (3.29)
shows that rj, =0 but Kjjx1 5 not restricted. If the material did not possess a
center of symmetry there would be no reason why the two orientations of the
crystal would lead to the same physical situation, and therefore neither Kj;
nor rij, would be required to equal zero. Thus we can conclude that the Kerr

electro-optic effect can exist in any material.

The Kerr electro-optic tensor relates the square of a vector (the electric
field) with a second rank symmetric tensor (optical indicatrix). Since the
order in which the electric fields are applied is immaterial and as discussed
above, the optical indicatrix is symmetric, we can write Kix = Kiak = Kix
Therefore the number of independent coefficients is reduced from 81 (for an
arbitrary fourth rank tensor) to 6x6 = 36. The quadratic electro-optic tensor
has the same symmetry properties as the electrostrictive tensor Q and each of
these tensors is of identical form for a given crystal class. In fact, the elec-
trostrictive tensor and the Kerr tensor have the same relationship to each

other as the piezoelectric and the Pockels tensors.

The symmetry of Qand K must be distinguished from that of both p,
the photoelastic (strain-optic) tensor, and c¢, the elastic tensor. All are fourth
rank, but each has distinct symmetry properties. The photoelastic tensor re-
lates two second rank symmetric tensors (the strain field and optical indica-

trix) and has 6x6 = 36 independent coefficients. Reversal of an applied stress

-81-

field will reverse the photoelastic response, whereas reversal of an applied
electric field leaves the Kerr response invariant, i.e., the Kerr effect is quadrat-
ic while the photoelastic effect is linear. Thus the forms of these two tensors
must be distinct. Meanwhile the elastic tensor c relates the stress field with
the strain field, also both second rank symmetric tensors28. But c has the ad-
ditional constraint that Cijkl = Ckliy $0 that only at most 21 independent coeffi-
cients remain. This last requirement follows by conservation of energy: The
elastic energy is U = CijkIeijeky and must be conserved under interchange of

strain components e;,. No analogous conserved quantity exists for the other

tensors considered.

3.4.2 Application to Photorefractive Effect

When a centrosymmetric photorefractive material is illuminated with
two interfering monochromatic plane waves, a spatially modulated intensity

pattern and a space charge field are formed, both having a grating vector k, =
2k sin@. Here k is the wavevector of the interfering beams and 6 is the angle

between the beams. The space charge field can be written

Eg(r) = fee leike r +e. cl}, (3.33)

The optical indicatrix is modified by

ABi(r) = gif €} (€x - 1) (er - 1) Escx(r) Esca(r). (3.34)

The quadratic form in E will clearly yield a term with spatial dependence 2 k g

as well as a constant term. The index grating thus formed cannot meet the

Bragg condition; this grating cannot cause coupling between the beams and is

-82-
useless from the standpoint of photorefractive coupling.

If a uniform externally applied field E, is maintained across the crystal
then the net electric field inside the material becomes E(r) = Ep + E,,(r) so that
the change in optical permittivity in (3.34) will have cross terms of the form
Eo KEsc1 ©xPl i Ky r]. These terms will be Bragg matched to the beams which
wrote the index grating and will be able to cause coupling between them.
Only these contributions to the index change are considered useful in the fol-

lowing sections.
3.4.2.1 Electric field applied along z

In a centrosymmetric material consider the experimental geometry as
shown in figure 2-4. The laser beams are symmetrically incident so that E,. is
directed along z. The applied electric field E, is also directed along z. The op-

tical indicatrix is modified by

ABij = 2 gij33 €6 €2 Ep Esc (3.35a)
ABi, = ABs. =2 212 &4 €2 Eo Ege (3.35b)
AB33 = 2911 €5 €? Eo Ege (3.35c)

So that the index ellipsoid becomes

+ + 2 212 € €? Eo Esc y*
No

4 + 2 212 €§ €2 Eo Bsc x2 4
No

+ + + 2 933 e6 €2 Eg Ege|z?2 = 1. (3.36)

no

Thus an ordinary polarized beam propagating along x (polarized along y) will

see an index grating given by

-83-

Any(z) = - ng 212 &% €2 Eq Ecc (3.37a)

while the same beam polarized along z will see

Anz) = - ng g11 & €2 Eo Ese. (3.37b)
Inspection of (3.35) reveals that the applied field has made the initially cen-
trosymmetric material uniaxial with the optic axis directed along the direc-
tion of applied field. We also note that the applied electric field has induced

in the material “effective” linear electro-optic coefficients given by

"r3" = 2 gir €§ €? Ey (3.38a)
33, = 2 811 &} 2 Eo. (3.38b)
If a space charge field is introduced in the x or y directions it is readily shown

that effective ry =r, coefficients have also been generated by the application

of E, along z with magnitude

vt

rq.) = "rs; = 2 244 €} 2 Eo. (3.38c)
Thus application of an external field along z makes a centrosymmetric mate-
rial with space group m3m (O,) exhibit effective linear electro-optic coeffi-
cients with the symmetry of a 4mm tetragonal crystal (see (3.23)). This result
stands to reason because many perovskites, including KLTN and BaTiO3,
have high temperature paraelectric phases with O, symmetry and undergo a
ferroelectric transition to a tetragonal 4mm state. The main difference be-
tween these two phases is the development of a uniform spontaneous polar-
ization in the material along the direction of the optic axis. The applied field
in the centrosymmetric phase serves to emulate the internal spontaneous

field so that the crystal behaves as if in the ferroelectric phase.

-84-
3.4.2.2 Electric field applied in an arbitrary direction

It has been shown above that a centrosymmetric material becomes uni-
axial when a field is applied along a crystallographic axis. When the field is
applied off axis, the situation is complicated by the multiple Cartesian compo-

nents of electric field. The symmetric geometry is considered again (Fig. 2-4).

The index ellipsoid in the presence of the field E = E(cosa, sina, 0) is given by

+ + Ki, Ef + Ki2 Ey| x2 +

4 + Ki2 E2+Kiy E; y?
No

No

+[-L + Kio (EX + Ej))2? +4 Kay ExEyxy = 1, (3.39)

No

We can immediately solve for n, by inspection

n, = No - 2 Kio E’. (3.40)

However the x and y terms of the ellipsoid have cross terms so we must find

a rotated set of coordinates x’ and y’ to diagonalize the expression. It is readily

shown that the axes are rotated by an amount @ where!9

tan{(20] = 4 Ka ExEy = 2 244

tan[2o]. (3.41)
Kui - Kao EZ - Ey $11 - $12

The semimajor and semiminor axes lengths in the xy plane (i.e., the ordinary
and extraordinary indices of refraction for beams propagating along z) can

then be calculated using a similar expression to that following (3.24):

-85-

Optic Axis Angle - 8
oO on
SG e N FB OW

o 9
= OO

oO WN

08 1 1.2
Applied Field Angle - a

Figure 3-5. Applied field effective c axis

In paraelectric materials the c axis is defined by the application of an
electric field. However, unless € = 2g¢4,/(1,-212) = 1 (isotropic case) the optic
axis is not parallel to the applied field. The relationship between the angle of
the optic axis and the angle of applied field is illustrated above for the cases €

=1/2, 1, and 2.

-86-

1 = ot. B- oy (3.42a)
ny
y’

where # is the optical indicatrix matrix and ois a unit vector in the direction
of the semimajor or semiminor axis, that is Oy = (cos8, sin, 0) and oy = (-

sin®, cos0, 0). We need not explicitly state the resultant indices of refraction to
conclude that the crystal is now biaxial with one optic axis along z and the

other at an angle @ to x in the xy plane. Thus the angle of the optic axis is not
generally equal to the angle of the applied field a. Figure 3-5 plots 0 versus
from(3.41) for several values of € = 2g44/(@1;-12). When € = 1 the optic axis is

always parallel to the applied field. The condition ¢ = 1 is well known as the

isotropy condition for cubic materials (Ref. 10 p. 323); for example, the num-
ber of independent elastic constants of a material drops from three for a cubic

system to two for an isotropic system when the condition c4q = (¢y-Cy2)/2 is
fulfilled. When gy, is larger than in the isotropic case, 6 2 a, so that the optic

axis is at a greater angle to the crystallographic axis than the applied field.
This occurs because g4, is the term responsible for rotating the index ellip-

soid. When gy, is smaller than in the isotropic case, the optic axis resists devi-

ation away from the crystallographic axis, i.e, 0

3.5 Summary

The basic elements of the electro-optic theory were presented, and their

-87-

application to the photorefractive effect discussed. The linear electro-optic
(Pockels) effect was shown to be forbidden in materials with a center of inver-
sion, and also in noncentrosymmetric cubic materials of symmetry 432. The
Pockels effect can be resolved into the true- and the indirect Pockels effects,
where the indirect term arises from a photoelastic/ piezoelectric contribution.
The quadratic electro-optic (Kerr) effect is allowed in all materials but does not
lead to a photorefractive (Bragg matched) grating without a uniform applied
field. The applied field serves to induce an effective linear electro-optic coeffi-

cient.

-88-
References for chapter three

[1] D. Brewster, “On the effects of simple pressure in producing that species
of crystallization which forms two oppositely polarized images and exhibits

the complementary colours by polarized light,” Philos. Trans. A 105, 60-64
(1815).

[2] J. Kerr, “A new relation between electricity and light: dielectrified media

birefringent,” Philos. Mag. 50, 337-348 (1875).
[3] W.C. Rontgen, “Ueber die durch elektrische Krafte erzeugte Aenderung
der Doppelbrechung des Quartzes,” Ann. Phys. Chem. 18, 213-228 (1883).

[4] W.C. Rontgen, “Bemerkung zu der Abhandlung des Hrn. A. Kundt:
Ueber des optisches Verhalten des Quartzes im elektrischen Felds,” Ann.

Phys. Chem. 19, 319-323 (1883).

[5] A. Kundt: Ueber des optisches Verhalten des Quartzes im elektrischen

Felds,” Ann. Phys. Chem. 18, 228-233 (1883).

[6] F. Pockels, “Ueber den Einfluss des elektrostatischen Feldes auf das optis-

che Verhalten piezoelektrische Kristalle,” Abh. Gott. 39, 1-204 (1894).

[7] F. Pockels, Lehrbuch der Kristalloptik , B. G. Teubner, Leipzig, Germany
(1906).

[8] B. Zwicker and P. Scherrer, “Electro-optical behavior of KH,PO, and
KD.PO, crystals,” Helv. Phys. Acta 16, 214-216 (1943).

[9] B. Zwicker and P. Scherrer, “Electro-optical properties of the signette-elec-

-89-

tric crystals KHjPO,4 and KD,PO,,” Helv. Phys. Acta 17, 346-373 (1944).

[10] A. Yariv and P. Yeh, Optical Waves in Crystals , chapter 4, John Wiley &
Sons, New York (1984).

[11] M. Born and E. Wolf, Principles of Optics , Pergamon Press, New York
(1965).

[12] Philadelphia, (1991).

[13] J. D. Jackson, Classical Electrodynamics , p. 302, J. Wiley & Sons, New
York (1975).

[14] ibid. p. 237.

[15] T., S. Narasimhamurty, Photoelastic and electro-optic properties of

Crystals , chapter 8, Plenum Press, New York (1981).
[16] ibid. chapter 2.

[17] S. Bhagavantam, Crystal Symmetry and Physical Properties , Academic
Press, New York (1966).

[18] H. Mueller, “Properties of rochelle salt, IV ,” Phys. Rev. 58, 805-811
(1940).

[19] see Ref. 15, p. 152.

[20] M. Segev, California Institute of Technology, Pasadena CA, private com-

munication (1992).

-90-

[21] D.F. Nelson, Electrical, Optic, and Acoustic Interactions in Dielectrics ,J.

Wiley & Sons (1979).

[22] J. Feinberg, D. Heiman, A. R. Tanguay, Jr. , and R. W. Hellwarth,
“Photorefractive effects and light-induced charge migration in barium ti-

tanate,” J. Appl. Phys. 51, 1297-1305 (1980).

[23] J. Feinberg, “Asymmetric self-defocusing of an optical beam from the

photorefractive effect,” J. Opt. Soc. Am. , 72, 46-51 (1982).

[24] K. R. MacDonald and J. Feinberg, “Theory of a self-pumped phase conju-
gator with two coupled interaction regions,” J. Opt. Soc. Am. , 73, 548-553
(1983).

[25] F. L. Wang and A. Y. Wu, “Analytical model for the quadratic electro-
optic effect of perovskites,” Phys. Rev. B 46, 3709 -3712 (1992).

[26] J. E. Geusic, S. K. Kurtz, L. G. Van Uitert, and S. H. Wemple, “Electro-
optic properties of some ABO perovskites in the paraelectric phase,” Appl.

Phys. Lett. 4, 141-143 (1964).

[27] FS. Chen, J. E. Geusic, S. K. Kurtz, J. G. Skinner, and S. H. Wemple,
“Light modulation and beam deflection with potassium tantalate-niobate

crystals,” J. Appl. Phys. 37, 388-398 (1966).

[28] R. P. Feynman, The Feynman Lectures on Physics , volume II, chapter 38,
Addison-Wesley, New York (1964).

-9]-

Chapter Four

Crystal Growth and Material Properties of

Potassium Lithium Tantalate Niobate (KLTN)

4.1 Introduction

As discussed in the introductory chapter of this thesis the potential of
volume holography for the construction of memory systems and computer
interconnects is well established. It is expected that the data storage density of
holographic media will reach 10!* bit cm-3 (in the diffraction limit), with
data retrieval rates up to 10Gbit/sec. Moreover, the special nature of holo-
graphic memories makes them especially attractive for implementing uncon-
ventional computing architectures such as associative memories and neural
networks. However, this potential has never been realized primarily due to
the absence of suitable storage media. The motivation behind the crystal
growth effort at Caltech in the last few years has been the development of a
new type of storage medium, the paraelectric photorefractives. These materi-

als were known to have three chief advantages: a) very high diffraction effi-
ciency!?, b) control of the photorefractive diffraction by an external electric

field!-4 , and c) special fixing mechanisms!.

Potassium niobate was the archetype of the crystal classes grown at

-92-

Caltech. It has been established as one of the most promising photorefractive

materials, but it cannot be operated at room temperature in the paraelectric
phase since its primary phase transition occurs at 716 K; below that tempera-
ture it is ferroelectric. In order to overcome this problem, tantalum was added

to partially replace niobium in the crystal. In a preliminary set of experiments
at Caltech, potassium tantalate niobate ( K,_,Ta,NbO, or KTN ) crystals were
grown®. The object was to add enough tantalum to the material to achieve a
phase transition at or near room temperature. This condition is desired be-
cause the paraelectric photorefractives operate most efficiently about 10K
above the phase transition (see chapter 3). Unfortunately, it was found that in
KTN the operating point could not be raised above 220K because the phase
transition becomes first order’ and the optical quality deteriorates substantial-
ly. It was therefore necessary to find a composition in which the phase transi-
tion can be raised without loss of optical quality. This composition was found
to be potassium lithium tantalate niobate (K,_yLiyTay_,Nb,O3 or KLTN).
(Since the inception of the project to develop KLTN, the growth techniques
for KTN have been improved and it is now possible to operate KTNs at ap-
proximately 280K, but the KLTNs grown are found to have superior optical
properties). This chapter describes the growth of KLTN crystals and their ma-

terial properties.

K,_yLiyTa,_,.Nb,O3 forms a solid solution for all values of x between 0
and 18, and for y < 0.13. Crystals with these compositions are transparent fer-

roelectrics which show very large electro-optic effects just above their Curie

temperature. They have the cubic perovskite structure and are readily doped

-93-
with transition metals. The Curie temperature increases with increasing nio-

bium or lithium concentration’. For very small values of the lithium con-

centration the material undergoes successive phase transitions on cooling,

from cubic to tetragonal, then orthorhombic, and finally rhombohedral.

KLTN crystals are grown in a non-stoichiometric flux containing an ex-
cess of potassium carbonate, since both potassium tantalate and potassium
niobate melt incongruently. The growing crystal nucleates on a cooled seed

which touches the flux.

In section 4.2 we describe the top seeded solution growth method
(TSSG), and in 4.3, the crystal growth system. In section four, the specifics of
particular growths and the compositions grown to date are enumerated.
Phase diagrams for the KLTN system and structural characteristics of crystals
grown with the TSSG method are discussed in section five. Optical absorp-
tion spectra are also presented in this section. In section six, the influence of
lithium in KLTN on the ferroelectric transition temperature and character is
described; the contrast with KTN is illustrated. The effects of niobium concen-

tration on the phase transition is discussed as well.
4.2 Top Seeded Solution Growth Method

The KLTN crystals are grown using the top seeded solution growth
method!-!5. First, the powder ingredients including an excess of an appro-

priate solvent, in this case potassium carbonate, are thoroughly mixed and
packed into a 100m] high form pure platinum crucible. The crucible is placed

in the center of the growth furnace and its contents heated and melted togeth-

-94-

er. The molten flux is soaked at a high temperature to ensure thorough mix-
ing. Then it is cooled to approximately thirty degrees above the anticipated
growth temperature. Meanwhile a seed crystal is attached to a ceramic pulling
tube with pure platinum wire, and is lowered into the furnace through an
opening at the top. After the seed and pulling tube are allowed to reach the
temperature of the furnace ( ~ 1 hour), the seed is touched to the surface of
the flux for about one minute. The seed is then raised to see if melting has oc-
curred. If so, the temperature is dropped ten degrees. If no melting has oc-
curred, the temperature is dropped two degrees. This process is repeated until
the rounded edges of the slightly melted seed are seen to sharpen up into a
square outline. This indicates that growth has started. The seed is then

redipped and is left undisturbed for at least twenty hours while the tempera-
ture is ramped down at 0.5-1.0°C/hr. Then if the crystal appears to have grown

properly, the pulling is started. Throughout the entire process, the seed is ro-

tated at about 20rpm to maintain a homogeneous mix in the flux.

The pulling is usually done at 0.5mm/hr until the crystal is free of the
flux (24-36 hours). Often the cooling rate is increased to 1-1.5°C/hr during the
pulling stage. After pulling the crystal is slowly annealed to room tempera-

ture at a rate between about 10°C to about 30°C per hour.

4.3 Crystal Growth System

The crystal growth system is composed of a high temperature furnace, a
seed support and pulling assembly, and driving and control electronics. A

cross section of the furnace used for crystal growth is shown in figure 4-1. The

-95-

\ Air cooled

fog ff FP PF LP LF FP ff Ff
we ig Ne ND PED PAO BE FIP. wae any
“oe
Sy ON ON SONOS ss NNN ON ON NS
Pr
sou ON NS SN ON SN NON ON OS ~~ Se S NOW NON
s~ % N ys NON
oes foes
ce 2 NWN
oe eee al
ey ws
yey fee
owes ws
rw fies
WS Ps
ea eles
ws Sats
fof re F
Sy VAS
aa 4 4 #4
ah a xs NON
“oe ee eal
SNS as
ay yes a
was . VN
rene Heating NON
‘rae’ aor
NON
ery ] bese
ay element ey
fg ff
ae lof 6 “Sey
va ve
wo ON a NON
fof 4 — oe ed
SON ON ~ NOS
fone be es
wo VAS
vie bees
Le Plati es,
Ran atinum aay
Cele g . ata
ia
eed crucible — LAS,
wa ve,
. ; ’
wos Seed VON
carrer ane
x OS
“oe eee
Wo crystal VAs
foe we Oe
wo SAS
ee ew,
ws NN x N ON
“oe fife
~~ ON ON sa NN
ee oy ee
Parr sy
foe “ote
SONS ss NON
fog fof
‘SON ON xa NON
vow eof #
~ oN ON ~ eS
“ie bees
rn as a NO
“of # fof #
wos AS
“oF fF
ON ON SNS
aoe foe
Sy Ss
ee fee
‘SON ~~ OS ON
ces eee
~ ON N a
fotog rons
~ OA ON eT
fod oe
VAS Ns
ff A woe re
wo SONS
bees be ee
Nos SAS
ff A
N 6
. *
cos " 7 er c eT
OOF fo A oo OP A A AP A A EP OP EE EP EEE AE A PE PPA LE PP EP LF PPP PPP PL FL LP La
a a i i i, i a a a a a a a a a a en a a er a a a a
of fo Fo oe Pe PA A A A A A Pe PP PAP PE AE EPA PA PE FP LE LP LA AHL HF PLL LP OLE
NON NON SON NON SON NN NS NAS NN NS NNN NN NN NN RS NS NR ON NS SON OS,
EECCA CC

Figure 4-1. The growth furnace

A cross section of the growth furnace is shown. Insulation consists of
concentric alumina ceramic cylinders. Viewing and illumination ports allow
observation of the platinum crucible and seed during growth. Six heating ele-

ments and the pulling tube are admitted through ports in the top plate.

-96-

furnace of figure 4-1 was used for the KLTN growths reported in this chapter.
(A second furnace has been constructed which incorporates several design
improvements to allow higher temperature operation and more uniform
mixing of the flux. It is not further reported on here.) The furnace consists of
a series of concentric rings of aluminum oxide ceramic material. The inner
rings are dense and insulate against high temperature radiation. The outer
rings are of a less dense “ash” material which insulate against heat conduc-
tion. The top and bottom of the furnaces consist of disks of the same materials
and cap the ends of the cylinders. Two diagonal holes are machined into the
top plates to allow illumination and viewing of the crystal growth process. A
central hole in the top plates allows the insertion of the pulling tube with its
attached seed. Six molybdenum disilicide heating elements are mounted in
the top plates in symmetrically placed slots around the center. The entire top
plate assembly is removable to afford access to the interior of the furnace.
Four 1/2” holes are machined in the bottom plates through which thermo-
couples are inserted. The inner dimensions of the furnace is 9” high by 9” in

diameter.

Several type S thermocouples (Pt/ Pt + 13%Rh) are used to monitor the
temperature inside the furnace. Their outputs are fed to a Eurotherm 818P
temperature controller which senses temperature to an accuracy of 0.1°C. The

temperature controller connects to a silicon control relay (SCR) which adjusts
the current to the heating elements. The SCR is driven by line voltage

stepped down to 30 volts by a variable transformer. The furnace operates at

up to 1600°C with a temperature stability of 0.1°C. The operating power is

-97-
900W at 1300°C.

The pulling assembly is mounted above the furnace. Its function is to
support the seed crystal in the flux during growth and provide a supply of
cooling air. It consists of a stand on which a motor driven translation stage is
mounted, and a rotatable stainless steel tube. An aluminum oxide ceramic
tube is attached to the steel tube and is inserted into the furnace. A stabilized
supply of cooling air flows through the stainless steel tube and through an
inner steel tube to the end of the ceramic tube. The cooling air flows back out
the ceramic tube and is vented in a fixture connecting the ceramic and steel

tubes.
4.4 Growth of KLTN
4.4.1 Sample Growths

The composition for a sample growth of KLTN ($H2271)with a low nio-
bium concentration is listed in table 1. The powder amounts listed in the

"Final Weight" column were packed into a 100ml platinum crucible. The

powder was then heated to 1300°C at 45°C/hr. After 15 hours of soaking at

1300°C, the furnace was ramped down at 45°C/hr to 1260°C. During the ramp,
the pulling tube with the attached seed crystal was slowly lowered into the
furnace, and allowed to come to thermal equilibrium. It was rotated at 25rpm
with the rotation direction reversed every 195sec; approximately 5liters/
minute of air flowed through the tube for cooling. The seed crystal was a

3x3x9mm° sample cut from a previously grown crystal with similar composi-

tion.

-98-

Material Formula Mole % Final Weight
Weight (gm)

KyCO3 138.21 50.0 44.0000

Li,CO3 73.82 7.0 3.2900

Ta,O5 441.90 24.7 69.5000

Nb 305 265.82 13.3 22.5100

2CuO 159.09 2.5 2.5300

V505 181.88 2.5 2.9000

Table 4-1. The flux composition of a KLTN:Cu,V ($H2271). The seeding tem-

perature was 1250°C. The resultant crystal had a composition of Kg 95 Lig gs}

Tag 367N bp 12993 9:CUg gq and a phase transition temperature of T, = 180K.

Material Formula Mole % Final Weight
Weight (gm)

K5CO3 138.21 53.0 38.00

LixCO3 73.82 5.0 1.9148

Ta,O5 441.90 11.10 25.4457

Nb 305 265.82 25.90 35.714]

2CuO 159.09 2.5 2.0633

V 905 181.88 2.5 2.3588

Table 4-2. The flux composition of a KLTN:Cu,V ($H2294). The seeding tem-

perature was 1142°C. The resultant crystal had a composition of Kj gj3Ligo44

Ta 635Nb_35903. and a phase transition temperature of T, = 310K.

-99-

When the system came to thermal equilibrium the seed was dipped
into the flux. It was observed to melt slightly after two minutes of dipping.

Then the temperature was lowered and the process repeated. Finally, growth
began at 1250°C. After five minutes of growth the seed was again raised out of
the flux to verify that the crystal was developing a sharp square profile. Then

it was left in the flux to grow while the furnace was cooled at -0.5°C/hour.

After 24 hours of growth a rectangular outline of the submerged crystal
was Clearly visible in the flux, and pulling was started. The pulling was done

at 0.55mm/hr, and lasted for 36 hours. When the crystal was out of the flux it
was annealed at 15°C/hr to room temperature. The entire growth process

took 9 days.

The resultant crystal was an oblong rectangle 7.60 x 15.60 x 20.75mm?
with subadamantine polish on all facets. It is shown in figures 4.2a,b (see also
figure 4-3 for an optical quality KLTN photograph). Small facets of [110]
growth surrounded the seed; otherwise all facets were [100]. The crystal was of
high optical quality under magnification, though slight striations were visible
with crossed polarizers. It weighed 16.33gm including the seed. Electron mi-
croprobe and atomic absorption analysis were performed on an identically
manufactured crystal, and the composition was determined to be Kg gsLig 9511
Tag g67NDp 129O3.9°CUg 994. Powder X-ray diffraction indicated a perovskite
structure, with lattice spacing of a=3.99A. The highest temperature phase

transition, i.e., the transition from the cubic to the tetragonal state, occurred at

180°K.

-100-

Figure 4-2a. Photograph of transparent crystal #71 as grown (no cutting or
polishing). It is viewed with transmitted illumination and is seen to be flaw-
less except for minor inclusions directly beneath the seed stump. The seed

crystal has been cut off.

-101-

Surface ZO ZT

growth
lines

—_a—_!_ 20.75mm———-

—a—_—§ 15.60mnr-—>

Figure 4-2b. Perspective drawing of crystal #71 with composition K g5Li 5
Ta gs7Nb 13g03:Cu. Surface growth lines are external features showing the ex-

pansion of the crystal during growth.

-102-

KLTN:Cu,v
$H2286

Figure 4-3. Photograph of transparent crystal #86 as grown (no cutting or
polishing). It is viewed with transmitted illumination and, again, is seen to be

flawless except for minor inclusions directly beneath the seed.

-103-

A growth of a KLTN ($H2294) with high niobium concentration fol-
lowing the same crystal growth procedure as set forth in the preceding exam-

ple was performed using the powdered ingredients listed in Table 4-2. For the
flux composition of Table 4-2, crystal growth on the seed began at 1142°C. The

crystal pulling and annealing was carried out under the same conditions as
set forth in the previous example. The resulting crystal was cube shaped and
of high optical quality. Its color was greenish yellow in the core and became

slightly yellowish green at the extremities. Its composition was determined to

be Ky 9331 go44 Ta 635Nb.35903. The phase transition occurred at 310K.

Using the same growth procedures as before, a KLTN ($H03101,

KLTN:Fe,Ti) crystal with the powder composition as in Table 4-3 was grown.

Crystal growth began at 1178°C.

4.4.2 Composition of Grown Crystals

Both electron microprobe analysis and atomic absorption were used to
determine the composition of the growths. The electron microprobe used was
a JEOL Superprobe 733, operating with a 15KeV beam of 50nA, 20 microns in
diameter. It was able to detect all elements but lithium, with an atomic con-
centration accuracy of ~0.003. The samples to be measured were polished and
then mounted on aluminum holders. Then a conductive carbon coating was

deposited.

The atomic absorption was performed with a Varian furnace AA ma-
chine. The samples were dissolved in boiling hydrofluoric acid, then diluted

with water to an approximate lithium concentration of 1-20 ng/liter. The

-104-

Material Formula Mole % Final Weight
Weight (gm)

KyCO3 138.21 57.0 45.00

Li,CO, 73.82 1.0 0.4217

Ta 05 441.90 14.0 35.3385

NbO5 265.82 26.0 39.4767

Fe,O3 159.69 1.0 0.9122

2T105 159.76 1.0 0.9126

Table 4-3. The flux composition of a KLTN:Fe,Ti ($H03101). The seeding tem-

perature was 1178°C.

Table 4-4. (next page) A partial list of the Ky_yLiyTay_,.Nb,O3:Cu crystals grown

to date. The first column is the crystal number. The next two columns show

the combined potassium plus lithium (K+Li) and tantalum plus niobium

(Ta+Nb) concentrations in the flux. The next two columns show the relative

concentrations of niobium and lithium where X=[Nb]/({Nb]+[Ta]) and Y=[Li] /

([Li]+[K]). The columns listed under "Crystal Composition" list the mole con-

centrations of the four constituents in the grown crystal, and the final column

is the growth temperature.

-105-

Xtl # Flux Composition Crystal Composition T seed
K+Li %Ta+Nb%_X Y i Ta ]
91 58 37 0.5 0.01724 0.992 .0025 0.830 0.17 1216
85 58 37 0.65 0 0.992 0 0.768 0.29 1177
86 58 37 0.65 0.0086 0.994 .0006 0.700 0.299 1179
99 58 37 0.65 0.01 00005 1174
84 58 37 0.65 0.0172 0.990 .0019 0.730 0.27 1188
98 58 37 0.65 0.03 1.016 .0023. 0.721 = 0.273 75
88 58 37 0.7 0 1.00 0 0.667 0.333 1160
97 58 37 0.3 0.0517 1.004 .0091 0.908 0.089 1250
96 58 37 0.35 0.0517 0.998 .0092 0.893 0.105 1241
74 58 37 0.45 0.0517 0.990 .005 0.857 0.144 1240
75 58 37 0.5 0.0517 0.980 006 0.827 0.176 1219
76 58 37 0.5 0.05417 1225
81 58 37 0.55 0.0517 0.990 .007 0.794 0.206 1197
82 58 37 0.6 0.0517 0.989 007 0.777 = 0.223 1185
83 58 37 0.65 0.0517 0.993 .006 0.687 0.314 1171
95 61.05 38.95 0.65 0.0517 1.020 .0044 0.072 0.275 1161
93 58 37 0.7 0.0517 1.031 .0052 0.644 0.347 1153
33 61 34 0.4 0.082 0.999 0.896 0.102 1193
87 57.6 374 0.65 0.0877 1.006 012 0.677 0.32 1169
94 58 37 0.7 0.0862 1.013 .0044 «(0.635 (0.359 1142
44 57 38 0.35 0.1228 0.95 0511 0.857 0.129 1238
71 57 38 0.35 0.1228 1250
73 57 38 0.35 0.1228 0.991 .0212 0.888 0.112 1243
90 58.5 36.5 0.65 0.1227 0.985 .015 0.708 0.293 1153
58 60 35.5 0.2986 0.2633 0.933 .035 0.894 0.095 1196
60 60 35 0.3 0.3 0.931 .032 0.881 0.108 1123
61 60 35 0.3 0.333 0.604 320 0.969 0.082 1188
69 63 324 0.3 0.333 0.951 .0645 0.975 0.032 1185
64 63 32.4 0.1429 0.333 0.950 .0601 0.979 0.029 1157

67 64.5 34.6 © 0.1428 0.36511 0.941 .0757 0.979 0.031 1217

-106-

standard was prepared from a 100ng/ liter aqueous solution of Li,CO3, and
was diluted 1:29, 2:28, ..., 5:25 with water to make a standard curve. The lamp
was operated with 5.0mA, and a slit width of 1.0nm was used. The results of
these measurements are illustrated in Table 4-4. The first two columns under
"Flux composition" show the combined potassium plus lithium (K+Li) and
tantalum plus niobium (Ta+Nb) concentrations in the flux. In all but one flux
(crystal number 95) the sum of the potassium plus lithium and tantalum plus
niobium concentrations was 0.95. The fraction remaining from 1.0 ( that is,
0.05) was split equally between the copper and vanadium dopants in the same
manner as set forth in Tables 4-1 and 4-2. The next two columns in Table 4-4
show the relative concentrations of niobium and lithium where
X=[Nb]/(INb]+[Ta]) and Y=[Li]/({Li]+[K]). The columns listed under "Crystal
Composition" list the mole concentrations of the four constituents in the
grown crystal. The temperature at which growth started for each crystal is list-
ed in the final column. Of particular interest is the contrast between the flux

and crystal compositions (Figure 4-4a,b).
4.5 Discussion of Growth Characteristics
4.5.1 Composition Considerations

The growth characteristics color, structure, and composition are found
to depend only on the flux composition, whereas the shape, size, and quality
of the as-grown crystal are functions of both flux composition and growth pa-

rameters. The effects of composition are discussed first.

The phase diagram for KLTN shown in Figure 4-5 plots the liquidus

and solidus curves of temperature versus niobium concentration. Of course,

-107-

[Li] = 0.052

0.15 - [Li]=.12 4

< OB F 4
oe)
O 0.25 - Z
2 0.2 - -
YD

0.1 l l l
0.3 035 04 045 0.5 0.55 0.6 0.65 0.7

Flux Niobium Concentration

Figure 4-4a, The niobium concentration in the crystal is plotted versus the
niobium concentration in the flux for two lithium flux concentrations: [Li] =
0.051 and [Li] = 0.123. The concentration of niobium in the crystal is approxi-

mately 1/3 the concentration in the flux.

-108-

0.02

0.016 - 4

0.012

0.008

0.004 +

Crystal Lithium Concentration

06 l ! l
O 0.02 0.04 0.06 0.08 0.1 O12 0.14

Lithium Concentration in Flux

Figure 4-4b. The lithium concentration in the crystal is plotted versus the
lithium concentration in the flux for a niobium flux concentration of [Nb] =
0.65. The lithium concentration in the crystal is approximately 1/7 of that in

the flux.

-109-

1260 | |
[Li] = 0.052

1240

1220

1200

Temperature - °C
rama
jaan
e2)
mn)

1160

[Li] = 0.123 -

1140 | | | | l | l
) 01 02 03 O04 05 06 0.7 0.8

Atomic Niobium Concentration

Figure 4-5. The phase diagram of K;_,LiyTa,_,.Nb,O3 showing liquidus and
solidus curves of temperature versus niobium concentration. The change in
these curves with increasing lithium concentration is illustrated by the differ-
ence in the curves for two lithium flux concentrations: [Li] = 0.052 and [Li] =

0.12.

-110-

the liquidus and solidus curves are also functions of the lithium concentra-
tion. The phase diagram has been plotted for two lithium flux concentra-

tions: [Li]=0.052 and [Li]=0.12. The niobium/tantalum ratios were found to
roughly obey the phase diagram for KTN!© Here, as shown in Fig. 4-6, if a

flux with a given niobium concentration X (X is defined in Table 4-4) is cooled
to the liquidus temperature, the crystal begins growing along the solidus with
a niobium concentration x ( concentrated to composition, say, X'> X with a resultant shift in crystal com-
position to x'( >x ). Across a 15gram cube shaped crystal grown in a 100mI cru-
cible with initial flux composition X= 0.65, a series of 200 microprobe mea-

surements determined a variation in x of 0.03; this agrees with calculations.

The phase diagram of KLTN for [Nb] = 0.65 in the flux, as a function of
lithium concentration is shown in figure 4-7. The deviations from a smooth
curve are caused by variations in the resultant crystal’s niobium content.
There is, in other words, a coupling between the lithium and niobium flux/
crystal concentrations so that a constant niobium flux concentration does not
yield a perfectly constant crystal niobium content if the lithium concentration
is varied. It is seen that the lithium/potassium ratios exhibit a solid solution
behavior for Y < 0.33 (Y is defined in Table 4-4), with yall cubic perovskites at room temperature. For Y 2 .33, however, the composi-
tion and structure abruptly changes. These growths, e.g., #61 (Table 4-4), are
dense and brittle materials, with the characteristic midnight blue color of oxy-
gen vacancies. The growth habit displayed [100] as well as [110] and [210] faces.

These materials were strongly pleochroic, and X-ray diffraction confirmed

-111-

13005 Top Seeded Solution Growth in
a KTN- like System

( 1260F
he
= 12204
fae)
Qu.
& 1180f
fb

1140}

|, Ly
0 xx! X X' 1

Niobium Concentration

Figure 4-6. The phase diagram for KTa,_,Nb,O; illustrating the top seeded so-
lution growth (TSSG) process. The flux with [Nb] = X is cooled to the liquidus,
and the crystal begins growing with composition [Nb] = x. As cooling drives

the growth, the flux and crystal compositions shift to X' and x' respectively.

-112-

1190 ! !
O ©
—O— Flux Composition
—85— Crystal Composition
1180 7]
Fon
©, 1170 4
a)
Fe
1160 4
1 150 | | i I I |

0 0.02 0.04 0.06 0.08 O.1 0.12 0.14

Atomic Lithium Concentration

Figure 4-7. The phase diagram of Ky_yliyTa,_,Nb,O3 versus lithium concen-
tration for fixed niobium concentration, [Nb] = 0.65, in the flux. Deviations
from the smooth curve are determined to be due to variations in [Nb] in the

crystal despite a constant flux concentration.

~113-

CPS 14.72 5.53 3.42 2.49 1.97 1.64 %
224 100
201 30
179 80
156

ray
BR
Nn
ee Se
Ob AN HOR ADAN HO

wo
DO BAKDUNAWNHRO
be

6 16 26 36 46 56
POTASSIUM TANTALUM OXIDE

2,822 | cubic

K TA 03
LITHIUM POTASSIUM TANTALUM OXIDE

23,1198 good fit for large d...

| i | - 2 di

Figure 4-8a. Powder X-ray diffraction of a cubic Ky_yLiyTa,_.N b,O3 with small
lithium concentration. The potassium tantalate standard is shown for com-
parison. 4-8b. Powder X-ray diffraction of a noncubic K3Li,(TaNb)O3. A lithi-

um potassium tantalate (K3Li,Ta,;O,5) standard is shown for comparison.

-114-

they were not cubic. The X-ray peaks are very similar to those in K3Li, (Taj_,
Nb,)compounds!” (Figure 4-8) which have the ilmenite tungsten bronze
structure. The measured composition, K ¢Li 39(TaNb)O3, is consistent with
this statement. This abrupt compositional and structural change is attributed
to an instability due to the size differential between the lithium and potassi-
um ions. It should be noted that the transition from cubic to tungsten bronze
growth could be forestalled somewhat by boosting the (potassium + lithium)
/ (tantalum + niobium) ratios, as in crystal #67, although these materials in -
variably grew with flux inclusions and were pyramid rather than cube

shaped.

The color of the KLTNs grown is due to the transition metal dopant,
lithium concentration, and niobium/tantalum ratio. Crystals grown without
dopants were colorless. In the copper/vanadium doped crystals, the copper
ion is present as either Cu!* or Cu**, with absorption peaks at 410 and 580nm
respectively. The relative strengths of these peaks is a direct indicator of the
relative concentrations of these oxidation states, according to Beer’s law!®. In

crystals with small niobium concentration, i.e., when X < 0.4, the color is
olive green. As X is increased, however, color banding is observed, from
greenish yellow near the seed to bright slightly yellowish green at the later

growing parts of the crystal. Optical absorption data confirmed these colors to

be the result of a progressively increasing [Cu2+]/[Cul] ratio (Fig. 4-9a), from 0

at the outset of growth to approximately [Cu2*+]/[Cu] = 0.15 near the end.

Additionally, when lithium is omitted from the flux composition, the crystals

grow a blue core surrounded by the yellow and green bands. The blue indic-

-115-

10 ¢ ! !
F o green :
r a yellow 7
GF |
\-)
oS L- 4
gle 7
Pa :
fe) ; 4
BS oF |
< F 1
0.1 i | | I
200 400 600 800 1000 1200

Wavelength - nm

Figure 4-9a. Absorption spectrum of an as grown Ky_yLiyTa,_,.Nb,O3 at two

positions along the growth. The color shifts from yellow to green as the con-

centration of Cut increases.

-116-

10 T l I

tot ti

proved

roporiil

Absorption - 1/cm

pont

200 400 600 800 1000 1200
Wavelength - nm

Figure 4-9b. Absorption spectrum of an as-grown KTa,_,Nb,O3 at three posi-
tions along the growth. The strongly absorbing band at 1000nm is due to oxy-
gen vacancies. As the growth continues, the environment becomes progres-

sively more oxidizing. First the oxygen vacancies disappear, then the concen-

tration of Cu2* increases.

-117-

cates oxygen vacancies, as evidenced by absorption data (Fig. 4-9b). These blue
crystals have been noted by other researchers!?22 growing KTN. As little as

0.5% of lithium in the flux prevents the blue color, and markedly improves

the growth quality of KLTN over that of KTN.

It should be mentioned that the vanadium present in the flux as a
dopant plays no role in the coloring, and is only present in minute quantities
in the growth. Since multiple growth attempts to grow KTN:Cu without the
presence of vanadium in the flux failed to produce high quality doped sam-
ples, we conclude that the vanadium assists in the introduction of copper into

the perovskite lattice.

KLTN crystals were also grown with iron/titanium doping. In these
crystals the iron is present as either Fe2+ or Fe>+. The optical absorption spec-

trum of KLTN:Fe,Ti (#03101) is shown in Fig. 4-10 The curves show the as-

grown spectrum as well as the spectrum after one and two thermal reducing

treatments. The thermal treatments were performed at ~800°C in an argon at-

mosphere and they serve to change the relative ratio of Fe**/Fe?+ from the

as-grown values. A detailed enumeration of the absorption characteristics of

iron doped KTN and the thermal reduction process can be found in Ref. 18.
4.5.2 Growth Parameters

Several parameters of the growth have considerable influence on the
quality of the grown crystal. The most critical condition for good growth is the
temperature. If seeding takes place more than a degree or so below the proper

growth temperature, rapid growth with associated flux inclusions results

-118-

10 ¢£ l
' Absorption of KLTN:Fe,Ti 03101

| ee ae ee Oe |

= after reducing treatments
St |
™ a

e |

3 8

E Tr 5 © as grown q
G [ a a reduced 1
< r ° reduced 2 1

0.1
200 400 600 800 1000 1200

Wavelength - nm

Figure 4-10. Absorption spectrum of an as-grown KLTN:Fe,Ti in the as-
grown state and after one and two thermal reduction treatments at 800°C in
an argon atmosphere. The thermal treatments serve to change the relative

ratio of Fe?+/Fe*+ from the as-grown values.

-119-

until the system reaches equilibrium. The maintenance of a temperature gra-
dient between the seed and the rest of the furnace is also crucial. If the air flow
is inadequate or the seed becomes loose during the growth, the growth will be
polycrystalline. Proper cooling always leads to a single nucleation site and sin-

gle crystal growth.

It is preferred that the flux ingredients remain as uniform as possible at
the seed junction because the proper growth temperature is a function of the
flux composition. The flux is stirred by rotating the seed. In order to maintain
a homogeneous flux composition and temperature profile, the seed rotation
direction is periodically reversed. But every time the rotation direction is re-
versed, growth accelerates briefly, causing striations. A possible explanation
for this phenomenon is that the tantalum tends to settle out, and that the ro-
tation reversal generates turbulence bringing the tantalum-rich mixture to
the surface. The increase in tantalum concentration (decrease in X) leads to an
increased growth temperature, which causes the growth rate acceleration. In
support of this theory, the longer the period of time between rotation rever-
sals, the more tantalum settles out, and the more violent the growth rate ac-
celeration on rotation reversal. Accordingly, it is preferred that more frequent
rotation reversals be used to reduce the growth rate acceleration. (The newly
constructed furnace mentioned at the beginning of this chapter has a facility
for rotating the crucible support that is designed to eliminate this problem en-

tirely.)

As an example, when the rotation direction was switched every 13min-

utes during the growth of crystal #87, flux inclusions resulted along the

-120-

growth line boundaries; when the rotation was reversed every 6 minutes (in
the same crystal) however, no inclusions were formed. For the same reason,
the rotation rate should be kept constant. In one instance, the rotation rate
was increased from 25 to 30 rpm causing flux inclusions to be deposited on

the face of the developing crystal for approximately one hour.

Figure 4-1la shows results of a series of microprobe measurements
taken perpendicular to the growth lines ina KLTN. The tantalum and niobi-
um concentrations are plotted as a function of position; the measurements of
tantalum and niobium are completely independent. Figure 4-11b represents
the sum of the tantalum and niobium concentrations along the crystal. It is
seen that the strongly anticorrelated fluctuations of the tantalum and niobi-
um concentrations individually does not carry over into their sum, which is
fairly constant at [Ta]+[Nb] = 1.002. The small variations in figure 4-11b lead to
an estimate of the accuracy of the data in figure 4-1la. The region displayed in
figures 4-Ila,b corresponds to 10-12 growth lines in the crystal. Thus the
semiperiodic fluctuations in the niobium concentrations occur in conjunc-
tion with the growth lines, lending further support to the hypothesis of com-

position variation in the flux due to incomplete stirring.

It was mentioned previously that the depletion of the nutrients in the
crucible leads to an increase in niobium concentration, and hence, lattice pa-
rameter, during the course of the growth. Thus, as the crystal grows onion-
style with one cubical shell wrapping around the previous one, the strain
gradually increases. This strain is visible under crossed polarizers as a bire-

fringence increasing from the center. Under transmitted light, a faint tetrahe-

-121-

—*— Niobium —o— Tantalum
- 0.28 growth ; ! 0.736
2 0.278 + direction \ i 40.734 s
by © a © >
§ 0.276 i © 40.732 =
1w} | O Yo
= 3
& 0.274} ‘s me Hi i 6 9.73 ©
) Hi | g <€
E 0.272 + eciimay ‘ mo + 0.728 A
yy . \ 0.726 =
0.268 0.724 3
0.266 : ! 0.722

56.6 56.7 56.8 56.9 57 57.1 57.2
Position - mm

Figure 4-1la. Atomic concentration of niobium and tantalum as a function
of position perpendicular to growth lines in crystal 84. The measurements are
strongly anticorrelated indicating fluctuations in the the flux composition.

The plot corresponds to 10-12 growth lines.

-122-

1.006

>.

1.004

O Gd OCDO

1.003 + tt | ; : uti

Tantalum + Niobium

COMO Cp

1.001 + ees a
0.999 | : !
56.6 56.75 56.9 57.05

Position - mm

Figure 4-11b. The sum of the tantalum and niobium concentrations is fairly

constant at the stoichiometrically expected amount. This verifies the accuracy

of the previous figure.

57.2

-123-

dral pyramid is visible, with its apex at the tip of the seed. If the twins are cut

apart most of the strain is relieved.
4.6 Influence of Lithium/ Niobium on Transition Properties

KTNs are ferroelectric materials which have a high temperature para-
electric phase and become tetragonal at a Curie temperature T.~850[x] K, (for
small [x]) where [x] is the niobium concentration. At least two more struc-
tural transitions occur as the temperature is lowered. The relative dielectric
constants versus temperature of a number of KTN and KLTN crystals are il-
lustrated in figures 4-12 through 4-16. The composition of these crystals and
the temperature of their highest symmetry phase transition is summarized
in table 4-5. Figures 4-13 through 4-16 show the relative dielectric constant
measured for both cooling and warming of the crystal. The cooling curve in-
variably exhibits higher peak dielectric constants and phase transitions at

slightly lower temperatures. Since the data were taken at cooling/warming
rates of approximately 0.2°C /min the temperature discrepancies of up to 12°C

between the cooling and warming curves are not experimental artifacts. In
figures 4-12a,b the low frequency dielectric constant of a K o5Li g4Ta gsaNb 1303
(crystal 44) and a KTa g-Nb 43 (crystal 18) are compared. In the KTN the transi-
tions from cubic to tetragonal to orthorhombic to rhombohedral are clearly
seen, in the KLTN only the first transition is distinct. Additionally, the transi-
tion temperature has been increased almost 50K in the KLTN, although the
niobium concentrations of the two crystals are equal. Fig. 4-13 shows the di-
electric response of a Kg ogol-i ggg Np 9977 9 17603 (crystal 75), and Fig. 4-14

shows that of a Kg o9g9 Li gg7 Nb 777T ag 99303 (crystal 82). In both of these plots

-124-

Xtl # Flux Composition Crystal Composition Te °C
xX x [K] [Li] [Ta] ___[Nb]
18 - - 1.00 0 0.86 0.13 -143
44 0.35 0.1228 0.95 0511 0.857 0.129 -95
75 0.5 0.0517 0.980 .006 0.827 0.176 -112
82 0.6 0.0517 0.989 007 = 0.777 = 0.223 -90
85 0.65 0 0.992 0 0.768 0.29 -41
86 0.65 0.0086 0.994 .0006 0.700 0.299 -28
84 0.65 0.0172 0.990 .0019 0.730 (0.27 -24
83 0.65 0.0517 0.993 .006 0.687 0.314 -22
87 0.65 0.0877 1.006 .012) 0.677 0.32 -2
90 0.65 0.1227 0.985 015 0.708 0.293 -21
93 0.7 0.0517 1.031 .0052 0.644 0.347 5
94 0.7 0.0862 1.013 .0044 0.635 0.359 20

Table 4-5. The flux and crystal compositions, as well as the phase transition
temperatures of several KLTNs are listed. The crystals are listed in groups of
increasing niobium concentration. Within each group the listing is in order
of increasing lithium concentration. The phase transition temperature in-

creases with either lithium or niobium concentration increase.

Dielectric const.

Dielect. const.

~125-

2.064
K gall e¢N
> oral g5tT g6N 13 rn
oo
a *
1.5644 val *
/ “Hae
a Ne
1.064 + “,
re
000.0 + ;
. wee”
Me
0.0 +
150 160 170 180 190 200
Temperature (K)
8E4
KT g7N
i, 87N 13
6E4 a [ a
fy \
{ 1
464+ } 5
‘ ‘
| \
2E4+4 | “ha
| sy
/ a
al nS ee
6)
100 110 120 130 140 150

Temperature (K)

-126-

Figure 4-12a. (previous page) Dielectric constant of the K gsLi 95 Ta gs7 Nb 479
O3 (crystal 44). The phase transition is approximately 50°C higher than a com-

parable KTa,_,Nb,O, (see figure 4-12b). Figure 4-12b. Dielectric constant of a
KTa g¢ Nb 13 O3 (crystal 18).

Figure 4-13. (following page) The dielectric response of a Kg ogol-i nog Nbo.g27

Figure 4-14, (following pages) The dielectric response of a Kp ggq Li gg7Nbp 777

Tag 97303 (crystal 82).

Figure 4-15a-f. (following page) The dielectric curves of Ky yLyTo.7N 0.3 Crys-
tals is illustrated with 0creases with increasing lithium concentration and, again, the temperature of

the highest transition increases with the lithium concentration.

Figure 4-16a, b. The dielectric response of two Ky yLyTo.65No.35 crystals.
Here, the phase transition temperature has been raised to approximately

room temperature.

Dielectric Constant

Dielectric Constant

-127-

Dielectric Constant of #2275

C0 ee ee
L Figure 4-13 J
30000 - +
20000 + 7
10000 - —_ ;
0 r it { itt | i on l ie ans | a | l | aan aes | [on oo | Jo l but |
-160 -120 ~ -80 -40 0
Temperature (°C)
Dielectric Constant of #2282
40000 [ TOT T T YT | PTT T TTT i TTT I TT T i TTT ] TTT | T FT "]
[ Figure 4-14 J
30000 + 4
20000 + . 4
10000 + 7
0 F id oe | ij 4 i | oot | 5 on oe ae { ja ol i} | Lu a
-160 -120 -80 -40 0

Temperature (°C)

Dielectric Constant

Dielectric Constant

-128-

Dielectric Constant of #2285B
4QQOQ

Figure 4-15a
30000

20000

T T T T T T T T T ] T T T T T T F t T
~ >
cae
1 L 1 | ‘3 i lL L I i i Ll i | i 1 i i

10000

pea toe bie be bt hl ta

-160 -120 -80 -40 0
Temperature (°C)

Dielectric Constant of #2286
40000 TTT ee pepe pees

Figure 4-15b
30000

T T T T T T T t T

20000 \

10000

L i i i 1 1 3 1 J 4. 1 i i | A. 1 I L

rr

0 peat rse bea bi ta a Pt

-160 -120 -80 -40 0
Temperature (°C)

Dielectric Constant

Dielectric Constant

-129-

Dielectric Constant of #2284

; Figure 4-15c ;
30000 F
20000 + / : 5
0 j tone Gee | jaan ner 2 i tot l | oan oe i it | oe | ‘i wen l itd | Lob
-160 -120 -80 -40 0
Temperature (°C)
Dielectric Constant of #2283
40000 [7 a r]
| Figure 4-15d |
30000 ~
20000 + |
10000 Er wa \ 4
/ oo J
0 EF Loud | ca a | | i | jot l i 1 | oe | ji | itt | ft |
-160 -120 -80 -40 0

Temperature (°C)

Dielectric Constant

Dielectric Constant

40000

30000

20000

10000

-130-

Dielectric Constant of #2287

| Figure 4-15e

t T T T ] T T T T

pee to ta a a

rrTyp rrr perp rer perry rrp ere yp rrr pr

ae
jo Va

pti

1 1. i 1

1 1 i i [ 1. L i 1 |

-160 -120 -80 -40

40000

30000

20000

10000

Temperature (°C)

Dielectric Constant of #2290

i eee es We es a wae ee ee a a a

T T TT

Figure 4-15f

Sees

T T T T T ¥ T Hi T T ¥ ¥ T T

Ltd babel be ltebbebot tot tut bear tee bo

mpd

cn |

i { 1. 4.

A. i. 1 L I 1 L 4 i I L 1 i L !

-160 -120 -80 -40

Temperature (°C)

Relative dielectric constant

Dielectric Constant

-131-

Dielectric constant of #2293

30000 LL T T t | 7 Li T i] T T Ul if t T T T T T T 4
20000 - y 4
15000 § \4
10000 — | X
s00F NNR UY

¢) po a La
-180 -140 -100 -60 -20 20

Temperature (°C)
Dielectric Constant of #2294

15000 C i SO OO
fF Figure 4-16b fo 3
10000 — AX “
5000 Fe 2
0 ; JL i L i | rl 1 i H { L L i i | 4. 1 L 1 | 1 Fl 1 i 4
-200 -150 -100 -50 0 50

Temperature (°C)

-132-

only one distinct transition is seen and the temperature of the highest transi-

tion is higher than would be expected for an equivalent KTN.

In figures 4-15a-f, the dielectric responses of Ky_yLyTo.7No3 crystals are
illustrated with 0phase transitions decreases with increasing lithium concentration and, again,
the temperature of the highest transition increases with the lithium concen-
tration. The lithium concentrations are read from table 4-5. Figures 4-16a,b
display the dielectric response of two Ky yLyTo.65No.35 crystals. Here, the
phase transition temperature has been raised to approximately room temper-

ature.

From figures 4-12 through 4-16 it can be concluded that in KLTNs with
low niobium concentration, the addition of lithium eliminates two of the di-
electric peaks corresponding to phase transitions. Only one very sharp peak
remains, at a temperature higher than expected for the cubic to tetragonal
transition in KTN. In KLTNs with higher niobium concentrations, the influ-
ence of the added lithium has a diminished effect. But in all cases the transi-
tion temperature increases with either lithium or niobium concentration in-

crease.

The phase transition temperatures for the KLTNs tabulated in table 4-5
is plotted. The phase transition temperatures of KLTN with a fixed lithium
concentration of [Li]=0.052 in the flux versus niobium concentration is shown
in figure 4-17. In figure 4-18. the phase transition temperature is plotted
versus lithium concentration for a niobium concentration fixed at [Nb] = 0.65

in the flux. The effect of lithium to raise the transition temperature saturates

-133-

an

| L |

0.15 0.2 0.25 0.3 0.35
Niobium Concentration in Crystal

Phase Transition Temperature - °C

an
Ny
oO

Figure 4-17. The phase transition temperature of K1.yLiyTa,_,.Nb,O3 with a

fixed lithium flux concentration of [Li] = 0.052.

-134-

c 0 ! 6

<8)

3 710 + 4

vo

Qu.

E20 + 0 O 4

= 30 7 :

E -40 ¢ |

= -50 ! ! !

0 0.005 0.01 0.015 0.02

Lithium Concentration in Crystal

Figure 4-18. The phase transition temperatures of Ky _yLi,Ta_,.Nb,O3 with
[Nb] = x = 0.65 in the flux versus lithium concentration. The deviations from
a smooth curve are determined to be due to variations in the niobium con-

centration of the grown crystal.

-135-

at y ~ 0.01 where T, is raised approximately 30°C. For KLTNs with lower nio-

bium concentrations the transition temperature was raised by 50°C in the

example given above.
4.7 Summary

A technique has been developed for the preparation of high quality
crystals of KLTN. The Top Seeded Solution Growth method is used, pulling
the crystal from a flux with an excess of potassium oxide. A range of composi-
tion of Ky_yLi,Ta;_,Nb,O3 from 0this method. The resultant growths are 10-20g crystals with no inclusions vis-
ible under magnification. The partial phase diagrams for (KLi)Ta,_,Nb,O3 for
lithium concentrations of 0, 0.05, and 0.12 are presented as well as the phase
diagram for Ky_,LiyTag 7gNbp 303. Growth quality was shown to be a sensi-

tive function of the flux composition and the growth parameters.

-136-

References for chapter four

[i] A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new pho-

torefractive material: Ky yLTy.N x Opt. Lett. ,17, 713-715 (1992).

[2] A. Agranat, V. Leyva, K. Sayano, and A. Yariv, “Photorefractive properties
of KTa;_,.Nb,O3 in the paraelectric phase,” Proc of SPIE Vol. 1148, Conference

on Nonlinear Optical Properties of Materials, (1989).

[3] A. Agranat, V. Leyva, and A. Yariv, “Voltage-controlled photorefractive
effect in paraelectric KTay_,.Nb,O3:Cu,V,” Opt. Lett. 14, 1017-1019 (1989).

[4] J. P. Wilde and L. Hesselink, “Electric-field controlled diffraction in pho-

torefractive strontium barium niobate,” Opt. Lett. 17, 853 -855 (1992).

[5] V. Leyva, A. Agranat, and A. Yariv, “Fixing of a photorefractive grating in
KTa,_,.Nb,O3 by cooling through the ferroelectric phase transition,” Opt. Lett.
16, 554-556 (1991).

[6] A. Agranat, V. Leyva, K. Sayano, and A. Yariv, “Photorefractive properties
of KTa,_,.Nb,O3 in the paraelectric phase,” Proc of SPIE Vol. 1148, Conference

on Nonlinear Optical Properties of Materials, (1989).

[7] C.M. Perry, R. R. Hayes, and N. E. Tornburg, Proceedings of the
International Conference on Light Scattering in Solids , M. Balkanski Ed.
, Wiley, New York, New York (1975).

[8] S. Triebwasser, “Method of preparation of single crystal ferroelectrics,”

United States Patent Office, patent number 2,954,300, (1960).

[9] R. Hofmeister, Amnon Yariv, and Aharon Agranat, “Growth and charac-

-137-

terization of the perovskite K,_. Li, Ta, .Nb,O,:Cu,” accepted for publication
P Ley y © S1-x0 8 8x3 P Pp

in J. Crystal Growth.

[10] V. Belruss, J. Kalnajs, and A. Linz, “Top-seeded solution growth of oxide

crystals from non-stoichiometric melts,” Matl. Res. Bull. 6, 899-906 (1971).

[11] P A. C. Whiffen and J. C. Brice, “The kinetics of the growth of KTay_

xNb,O3 crystals from solution in excess potassium oxide,” J. Crystal Growth

23, 25-28 (1974).

[12] H. J. Scheel and P. Guenther, “Crystal growth and electro-optic properties
of oxide solid solutions,” Crystal Growth of Electronic Materials , Elsevier

Science Publishers, (1984).

[13] W. A. Bonner, E. F. Dearborn, and L. G. Van Uitert, “Growth of potassi-
um tantalate-niobate single crystals for optical applications,” Am. Ceram. Soc.

Bull. 44, 9-11 (1965).

[14] J. Y. Wang, Q. C. Guan, J. Q. Wei, M. Wang, and Y. G. Liu, “Growth and
characterization of cubic KTa,_,Nb,O3 crystals,” J. Crystal Growth, 116, 27-36
(1992).

[15] D. Elwell, in Crystal Growth , p. 185ff, Ed. B. R. Pamplin, Pergamon press,
Oxford (1975).

[16] A. Reisman, S. Triebwasser, and F. Holtzberg, “Phase diagram of the sys-
tem KNbO; KTaO; by the methods of differential thermal and resistance anal-
ysis,” J. Am. Chem. Soc. 77, 4228-4230 (1955).

[17] T. Fukuda, Jap. J. Appl. Phys. 9,599 (1970).

-138-

[18] V. Leyva, Ph.D. thesis, California Institute of Technology, Pasadena CA,
unpublished, (1991).

[19] S. H. Wemple, “Polarization effects of magnetic resonances in ferroelec-
tric potassium tantalate,” Mass. Inst. Technol. , Res. Lab. Electron. , Tech..

Rept. #425, (1964).

[20] W. A. Bonner, E. F. Dearborn, and L. G. Van Uitert, “Growth of potassi-
um tantalate-niobate single crystals for optical applications,” Am. Ceram. Soc.

Bull. 44, 9-11 (1965).

[21] W. R. Wilcox and L. D. Fullmer, “Growth of KTaO - KNbO3 mixed crys-
tals,” J. Am. Ceram. Soc. , 49, 415-418 (1966).

[22] P W. Whipps, “Growth of high-quality crystals of KTN,” J. Crystal
Growth, 12, 120-124 (1972).

-139-

Chapter Five

Electric Field Control of The Photorefractive

Effect in Paraelectric KLTN

5.1 Introduction

The use of photorefractive materials to store volume holograms for

optical computing and optical memories has long been an active area of re-

search'-3. Recently, the use of photorefractive materials in the paraelectric

phase has also received attention”. As discussed earlier, the photorefractive
response in the paraelectric phase is controllable with an externally applied
electric field. The dielectric constant for these materials obeys the Curie-Weiss

law above the phase transition. This leads to very large values for the dielec-
tric constant just above the transition (c = 104 - 105). Since the magnitude of

the voltage controlled index grating is proportional to the square of the dielec-
tric constant, the optimum operating temperature is just above the phase
transition. Crystals operated at this temperature exhibit a strong photorefrac-

tive response and achieve extremely high diffraction efficiencies.

In the experiments described in this chapter the crystals were main-
tained just above the para/ferroelectric transition. Here the material's pho-
torefractive properties are described by the quadratic electro-optic effect. The

experimental geometry used is the symmetric one (Fig. 5-1 and 5-4), with both

-140-

3 axis

detector

1)

detector

1,0

Figure 5-1. The symmetric geometry used for the two beam coupling experi-
ments. The space charge field and the applied field are along the 3-axis. The

beams are extraordinarily polarized.

-141-
the space charge field and the external applied field directed along the x (3)

axis. The change in the refractive index due to E,.(x) was derived in chapter

three and is given by

Anj(Eo+Ese) - Any(Ep) = 22. 25 2((E.. + Eg? - EF] 5.1
nij( ot Esc) nij( 0) 5 £ij33 (E30) ( sc + Ko O} (5.1)

Here E, is the applied electric field and g. jk] are components of the quadratic
electro-optic tensor. np is the nominal index of refraction and €3 is the rela-

tive dielectric constant along the x axis. The useful, Bragg matched part of the

index grating is

Anij = n§ 8ij33 (€3€0)" Esc Eo. (5.2)

Thus the Bragg matched contribution to the index grating is present only

with, and is proportional to, an applied electric field.

In the following sections voltage control of the photorefractive effect in
paraelectric materials is demonstrated. Voltage dependent two beam coupling
and voltage controlled diffraction results are presented. A theory is devel-
oped to describe the diffraction versus applied electric field off a grating writ-
ten with or without an applied field. The theory is applied to the diffraction

results with good qualitative agreement.
5.2 Beam Coupling

When two beams 1, (0) and I,(0) are symmetrically incident on a pho-
torefractive material and write an index grating, intensity coupling occurs be-

tween the beams. The output intensities are given by (chapter two)

-142-

I,(z) = eal Ti(h +b) (5.3a)
I, +I etl
ce (li +b)
— eal
In(z) = e@ Lela (5.3b)

where [= 2g sing is the power coupling coefficient and we define I, =1,(0)

and I, =1,(0). When I, >>I, we can approximate equations (5.3) as

I}(z) = eel J, (5.4a)

In(z) = e@-ML J, (5.4b)

In paraelectric materials the coupling constant g is (conventionally)

zero in the absence of an applied field, so that no intensity coupling occurs. In
the presence of an electric field the coupling constant is given by

g = TAL = E[2n§ gi (0e)” Bok sco) (5.5)

where nj is the peak to peak index change when I, =I,, and E,.9 is the space

charge field obtained with unity modulation depth, i-e., when I, =1,. This

convention is required to ensure that g is purely a material parameter, inde-

pendent of the beam intensity ratio. From the results of chapter two, the

space charge field has a dependence on the applied field given by

2 2
Eq = m En a/ Eq + Eo cos[20] (5.6)
E6é + (Eq + En)”

where the diffusion and trap limited fields are Ey = kTK/e and Ey =
N ae/(eK). The modulation depth of the intensity pattern is m = 2 (I, 1)1/2/1

(not to be confused with the variable m in (2.5)) and the half angle of beam

-143-

intersection within the crystal is a. The beams are polarized in the plane of

their intersection within the material. From the same equations one can de-

rive an expression for the phase of the grating

- [E§ + Eg(Egt+En)] (5.7)

sind = .
V (EJ+E3) (E3 + (EatEn)?)

Thus the phase 6 = n/2 (by convention) when no field is applied. (In the next

section a new photorefractive mechanism is discussed which yields an index
grating with zero applied field. In this case the phase is equal to zero when no
field is applied. When a field is applied, the electro-optic grating and the new
grating are added in quadrature to obtain the net phase of the sum of the grat-
ings. In this section the new photorefractive mechanism is ignored.) From

(5.6) and (5.7) we obtain

E§ + Eq(Eg+En)

5 5 cos[2o] (5.8)
Eo + (Eg+En)

[= “on nd 211 (€o£)” Eo En

with parameters defined as in chapter two. (5.8) gives the intensity coupling
coefficient when an index grating is written in a material with an applied
electric field. When Ey << Ey,E, the intensity coupling coefficient is approx-

imately linear in Ey. The intensity coupling vanishes when Ep = 0.

Equations (5.6-8) are modified if the photorefractive grating is written
with zero applied field and the two beam coupling is subsequently measured
with applied field. In this case (5.6) and (5.7) reduce to their zero applied field

values so that

-144-

Ex = m—ENEa cost 20 | (5.9)
Eg + En

and sind = 1. Thus (5.8) becomes

T= 2m 1) 211 (oe)? Eo aut cos[2a]. (5.10)
d N

Figure 5-1 shows the experimental geometry used to characterize the
two beam coupling properties of the KLTN sample with an applied electric
field. An index grating was written in the material with zero applied field. A

field was applied and the beam coupling was measured. The beam amplifica-
tion was defined as e!/ = I,(L,Eg)/I,(L) where I,(L) is the output intensity of

I, when no index grating exists in the crystal and I,(L,Ep) is the output intensi-

ty when a grating has been written and a field is applied. Initially, the value
of Tis given by (5.10). But after several seconds, the beam coupling caused by
Ey modifies the magnitude and phase of the index grating as described by
equations (5.6,7) so that F evolves to the value dictated by (5.8). As T ap-

proaches its steady-state value, I,(L,E,) changes accordingly.

The crystal used for the following experiment was a copper doped

KLTN ( #94) with composition K 913Li ggqqNb ¢35Ta 35903. Its phase transi-

tion temperature was T, = 20°C where its dielectric constant reached € = 13600.

The experiment was performed at 24°C in open air. During writing I, was ap-
proximately 28 times greater than I,, and the total intensity incident on the
crystal was | ~200mW at 488nm. The two beams I, and I, were incident on

the crystal and wrote an index grating with no field applied. Then a field was

-145-

applied and the beam coupling was immediately measured. After approxi-
mately 20 seconds the beam coupling was measured again. During this time

the grating evolved to its steady-state value with the applied field.

In figure 5-2 the values of the measured intensity coupling are plotted
versus applied field. The lower set of data points refers to measurements
made immediately after the field was applied. In this case (5.10) applies. The
upper data points give the intensity coupling after the grating has evolved to
its steady-state value (5.8). The curves are the relevant fits to the theoretical
coupling constant. The results for the long write time case do not agree well
with the theory, especially for small values of the field. This is possibly ex-
plained as follows: It is well known that ferroelectric oxides exhibit substan-
tial frequency dispersion of the dielectric constant down to millihertz fre-

quencies” so that the effect of an applied electric field increases over a period

of time on the order of several minutes. This effect was observed in a sepa-
rate series of experiments in KLTN whereby the quadratic electro-optic coeffi-
cients (described in chapter six) were determined. Here the birefringence was
determined as a function of applied field; with application of a field, the bire-
fringence instantaneously rose to a certain value and then slowly increased
over a period of ten seconds or so to a value 10-50% higher. In the experi-
ment of Fig. 5-2 this would indicate that the short write time data was ob-
tained with a smaller effective dielectric constant than the long write time

data.

The results in Fig. 5-2 demonstrate linear voltage control of the beam

intensity coupling coefficient. Although the magnitude of the beam cou-

-146-

100 I
© Long write time
5 QO Short write time
Po
fev
& 10+ -
oS
on
106 l ! l l l |
0 200 400 600 800 1000 1200 1400 1600

Applied Field V/cm

Figure 5-2. The beam amplification caused by the intensity coupling coeffi-
cient I for a paraelectric KLTN is plotted versus applied electric field. Since
the material is centrosymmetric the coupling coefficient is exactly zero with

no applied field.

-147-

pling is smaller than published results for other materials,? the coupling is

controllable. In addition, the KLTN crystal displays little beam fanout even
with large applied fields. Beam fanout is often a severe problem for most

other materials, including BaTiO; and LiNbO3.
5.3 Electric Field Control of Diffraction
5.3.1 Solution of Coupled Equations

In this section we calculate the diffraction of an incident beam off a dy-
namically written index grating in a paraelectric material as a function of the
applied electric field. The grating is written with an arbitrary applied field.
The reading and writing fields are generally not equal. We include the effects
of the zero phase ZEFPR grating (see chapter six for details of the ZEFPR ef-

fect).

The photorefractive response of paraelectric KLTN is described by the
quadratic electro-optic effect in conjunction with the ZEPFR effect!®". The
ZEFPR gratings are unique in that they are always n/2 out of phase with the

electro-optically induced index grating. In addition, the ZEFPR index grating

is proportional to the space charge field. Thus we can write the nominal
index grating as An(z) = Ango(z)+ Anzz) which has the contribution from

both the conventional electro-optic grating as well as the ZEFPR grating.

An(z) = Ege(yEgcos(Kz + Og) + Yzr sin(Kz + o)}

= ExeV (yEo? + y2sin(Kz + dg + &) (5.11)
=n, Im{exp[i(Kz + )] }

-148-

where o = tan" (yE,/yz). Note that (5.11) is simply the nominal index grating;

the dynamically written grating is modified by the beam coupling (see 5.14).

Again E, is the applied uniform electric field and E,, is the photorefractive

space charge field. yis the effective linear electro-optic coefficient induced by
the presence of the applied field. It is given by (chapter three) y = -n,?

g(e,)°E,. Yz-¢is a coefficient which relates the index grating due to the ZEFPR

effect with the magnitude of the space charge field which inevitably forms in

conjunction with the ZEFPR grating. It is experimentally determined by the
diffraction observed with zero applied field. Finally o, is the phase between

the intensity and electro-optic gratings (see (5.7)) as dictated by the Kukhtarev

solutions of the band transport model in chapter two. We have

OE = tan’!

Eo En

where E, and Ey are as defined above. Reference to (5.11) shows that the ad-
dition of the ZEFPR grating modifies the net phase of the grating by a = tan”

'(yE,/¥z~, so that the net photorefractive phase 6 = dp + & is not equal to the

phase between the intensity and the electro-optic grating. Finally, equation

(5.11) gives the coupling constant (contrast with (5.5))

gE = . Esc,0 V (y Eo)” + yz

y Eol

vr +1. (5.13)

=8

Here E,.,, is the space charge field for unity modulation depth. The applied

-149-

field dependence of the space charge field is given in equation (5.6).

Therefore, dynamically written gratings in a paraelectric photorefractive ma-

terial such as KLTN are characterized by a net photorefractive phase © = dp +

a and a coupling constant given by (5.13).

The two beam coupling of two incident copropagating beams was
solved in chapter two. The solution of the beam equations yields the index

grating in the material. The result is

An(z) = ni Viq Io (I e F224 Ip et P22 joo! (5.14)
where J, and I, are the incident beam intensities. If the grating is written with

a field E,, applied (“w” = writing), the photorefractive phase 6 = ,, follows

from (5.12), and the intensity coupling coefficient [=2g,, sing, follows from

(5.12) and (5.13). ny is obtained from (5.11) or (5.13).

The diffraction of an incident beam off the grating of (5.14) is given by

the coupled equations
A(z) cosB = ig,VIy Ib eti% x (5.15a)
(I eT? + Ty etT22}**" 1 BG). & A)
B(z) cosB = ig, VI; bh e-it x (5.15b)

(Ip eT42 4 Ip etP2} "1" l Az). & Biz)
1 2 ,)

where a is the optical absorption coefficient, and B the half-angle of beam in-

tersection inside the material. g,, the coupling constant during reading, fol-

-150-

lows from n of (5.14) using the reading electric field as the applied field,
whereas 6,,, and I are calculated using the writing electric field. In other

words, the photorefractive phase and the intensity coupling coefficient in
(5.14) are determined by the writing field, but the material constant nj is a
function of the reading field. This fact becomes clear on inspection, and it is
the replacement of g,,, with g. in (5.15) which is responsible for the complexi-

ty of the solutions of (5.15).

Equations (5.15) are solved along similar lines as equations in chapter
nine. The motivation for the approach used here is given in chapter nine
and a reading of chapter nine is probably required to fully understand the
mathematical hoops of the following paragraphs. Since we will only calculate

the diffraction off the grating, i.e., only one beam will be incident on the grat-
ing, we will hereinbelow ignore the e!® terms in the coupled equations. This

term can easily be carried through the equations to show that it has no effect.
The optical absorption terms are eliminated in the standard fashion (chapter
two). On the resultant equation we perform the independent variable trans-

formation

é 2 JT gw sind,

= 2tan)[Vb/ et P22]. (5.16)

Equations (5.15) become

=ic sinfé]~7'" bE) (5.17a)

' _ i gr I - 2in 9) - 2in
b«) 2 sindy Sw 1b al a(S)

=ic* sin[E]*7!" a(€). (5.17b)

Here 7 = coto,,,/2 andc=g./g,, 21 Iy)1/2 / (2 sing,,,). The change to lower
case symbols “a” and “b” is used to highlight the change of independent vari-

able from z to € New functions T(§) and V(&) are defined with a(&) = T(é)

sing" and b(&) = V(&) sing. Second-order equations are obtained with

T(E) + ee TE) = 0 (5.18a)
L sin?é

V"() + |Iq2- ne V(E) = 0. (5.18b)
f sin’é

It is noted that the quantity Ic| 2. n? is given by
2 y2=1 SF - Bw 2
I 7 = + Se 4 B (5.19)
4 4 sin*6 g2,
so that when g. =g,,, B as wellas Icl?- ? reduce to 1/4. The solutions for

the diffraction in this special case are obtained in chapter eight; they are com-
paratively simple. When g, # g,, the solutions are substantially more com-

plicated.

-152-

Following a procedure similar to that outlined in chapters eight and

nine we obtain solutions for T(€) and V(6), i.e., a(€) and b(&), of the form
a(E) e@= a; sin’ 7"E cosé 2Fi[ - i +1 -B, - iS Nay +B; 3 a in; sin’& ] +

ao cosé oF fit +L jl4l4g.14 5 sin? 5.20a
20086 2Fili} +1 -B, ih +t +B; 1 + in; | ( )

b(E) e&@= b; cost Fila} + -B, - ity +B ; ; - in; sin2&] +

bz sin'**NE cos 2Fi[ i + 1-8, 3 +1 +B; 5 + in; sin2& ] (5.20b)

where 5F,[ a, b; c; z] is the hypergeometric function and aj, a ,b,, and by are

constants determined by the original coupled equations and the boundary

conditions. Since we are considering diffraction we take b(§p) = bg and a(&) =

0, where &, is the value of § when z = 0.

Eq = 2 tan! [VIo/Iy J. (5.21)
Using the Gauss transformations and the relations for hypergeometric func-

tions in equations (9.11) and (9.12) we obtain, after some manipulation

-ic(1+4n2) bo

aj = X 5.22a
1-2i1nN cos&
Fifi - B +1, 2+ pel L tin; sin°€o/D
2Fil i, - B +, 1, + +35 tin Fo]
ay = +icsin'-2"&) (1 + 2in) CO x (5.22b)
cost

9F[ - i} BH, il 5 +B +1; 57 in sin’Eg |/D

-153-

by = (14+4n2) Oo x (5.22c)
cost
5Fi[ iD -B +) 0 +B +53 , +in; sin’E]/D
by = sin! Eq (n2 + 48’) x (5.22d)
cos wosks

Fi - 7! -B, - ath +B; 3 x in; sin7Eo]/D
where the common denominator D is given by

= (1+ 4n2) oF, [it - B+, 1 + Bl; L sin: sino] x
( ee

9Fy[ -i- - 7 Bt i} B+: 5 -in; sin’Eq] +
sino (2 + “oh >F iI i} - B+1, i} + B+; > in; sin’&o] x
oF il iO - B+, oy + B+; 5 +in; sin2&o]. (5.23)

The equations (5.20a,b) with constant coefficients defined in (5.22a,b,c,d) and

(5.23) determine exactly the amplitude and phase of the transmitted beam b(&)

and the diffracted beam a(&). The solutions can be converted back to the inde-

pendent variable z using (5.16) where

sing = 2v1ik (5.24a)
-P2/2 . T2/2
cos& = (lie he) (5.24b)

(Ty e- 2/2 + L, e Pz/2)

-154-

sin€y = 22 (5.24c)
T, + 16)

cosy = i -h. (5.24d)
I, + 1)

Note that when I, =I, the coefficients (5.22) blow up and the eigenfunctions
(5.20) go to zero at z=0. This case must be treated separately because from the
outset the equations assume a different form (equations (5.15)). In practice it
is easier to take the limit as I, ->I5. Another caveat: if I, >1, the coefficients
(5.20) are only valid for z 20, and vice versa when I,<1,. At the point z=0
the coefficients a; and a, change sign. The reason that z cannot be chosen ar-
bitrarily is that the original index grating (5.14), from which (5.15) was de-

rived, was written with beams incident at z=0.

Finally, the reflectivity of the grating, i.e., diffraction off the grating, is
given by R= |A(L)1*/1B(L)I2 . Thus we have solved for the diffracted power
of a beam off a grating as a function of the applied (reading) electric field. The
grating is written dynamically with an applied (writing) electric field. When
the reading field is equal to the writing field the solution simplifies consider-
ably and is given in chapter eight. This previously obtained solution can be
used as a check in numeric calculations. In the next section we repeat the
above calculations for the special case of zero applied writing field during

grating formation.

5.3.2 Solution of Coupled Equations with E,, =0

When the writing electric field is zero, the photorefractive phase 6 is

also zero. In this case the index grating of (5.14) simplifies to

-155-

An(z) = ny are In exp[ ig z(I2-1,)/T. (5.25)
Here n, and g are the zero field values of those parameters (use (5.13) and the
definition g = mn,/A) and | is the total incident intensity. The coupled mode
equations (5.15) become

A(z) cosB = ig, V2 gia: Biz) - & A) (5.26a)
I; + I, 2

B(z) cosB = ig, otL 2 Ip gid A(z) - & Biz) (5.26b)

1+Iy

where A =g (I, - 1,)/(, + I,) is a Bragg mismatch term caused by the phase

coupling each beam exerts on the other. The solution of (5.26) is relatively

straightforward. The optical absorption terms are eliminated by allowing A(z)

az/2 az/2

= A(z)e and B(z) = Bizje . The equations are simplified by the coro-

tating transformation a(z) = A(z) etA2/2 and b(z) = B(z) etiAz/2 TF the
boundary conditions are taken as B(0) = by and A(0) = 0, i.e., diffraction of by
off the grating, the solution is easily shown to be

A(z) e&/2= bo e} 42/2 Lk sin{sz] (5.27a)

B(z) e%/2= bo e i 47/2 | cos[sz] + LA sin[sz] (5.27b)

where s is defined by s? = «2 + A?/4 and K=g, (1, 1,)!/*/1. The diffracted in-

tensity off the grating is

-156-

h@) eax — (<} sin’[sz]

bo
(YEo/yzt) +1 4ub
- 2 41 2
1 + (yEo/yzp) ——1-2- * (I) + I)
I, + Ip)
in? & 2 4b |”
sin) 2} 1 + (YEo/yzp) | 2 (5.28)
2 (I) + Ip}

Equation (5.89) is considerably simpler than the solution given by equations
(5.20) and (5.22). This is a direct consequence of the fact that the index grating
is constant amplitude. In the succeeding paragraphs we compare the theoreti-

cally expected reflectivity with experimentally obtained data.

5.3.3 Diffraction Experiment Results

The sample used in the following discussion was a 6.8x 4.9x 2.9mm?

piece cut from a crystal grown by us using the "top seeded solution growth

method." The crystal was pale olive green and had only weak striations. Its as-
grown weight was 11.6gm. Using electron probe and atomic absorption anal-

ysis, the composition was determined to be K gsoLi og Ta gsgNb 499 O3: Cu ggg:

Figure 5-3. shows the absorption spectrum of the as-grown sample with

peaks at 370nm and 585nm caused by Cu!+ donors and Cu?* traps respectively.
The concentration of the impurities Cu!+ and Cu?* can be determined from
the absorption peaks using Beer's law!*. We calculate [Cu!*] = 6.0x10!9/cm3

and [Cu2*] = 2.1x10!8/cm3. The optical absorption in the material gave expl-

-157-

aL] = 0.79.

White light birefringence measurements between crossed polarizers

were used to measure the effective quadratic electro-optic coefficients. The co-

efficients were determined g,)-8)5 = 0.186 m4 C 2 to an accuracy of 5%.

The diffraction experiments with applied field were performed as in

figure 5-4a with the sample maintained at a temperature of 15°C above the

transition (T,= 178K). The writing argon laser beams were at either 488nm or

514nm. They were ordinary-polarized to minimize beam interaction. The
diffraction efficiency of the grating thus written was monitored with a weak
extraordinary-polarized HeNe beam at 633nm. The 633nm beam was incident
at a higher angle of incidence to be Bragg matched to the index grating. The
633nm beam was verified not to erase the grating at the temperature of the ex-
periment. The writing continued until the maximum diffraction was

achieved.

Figure 5-4b illustrates a “momentum matching” diagram used to cal-
culate the Bragg matching angles. Two beams with wavevector k, interfere to

create an index grating with wavevector K,. By the Bragg condition of scatter-

ing from a volume hologram, the angles of the incident and reflected beams

must be equal, hence K, = 2 k, sin@,. the same condition applys for the read-

ing beam at 633nm: K, = 2k» sin05. The angles are graphically determined by

the intersection of two circles of radius k, and ks as shown in the figure.

After the gratings were written, the crystal was cooled to a few degrees

-158-

above the transition, and the diffraction efficiency determined as a function of
applied field. The results are illustrated in figures 5-5, 5-6, and 5-7. Figure 5-5
shows the diffraction as a function of applied field for a grating written with
zero applied field. The solid curve is a fit of the theory developed above to
the data. The best fit occurred for g = 1.3/cm. Figures 5-6 and 5-7 show diffrac-
tion versus applied field for gratings which were written at +1450V/cm and -
1450V/cm. The solid curves are fits to the data using the same parameters as
in figure 5-5 except with g = 3.0/cm, and g = 4.0/cm respectively. When a field
is applied during writing an internal compensating field of +300V/cm seems
to be set up. This offset is manually inserted into the theoretical curves for
reasons explained below. In all cases, when the grating was optically erased
with the argon beams, some residual diffraction (~1%) remained which was
not optically erasable and could only be removed by heating the crystal to near

room temperature.

The highest diffraction efficiency observed for 488nm writing beams
was 75% for a sample of thickness 2.9mm, where corrections were made for
fresnel losses. For 514nm writing beams the maximum value was reduced to
30%, and at 633nm, the diffraction was almost undetectable. The writing

times for maximum diffraction roughly followed 1 ~ 6sec*-cm2/J, inci-

write

dent intensity on the crystal.

From the calculated index modulation of n, = 1.7x10°4, we determine

the space charge field to be E,, = 150V/cm. Using the writing time of 180sec at

beam intensities adding up to 27.2mW, we estimate the sensitivity for this

-159-

N Ww
on fo)

fo)

ro)

Absorption (1/cm)
on

Sa)

350

450 550 650 $750 850
Wavelength (nm)

animes eel
950 1050

Figure 5-3. Absorption spectrum of the as-grown KLTN:Cu.

-160-

vacuum oO

488nm
cryostat

diffracted
beam

Figure 5-4a. Experimental setup for measurement of diffraction efficiency. A
photorefractive crystal is mounted in vacuum cryostat. 488nm light beams
write an index grating which is read with the Bragg matched 633nm beam.
Diffraction is measured versus applied voltage V,. Figure 5-4b. A momen-
tum matching diagram illustrates the angles required to achieve Bragg match-

ing of a grating to more than one wavelength of incident light.

-161-

KLTN of 7.30x10-6cm3/J for an applied field of 1.6kV/cm. Following the pro-
cedure outlined above, the erase time near Twas up to two orders of magni-

tude longer than the write time at T, + 15°.

5.4 Discussion of Experimental Results

Several points about figures 5-5,6,7 need to be addressed. The first is
the change in the best fit value of g in the three figures. Since g is a material
parameter it should be independent of the field applied, however, the best fits
were g= 1.3/cm , 3.0/cm, and 4.0/cm. The particulars of the experimental con-
ditions may be able to explain the variations. The diffraction experiments
were done using a vacuum chamber which was not well isolated against
acoustic vibrations; it was determined that good data could only be taken late
at night while standing motionless next to the optical bench. Therefore the
gratings were written with an intensity pattern which rapidly flitted back and
forth a fraction of a wavelength. This artificial modulation of the effective
photorefractive grating phase caused beam coupling which was visible as a 10-
30Hz flickering of the output intensities. The varying beam coupling affected
the form of the dynamic grating, i.e., smeared it out. This might have caused

the variation in the measured value of g for figures 5-6 and 5-7.
As shown in chapters seven and eight, the susceptibility of a grating to
vibration is strongest for zero phase $= 0 gratings (more beam coupling oc-

curs) and decreases as $ -> m/2. Thus when a field is applied during writing

the grating, as in figures 5-6 and 5-7, the nonzero phase should decrease the

effect of vibration on the beams. In this way a stronger grating can be written

-162-

80

on
an)

Diffraction Efficiency - %
§ s
| ]

-2400 -1600 -800 0 800 1600 2400

Applied Field - V/cm

Figure 5-5. The measured diffraction efficiency for an index grating written at
T, + 15° with zero applied field. The crystal was cooled to near T, and the
diffraction monitored versus applied field. The solid curve is the best fit to

the theory with g = 1.3/ cm.

-163-

80

oO

Diffraction Efficiency - %
S S

0 | I i
-2400 -1600 -800 0 800 1600 2400

Applied Field - V/cm

Figure 5-6. The measured diffraction efficiency for an index grating written at
T, + 15° with an applied field of +1450V/cm. The crystal was cooled to near T,

and the diffraction monitored versus applied field. The solid curve is a fit to

the theory with g = 3.0/ cm.

-164-

Ce lo, io)
om) © ©

Diffraciton Efficiency - %

0 l |
-2400 -1600 -800 0 800 1600 2400

Applied Field - V/cm

Figure 5-7, The measured diffraction efficiency for an index grating written at
T, + 15° with an applied field of -1450V/cm. The crystal was cooled to near T,

and the diffraction monitored versus applied field. The solid curve is a fit to

the theory with g = 4.0/ cm.

-165-

than in the case of zero applied field, so the value of g measured will be high-

er.

The second point to be discussed is the positions and magnitudes of the
maximum diffraction efficiency which seem to have behaved strangely at first
blush: The highest diffraction achieved occurred in figure 5-5 with zero ap-
plied field during writing (g = 1.3/cm), but at a higher value of the applied
reading electric field than in figures 5-6 and 5-7. Also, the maximum diffrac-
tion in both figures 5-6 and 5-7 occurred for a net applied field of approximate-
ly 1600V/ cm even though the coupling constant g was different in the two
cases. Finally, the value of the maximum diffraction efficiency decreased as g
increased, and the diffraction in the case of non-zero writing field was no
longer periodic with the readout field, that is, with increasing readout field
the diffraction would not return to zero. All these characteristics are expected
from the theory presented above and are illustrated in the theoretical curves.

These results of the theory can be explained fairly intuitively.

When = 0 (writing electric field = 0), the diffraction is given by (5.29),

and its maximum value is independent of g, and depends only on the beam
intensity ratios. (To recapitulate, g and g..,,4(E) are defined in (5.13).) Thus
when g is increased, the position of maximum diffraction will shift to a lower

value of the reading electric field, but the magnitude of the maximum diffrac-
tion will be unchanged. When > 0 (non-zero applied writing field) the

diffraction maximum decreases with increasing g because the beam coupling
causes grating apodization. The higher the value of g, the more apodization,

so that the “integral” of the index grating throughout the volume of the crys-

-166-

tal decreases. Nevertheless, the position of the diffraction maximum is
roughly independent of g over certain ranges (see Figure 5-8) because the de-
crease of the index grating “area” inversely proportional to g must be com-
pensated by a linear increase in g,,,4g to obtain maximum diffraction. The ap-
propriate increase in g,,,4 occurs at approximately the same value of E for all
g. Meanwhile, the maximum value of the diffraction decreases with increas-
ing g because the grating is apodized in phase as well as in amplitude; thus a
highly apodized grating does not Bragg match well to an incident beam. This

lowers the maximum value of the diffraction.

In figure 5-8 the theoretical diffraction versus applied reading field is
shown for gratings written with an applied field of 1150V/cm, including opti-
cal absorption as in the experiment above, and for g = 2.0/cm, 3.0/cm, and
4.0/cm. The last two curves are the same as in Figs. 5-6 and 5-7. It can be seen
that the maximum diffraction occurs at roughly the same value of the read-
ing field (~1600V/cm) in all three cases, but the diffraction maximum decreas-
es with increasing g. As a final note, the diffraction never returns to zero
with increasing applied electric field during reading. This is again due to the
phase apodization of the grating which prevents the coherent buildup of the
diffracted wave. The grating written with zero applied field has no apodiza-

tion and does not suffer from either of these two effects.

When a field was applied during grating formation as in graphs 5-6 and
5-7, a compensating internal field of t300V/cm was formed. The compensat-
ing field is most likely caused by nonuniform illumination of the crystal.

When an external field is applied, photoexcited charges drift in a direction de-

-167-

wo

I i! J ]

—s— g=2.0 E)=1150

—e— g=3.0

oO

Diffraction Efficiency - %
5 S

-3000 -2000 -1000 0 1000 2000 3000
Applied Field - V/cm

Figure 5-8. The theoretical diffraction efficiency for an index grating written
with an applied field of 1150V/cm for g = 2/cm, 3/cm, and 4/cm. The maxi-
mum diffraction decreases with increasing g. However, the maximum
diffraction occurs at a reading field of approximately 1600V/cm for g 2 3/cm.
The diffraction is nonperiodic in the applied field, it does not return to zero

when the applied field is increased.

-168-

termined by the field before being retrapped. This preferential drift is mani-
fested as a photocurrent proportional to applied field. After repeated excita-
tion and trapping, charges may end up many wavelengths removed from
their origin. If certain areas of the crystal are dark, charges will become
trapped there, yielding a more or less uniform field opposing the applied
field. This explanation of the field offset seems consistent with the theoretical
results because the theoretical diffraction matches the experiment much bet-
ter with a net applied field of 1150V/cm rather than 1450V/cm. If 1450V/cm
is used in the theory then the position of maximum diffraction occurs at
higher electric fields than were observed (that is, at 2400V/cm rather than
1600V/cm), so the theoretical results indicate that the field is more like

1150V/cm as expected.

From the arguments presented we see that the qualitative features of
the experimental data are explained by the theory. However, several discrep-
ancies remain. The shape of the experimental and theoretical curves in fig-
ure 5-5 do not agree well. Also the diffraction in the case of an applied field
during writing does not exhibit the ZEFPR effect. This can be inferred since
the diffraction efficiency is zero for some value of the applied field (+300 V/
cm ) whereas it has been shown above that the ZEFPR grating is z/2 out of
phase with the electro-optic grating and the two gratings should add in
quadrature so that the diffraction efficiency never drops to zero. Apparently
the strong internal field during writing seems to inhibit the ZEFPR effect.
These two anomalies indicate a possible coupling between the ZEFPR grating
and an applied field which is not described by the theory presented above.

Nevertheless, despite this possible coupling and the fact that the experiment

-169-

was performed near the phase transition temperature (where bizarre effects
are typical), the theory still provides good qualitative agreement with the re-

sults.
5.5 Summary

The photorefractive effect of paraelectric KLTN material with an ap-
plied field was discussed. The photorefractive response has contributions
from both the conventional electro-optic index change as well as from the
ZEFPR response. In the absence of an applied field no two beam coupling oc-
curs although diffraction is observed. The diffraction of a single beam as a
function of applied field off a photorefractive grating was solved analytically
using the techniques described in chapter nine. Good qualitative agreement

was observed between this theory and experimental results.

-170-

References for chapter five

[1] PJ. Van Heerden, Appl. Opt. 2,393 (1963).

[2] D. Gabor, IBM J. Res. Dev. 13, 156 (1969).

[3] D. Von der Linde and A. M. Glass, Appl. Phys. 8, 85 (1975).

[4] A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new pho-

torefractive material: Ky yhyT Ny” Opt. Lett. ,17, 713-715 (1992).

[5] A. Agranat, V. Leyva, K. Sayano, and A. Yariv, “Photorefractive properties
of KTa;_.Nb,O3 in the paraelectric phase,” Proc of SPIE Vol. 1148, Conference

on Nonlinear Optical Properties of Materials, (1989).

[6] A. Agranat, V. Leyva, and A. Yariv, “Voltage-controlled photorefractive
effect in paraelectric KTa1_,Nb,O3:Cu,V,” Opt. Lett. 14, 1017-1019 (1989).

[7] J. P. Wilde and L. Hesselink, “Electric-field controlled diffraction in pho-

torefractive strontium barium niobate,” Opt. Lett. 17, 853-855 (1992).

[8] M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics
and Related Materials , Clarendon Press, Oxford (1979).

[9] F. Laeri, T. Tschudi, J. Albers, “Coherent cw image amplifier and oscillator

using two wave interaction in a BaTiO; crystal,” Opt. Comm. 47, 387 (1983).

[10] R. Hofmeister, A. Yariv, S. Yagi, and A. Agranat, “A new photorefractive
mechanism in paraelectric crystals : A strain coordinated Jahn-Teller relax-

ation,” Phys. Rev. Lett. , 69, 1459-1462 (1992).

[11] RK. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, “Simple methods of

-171-

measuring the net photorefractive phase shift and coupling constant,” to ap-

pear Opt. Lett. April 1, 1993.

[12] V. Leyva, Ph.D. thesis, California Institute of Technology, Pasadena,
California, unpublished (1991).

-172-

Chapter Six

Zero External Field Photorefractive (ZEFPR) Effect in

Paraelectric Materials

6.1 Introduction

As indicated in chapter one, the field of photorefractive studies has
been thwarted by severe material limitations since its inception some twenty

years ago. The photorefractive properties of potassium tantalate niobate
(KTN), for example, were reported more than fifteen years ago! but the diffi-
culty of growing high quality crystals hindered their study until recently. The
photorefractive materials development program at Caltech was conceived
largely in response to this demand for better and more cheaply available ma-
terials. The program focussed on the growth of a new type of material - the

paraelectric photorefractives. These materials had been proposed? because
they possessed the advantages of very high diffraction efficiency and control
of the photorefractive diffraction by an external electric field. Both of these
predictions were borne out in experiments documented in earlier chapters*”.
But in these studies another effect surfaced which could not be explained by
conventional theories of photorefractives. Investigations of these paraelectric

crystals have revealed a photorefractive response which cannot be explained

-173-

by the conventional electro-optic theory. The effect was discovered serendipi-
tously in experiments where the conventional effect was forbidden, that is,
when no electric field was applied. This new effect is expected to occur in
most transition metal doped perovskites, and possibly other materials as well.
But in these materials, the conventional photorefractive effect will mask the
new one. Possibly this explains why the effect has been unreported until re-

cently.

Perovskite oxide photorefractives operated above the phase transition
temperature (paraelectric and centrosymmetric phase) lack a linear electro-
optic coefficient. Instead, the photorefractive mechanism in these symmetric
materials arises through the quadratic electro-optic effect. Here the index grat-

ing due to a spatially periodic field E(x) in the presence of an applied electric

field Ey can be written

Anjj(E+Esc) - Anij(Eo) = oe gi (e3€0 ( (Ese + Eo)” - ES) (6.1a)

Ani; = N§ 2ij33 (€3€0)" Esc Eo (6.1b)

where i333 = &m3 18 the relevant quadratic electro-optic coefficient when all

fields are nominally along the z, i.e., the “3,” axis, Ny is the index of refrac-

tion, and Ani; is defined as the term on the right-hand side of equation (6.1a)
which leads to a Bragg matched grating. We assume that the polarization is

linear (P = €E ) and that the dielectric constant € >> 1. The externally applied

field E, thus induces an effective electro-optic coefficient T o¢¢ m =2 8 m3 (e€,)*

E,. Therefore the conventional photorefractive effect is zero in the absence of

-174-

a spatially uniform electric field (see chapter three for details).

Nevertheless, photorefractive experiments reveal the existence of a

zero electric field photorefractive (ZEFPR) effect in paraelectric KTN and
KLTN at temperatures at least 120°C above the phase transition where the
crystal is nominally symmetric. No effect is seen with undoped crystals or
with thermally reduced samples. The diffraction expected from absorption
gratings®® is three orders of magnitude too weak to explain the effect. The

studies of the phenomenon which are described here, reveal a new photore-
fractive mechanism. In this chapter we describe the effect and present what
we believe is the most plausible explanation. The experimental results sup-
porting the explanation follow and, finally, a theory of the Jahn-Teller relax-

ation is given.
6.2 The ZEFPR Effect

The zero external field photorefractive (ZEFPR) effect was first noticed

by us? in KTN and KLTN crystals. This photorefractive effect was attributed

to the presence of a growth induced strain. In addition, Yang et al.? have cited

an “extremely small” effect in the absence of an applied field, but without ex-
planation. In KTNs and KLTNs grown at Caltech with high niobium concen-

trations and of high optical quality, the effect is greatly enhanced. Under cer-

tain conditions of crystal preparation zero field index gratings with An ~

1.7x10°° , and diffraction efficiencies of over 20% have been produced in a

4.15mm thick sample using 488nm argon laser beams.

-175-

6.2.1 The Jahn-Teller Distortion

In the KTN and KLTN crystals which displayed a strong ZEFPR effect,

the photorefractive dopant was copper, which is stable as either Cult or Cu**.

The Cu2* ion is known to cause large Jahn-Teller!0"! (J-T) distortions, espe-

cially in octahedral symmetry. The electric field of the six oxygen ligands

splits the 3d orbitals into an orbital doublet and a triplet, with E,- T 5 sym-
metry. Since the copper ion has 3d° configuration the distortion allows, by re-

moving the triplet degeneracy, the vacancy of the tenth electron to go to the

highest energy level of the former triplet state, thus lowering the energy of
the system. The Cu!+ ion by contrast, has a 3d!9 configuration, and conse-

quently has no tendency to distort. The illumination of the crystal by the pe-

riodic intensity pattern of the optical field leads to a mimicking spatially peri-

odic Cu2+/Cult ratio due to excitation of electrons (from Cu!*) and trapping

by Cu2+. This, as explained above, gives rise to a spatially periodic distortion.
Since the copper concentration is relatively small in the KLTNs we do not ex-
pect strong cooperative ordering of the distortions; rather, their orientation
should be random, or display partially ordered regions which are small com-
pared to a wavelength of light. But when a macroscopic (growth induced)
strain is present, as is the case in the crystals studied, the distortions will ori-
ent preferentially in order to minimize that strain. The result is a spatial
modulation of the strain field in phase with the intensity which leads to a
corresponding modulation of the index of refraction ( index grating ) via the
photoelastic effect. In the absence of a strain field, J-T distortions will still

cause an index grating because the distortion changes the volume of the unit

-176-

cell.

Although the J-T effect has previously been reported in photorefrac-
tives!2-16 and it has been shown to generate birefringence!7, the ZEFPR effect

has not been reported. As explained above, this is probably because the con-
ventional effect obscured any ZEFPR effect which may have existed in those
materials. In Refs. 13-16 the possibility of ionization state dependent distor-
tions are considered, and in Refs.13-15 it is shown how the ionization of pho-
torefractive centers may lead to a modulation of the spontaneous polarization
and thus a contribution to the photorefractive effect in the polar (noncen-
trosymmetric) phase. However, the possibility of a ZEFPR-type photorefrac-
tive effect in centrosymmetric materials is explicitly disallowed (no macro-
scopic spontaneous polarization exists in centrosymmetric materials). These
erroneous conclusions were reached by ignoring the possibility of unit cell
volume change under J-T distortion and ordering of J-T distortions by a
macroscopic strain field which can occur in any material, including a cen-

trosymmetric one.
6.2.2 Strain Dependence of ZEFPR Effect

The strain in the crystals is due to the particulars of the growth process;
it is only briefly described here since it was already discussed in chapter four.
The crystal grows as a series of cubical shells, expanding from the seed.
During the growth the composition of the crystal changes which is attendant
by an increase in the lattice constant. Thus each face of the cubical shell of the
growing crystal must be compressed slightly to mesh with the previous cubi-

cal shell. In this way there arises a compressive strain in the plane of each

-177-

Seed crystal

Cubical growth

Grown crystal planes

Internal stress
vectors \

Unconstrained
crystal position

Plane of VA

zero strain

Figure 6-1. Origin of the Growth-induced Strain

A cross section of the grown crystal shows how it grows as a series of
cubical shells, expanding from the seed, with a progressively increasing lattice
constant. Each face of the cubical shell of the growing crystal must be com-
pressed slightly to mesh with the previous cubical shell (the innermost shells
are stretched). The dotted line shows the plane of zero stress. In this way
there arises a birefringence in the plane of each face of the cube which increas-

es with distance from the seed crystal.

-178-

face of the cube which increases with distance from the seed crystal. Similarly,
there is also a tensile strain perpendicular to each face of the cube which also
increases with distance from the seed (Figure 6-1). These strains induce a lin-
ear birefringence which is readily apparent when the crystal is viewed
through crossed polarizers. When a small sample is cut from the grown crys-
tal near the center of one of the cube faces, the strain in the sample will be ho-
mogeneous: uniformly compressive in two directions and uniformly tensile

in the third.

To test the validity of the theory of J-T relaxation we investigated the
dependence of the index grating on the strain present in the crystal. The ex-
perimental setup for performing diffraction experiments is illustrated in fig-
ure 6-2. Two extraordinary beams at 488nm symmetrically incident, created an
optical intensity standing wave inside the sample. After several minutes, one
of the beams was blocked with an electronic shutter for 50msec. While the

shutter was closed we measured the optical power which was diffracted by the
grating into the direction of the blocked beam. The diffraction efficiency, 1, is

defined as the ratio of the diffracted power to the power incident on the crys-

tal. We corrected for losses from facet reflections.

Unless otherwise indicated, all measurements were performed at room
temperature on several KTNs and KLTNs with phase transition temperatures
from 180 K - 280 K. It was determined that in homogeneously strained sam-
ples the diffraction efficiency increased as the square of the interaction length
and was independent of intensity. A two-dimensional vise was constructed

(figure 6-3) to allow compensation of the growth induced strain with external

-179-

Photorefractive Detector
488nm crystal

ND V,, (optional)

Diffracted
beam

External
trigger

Figure 6-2. The setup used to measure diffraction of an index grating written

in paraelectric KTN/KLTN crystals. The beams are extraordinarily polarized
and converge at an angle 8, on the crystal. 8, can be adjusted by moving the

crystal assembly along the bisector of the two beams. A shutter is coordinated
with a detector to measure the diffraction off the grating. A variable attenuat-
ing filter ND allows the setting of the beam intensity ratios. The applied volt-
age is only used when measuring the quadratic electro-optic contribution to

the index grating.

-180-

P = polarizer

A = analyser

X = screw, x-axis clamp

Y = y-axis clamp
(electrode)

S = sample

G = glass blocks

W = white light source

P = polarizer
A = analyser
X = screw, x-axis clamp

“4 S = sample
QQ G = glass blocks
oN x W = white light source
E = electrodes

Figure 6-3a,b. Two Axis Strain Compensation Device

A KLTN sample S is mounted between glass blocks G and brass elec-
trodes Y which allow the crystal to be squeezed in dimensions X and Y inde-
pendently. A white light source and crossed polarizers oriented at +45° to the
crystal axis are used to monitor the birefringence. By adjusting the pressure
applied to the X and Y axes, most of the internal strain in the material can be

compensated. In figure 6-3b, stress is only applied along one axis.

-182-

pressure applied in two dimensions. With the strain nominally compensat-
ed, diffraction experiments were performed with the setup in figure 6-2. The
diffraction efficiency in a 2.85mm thick sample was reduced by 40% when the
internal strain was minimized. Additionally, the effect is reduced when the

sample is exposed to heat treatments which reduce the internal strain but
which leave the conventional photorefractive properties unchanged!®.

Although this evidence shows that the ZEFPR effect relies partly on the

macroscopic strain, it remains to be shown that the strain does not induce a
morphic lowering of the crystal symmetry !%20 (see also chapter three), allow-
ing a linear electro-optic coefficient. We tested for the existence of a linear
electro-optic coefficient in a 2.85x5.00x9.50 mm? sample of K 993Li o06

Ta ¢g7Nb 3,403 by measuring the birefringence induced under application of

electric fields in various directions. A Soleil-Babinet compensator between
crossed polarizers oriented at 45° to the crystal axes was used to measure the

birefringence at 633nm. The results were fitted to a third order polynomial
but the best fit was purely quadratic to the resolution of the experiment. The

best fit (see figure 6-4) was determined to be

An = 9.47x10 > + 2.56x107E - 1.07x10 > E? - 8.07x10° E° (6.2)

where E is in units of 10°V/cm and individual measurements were repeat-

able to within (An) < 5x10°7. This indicates an almost perfect quadratic elec-
tro-optic effect with 21,-2)= 0.123m4* C2, which agrees with previously pub-

lished values for KTN?2!. The constant term of the birefringence is from the

-183-

ry

Birefringence An - (x 10° )

-2 | l l i l | l
-4000 -3000 -2000 -1000 O 1000 2000 3000 4000

Applied Field - V/cm

Figure 6-4. The birefringence of a Kg 993L1 pg9gNbo 6977 a9.31403 at room tem-
perature versus applied field. The birefringence is quadratic with no linear
component, thus there is no linear electro-optic coefficient and the crystal is
centrosymmetric. The birefringence with zero applied field is due to the uni-

form growth induced strain.

-184-

growth induced strain. The small linear term is effectively zero to the resolu-
tion of the experiment. It should be noted that the absence of a significant
third order term indicates little or no polarization nonlinearity. We conclude
that the strain does not induce a linear electro-optic coefficient and leaves the
crystal centrosymmetric. This data also disproves the possibility of an inter-

nal polarization which would render the crystal noncentrosymmetric.

6.2.3 E,. Dependence of the ZEFPR Effect

Next, a series of experiments were performed to verify anticipated
characteristics of the ZEFPR effect. For these experiments a 8.69 x 4.45 x 4.15

mm? sample was used with composition K gogh-1.ao19 Ta.739NP 27993: Cu gg15-
Using optical absorption data* we determined [Cu2*] = 3.1x10!8 cm°3. The

crystal was centrosymmetric above its phase transition at Ty = -23°C.

First, since the ZEFPR effect is due to a strain “grating” which modu-

lates the refractive index via the photoelastic effect, we expect at most weak
dependence on the DC/low frequency dielectric constant e. We confirmed
this by measuring the dependence of the ZEFPR effect on temperature near
the phase transition ( here e obeys the Curie-Weiss law). Using the setup as
in figure 6-2, two interfering 488nm beams with equal intensities of ~500mW
cm"? uniformly illuminated the crystal. No field was applied during writing

of the diffraction grating. After a writing time of 60 seconds, one beam was
blocked and the other beam attenuated to minimize erasure, and the resul-

tant diffraction was measured. After the diffraction due to the ZEFPR effect

-185-

was determined, a field was applied to determine the index change due to the
quadratic electro-optic effect (equation (6.1b)). Finally, after each measure-
ment, the gratings were completely erased by flooding the crystal with uni-
form illumination and raising the temperature, if necessary. The results for

various temperatures are illustrated in figure 6-5. The quadratic effect in-
creases dramatically as the phase transition temperature (= -23°C) is ap-

proached because of the concomitant increase in dielectric constant. The
phase transition temperature was measured with a different temperature
probe from that used to control the diffraction experiment, thus the two tem-

perature scales may have a relative shift between them of a few degrees. The

extremely high diffraction efficiencies with a sin? rollover at high fields are

characteristic of the quadratic effect in paraelectric KLTN*° (and see chapter
three). The ~1% diffraction efficiency observed at zero electric field is caused
by the ZEFPR effect. It is independent of the dielectric constant since it is
nearly a constant for the nine temperatures investigated which range through
the ferroelectric transition at -23°C. Although the ZEFPR effect diffraction is
weak in the configuration used for this experiment, the same sample yielded
20% diffraction efficiency at a higher angle of beam incidence. The fact that
the ZEFPR index grating has no noticeable dependence on the dielectric con-
stant proves that the effect is distinct from the quadratic electro-optic effect or
the dielectric photorefractive effect?3-25. The ZEFPR effect is a new phe-

nomenon which has nothing to do with the polarization caused by a space

charge field (P = eE).

The model stipulates that the magnitude of the ZEFPR index grating

-186-

I ] I
iS Oo T2107. - 20006
~ B-- =0.0
—@— =-10.8 ro:
en. rr, oe A-- =-14.9 ae
i —o— =-218 OLS
Nha a
= he 25.8
2 vy aa A
20+ aN / “
, Oo NY ee 7
-1500 -1000 -500 0 500 1000 1500

Applied Field E. - (V /cm)

Figure 6-5. The diffraction efficiency versus applied field for an index grating
written with zero applied field. Measurements are taken at several tempera-
tures from room temperature to T = -29.19°C. The small zero field value is rea-
sonably independent of temperature although the quadratic electro-optic con-
tribution increases by more than an order of magnitude as the sample is
cooled to the phase transition. The lines connecting the data points are only

guides to the eye.

-187-

depends linearly on the spatial modulation of the Cu2* ions. Since the Cu2+

modulation is equal to the modulation of the photoexcitable charge carriers,
we expect the ZEFPR index grating to have the same functional dependence
as the light induced space charge electric field caused by the charge modula-
tion. From the basic Kukhtarev model for the space charge field induced by

the extraordinary polarized interfering beams we have?®

j Wik _EpEN _ cos 29

E =
SC "ly + b Ep + En

j Vik KTk/e cos 20 (6.3)
I + Io 1 + K*kTe/(e2Naq)

where @ is the angle of beam incidence in the crystal, I, and I, are the beam

intensities, and K = 2k sin®@ is the index grating wavevector. The dependence

on E,, was tested, again using the setup of figure 6-2, by monitoring the
diffraction efficiency as a function of the grating wavevector and the beam in-
tensity ratios. We determined a relation between the index change and the
space charge field for the ZEFPR effect from the E,=0 data of figure 6-5 to be

Anzeppr = 5x10°°E,, in cgs units. Figure 6-6a shows the diffraction efficiency

versus K. Peak diffraction efficiency of 6.1% was observed, corresponding to

An = 9.1x10°°. Figure 6-6b shows a plot of K,/(An cos(2@)) versus K? with a fit

to equation (6.2). The best fit occurs for N, =3.0x10!8em™, which is near the

value (3.1x10!8 cm-3) obtained from absorption data. Figure 6-7 plots the

diffraction versus beam intensity ratios, along with the theoretical curve from

equation (6.2). The good correspondence of the data in figures 6-6 and 6-7

-188-

7 | T I
SL ° 0.9W . :
Oo 1.8W 5 (89 O
= 5 | te) e _]
ra) e
aa O°
c e
& 3° °° =
a) e
fae)
o 2 4
Fs .*
AL O6 1

eo °
0 | | l
0 5 10 15 20

Grating Wavevector K - (x1 0° m! )

fF

a _

S i —®—99W N_.=3.5 1074 m? |
A 0.5 eff

S --O--7.8W N_.=2.3 1074 m?3

oO eff |

ed, 0.1 0.2 0.3

K? - (x10'°m”)
Figure 6-6a. The diffraction efficiency versus grating wavevector (a function
of beam incidence angle). Figure 6-6b. The same data replotted to allow a fit
to the effective donor concentration Np o¢¢-= 3.0x10!8 em’3, in agreement with

results from optical absorption data.

-189-

6 T
TS :
ms
Q 4 7
oO
m3 i
——
g 2 ;
A 4 a
0 | ] l | |

0 0.2 0.4 0.6 0.8 1 1.2 1.4
Beam Intensity Ratio

Figure 6-7. The diffraction efficiency of a KLTN crystal versus the beam in-
tensity ratios. The curve is the theoretical dependence of the space charge

field on the modulation depth.

-190-

with equation (6.2) allows us to conclude that the index change of the ZEFPR

effect varies as the space charge field.

The final test of the ZEFPR effect model was a measurement of the
phase of the ZEFPR index grating relative to the intensity grating of the writ-
ing beams. Since the index grating is modulated by the local Cu2* concentra-
tion, we expect it to be in phase with the intensity, so that no two-beam cou-
pling (power exchange between the two writing beams) will occur. This con-
dition was verified by testing for two-beam coupling with the setup of figures
5-1/ 6-2. No power transfer was observed even though strong diffraction oc-
curred when either beam was blocked. Two-beam coupling was observed,
however, when an electric field was applied, the sign of the coupling coeffi-

cient changing with the direction of the applied field. (Beam coupling with

applied field?” in ZEFPR materials is discussed in chapter five and detailed

phase measurements”® are performed in chapter eight. These two experi-

ments establish, without a doubt, the in-phase condition of the ZEFPR index
gratings). The zero phase of the ZEFPR gratings further confirms the distinc-
tion of the ZEFPR effect from electro-optic mediated photorefractive effects
where the phase must be nonzero because it is due to the intrinsically nonlo-

cal space charge field.

The tests described above form the basis for our conclusion that the
ZEFPR effect is caused by a Jahn-Teller relaxation in conjunction with the
photoelastic effect. In what follows we present a simple theory of the

interaction between the local J-T distortions and a macroscopic strain field.

-191-

6.3 Theory of Distortion Coordination

To model the effect of a macroscopic strain on the coordination of J-T
distortions consider a crystal with a growth induced tensile strain along the x-
axis (u,,> 0) and with compressive strains along y and z axes (Uy, = U3 < 0).
All other strain components are zero and by the cubic symmetry where only

elastic constants C444), Cy99, Cy 912 are nonzero, it is readily determined that

_ C1122 _
u33 = ull = -O U1 (6.4)
Cy122 + C1111

where o is Poisson’s ratio. The elastic coefficients were determined2? for a

KLTN with similar composition to that considered above to be cy

WW

7.28x10!9N /m?, cy, = 30.93x10!0N /m2, and ¢qy = 10.37x10!0N/m2 so that o

0.19. The energy of the strained crystal per unit volume is given by

~lowuay. = 2 2
E = aij = youn (uz, + 2u3,} + C1429 (2uy1u33 + u33)

= + or -2¢126] (6.5)

where the convention is that Uyqy = Uy, Un3 = Ug, Cqq74 = qq, C7122 = Cz, ete.

The J-T distorting centers distributed randomly throughout the vol-
ume of the crystal will alter the strain so that u; =u, + Au; where Au, is the
change in macroscopic strain due to the summation of the individual J-T dis-
tortions. The individual distortions are elongations of the oxygen octahedra
in one of three orthogonal directions. We denote y as the fraction of distort-

ing centers oriented along the x axis, and (1 - y) as the remaining fraction dis-

tributed along either the y or the zaxes. We readily determine

-192-

Auily] = Auj[1][(1 +o) y-o] (6.6a)

Aus[y] = Aus[y] = Sui

[l-o “Ul +o)y] (6.6b)

where Au,[1] is the strain change when y=1 (complete ordering). We neglect

Au; for i=4,5,6 and dipole effects, that is, interactions between local distortions
are ignored. Since local correlations of distortions have been observed in
KTN®, this approximation will not be strictly accurate in the presence of an
applied strain field. However, it is not expected that correlation of distortions
will affect the value of y in the absence of such a field. The effect of distortion
correlation will average out to zero on a macroscopic scale ( the wavelength
of light) unless a macroscopic ordering field is present. This follows from the

definition of a centrosymmetric material.

The decrease in elastic energy per unit volume resulting from the re-

duction in strain Au,[y] and Au,ly] is obtained as

AE[y] = -u; Aufl} [(1 +o) y - o] [en - 2c120). (6.7)

The entropy change of the ordering follows readily by counting the
number of configurations of the distortions. There are (ny) distorting centers
oriented along the x axis per unit volume, and n(I-y) oriented along either
the yor zaxes. The number of configurations W is given by the combinatori-
al equation

n(1-y)

we (B)E PO”) -(g)m 68

-193-

Using Stirling’s approximation, the entropy change (AS = k InW) is found to

be

ASly] = -nk {y In[y] + (1-y) in| | (6.9)

Now the free energy G = AE- TAS can be minimized to determine the
temperature dependence of the ordering parameter. We calculate that
y(p) = 2 (6.10)
ebU +2

where the “strain alignment energy” per distorting center U = AE[1] (1+ 6)/n

and B = 1/(kT). Thus

BU
Aujly(B)] = Auj[{1] psa” - ol. (6.11)

The final equation is plotted in figure 6-8 as the amount of strain compensa-

tion Au,[y] along the x axis versus BU, for o =0.2. From it we conclude that

large macroscopic strains lead to ordering of distortions.

A plausible value for the photoelastic coefficient?! is =(0.1. If we
Pp P Pu 3S Py

take values for the strains consistent with the observed index changes we ob-
tain uy = 3x 1074, Au,[1] = 10°74 and BU = 0.2. Since we have not included the

effect of spontaneous cooperative ordering due to elastic or electric interac-
tions between the distorting centers, the estimate above is a lower bound for
the amount of ordering expected for the strain values considered. The experi-

ment showed that the diffraction was reduced by ~40% when the strain was

-194-

ae) 0.8 _
<5
S A al
o)
ol 06 4
an =
co.
2 Ss
es 0.4 4
o<
z 0.2 4
os
NY
0 l | | | l

0 1 2 3 4 5 6
Alignment Parameter - BU

Figure 6-8. The relative strain compensation along the x axis. When the

strain alignment energy U is greater than several kT, almost complete align-
ment occurs. Since o = 0.2 in KLTN, the distortion causes a substantial vol-
ume change of the unit cell, so that strain compensation, and therefore index

change, is seen in the absence of any alignment, i.e., Auz[y(B)]/ Au,[1] = 0.2

when BU = 0.

-195-

compensated in the material. This indicates that roughly 20% or more of the
index grating is coordinated by the strain field, the remainder being due to the
volume change of the unit cell under distortion. The 20% experimental fig-
ure is a lower bound because the strain compensation vise used in the experi-
ment may not have compensated the strain completely. Nevertheless this
measured value is reasonably close to that predicted by equation (6.10), so the
noninteracting theory described above is probably adequate if the experimen-

tally measured lower bound is accurate.
6.4 The Photoelastic Photorefractive Effect

When the distortions are ordered as described above, there is a photoe-
lastic change in the index of refraction proportional to Au,. If interfering light

beams are subsequently incident on the medium, the optical standing wave

will create a spatially periodic concentration of distortable ions, in our case,
Cu*+ ions. Since the relaxation strain Au; was assumed proportional to the

number of distortable ions, there will be a periodic modulation of the total

strain of

S(Auily] ) = 8{Cu**] / [Cu**] Aujly] (6.12)
where 8{Cu2*] is the modulation of the atomic Cu2* concentration. Thus the

modulated strain field is proportional to the number of displaced electrons ( =

§[Cu?+] ), and therefore is proportional to the electronic space charge field.

The change in index of refraction due to the strain modulation can be

written

-196-

Anily] =- FP SCD (pi2 (1-6) - 6 pir + y (146) (pir - pz) (6.13a)

An3ly] = - a 8(ui{1]) (pri(1-o) - (1-30)pi2 - y+0)(Pi1 - Piz) (6.13)

where pj; are the photoelastic constants. Here we have ignored local changes
in y due to 6(Au,[y]), ie., the modulation of the Cu2+ concentration modu-

lates the number of distorting centers per unit volume thus modifying y.
6.5 Summary

A new photorefractive mechanism has been demonstrated in
KTN/KLTN:Cu_ with An ~ 2x10°° which arises from a periodic strain grating.

A Jahn-Teller spatially periodic distortion was shown to be responsible. The J-
T distortions are at least partially ordered by a relaxation of a macroscopic
strain field. The periodicity of the distortion is due to a spatially periodic
Cu?*/Cu!l* ratio created by charge redistribution in a spatially periodic optical
intensity pattern. This mechanism forms an index grating through the pho-
toelastic effect and causes diffraction in paraelectric photorefractive materials
without an applied electric field. This phenomenon is expected to be quite
general, although only noticeable when the conventional photorefractive ef-
fect is forbidden. The most striking characteristic of this effect is that it is in-
phase with the intensity grating. This is in contrast to any phenomenon
which is mediated by the nonlocal space charge field. This characteristic is
useful for implementing linear optical phase detection devices (chapter

seven).

-197-

The coordination of the Jahn-Teller distortions in the macroscopic
strain field was modeled as a function of temperature. As expected, for strain
alignment energies greater than several kT, almost complete ordering occurs.
Experimental evidence indicates that the ordering is only partial. Also, the
amount of ordering estimated from the parameters of the noninteracting dis-
tortion center theory is reasonably close to that observed experimentally. This
indicates that the interactions between the J-T centers can probably be safely

neglected in this first-order treatment. The index grating caused by the spatial

modulation of [Cu2*] ions was calculated.

-198-

References for chapter six

[1] F. S. Chen, “A laser-induced inhomogeneity of refractive indices in KTN,”

J. Appl. Phys. 38, 3418 (1967).

[2] A. Agranat, V. Leyva, K. Sayano, and A. Yariv, “Photorefractive properties
of KTa,_,Nb,O3 in the paraelectric phase,” Proc. of SPIE Vol. 1148, Conference

on Nonlinear Optical Properties of Materials, (1989).

[3] A. Agranat, V. Leyva, and A. Yariv, “Voltage-controlled photorefractive ef-
fect in paraelectric KTa1_.Nb,O3:Cu,V,” Opt. Lett. 14, 1017-1019 (1989).

[4] A. Agranat, R. Hofmeister, and A. Yariv Technical Digest on
Photorefractive Materials, Effects, and Devices , 1991 (Optical Society of
America, Washington DC, Vol 14, 6-9 (1991).

[5] A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new pho-

torefractive material: Ky -yLT1.Ny” Opt. Lett. , 17, 713-715 (1992).

[6] A. A. Kamshilin, “Simultaneous recording of absorption and photorefrac-

tive gratings in photorefractive crystals,” Opt. Comm. 93, 350 -358 (1992).

[7] R.S. Cudney, R. M. Pierce, G. D. Bacher, and J. Feinberg, “Absorption grat-
ings in photorefractive crystals with multiple levels,” J. Opt. Soc. B 8, 1326 -
1332 (1991).

[8] T. Jaaskelainen and S. Toyooka, “Analysis of absorption reflection grat-

ings,” Opt. Comm. 71, 133 - 137 (1989).

[9] Changxi Yang, Dadi Wang, Peixian Ye, Qincai Guan, and Jiyang Wang,

-199-

“Photorefractive diffraction dynamic during writing in paraelectric KTN crys-

tals,” Opt. Letters 17, 106-108 (1992).

[10] H. A. Jahn and E. Teller, “Stability of polyatomic molecules in degener-
ate electron states. I - Orbital degeneracy,” Proc. of the Royal Soc. , A161, 220-
235 (1937).

[11] A. F Wells, Structural Inorganic Chemistry 4th Ed. , Clarendon Press,
Oxford (1975).

[12] G. Chanussot and C. Thiebaud, “Investigation of the [1,0,0] polarization
of iron-doped barium titanate under irradiation, part I. Experimental,”

Ferroelectrics 8, 665-670 (1974).

[13] G. Chanussot, “Static pseudo Jahn-Teller effect at point defects in irradiat-
ed ferroelectric crystals (perovskite structures), part II. Theorectical,”

Ferroelectrics 8, 671-683 (1974).

[14] A. Quedraogo, B. Dehaut, et G. Chanussot, “Les propriétés photofer-
roélectriques de BaTiO, dopé avec cuivre,” J. de Physique Lettres 39, 179-182
(1978).

[15] G. Chanussot, “Physical models for the photoferroelectric phenomena,”

Ferroelectrics 20, 37-50 (1978).

[16] S. M. Kostritskii, “Photoinduced structure distortions of the oxygen-octa-

hedral ferroelectrics,” Ferroelectrics Letters 13, 95-100 (1991).

[17] S. L. Gnatchenko, N. FE Kharchenko, V. A. Bedarev, V. V. Eremenko,

“Photoinduced linear birefringence and cooperative Jahn-Teller effect in

-200-
manganese-germanium garnets,” Fiz. Niz. Tem. 15, 627-632 (1989); also see

Sov. J. Low Temp. Phys. 15, 353 (1989).

[18] V. Leyva, Accuwave corporation, Santa Monica CA, personal communi-

cation (1992).

[19] T. S. Narasimhamurty, Photoelastic and Electro-optic Properties of
Crystals , chapter 8, Plenum Press, New York (1981).
[20] H. Mueller, “Properties of rochelle salt, IV ,“ Phys. Rev. 58, 805-811

(1940).

[21] F. S. Chen et al. J. Appl. Phys. 37, 388 (1966).

[22] V. Leyva Ph.D. Thesis, California Institute of Technology, Pasadena,
California, unpublished (1991).

[23] A. Agranat and Y. Yacoby, “Dielectric photorefractive crystals as the stor-
age medium in holographic memory systems,” J. Opt. Soc. Am. B 5, 1792 -
1799 (1988).

[24] A. Agranat, Y. Yacobi, “The dielectric photorefractive effect - a new pho-
torefractive mechanism,” IEEE Ultras. 33, 797 -799 (1986).

[25] A. Agranat and Y. Yacobi, “Temperature dependence of the dielectric in-

duced photorefractive effect,” Ferroelec. Lett. 4,19 -25 (1985).

[26] N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L.
Vinetskii, “Holographic storage in electrooptic crystals. I. steady state,”

Ferroelectrics 22, 949-960 (1979).

-201-
[27] R. Hofmeister and A. Yariv, “Vibration detection using dynamic photore-

fractive gratings in KTN/KLTN crystals,” Appl. Phys. Lett. , 61, 2395-2397
(1992).

[28] R. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, “Simple methods of
measuring the net photorefractive phase shift and coupling constant,” to ap-

pear Opt. Lett. April 1, 1993.
[29] J. Werner and R. Hofmeister, undergraduate senior thesis project (1992).

[30] J. Toulouse, X. M. Wang, L. A. Knauss, and L. A. Boatner , “Dielectric
nonlinearity and spontaneous polarization of KTa,_,Nb,O3 in the diffuse

transition range,” Phys. Rev. B 43, 8297-8302 (1991).

[31] A. Yariv and P. Yeh, Optical Waves in Crystals , chapter 9, John Wiley &
Sons, New York (1984).

-202-

Chapter Seven

Applications of the ZEFPR Effect

7.1 Introduction

As discussed in the previous chapter, gratings written with the Zero

External Field PhotoRefractive effect (ZEFPR) exhibit a unique phase relation-
ship (@=0) between the phase and intensity gratings. This feature will be

shown to enable implementation of numerous interesting devices, and, in
particular, an all-optical self-aligning holographic vibration sensor. Under the
zero-phase condition, no intensity coupling of two beams occurs. However,
the transmitted intensity of two interfering beams varies linearly with dis-

placement of the medium along the grating vector or with phase shift of one
of the interfering beams. The phase shift must be 0 << m/2, with a frequency

much higher than the inverse grating rewrite time, and the average position
of the phase must be zero or very slowly varying. Figures 7-1 and 7-2 illus-
trate a vibration sensor utilizing this principle. Optical fiber is used to deliver
the interfering light beams to the sensor, and to collect the transmitted light.
One beam is phase modulated by reflection off a vibrating membrane. In this
way, vibrations can be sensed remotely without any electrical signals in the
vicinity of the sensor. In addition, since the detecting element is a holograph-

ic grating which is continually rewritten, the device is self-aligning and resis-

-203-

tant to mechanical shocks. The device is useful in applications where electric
fields cannot be permitted into the sensing environment, or where electronic

wiretapping is a problem.

Other devices are also discussed but are not demonstrated here.
Specifically the application of the ZEFPR zero-phase response to performing
image subtraction and to implementing a high-speed interferometric data

link is described.
7.2 Self-Aligning Vibration Sensor/ Microphone

7.2.1 Introduction

All-optical sensors have received considerable attention recently! for
use in applications where electrical signals cannot be used or are impractical.
These include aqueous, explosive, corrosive, and electromagnetically sensi-
tive environments. Traditionally, intensity modulating sensors have been
preferred over interferometric methods®® because of their relative ease of
alignment. Unfortunately, even these techniques require precise fiber posi-
tioning in order to gain sensitivity>. Thus they are not robust. There is a

need for an all-optical sensor which combines sensitivity with robustness.
Here an interferometric vibration detector and microphone is described
which overcomes the alignment and stability problems of conventional opti-

cal devices.

As discussed before, in the KTN and KLTN crystals which displayed a

strong ZEFPR effect, the photorefractive dopant was copper, which is stable as

-204-

Differential
Amplifier

Output

yi Det on J
dd : Crystal

Input

Mirror

iy
my
rats

Membrane

Prism

488nm

Lens

Fiber

Figure 7-1. The setup of the ZEFPR microphone. The light is delivered with
a single-mode fiber and is split with a prism. Vibrations of the membrane

cause two-beam coupling differentially detected Det at the exit facet.

-205-

Collimated Fiber
Detectors

Zefpr Photo- —_, ~
Refractive Crystal 5

*. Index
Fiber — Grating
Collimators
To Differential
Detectors

Fiber
Coupler

ne,

— Input Fiber

Figure 7-2. A ZEFPR material all-optical microphone. The light is delivered
with a single-mode fiber and is collected with multimode fibers at the exit
facet of the crystal. In this way, all electrical devices and signals can be re-

moved to a remote location; the sensing device is then all-optical.

-206-

either Cu!+ or Cu2+. The Cu2* ion is known to cause large Jahn-Teller (J-T)
distortions. The Cu!* ion, by contrast, has no tendency to distort.
Illumination of the crystal by the periodic intensity pattern of the optical field
leads to a mimicking spatially periodic Cu2*/Cul!* ratio due to excitation of

electrons from Cu!* and trapping by Cu?*. This, as explained above, gives rise

to a spatially periodic distortion. The result of this is a spatial modulation of
the strain field in phase with the intensity which leads to a corresponding
modulation of the index of refraction ( index grating) via the photoelastic ef-

fect.

Since the index grating is modulated by the local Cu2* concentration,

we expect it to be in phase with the intensity (@ = 0) so that no two-beam cou-

pling (power exchange between the two writing beams) will occur. When ei-

ther the phase of the index grating or of the interfering beams is modulated
with 6 « 2/2, however, a power exchange, proportional to 0, takes place be-
tween the two beams. If the vibration frequency is much higher than the in-
verse of the grating rewrite time, and if the time averaged value of © is zero,

the vibration will not erase the grating.
7.2.2 Experiment

The setup of the fiber microphone reported on here is illustrated in fig-

ure 7-1. A single mode fiber was used to deliver the 488nm light to the de-
vice. The emerging beam was collimated and split by a 90° prism. This

method of beamsplitter was chosen over a fiber coupler, because couplers at

-207-

this wavelength are expensive and problematic. One beam was reflected off a
mirror, and the other off a reflective membrane. The equal intensity beams
interfered in the volume of the crystal, and the outputs of the two beams
were detected differentially. Differential detection improved the signal to
noise ratio substantially. Differential detection normalized to the sum of the
intensity (see (7.4)) would, in addition, eliminate noise due to intensity fluc-
tuations of the source; this was not done in the following experiment. Most
of the light was blocked in order to prevent saturation of the detectors. The
beams were detected locally for convenience; in an operational device the
outputs would be coupled into fibers for remote sensing. Since the beams are
not deflected, but only intensity modulated, we conclude that it will be rela-
tively easy to couple the output light into multimode fibers so that the experi-

mental shortcut described above will not alter the performance.

The membrane was a uniformly stretched circular film of reflective
mylar. Since both sides were open to the air, it operated as a pressure gradient
sensor, with resultant poor low-frequency response. The first resonance of the
membrane was at 900Hz. The usable range of operation was from 1.2 - 25kHz

(the limit of the spectrum analyzer).

The detector outputs were processed with an audio spectrum analyzer.
The performance of the microphone was evaluated for both single ended and
differential detection. Figure 7-3a shows the output of a single detector with a

58dB sound pressure level (SPL relative to 0.0002ubar) signal at 4.9kHz inci-

dent on the microphone. The background noise floor is also plotted. From the

figure, we measure a noise equivalent power (NEP) in a 10Hz bandwidth, of

-208-

pid

"omy Ea

Figure 7-3a. The output of one detector with a 58dB signal at 4.9kHz incident
on the microphone. The lower trace is the noise floor without the signal. The
NEP is 38dB. Figure 7-3b. The differential output of both detectors under the
same experimental conditions. Compensation of intensity fluctuations lowers

the noise floor 26 dB yielding NEP = 12dB.

-209-

38dB. Figure 7-3b shows the same incident signal detected differentially. The
compensation of intensity fluctuations (equation (7.3)) with differential detec-
tion lowers the noise floor by more than 30dB, leading to an NEP of 12dB SPL.
Between the range of 1.6 - 15.5kHz we determined the NEP < 15dB (Fig. 7-4).
The useful upper limit of the microphone (determined by listening to the
output) was approximately 95dB SPL, giving a dynamic range of about 80dB.
Above 95dB, the third harmonics of the input signals became noticeable to

the ear.
7.2.3 Coupled Wave Analysis

When two beams I, (0) and I,(0) are symmetrically incident (Fig. 5-1) on

a ZEFPR ( = 0) material it can be shown (chapter 8) that the resultant index

grating is given by

n(r,t) = no +a |A1(0)| |A2(0)| e i42e “Wr -k2)2 + 2c, (7.1)

where A =(g/1,)(1,(0) - 1,(0)), g = mn1/A, and n, is the peak-to-peak index mod-
ulation for 1,(0) =1,(0). Ig is the total intensity and A;(0) are the complex am-

plitudes of the input beams. If this index grating is considered fixed in the

crystal and one beam is phase shifted by 6 = 0, sin wt, where 6,« n/2 and 1/@ «

t then the coupled mode equations can be solved to yield

write
L@=h 1 - © sin’[sz] + Ip © sin’[sz] -& VIjh sin[2sz] sin
S S

- KA VIiI2 sin’[sz] cos (7.2a)

-210-

= 30
QO.
2 95 b Noise Equivalent Power _
~~ Spectrum
& 20 |
m15 6
a 10-F
= 5+
aa
a OF [ Diaphragm 7
oA Va Resonance
5 l l l ! l l

0 2 4 6 8 10 12 14 ~~ 16
Frequency - kHz

Figure 7-4. The measured noise equivalent power (NEP) of the microphone
across the audio spectrum. In the range of frequencies 1.6kHz < vs 16kHz, the

sensitivity of the device was NEPs 15dB SPL. Two mechanical resonances of

the membrane are marked.

-211-

L(z) =b f Bo sin”[s2]] +h, Ke sin2[sz] +£VT\hp sin[2sz] sin
S S

+ KA Ilo sin?[sz] cos (7.2b)

where s? = Kk? + A*/4, and k =g A,A,’/I,. We defined I,(0) =I, and 1,(0) =].
When the beam intensities are equal I, =I, = I the solution for arbitrary phase

o is given by (chapter eight)

I\(z)/l = 1 -tanh[Tz/2}cos0 - ae re sind (73a)
L(z)/I = 1 + tanh{I'z/2]cos® + SSDP sin® (7.3b)

where T= 2g sing. When ¢= 0 as in (7.2) and the equal intensities are detect-

ed differentially, the signal is
In(z) - I(z) = S = 2 I sin[g z] sin (7.4)

which is linear in 6 (for small 6).

The same calculation can be performed for a conventional photorefrac-

tive material (¢=7/2 ). The result for arbitrary incident intensities is
I(z) = 1, [cos?[y-6] + C*sin’[y-8] - © sin[2(y-8)] cos0] (7.5a)
I,(z) = I; [C?cos?[y-8] + sin’[y-5] + ¢ sin[2(y-8)]cos0] (7.5b)

where C2= 1,(0)/1,(0), y= Tan HC eX4) and 5 = Tan“![C]. When the intensities

are equal, I, =1, = I, the differentially detected signal may be obtained from ei-

-212-

ther (7.3) or (7.5) as

In(z) - I1(z) = 2 I tanh[g z] cosé. (7.6)

The details of these calculations are performed in chapter 9 for arbitrary

phase . Only the two extremes o® = 0 and ¢ = n/2 are considered here be-
cause they are sufficient to illustrate the point made. In figure 7-5a the output
intensities for the = 0 case (equation 7.2) are plotted for @, = 0.1rad, for z=
0.5, and A,(0)/A,(0) = 1.0 and 2.0. Figure 7-5b plots the output signal of equa-
tion (7.5) for the same experimental conditions as in figure 7-5a. The signal
amplitude is much weaker than in the o = 0 case and is at twice the modulat-

ing frequency. From these figures it is clear that only a material with zero-
phase gratings is sensitive to small phase fluctuations. The sensitivity of a
conventional photorefractive material is quite low and peaks near gL =
1.0/cm; beyond that value it decreases with increasing coupling constant. In
addition, the zero-phase grating yields a linear response whereas the conven-
tional photorefractive gives the second harmonic. Calculations performed in
chapter eight show that for a material with 0 < 6 < n/2 the output is a combi-
nation of the linear terms and the second harmonic. However the phase
must approach zero very closely before the second harmonic term becomes
negligible. Thus only an identically zero-phase material is suitable as an all-
optical vibration sensor (though a conventional material can be made to ap-

proximate the zero-phase condition by applying a field, introducing birefrin-

gent plates’, or frequency shifting one of the beams).

-213-

Signal - 1 (L)/ L (0)

ot - Modulation

Figure 7-5a. Intensity modulation of beams interfering in a ZEFPR material
in the geometry of figure 5-1 for a sinusoidal input phase modulation of
0.1rad. The output approximates a sine wave at the same frequency even for

large amplitudes of the signal modulation.

-214-

Kee KG WG CG CO
Ke Sa COG Sa rr"
S 1.45 OOS COS COE CORE
—_
44
—~ L— =
=) —o— gL = 1.0
meal
“135 + —o— gL =0.5 -
can
ob
va) 13 47 7

0 3.25 6.5 9,75 13
wt - Modulation

Figure 7-5b. Intensity modulation of beams interfering in a diffusion limited
conventional photorefractive (® = n/2) under the same conditions as in Fig. 7-
5a. The output is a weak modulation (different y-axis scale) at twice the input
frequency. Thus a material with o = 7/2 is insensitive to vibrations and gives

a second harmonic response of intensity fluctuations to input phase fluctua-

tions.

-215-

7.2.4 Discussion of Results

A sensitive all-optical microphone/vibration detector has been demon-
strated utilizing the ZEFPR effect. The response was measured from 1.2kHz -
25kHz, limited by the membrane at the low end and by the spectrum analyser
at the high end. Since the microphone relies on physical motion of the mem-
brane, the ZEFPR microphone is expected to have the same fundamental lim-
itations to flatness of response as a conventional microphone. Nevertheless
it may not be advisable to simply place the membrane of a good capacitive mi-
crophone into the ZEFPR setup of figure 7-1, for several reasons. First, the
membrane would be metallic which may be undesirable in a corrosive envi-
ronment. Also a nonmetallic membrane would be lighter - a substantial ad-
vantage. Second, conventional membranes derive flatness of response by fol-

lowing the air motion of the acoustic wave. This requires large excursions at

low frequencies®!°. The ZEFPR device must stay within the linear regime

where the motion d of the membrane obeys d<membrane is tuned with regard to the average response of the membrane: the
response at any one point is generally unknown!!. The ZEFPR microphone

does not require a large membrane; it senses phase modulation only where
the membrane is illuminated. It is unclear whether the solution of this engi-
neering problem would be more or less difficult than that of a conventional

microphone.

The dynamic range of the device tested was only 80dB. This is substan-
tially less than that of conventional microphones. However, no attempt was

made to stabilize the laser intensity or the coupling into the fiber; thus the in-

~216-

tensity fluctuations were quite large, also, the signal was not normalized to
eliminate the common mode noise from fluctuations in intensity (7.4). It is
expected that the fundamental sensitivity of the optical microphone will be

much higher.

Conventional microphones are limited by shot noise in the sensing
unit and by corruption of the microvolt signals before preamplification. Thus
they are quite sensitive to external electrical signals which cause noise and
feedback. A ZEFPR microphone, by contrast, is immune to electrical noise be-
cause the detected signal (proportional to intensity) will always be on the
order of volts. Also the electrical part of the device is remote and can be
placed in an electrically isolated environment. The fundamental limit of the
ZEFPR device is given by Brownian motion of the membrane. The noise
equivalent power (NEP) of Brownian motion will scale inversely with mem-

brane area, but as stated above we are free to use a large membrane and only
sample a small part of it. In any case, the ZEFPR microphone with a 1em?

membrane should have an NEP of about -15dB SPL which is much better

than any electronic microphone. This would yield a dynamic range of about

110dB, which would be acceptable for most (music) recording purposes!!.

Finally, it is noted that the upper limit of detection can be raised from 95dB
SPL to any value simply by partially isolating the microphone from the acous-

tic source. The dynamic range would be unaffected.
7.3 Other Applications of the ZEFPR Effect

7.3.1 Interferometric Data Link

-217-

Materials displaying the ZEFPR effect perform a linear phase to intensi-
ty transduction when one beam scattering off a dynamically written index
grating is suddenly phase shifted. In the previous section this phase shift was
achieved by reflecting one beam off a vibrating membrane, but any method of
phase shift will yield the same result. In this section the phase shift is accom-

plished with an electro-optic modulator.

A conventional high-speed fiber optic data link is illustrated in figure
7-6a. A laser is coupled into a fiber optic Mach-Zender interferometer. Data is
impressed onto the signal beam with an electro-optic phase modulator. A sec-
ond fiber coupler is used to complete the interferometer. The outputs are de-
tected differentially and normalized. Inevitably, the fiber lengths will fluctu-
ate due to thermal changes or stretch, so the outputs of the interferometer
drift over time if the lengths are not actively stabilized. The length changes

can be tracked and corrected with a feedback device. Usually a low-frequency
(10° - 106 Hz) reference oscillator is employed to dither the reference beam

(the fiber is stretched with a piezoelectric device). The transmitted optical sig-
nal is multiplexed with the reference oscillator, and the time integrated prod-
uct is used to generate a feedback signal which is amplified and directed to a
fiber stretcher in the signal arm. This procedure is complicated and undesir-
able: Loss of phase lock is a common problem. Often acquisition of lock re-
quires different feedback parameters than maintaining lock. The output in-
tensities must be rectified in the time averaging process, and the frequency of
the dither places constraints on the lower frequency limit of the signal.
Usually the signal must be placed on a high-frequency carrier, otherwise a

long string of “0” or “1” bits will lead to loss of phase lock.

-218-

Signal in Dither

Fiber stretchers Detectors

Control

Signal in Signal out

° EH:

ZEFPR crystal / Detectors

Signal out

Figure 7-6a. A conventional high-speed fiber optic data link is illustrated in
figure 7-6a. A laser is coupled into a fiber optic Mach-Zender interferometer.
Data is impressed onto the signal beam with an electro-optic phase modula-
tor. A second fiber coupler is used to complete the interferometer. Path
length changes are corrected with a feedback device and fiber stretchers.
Figure 7-6b. The second fiber coupler is replaced with a ZEFPR crystal. The
dynamic zero-phase response of the crystal automatically tracks the path

length changes eliminating the need for a feedback control system.

-219-

In figure 7-6b a schematic of a self-aligning fiber-optic interferometric
data link is shown. The second fiber coupler of figure 7-6a is replaced by a
ZEFPR crystal. The intensities transmitted through the crystal obey equations
(7.2) or (7.3). These intensities are detected differentially and normalized to
the intensity as before. Any variations in the lengths of the fibers will be dy-
namically compensated by the ZEFPR crystal as long as the rate of change of
fiber length in units of wavelength is small compared to the grating rewrite
time. For a 1km data link with the fibers cabled together, the averaged tem-

perature difference between the two fibers is normally controllable to better
than 0.1°C/hr. This rate will give a fringe velocity of approximately one

wavelength in 10 seconds. For moderate intensities (10-100mW), a ZEFPR
crystal can track this motion. In addition, the ZEFPR material has no upper
limit to the path length mismatch it can accommodate; it is only limited by
the laser coherence length. A fiber stretcher, by comparison is usually limited
to 0.5% stretch. For certain applications, the ZEFPR interferometer would
seem to be a drastically simpler solution than the conventional interferome-

ter
7.3.2 Phase Image Subtraction and Phase to Intensity Conversion

This section describes how the characteristics of the ZEFPR gratings can
be used to perform image subtraction and transform phase transparency im-
ages into intensity modulated images. As discussed above, two beams inter-
fering in a ZEFPR material show linear intensity coupling with relative phase
change between the beams. This concept can be extended to include many

beams within the same crystal (one for each pixel of an image). If the individ-

-220-

ual pairs of beams are not overlapping within the volume of the crystal, they
will not affect each other and the phase to intensity relationship will hold for

each pair of beams individually. Figure 7-7 illustrates the setup.

The device in figure 7-7 has two input phase images, T,(x,y) (28) and
T (x,y) (30). These images can be provided, for example, by a liquid crystal tele-
vision with the polarizing sheets removed. Two beam splitters (38) and (40)
provide outputs. The operation is as follows: Two input phase transparencies
are imaged onto a ZEFPR crystal (12) in exactly counterpropagating geometry
with lenses (32) and (34). The focal depth of the lens configuration must be
longer than the interaction region within the crystal. Because of the counter-
propagating geometry of the setup, each imaged pixel from one transparency
interacts only with one imaged pixel of the other transparency. Thus each
pair of pixels can be considered independently. When one pixel’s phase is
changed, the output intensity of one pixel increases at the expense of the
other. Thus the instantaneous output intensities are given qualitatively by
a(T,(~%y)-Tyy)) +C, and b(T2(x,y)-T;(x,y)) +Cz (18), where a, b, Cy, and Cy are
determined by the specifics of the crystal and relative beam intensities. The
device acts as a “novelty filter” because if the images remain fixed the outputs

evolve back to zero as the hologram is rewritten.

The device can be used to calculate the derivative of an image. For ex-
ample if T,(x,y) is the same image as T,(x,y) but spatially shifted, the subtrac-
tion performs a gradient operation. If T,(x,y) is the same image as T,(x,y) but
delayed in time, the time derivative is calculated. One application for novelty

filters which is often bandied about but never implemented is quality control

-221-

Figure 7-7. A schematic of a ZEFPR crystal based phase image subtracter and
novelty filter. Images T,(x,y) (28) and T,(x,y) (30) are imaged on the ZEFPR
crystal with lenses (32) and (34). in exactly counterpropagating geometry, so
that each pixel from T, interferes with only one pixel of T>. The beam cou-
pling between each pair of pixels is governed by the relative phase between
them. Outputs (18) are picked off with beamsplitters (38) and (40). The device
act as a novelty filter because the gratings in the material evolve back to the

zero-phase condition if the images are stationary.

-222-

for complicated manufactured parts. For example, a photographic image of a
semiconductor die ( T,(x,y)) can be compared to a reference image for the
same part (T»(x,y) ). Any discrepancies between the two images will stand out
as a bright spot in the output. The subtraction operation can also be used to

calculate the difference of the images in the Fourier plane.
7.4 Summary

The ZEFPR effect is unique in that the phase between the index grating
and the intensity grating is identically zero. This quality cannot easily be
achieved with an electro-optic grating. The applications of this effect arise
naturally from the linear phase to intensity transduction of beams interfering

in the ZEFPR material.

The use of a crystal of paraelectric KLTN as a microphone and vibration
sensor was established. The device is expected to have practical applications.
The use of a ZEFPR crystal in a self-aligning data link and in an image sub-
tracting “novelty filter’ was discussed. The latter two applications have not

been demonstrated experimentally.

-223-

References for chapter seven

[1] W. J. Bock R. Wisniewski, T. R. Wolinski, “Fiberoptic strain-gauge
manometer up to 100MPa,” IEEE-Instru 41, 72-76 (1992).

[2] R. I. Macdonald and R Nychka “Differential measurement technique for
optical fiber sensors,” Electr. Lett. 27, 2194-2196 (1991).

[3] PY. Chien and C. L. Pan, “An active fiberoptic interferometric sensor
based on a low finesse fiberoptic Fabry-Perot,” J. Mod. Opt. 38, 1891-1900
(1991).

[4] W. A. Gambling, “Optical fibers for sensors,” Sens. Actu. A. 25, 191 -196
(1991).

[5] D. Garthe, “A fiberoptic microphone,” Sens. Actu. A, 26, 341-345 (1991).
[6] D. Garthe, “A purely optical microphone,” Acustica 73, 72 -89 (1991).

[7] I. Rossomakhin and S. Stepanov, “Linear adaptive interferometers via
diffusion recording in cubic photorefractive crystals,” Opt. Comm. 86, 199-204
(1991).

[8] J. W.S. Rayleigh, The Theory of Sound , Vol I chapter 9, Dover, New York
(1945).

[9] B. A. Auld, Acoustic Fields and Waves in Solids , Vol 1 chapter 6, Wiley

Interscience, New York, New York (1973).

[10] I. Malecki, Physical foundations of technical acoustics , Pergamon Press,

Oxford (1969).

-224-

[11] J. Boyk, California Institute of Technology, Pasadena, California, private

communication (1992).

-225-

Chapter Eight

Determination of Photorefractive Phase

and Coupling Constant

8.1 Introduction

It is clear from the analysis of the photorefractive effect in chapter two
that the most important parameters in describing the photorefractive cou-
pling are the photorefractive phase @ and and the coupling constant g. In
fact, if one ignores the ill-quantifiable effect of fanout, these two material pa-
rameters, in addition to the optical absorption coefficient, are the only ones
needed to calculate the interaction of two or more coherent beams in a pho-
torefractive material in the steady state. This conclusion follows directly from
the coupled mode equations. In light of this it is surprising that the exact de-
termination of these parameters has received little attention to date. One
exact treatment, the 1975 paper by Vahey! follows a similar analysis to the one
described in this chapter. Unfortunately, Vahey’s paper is of limited useful-
ness since he assumed an intensity dependent term describing the index

change in the coupled equations. Vahey’s paper predated the intensity inde-

pendent Kukhtarev model? of the photorefractive effect which is now the ac-

cepted formulation. In another paper, Kondilenko et al.3 present a theory to

evaluate the phases of fixed holographic gratings. Since that time, all pub-

-226-
lished reports have been content to describe approximate methods of phase
and coupling constant determination*®. The general trend has been to as-
sume beam coupling in which one beam (the “signal beam”) is much weaker
than the other. In some cases even this approximation is treated incorrectly”®
(see section 8.8). Here an exact solution of the coupled equations is reported
describing the evolution of two arbitrary beams incident at the Bragg angle on
a dynamically written photorefractive grating. The incident beams need not
possess the same phase nor the same intensity as the beams which wrote the
grating. In this analysis, all the beams are of the same frequency, but the treat-
ment could easily be extended to beams not of the same frequency (see chap-

ter nine) if a reason for such an extension arose.

Two coherent beams are symmetrically incident on a photorefractive
crystal. They induce a spatially periodic space charge field E,, which is phase

shifted with respect to the intensity interference pattern by the photorefrac-
tive phase 6. A dynamically written refractive index grating is the result. The

writing beams are then replaced with two beams of arbitrary intensity and
phase and of the same frequency, incident along the same directions as the
initial beams, i.e., at the Bragg angles. We calculate the instantaneous beam
coupling experienced by these new beams off the dynamically written grating
which for the short initial period, is considered fixed. The time dependent
formation of dynamic gratings written by the new beams is ignored. This
condition is only valid for a time period on the order of a second, depending
on the intensity, but this is much longer than is required to obtain the neces-

sary data. If the new beams continued to illuminate the crystal for a longer

-227-

time period, they would first generate secondary gratings caused by diffraction
off the original grating?!9, and eventually they would completely erase the

original grating, replacing it with a new grating.
8.2 Formulation of the Problem

The starting point is calculation of the two beam coupling of two inci-
dent copropagating beams with amplitudes A(z) and B(z) in a photorefractive
material (Figure 8-1) as in chapter two. The well known coupled beam equa-

tions follow from (2.35a,b)

A(z) cosB =i g ei? Bi)” A(z) - & A(z) (8.1a)
I(z) 2
B(z) cosB =i gel? A(z)" B(z) - & B(z) (8.1b)
I(z) 2

where A is the wavelength of the interfering beams, o the optical absorption
coefficient, and 8 the half-angle of beam intersection inside the material. The
definitions g = mn,/A and I(z) as sum of the intensities I(z) = | A(z)|* +
|B(z)|* are employed as usual. Here 6 is the photorefractive phase between
the optical intensity grating and the induced index grating; K = 2k sin is the

nominal grating wavevector with k = 2nn,/A. In obtaining (8.1) the coupled

equations of chapter two are used with
n(z) =No + 5 (An(z)eisei Kz + c.c.} (8.2)

where An is formed dynamically by the writing beams:

-228-

An(z) = mA(z) B’@) MC) (8.3)
where I(z) is the total intensity, and n, is the peak-to-peak amplitude of the

index grating when A(z) = B(z).

I,(z) = eb (hi +b) (8.4a)
I, + lh et!

Ih(z) = eo hh (t+ b)

(8.4b)
I; elz 4 In

where A(z) = (1,2) !/expliGy] and B(z) = [1,(2)]'/*explis], andl = 2g sino is

the power coupling coefficient. The variable z is, for convenience, taken to be

the unit of length in the propagation direction. L is the effective thickness of
the crystal: L = d/cosB. We have defined I, =1,(0) and I, = 1,(0). The phases of

the two beams are given by (see chapter two)

Ci(z) = 5 coto In{I, + Ip e+! 4] (8.5a)
C2(z) = -g cosd z - coto In[I; + Ip e*! 7). (8.5b)

The index grating in the material follows from equations (8.3), (8.4), (8.5), and
the definition of A(z) and B(z) in terms of phase and intensity. It is given by

the surprisingly simple form

An(z) = m VIy Ib (Ly eT #24 Ip eth? OO =) (8.6)

Thus we have used the coupled mode equations to solve self consis-
tently for the amplitudes and phases of two dynamically coupled beams in a

photorefractive material. Dynamically coupled beams are those which yield

-229-

an index grating given by (8.3). The solution for An ( or A(z) and B(z) ) given

above can be reinserted into equations (2.35) ( or equation (8.1) ) to verify that

the solution is self-consistent, as advertised.

The next step on the way to calculating the coupling constant and
phase is to solve for the beam coupling of two arbitrary intensity and phase
beams incident at the Bragg angle off the grating given by (8.6). When the
new beams are the same intensities as the old ones but with different phases
it is equivalent to simply phase shifting one of the original beams. When, in-
stead, the phases of the new beams are unchanged with respect to the original
beams but the intensity of one new beam is reduced to zero, we are calculating
the diffraction off the grating. The general case when both the phases and/ or
the intensities are different in the new beams yields the response of the holo-
graphically coupled beams to fluctuations in the interferometer system. This
last point would be of importance in calculating how phase and intensity fluc-
tuations at the input of a photorefractive interferometer are magnified in the
outputs. The solutions for all these problems are developed in this chapter,
however we are only concerned with the application of the solutions to the

determination of the photorefractive phase and coupling constant.

8.3 Solution of Beam Coupling

We calculate the beam coupling of a new set of Bragg matched beams
T(0) =[P,]'/* expliy,] and V(0) = [P,]/2 expliy,] off the index grating of equa-
tion (8.6). In analogy to equations (8.1a,b) and (2.35a,b), the coupled mode

equations are written

-230-
T(z) cosB = ig VI; Ih eti@+® x

(Ip eT#2 4 1p e the2}t ool yyy o T(z) (8.7a)

V(z) cosB = ig VI; Ip ei @+) x

(Ip eT#2 4 Tp etl2} 10-1 riz). o Via) (8.7b)

where I, and I, are the intensities of the writing beams, not to be confused
with the new intensities P, and P,. The phase 6 = C,(0)- €,(0) - w,(0) + wp(0)
is the phase difference between the intensity pattern of the beams which
wrote the grating and the intensity pattern formed by T(z) and V(z). If 6 =0
and P; =I; for i=1,2, it is easy to see how (8.7) reduces to (8.1). Since the pho-
torefractive effect is intensity independent, it is also true that if @ =O and P; =c

I; for i = 1,2 where c is an arbitrary constant, the solutions are essentially un-
changed except that all intensities are scaled by the constant factor c.
Equations (8.7a,b) ignore the new dynamic grating which is written by beams

T(z) and V(z) in the case 6 # 0. In addition, as intimated in the introduction

to this chapter, if the new beams are allowed to continue illuminating the
crystal, they will eventually overwrite the existing grating with a new one

given by

An(z) = mT(z) V'@)/P(@) (8.8)
where T(z) and V(z) are different beams than the original A(z) and B(z) al-
though both sets of beams propagate along the same directions. It is clear now
that this new grating will have the identical form of equation (8.6) with the

change I; -> P;. The new grating would also be shifted in phase by an

-231-

amount 8. In the experiments which follow, however, the data will be collect-

ed immediately after introducing the new beams T and V. Therefore the
beams will have no chance to alter the existing grating and we are justified in

applying equations (8.7).

The solution of (8.7) is outlined in detail in chapter 9. After a whole

bunch of work the solutions are reduced to the following form

T(z) e2 = Cy (bth eT"? 4+ Co (bette 4+ 1)? (8.9a)

Viz) eM? = C3 (+ hel"? + Cylbet+ 1)" 1? (8.90)
where ) = coto/2, and C; are constants which are determined by the coupled
equations and the boundary conditions.

Equations (8.9) are the solutions for beam coupling off a dynamically
written grating; it only remains to determine the coefficients Cj for particular

special cases. In this way, methods for determining the coupling constant and

phase will be established.

8.3.1 Harmonics Method

We solve the coefficients G for the special case where one or two of the
reading beams are phase shifted relative to the recording beams but their in-
tensities are unchanged. This can be accomplished in practice by merely in-

serting a phase shift in the path of one of the recording beams. From equa-

tions (8.5a,b) and the definition of @ in equations (8.7a,b) we use the boundary

conditions T(0) = 1,!/7, +1,!1 and v(0) = I,'/2, +1,1. It should be

-232-

stressed again that the term [I, +I,] raised to +in is purely a phasor which is
required to ensure that the index grating have the proper form. With simple
differentiation and insertion into the coupled equations (8.1) we determine

the coefficients as

C, = Vi/ith) b (1 - ei®) (8.10a)
C= Vh/di+h) (hh + he’) (8.10b)
C3 = VIo/Mi+l2) (In + ei I) (8.10c)
Cy = VIo/(it]y) hy (1 - e®). (8.10d)

The combination of equations (8.9) and ( 8.10) completely determines the out-

put amplitudes A(L) and B(L). The intensities are given by

ith In et!2 4+],

[ 215” e+! (1 - cos6} + (Ii? + 12+ 2hbcos0) + 2 b etl 2/2 x

{ (li - In) { (1-cos®) cos[ gz coso]) - (I; + Ip) sin@ sin[g cosd z] } ] (8.11a)

. I ]
li+ly I, e14+1,

[ 2° et (1 - cos6) + (1, + 174+ 21,locos0) + 2],eT22 x

{ (In - 1h) ((1-cos®) cos[gz coso]) + (I; + In) sin@ sin[g cos z}} ]. (8.11b)

When I, =I, = I the transmitted intensities reduce to a simple form given by

-233-

Piz) = e-% {1 - tanh[T'z/2]coso - CE £080 2) 5 12
(z)/l= e | tanh[I-z/2]cos@ coshiT'z/2] sin8 (8.12a)

Po(z)/l = e -%

1 + tanh{T'z/2}coso + SINE C080 Z) sind} (8.12b)
cosh[Tz/2]

The approximate solutions for the case I,<

tions (8.11) with

P1(z) = |e a2 I, (8.13a)

Pata) =| Ine! +21, e!%? sinO sin[g cosd z]
WAZ ee

| (8.13b)

+ 21)(1 -cos6} {1 - e' #2 cos[gz coso]}
We consider the effects of phase shifting one of the interfering beams
by an amount @ using a piezoelectrically driven mirror. The experimental

configuration is as in figure 8-1. When the mirror is driven sinusoidally we

have O(t) = 8 sin(a@t). If @>>1/t1, where tis the writing time of the grating,
and 0) << n/2,and I, =I, then inspection of (8.12) shows that the optical pow-

ers of the transmitted beams at DC, w, and 2 frequencies are related by

Ppc} — {coth[g sind z] + 1) (8.14a)
P20 05

Le _ 4 sin[g cosd z] (8.14)
P2| 98 sinh[g sind z]

where we have used the Taylor expansions for the sin and cos functions. The
powers in (8.14) are taken to be peak-to-peak values, so that Ppc is actually
twice the power of the beam coupling. This convention is used to comply

with that of many spectrum analyzers. The “+” in equation (8.14a) refers to

-234-

detector

488nm

Figure 8-1. The setup for determining the photorefractive phase via beam
coupling and diffraction measurements. Two beams incident on a photore-
fractive material write an index grating spatially shifted by a phase ¢ from the

intensity pattern. A voltage applied to the piezoelectric mirror (PM) phase
modulates one of the incident beams, shifting the position of the intensity
pattern. The response of the beam coupling to this phase shift can be used to

determine the photorefractive phase.

-235-

gO

oe -10 F

5-20 +

Ee

qi

ao} -30 [~

ar

«© ‘

S -40 +
e)
C)

-50 | | | | | |

-2 -15 -1 -05 0 05 1 #15 2
Photorefractive Phase 6 - rad

Figure 8-2. The intensity I,(L) of figure 8-1 at frequencies @ and 2@ relative to
the DC power ( = 0 dB ) when one of two interfering beams is phase modulat-

ed at @, as a function of the photorefractive phase 6. The results are plotted for
various coupling constants: gL = 3,5, 10. For positive values of 6, I,(L) is
deamplified so the wand 2 terms are relatively large compared to the DC

term; for @ <0 the reverse is true.

-236-

the beam which is amplified, and the “-” to the beam which is attenuated.

Figure 8-2 illustrates the relative powers of the DC, a, and 2m. Inspection of
(8.14) reveals that they are equations with independent variables g cosd and g

sind. As mentioned earlier, g and never appear individually as parameters
in experimental measurables. Two equations in two variables have been ob-

tained. Thus g and $ can be determined from (8.14).

To recap, we write a grating in a photorefractive material. One of the
writing beams is phase modulated with a small amplitude, high frequency, si-
nusoidal modulation. The intensity of either of the transmitted beams is de-

tected and spectrally separated into a DC component and components oscillat-
ing at @and 2m. The relative powers at these frequencies are used in (8.14) to
obtain the coupling constant and photorefractive phase. This procedure has
been attempted before for the case I, << 1, but has not been treated for the
general case. The results presented above reduce to simpler expressions (after
some work) when the phase 6 = 0, x/2, and are given in chapter seven equa-

tions (7.2) and (7.5) or in Ref. 11.

8.3.2 Beam Coupling/ Diffraction Method

Another method of determining g and @ follows from an examination

of the photorefractive beam coupling and the diffraction off a dynamic grat-

ing. The formula for beam coupling is well known (8.4). But this only yields
the value of g sing. In order to determine g cosd we need to consider diffrac-

tion of a single incident beam off the grating. Here, we take B(0) = 0 and A(0)

-237-

as before. In this case we follow the same steps as leading to (8.10) to obtain,

C; = V1,/i4+I2) Io (8.15a)
C3 = v1o/Uith) I; = -Ca. (8.15c,d)

Equations (8.9) and (8.15) are combined to yield the transmitted and diffracted

intensities

[13 elz+It+21, hel2 cosl g cosh z]|

I(z) = e-w —h (8.16a)
I, + Irthelz
2 Tz 2 V2/2
b(z) = e-% rao le 74+1-2e cos[ g cosd z)) (8.16b)

I, + Ip I,+lhe
Again, for the condition I,=I, the transmitted and diffracted intensities re-

duce to a simple form

ard

Pi(z) e% = > (1+ cos(g cosd z)/cosh(g sind z)} (8.17a)
P(z) e % = (1 cos(g cos z)/cosh(g sind z)). (8.17b)

Equations (8.17) give an expression functionally dependent on both g sing

and g cosd. But g sind can be determined directly from the expression for
beam coupling (8.4). In this way, (8.4) and (8.17) can be used to determine g

and odin the material.

8.3.3 Phase Ramping Method

Returning to the harmonics method of section 8.3.1, if one beam is

-238-

shifted linearly in phase instead of sinusoidally, information about g and 6

can be obtained from the beam coupling. When one beam is shifted through
m radians, the phase positions during the ramp corresponding to the mini-

mum and maximum of the transmitted intensities can be used to determine

gand 9. In particular, the maximum of P,(z) from equation (8.12b) occurs at

-1

sin[g cosd z] | (8.18)
sinh[g sind z]

The harmonics method and the beam coupling/diffraction method
constitute two independent methods of measuring g and 4 in a photorefrac-
tive material. Equation (8.18) is a third method which cannot independently
determine both g and 9, but can be used to verify the results obtained with the

previous two methods. The experimental use of these methods is described

in section 8.5.
8.4 Photorefractive Coupling and Phase in Paraelectric Materials

In this section a theory is developed to describe the photorefractive pa-
rameters of paraelectric KLTN under an applied electric field. The photore-
fractive response of paraelectric KLTN is described by the quadratic electro-
optic effect in conjunction with the Zero Electric Field Photorefractive
(ZEFPR) effect!2. The ZEFPR gratings are unique!* in that they are always n/2
out of phase with the electro-optic induced index grating (see chapter six). In

addition, the ZEFPR index grating is proportional to the space charge field.

Thus we can write An(z) = Ango + An7zz) which has the contribution from

-239-

both the conventional electro-optic grating as well as the ZEFPR grating.

An(z) = Ege'yEgcos(Kz + O£) + Yzr sin(Kz + O¢))

= EgeV (yEo)? + ye sin(Kz + og + @). (8.19)
Here E, is the applied uniform electric field and E,, is the photorefractive

space charge field. yis the effective linear electro-optic coefficient induced by

the presence of the applied field. It is given by (chapter three) y = n° e(e8,)*,
where n, is the refractive index, g is the relevant quadratic electro-optic coef-
ficient and €,, is the dielectric constant. Yz¢ is a coefficient which relates the

index grating due to the ZEFPR effect with the magnitude of the space charge
field which inevitably forms in conjunction with the ZEFPR grating. It is ex-

perimentally determined by the diffraction observed with zero applied field.
Finally o, is the phase between the intensity and electro-optic gratings as dic-

tated by the Kukhtarev solutions of the band transport model in chapter two.

We have

Of = tan"!

Eo En

where Eq and Ey, are the photorefractive diffusion and maximum charge

fields. The addition of the ZEFPR grating modifies the net phase of the grat-
ing by o = tan |(yE,/yz~), so that the net photorefractive phase $ = og + o is
not equal to the phase between the intensity and the electro-optic grating.
Finally, the coupling constant is given by g=2 [(yE,)* + ze)’ Foc o/h- Here

Exe is the space charge field for unity modulation depth. The applied field

-240-

dependence of the space charge field is given by!4

I Ent Eq - 1E9

Therefore, with an applied field, a paraelectric photorefractive material such

as KLTN should exhibit beam coupling characterized by a net photorefractive
phase = 6g + a and a coupling constant described by (8.21) and the value for g

given above. The following section provides experimental results which cor-

roborate this theory.
8.5 Experimental Results

The methods described above were used to experimentally determine g
and $ for an iron doped lithium niobate crystal, an iron doped barium ti-

tanate, and a paraelectric potassium lithium tantalate (KLTN) crystal doped
with copper. We note that the use of independent methods (the harmonics
method and beam coupling/diffraction method) was instrumental in obtain-
ing accurate data. Before the experimental setup was sufficiently isolated
from air currents and vibration, each method yielded different results which,
however, were remarkably consistent among themselves. This would indi-
cate that any experiment which used only one method to determine the pho-
torefractive parameters and which relied on consistency of data to ascertain
the reliability of the measurement, would consistently and radically underes-
timate the error. The experimental setup is illustrated in figure 8-1. All the

experiments were performed at room temperature. The crystal dimensions

were 3.82 x 5.49 x 2.34 mm? for the KLTN, 5.0 x 5.0 x 2.20 mm? for the LiNbO 3,

-241-

and 5.0 x 5.0 x 5.23 mm? for the BaliOz, where the last dimension is the thick-
ness of the sample. The KLTN was grown and prepared as described in chap-
ter four. The laser beams were at 488nm with polarization in the plane of
their intersection (extraordinarily polarized). The c-axis of the samples was
perpendicular to the bisector of the light beams and in the plane of polariza-
tion. For the experimental geometry used, the grating wavevector was K =
1.7x107/m. The sinusoidal phase modulation was performed with 0, =
0.0613rad at 10.6kHz. The phase was not modulated during writing of the
grating, although the effect of concurrent modulation and writing was almost
negligible. When the phase was ramped through z radians to determine the
phase positions of the intensity extrema, a similar procedure was used. Data
was taken using a Stanford Research SR760 spectrum analyzer and an
HP7090A x-y plotter. The results for the samples tested are shown in table 8-
1. The lithium niobate had © = 0.41 so that, despite a coupling constant of g =
13.3/cm, it showed fairly weak beam coupling. For the barium titanate the
phase was much closer to m/2 as is expected for a material with a weak photo-
voltaic effect. Paraelectric KLTN is forbidden from having a conventional
photorefractive response without an applied field, hence it only displays the

ZEFPR effect in this case!3. As predicted in chapter six, the ZEFPR gratings
show 6 = 0 to within the accuracy of the experiment.
The photorefractive parameters of the paraelectric KLTN were deter-

mined as a function of applied electric field. The data were taken in the man-

ner described above and the results compared to the theory of the previous

-242-

Material g/cm
1 2 Best
Fit
LiNbO, 12.75 13.67 13.3
BaTiO, 2.25 2.33 2.30

KLTN 241 2.45 2.43

KLTN(2.2kV/cm) 7.18 7.0 7.1

Table 8-1. Values of the coupling constant g and the photorefractive phase $

for the crystal samples tested. Column 1 is data from the beam
coupling/diffraction method, column 2 is from the harmonics method, and
column 3 is the best fit to data. The KLTN is paraelectric so its photorefractive

response in the absence of an applied field is due to formation of zero phase

(o= 0) ZEFPR gratings. When a 2200V/cm field is applied (last row), the elec-

o - rad

1 2 + Best

Fit

0.40 0.42 0.41

15 1.46 1.47

0.0 0.00 0.00

1.28 1.00 1.1

tro-optic gratings dominate, and the phase is far from Zero.

-243-

section.

The sample used had composition Kg g9Lig 97 Tag 73 NBp 9O3:Cug ggg.

The acceptor concentration was determined with absorption data and grating

spacing dependence of the space charge field to be Na = 1.5x10%4m “3.

Capacitive dielectric measurements yielded € = 12000 at 24°C (the temperature
of the experiment). The crystal was paraelectric above its phase transition at

15°C.

Experimental results and theoretical curves are shown in figures 8-3a

and 8-3b. There is good agreement between the experimentally determined

values for g and 6 versus applied field with the theory described above.

8.6 Discussion of Results

As noted earlier in this chapter, the solutions of beam coupling and
diffraction off dynamically written holograms are generally considered in the
nondepleted pump approximation where one beam is much weaker than the
other. This requirement may be undesirable because the photorefractive ef-
fect may be weak and difficult to observe in this limit. The reason this restric-
tion is often acceptable is that in materials such as LINbO, and BaTiO; fanout
is such a problem that it dictates the use of the crystal in the weak signal limit.
Since the KLTN family of materials does not exhibit fanout without an ap-
plied electric field (and only weak fanout with an applied field) there is no

reason to use the weak signal limit.

Second, having solved the equations in the general case, it is easily de-

-244-

1.2

Phase © - rad
o 9°
Oo we

Od l L |
0 500 1000 1500 2000 2500

g 1/cm

0 lL ! | !
0 500 1000 1500 2000 2500

Applied Field V/cm

Figure 8-3a. The photorefractive phase of gratings written in KLTN versus
applied electric field. Solid line is the theoretical curve describing the interac-
tion between a ZEFPR- and an electro-optic grating. 8-3b. The coupling con-
stant g of KLTN versus applied field. Again, the data are in accord with the

theory (solid line).

-245-

termined which experimental configuration will yield the best quality data.
Comparison of equations (8.12) and (8.13) show that the beam coupling re-
sponse for I, =I, yields a much more sensitive measurement of the photore-

fractive phase than the case I, << I, when the phase is near zero. This results

from the coefficient of the cos@ term approaching zero as sind -> 0 in (8.12),

ie., the second harmonic of the output vanishes when = 0. In the weak sig-
nal limit (8.13), however, no such sensitivity is exhibited. In fact, depending
on the value of g cos@z either the first or the second harmonic of the phase

modulation can dominate the detected signal, i.e., the thickness of the crystal
is a major factor in determining the response. We conclude that the equal in-
tensity solution to the diffraction and beam coupling is more experimentally
useful to the determination of photorefractive coupling constant and phase,
especially when the phase is near zero. In any case, the existence of the gener-
al solution allows the calculation of the photorefractive parameters for arbi-
trary input intensities, and allows the optimization of the experimental setup

to suit the particular material being characterized.

The effect of a phase shift of one beam on a dynamic hologram has ap-

parently been misconceived on occasion. It has been proposed®, for example,

to measure the photorefractive phase in the following way: The intensity
coupling of two beams in a photorefractive material in the weak signal ap-
proximation I, << I, is roughly given by

In(z) = 1n(0) e 28 sind z, (8.22)

If one of the beams is shifted by a phase 6 then the beam coupling equation

-246-

will be transformed to

"Io(z) = 12(0) e 2g sin(o+5)z" (8.23)
where the “+” and “-” depend on which beam is phase shifted, i.e., the sign of
the phase shift. The quotation marks are to remind the reader of the dubious

nature of the above equation.

This approximation is not very useful. Comparison with the correct
calculations show that it is not accurate under most conditions. The reason-
ing behind (8.23) is specious because it assumes that the weak signal beam am-
plification coefficient is proportional to the phase difference between the in-
tensity gratings and the phase gratings. Since this relative phase can be man-
ually adjusted by phase shifting one of the beams, the effective photorefrac-
tive phase is altered and the beam coupling is expected to follow suit by (8.23).
What this argument neglects is that (8.22) only holds for dynamically written
gratings! and dynamic gratings have a spatial phase and intensity profile
which depends on 6. The law of exponential gain of one beam at the expense
of another (8.22) derives from two effects. The first is that the intensity cou-
pling of one beam to another is governed by the phase 6. The second is that
the efficiency of the coupling is modified by the amplitude of the index grat-
ing, which is itself a function of the beam coupling. It is the combination of
these two effects which leads to (8.22). By varying the phase manually, we are
changing one of the parameters while leaving the other (the grating) fixed.
This approximation is particularly misleading because in the weak signal
limit the grating is most strongly apodized and is a strong function of the

photorefractive phase.

-247-

8.7 Summary

In summary, the problem of two-beam coupling and diffraction off a dynamic
photorefractive grating written in the copropagating geometry has been ana-
lyzed. The solutions allow the determination of the coupling constant and
phase of the photorefractive grating with several methods. These parameters
were measured in several crystal samples with no applied field, and in KLTN
as a function of applied field. The latter data are in agreement with a theory
describing the coherent addition of a normal electro-optic grating and the

ZEFPR grating.

-248-

References for chapter eight

[1] D. Vahey, “A nonlinear coupled-wave theory of holographic storage in

ferroelectric materials,” J. Appl. Phys. 46, 3510-3515, (1975).

[2] N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L.
Vinetskii, “Holographic storage in electro-optic crystals. I. steady state,”

Ferroelectrics 22, 949-960 (1979).

[3] V. Kondilenko, V. Markov, S$. Odulov, and M. Soskin, “Diffraction of cou-
pled waves and determination of phase mismatch between holographic grat-

ing and fringe pattern,” Opt. Acta. 26, 239-251 (1979).

[4] R. M. Pierce, R. S. Cudney, G. D. Bacher, and Jack Feinberg, “Measuring
photorefractive trap density without the electro-optic effect,” Opt. Lett. , 15,

414-416, (1990).

[5] R.S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, “Measurement of
the photorefractive phase shift,” Opt. Lett. 17, 67-69, (1992).

[6] W. B. Lawler, C. J. Sherman, and M. G. Moharam, “Direct measurement of
the amplitude and the phase of photorefractive fields in KNbO3:Ta and
BaTiO,” J. Opt. Soc. Am. B, 8, 2190-2195, (1991).

[7] P. M. Garcia, L. Cescato, and J. Frejlich, “Phase-shift measurement in pho-

torefractive hologram recording,” J. Appl. Phys. 66 (1), 47-49, (1989).

[8] M. Z. Zha, P. Amrhein, and P. Glinter, “Measurement of phase shift of
photorefractive gratings by a novel method,” IEEE J. Quan. Elec. 26, 788-792,
(1990).

-249-

[9] C. Gu and P. Yeh, “Diffraction properties of fixed gratings in photorefrac-
tive media,” J. Opt. Soc. B 7, 2339-2346, (1990).

[10] M. Segev, A. Kewitsch, A. Yariv, and G. Rakuljic, “Self-enhanced diffrac-
tion from fixed photorefractive gratings during coherent reconstruction,”

Appl. Phys. Lett. , 62, to appear March 1, 1993.

[11] R. Hofmeister and A. Yariv, “Vibration detection using dynamic photore-
fractive gratings in KTN/KLTN crystals,” Appl. Phys. Lett. , 61, 2395-2398,
(1992).

[12] R. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, “Simple methods of
measuring the net photorefractive phase shift and coupling constant,” to ap-

pear Opt. Lett. April 1, 1993.

[13] R. Hofmeister, A. Yariv, S. Yagi, and A. Agranat, “A new photorefractive
mechanism in paraelectric crystals : A strain coordinated Jahn-Teller relax-

ation,” Phys. Rev. Lett. , 69, 1459-1462, (1992).

[14] J. O. White, S. Z. Kwong, M Cronin-Golomb, B. Fischer, and A. Yariv, in
Photorefractive Materials and their Applications I] , Eds. P. Gunther and J. P.

Huignard, chapter 4, Springer-Verlag, Berlin, (1989).

-250-

Chapter Nine

General Solutions of Coupled Mode Equations with

Applications to Fixed Holographic Gratings

9.1 Introduction

In the scientific disciplines of electromagnetics and optics it is often
necessary to solve sets of coupled differential equations. Only a tiny propor-
tion of the equations typically encountered have been solved. The complexity
of the solutions for even relatively simple differential equations can be dis-
couraging; usually solutions are performed numerically. Unfortunately, nu-
merical solutions are more difficult to verify than analytic ones, and the
methods in which they are computed can easily lead to errors!?, Also, a nu-
merical solution requires a great deal of computation so it may take an unac-
ceptably long time to map out a family of solutions for a set of equations with
several varying parameters. Finally, with numerical computation all sense of
intuition about the nature of the solution in terms of its variables is lost. For

these reasons, analytic solutions are generally preferable to numerical ones.

In this chapter, attention is devoted to a general solution of a specific
type of linear coupled differential equation. The usefulness of the solution of
this type of equation will then be demonstrated by its application in several

problems of beam coupling and diffraction off fixed and dynamically written

-251-

holographic gratings. The results of these applications are used in chapters

two, five, seven, and eight of this thesis (and also this chapter).
9.2 Solution of Coupled Differential Equations

In the study of beam coupling and holography in photorefractive mate-

rials one often encounters equations of the form

A(z) = i« f\(z) ef 82 B(z) (9.1a)

B(z) =+ ik* f(z) e182 A(z). (9.1b)
Here we define both f,(z) and f(z) to possess the same sign, usually f,(z) > 0.
The typical method of solving this type of equation set is to differentiate ei-
ther equation, obtaining a second-order equation, and then substitute the re-
maining equation in to obtain a second-order equation which contains terms
of only one dependent variable, i.e., A(z) and A’(z), or B(z) and B’’(z), but no
mixed terms. As seen in chapter two, the solution of the archetype of this
family of equations (equation 2.29) leads to eigenfunctions of A(z) and B(z)
which are sin and cos when the “+” is used in (9.1b), and are sinh and cosh
when the “-” is used. These eigenfunctions are plugged into the original set
of equations and, in conjunction with boundary conditions, yield the specific

solutions.

The presence of the 6 term and the f(z) and f(z) terms in (9.1a,b) com-

plicate the use of this method since they contribute terms to the derivative.
In general, after following the procedure outlined above, we are left with sec-
ond-order equations containing terms of A(z), A’(z), and A”(z). The difficulty

in solving the equations is that the coefficients of these terms are functions of

-252-

z. In order to overcome this problem we seek a simplifying transformation of
the dependent variables of (9.1a,b) which will enable the equations to be
solved. The method developed in the succeeding paragraphs is capable of
solving a few important cases, but in instances where it might fail it is not to
be construed that the equation is intractable: the solution presented below is
not a general solution for all such equations with arbitrary f(z) and f(z),
however, it has been able to solve every naturally occurring set of linear cou-

pled equations put to the test so far.

First we seek a transformation which enables the expression of (9.1a,b)
to be expressed as two second-order equations without A’(z) or B’(z) terms.

We perform the transformation

A(z) = a(z) e! F@) (9.2a)
B(z) = b(z) e# F@) (9.2b)
where F(z) is a function to be determined. Equations (9.1a,b) become (using

the “-” sign)

a'(z) =i « b(z) e' ©2-2F@) f,(z) - i F(z) a(z) (9.3a)
b'(z) = - i «* a(z) e! CF) - 52% f,(z) +1 F(z) b(z). (9.3b)

When (9.3a) is differentiated and (9.3b) is used to eliminate terms containing

b(z) and b’(z) we obtain

a"(z) =[i& - 2iF(z) + f)'(z)/f\(z)] az) +

IIx? f(z) f(z) - 8 F(z) + [F@ + i F@) fi'@)/f(z) - iF"@] a@). 9.4)

-253-

The a’(z) term vanishes when F’(z) = 6/2 + f,’(z)/ (21 Fy (Z) ) or

F(z) = &z 4 Il f(z) ) 9.
(Z) 5 + OG (9.5)

An analysis of b’’(z) similar to that leading up to (10.4) gives the condition

F(z) = Zz _ Inf f2(z) ] ;
(Z) 5 5 (9.6)

in order that the b’(z) term be zero. Comparison of (9.5) and (9.6) shows that

the two conditions can be met simultaneously if and only if

f(z) = —L. (9.7)

When f,(z) is purely imaginary, the stipulation of (9.7) reduces to f,(z) =
fy (2). When the condition (9.7) is satisfied, equations (9.3a,b) can be written

as

a"(z) = |KP - [FQ - iF"@)) a) (9.8a)

b'(z) = (IK? - [F(@) + iF"@)] bt). (9.8b)
If we had taken the “+” in equations (9.3), then (9.8a,b) would be modified by
replacing |«|* with -l«l?. The transformation described above will be re-

ferred to as the “corotating transform.”

The advantage of expressing the coupled equations in this way is that

many second-order equations of the form of (9.8) have been solved and their
solutions can be referenced?. In the following sections it will be shown that

when coupled equations describing photorefractive beam coupling are sub-

-254-

jected to the procedure described above, the two second-order equations

which result are often of the form

a) K(K-1) — A(+1)

+ (21 -1)7| wr). (9.9)
sinh?(ar) cosh?(ar)

yw'r) = a

This form is the second Péschl-Teller equation which has been solved* using

a modification of the factorization method>. The equation is used to describe

the bound and scattering states of diatomic molecules, hence the quantum

mechanical notation.

The method of solution will not be described here (see reference 2) in-

stead we merely list the solutions. Two eigenfunctions are solutions of (9.9);

they are
wi(r) = sinh! *(@ r) cosh**(ar) x
2F | [Ack -I+1)+ rt [Ask +I} + Pi 7 ssinh?(or r)| (9.10a)
wa(r) = sinhS(a r) cosh**(ar) x
2F | [At -1+ 1) (Ack +1} Ls « ;-sinh2(o r) | (9.10b)

where >F,[....] is the hypergeometric function, defined by
2Fi[a,b;c;z] = 2Fi[b,a;c;z] =

Tc) y T(atn) M(b+n) za

11
T(a)T(b) I'(c+n) n! O11)

n=(0
The particular solutions of the first-order coupled equations (9.1) which led to

the second-order Péschl-Teller equations will consist of linear combinations

-255-

of the eigenfunctions y, and Wo, with coefficients determined by boundary
conditions. Calculation of the coefficients is generally tedious. Use is made of
the Gauss transformations for hypergeometric functions®, the differential re-

lation

“yo ilas b; c; z] = ab 2Fylat1, b+1; c+]; z], (9.12)

and the linear transformation

oF \[a, b; c; z| = (1-z)"#°? oF i[c - a, c - b; c; z\. (9.13)

The handbook by Abramowitz and Stegun (Ref. 6) is a useful reference for

manipulation of these functions.

The following sections will describe several applications of the mathe-
matical formalism elaborated above. These examples will illustrate exactly
how the procedure is used, and will clarify any abstractions in the foregoing

discussion.
9.3 Spectral Response of Fixed Holographic Grating Interference Filters
9.3.1 Introduction

Fixed photorefractive gratings have received interest recently for appli-

cation as narrow-band optical interference filters and wavelength multi-

plexed optical memories”®919. The advantages of photorefractive filters
over conventional methods are ease of fabrication and extremely narrow
spectral response. We consider Bragg gratings which are written in a photore-

fractive material in exactly counterpropagating geometry (reflection geome-

-256-

try) as in figure 2-4. This method allows tuning the bandpass of the filter by
adjusting the frequency of the laser writing beams. The index gratings writ-
ten by the beams are then considered to be fixed by either thermal fixing or
other methods. We calculate the spectral response of these filters to broad-

band or tunable incident radiation.
9.3.2 Theoretical Investigation, Lossless Case

We start by calculating the two beam coupling (see also chapter 2) of
two incident counterpropagating beams A(z) and B(z) in a photorefractive
material. The well known coupled mode equations in the absence of optical

absorption are (chapter two)

A(z) = ian e'® B(z) (9.14a)
B(z) = “im An” ei A(z) (9.14b)

where A is the wavelength of the interfering beams, and the index of refrac-

tion is
n(z) = ng + (An(z)ei%ei Kz + c.c.), (9.15)

Here 6 is the phase between the optical intensity grating and the induced

index grating. This grating phase is a property of the material; it can be altered
with applied electric fields but is independent of the illuminating beams. We
require, without loss of generality, that B(z) be the amplified beam. That is,
the propagation direction of B(z) is nearer the z-axis (optic axis) than that of

A(z). Generally B(z) propagates parallel to the optic axis and A(z) propagates

~257-

antiparallel to the same. This is equivalent to restricting to the range [0, m].

Since the index grating is formed dynamically by the writing beams we have

An(z) = nyA(z) B’(z)A(z) (9.16)
where I(z) is the total intensity, and n, is the peak-to-peak amplitude of the

index grating when A(z) = B(z) (see figure 2-5). Thus in the case of dynamic

holography, the coupled mode equations can be simplified to

A(z) =i g |[B(z)? A(z) ei? A) (9.17a)
B(z) = -ig |A(z)|? A@) ei? Biz) (9.17b)

with g=1n,/A. We postulate solutions of the form
A(z)=a(z)e! B(z)=b(z)e*¥2 (9.18)

where a(z) and b(z) are real. Equations (9.17a,b) can be separated into two

equations each describing the evolution of the amplitude and phase of the

two beams.
a(z) = - g sing [b(z)| */I(z) a(z) (9.19a)
b(z) = - g sino la(z) 712) b(z) (9.19b)
wiz) = g cosd |b(z)]? /I(z) (9.20)
W2(z) = - g cosd |a(z)| * (2). (9.20b)

Equations 9.19a,b are solved by converting them to equations for inten-
sities using I, = (a2) =2aa’,and similarly for I,. Simple Bernoulli equations

are obtained with solutions

~258-

L 1. i i | L i i L

1.5
DH

0.5

0 0.04 0.08 0.12 0.16 0.2
Position z (cm)

Figure 9-1. The effect of the photorefractive grating phase $ on the intensity
coupling of two equal intensity input beams. For $ = 0, the intensity coupling

is zero, and the beams are purely phase coupled. As $ deviates from zero, the

beams are coupled more and more strongly near the entrance face of the crys-

tal. The coupling constant is g = 20/cm, and the crystal length is L = 0.2cm.

-259-

I,(z) = 7 (Vot+v2eTz + c} (9.21a)
In(z) = 5 (Yorv2ebz - cl. (9.21b)

Here we have used the constants I = 2gsing, c =1,(z)-I,(z) and v2 = 4 I, (0)
1,(0) = 41,(z) I,(z) e'2. Also I,(z) = | A@)12 and I,(z) = | B(z)|2. In the case of
equal intensity inputs (I,(0) = I,(L) ) these constants are given by v? = 4 1,2(0)

Exp[ PL/2]/ Cosh{TL/2] and c¢ = -I,(0) Tanh[ TL/2]. We note that c as it is

defined is a constant by conservation of energy. This follows immediately
from the coupled equations whereby any decrease in intensity of one beam is

added to the intensity of the other beam - propagating in the opposite direc-

tion! The intensities are plotted in figure 9-1 for the case of equal intensity
inputs, for g = 20/cm, L = 0.2cm, and several values of @. The intensity cou-

pling increases as the photorefractive phase deviates from zero. From the

solutions in equations (9.21) we readily solve equations (9.20a,b) to give

Wi(z) = Lg cos z - cotod coth {V7 1+(v/cYeT]} (9.22a)

woz) = - L[g coso z + cord coth[V 14(v/c)eT }}. (9.22b)

Thus equations (9.22), (9.21) and (9.18) completely describe the beam coupling
in the counterpropagating geometry for arbitrary input beams and arbitrary
material grating phase ¢. The index grating in the material follows from

equations (9.15) and (9.16) and is given by

-260-

S 2S nd 2
nN w cy uw

Magnitude of Index Grating

0 0.04 0.08 0.12 0.16 0.2
Position - z (cm)

Figure 9-2. The magnitude of the index grating |An!/n, formed by the in-
tensity coupled beams from figure 9-1. For = 0, the index grating is constant,

and as $ increases, the index grating is more strongly apodized with its maxi-

mum at the entrance facet of the crystal.

-261-

Anei?=n, v_ e272 +18 cos z+ 10 (9.23)

Vc24+y2eTz

The functional form of the index grating is simplified by the partial cancella-

tion of the beam phases in equation (9.22). The spatial variation of the mag-

nitude of the index grating is plotted in figure 9-2 for the case of equal intensi-

ty inputs and for several values of 9.

We calculate the frequency reflectivity of a new incident beam A(0) off
this index grating. In analogy to equations (9.14a,b) we write the coupled

mode equations

. -Tz/2 + i g cosd z + id ‘ y
A(z) = igY¥& e271 4B z Biz) (9.24a)
*2 Vc24+v2eT%
' . -Tz/2 -ig cosh z - id :
B(z) =-ig¥& et2i 4B z A(z) (9.24b)

where AB =(@,-@) ng / c_ is the frequency mismatch between the beams
which wrote the grating and the one undergoing Bragg reflection. We have
ignored the new dynamic grating which is written by the interference of the
incident and reflected beams!!5 since we are calculating the filter response to
broadband illumination. Any such secondary grating would be bleached by
the majority of light which is not reflected. In equations (9.24), B(z) is the new
reflected beam so that we take B(L) = 0, where L is the length of the
crystal.The analytic solution of equations (9.24) subject to this boundary condi-

tion follows.

We first note that equations (9.24a,b) can be written in the form

-262-
A(z) = i g(z) f(z) B(z) (9.25a)
B(z) =- i [g@/(z)] A@) (9.25b)
where f(z) = exp[-2iABz + igcosoz + id] and g(z) = gv expl-Tz/2]/f2 [c?

+ v*exp[-Iz] ]!/2}. We can eliminate g(z) by performing the independent

variable transformation of z to € where &/sind = Jg(z) dz

e=-Tv en? dz = sinh Ye Te? J, (9.26)
2¢ | WV 14(v/c)el .

Inverting, we obtain e!?/2 = (c/v) sinh[é]. Applying this transform to

(9.25a,b) yields a set of coupled equations in &:

a'(§) = ix f(€) b(&) (9.27a)

b'(E) = -ix* f(E) a(€) (9.27b)
where

£() = (Csinh(})*' api (9.28)

kK = - e! $/(2 sino) (9.29)

and AB = AB - (g/2) coso.

Equations (9.27a,b) are clearly of the form (9.1a,b) with 6 = 0, and can be

transformed with the corotating transform of the last section. We define

a(é)= Te) lO and p(& = vie) e IFO) (9,30)

-263-

where

F(E) = 1/(2i) Ln[f(€)] = AB /(g sind) Lnfsinh[E]]. (9.31)

In analogy to (9.9) we obtain the simple second-order equations

T'&) = [p?- (FOP - iF’) TE) (9.322)
V'© = |e? (F@P+ FO] VO. (9.32b)
Here F(é) was chosen to eliminate the first derivative terms T’(§) and V’(&) in
the equation (9.32). From (9.31) we can write F(E) = AB/ (g sind) coth[é]

and F’’(&) = - AB /(g sing sinh2(E] ) so that

T'(é) ¢{—=L_ 4 n2 - MOND) pe) = (9.33a)
in ) sinhE

V'(é) + {L492 - Man ahve (9.33b)
ian “) sinh€

where we define 7 = 2AB’/T. Comparison with equation (9.9) confirms that

(9.33a and b) are examples of the second Péschl-Teller equation where A =0, @

= 1, and the “quantum” parameters are

pal al gas + 1 (9.34)
2 4sin7 2

Kr = 1417 (9.35)

Ky = iN. (9.36)

The parameters ky and ky are those required for (9.33a) and (9.33b) respective-

-264-

ly to match the notation of (9.9). The quantity “] ” is the angular momentum

quantum number in the diatomic system. Here it assumes not only noninte-
ger values, but, for AB’ > g sind/2, becomes complex. Thus little intuition can

be drawn from the quantum system to help us in this application.
Fortunately, the solution of the Péschl-Teller works with complex parameters

and can be utilized as is.

Using (9.10a,b) we can directly write solutions for T(E) and V(&):

T(&) = t) sinh NE cosh& Fil - Py -B,+- 5 +B; 7 in; -sinh’E] +

to sinh’*!"E cosh& Fly - B+1, it 5 tB+1: 3 5 2 +in; -sinh’€] (9.37a)
V(E) = v, sinh! !%E cosh€ >F,f1 i} - Bil “ip +B; 3 -im; -sinh2E] +

v2 sinh'"E cosh& 2Fyfi 3 Bt iD i 5 tBHS J oe) 1 4in; -sinh7€] (9.37b)

where t,, ty, vj, and v> are arbitrary constants and we define

B= 5,)/ wat 5, (9.38)
2 4sin7 2

Using equations(9.26), (9.30), (9.31), and (9.37), the expressions for A(z) and

B(z) are obtained

-265-

A(z) =4/1+(@%y’e™ x

1_j2_g 1 i148. 1 jn. yer
[Cr 2Fil,- i, - B. 5 i +Bs 5 -ins eT) +

Co e-(1+2in)ra2 Filia -B +1, id +B +1; 5 tin: yet) | (9.39a)

B@) = 4/1+@y'e™ x

-(1-2in) 2/2 ao a -3 Lin: _(V.)%e-Tz
fe e 2Fi[ 5 B +1, i+ Btls 5 in; ye ]+
Cy oF fil -p +h i+ B 4d; 1 tin; -)%e 9.39b
4 ili, -B+,.15 +B +55 +m 1] ( )

where C,, Co, C3, C4 are constants. The constants are determined from the

boundary condition B(L)=0 and equations (9.24), using the hypergeometric
function identities (9.12) and (9.13). The result is

C, = A(0) 4sin2o (1+4n2) e(l-2in)TL/2 x

V1+(v/e) |
oF y[ iD -B + 5 + Bre 5 +in; (“fe L/D (9.40a)
-A(0 2
Fil i - BH, i} + Bets 3-ins {XP eM YD (9.40b)

C3 = - A(0) | v (2+4in) sing | e(l-2inyTL22 x
ic eld V1+(v/c)*

oF if iD -B +, 0 + Bes 7 +in; {¥/ e TLYD (9.40c)

-266-

Cy = A0)| v (2+4in) sino |
ic eld V14(v/c)*

oF [ i} BH, i} + B+; 57 in {x p e TLY/D (9.40d)

where the common denominator D is given by

D = 4sin7o (14472) e (2m) TL2 x

2Fil 5 - B+s. es + Bre: 5 tin: (YPetM] x

Pil iy - B+, H+ Beds bins O°]

v2 lie mul -3 in. (¥)% e TL

© oF i[ rs B+1, i, + Btls 5 in; Cs) etl] x
F,[ it - B41, i + B41; 3 +in; -(%?1. 9.41
2 it, B zm) B 5 in Co] (9.41)

The equations (9.39a,b) with constant coefficients defined in (9.40a,b,c,d) and
(9.41) determine exactly the amplitude and phase of the incident beam A(z)

and the reflected beam B(z).

To recap, the reflected wave is generated by interaction of A(z) with the
previously dynamically written hologram defined by the index grating of
(9.23). Since B is a reflected wave, the solutions obey the boundary conditions

B(L) = 0. Both A(z) and B(z) are expressed in terms of, and are proportional to,
the input beam A(0). The reflectivity of the grating is given by R= | B(0)12/
|A@)I* . It is plotted in figure 9-3 for the case of equal intensity input beams

(during the writing phase), and for several values of the grating phase 6. In

general, the reflectivity maximum occurs at a different frequency than that of

-267-

fii» 6-0
0.8 —O— o=n/32 | 7
~ . —O— o=n/16
5 0.6 + " a b=n/8
oan —o— g=n/4
esi ; —B— $=3n/8
v 0.4 - X —@— =n /2 a
(J
(.)
\\
0.2 + Q ; 4
0 10 20 30 40 50

Frequency Mismatch Ap - (cm' )

Figure 9-3. The reflectivity from the fixed index gratings of figure 9-2. The

reflectivity maximum occurs at a frequency mismatch given by AB = (g/2)

cosd. The overall reflectivity as well as the sidelobes are reduced by the grating

apodization for > 0.

~268-
the writing beams; it occurs at a frequency mismatch of AB = g cos/ 2, i.e., it
occurs at AB’ = 0. This mismatch arises simply because the two writing beams
influence each other’s phase (as well as intensity) by a total amount Ag(z) = g

cosd z.(see (9.22a,b) ) This increases (or decreases) the spatial frequency of the

index grating, altering the frequency at which maximum reflectivity is

achieved. The reflectivity curves plotted in figure 9-3 are reflected about the

line AB =0 when dis reflected about 1/2, i.e., when cos -> -cosd the reflectiv-
ity transforms as R[AB] -> R[-AB ]. For this reason the reflectivity versus fre-
quency mismatch has only been illustrated for 0 < ¢< 7/2. We note that the

grating apodization caused by intensity coupling for @ # 0 or m leads to reduced

sidelobes in the reflectivity at large values of frequency mismatch, as well as

lower overall reflectivity. This is illustrated in figure 9-4 where the reflectivi-

ty of the 6 = 0 and $= 2/2 cases are compared at large values of mismatch.

We now consider how the previous equations are simplified under
special conditions. For »=n/2 note that the beam coupling consists purely of
intensity coupling rather than phase coupling. This leads to a simpler form
of the index grating (Equation (9.23)), however, little simplification of the
final solutions occurs. Since the phase coupling is absent in this case, the re-
flectivity maximum of the grating occurs at the same frequency as that of the

writing beams.

The case =0, is completely different. Here, there exists no intensity

-269-

1.0
-2
~° 10
~~
ae -4
5 1
rs}
a)
mM 106
10° | l | l

0 500 1000 1500 2000 2500
Frequency Mismatch AB - (cm)

Figure 9-4. The reflectivity from the fixed index gratings for 6 = 0 and 6 =1/2
at large values of frequency mismatch. The $ = x/2 grating reflectivity has re-

duced sidelobes whose peaks are approximately 8dB lower than the @ =0 case.

-270-

coupling while the phase coupling is maximum. Hence the magnitude of the
index grating is a constant throughout the volume of the crystal. We obtain,

in analogy to equation (9.23),

VIO) oigz (9.42)

An =n,

This grating is fixed and the equations describing reflection off it are

A(z) = ik ei-248)z Biz) (9.43a)

B(z) =-ik ei @-248)z A(z) (9.43b)
where

Ka meee). (9.44)

The reflectivity is easily shown to be

«sinh*[sL] + s2

where

s = VK2.67/4 (9.46a)
5 = g-2AB. (9.46b)
Equation (9.45) has the simple form characteristic of Bragg reflection from a

constant amplitude grating!4. The only difference is that the maximum reflec-
tivity occurs at a frequency shift of AB = g/2 rather than AB = 0. Again, this is

due to the mutual phase coupling of the two beams which shift 6 by an

amount g. Also the parameter s is complex for large values of the frequency

~271-

mismatch; this is no problem analytically, but may cause problems during nu-
meric calculations. If this occurs, s can simply be redefined as s -> is, so that

sinh -> i sin, and so on.

A similar case occurs for arbitrary phase @ under the special condition

c=0, that is, when the coupled beams are everywhere of equal intensity ( see
figure 9-1). Starting again from equations (9.17a,b), we derive in analogy to

equations (9.21) and (9.22), intensities and phases of the coupled beams

I\(z) = lh(z) = iy eT22 = 1,(0) e Tx (9.47)

Wiz) = -We(z) = cos g/2z (9.48)

where v is as defined following equation (9.21a,b). From these we derive the

index grating to be a constant magnitude

An(z) = rs e@ 1g cosd 2. (9.49)

In comparing (9.49) with (9.42), note the important difference between the two

cases c=0 and 6=0. Here the beams are intensity coupled with a gain coeffi-

cient of g sing; this reduces the phase coupling coefficient from g to g cos@.
Also note that since the beams are everywhere of equal intensity, the term
[1,(0) 1,(]}!/2/ I from (9.42) reduces to 1/2 in (9.49). Since the formation of the
index grating is intensity independent (9.16), the index grating is constant al-
though the intensities of the beams are not. In the case @ = 0 the beams need

not be equal intensity because their intensities are constant throughout the

-272-
volume of the photorefractive medium.

The grating (9.49) is fixed in the material. The coupled mode equa-

tions describing reflection become
A(z) = i e 9 ¢ i (Ecos - 2.48) z Biz) (9.50a)

B(z) =-i eid ei (g cos - 2. AB) z A(z) (9.50b)

these equations are readily solved to yield a reflectivity identical in form to

that in the case » = 0 (equation (9.45) ) with the modified definitions Kk =g/2

and 5=gcos@ - 2AB.
9.3.3 Investigation, Lossy Case

The previous section investigated the problem of frequency dependent
reflectivity off a dynamically written photorefractive hologram. The effects of
optical absorption were ignored. When the loss in the material is considered,
the equations become substantially more complicated. Fundamentally, the
problem arises because the absorption changes the relative beam intensities as

a function of z. (This would not occur if the beams were copropagating.)
Analytic expressions are only obtained for =0. Reflectivities of the fixed

holograms are calculated numerically. We start with the dynamic coupled

mode equations as in equations (9.17a,b).

A(z) =i g |B(z)? /(z) e!® A(z) - o/2 A(z) (9.51a)

B(z) =-ig |A(z)? A@) ei B(z) + a/2 BY). (9.51b)

-273-

Performing the same transformation as in equation (9.18) we are led to the

following equations for the phases and magnitudes of the coupled beams.

a(z) = - sing g [b(z7/I(z) a(z) - a/2 a(z) (9.52a)
b(z) = - sing gla(z)|7M(z) b(z) + a/2 b(z) (9.52b)
Wi(z) = cos og |b(z)|* A(z) (9.53a)
Wo'(z) = -cos 6 g |a(z)|* Iz). (9.53b)

Again, the special case $=0 yields considerably simplified formulas. In this

special case, analytic solutions of equations (10.52) and (10.53) are obtained eas-

ily. They are

ig.

A(z) = A(0) cone BOE AG) bs (9.54a)

B(0)2+A(0)?

‘ig,

B(z) = B(0) evo BON eA Ore (9.54b)

B(0)2+A(0)2

Thus the index grating is given by

An(z) = ny, ——A() BO) e® (9.55)

A(0)? e + BO)? em

When this grating is fixed, we can formulate the coupled equations describing
reflectivity in analogy to (9.14a,b and 9.24a,b), using equations (9.51a,b). These
equations can be manipulated following identical steps as in the analysis up

to equation(9.32a,b). In this case, however, instead of the Pdschl-Teller

-274-

(Equation (9.33a,b).), the second-order equation obtained is of the form

M? + K?-1- 2MK cosx

" 4 2,]
y +|0o+K*+}ly (9.56)
sin?x | al

which is a symmetric top equation. Although its solution is documented?,

we do not present it here, since it is useful only for o=0.

Alternatively, equations (9.51a and b) have been solved numerically for
arbitrary photorefractive phase in order to obtain the index grating which is

fixed in the material. This calculated index grating is then fitted to a high
order polynomial (usually 9th order), and that polynomial approximation is
then used in the reflectivity equations. In solving the dynamic equations we
specify boundary conditions at a given value of z, usually at z = 0 (i.e. , A(O)
and B(0) ). Since the physical inputs to the crystal are A(0) and B(L), B(0)
must be iteratively modified by trial and error to obtain the desired value for
B(L). This procedure yields the dynamic index grating for inputs A(0) and
B(L) in a material with optical absorption. This grating is considered fixed

when the reflectivity is calculated. Here, we use the standard procedure of fix-
ing B(L) = 0 and A(L) = 1!2 and working backwards toward z=0. Finally, the

reflectivity is defined as R = |B(0)/*/1A@)1* where |A(0)| > 1. The addition

of a nonzero optical absorption coefficient has two noticeable effects on the re-

sults. First, the reflectivity of the filter is now strongly nonreciprocal; that is,

the reflectivity is different if the crystal is flipped by 180°. Second, since energy

is no longer conserved, the quantity | A(z)! ?-|B(z)|? is no longer constant. In

-275-

the previous (lossless) case, the intensity difference was defined as c. This pa-
rameter appeared in the final solution. In this case, c is no longer constant,
and may change sign (compare figures 9-5a and 9-5b). When c = 0, the intensi-
ties are equal; this may happen at any arbitrary point within the volume of
the crystal. Thus the index grating may have its maximum at any point in
the volume of the crystal, rather than only at the entrance or exit facet (com-

pare figure 9-6 with 9-2).

In the following calculations we have used a coupling constant g =

20/cm, a crystal length of L = 0.2cm, and we have assumed equal intensity in-
puts A(O) and B(L). The loss coefficient is taken to be a = 6/cm. First the

equations for the dynamically coupled beams A(z) and B(z) are computed

from equations (9.5la and b). The results for various values of the grating
phase $ are shown in figures 9-5a and b. The nonzero loss mostly affects the
growth of the amplified beam, B(z), incident from z = L. This occurs because
the beam coupling and loss mechanisms are opposed. Thus B(z) has a mini-
mum at the position where the material loss is balanced by the beam cou-
pling. Also, as discussed above, the loss allows the two beams to have equal
intensities at a point within the volume of the crystal. From equations (9.52)
we determine that this condition is allowed roughly when g sind < a/2. The
index grating formed by the two beams (Figure 9-6.) illustrates this effect. For
= 0, it is clear that the index grating is maximum in the center of the crystal
when the input intensities are equal since only absorption, and no intensity

coupling occurs.

-276-

] T T T ] T ¥ T T T T T T T 6=0
r —O— 6 = 1/16
0.8 F —— o=17/8
I —O— o=n/4
2B F —™— 6 =31n/8
B 0.67 7
g - —8B— o=1/2
Sf 7
0.4 > _
0.2 7
0 i 1 i i 4. 4. 1 | 1 1 1 i st +t
0 0.04 0.08 0.12 0.16 0.2

Position z - (cm)

Figure 9-5a. The intensity of beam 1, A(z), incident at z=0 for various values

of , for a loss constant of « = 6/cm, and for equal intensity inputs. The cou-

pling constant is g = 20/cm, and the crystal length is L = 0.2cm. When 6 =0,
the decrease in amplitude of A(z) is due entirely to the optical absorption. As
o increases, progressively more beam coupling occurs (which depletes A(z))

and the beam is attenuated more rapidly.

-277-

1 T
0.9 i
0.8 4
ay
B07 7 —o— 9 =0
E nel: —O— 6 =n/16
—t— o=n2/8
0.5 —O— o=7/4
—— 6 =3n/8
0.4F —a—g=n/2
0 0.04 0.08 0.12 0.16 0.2

Position z - (cm)

Figure 9-5b. The intensity of beam 2, B(z), incident at z=L under identical

conditions as figure 9-5a. When = 0, the decrease in amplitude of B(z) prop-

agating toward z=0 is due entirely to the optical absorption. As 6 increases,

progressively more beam coupling occurs (which amplifies B(z)) counteract-
ing the absorption and eventually causing net beam amplification. The com-

petition between the optical absorption and the beam coupling gain makes

the amplitude of B(z) a complicated function of in the range z=0.00 - 0.08.

0.5

0.4

0.3

0.2

0.1

Magnitude of Index Grating

Figure 9-6. The index grating formed in the crystal by the coupled beams in
figures 9-5. One effect of the loss is to eliminate the “no crossing” rule for the
two beam intensities, i.e., with zero loss, c = constant, so that the two beam in-
tensities cannot cross. In the figures above,when g sind < a (roughly), the in-

tensities can be equal within the volume of the crystal. This leads to a maxi-

mum in the magnitude of the index grating located away from the edge of the

crystal.

-278-

—— o=0

—O— o=n7/16
—A— o=n/8
—B— g=n/4
—a— = 32/8
—S— o=n/2

H {

0.04

0.08 0.12
Position z - (cm)

0.16

0.2

-279-

Next we calculate numerically the reflectivity from the index gratings
of figure 9-6. Figure 9-7. shows the reflectivity for a beam incident on the z=0
side of the crystal, and figure 9-8. shows the reflectivity under the same con-
ditions for a beam incident from z=L. Since the index grating is, in general,
stronger near z = 0, the reflectivity is higher for a beam incident at z = 0. The

difference between the two cases is most pronounced when the grating phase
dis near x/2. This follows from the fact that the case = 2/2 results in the

strongest spatial variation of the index grating. When the c-axis of the mate-
rial is defined as the direction of power transfer of symmetrically incident
beams in a two-beam coupling experiment, we can say that a beam incident

antiparallel to the c-axis is reflected more strongly. Also we point out that the
behavior of the reflectivity as increases is more complex in the case of reflec-

tion from the z=0 side. The reason is that the index grating near z=0 (the

most efficient region of reflection for a beam incident at z = 0) first increases,
then decreases with increasing photorefractive phase . This is funda-

mentally due to the competitive interplay between the intensity coupling and
the absorption. We can see how the effect arises by an inspection of the shape

of B(z) in figure 9-5b. The concept of nonreciprocal reflection has recently

been employed to determine the photorefractive phase in LiNbO,!4

9.3.4 Summary of Spectral Response of Fixed Holographic Grating Inter-

ference Filters

In summary, an analytic solution for the frequency response of interfer-

ence filters written with counterpropagating beams in a photorefractive mate-

-280-

Reflectivity

Frequency Mismatch AB - (cm)

Figure 9-7. Reflectivity for the case a = 6/cm, versus frequency mismatch, for

a beam incident at z=0 on the index gratings of figure 9-6. The behavior is
complex because the magnitude of the index grating near z=0 (the most effi-

cient region of reflection) first increases then decreases with increasing pho-

torefractive grating phase 9.

-281-

0.5 | |
0.4 7 |
> O x
B03 |
& D
LS As
0.2 7 |
>:
0.1 as . /
TT) OQ 2.
0 | um rth, Se
0 10 20 30 40 70

Frequency Mismatch AB - (cm” )

Figure 9-8. Reflectivity for the case a = 6/cm, versus frequency mismatch,

for a beam incident at z=L on the index gratings of figure 9-6. Note the strong
non-reciprocity compared with figure 9-7. The behavior for this case is sim-
pler than that of figure 9-7. because the magnitude of the index grating near

z=L (the most efficient region of reflection) decreases monotonically with in-

creasing photorefractive grating phase 9.

-282-

rial has been described. Solutions were performed for arbitrary photorefrac-
tive grating phase 6. These solutions were obtained from the coupled equa-

tions by application of the corotating transform and subsequent reduction to
second-order Péschl-Teller equations. A significant result of the analysis was
that the maximum of the frequency reflectivity occurs at a frequency different
from that of the writing beams. This frequency shift is due to the phase cou-
pling which the two beams enact on each other. In other words, when two
beams interfere to write a grating in a photorefractive material, that grating is

generally not Bragg matched to the beams.

In addition, numeric solutions of the identical procedure with a nonze-

ro optical absorption in the material were presented. The chief differences be-
tween this case and the previous case are that with a #0, (1) the index grating
can have its maximum at any plane in the volume of the crystal, rather than
only at the entrance or exit facets and, (2) the reflectivity of the grating is non-
reciprocal. Generally a beam incident antiparallel to the c-axis is reflected

more strongly than a beam incident parallel to that axis.

9.4 Response of Fixed Holographic Gratings Written in the Copropagating

Geometry
9.4.1 Introduction

In the previous sections the response of gratings written in the anti-
symmetric (counterpropagating) geometry were investigated. In particular,
we examined the spectral response of the reflectivity of fixed gratings. In this

section, a similar analysis is performed for fixed gratings written in the sym-

-283-

metric copropagating geometry (figure 2-4). Here the emphasis is not on the
spectral response, but on the response of the beam coupling to phase modula-
tion of one of the beams. In addition, the diffraction off dynamically written

gratings is discussed. Diffraction is simply the reflectivity of the grating when
AB = 0, that is, when the beam being reflected is the same frequency as the

beams which wrote the grating. The results of this section are applied in
chapters two, five, seven, and eight. In chapter two the results are used to
characterize the nature of the photorefractive two-beam coupling, and to de-
scribe the diffraction. In chapter five, the following results enable the expla-
nation of diffraction phenomena in paraelectric KLTN crystals. In chapter

seven, they are used in the development of methods for determining the
photorefractive phase shift o. Finally, in chapter eight, the results are applied

to vibration detection in materials displaying the zero external field photore-
fractive (ZEFPR) effect; further applications to image processing and coherent

data links are discussed there.

As illustrated in figure 2-4, two beams are symmetrically incident on a
photorefractive crystal. A dynamically written refractive index grating is the
result. This dynamically written grating is assumed to be fixed against optical
erasure by some means. In practice, the grating will not actually be fixed, but
the experimental conditions will allow this approximation. The writing
beams are then replaced with two beams of arbitrary intensity and phase and
of the same frequency incident along the same directions as the initial beams,
i.e., at the Bragg angles. We calculate the beam coupling experienced by these

new beams off the dynamically written grating now considered fixed. We ig-

-284-

nore the time dependent formation of dynamic gratings written by the new
beams. This condition is only valid for a time period on the order of a sec-
ond, depending on the intensity, but, as will be seen, this is much more than

is required to obtain the necessary data.

9.4.2 Formulation of the Problem
The starting point is calculation of the two-beam coupling of two inci-
dent copropagating beams with amplitudes A(z) and B(z) in a photorefractive

material. The well known coupled beam equations are given again as

A(z) cosB = i7An ei? B(z) - e A(z) (9.57a)
B(z) cosB = im An” ec A(z) -& B@) (9.57b)

where A is the wavelength of the interfering beams, « the optical absorption

coefficient, and B the half-angle of beam intersection inside the material.

Since these equations describe copropagating beams, the equation describing
B’(z) is multiplied by the factor -1 as compared to (9.14b). The index of refrac-

tion is
n(z) =no + 5 (An(z)ei%ei Kz + c.c,), (9.58)
Here 6 is the photorefractive phase between the optical intensity grating and

the induced index grating; K = 2k sinB is the nominal grating wavevector

with k = 2mn,/A. Since the index grating is formed dynamically by the writing

beams we have as before

An(z) = n; A(z) B’(z) I(z) (9.59)

-285-

where I(z) is the total intensity, and n, is the peak-to-peak amplitude of the
index grating when A(z) = B(z) (see figure 2-5). Thus in the case of dynamic

holography equations (9.57a) and (9.57b) can be rewritten

. 2
A(z) cosB =i g ei? IB)” A(z) - & A(z) (9.60a)
I(z) 2
B(z) cosB =i g ei? JAZ)" B(z) - @ B(z) (9.60b)
I(z) 2

where the coupling constant is defined as g = mn,/A and I(z) is the sum of the
intensities I(z) = | A(z)|* + |B(z)|2. Again, since g is defined in terms of ny
rather than An, it becomes a pure material parameter, the dependence on the

relative beam intensities being factored out.

The method of solution of equation (9.60a) and (9.60b) is straightfor-

ward. First the optical absorption term is eliminated by the change of inde-
pendent variable A(z) = A(z) exp[ az/2] and B(z) = B(z) exp[ az/2]. Then we
postulate solutions of the form

A(z) = a(z)eib B(z) = b(z)e' (9.61)

where a(z) and b(z) are real. Equations (9.60a, b) can be separated into two
equations each describing the evolution of the amplitude and phase of the

two beams:

a(z) = - sind g b(z)* /I(z) a(z) (9.62a)

b(z) = + sing g a(z)? / I(z) b(z) (9.62b)

-286-

Ci(z) = cosd g blz)” / I(z) (9.63a)

Co(z) = cosd g a(z)* /I(z). (9.63b)

Equations (9.62a,b) are solved in the usual way by converting them to
equations for intensities using ly = (a2) =2aa’ , and similarly for I,. Note that
these “intensities” are not the true optical intensities, but are related to them

by the multiplication of exp[ az ]. We use the identity b(z)? = I(z) - a(z)? in

(9.62a) and b(z)? = I(z) - a(z)?_ in (9.62b) to obtain simple Bernoulli equations!

which are readily solved to yield

i) = ew Hh +h) (9.64a)
I, + Ip etlz

b(z) = eo 2th +b) (9.64b)
I, elz + Ip

where A(z) = (1, 2)]}!/*explig)] and B(z) = [1,(z)]"/*expliGy], andI= 2g sing is
the power coupling coefficient. The variable z is, for convenience, taken to be

the unit of length in the propagation direction, so that z = Z,jg/cosd. L is the

effective thickness of the crystal: L = d/cosB. We have defined I, = 1,(0) and I,

=1,(0) for convenience. The phases of the two beams are readily determined

from equations (9.64a,b) and (9.63a,b) to be
C(z) = 5 coro Inf] + Ip e*? 2] (9.65a)

C2(z) = -g cosd z - 5 coto In[I; + Ip e+! 2]. (9.65b)

Note that the phases do not go to zero at z= 0! This condition is necessary for

-287-

equations (9.58/9.59) to hold, the phase difference (, - ¢, corresponding
roughly to the phase difference between the intensity grating and the index
grating (see equation (9.66) below). It is imperative to remember this condi-
tion when performing calculations, especially numeric ones; the results oth-
erwise will be completely erroneous. The index grating in the material fol-
lows from equations (9.59), (9.64), (9.65), and the definition of A(z) and B(z) in

terms of phase and intensity following (9.64). It is given by the form

An(z) = ny VIq lb (I; eT 424 Ty et Pei2 cote =} (9.66a)

when = 0, we either take the limit with care in (9.66), or backtrack to equa-

tions (9.63). The index grating in this case reduces to

An(z) = ny ore exp[ ig z(In-1,)1]. (9.66b)

Thus we have used the coupled mode equations to solve self consis-
tently for the amplitudes and phases of two dynamically coupled beams in a

photorefractive material. Dynamically coupled beams are those which yield
an index grating given by (9.59). The solution for An given above can be rein-

serted into equations (9.57) or (9.60) to verify that the solution is self-consis-

tent, as advertised.

The dynamic grating calculated above is considered fixed and we solve
for the beam coupling of two arbitrary intensity and phase beams incident at
the Bragg angle. When the new beams are the same intensities as the old
ones but with different phases it is equivalent to simply phase shifting one of

the original beams. When, instead, the phases of the new beams are un-

-288-

changed with respect to the original beams but the intensity of one new beam
is reduced to zero, we are calculating the diffraction off the grating. The gen-
eral case when both the phases and the intensities are different in the new
beams is equivalent to calculating the response of the holographically cou-
pled beams to fluctuations in the interferometer system. This application
would be of importance in determining how phase and intensity fluctuations

at the input of a photorefractive interferometer are magnified in the outputs.
9.4.3 Solution of Beam Coupling

We calculate the beam coupling of a new set of Bragg matched beams
TO) = [P,]'/2 expliy,] and V(0) = [P,]'/2expliy,] off the index grating of equa-

tion (9.66). In analogy to equations (9.57a,b) we write the coupled mode equa-

tions
T(z) cosB = igvIj Ib eti@+®) x
(Ty eT#2 4 Ip ethui2}* ome l yyy. ° Te) (9.67a)
V(z) cosB = ig VI; Ip ei @+®) x
(I) eT#2 4 1p etl?) 1-1 Ty) - “ V(z) (9.67b)

where I, and I, are the intensities of the writing beams, not to be confused
with the new intensities P; and P,. The phase @ = €,(0)- €,(0) - y,(0) + w,(0)
is the phase difference between the intensity pattern of the beams which
wrote the grating and the intensity pattern formed by T(0) and V(0). If @ =0

and P, =], for i= 1,2, it is easy to see that (9.67) reduces to (9.57) with the

-289-
index grating as defined in (9.66). Since the photorefractive effect is intensity

independent, it is also true that if @ = 0 and P, =cl, for i= 1,2 where c is an ar-

bitrary constant, the solutions are essentially unchanged except that all inten-

sities are scaled by the constant factor c. Equations (9.67a,b) ignore the new dy-

namic grating which is written by beams T(z) and V(z) in the case 6 # 0.12

In order to solve (9.67a,b) we first eliminate the optical absorption term
by the transformation T(z) = T(z) exp[ az/2] and V(z) = V(z) exp[ az/2 ], and
then proceed along the same lines as before. (9.67a,b) are clearly of the correct

form (9.25) to perform the independent variable transformation (9.26) to § =

Jg(z) dz. By inspection of (9.67a,b) we write

b= | VIjIo g sind dz

I, e-F#24I,etTar

= 2tan![ Vo, e+ 7/2 ] (9.68)

which yields, after inverting,
elz2 = vVIj/Ip tan(E/2]. (9.69)

This transformation is applied to (9.67a,b) and gives

“E) = i i (+6) 2in { 2 PM
a(§) Tsino © vIy Ip al b(§) (9.70a)
(£) = —1 -i (040) J, 7, -2in {2 2 70b
b(§) 2 sind e Ty ly Fars a(§) (9.70b)

where 1 = cotd/2. The lower case variables a(§) and b(§) are used to repre-

-290-

sent the functions T(z) and V(z); the change in notation is simply to empha-
size the change in independent variable from z to €. Equations (9.70) are rec-

ognized to be of the form amenable to the corotating transform, where, by

comparison with (9.1-8), we identify f,(§) = sin(é)72iN so that

F(€) = -n Inf sing ]. (9.71)

As in equation (9.30) we perform the transformation
ae) =S() FG) and be = we etFO), (9.72)

Using (9.8a,b) with the previous two equations yields the second-order equa-

tions
S"(E) + ar S(E) = 0 — (9,73a)
4 sinh?[ if]
webs re We) = 0. (9.73b)
sinh’[ if]

Numerous manipulations were used to obtain the above results, including
the identity sin’x = -sinh2[ix]. Comparison with (9.9) shows that the above
equations are again examples of the second Pdéschl-Teller with parameters

given by A4=0, a =i, and

l= 5 Kp = in Ky = 14in. (9.74)

The parameters Ky and Ky are those required for (9.73a and b) respectively to

match the notation of (9.9).

-291-

We apply the recipe (9.10a,b) to solve (9.73) and obtain

S(E) = t; sin’ ing cos& 2F)[ 3. ue ri - ms 5 -in; sin? ] +

rane)
to sin'"E cosé Fy [i a +5 es +4: 7 +in; sin’é ] (9.75a)
W(é) = vi sin'"E cost Pilih, Fis him sin? ] +
v2 sinkE cosé Filip +4, 0 +3; 5 tin; sinZ& | (9.75b)

which, using (9.72) is transformed back to

a(&) = a, sin’ E cost PP - iD 2 its 5 - in: sin’& ] +

2’ 4 2
a2 cos& Pili + iD + 5 +in; sin’ ] (9.76a)
b(E) = b; cost Fld - Os ; - re -in; sin2&] +
bo sin!*7!NE cosE oFi[ (i +3, iO +5; 3 +in; sin2& ] (9.76b)

where t1, ty, Vy, Vo, are arbitrary constants and aj, a9, b,, b, are constants to be

determined. To simplify (9.76) we apply the hypergeometric identity®

Fila, atl: 2a; 2) = 201 Ue Ve) (9.77)
114, ~ ’ _ .
2 V1-z
from which we readily obtain
a(E) = a) sin’ *"E (14cosé)? + a'y (1tcos&)¥?-™ (9.78a)
b(E) = by (1tcosE)*!? + a'y sin!*?ME (1+cosé)y 2°, (9.78b)

-292-

Finally we convert back to the independent variable z using (9.68) and (9.69),

and with some manipulation arrive at

T(z) 2 = Cy (b+, eT"? + Cy (bette4 1, 7 (9.79a)
V(z) e2 = C3(b+ let"? + Cylbettz@4 ny"? — (9.79b)

Here CG are constants which are determined by the coupled equations and the
boundary conditions. After comparing with equations (9.66) and (9.67) we see
that the eigenfunctions of (9.79) seem quite plausible, and might perhaps
have been arrived at without the cumbersome machinery of the intervening
mathematics. But this retrospective argument is based on the knowledge that
a solution actually exists, whereas that fact might not have been obvious be-
fore the mathematical analysis. The method described above, although te-
dious, is quite straightforward, and is guaranteed to yield the solution for
problems which reduce to one of several forms of second-order differential

equation.

In addition, the verification of solutions arrived at by guesswork is not
completely trivial either. If one were to test the equations (9.79) in the origi-
nal set of coupled differential equations, the result would be (using boundary
conditions) four linear equations in the variables C; (i = 1,4). The solution
would be verified by confirming that the “constants” C; were actually con-
stant for several values of z. This procedure and many others were employed
using the software package Mathematica to verify every equation in this chap-
ter both analytically and numerically. It should be noted that the use of
Mathematica was indispensable in debugging the solutions given through-

out this thesis.

-293-
9.4.4 Determination of Coefficients

The equations for beam coupling off a dynamically written grating
have been solved and it only remains to determine the coefficients C; for the

special cases of interest.

We solve the coefficients CG for the special case where one or two of the
reading beams are phase shifted relative to recording beams but their intensi-
ties are unchanged. This can be accomplished in practice by merely inserting

a phase shift in the path of one of the recording beams. From equations
(9.67a,b) and the definition of 8 we use the boundary conditions T(0) = 1,
fl, + 3" and VO) = 1,!/ [l, +1,/™. It should be stressed that the term 1, +

I,] raised to tin is purely a phasor which is required to ensure that the index

grating have the proper form. With simple differentiation and insertion into

the coupled equations (9.67) we determine the coefficients as

Cy = V/M th) b (1 - e'®) (9.80a)
Co= VI/M4+h) (hh + hb e'®) (9.80b)
C3 = VIo/(ith) (Ip + eI) (9.80c)
Cy = VIp/(11+I2) I; (1 - e“). (9.80d)

The combination of equations (9.79) and (9.80) completely determines the out-

put amplitudes T(z) and V(z). Combination of the two equations yields

T(z) e&#/2 = V1,/(y4I) [ ly (1 - 9) (Ip 41, etztin 2 4

(I) + Ip e'9) (Ip etl + 1/7 7 (9.81a)

-294-

V(z) e%/? = VIo/(1, +12) [ (In +e I) (n+l) ety? 4

I) (1 -e%9) (pe? 4 1)" 7. (9.81b)

From (9.81) the output intensities are calculated as

Pi(z) = e-% hy 1
Ii+I2 I, et!7 +],

[ 21? eT (1 - cos6) + (1h? + In? + 2lIncos6) + 2 Ip e*F#? x

{ (Ii - In) ( (1-cos®) cos[gz coso)) - (I; + Ip) sin@ sin[g coso z] } | (9.82a)

. I |
Px(z) = ea 2
1,412 I, elz + Ip

[ 21,7 eT (1 - cos®) + (1,2 + In7 + 2Iocos@) + 21, eT 22 x

{(I2 - hh) ((1-cos®) cos[gz cos6]) + (I; + Ip) sin® sin[g cos z}} ]. (9.82b)

To summarize the work leading up to (9.82), we use two inputs A(0) and B(0)

to write a dynamic grating in a photorefractive material. One beam is sud-
denly shifted in phase by 8, and the beams are now designated T and V.
Equations (9.82) represent the output intensities P,(z) = IT(z)|? and P,(z) =
| V(z) |? as a function of @ and 6. When @ = 0, (9.82) reduce to the simple beam

coupling equations (9.64).

In the special case 1, = 1, =I the transmitted intensities reduce to a par-

-295-

ticularly simple form:

Pi(z)/I = 1 - tanh{T'z/2]cos0 - SCE £980 2) cing (9.83a)
cosh[T-z/2]

P2(z)/I

1 + tanh{I'z/2]cos0 + SIMS £089 2) cing, (9.83b)
cosh[Tz/2]

The results for the cases 6=0 and 1/2 also assume a particularly simple

form for arbitrary I, and 1,16. For 6 = 0 we obtain

Pi(z) = | t - sin’[sz]|

+ I> S sin’[sz] - * I,I2 sin[2sz] sin@

- KA Inn sin?[sz] cos (9.84a)

Po(z)= [1 as sin’[sz]} + i; & sin*[sz] + Ip sin[2sz] sin®
S S

+ KA V1yIo sin*[sz] cos® (9.84b)

where s* = x* + A*/4, and k=g A,A,'/I,and A =(g/I,)(1,(0)-1,(0)).. When 6

= 1/2 the result is

Pi(z) = I, [cos?[y-8] + C?sin*[y-8] + C sin[2(y-5] cos6] (9.85a)

Po(z)

1, [C?cos?[y-8] + sin*[y-8] - C sin[2(y-5)]cos6] (9.85b)
where C= 1,(0)/1,(0), y= Tan“! [¢ e%4) and5= Tan“}[C}.
The other special case considered is diffraction of a single incident

beam off the fixed grating. Here, we take V(0) = 0 and T(0) as before. In this

case we obtain, after following the same steps as leading to (9.80)

-296-

C3 = vio/dith) I) = -Cq. (9.86c,d)

It can be shown easily that a phase shift @ of the incident beam does not

change the diffracted intensities, it merely shifts the diffracted beam by 8.

This, of course, is in stark contrast to the case of beam coupling where the
magnitude and sign of the power transfer depend strongly on a beam phase

shift.

Combining (9.79) with the coefficients (9.86) yield the transmitted and

diffracted intensities. They are

[13 elz417 +21; hel? cos g cos z]|

Pi(z) = e-% h (9.87a)
I,t+h the!

P,(z) = e-0 Gb [el 4+1-2e!% cos g cosd z]| (9.87b)
If+hL IL+hel

Again, for the condition I, =I, =I the transmitted and diffracted intensities

reduce to a simple form

Pi(zyl = & 7 (1+ cos[g cos z]/cosh[g sind z]) (9.88a)
P,(z)/I = a (1- cos[g cos z]/cosh[g sind z]). (9.88b)

To recap, the analysis leading to (9.81) entails writing a dynamic grating
in a photorefractive material using two inputs A(0) and B(0). B(0) is suddenly

blocked so that only A(0) illuminates the crystal. (These beams are renamed

-297-

T(O) and V(0). ) Equation (9.87) represents the transmitted intensity P,(z) =
IT(z)|* and_ the diffracted intensity P,(z) = | V(z)|? as a function of 6. This
quantity is fundamental to many experiments in beam coupling; for example,

one figure of merit of a photorefractive material is its diffraction efficiency in

a given length.
9.5 Summary

A general mathematical formalism was developed to solve certain lin-
ear first-order coupled differential equations which commonly occur in opti-
cal beam coupling analyses. The formalism entailed the reduction of first-
order coupled equations to two second-order equations - either second Péschl-
Teller equations or symmetric top equations - by application of the “corotating
transform.” Reductions to other types of solvable equations might also occur,
although none were encountered in the preceding studies. The formalism
was applied to beam coupling and diffraction off dynamically written fixed
holographic gratings in both the copropagating and the counterpropagating

geometries. The spectral response of fixed holographic interference filters was
examined. A numerical study of the case a # 0 was also described and com-

pared to the lossless analytic solution. The coupling of beams phase shifted

with respect to a dynamically written grating was discussed.

The mathematical tools derived in this chapter should be applicable to
a variety of problems in the study of beam coupling. Several examples which
were not addressed in this chapter are the response of counterpropagating

beams to phase fluctuations and the frequency response of copropagating

-298-

beams with a grating written at a reference wavelength. In chapter five the
methods described above are applied to solve the diffraction in a photorefrac-

tive material material with an electric field controlled coupling constant.

-299-

References for chapter nine

[1] S. Wolfram, Mathematica , 2nd ed. , pp. 684ff, Addison-Wesley, Redwood
City, California, (1991).

[2] R. J. Burden, J. D. Faires, A. C. Reynolds, Numerical Analysis , 2nd ed.,
chapter 5, Prindle, Weber, & Schmidt, Boston, Massachusetts, (1981).

[3] D. Zwillinger, Handbook of Differential Equations , Academic Press, New
York, New York, (1989). This is an excellent reference for solving differential

equations.

[4] A. O. Barut, A. Inomata, and R. Wilson, “Algebraic treatment of second
Péschl-Teller, Morse-Rosen, and Eckart equations,” J. Phys. A: Math. Gen. 20,
4083-4090 (1987).

[5] L. Infeld and T. E. Hull, “The factorization method,” Rev Mod Phys 23, 21-
68 (1951).

[6] A. Abramowitz and I. Stegun, Handbook of Mathematical Functions ,
chapter 15, Dover Publishing, New York, (1972).

[7] G. Rakuljic, A. Yariv, and V. Leyva, “High resolution volume holography
using orthogonal data storage,” Photorefractive Materials, Effects, and Devices

Conference of OSA, July 29-31, 1991, Beverley, MA, paper MD-3.

[8] G. Rakuljic, V. Leyva, and A. Yariv, “Optical data storage using orthogonal

wavelength multiplexed volume holograms,” Opt. Lett. Oct 15, 1992.

[9] V. Leyva, G. Rakuljic, and A. Yariv, “Volume holography using orthogo-

nal data storage approach,” OSA annual meeting Nov. 3-8, 1991, San Jose,

-300-

California, paper FU-7.

[10] G. Rakuljic, V. Leyva, K. Sayano, and A. Yariv, “Comparison of angle and
wavelength multiplexing in holographic data storage,” (invited paper) OSA

annual meeting Sept 20-25, 1992, Albuquerque, New Mexico, paper WE-2.

[11] J. O. White, S. Z. Kwong, M Cronin-Golomb, B. Fischer, and A. Yariv, in
Photorefractive Materials and their Applications I , Eds. P. Giinther and J. P.

Huignard, chapter 4, Springer-Verlag, Berlin, (1989).

[12] C. Guand P. Yeh, “Diffraction properties of fixed gratings in photorefrac-
tive media,” J. Opt. Soc. B 7, 2339-2346 (1990).

[13] M. Segev, A. Kewitsch, A. Yariv, and G. Rakuljic, “Self-enhanced diffrac-
tion from fixed photorefractive gratings during coherent reconstruction,”

Appl. Phys. Lett. , 62, to appear March 1, 1993.

[14] A. Yariv, Optical Electronics 4th ed. , p. 496, Saunders college publishing,
Philadelphia, (1991).

[15] J. Mathews and R. L. Walker, Mathematical Methods of Physics , chapter
1, Addison-Wesley, Redwood City, California, (1970).

[16] R. Hofmeister and A. Yariv, “Vibration detection using dynamic photore-
fractive gratings in KTN/KLTN crystals,” Appl. Phys. Lett. , 61, 2395-2398
(1992).

-301-

Chapter Ten

Summary and Future Directions

10.1 Summary

Both research and applications in the field of photorefractive materials
have been hindered by a severe dearth of high quality, readily obtainable ma-
terials. The photorefractive crystal growth effort at the California Institute of
Technology was initiated to respond to this need, and it was the aim of this
thesis to develop a new photorefractive material and explore applications par-

ticular to its characteristics.

Photorefractive potassium lithium tantalate niobate (KLTN) was
grown using the top seeded solution growth method. The composition of the
material was tailored to yield a phase transition at a predetermined tempera-
ture within the range 180°K < T.<310°K. Various transition metal dopants,
most notably copper and vanadium, were admixed in the flux to provide the
photorefractive donor species. These photorefractive crystals were of exem-
plary optical quality over the composition range described, and ranged in size

from 4-20gm.

The photorefractive properties of the KLTNs were investigated in the

paraelectric (centrosymmetric) phase as functions of a uniform applied elec-

-302-
tric field. In this high temperature phase, the linear electro-optic effect is for-
bidden by the symmetry of the material, and the photorefractive response is
described by the Kerr effect. It was shown that the applied electric field in-
duces effective linear electro-optic coefficients that are proportional to the ap-
plied field. This property was harnessed to demonstrate electric field control

of two-beam coupling and diffraction in the material.

When no electric field was applied, however, the photorefractive re-
sponse did not vanish as expected by the conventional electro-optic theory.
This zero electric field photorefractive (ZEFPR) effect was proved to result
from local Jahn-Teller distortions of ions whose concentration in the material
was spatially modulated by an incident intensity pattern. The distortions cre-
ated a photoelastic strain index grating that was identically in-phase with the
intensity pattern. The ZEFPR effect has not been previously reported, proba-
bly because it is weaker than the conventional effect and would be obscured in

any material in which the conventional effect is not forbidden.

The identically in-phase condition of the ZEFPR index grating lends
unique properties to this effect. No two-beam intensity coupling occurs with
such a grating, however, if the phase of the grating is suddenly and artificially
shifted, beam coupling occurs in linear proportion to the amount of phase
shift. This allows the implementation of numerous self aligning (robust) and
yet extremely sensitive linear phase-to-intensity transducers. An all-optical
microphone was demonstrated with this principle and several other devices

were proposed.

A mathematical theory of the response of beams incident on a photore-

-303-
fractive grating was developed to explain the results observed in the ZEFPR
materials. The mathematical machinery was used to prove the linear vibra-
tion response of the ZEFPR material and the quadratic response of a diffusion
limited photorefractive material. Subsequently it was expanded to determine
the response of a material with arbitrary phase shift. With these techniques,
the photorefractive phase and coupling constant of photorefractive materials
could be measured. The diffraction properties of paraelectric KLTN were also
described using the new mathematical tools. Finally, the frequency response
of reflection of an incident beam off a fixed holographic grating was derived
for the case that the grating was written with counterpropagating beams. The
mathematical framework established in solving these beam coupling prob-
lems was presented in a sufficiently general manner to be easily applicable to

other problems.
10.2 Future Directions

The most clear-cut projects for future work are in crystal growth.
Extensive effort has been expended in the design and construction of a new
crystal growth system which incorporates numerous improvements over the
old system. The new system will have a more uniform temperature profile
and better insulating components. A ceramic spindle has been added to allow
crucible rotation and thus a more homogeneous mixing of the flux during
growth. The process parameters will all be computer controlled to provide
greater flexibility of the growth variables. The improved system will be used
to grow KLTNs with previously untried dopants including certain lan-
thanides, as well as an entirely new class of ferroelectric oxide material which

will require the higher operating temperature capacity of this new system.

-304-

Substantial research remains to be done to characterize the photorefrac-
tive response of paraelectric centrosymmetric materials near their phase tran-
sition. Hologram fixing has been observed in KLTN near the phase transition
but has not yet been quantified sufficiently to merit disclosure. It also re-
mains to be seen whether such fixed gratings can be obtained at room temper-
ature. This will likely depend on the photorefractive dopant used. Thermal
fixing is being investigated as an alternate means of permanent data storage.
This may allow the electric field controlled readout of fixed holograms when

the crystal is operated near its para/ferroelectric phase transition.

Finally, the applications of the ZEFPR effect have been demonstrated
only in an inchoate form, or not at all. The fundamental limits of sensitivity
of the all-optical ZEFPR microphone have not yet been explored.
Development of the microphone should be pursued to the prototype stage.
Other applications of the ZEFPR effect, the self-aligning data link, for exam-

ple, should be demonstrated in the lab, to prove their feasibility.