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Aperiodic Structures in Optics and Integrated Optics and the Transverse Bragg Reflector Laser
Citation
Shellan, Jeffrey B.
(1978)
Aperiodic Structures in Optics and Integrated Optics and the Transverse Bragg Reflector Laser.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/662b-ca80.
Abstract
The first part of this work describes the uses of aperiodic
structures in optics and integrated optics. In particular, devices
are designed, fabricated, tested and analyzed which make use of a
chirped grating corrugation on the surface of a dielectric waveguide.
These structures can be used as input-output couplers, multiplexers
and demultiplexers, and broad band filters.
Next, a theoretical analysis is made of the effects of a random
statistical variation in the thicknesses of layers in a dielectric
mirror on its reflectivity properties. Unlike the intentional
aperiodicity introduced in the chirped gratings, the aperiodicity in
the Bragg reflector mirrors is unintentional and is present to some
extent in all devices made. The analysis involved in studying these
problems relies heavily on the coupled mode formalism. The results
are compared with computer experiments, as well as tests of actual
mirrors.
The second part of this work describes a novel method for confining
light in the transverse direction in an injection laser. These
so-called transverse Bragg reflector lasers confine light normal to
the junction plane in the active region, through reflection from an
adjacent layered medium. Thus, in principle, it is possible to guide
light in a dielectric layer whose index is lower than that of the surrounding
material. The design, theory and testing of these diode
lasers are discussed.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics)
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:
28 February 1978
Record Number:
CaltechTHESIS:07212014-142218548
Persistent URL:
DOI:
10.7907/662b-ca80
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No commercial reproduction, distribution, display or performance rights in this work are provided.
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8576
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APERIODIC STRUCTURES IN OPTICS AND
INTEGRATED OPTICS
AND
THE TRANSVERSE BRAGG REFLECTOR LASER
Thes is by
Jeffrey B. Shellan
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1978
(Submitted February 28, 1978)
-ii-
This work is dedicated to the memory of my
mother, who, after all, started me on "my numbers."
-iii-
ACKNOWLEDGMENT
I would like to express my sincere appreciation for the fine
guidance I received from my thesis advisor, Professor Amnon Yariv.
His support, counseling and expertise helped make the last three
years at Caltech enjoyable for me.
His fine instruction in quantum
electronics, which I received as a senior, was a major factor in my
choosing his group to work in two years later.
During my three years as a graduate student, I have had the
privilege of working with a large number of people in our group.
would especially like to thank C. S. Hong, Pochi Yeh, Willie Ng and
Pinchas Agmon for their cooperation and guidance during various
phases of my work.
The advice of Alexis Livanos of Hughes Research
Center and Al Cho of Bell Labs was also invaluable.
It is a pleasure to acknowledge the skillful technical assistance
of Desmond Armstrong, as well as the fine typing job of Ruth Stratton
and Verona Carpenter in preparing the manuscript.
Generous financial support from the John and Fannie Hertz Foundation is especially appreciated, and allowed me total freedom in completing my Ph.D.
Finally, I wish to thank my parents for their financial support
during my undergraduate years at the California Institute of Technology.
More important than financial support, however, was the constant moral
support and encouragement I received from my father over the last
seven years.
His character and example will always be admired.
-ivABSTRACT
The first part of this work describes the uses of aperiodic
struct ure s in optics and integrated optics.
In particular, devices
are designed, fabricated, tested and analyzed which make use of a
chirped grating corrugation on the surface of a dielectric waveguide.
These structures can be used as input-output courlers, multiplexers
and demultiplexers, and broad band filters.
Next, a theoretical analysis is made of the effects of a random
statis tical variation in the thicknesses of layers in a dielectric
mirror on its reflectivity properties.
Unlike the in tentiona l
aperiodicity introduced in the chirped gratings, the aperiodicity in
the Bragg reflector mirrors is unintentional and is present to some
extent in all devices made.
The analysis involved in studyin g these
problems relies heavily on the coupled mode fo rmali sm.
The results
are compared with computer experiments , as we ll as tests of actual
mirrors.
The second part of this work de scribes a novel method for confining light in the transverse direction in an injection lase r.
These
so-ca ll ed transverse Bragg reflector lasers confine light normal to
the junction plane in the active region, throuqh reflection from an
adjacent layered medium.
Thus, in principle, it is possible to guide
light in a dielectric layer whose index i s lower than that of the surrounding material .
The design, theory and testing of these di ode
1asers are di scussed.
-v-
TABLE OF CONTENTS
Part I
APERIODIC STRUCTURES IN OPTICS AND INTEGRATED OPTICS
Chapter 1.
General Introduction
Integrated Optics and Optical Communication-Introduction
Periodic Structures
1.3 Aperiodic Structures and Outline of Thesis Part I
1.1
1.2
Chapter
References
15
Appendix 1-A - Solution to Electromagnetic Propagation
Inside a Dielectric Slab Waveguide
17
Appendix 1-B - Coupled Mode Theory
22
Broad Band Filters
40
2.1
Introduction
40
2.2
Theory of Broad Band Filters
40
2.3
Fabrication of the Broad Band Filter
51
2.4
Testing and Evaluation
52
2.5
Conclusion
55
Chapter 2.
Chapter 2 References
Appendix 2-A - Exact Solution of Coupled Mode Equations
for Broad-Band Filters
Appendix 2-B - Linear and Enhancea Se11sitiv1ties of the
Shiolev AZ-l350B Photoresist
59
60
67
Chirped Gratings Used as Input-Output Couplers
76
3.1
Introduction
76
3.2
Grating Fabrication Considerations
76
3.3
Wave~uide
Chapter 3.
Coupling
84
-vi89
3. 4 Calculation of Power Output Distribution for
Chirped Gratings
3.5 Experimental Results
105
3. 6 Con c 1us ion
106
Chapter 3 References
109
Statistical Analysis of Braqq Reflectors
110
4. 1
Introduction
110
4.2
Low Reflectivity Limit
11 2
4.3
Coupled Mode Theory
112
4.4
126
4.6
Connection between E2I 2 , ~ . and a(O). b(O),
a 2 of the slab reflector considered in the low
reflectivity limit section
Computer Results
A Phenomenological Expression for p(I PI 2 )
4.7
Experimental Results
148
4.8
Conclusion
151
Appendix 4.A
152
Appendix 4.8
154
Appendix 4.C
156
Appendix 4.0 - Computer Program
157
Appendix 4.E
158
Appendix 4.F
161
Chapter 4 References
164
Additional Uses of Aperiodic Structures
166
Chapter 4.
4.5
Chapter 5.
133
145
5.1 Introduction
166
5.2 Effects of a Tapered Coupling Coefficient
166
5 .3 Perturbation Solutions to Aperiodic Braqg
Reflectors
171
-vii5.4
Pulse Compression
178
5.5
Use of Aperiodic Dielectric Mirrors to Reduce the
Electric Field Intensity
Conclusion
182
Chapter 5 References
188
5.6
187
Part II
THE TRANSVERSE BRAGG REFLECTOR LASER
Introduction
190
Chapter 1 References
192
Theory of Bragg Waveguides
193
2.1
Introduction
193
2.2
Design of Structures
193
2.3
Calculation of the Loss Constant
200
2.4
Conclusion
218
Chapter 2 References
219
Fabrication and Experimental Results
220
3.1
Introduction
220
3.2
Fabrication and Testing
220
Chapter 1.
Chapter 2.
Chapter 3.
3.3 Conclusion
Chapter 3 References
232
233
-1-
PART
APERIODIC STRUCTURES IN OPTICS AND
INTEGRATED OPTICS
-2-
Chapter 1
GENERAL INTRODUCTION
1.1
Integrated Optics and Optical Communication--Introduction
The invention of the laser almost twenty years ano brouaht with it
the possibility of optical communication.
problems to overcome.
There were, however, many
The optical communication systems, as envisioned
at that time, consisted of bulky, heavy components requirinq careful
alignment and protection from temperature fluctuations and vibrations.
The most difficult problem to overcome was that of the high attenuation
of light propagatin9 in the atmosphere or in existing alass fibers.
The
major breakthrough came with the chemical vapor deposition techniques that
enabled Corning to produce fibers with losses as low as l or 2 dB/km.
The alignment and vibration problems have been largely overcome by the
techniq ue of integrated optics; that is, fabricatinq optical components
on a small chip where liqht is wave9uided in a thin film from one fixed
optical component to the next.
There are a number of advantages in optical communication. There are
larqe savin9s in size, weight, power consumption and cost; silicon is much
liqhter and cheaper than copper. Furthermo re, no around loop problem exists and the system is free from electromaqnetic interference.
These
advantages are important for applications ran9ing from aeronautics and
avionics to the telephone industry .
The greatest advantage of optical communication, however, is the
extremely large bandwidth and high data rates possible.
The evolution
of increasing carrier frequency started with AM radio in the kHz range,
proceeded to FM transmission in the MHz range, and on to microwaves in
the GHz ranqe.
The carrier frequency available if optical methods are
-3used is almost one million GHz.
That is, if methods can be found to
modulate, transmit and process the optical signals, a factor of almost
one million can be gained in the data rate over that of microwave communication.
There are many problems to be overcome if such high data rates are
to be realized.
One of the principal problems is that of pulse spreading
or broadening in a fiber, which is a result of multimode group delay or
material dispersion. Figure 1.1 shows three kinds of fibers which can be
used. 1 Figure l.lb is a multimode fiber and a pulse will typically spread
at a rate of 50 nsec/km as a result of multimode group delay.
This sets
an upper limit of about 20 MHz in pulse rate for a one kilometer long
fiber.
If the index of the multimode fiber is graded as in Figure l.lc,
the pulse rate can be increased to about 2 GHz.
For maximum data rates,
however, a single mode fiber as in Figure l.la must be used.
The main
source of pulse spreading in this fiber is material dispersion, and for a
single mode injection laser with a frequency width of 1~, 100 GHz data
rates should be possible over a one kilometer long fiber.
The li ght source for the system should have adequate output power,
long lifetime, high efficiency, ease of modulation, low cost and fiber
compatibility.
The double heterostructure diode laser is superior to the
LED and solid state laser in these requirements.
The rise time of the
injection laser is a fraction of a nanosecond (due to the finite carrier
recombination time), thus making direct modulation of up to several
hundred megahertz possible.
This may be sufficient for multimode fibers,
but in order to take advantage of the small pulse spread in single mode
fibers, external modulators may be employed .
(b)
-a
{c)
tiber, (b) multimode step index fiber, and (c) multimode graded index fiber.
Fig. 1 · 1 Cross-sectional sketches of principal fiber tyres and their refractive index profiles for (a) single-mode step index
·-a
{a}
n,
n2
--•..--,· n2
·. :·.:~~t\:~:6~~?:?>
. . ......,.~1·"1'"''-' ...
. • ·{l~~~\1;,:.•. .
.. · ..
.Po
-5-
One of the best materials to use for modulation, switchinq and
guiding of light is LiNb0 3 .
It has a high electrooptic coefficient. and
directional couplers can be used to efficiently switch light from one
channel to another.
Guides operating near cutoff can be made to guide
light or radiate liqht, depending on whether an external voltage is
applied.
Lens, prisms, beam splitters and gratings can all be fabri-
cated on the LiNb0 3 processing chip.
A large part of this thesis inves-
tigates the uses of gratings and in particular chirped 9ratings in the
processing of optical signals.
1.2
Periodic Structures
The study of periodic structures in nature. as well as man-made
structures, has occupied scientists since before the time of Lord
Rayleigh in the 19th century. 2 The interaction of various kinds of
waves, from sound to electromagnetic to quantum mechanical, with periodic structures plays an important part in our understanding of nature.
The special properties of periodically stratified media have been used
to make devices ranging from electric filters to linear accelerators to
distributed feedback lasers.
Our study and understanding of crystals are largely based on the
interaction of these crystals with x-rays. In 1928 Bloch 3 generalized
the results of Floquet and formulated the basis of a theory of electrons
in crystals.
The mechanical and structural engineer must understand the properties of periodic structures if he is to understand the interaction of
a bridge or skyscraper with its surroundings. 4 • 5 Even the biologist
-6-
encounters layered media when he studies nature.
The cornea of a horse-
fly eye is coated with a periodic set of layers and the rhabdom of a
Buckeye butterfly eye or the rhabdom of a skipper eye has reflecting
filters. 6
However, it is the man-made device or technological application
of periodic structures which most interests an applied physicist or
engineer.
The material scientist may work with zeolite crystals and
7-ll
superlattices which can now be grown with molecular beam ep1taxy.
The high energy physicist will corrugate a linear accelerator in order
to slow the microwave to a velocity comparable to that of the particle
being accelerated. 12 In order to perform the complementary function and
remove enerQY from a beam of particles and convert it to electrical
energy, the electrical engineer uses the travelinq wave tube. 13 Electrical filters, pulse compressors, and antenna arrays are further applications of periodic structures in electrical engineering.
Perhaps no other field makes greater use of layered media than
optics where reflectors, filters, antireflection coatings, polarizers,
pulse compressors and beam splitters are used extensively . 14
In
integrated optics and integrated surface acoustics the distributed
feedback laser (DFB), 15 - 17 distributed Bragg reflector laser (DBR), 18 • 19
second harmonic generator, mode converter, 20 • 21 qrating coupler,22,23
def1ector,
24 25
transducer a nd mo du1a t or 21 •26 are a 11 f am1·1 1ar
devices.
There are several methods of analysis available for the propaqation of waves in periodically stratified media. They include the use of
27
the Floquet theorem,
Hill and Mathieu functions and differential equa6' 28 t he transfer matr1x
. and matrix multiplication, 14 , 29 and the
t 1ons,
coupled mode formalism. 30,3l
-7Because of the ease and versatility of the coupled mode equations,
these equations will be used extensively in this work.
They are easily
adaptable to the study of aperiodic structures and, although they do not
give exact solutions as do other methods, they give accurate closed form
solutions in many cases of practical interest.
An outline of this impor-
tant method is given in Appendix 1-B.
1.3
Aperiodic Structures and Outline of Thesis Part I
As can be seen from the previous survey, the field of periodic
structures has been studied extensively for over one hundred years, and
thousands of articles, books and theses have dealt with the topic.
By
contrast, aperiodic structures, or almost periodic structures, have been
largely ignored until quite recently.
One of the main reasons for the
lack of literature on the subject is due to the mathematical difficulty
in solving problems involving aperiodic devices.
Most of the previously
mentioned methods cannot be used, except perhaps if perturbation techniques
are employed.
It is the purpose of Part I of this thesis to study practi-
cal aperiodic devices in optics and integrated optics.
Aperiodicity in a
device can be of two varieties; intentional or unintentional.
In the
class of intentional aperiodicity, we will describe the use of chirped
gratings on the surface of dielectric waveguides.
A chirp is simply a
monotonic variation in the period of a grating.
The concept of chirping
is familiar to electrical engineers who have studied radar.
It was used
in Great Britain during World War II to measure weak reflected radar signals reflected from distant targets.
In Figure 1 .2a,b,c a chirped pulse is emitted, reflected from a
distant target and enters a dispersive element.
If the lower frequency
-8-
a)
----4·~
EMITTER
b)
RECEIVER
c)
OUTPUT
INPUT
DJSPERSIVE
ELEMENT
Fig. 1.2
a) Chirped radar pulse being emitted. b) Pulse after reflection
from tar~et. c) Pulse amplitude is increased after propagation
through a dispersive element.
-9-
longer wavelength part of the signal travels faster in the dispersive
element than the shorter wavelength, then the pulse is compressed and
the increased amplitude can be detected more easily.
Figure 1.3a shows light guided in a dielectric waveguide.
The
light is confined inside the guiding layer by total internal reflection
from the dielectric interfaces.
A summary of the analysis of this guid-
ing is presented in Appendix 1-A.
Figure 1 .3b shows a chirped orating
which has been etched on the air-quide interface of the structure.
The
possible effects of such a corrugation are shown in Figures 1 .4 and 1 .5.
In Figure 1.4a the light is coupled out of the auide and focused along
a line normal to the plane of the paper. 32 In Figure 1.4b the li CJht is
reflected straight back and remains inside the qu id e. 33,34
It should
be noted that an output coupler similar to that in Figure 1 .4a, but with
uniform gratinq, couples out a plane wave rather than a focused wave.
Similarly a reflector such as that shown in Figure 1 .4b, but with uniform corruoation, will also reflect liqht, but over a much narrower
bandwidth; that is, the use of chirped qratinq produces a broad band
filter and will reflect liqht over a wide frequency range, while the
uniform grating will only reflect a tiny frequency range proportional
to the reciprocal of the length of the corrugated region.
The design,
fabrication, and analysis of these two devices is presented in Chapters
2 and 3 .
Fiqure 1.5 indicates a third use of such a device.
Here the
normal to the qrating makes an angle a (a~ 45°) with respect
to the sides of the waveguide.
When light is coupled into the guide
parallel to the edge it is reflected in the plane of the guide as shown
in the figure.
Since the grating is chirped~ different wavelengths of
-10-
a)
b)
Fig. 1.3
a) Light quided inside a dielectric waveguide.
b) Chirped grating etched on the surface of a dielectric
waveguide.
-11-
a) Output Coupler
b)
Fig. l .4
Reflector or Broad Band Filter
a) Chirped grating used as an output coupler.
b) Chirped grating of smaller period being used as a broad
band filter.
-1 2-
(1)
><
(1)
0..
·..+->
:::1
,.:.:.'
,.-
(V)
-o
.. <
r:#""<
(V)
.,•..
r-
r-
0..
......
+->
(j
...r:::.
Vl
<44(1)
II
+>
0'>
(1)
(1)
ro
U')
........
.--
CT>
•r-
l.1..
-13-
light will be reflected from different locations along the grating;
short wavelengths are reflected from the left side where the grating
period is small, while longer wavelengths are reflected from the right
side where the grating period is longer. 35 The condition
A= 2neff Acos a is simply the Bragq condition and is familiar to those
who have studied x-ray diffraction.
plexer or demultiplexer.
The device thus acts as a multi-
If a multiplexed siqnal is coupled into the
left side of the quide, the various components, A1 , A2 , A3 , will separate.
Conversely, if several frequencies are coupled into the guide from
above at various positions, they can be combined and coupled into one
optical fiber joined to the left edge of the guide.
In Chapter 5 additional aperiodic structures are presented, including a review of the work of others, notably Streifer and co-workers,
Kogelnik, Kock, Cross and Apfel. Topics covered are the effects on the
gain in distributed feedback lasers of chirped gratin9 or, what is equivalent, the taperina of the thickness of the auidinq layer. Also
considered are the effects of taperinq qratinq depth (constant period)
and period variation in dielectric mirrors so as to reduce the electric
field intensity inside hioh power laser mirrors.
The second class of aperiodic structures consists of "unintentional"
period variations.
For example, ~dielectric layered medium made by
man is aperiodic.
It is not possible to manufacture perfect structures,
and the effect of these imperfections is analyzed in Chapter 4
The
coupled mode equations are used in this statistical analysis of Braqg reflectors and the results compared with a computer experiment in which
-14-
1500 different mirrors are analyzed and average properties determined.
As expected, the peak reflectivity decreases and the bandwidth widen s
for these imperfect structures. 36
-15Chapter 1 References
1.
H. Hodara, Fiber and Integrated Optics (Crane, Russak and Co., Inc .•
New York, 1977).
2.
L. Rayleigh, Phi 1. Maq. 24, 145 ( 1887).
3.
F. Bloch, Z. Phys. 52, 555 (1928).
4.
M. A. Heckel, J. Acoust. Soc. Amer. 1§_, 1335 (1964).
5.
D. J. Mead, Shock Vibr. Bull. 35, 45 (1966).
6.
An excellent and comprehensive review article on periodic structures
is found in the work of Charles Elachi: C. Elachi, IEEE Proc. 64,
1666 ( 1976).
7.
L. Esaki and L. L. Chang, Phys. Rev. Lett. 33, 495 (1974).
8.
R. Dingle, A. C. Gossard and W. Wiegmann, Phys. Rev. Lett. 1!· 1327
(1975).
9.
J. P. Vander Ziel et ~. Appl. Phys. Lett.~. 463 (1975).
10.
K. Fischer and W. M. Meier, Fortschr. Mineral 42, 50 (1965).
11.
W. M. Meier, SCI Monograph on Molecular Sieves~ (1968).
12.
P. M. Lapostolle, Linear Accelerators (North Holland Pub. Co.,
Amsterdam, 1970).
13.
J. R. Pierce, Traveling-Wave Tubes (Van Nostrand, New York, 1950).
14.
M. Born and E. Wolf, Principles of Optics 4th ed. (Peroamon Press,
New York, 1970).
15.
H. Kogelnik and C. V. Shank, Appl. Phys. Lett.~. 152 (1971).
16.
C. V. Shank, J. E. Bjorkholm and H. Kogelnik, Appl. Phys. Lett.~.
395 (1971).
17.
J. F. Bjorkholm and C. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
18.
F. K. Reinhart, R. A. Logan and C. V. Shank, Appl. Phys. Lett. £I,
45 (1975).
-1619.
H. C. Casey, S. Somekh and M. Ilegems, Appl. Phys.
(1975).
20.
L. Kuhn, P. F. Heidrich and E. G. Lean, Appl. Phys. Lett.~. 428
(1971).
21.
Y. Ohmachi, Electron. Lett.~. 539 (1973).
22.
T. Tamir and H. L. Bertoni, J. Opt. Soc. Am. £1, 1397 (1971).
23.
S. T. Peng, T. Tamir and H. L. Bertoni, Electron.
24.
L. Kuhn, M. L. Dakss, P. F. Heidrich and B. A. Scott, Appl. Phys.
Lett. lZ_, 265 (1970).
25.
M. Lurikkala and P. t1erilainen, Electron. Lett. lQ_, 80 (1974).
26.
J. N. Polky and J. H. Harris, Appl. Phys. Lett. ~. 307 ( 1972).
27.
C. Kitte l, Introduction to Solid State Physics 5th ed.
York, 19 76) .
28.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGrawHill, New York, 1953).
29.
P. Yeh, A. Yariv and C. S. Hong. J. Opt. Soc. Am. 67, 423 (1977).
30.
D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press,
New York, 1974).
31.
A. Yariv, Quantum Electronics 2d ed. (,John Hiley &
32.
A. Katzir, A. C. Livanos, J. B. Shellan and A. Yariv , IEEE J. Quant um
Electron. ~. 296 (1977).
33.
J. B. Shellan, C. S. Hong and A. Yariv, Opt. Comm. 23 , 398 (1977).
34.
C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv and A. Katzir,
Appl. Phys. Lett. ll• 276 (1977).
35.
A. C. Livanos, A. Katzir, A. Yariv and C. S. Hon g, Appl. Phys. Lett.
30, 51 9 ( 19 77) .
36.
J. B. Shellan, P. Agmon, P. Yeh and A. Yariv, J. Opt. Soc . Am., to
be published January 1978.
Lett. ~.
Lett.~.
142
150 (1973).
(\~iley,
New
-17-
Appendix 1-A
In this appendix we outline the solution for the electric field
and propaqation constant inside a dielectric slab waveguide.
We consid-
er only the case of TE waves (Ex = E2 = 0) for confined modes and air
radiation modes, since these solutions will be used elsewhere in this
thesis.
The case of TM waves, as well as substrate radiation modes and
leaky waves are solved by A. Yariv and D. Marcuse. 1 • 2
Consider the slab structure shown in Figure 1-A.l.
The field must
satisfy the wave equation
~;iE - ;- E
(1-A.l)
E(x) e i6z e-iwt
(1-A.2)
We assume it has the form
E( x,z,t )
Combininq equations (1-A.l) and (l-A.2) we find
(l-A.3)
k - w
- c
This equation must hold in all three regions of the guide.
If we first
consider the case of a confined wave we get
E(x) = A e-ox
= A[cos KX - -0K s1. n KX]
Kd] ey(x+d)
= A[cos Kd + §_sin
for x .;: 0
(l-A.4)
for 0 _:: X _:: -d
(l-A. 5)
for X ~ -d
(l-A.6)
-18-
Fig. 1-A.l
Geometry of dielectric slab waveguide
-19K = (nl2 k2 - 62)1/2
(62
n2 k2)l/2
2 k2)l/2
(62
n3
where
(l-A.7)
(l-A.8)
(l-A.9)
It is easily verified that these equations satisfy equation (1-A.3)
for the three regions and the form was chosen so that E(x) is continuous
across x = 0 and x = -d, and aE; ax is continuous across x = 0.
Requir-
ing that aE; ax also be continuous across x = -d, we can combine equations (5) and (6) and find an eigenvalue equation for the propagation
constant 6 ,
tan Kd = K(y+o)
K - yo
(1-A . 10)
For a guide which carries a total power P, where
00
p =
(E x H)
(1-A.ll)
dx
- 00
00
2w~
I EY(x) 1
0 -ooJ
dx
( 1-A. 11 )
we find a value for the remaining constant A
A2 =
4K2 WlJ 0 P
2+ o2J
IBI[d + ly + l][K
(l-A.l2)
Upon examining equations (l-A.4) through (l-A.9) we see that in
order to have confined modes we require
(l-A.l3)
-20If kn 3 < lsi
kn 2 , substrate radiation modes exist;
light is not confined but radiates into the substrate.
that is, the
For the case
o < lsi < kn 3
(l-A.l4)
air radiation modes will exist for which light will radiate both into the
substrate and superstrate (air).
We again use the Helmholtz equation qiven in equation (l-A.3) with S
in the range given in equation (l-A.l4).
The solution is
E(x) = G cos t.x + H sin t.x
for x ;:. 0
(l-A.l5)
= L cos ax + M sin ax
for 0 ~ x > -d
(1-A. 16)
= N cos p(x+d) + p sin p(x+d)
for X < -d
(l-A.l7)
k2 - S2)1/2
a = (nl2 k2 _ 62) 1/2
(l-A.l8)
p = (n22 k2 - s2) 112
(l-A.20)
!5. = (n2
(1-A. 19)
Again we match boundary conditions at x = 0 and x = -d and are
able to eliminate four of the coefficients in equations (15), (16) and
( 17) .
We find
E(x) = Cr[cos t.x + (~) Fi sin t.x]
for X ;:_ 0
(l-A.2l)
= Cr(cos ax + F.1 sin ax )
for 0 ;:_ X > -d
(l-A.2?)
= Cr[(cos ad- Fisin a d)cos p(x+d)
for X~ -d
(l-A.2 3)
+ ap (sin ad + F.1 cos ad) sin p ( x+d)
-21where Cr is again determined by the power carried by the mode, just as
in the guided mode case, and F.1 can be chosen arbitrarily. Expressions
for Cr are given in the main body of the thesis.
References for Appendix 1-A
l.
A. Yariv, Quantum Electronics, 2nd ed. (John Wiley and Sons, New
York, 1975).
2.
D. Marcuse, Theory of Dielectric Optical Haveguides (Academic Press,
New York, 1974).
-22Appendix 1-8
In this appendix we introduce the coupled mode equations in their
general form which will be used extensively in this thesis.
We start
with Maxwell's equations for a non-magnetic bulk material with no external charges or currents.
\] X
E = -lJO ()H
ff
(1-8.1)
\] X
H ~a ( c E)
(1-8.2)
v . H= 0
\]
. (c E) = 0
(1-8.3)
(1-8.4)
where
electric field
H = magnetic field
0 =free space magnetic permeability
permittivity of dielectric
After taking the curl of equation (1-B.l) and using equation
(1-8.2) we find
(1-8.5)
In arriving at (1-8.5) we have assumed c is constant in ti me and
If we next use
the time dependence of the electric field is e -i wt
equation (1-8.4) we find
v2E +V [E · "9~] + n2k2E = o
(1-8.6)
-23-
where n2k2-=~ 0 E:w2 , k = cw
Finally, we will consider, for simplicity, the case of the TE
wave, that is E = E y and an c = c (x) dependence.
shown in figures (l-B.2a) and (l-B.2b).
The geometry is
With these restrictions eq.
(1-8.6) reduces to the familiar Helmholtz equation.
(1-8.7)
n = n(x) = index of refraction
Next we take the electric field and index of refraction to have
the following form
E(x,z) = R(x)eiBz e
i(n/ A0 )x
io
+ S(x)e ~z e
x-
n (x) = n 2 + n 2 cos (2n x + ~(x))
Q.cE_ < 2n
dx
n 2 «
-i (n/A 0 )x
(1-8.8)
{1-8 . 9)
n 2
Since in this derivation we have taken a time dependence e -i wt
R(x) represents the amplitude of the forward traveling wave, while
S(x) is the amplitude of a wave traveling in the backward direction.
The form of the index of refraction indicates a stratified media with
variation in the x direction.
For ~ = 0, we have a periodic structure.
Before proceding, a further comment is due regarding the assumed
form of the solution taken in equation (1-8 .8 ).
From the Floquet theorem
-24-
we expect an E field of the following form for ¢(x)
0.
(l-8.10)
If this is substituted into the Helmholtz equation (l-8.7) and
terms of similar x dependence are equated we find
(l-8.11)
Furthermore, we are free to restrict -
x'If
~ K~
x- in order to
1T
obtain a unique solution for (l-8.10).
The sum in (l-8.10) represents
an infinite sum of forward and backward traveling waves (space harmonics)
and equation (l-8.11) indicates that Ep is only coupled to Ep-l and
Ep+l to first order in (n 1 ). The coupling between Ep and Ep_ 2 and
Ep+ 2 is proportional to (n 12 ) 2 and so on. Thus to first order in n1 ,
we need only consider a single interaction; that is, E interacts with
Ep+q and Ep-q only for q=l.
For coupling between a forward and reverse
traveling wave, this requires interaction between the smallest positive
2TI
2TI
value of (-- p + K) and the largest negative value of (x- (p-1) + K).
For K positive and thus p=O, interaction between the forward traveling
harmon1c
. e i ( K - 2n I fl.o) x
. e i KX an d th e reverse t rave l 1ng
space harmon1c
is dominant.
Thi s is the only interaction between a forward and reverse
traveling wave to first order in (n 2 ).
Using equation (l-8.11), taking p = - 1 and ignoring the noninteracting E_ 2 term, we find
-25k
E_1 =
n1
(1-8.12)
Similarly for p = o equation (1-8.11) becomes
2 2
k n1
E_l
(1-8.13)
For maximum coupling, the denominators in equations (1-8.12) and
(1-8.13) should both approach zero.
- K
or
This can only happen when
- n
- Ao
This condition results in maximum interaction between the forward
inx/A
-inx/ A
traveling wave - e
and the reverse traveling wave - e
and
is the reason for the assumed form of equation (l-8.8).
After combining equations {1-8 .7) and {1-8.8) we find
-26-
1T 2 s2 )(R(x) eiSz e i(1TX/A0 ) + S(x) ei Bz e -i( 1TX/A0 ) )
(k 2 n 2 ---o
A2
k2 n 2
i(2,./A )x + ¢(x))
-i(21Tx/A ) + ¢(x))
·s ei (TI/A 0 )x
21 (e
+ e
)(R(x)e, z
(1-B.l4)
d ~~ << 12
~~~and\ ::~H
A0 )~
We have assumed!
( n/ A0 )
2 (n/
and thus
, dx
neglected these terms in equation (1-B.4). An inspection of equation
i( 1TX/A)
-i(1TX/A0 )
We
(l-B.14) indicatesterms with x dependence e
and e
can thus separate equation (1-B.l4)into two equations, each equation containing only coherent terms. This is the coupled mode approximation.
~~- i 8 R = - n s ei ¢
(1-B.l5)
dS
dx
-- + i 8 S = -
n* Re - i ~~
-27-
(k n
with
2 - -n 2 - 8 2 )
!1.2
0 - _ _ _ __;0~--
( 1 -B. 16)
n - -i
(l-B.l7)
K ;;
These are the coupled mode equations.
The term o is often
referred to as the phase mismatch term and must be near zero for
effective interaction between the forward and backward traveling waves.
The quantity n is the coupling constant and depends on the amplitude
of the modulation of the refractive index.
If we make the substitution
R(x) _ R'(x) ei ox
(l-B,l8)
S(x) :: S'(x) e -i ox
the equations can be put in a more compact form which i s often easier
to use
(1-8.19)
dS'
_ n* R'
dX;;
e-i( ~ - 2ox)
-28-
This form also provides an obvious physical interpretation of lnl
lnl =
dS'/dx
R'
amplitude reflected/unit length
amplitude incident
From equation (l-B.l3) we have for n~ << n~
n 2
n(x) "' n + 2 ~
and
cos ( 2n x) (l-B. 20) (l-B. 21) /\.0 It is well known that light incident with electric field parallel t o (l-B.22) Where 8 is the angle of incidence and quantities with L and R For small discontinuities e(]uation (l-8 .22) can be replCice d 2n cos e 2n cos 28 (l- B. 23) Snell's law ( n sin e = constant) was used for the ri ght side of - 29- dn ( x) d If this is summed over one period, with the phase factor included, we find an expression for the amplitude reflected/unit length 2n cos 2e dx 1r n12 In arriving at equation (1-8.24) we have taken the x component of lnl = -30- cos 8 = kx k n~ 2 kn 0 8 = (k~- A.~) 27T Fiq. 1-B.l Wavevector geometry and relationships -31- The case of finding lnl for a slab structure with actual dielec- tric discontinuities is straightforward. (See Fi,ure l-8.2b.) If the reflection from each dielectric interface is r 1 ~ then n 2 If we take nl = n0 + 2 ~ 0 and nR n0 nl (l-8.26) Taking the ratio of lnl given in equation (l-8.24) and {l-8.26) This was to be expected, since the slab structure can be Fourier Again the co upl ed mode equations provide an excel- lent means of analyzing the phenomenon. Expressions for the coupling -32 ·.·:·. J{ . :;:;.:-· X a) .··.. Index profile for a b) Index profile for a c) Cross section of a d) Cross sect ion of a Fig. l-B. 2 Slab and guiding structures for which t he coupl ed mode -33 - constant n have been qiven by several authors. A. Yariv 1 ha s found that for well confined modes and a "square wave" surface corru gation (~t ) [ l + __]_ -" (Va) + _ 3_ (l-8. 28) where = waveguide thickness .t is an integer given by ~ = .tn with s m the propagation 1\o constant of the m!h order mode (m = l ,2, ... ) has arrived at the fol- lowing equation for the contradirectional coupling constant for the m!h mode for a square wave surface perturbation . -34- (n - n ) Em(x) dx -a 2 2 sin(2 Km a) (l-B.29) t + l + 10 Km' om, Ym defined in Appendix 1-A. If a << t we can find an approximate expression for lnl to fir st order in a as t = 1 , 3 ,5 (l-B .30) In a similar manner, for a s inusoidal corrugation (Figure l-B.2d), has shown a Km (l-B.31) -35- Once again, if we take the ratio of nsinusoidal to nsq uare we find 7T (l-B.3~) = lf Streifer et al. 4 have worked out the calculation of lnl for an (l-8.33) -36- The solutions to the equations for the case ~(x) = 0 are S' ( x) -in* e+iox sinh[T(x - L)] -iox R'(x) = o ~inh(TL) +iT cosh(TL) {- 0 sinh[T(x- L)] (l-8.34) (l- 8 · 35 ) + i T cosh[T(x - L)H where K ::: - in = real number For waveguide structures with the periodicity along the length ~(x) being a very small perturbation, 6 equations(l-8. 15) cannot be They can, however, be transformed into a first order means. We first make a change in variables -37- z=O z=L ~A~ iuu--u ·-u--uu-Ut_j waveguide 1Ei(O)j 21 KL=I.O : jEi(L)j 2 eJP _::z_.. L/4 L/2 3L/4 KL = 4.0 L "z :IEi(L)j 2 Fig. 1-8.3 L/4 L/2 The behavior of IE;(z)l 3L/4 and 1Er(z) i in a periodi c waveguide with KL = 1.0 and 4.0 (68= 0). (From reference -38- (l-8.36) (l-8.37) After combining this with equation (l-8.5) we find dX (1 -8. 38) The boundary condition on equation (1-8.38) is p(L) = 0 and the refle ction coefficient p(O) is the quantity of interest. Some of the important Additional details can be found in the -39- References for Appendix 1-8 A. Yariv, Quantum Electronics, 2nd ed., (John Wiley and Sons, (2) H. Yen, Ph.D. Thesis, California Institute of Technology, (1976), p. 23. (3) D. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl. Phys. (4) W. Streifer, D. Scifres, and R. D. Burnham, IEEE J. Quantum Electron. (5) See appendix (6) W. Streifer, D. R. Scifres, and R. D. Burnham, J. Opt. Soc. Am. 66, (7) H. Kogelnik , Bell. Syst. Tech. J . .§2_, 109 (1976). (8) A. Yariv, IEEE J. Quantum Electron. 2_, 919 (1973). (9) H. Kogelnik, Bell Syst (10) H. Kogelnik and c. v. Shank, J. Appl. Phys. 43, 2327 (1972). (ll) D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969). (12) D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, Tech. J. 48, 2909 (1969). New York , l 974 ) . J. B. Shellan, P. Agmon, P. Yeh, and A. Yariv, (to be published Jan. 1978 (14) A. Katzir, A. C. Livanos, J. B. Shellan and A. Yariv, IEEE J. -40Chapter 2 Introduction chirped corrugation with periods approximately 2000~ are studied. The period of the grating is such that guided light is retroreflected in the in the grating has the effect of reflecting a broad band of fre~uencies. More exact results follow if asymptotic expansion of the parabolic cylinder functions are used. Theory of Broad-Band Filters be represented by E(z) = R(z)e-i(n/A(O))z + S{z)ei(n/A(O))z ( 2. 1 ) Rand S are the complex amplitudes of the forward and backward traveling If -41- the guide is multimode we assume that the corrugation is such that the ~~- i oR- - nSe+iyz dS + i oS = - n*Re-i yz (2.2) (2.3a) n- iK, K is real the coupling constant which depends on the amplitude of the grating perturbation. The period of the grating can be approximated by _ 2TT (2.4) i\fZT - i\TOT - 2y z where A is the grating period and y is the chirp factor of the grating. B0 is the propagation con- stant of the unperturbed guide. In uniform grating o must be small if we are to have substantial interaction between the forward and backward This is not necessary for chirped gratings. For large chirps we expect the incident wave to be coupled into the backward running wave for a large range of wavelengths and for many cases SettingS= 0 in equation (2.2) qives R = e+i oz (unit intensity). We now replace Ron the right side of equation Oa) with e+i oz and letS' ~ Seioz . This results in (2.3b) -42Before integrating this equation, we will modify it to include attenuation by replacing o with o0 + iz , where% is the attenuation factor for the wave amplitude. 1 = n*e dz +2i ( 0 + i ~) z (2.3c) We take our grating region between z = 0 and z = L and use the boundary I ( 0) = n*{z )e -i Yz 60 I ( 0) (2.5) dz 60 -('1;-- 2 e+2i( o 0 + i~)z e -a . 2 (2.6) 60 t~ost of the contribution will come from a region of width - - 1- l"Y 2 2 centered about u = 0. We thus expand e-au as l -au+ a u and then I ( 0) _ Reflectivity -2az8 (1 + ~) l6y (2. 7) for 0 < z8 < L -43- In almost all cases of interest a 2 << 1 and that factor can be dropped in equation (2. 7). Alternatively we may write equation (2.7) as: Refl ecti vi ty e -2az o (2.8) where Note that z 0 < ZB for the case of losses (a > o) and zo > zB for the and are presented in Appendix 2-A. By matching the boundary conditions Reflectivity = 1 - e (2.9) This asymptotic expression is good for cases when the Bragg point is -2 az are included. equation (2.9) must be modified by adding the factor e "o n/ - N2 (2.10) -44- nc = index of guide cover A0 = ~: =wavelength of li ght in vacuum Weff = t + lqo + _1Po_ a = 350 A Ao = 6,000 A .75 ).l we get K ~ 2300 m- 1 . If we take a chirp of A 1960 to 2060 A over a 1 .n em l ength y = -45- From equation (2.9), reflectivity= . 19; equation (2.8) (a= 0) qives If we do not (2.11) where X f - L cos(cp + ~) cos( cp - -z) L cos (cp + ~) tan -1 ( 2fd) Equation (2.11) follows from the well known result that the grating 82 The quantities e, xt• zf, d and fare all defined in Figure 2.1 xt 2 which is the linear approximation. "' '-¢ ',~ p ( xf , zf ) ',"·, " - Record ing Plate ''1 '\ Fig . 2. 1 Recording arrangement and geometry for the fabrication of chirped Cyl indr ic al _j """ '\ Xylene +=> 0'\ -47- dR' - nS' e -i I¥ (2.13) dS' - n*R' e+'I¥ (2.14) ~(z)=J We will, for simplicity, neglect losses in the following derivation. After replacing R' with 1 in equation (2.14) we obtain S' (0)=- (2.15) n(z) eW(z) dz We can evaluate equation (2. 15) hy the method of stationary phase (z- z') 2 d 21!'(z') l!'(z) "' I¥ ( z s' ) + ---:::-....:;B;.__ di where (2.16) -48- The quantity zB is the new Bragg point and from equations (2 . 11) and 2'JT [, 2'JT [, /..So - ('IT- 9 2] 3/2 (zs - zf)2 (2.17) - sin -z) zf) xf After integrating equation (2.15) we get I ( = Reflectivity = 0) 2'TT K (zg) d21¥(zB) 'IT K ( Zg) = --~:..,..-/..-=-8o----8-2----.--3/-,-,;:2 ( 2 . 18) for 0 < ZB < L = 0 otherwise where Y = ~ AXf As pointed out in the last section, for effective conversion of At the point -49- we have the desired coupling, but at the points ~Cz) ~ B0 + kns If 2TI (2.20) then clearly zs > z8 for effective coupling, but if zs is too close ;y the point z8 . In fact at the point ~ ~ - v(_ .;y beyond z8, the reflected intensity is down to 10% of the value, far to the left of the Bragg 2TI( A(z ) Thus the condition for the radiation roodes not to interfere \'lith or This condition is easily satisfied by our gratings, but could present -50- Finally for the lossless case and for small reflections we The result is Reflectivity = 2!. (2. 22 ) p=O Typically the ratio of two successive terms in the sum in (2. 19) is where we have taken y ~ 1~2 and L ~10- 2 m as in our samples. Thus, -51- 2.3 Fabrication of Broa~-Band Filters layer of Corning 7059 glass on glass microscope slides. Two parts of Shipley AZ-1350 photoresist were diluted with one part thinner of resist 1700 A thick. AZ-1350 resist was used because of extensive studies which the author had conducted previously on its properties. An Ar laser (4579 A) was used, but in order to create the 2000 A period grating it was also necessary to i~~ ~ 3053 A. the converging beam was 102° and thus resulted in a grating whose period was 1950 A in the central portion. Using the results of Appendix 2-8, 6 the samples were exposej for one minute and developed for 7-~ with argon ions of energy 2 keV. For maximum efficiency -52- in grating transfer, it is necessary that the etch rates of the resist not be substantially higher than the glass underneath it. This in fact occurs at low ion beam energies and it was found that the Figures 2. 2 are SD1 photo- graphs of a typi ca 1 structure. Testing and Evaluation Light from a tunable dye laser (linewidth ~ 1 A) is coupled through a high By changing the frequency of the light output from the dye laser, the shifting Bragg point or point from which the light is reflected is readily evident. As mentioned earlier, in order to avoid excessive losses into the substrate, it was necessary to couple the Thus longer wavelengths penetrate further into the grating and undergo larger The loss factor a was determined -53- IL Fig. 2.2 .... SEf,1 photo~raph of typical vlavegu·ide and sudace con·ugatio n. Guide tlli ckncss is . 77 11 and conugati on depth is 350 A. Tunable Waveguide (After reference 14 . ) Prism Fig. 2.3 Schematic of filter evaluation setup. Reflected De tee tor (J1 +=> -55- experimentally and the observed reflectivity at A was multiplied by The measured reflectivities, after being corrected for waveguide losses, The spectral response for the three filters are plotted in Figure 2.4 with t1e response of the The broad-band filters with a typical response of 300 A have Both the general and analyzed this way. linear chirped gratings are For the case of the linear approximation, the results compare favorably with the more exact theory based on asymptotic expansions of the parabolic cylinder functions. For the case of a combination smooth taper and large chirp,the response depends If n 5 -1.51, TABLE 2.1 n 1 -1.54, lmm 4ooK 350 A Corrugation 0.80JJ.m 0.85JJ.m 0. 77 }J-m Wavegu1de 1.519 1.524 1.52 4 Effective index of Summary of data obtained from three grating filters. n0 - l Filter 3 a. 1955 A Un1 form IOmm 1925~ - 1975 K Chirped Filter 2 10 mm 1905 ~- 2005 A Ch1rped (L) (.,.\) F1lter I Length Period ~- 4~ 150 ~ 300 ~ Bandw1dth (After referencE' 14. ) 5946 1i 587oA-6o2oK 581 0 A- 6 I I 0 ~ Wavelength 80% 40% 18% ReflectiVIty 0'\ (Jl -57" 100 10 ·-> ...... 0::: 5800 5900 6000 6100 Wavelength (~\) Fig. 2.4 Reflectivity vs wavelength for three grating filters. Circle -58- higher order terms are kept inthe loss, the effect can be interpreted a!. a -59Chapter 2 1. References A. Yariv, Quantum Electronics, 2nd ed. (John Wiley and Sons, Inc., N.Y., 1975), p. 526. M. Matsuhara, K. 0. Hill, and A. Watanabe, Opt. Soc. of Amer. §5, 804 3. Robert B. Smith, J. Opt. Soc. Amer. 66,882 (1976). 4. D. C. Flanders, H. Kogelnik, R. V. Schmidt, and C. V. Shank, Appl 5. A. Katzir, A. C. Livanos, J. !3. Shellan, and A. Yariv, IEEE J. 6. A. C. Livanos, A. Katzir, J. B. Shellan, and A. Yariv, Appl. Optics 7. H. Garvin, Solid State Tech 8. P. G. Gloersen, J. Vac. Sci. Techno1. j_£, 28 (1975). 9. W. Laznovsky, Research Development 47 (August, 1975). 10. R. Ulrich and R. Torqe, Appl. Opt . ]1_, 2901 (1973). 11. R. Th. Kersten, Optica Acta~. 503 (1975). 12. G. B. Brandt, Appl. Opt. }i, 946 (1975). 13. J. B. She1lan, C.S. Hong and A. Yariv, Opt. Com. 21, 398 (19//). 14. C. S. Hong, J. B. Shellan, A. C. Livanos, A. Yariv and A. Katzir, (0 -60- Appendix 2-A EXACT SOLUTION OF COUPLED MODE EQUATIONS FOR BROAD-BAND FILTER (2-A. l) (2-A.2) (2-A. 3) = Se+i6z (2-A.4) S' (2-.11.5) (2-A.6) Equations (2-A,5) and (2-A.6) can then be combined. giving the defining (2-A. 7) -61- where x - z - z8 is rea 1 The solution to equation (2-A.?) is (2-A.8) where K = - -} + i -;:y , l.l = l/4 s· = -1 e i( yz~ - 2oz 8 ) -3/2 l x2 • i 2 + H·l(- .!__ 2 K)W (2- A.9) -62- After using the approximation Yx 2 +oo in equations (2-A.8) and (2-A. 10) 0) = Q(x) R1 (X) + P(x) * S 1 (x) (2-A.ll) I ( where P(x) ={ (2-/\. 12) Q(x) = -1 e { rn 2} 2 1 4 iyz~ (2 -A .l3) c J K20 .;y f( i 4y) After using the initial conditions R1 (0) = 1 and S 1 (L) = 0 TIK =e (2-A.l4) TIK Is I ( 0) I = reflectivity = 1 - e (2-A .15) -63- It should be pointed out that in order to use the asymptotic < L - 5K (2-A.l6) In other words, the Bragg point or region of maximum interaction The key result is Fiaures 2-A. 1, 2-A.2 and 2-A.3 are plots of the power reflected at various points along the guide for increasinq Because the para- bolic cylinder functions are difficult to work with directly, except in It should be noted from the fiqures that althouqh the reflectivity (at z = 0) -2 LL 10 Reflected power as a function of distance from Bragg point for a filter with small Z VALUE FROM POINT OF MAXIMUM COUPLING IN UNITS OF 1/JY (Z=Z 8 =~) -4 ~ 0.2-+ f- 0 0.4+ 0.... ~ 0.6+ 0::: -8 1.2 ~ 1.0 f- Fig. 2-A.l -10 ll: 7TI(2 REFLECTIVITY= 1-e-Y = .0841 ~ : .0280 0'1 ..j:> 7rl(2 LL 0:: 1- (Z=Z 8 = ~) Z VALUE FPO~ PO ~T OF MAX MUM COUPLING IN UNITS OF 1/./Y YL REFLECTIV ITY = 1-e-Y = .6232 = .3107 10 Fig. 2-A.2 Reflected power as a function of distance from Bragg point for a filter with moderate -10 K2 Q) U1 7TK2 -8 -6 -4 -2 (Z = Z8 =~) w 1.0 0:: Z VALUE FROM POINT OF MAXIMUM COUPLING IN UNITS OF 1/.fi YL = 8.751 REFLECTVITY = 1-e-y- = 1.000 : 5 . 223 _j LL Fig. 2-A.3 Reflected power as a function of distance from Bragg point for a filter with large -10 Ky f- 1.2 10 0'1 -67Appendix 2-B The properties of this positive acting photoresist exhibits strong nonlinearity, especially for etch depth ranging from 0.05 to .2 lJm, a different developer, namely the AZ-303A, used with the AZ-1350J Linearity and speed or sensitivity are always of practical interest. 7 • 8 The AZ-303A developer has not been used in conjunction with the AZ-13508 photoresist because of the unacceptable etch rate of unexposed AZ-1350J resist, namely 150-200~ per It is the purpose of this appendix to show that the AZ-303A developer can be used with the AZ-13508 photoresist, resulting in improved has shown that for a positive acting photoresist the fo 11 ~ing re 1 ati onshi p exists between etch depth 1\d and exposure E (in units of energy per unit area}: (2-B .1) -68where T is the development time in seconds, c is the exposure constant ~ ~r (2-8.2) TeE + r 2T In this work the parameters involved in equation (2-8.2) are determined for the Shipley AZ-13508 photoresist used with the AZ-303A developer. The samples used were N0-3010 microscope slides made by Clay Adams, which, cut in half, resulted in a size of 38 mm x 25 mm x 1 mm. The samples were cleaned according to the method presented in Ref. 6, The AZ-13508 photoresist was then deposited in a single .ayer, and after 30 seconds it was spun at 6000 next for 30 min at 125°C. The first experiment involved the determination of the etch For this purpose the samples were half immersed in the developer for the required time, rinsed with deionized water for 2 minutes, and then The step size was meas ured using a Sloan Dektat instrument and the results are shown in Figure In all the experiments a 6:1 dilution was used, since it -69- 0.3 DEVELOPER TEMPERATURE 25°C ::t. 0 . 2 r0.... ~ 0.1 (8: I) 10 20 30 40 50 60 DEVELOPMENT Tl ME (sec) Etch depths in ~m of unexposed AZ-13508 photoresist as a function of development time in seconds, for various dilution -70exhibited a linear behavior and an acceptable etch rate of r 2 = 35~ ±5~ The development time chosen for the subsequent experiment was 10 sec. This developer is free from trace-metallic elements and is commonly used in the fabrication of photoresist gratings in Figure 2-8.2 shows the unexposed etch rate as a function of development time in minutes for the manufacturer's In this case r thickness is small, if long exposure times are acceptable. and if The back surface of the sample was coated with black paint to avoid interference fringes. The intensity distribution of the laser was better than 5% across the surface of the sample. The photoresist was exposed for a given time, and then developed for ten seconds in the Figure 2-8.3 shows the thickness change as a function of exposure energy. It can be clearly seen that the behavior In contrast. the MF-312 developer gave very small thickness changes for In order to verify that the stylus of the instrument was not scratching the surface, the same samples were aluminized 0.2 MF -312 DEVELOPER TEMPERATURE 25°C 10 Fig. 2-8.2 Etch depth in ~m of unexposed AZ- 13508 photoresist as a function of DEVELOPMENT TIME (min) 0~----~----~----~----~----~ G o.1 0... J-- :t 0.3 '-J k/ /! 10 20 EXPOSURE E (mJ/cm 2 ) • AZ-303A (6: I) 30 Fig. 2-8 . 3 Thickness change L ~ in ~m of AZ-13508 photoresist as a function of exposure z~·05 (f) ~ .I <.9 "'0 :1.. E .2 '-I -73To demonstrate the feasibility of using AZ-1350B photoresist with Photoresist was spin-coated on samples at 3000 rpm, resulting in a resist thickness of about 3.1 ~m. The gratings were generated by exposing the samples to the sinusoidal The exposed samples were developed in AZ-303A developer, baked under Figure 2-8.4 shows the absolute efficiency of the gratings as a function of exposure It is clear that high efficiency 23% resulted by developing the samples for 10 seconds. To verify the theoretically predicted period and the peak to trough height, a scanning electron microscope was used . The period was measured to be ~ 35~/sec and significantly increases the sensitivity and linearity of 10 10 AZ-303A 20 30 40 (lOsee) DEVELOPER TEMPERATURE 25°C Fig. 2-B .4 Grating efficiency (absol ute) as a function of exposure E in mw/cm~ for 10 and 20 EXPOSURE E ( mJ/cm 2 ) ~~ OL_~~L---~~--~~----~~--~~--E3 r-- LL >~ 20 ...__. -~ 30 -....,J +:> -75Appendix 2-B References H. L. Garvin, E. Garmire, S. Somekh, H. Stoll and A. Yariv, Appl. 3. s. Austin, F. T. Stone, Appl. Opt . .!_?_, 1071 (1976). 4. R. A. Bartolini, Appl. Opt. _ll, 1275 (1972). 5. R. A. Bartolini, Appl. Opt. ]]_, 129 (1974). 6. 7. s. L. Norman, M. P. Singh, Appl. Opt. Ji. 818 (1975}. 8. F. J. Lop rest, E. A. Fitzgerald, Photogr. Sci . Eng.~. 260 (1971). 9. A. C. Livanos, A. Katzir and A. Yariv, accepted for publication in 2. Opt. Coi11Tlun. -76Chapter 3 CHIRPED GRATINGS USED AS INPUT-OUTPUT COUPLERS Introduction used as broad-band filters . The grating period had to be of the order of 2000~ for reflection. If, however, devices are made with longer periods,s ay With conventional,uniform grating output couplers the light is coupled out as a plane wave. If chirped grating is used, however, the light is coupled out as a converaing or diveroinq wave and can be Con- versely, external line or point sources of light may be coupled to waveguides more efficiently using chirped input couplers thanuniform gratin g. The relationships among key parameters such as period variations, F number of lens, length Next, the coordinates of the point where light coupled out of the guide is focused Finally a comprehensive theory is presented which relates power output to grating and guide parame ters . Grating Fabrication Considerations grating, except that now the xylene filled prism i s not needed, s ince t he -77- period is larger. Fiqure 3.1 is a schematic diagram of the setup used. The The interference pattern is recorded over a distance L on the recording plate. The converging wave is generated by a cylindrical lens of focal length f and xf = -L cos(¢+ ~) 2 s i n2 and = L cos(¢+ 2 sin 2¢ t) ( 3. 1 ) t) sin(¢ - t) (3.2) cos(¢ - where We note that in Eq. (3.1) xf is always negative, while zf can take negative or positive values depending on (3.3) where k = 2n/"A is the wave number for theincident field, A and a th e If we as s ume -78- Cylindrical Lens Fig. 3.1 Recording arrangement and geometry for the fabrication of (After reference 1.) -79that the transmission function of the recordin g medium t is proportional (3.4) The period A for this particular grating is given by: 1\ ( z) = - - - " " ' - - - - - - - z-z sin(0/2) + . f2 ~ v ( z-zf) +xf . l approx1mat1on xf 2 Eq.(3.4) and (3.5) reduce to: B G+ cos(-f-- z2 + (k sin(0/2) - kzf )z + (3.6) The corresponding expression for Eq . (3.5) is therefore (3.7) z-z sin(8/2) xf /\(z) depends on the F number of the lens (F- f/d), 0 the angle sub- tended by the collimated beam and the bisector of the converging beam is set at 60° and the grating has a total For various F numbers period vari ations from0.8 0.4 )lin are obtained . ]lnl The lower the F number the greater the period to -80- 1.0 _ _ _ _ _ 1_ _ _ _ - - , - - r----- 8 = G0° <: 0.4 0::: Q_ 0.2 0.0 L::,-----~,--- _ _ _ _ L 0.6 _ L 0.8 ---- J 1.0 z (em) Period variation as a function of the F number of the converging l.) The angle e = 60° is suitable for variations 0.8 lJffi to 0.4 pm over a distance of l em. (After reference -81- - -, --- -- -- - --,-" f- -------5.0 0:: _ _ _ ?..0 --- ------===~- -- l.~> 0.2 0.2 0/l O.G l__~ o.s j __ r.o z (em) Fiq. 3.3 Period variation as a function of F number. The angle o ~a s a value of 90° and the range of period vnriation is from .45 (After reference l.) -82- r- - - 1.0 0.8 ::j_ ::::: 0.6 0/l 0:: 0.2 0.2 _ _I ---- 0.4 l_ 0.6 L __ 0.8 .l 1.0 z (em) Fig . 3.4 Period variation as a function of o (the angle betvtecn the plane •.-Jave and The rc- (After reference 1 .) -83- 0.5 --~---- -- 1- f =1.33 ::i_ 0.11 -- --~ ..._... o::w - - -L =1.0 0.2 _ - - - - - '-="------ .L 02 .L _l 0.6 0.8 j_ l.O z (em) Period variation as a function of recording distance L. th~ s arne for a 11 curves; the line a rit.y of variation is seen to improve for large values of L. The The is 90 degrees. the F number is l. 33 and the vJave l cngth is .4579 lJnl. (After reference l.) - 84variation; higher F numbers result in smaller and more linear period variations. In Fig. 3. 3 the angle a has the value of goo . Here the maximum period variation is for an F=l lens and it extends front .45 ~m to a distance of l em. . 28~m It is noted that large values of o produce smaller period variations. particular point is illustrilted in Fig. 3.4 where an F= 1.33 lens was chosen and e was varied from 45 ° to 120° . Again the grating It is seen that with e = 120° the period varies only by 0.05 ~m . while for o = 45° the period variation is It should be noted that the beginning and end period is identical for all values of L. Again the F number is 1.33 and the angle o is goo . When the guided mode is propagating unperturbed in the waveguide its z dependence is given by -85- .~- L ---~ Fig. 3 .6 Geometry for a chirped grating etched on the top surface of a The substrate has an index n2 , and n3 is the index of refraction of air . A waveguide mode will focus at point P(xA,zA) depending on the chirp of the grating and the (After reference 1.) -862n ( 3. 8 ) B - /\TOT and at z = L Equations (3.8) and (3.9) simply express conservation of momentum Referrinq to Fiqure 3.6 we have k (z=O) tan 00 = kz(z-0) = tan eL = kz(z=L) = 2 2 2 Jk n - k (L) k - w The equations for ray l and ray 2 are = Z tan 80 X = (z-L) tan 8L The point of focus P(xA,zA) is thus found to be L tan eLw (3. 10) -87- and The focusing effect and especially the variation of the focus as = 1.565, n2 = 1.51, n3 = 1.0 and a waveguide t~ickncss of d = 1.35~m the eigenvalue equation forB was solved for wavelen gths ranging from 4500A to 6500A. Having thus determined B for the unperturbed waveguide we calculate k2 (0), kz(L) for various ranges of period variation. - 88- 10.0 • .6328 f-Lnl 8.0 i( ?.:9- ......................... . .. .······· ··· ·········· • ···················+ ,..-...E ...__... ~--- --K LI.J ...- ------ ;--/-- Chirp •I ~-------------L -?.0 -1.0 2.0 ------------ 0.0 - .33-.30/Lm 2.0 1.0 z (em) of the foci of various wavelengths for different chil~p s . at x = 0. The grating is located between z = 0 and z- 1 . 0 em The \'laveguide mode is traveling in the positive z direction. -89- 3.4 Calculation of Power Output Distribution for Chirped Gratings and some of their properties were discussed. To complete our theoretical discussion we present a calculation of the actual power radiated into air but the notation and method are simila r to Marcuse's. -ox for x > 0 (3. 12) = A[COS KX - -6K SlnKX for 0 >- X >- -d (3.13) = A[COSKd ~K sinKd]ey(x+d) for x < -d (3.14) ~y = Ae where - ( n, 2k2 -s2)1/2 (3.15) (s2-n/k2) l/2 (3.16) (S2-n 2k2) l/2 (3.17) 6 - -90- z=O x= 0 - - - - - - - - x=O----.~ n 1 Waveguide x=-d-------------n2 Substra1e x=-d-------------- (a) (b) Geometry for a dielectric waveguide. (After reference 1.) -91- where n1 , n2 , n3 are the indices of refraction of the waveguide, of the electric field. eiwt e-isz has been suppressed in Eqs. (3.12)-(3.14). the constants K, y, and It should be noted thi'IL the factor 6 can be determined by the eigenvalue equation: (3.18) The amplitude of the electtic field A is related to the power A2 = (3. 19) IBJ[d+l/y+l/o](K 2+o?) p = J l~y 1 dx 2w).l where P is the power carried by the mode, d is the thickness of the guide, For the region kn 3 < lsi 2 kn 2 the substrate modes exist, and finally in the region kn 3 the TC air modes oft~~ continuum occur. For the purposes of this discussion we consider the air mode since v.Je -92- Cr (cosax+F.sinox) Cr[(coso d - Fisinod)cosp(x+d) for x >- 0 (3.20) for 0 > x~-d (3.21) for x <- -d +~ ( 3. 23; (n 2k2_ 8 2) 112 (3.24) P = (n 2k2- S2 )l/2 (3 . 25) 0 = where Cr is again related to the power carried by the mode c2 4w)JOP lsi 6 l 62 (3.26) and Fi can be chosen arbitrarily. ClE Matching boundary condi- four constraints on the solution. A fifth constraint is obtained by the normalized power condition. Since F.1 coefficient. For the guided -93- waves, however, the field must decay toward zero as x ~ ±00 • These two added constraints give a total of seven constraints involving six An arbitrary continuum mode can then be given by a linear combination of the F1 and F2 type modes. Following the conventional procedure,F 1 and F2 are chosen so that the two radiation 2 2 + [(o -p ) 2 2 (3.27) where 00 r ,. . a "a* P o(p-p') = -oo Again the factor eiwte-iBz has been suppressed in equations (3.20)(3.22); in this work B is an inherently positive quantity. I: Cn(z){n+ k~ h(p,z)~(p)dp even (3.28) -94where~ are the discrete guided modes, given by equations (3. 12), (3. 13), (3. 14) for then values of B determined from the eigenvalue Similarly ~ a are the air modes qiven aqain by equa- tions (3.20), (3.21) and (3.22), where even and odd refer to t he choice The calcu- lation is simplified due to the orthogonality of the modes as a re sult < B < kn 2 , and for air modes 0 < B < kn 3 (B inherently positive) az The above differential equation is easily converted to an integral h( p,z) = Q( p) + R( p)ii Bz + 2~ 8 J [e2i B(z-l;;} - 1]H( p ,l;;}d~;; (3 . 29 ) -95- 00 H(p,z) = 2~:p [ ~Cn(z) J f*a(p)iln2fndx kn2 oo -oo dp'g(p ' ,z) J&*a(p)L\n 2~s(p')dx + J k(n~-n~)~ -00 oo dp'h( p ',z) J B.*a( p).6nt,a( p ')dx k(n~-n~)~ - 00 (3.30) where .6n 2 describes the deviation of the corrugated guide dielectric h(p,z) = Q(p) + R(p)e2iBz When this is multiplied by~, which contains a z dependence This, of course, was to be expected. To solve equation (3.29) we use the Born approximation. In other In the next step we assume that the perturbation of the gu i de from -96- (3.32) 28j = [R + -~ Je-2i Bsll(p r.)dr l / fBz 21!3 such that (3.33) ,, ~- (3.34) h = h+ + hRecalling equation (3.28) we note that the contribution to the total h(p,z)~}( p ,z) = h + e- ·s + [R+ 2 1;~ J (3.35) Then we can associate the h+ part of the wave with the amplitude of the l>)radiated = L (3.36) CX) -97- The term involving the integration with respect to x gives the fraction Furthermore, the boundary conditions require that 1 b ( 3. 37) (3.38) Using the above conditions and equations (3.32). (3. 12) and (3.20) we get h-(p,O) = k (n 2- n 2 ) 4iw~P where cj>+ = cj>+(R,L) = t ( z) AC (3.39) (3.40) i(B-B )z (3.41) -i(B+G 0 )z ( 3. 42) and cj>_ = cj>_(B,L) = f(z) e dz -98- Using equations (3. 19) and (3.26) we can calculate h+ (p,L), namely [(cosod- Fisinod) 2 + 0 2 (sinod + F;cosod) (3.43) where K , Y , 60 refer to the zero order mode solutions for equations (3.15), (3.16), (3.17), and (3.18). Simi 1arly (3.44) Finally, to calculate the fraction of the air mode radiated into F.2 --;;1 = _ _..::::___ _ ~-- 2 a2 2 where 82 (3.45) Y; = cos ad - F;sin oc (3.46) W; = ~ (sinod + F;cosod) (3.47) -99Now using (3.45), (3.44), and (3.43), equation (3.36) becomes 6P) P radiated into jA-_j2] u. .I i=l [(v.) +(w.) +u.]·[(v.) +(w 1. ) + ~P u1.] (48) {3.48) where = l+S!_F. 2 (3 .49) and Equation (3.48) shows the fractional power radiated per unit beta for Once the perturbation is given, then cp+ and cp_ can be calculated. (3.5 1) f ( z) -= a s i n ( az + Yi) Direct substitution into equations (3.41) and (3.42) and using the Y>O,a+B-8 0 < 0, anda+B- B+2YL (a) or Y < 0, a+f3 - S 0 > 0 , and a+f3 - S0 + 2YL < 0 (b) -100otherwise + = 0. Similarly, rra 2 (3.53) if y > 0, a-8 - 80 < 0, and a-8-8 0 + 2yl > 0 (a) or y < 0, a-8-8 0 > 0, and a-B-8 0 + 2YL < 0 (b) or y < 0 a+8+8 0 > 0, and a+B+B + 2yl < 0 (c) otherwise l =0 These equations, (3,52a,b,c) and (3.53a,b,c) give the range They can be interpreted simply as conservation of momentum equations for Umklapp processes. For forward scattering, an Umklapp process requires _ 0 + 2TT ( 3. 54) where A(z) is the period of the grating at point z, but 'i\('Z} = a ( a z + y z 2 ) = a + 2y z Thus, 2yz ( 3. 55) (3.56) Equation(3.56) must hold somewhere in the grating region (O Considering the separate cases y > 0 and y < 0, as well as the two sign possibilities in equation(3.56) results in the -101- -8 (3.57) The above equation results in (3.53a,b,c). = 1.55. The sub- strate index of refraction is n2 = l.52,and that of air is taken to be The film perturbation is of the form of equation (3.51) and a was chosen to be 1.505 100ft. Jhe figure illustrates the fractional power output in the air per unit B for various chirps. We see from the figure that the lower the chirp (curve4) the narrower the range of B distribution. In For hiqh chirp we have a wide range of B distribution extending over most of the The total power output radiated into air is the area under the curves. For Figure 3.9 it ranges Consider, for example, curve 4 of Figure 3.9, for which A ranoes from .35~ to .4 ~m. 8o _ 2n (3.58) where B' = z dependence of field after interaction with the qrating. These values are simply the -102- ___ ____ _ 0.75 Gro1ir HJ f 'c·r1od llJ I- <( oco (L (L 0.5 0 f3o =1.5051 ' 10 1 (rn 1) ~()_, ()_ o - 100/: o <( n? - 1.~)? 0::: LL n1 - 1 5 ~ > n3 = 1.0 - - j _____________ 0.25· ,--------~-- - - 2 - - -~--- <( (L ___ LL 0.00 ]_ - - __ It -- L - !_ _ _ _ _ __ 10 - {3 x 10 6 (rn- 1) Fiq. 3.9 52° Fraction of mode power radiated into air per unit B as a -103- range of permissible S values and mark the two sides of curve 4. hP/P is calcu- lated from the first section and then it is subtracted from the total P. This enables us to handle large power coupling and not be limited by the first Born approximation. This particular point is illustrated in The total power output radiated into air is greater than 45 percent, due to the larger perturbation. In Figure 3.10 the film thickness is 1.35 ~m and its index of refraction n = 1.565. The substrate The different curves represent the fractional power output per unit S for various wavelengths. It can be seen from tnis f1gure that different wavelengths radiate over different and non-overlapping -104- --,--- ----- 1- WovelcrlCJIIJ ow /H:3 80,um <( <(' 0:: 0:: a = 500A o_l 1- n2 = 1.51 L = 1.1 em 1- I - (_) <( 0:: Fig. 3. 10 Fraction of mode p01ver J'ad i atcd into a ir per unit !3 as a .3 3 to . 295 ~m and various Th e amplitude of the chirp i s set at sooR , and the corrugation extends over a l engt h L of 1.1 em . (After -1053.5 Experimental Results chapter, by sputterinq Corning 7059 qlass with a Technics MIM Model The refractive index of the sputtered glass was 1.565 and the film thickness of all samples, as measured After prebaking, the photoresist was exposed to the interference pattern of a collimated laser beam with As detailed above, such interference pattern gives rise to chirped gratings. The A = 4579~ line from an Ar+ laser was used under the following conditions e = 94.5, F = 1 .33, L = 1.2 em, Gratings of high efficiency were obtained using AZ 303 developer, and 10 sec development time. The gratings thus fabricated in the glass had a peak to trough height of -106In the focusing experiment light was coupled from an argon laser The li9ht entering the cor- rugated section was focused outside the waveguide. The position of the focal point (xf,zf) was measured experimentally for various lines of the The experimental points are shown in Figure 3.11, along with the theoretical predicted curve for this particular waveguide. The light intensity which was coupled out was measured for two cases: a) light going tnrough the corrugated region, and b) going through a neignboring uncorrugated region. light The ratio betweer~ the intensities in case (a) and case (b) was found to be 1:10. The fabrica- t1on and experimental ~ork was done by Alexis Livanos and A. Katzir. Conclusion have been studied. Expressions giving the qratinq period variation for various geometries and recording conditions have been presented. The thick- ness of this waveguide, and the chirp were chosen so as to focus the The theoretical calculations, which were verified experimentally, show that the focal point moves by The chirped grating structure therefore separates very we ll between propagating beams of different wavelengths, while focusin9 them outside the -107- lOr-------~------~------~--------~------~------~-. 4765A ------------~ --E 4965A 45 79 A . .....__ 4- 1.1 em - - - - - - - ' - - - - - - _j 0.2 0.4 0.6 0.8 1.0 1. 2 z (em) Fiq. 3,11 Experimental and theoretical results of the focusing of the The solid 1 inc rcpl~esents tile theoretical position of the focus as a func tion of wavelength. The large circles are the experimental points for these v~avelength s as measured with a two-dimensional translation probe. (After reference l .) -108- Finally, a coupled mode theory was presented which predicted the -109Chapter 3 References 1. A. Katzir, A. C. Livanos, J. B. Shellan and A. Yariv, IEEE J. 2. R. A. Bartolini, Appl. Opt . .J_l, 129 (1974). 3. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969). 4. D. Marcuse , Theory of Dielectric Optical Waveguides (Academic Press, 5. N. Streifer, R. D. Burnham and D. R. Scifres, IEEE J. Quantum Elect. -110Chapter 4 We now turn our attention to studying the effects of a random, statistical We will no lonqer be able to predict the reflection properties of any single sample, but only the Analytic expressions are obtained for ~P' and The results of the computer experiment are presented in the form of p(pp*), the probability distribution function of a statistical Bragq reflector. Finally, simple phenomenological expressions are presented for the reflectivity probability distribution. Introduction Extensive studies have been made of the reflection of light from Other proposals involve the use of these structures for phase matching in nonlinear optical applications 3 •4 • 5 and for obtaining optical birefringence -111cially made mirrors is typically 2% when monitored optically, and even It is the purpose of this paper to study the effect on reflectivity of a random fluctuation in layer thickness about an ideal thickness. There are many additional causes of a less Among them are absorption, index variations, and systematic errors in the manufacturing of reflectors. Next a perturbation solution to the coupled mode equations is presented which gi ves results for arbitrarily large reflectances. Finally, a computer study is presented which uses the formalism of the matrix and translation operator developed by Yeh, An analytic expression is then presented for p( IPI ) which aqrees well with the results from the computer experiment . -112- 4.2 Low Reflectivity Limit the limit of low reflectance. Assuming a constant incident wave of unit amplitude we obtain for the reflected wave: 2i(k 1xa 1+ k2 b1 ) + /i[k 1x(a 1+a 2+ ... aN)+ k2x(b 1+b 2+ · .. + bN)]] ( 4. 1 ) where k.lX -n. = 1,2, w is the radian frequency of light, cis the velocity of liqht, and n is the index of refraction in a layer of material 1, and n2 is the index N is the number of unit cells and the number of dielectric interfaces is 2N+l,with r representino the magnitude of index n 1 in the pth cell, and bp is the thickness of the layer of index n2 p = 1 ,N bp = b(o) + v a(o) ,b(o) = ideal thickness of layers (4.2) u ,v = random variables with assumed Gaussian n, Fig . 4.1 .-b, n2 n, b2 ~ 02 ..... n2 • • • • • • • • Geometry of reflection with n cells in low reflectivity case n, +-On-. n, n2 --' -114In the process of taking the ensemble average of r we use the following theorem: If G is a random Gaussian variable with average value zero and standard deviation aG, then the ensemble average of eiG = 2i[k 1x(a 1+···+ap) + k2x(b 1+· ··+bp)] a(o)+ k b(o)] 1x 2i[k 1x(u 1+. ·.+up)+ k2x(v 1+· · ·+vp)] (4.3) The ensemble average of this term is 2 2 2 where e2ip ki\ 2 2 e-2pk a k2 We thus find the expectation value of pis rl 22 {l 2 2 - e 2i (N+l )ki\ e -2(N+l )k a 2i ki\ -2k a 2ik 2x b(o) e- 2 2 (4.4) The magnitude of this quantity is plotted in Figure 4.2 for the as k b(o) = k a(o). It can be seen in Figure 4.2 that the nonzero 7T(1 -~ ) 7T(1 - ~) 7T (1 -t) kA =klx 7T 7T(I +t) 7T(I +ft) 7T(I+ ~) 0(0 ) + k2x b(O) 7T(I +*) 0 ~----~-----L----~----~------~----~-----L----~ 7T (1 -~ ) kCT =. I Fig. 4.2 Average reflectance as a function of layer standard deviation and shift from n:: u... _j 1- <{ II '-./ ko- = .0 5 kCT = 0 U1 __. 0:: lL _J )~ ~- 8 = 1.00 8: .01 8 = 1.05 8: 1,10 8: .50 Fig. 4.3 Reflectivity profile for a chirped slab structure FREQUENCY 0 ~------------~------------~----~~~--~~----------~----------~ 0.5 0 r-------------~------------~------~~L-~~~------.0 0.5 ot r-- r-- >- 0.5 .O r-------------.-------------~--------------.-------------.-------------, 0\ ...... ...... -117- it, as expected. For comparison Figure 4.3 is a plot of a chirped dielec- tric mirror containing a total of 51 layers of index 3.4 and 3.6. The The lenqth of the period on the end of the structure nearest the incident liqht is A / o while that on the other end is A o, with all l~yers one extreme to the other. The frequency scale is normalized to wA 0 / c. The between these varyinq qeometrically from qualitative effect of the aperiodicity is quite similar to that of the Figure 4.4 is a plot of as a function of ka for various values of Nat the condition kA = n , on a for large N values. The parameters of each structure have been chosen to give a 10% reflectanfe for In the limit of Nk 2a 2 << l, N >> l, and e 2ik b o) = - 1 expression (4.4) reduces to (4 .4a ) = (4.5) where vibration .09 .05 0 .2 0 .4 0 .8 1.0 1.2 1.4 1.6 1.8 N =50 2.0 ~ N =20 2.2 2.4 N=S N=2 N =I Fig . 4,4 Average amplitude reflectance in low reflectance limit as a function of cell standard deviation and the number of cells. Note that the parameters of each st ructure have been chosen to .01 > .02 0::: (9 a: .04 I.J.... _j u .07 II '-/ .08 Q_ .;"".. .10 co ...... ...... -119- Note that (4.5) does not depend on N, the number of atomic layers, while If we con- s i der a structure for which the thickness of each layer can be contra ll ed exp[-Bn2 cos 2e], where n is the reflection from a smooth surfa ce , the surface from its averaqe, and 0 is the anole of incidence . Anain, taking the case where k2xb(o) = k xa(o) = n/2 we find that the reflection where r. is the relative standard deviation, ora= a(o) for layers of index n 1 arb = ~ for layers of index n2 = l [e - 7 °r a + e which is independent of N. (4.6) -120- The quantity similar to . If we take, for simplicity, the case kA = n, we arrive at the rather J[ 2 2~ [ x [ b(o) )(e2x 1 2k2 2 2 2xab+ e- 2 2 2k2 2 ,-1~ (4 .7) (l-Ne2(N-l)k a+ (N-l)e2Nk a )+N~ ( In the 1imit of Nk 2a 2 « 1, ( 4. 7) reduces to Equation (4.7) is plotted in Figure 4.5 for various values of N under the For large ka values, the reflections worn each interface are not correlated and the 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2.0 f- ~~ >- .007 r -N = 20 II 2.2 2.4 N =S ko .001 l9 .003 1.8 N= 2 _, LL ~ .004 _j w .005 > .006 f- ---j ~. 008 *Q_ /".. .009 . 0 I ~-----,--------,----,------,-----,-------,----,-----,-~ -1224.3 Coupled Mode Theory vities and the methods of the previous section are inappropriate. This We take the dielectric constant as varying according to cos(~nz + E ~(z)) where A is the ideal period and E~(z) describes the perturbation or deviation from this ideal R' = iK ei£~ S (4.9) S' (4.10) where the prime denotes derivative with respect to z. In order to keep the results fairly simple we will solve the rroblem at the Rragq condition only. K is the coupling constant. It is seen from equation (4. 10) that -123The boundary conditions are (4.11) 1 L) = 0 (4. 12) The filter function or reflection coefficient is defined as p(- 2) = S(- 2) Equations (4.9) and (4. 10) can be combined to give When this is substituted in equation (4.13) and powers of £ are equated, I I Rl 2R 0 = 0 ( 4 .15a) ,., K R1 i ¢ 1 Rl0 (4.15b) K R2 = ¢ 1 Rl (4.15c) 2R n ¢ 1 Rl Rl I n-1 n ~ 1 ( 4 ·15d) These equations) subject to the boundary conditions of equations -124- cosh(K(~- z)) (4.16) ( 4. 17) - KC sinh[K(~ + z)] L/2 J cp'(EJ cosh KL -L/2 The reflection coefficient p (-~), is given by -1 =- e - iE: cp ( _1_) (4.18) If we consider an ensemble of these structures,each will have In order to proceed we must consider the auto-correlation function of cp'( z) which we will take as R~ (z ) - <¢ '(z) cp• (z + z ) > = 1lm 2W z = -W cp'(z) cp • (z + z 0 )dz t (1 _ I~o I) -125- Expression (4. 19b) is an assumed form for the autocorrelation function . 2 = <¢ ' 2> is the 2.. Also we assume <¢ ' / 0. This will be discussed further in the next section. following results for = 1e -i t:¢ ( _.!::_) (4.20) ( tanh KL - L p*(-2)
(4. 21) Where Sn- sinh(nKL), en ~ cosh(nKL). In the low reflection limit(4. 20) and (4.21) reduce to . - i E ¢ ( -~) = 1 e KL(l - t: t4 L.t) (4.2 2) In the high reflection limit -126- (4.24) L p*(:z) 2 1.2 .e. -2Kl (4.25) - e-2KL(4 + E:2~ .e_ (Kl- })J reflector considered in the low reflectivity limit section. 2n 2nz + f ¢(z) 1\.0 (4.26) ( 4. 27) where 8/\.(z) is the local period variation. (4.28) From equation 27) we obtain otherwise -127- 4 2 <¢'(z) ¢' (z + z ) > = _!_ <6 A(z) 6A (z + z ) > A4 (4.29) .e. ( 4.31 ) .e. Next we relate .e. to A ~ a(O) + b(O) and s2 to a2 . As shown in Appendix 4-D the autocorrelation function for a slab ~ (1 - :tJ <~ t(z) ~t(z + z ) > = a ( 4.32) The quantity ~ t(z) is the deviation in the slab located at z In -128- (4.33) a2 and A0 /2 whic~ are assumed Thus we have related the quantities known for our slab Braqq reflector to the quantities £ 2 [ 2 and £ which A /2, equations (4.20) and (4.21) become = 1e -i £¢(- h)~ tanh KL (4.35)
2{ i)>= 4 -2 KL (4.36a) - tanh X X = KL (4.36b) -129- G(x) l~_l_ (l s - - x - xsl + c4 f sls4~ The function G(x) is plotted in Figure 4·.6. Forsmall x, G(x) Alternative Derivation Using Coupled Mode Theory The procedure is outlined below = i Ke i e:~s ( 4.37) R' ( 4.39) ( 4.40) Z/ A (4.41) The boundaries of the reflector are between z = 0 and z = L, and ui and -130- <;;!" L[) r0 o. r0 <..!:' t:: L[) (\j t:: ::::! 4- _j (\j Q) ..s:::::. 40 L[) c... 1.0 <::t Q) ::::! cr ·...- LL L[) c5 ( l >1) 9 -131- (4.42) s = so + sl + s 2 + ( 4.43) with boundary conditions R (0) = 0 ( 4.45) R'n = iKS + xsn -1 n ~ ( 4A6) s• = -iKR + x* R n ~ ( 4. 47) This system of equations can be solved iteratively for increasing We will only consider first order and take the region of space containing the reflectors between z=O and z=L. s (O) + s1 (o) P~ o R (o) = so(O) + sl(o) ( 4.48) Solving equations (4.44) and (4.45) we find K (L-z) ( 4.49) = + i sinh K {L-z) ( 4 .50) c, c, Next we combine equations (4.46) and (4.47) to find an equation for s 1 -132- (4.51) cz1 J [!e-i £0_1) cosh <(L-z) S (0) = iK (4.52) - (e 1 c¢ - 1 ) sinh K( L- z JJdz /SbO) + s 1 (0) / 2 % /S / 2 + s s * + S *s 1 0 (4.53) After using equations (4.50) and (4.52) we arrive at the result f (2 _ ei ~¢ (z)_e-i c¢(z))dz (4.55) Next we averaqe the above equation, recalling (see eq. (4.41)) that w z _ 2z'l' - w 2c c3 (4.57) + n2 ob] -2 K sl (4.56) 2K S1 A [~~ +~e-N~-1] (4.58) -133- Finally the fractional decrease in average refl ecti vi ty is -2KA tanh KL cosh 2KL [N~ + :-N~_ ~ ( 4.59) Although this formula is simpler than equation (4~36b), it is not as Greater accuracy could be obtained by including higher order terms in S. Computer Results culated reflectivity values of a large number of computer si mulated The multilayer samples were 11 fabricated 11 s uch that the thickness of each layer was a random variable ass umin g Gaussian (For a detailed dis- cussion of the method, the reader i s referred to ref. [2]). The refle cti- vity was calculated for each sample every 5 cells, giving reflectivity values of stratified media of 5, 10, 15, 25, 30, 35, 40, 45 and There were 1500 such samples. calculations were: n1 = 3.6 0.5143A (A _ a(o)+b(o) 1T ' w 2n a(o) , and normal Each sample structure has a different reflection, but all are -134- o = 0. The average reflection ffr The value N = 1500 was taken to insure sufficient accuracy in ~ R > . The Valuesof or were used. It can be seen that there is excellent agreement between the co~puter results and the second order theory using ~ · for The first order theory using ~ also gives good results. P(R)dR gives the probability of a structure having reflection between Rand R + dR. The vertical axis on the left hand side qives P(R)dR with dR specified. -135- In Figures 4.12, 4. 13, and 4.14 we take a structurecr 25 cells The same scale is used in these three figures and the broadening of P(R) with standard deviation is readily seen. 1- r-- .98691 .97694 50 Table 4.1 .65870 .47785 -r- f - .07695 -- -- - .98580 .97523 .95695 .92573 .87343 ()r = .02 9.22 XI0- 3.69 XI0- 3 5.0 I X 10- 3 I.04xlo- 3 n2 = 3.4 n1 =3.6 1.83X I0- 3 1.68XI0-~2.63 XIO==-- 2.50 XI0- 3 -+--I --- 3.79 X 10- 3 0" _. 9.41 X 10- 3 8.00X 10- 3 4.67xlo- 3 Rp 7.45 X 10-3 2.68 ±.04XI0-3 -+- 3.97 ± .06XI0- 3 5.66 ±.II xi0- 3 7.60±.15XI0- 3 9.47XI0- 10,6 X 10-3 9.75 X 10- 3 6.11XI0- 3 ....--- +- 1.75 ± .04xlo- 3 -- . 9.47 ±.20xlo- 3 _____L .---+- --+ 10,4±.20XI0-3 ·~ -~ - ·- 9.75 ±.19 X 10- 3 ~...-- Rp I RP - 6.20 ± .20 X 10- 3 I Rp RP - Comparing results of the computer experiment with the two analytic expressions. Note .78865 ---- ~- -.92942 .87840 .79469 45 -- -- .66500 .48289 .95952 -~ .26680 .07743 40 35 30 25 20 15 Rp Rp = pp* for perfect structure ;--- f--- 1--- ~ -- 10 Number of cells 20 25 30 n2= 3.4 n1 = 3.6 35 40 Figure 4.7 Plot of the data given in Table 4.1 (or= 2%) 45 50 ~--------~ x~-c..,, ... ,"{) ~~~',,, = .02 N = NUMBER OF CELLS 15 00 10 0:::1 v 0::: 4 ', ~u.',, 2nd order theory with cp' ~~ ~','0.,, ~, ~'' IQ_I " /,' /// _,..o---- -.! a:lo...5 AI -0 r<) ~~f~ -....J -' CTr 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Figure 4.8 Comparison of computer experimental results and theory for a 10-cell structure with 0.04 0.08 Q_ .....__... OLH, 0.125 (i 0.250 -o (L 0.375~ 0.500 25.5 Fig. 4.9 REFLECTIVITY (%) 26.0 ~~ 00 ~ NUMBER OF CELLS :10 CTr =2.0% 26.5 Q_ (L Q_ \.0 10 250 jsoo ~ 2 ~750 1000 CL "--"' 3.00 01 0.250 0:: 0.500 'D 0::: 0.075~ 0.100 76.50 Fig. 4.10 78.25 ~90~ REFLECTIVITY (%) 74.75 NUMBER OF CELLS : 25 o-r =2.0% CL 80.00 10 jsoo ~ , ~750 c£- -1250 1000 CL o_ "-- 01 0.05 0.10 0.15~ 0.20 o-o 98.2 r~!(] 98.4 Fig. 4.11 REFLECTIVITY (%) =-o:o:o=:a NUMBER OF CELLS :50 CJr = 2.0% \-I CL 98.6 (L 1\ I0 250 500 ~750 1000 ...... Q_~ ....-- Q_ ...__.... 01 0.0625f- (2 0.12501- 0:::: 0.1875f- 20.0 Fig. 4.12 60.0 of IQ Ill o::~ll 12.50 18.75 25.00 80.0 00;7 IV REFLECT IVITY (0/o) 40.0 oool o NUMBER OF CELLS :25 o-r=5.0% 0.2500. . . . - - - - - - - - - - - - - - - - - - - - - Q_ --... --' 0::: Q_ '--"' 0.0625 ~ 0.1250 -o 0.1875~ 0.2500 7.5 % 20.0 Fig. 4.13 REFLECTIVITY (%) 40.0 NUMBER OF CELLS :25 CTr = 60.0 Q_ ~6.25 80.0 ~ 11250 ~ 18.75 25.00 CL Q_ ...__ NUMBER OF CELLS :25 CTr =10.0% 20.0 Fig . 4.14 REFLECTIVITY (%) 40.0 0 ~ oooao 0 0aaa 0 ooo0 0 600oo%oo 00olliw o o 9 °o~ 0.0625 C2 0.1250 -o 0.1875~ 0.2500 /oo\ I1 0~ 0 IV //o 80.0 UO -H6.25 12.50 ~ 18.75 25.00 Q_ ~ ---. -145- 4.6 A Phenomenological Expression for p(lpl ) the value of important parameter of the reflector, does not describe the spread If It also the value of a is allowed to increase, how many structures will no longer satisfy a given required reflectivity. For example from Figure 4.14 we see that although the reflectivity from a perfect structure is almost Also i t may be desirable to relax the manufacturing tolerar1ces if thi s does not lead to a large increase in the number of "bad" mirrors . cq-1 r -r ( 4. 60) (r -r)q where r = pp* is the reflection, r is the gamma function, rp i s the c= (r - - ( 4. 61) (4.62) -146From equation (4.55) we have (4 .63) f - J [2 - ei E~ (z)_ e -i L~ (z)]dz -N'I' 'I' -l] (4.64) We need only use equation (4.55) to find Before proceeding, an In the derivation of (4 . 58) we neglected terms of order Sn a (ei E~-l)n ~ (i r~ )n where n > 1. The solution (4.58) is, however, of order ~ 2 • indicating that terms of s2 sho uld have $1 = --::--'-- C6 ( 4. 65) Using the results of Appendix 4-F we find Combining equation (4.58) and (4.57), using the definition of a, and -147- (4.67) - We thus arrive at values for the parameters q and C ( 4. 68) q = 4 C = t :~1 NL (~)2 (1\~2) 2 ( 4. 69) which when used in (4.60) give r -r p( r) = cll/4 e p r( l) t' % 1. 608 ( 4.70) The function p(r) is indicated in Figures 4.9-4.14 by the solid or rp - rpeak -148- 4.7 Experimental Results manufactured by Spectra-Physics. The structure i s designated by S, It is a quarter wave stack for A0 = 4500 A (except for the double nl stack) with a refl ection -149- n 5 = I .4 7 = I .46 nL -------------= 1.46 nL nL = 1.46 nL = 1.46 nL = I .46 '7 7 ) ' ' ; / t1H ~ 2 ..3B / . /_/ / / , // 1.0 Dielectric mirror tested by Spectra-Physics -150- Reflectivity measurements are accurate to .003 and the following For example, the index of refraction may increase slightly as the sample is grown due KL = 2.34 2 2 we find KLN or ::: J\o/2 TT KL cosh KL 4 . 12 (4.73} We next check to see if this is consistent with equation (4.66) and (4.66) and J From aI 1\. 012 "' .10, in approximate -151- 4.8 Conclusion thickness of layers in a Bragg reflector has been studied using the Closed form expressions were obtained for the reduction in reflectivity, which agreed with a computer experiment. These expressions are accurate for small values of a which is typical for most cases. A phenomenological expression for the reflectivity distribution function p(r) was presented which also Results for arbitrarily large a values \Jere obtained for low reflectivity reflectors~ The formulas were then applied to a structure which was grown and results were -152- Appendix 4-A and 2R ,., RI I KLR = 0 (4-A.l) i ¢ I RI , n 1, 2 (4-A.2) subject to (4-A.3) 2 L)= and (4-A.4) R = R0 + cR 1 + c2R (4-A.S ) S(t L) where S = - ie-i c¢ (~) (4-A.6) solution is R ( z) - The boundary conditions on R are , from (4-A.4) and (4-A.6) R (i) =0. Since these must hold for all values of c we have Rl (-i) = R2(-i) = Rl (i) = R2(i) = O (4-A.8) Bn sinh[K(i + z)] The particular solution is given by zJ i ¢ 1 (n) R~_ 1 (n) sinh[K(z-n)]dn (4-A.9) This can be confirmed by differentiating (4-A.9) and substituting in -153- The total solution is thus Rn(z) • Bn sinh[<(~+ z)] + ~ zf O'(n)R~_ 1 (n) sinh <( z- n)]dn The boundary solution at z = -L/2 is also satisfied and Bn i s The result -154Appendix 4-B ~) (4. 16), (4. 17), and (4. 18), L/2J <¢ 1(n) > R~(n) cosh[K( 2 - n)]dn ; 0 (4-R.2) since <¢ (n )> ; 0 -L/2 L/2J c1 -L/2 The result is 2(-~) ~I ~=-2 <4>1 (~) (4-8.4) n;-2 • cosh[K(~- n)] sinh[K(~- n)]d~dn K3 L/2f L/2f r,= -2 <¢ 1 ( ~ ) 4> 1 (n) > · n; -2 • cosh[K( 2L - ~ )] cosh[K( 2L + r, )] cosh[K( 2L - n)] slnh[ we haveR ( ~ -n)' = < The integration is quite involved unless we make the approximation R .e.t o( ~ - n) (4-8.5) where o(x) is the Dirac delta function. << 1. -155- After using for a < 0 < b (4-8.6) for b > 0 (4-B. 7) and J o(x) f(x) dx = -} f(O) we find (4-B .8 ) When this is combined with (4-B. 1) a~d equation (4. 18) we arrive at -156- Appendix 4-C similar to (4-C.2) E2 z = -2 All terms of order E do not contribute since <~ > 0 (See Appendix 4.8) ~ L/2f L/2f <~' (n)~'(s)> -2 (4-C.3 ) -L/2 L ~ ) ]dnd( c4 f .t (--1 (4-C.4) After combining (4-C.2), (4-C.4) and 4-8.8) we arrive at equation -157- Appendix 4-0 .. lim w-jo()O 2~ f ~t(z) ~t(z + z )dz. ( 4-0. l) -W <~t(z) ~t(z + zo ) > = o 2a ( 4- 0. 2) where Pn.(z 0 ) is the probability that two points separated by a di s tance z 0 will both be in the same cell of index ni. - ao (1- nl - 1\.0 ~) for l z 0 1 < ao bo 2 = 1\.0 (4- 0. 3) (1- ~) for l z0 1 < bo otherwise = 0 oa ob where we have assumed -- , ~ << l. and (4-0.3) we obtain <~t(z) ~t(z + z0 ) > = ~ (1 - ~) otherwise. -158- Appendix 4-E l"IAY .TY RRA.F4 -159- 1 ;-, A f' C 1- P T 4 H , r IT V • h: M r N •· 1: 1'1 (l X v ~ :to'! () X • X tJP I· I I ; • N () Mr· KX::::l"l I \J*F ::> I 2 I T ~.,.I • N'~;Mr· R A:::: I< (d h: I f I ' l ()~; 49 "'• FDHMAT ( :1. X • 'ri():::' • r ,~, . '," ' <' If: PF' ':ir)MPI F•:; ·::' • I 4" ' > · ·, ~' : ' CON T I NIH 13 o::; :::, l ·1 o nIV XPK=RPK/NSMF'*100. 1H F ... , ·,"''I · 1 1.. •:; ' • -160- F F< P :::: F< 1:.· 70 I I 0 '.:> I' F N C Cl [I F ( ::> 4 ~ 7 0 I • ' ; I\ A I·' ) I· h' F· • F 1:.: A r· r 1'\T=O 300 Y XF'(l)::::FW nr· • 44 -161- Appendix 4-F J (2 - ei c~( z)_e-i L~ (z))dz ( 4-F .1) c ~(z) = c n1 II. u. +~n ; =1 L: 2 i=l v. ( 4- F. 2) The variables ui and vi are Gaussian distributed as de scribed LL = 4L 2 - 8L J + 2 I J + 2 J J (4-F.3) 0 0 In arriving at (4-F.3) we used the fact that -162L -8L -8L - 8L 2 2 '¥ e --,;_zdz = 8L (e-Nif'_l) ( 4- F. 4) N'¥ We now examine II 0 0 Over the region of integration where z' > z we can write ~ are independent and ~ (x) has the same probability distribution Thus (4-F.S) with a symmetric expression for z > z'. II 0 0 In a similar manner we find After integra ting this third -163- 2 J J - :l!.(z-z') After combining our results and expanding all expressions in a -164Chapter 4 References M. Born and E. Wolf, Principles of Optics, (MacMillan, N.Y. 1964). 2. P. Yeh, A. Yariv and C. Hong, J. Opt. Soc. Am. 3. A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13 Novem- ~. 423 (1977). ber 1961), (unpublished). N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 5. C. L. Tang and P.P. Bey, IEEE J. Quantum Electron. 6. S.M. Rytov, Zh. Eksp. Tear. Fiz. 29, 605 (1955). [Sov. Phys.- 17, 483 (1970). 9 (1973). JETP 2, 466 (1956).] J. P. Vander Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. 8. Private communication with Wess Icenogle of Spectra Physics, 9. Private communication with Steve Silver of OCLI, Santa Rosa, Cali f ornia. 10. K. 0. Hill, 11. A. Katzir, A. C. Livanos, J. B. Shellan, and A. Yariv, IE EE J. Quantum 12. Appl. Opt. J..l, 1853 (1974). QE-13, 296 (1977). C. Hong, J. B. Shellan, A. C. Livanos, A. Yariv, and A. Katzir, 13. IL· 276 (1977). C. Kittel, Introduction to Solid State Physics (Wiley, N. Y. 19 71), p. 85. 14. Petr Beckmann, The Scattering of Electromagnetic Waves From Rough 15 . H. Kogelnik, Bell. Syst. Tech. J. ~. 109-126 (1976). 16. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973). -165- 17. W. Streifer, D. R. Scifres, and R. D. Burnham, J. Opt. Soc. Am. 66, 1359 (1976). H. W. Harman, Principles of the Statistical Theory of Communication, -166- Chapter 5 The use of a tapered grating to suppress the side lobes of the reflectivity spectrum in a broad-band The analoqy between this device and the method of side-lobe suppression in the Sole filter is also covered. The gratings are corrugations on the surface of a dielectric waveguide as we considered in previous chapters. Figure 5.1 is a typical result of the broadening of the reflectivity spectrum with increasing chirps. 1 The notation -167- 70 •Cl. 60 Cl. >1-> _J u.. cr: 30 20 NOR:0..1ALI7EO F REOUENCY bl Fig. 5.1 Reflectivity spectrum of a chirped grating used as a -168- edoes is qiven by 1 - e-(nK )/y. The values of the reflectivity in L are in agreement with the above formula. The plot is also in qualitative aoreement with Fioure4.3 which is a plot with T being an adjustablP rarameter; a positive T value represents a The orat- The key result of Figure 5.2 is that the side-lobe levels of the response characteris1ics For quadratic tapers the side lobes are strongly suppressed if the incident light first encounters a This effect, which results in lower- inq side-lobe response, occurs in other devices as well. For example, -169- 100r-----------------------------------------------------~ 1< 0 L=JT/2 ,, 70 C.' Q. >t- so I- l.U u.. a: 10 10 Fig. 5 . 2 Reflectivity spectrum of a tapered grating used as a reflector in a dielectric waveguide. (After reference 1 .) -170- it is well known that light incident on a slit will produce a far field However, if the slit transmission function is tapered, for example by using a variable transmission filter with maximum transmission near the center and decreasing transmission near the Sole in his study of the Sole filter. The original Sole filter con- sisted of a periodic stack of birefringent elements of equal thickness If the optic axis of the nth layer was rotated by an anqle (-l)na with respect to then-1st layer, then the rlevice would transmit only one frequency of light. The bandwidth of the filter narrowed as the number of layers was increased, but the response These side lobes .., were later suppressed by Sole when he "tapered the anqles" through which That is, by destroying the periodicity of the filter and rotating the birefringent elements near the center of IS'(O)i = J K(z) e+i 2oz dz (5.2) - oo IS'(O)i, which is the reflectance, is simply the Fourier transform of If K(z) is a constant between z = 0 and z = L, and zero outside this reaion, then IS'(O)i will be a sine function -171with side lobes. On the other hand, if K( z) is a Gaussian, then /S '(O) / will be the Fourier transform of a Gaussian which is another Thus, this interesting effect is readily apparent in the low reflectivity limit. This was demonstrated in Chapter 4 , where a statistical analysis of Braqo reflectors was presented. Streifer and co-workers 4 have also used this We start with the coupled mode equations given in equation (1-8.5) of Appendix 1-B, but include the possibility of gain or loss for (1-8.5) must be modified by replacing o with o +Lt- i JJ where 2a is the power loss or gain per unit length with a / 0 for loss. where the primes denote differentiation with respect to z. The orating is in the reqion 0 < z < L and the period is assumed to vary according cos( 2 ~z + ¢(z)) (5.4) -172- The variables in equation (5.3) are taken to vary as K = K0 + K 1 ( z), R = R0 (z) + R1 (z), where the terms with the subscript one are assumed small and the R1 term represents the forward travelin g wave (5 . 5) where ul = 2K0 KlR 0 + (2~ 1 ~o + i ¢ ' + -)(R'0 + ~ 0 R0 ) Ko n = y cosh yl + ~ 0 sinh yl k {y cosh[y (L-z)] + ~0 sinh[y (L-z)]} R0 (z) is, of course, the solution for the case ¢ = 0. Figure 5. 3 is a plot of the filter response for simultaneous l i near ( 5.6) Notice that for positive £ , there is a larger response for positive This is easily explained. The orating -173- KoL = 1.0 . _, ," ..... ( = -0.24 2.0 - '' 1.0 0.6 ,_, 0.4 ~~--~--L-~___ L--L -4.0 - 2.0 \~/l --~~--~--~--~~---~- 0.0 2.0 4.0 6.0 oL Fig. 5.3 -174- depth is larger in regions where the qrating period is smaller, and thus The opposite effect occurs for negative c values. tors. IE 5(0)12 (reflectivity= RTOT The grat1ng per1od 1s taken to vary as ATZT = ATOT + ~ z . oscillation condition The for a DFB laser is that the reflectivity be in- finite (no input and a finite output) and this condition on the qa in seen that, to first order, the poles do not shift as the chirp is It is readily apparent, however, that the contours are pulled in closer to the poles as the chirp factor t is increased . This is to be expected, since for a chirped structure the refl ectivity will drop. In the latter case, for a guide whose thicknes s is given by w + qz, the appropriate E. va 1 ue t o use 1s (5. 7) where neff is the effective guide index, nf is the quide index, weff is _J -7.0 1.0 2.0 5.0 6.0 7.0 1.0 600.0 .ooo 4000.0 ooooo.c 3000C.C 15000.C eooo.c 2000.0 3000.0 8L 2.0 3.0 4.0 50 NO CHIRP £ ' 0.0 6.0 7.0 Fig. 5.4 Contour plots of the reflectivity as a function of gain and frequency for uniform grating -6.0 10.0 0.1 0.2 E (0) The reflect1on gam ~ , as o funct.on of 8L and al for KL = 0.4 ...... (.11 "'-J _J -7 .0 -6.0 OOO.'J 100.0 -5.0 0 2 -4.0 -2.0 - 0 0.0 .0 2.0 ~c~~A~ - 3.0 -------- ·5000o 3.0 1E (0)1 4.0 .£ 5.0 CHIRP € = -0.2 6.0 Fig. 5. 5 Contour plots of the reflectivity as a function of gain and frequency for grating with 3. The reflectJon gom 0'1 -J ...., tl ..... - e .\.i ·5.0 -40 .... 30000.C 0 2 2000.J 0.1 IE - 2.n 1.0 SL I~' 2.0 3.0 5.0 CHIRP 6.0 € = -I 0 E:=-1.0 Fig. 5.6 Contour plots of the reflectivity as a function of gain and frequency for grating with ~.v 6.0 70 •e (0)12• os o funct'on of SL and OL for Kl.. = 0.4 _. ....... ....... -178the effective guide thickness, and A0 is the wavelength of light in the period of the grating is For a more accurate analysis, we break up the nonuniform re- qion into a series of nine approximately uniform regions. The transmis- sion and reflection properties of each of these nine uniform reqions is After the transfer matrix of each region is determined, the matrices are multiplied together, and the oscillation condition thus determined. It is seen that the gain condition for the lowest order mode increases, while the gain As the chirp is increased, the modes become intermingled. For a DFB laser 500~ long and A = 3500~, the riqht side of the plots in Pulse Compression signal. Referring to Figure 5.9, which is simply the broad-band filter discussed in a previous chapter, we see that if a chirped frequency For example, if the high frequency part of the siqnal is leading , it will 3.0 2.0 240 '--r rp 36.C 6~" 72 c !. ' z- L' - ' 1z- Cl 48J = 8€ sao 96.0 KL'0.4 f ch rp Fig. 5.7 Gain required for oscillation as a funct ion of chirp for KL = 0.4 _J Ga•n reqNed for ocdlo!lon of first four modes os o ft..nct or ""-J U) 48.0 60.0 !. 720 Ch~rp '" u!"lots of 2rrL A(z'L: -f.(z 36.0 84.C 96.0 Fig. 5.8 Gain required for oscillation as a function of chirp for KL = 2.0 <.:> ....J KL' 2.0 Goon reQuored for ocollotoon of forst four modes os o funcroon of chorp _, -181- ·......., s... 0) -o 0.. s... ·.- ..c: s:: ...., s:: Vl 0.. -o 0.. s... ..c: en ·.l.J.. -182- reflected from the closer left side of the reflector. A narrow pulse which has been broadened after traveling through a long dispers ive Some parts of the incident pulse must travel a greater distance before returning to the point of focus, and thus the Since all areas reflect the sinqle-frequency waves back toward the focal point, all parts of the long reflected pulse can be A similar effect can be achieved with a uniform grating, but in that case a chirped li ght signal Several theories have been proposed, includinq dama~e resulting from thPrmal induced stress HORN ROTATING (c) (b) ~- ECHO Fig . 5.1 0 (i) l _!~S ___ - _ - - - - - - - - -T~~~~==~ TIME PULSE SWEPT HORN TIME 1.1. 1.1. ECHO >zw a:: PULSE a: >- TO RECEIVER (X) -' -184- within the film 9 and the heating of the material to the point of melting. 10 Thus if a method can be found to suppress the peak or maximum value of the electric field intensity We assume that the layer with the intensity profile shaded is the This is The first added layer of index n1 is slightly thicker than the quarter-wave thickness, The modified structure with resul t ing intensity The con- straints which Apfel places on the two added layers in order to calculate their thicknesses are that the reflectance of the total multilayer This second con- dition is evident in Figure 5.12, where it is seen that the heiqhts of the -E2 Fig. 5.11 AIR \I ~ \ I J..... I I I I SUB. ;After reference 11 . ) Intensity distribution inside a quarter-wave stack dielectric mirror. \I 1:/J. -J Ol 00 -E2 Fig . 5.12 AIR \11/ji \I~ I L \ I I I I I I I SUB. additional layers on the surface . (After reference 11 . , Intensity distribution inside a quarter-\'Jave stack dielectric mirror with two \1!1 ,,(\ l I 0"1 co -' -187- first two shaded regions are identical. For even greater field reduc- tion, additional layers may be added. Conclusion covered. The possibility of using tapered gratings to suppress the s i de lobes of the reflectivity spectrum in broad~band filters was discussed. It was found that the gain condition was altered more in some modes than others. The process may be very useful for mirrors designed to operate with intense laser pulses. -188CHAPTER 5 REFERENCES l. H. Kogelnik, The Bell Syst. Tech. Jour.~. 109 (1976). 2. M. Matsuhara and K. 0. Hill, Appl. Opt. ll• 2886 (1974). 3. I. Sole , J. Opt. Soc. Am.~. 621 (1965). 4. W. Streifer, D. R. Scifres, and R. D. Burnham, J. Opt. Soc. Am. 66 , 5. A. Yariv, Quantum Electronics, 2nd Ed. (John Wiley and Sons, Inc., 6. R. V. Schmidt, D. C. Flanders, C. V. Shank, and R. D. Standley, Appl. 7. W. E. Kock, Proc. IEEE (Lett.) 58, 153 (1970). 8. J. H. Apfel, J. S. Matteucci, B. E. Newnam, and D. H. Gill, The Role A. L. Bloom and V. R. Costich, Design for High Power Resistance 10. B. E. Newnam. Damage Resistance of Dielectric Reflectors for Picosecond Pulses (U.S. Government Printing Office, Washington, D.C., 11. J. H. Apfel, Appl. Opt.~. 1800 (1977). -189- PART II -190CHAPTER It is possible, however, to confine the liqht within a lower index reqion if the material adjacent to the low index In the Gragg structure, the light is reflected coherently from successive dielectric This mechanism of guiding will be referred to as Bragg waveguiding. Fox analyzed the problem by finding a transfer matrix relatino the field in one layer to This aeneral analysis in- eluded as a special case the propagation of liqht in a low index quidinq -191- great flexibility and accuracy in the growth of layered media. 7 The Thus it is possible that the Transverse Braqq Reflector Laser (TBRL) would provide It should also be pointed out, that unlike the conventional high index guide, the TBRL structure makes it possible to guide light * Actually, lossy (leaky) guiding is possible in a low index (compared -192CHAPTER 1 REFERENCES E. A. Ash, "Grating surface wave waveguides," presented at 2. A. J. Fox, Proc. IEEE 62, 644 (1974). 3. P. Yeh, "Optical Waves in Layered Media," Caltech Ph.D. Thesis (1978). 4. P. Yeh and A. Yariv, Opt. Comm. ~. 427 (1976). 5. P. Yeh, A. Yariv and C. H. Hong, J. Opt. Soc. Amer. 67, 423 (1977). 6. A. Y. Cho, A. Yariv and P. Yeh, App1. Phys. Lett. 30, 471 (1977). 7. A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry, Vol. 10, -193- Chapter 2 Introd uction transverse Bragq reflector laser are studied. The condition for maximum confinement of light inside the active reaion is used in order to minimize radiation losses into the substrate; thus a quarter wave stack is The substrate losses are then determined in order to find an expression for the imaginary component of th e index Design of Structure The use of periodic structures in injection lasers is not new. Distributed feedback (DFB) 1 •2 • 3 and distributed Bragg reflector (DBR) 4 The use of a peri oci c structure perpendicular to the junction plane to provide confinement and < n . The case of a symmetric Bragg waveguide It has been shown that, for maximum confinement of light,the electric field or the rate of change of no Fig. 2.1 nl n2 n2 n2 A Bragg reflection (slab) waveguide (a;ay =O) . no L. : ~ . ~~.il.....:...:...~ ~14~~1-::. h1. ~~"/o/""/ ,./ -;-,~"/, -........ ;.;:/~} /d-~1 ',~% ',. lLx ~~~/~1'.,/~,0-~"... (Afier reference 7. ) n2 -' 1.0 '\:: '"" ,~ ... Fig. 2.2 Transverse field distribution of the fundamental modes of a typical Bragg reflection t-- (/) Q.l (/) Q.l Q.l "'0 "'0 .~ ..- .D ;:) ..- ~~----------------~ no Q.l n2 ~A-l no ..u Q.l "'0 Q.l )( ~_tiJ r--d2 ....... (.J'1 <.0 )( "-"""' """" -f \.J no :: ' ' 11/ 1: I ' 'I 'I ' ' /1/i//f/ it'fi fffr ////1)7~ 1,'1/1/l ;1:~1/ ' (I!/ J 'r' ,/ ·I ' I ;/fl7/% y'Lz _!i_ -n2 - - - - - - ...... Fig . 2.3 Transverse field distribution of the fundamental mode of a typical Bragg reflection Q) '- .! "'0 i; I f) '- 21 a:: '- "'0 __. 1.0 Fia. 2.4 )( no nl ~t1~t2--J t g . ,, . no I~ Field Distribution of Transverse Bragg Ref Iector Laser a:: Q) <.> Q) -c: "'0 Q) II _, -....) ID -198- the electric field with respect to the x direction must be zero. Referring to Figure 2.5, the conditions for maximum refle ction on radiation EY(x= interface, type A) = 0 hi9h index on left Using these conditions and referring to Fiqure 2.4 • we have for -t q < X < 0 < X < -t (2. 1) C1 (2 . 2) kgx = 2n neff (2 .3) kax = 2n n2 (2.4) "o = 2n = wavelength in vacuum We have suppressed the ei Bz factor and for the moment will concentrate only on the field in the guide and cover (superstrate) regions. (2.5) ng , na and >. 0 have been specified, then the easiest way to design an optimum structure is to choose an arbitrary neff' -199- B) A) E(x=O)=O Fi9. 2.5 dE (x = 0) = dx The field condition at dielectric interfaces -200- na < neff < n , and use equation (2.5) to solve for tq. Next we must It is well known that maximum reflectivity i s obtained with a quarter-wave stack, and now that the dynamical variable neff has been lX TI i = 1,2 lX ;,.o jn~ (2.6) neff (2. 7) Note that eigenvalue equation (2.5) is the result of considering the case ng > n1 . For the case ng < n 1 we must have -t g < X < 0 X < -t (2.9) with resulting eigenvalue equation (2.10) The case where ng > n1 is of greater importance in the work to follow, The steps involved in de- signing the laser are summarized in Figure 2.6. Calculation of the Loss Constant in the reflector region, but is oscillatory under an exponentially decaying envelope. For any real structure with a finite number of periods, some of the field will leak out into the substrate, and in order to Fig. 2.6 ng 1g ng > n2 > n1 AQ V neff - na 1x AQ V eff In?- n2 i =I 2 k·1 t ·1 = JL 3) Find t 1 and t2 from requirement 2) Find a t value to satisfy C D. I) Choose an arbitrary n0 < neff < ng 0x- k - 271' _ 271' I 2 Region a : E = Aekoxx (Since ~; = 0 for maximum reflection) B. Region g: E = cas kg/ Design of Laser Structure for Maximum Confinement of Light ka X C. Match Boundary Conditions A. 0__. -202satisfy a steady state condition, it is required that enough qain be In other words, the index of refraction of the guiding layer must be complex in order to compensate for the substrate losses. v2Ey + (k 20 n2 + k20 on' 2 )E y R(x) ei(K/2)x eiSz + S(x) e-i(K/2)x ei6z (2.11) (2.12) where c = velocity of light in vacuum n2 + n2 A.= period of reflector 2rr/fo. on' 2 = index variation = -( on2 ) sin(Kx) amplitude of incident field (field travelino toward the S(x) = amplitude of reflected field -203- If we combine (2.11) and (2.12), make the usual coupled mode approximations (ignore R"(x) and S"(x)) and collect "coherent terms", we i oR = -ns + i oS = (2.13) (2.14) -nR with T) (2.15) (2.16) L = length of reflector region = NA (2.17) i o . h[J2 J n2- 02 1l 1l cosh[ Jr/- ( 2 (x-L)] These solutions are valid for x ~ 0. The solution at x 0 is - S(O) R(O) The solutions in region g (guide) and region a (cover region) are, respect i ve 1y , -i k gx + R e +ik gx ) e i Bz , -t < X< 0 (2 . 20) -204- (So e ik -ik t -t (2 . 21) After matc hing aE; ax at x = - tq , we find r = (S /R ) e 2ik qx q + ax where from equations (2. 17) and (2.17) (2 . 23) = 1 - E: (2.24 ) For most cases of practical interest the reflection from the mu l tilaye red For E:= 0 , n i s pure re al, and to first order in E: we may take gx (2 . 25) where kgr is a real number and s i s to be determined. Comhi ninq equa- ti ons (2.22), (2.24) and (2.25) and equating parts of s imilar orde r in r tan(k t ) = ka x "' tan(k t ) Thi s is just the ei9envalue equation given by equation (2.5 ) . (2 . 26) -205- s = 1st order 2 si n 2( k -1 1- + _t""*~- (2.27) -1 (2.28) }( gr 2t + --,.-:-a_x--,.,--( k2 + k2 ) gr ax Equation (2.26) was used in arrivinq at (2.2R) from (2.27). Refer- rinq to equation (2.24) E: "" E: "" l+ E: "" sinh( Vn 2- o2 L) cosh( / n2-o 2 L) - -1·o sinh (if2 Jn2- 62 coth(Jn 2- 02 L) - 1] - -i o (2.29) Thus after combining (2.25), (2.28) and (2.29), we have gr + ik • o/n k'or = kqr kgi = with j 2 2 - L n ~6 (2.30) k + k2] coth( Jn 2-o 2 L) - 1...JI ( 2. 31 ) ax tan(k gr t g ) = kax /k gr· 1ate , the index of the guide must be comp 1ex and -206- gr n . (2.32) n . is the imaginary part of the index in the guide. kqr ~ kgx is determined by equation (2.26). ngr ~ nq is the real part of the index, and If we examine (2.31) in the limit of hiqh reflectivity (larqe nL) e -2nL 01 (2.33) or ng1. -k qr e -2nL (2.34) k2 k2 + qr In Figure 2 .7 the imaginary part of the index of the active reaion is The guide parameters were those of the laser which was tested, and are given in Fiqure 2.12 and equation (3. 1). For the waveguide con- sidered Sx(x=L) (2.35) J Sz(x) dx where S = Poynting vector = '2 Re(E X H*). Physically, a is simply the -207- -.... ::;, ::;, <1> U) <1> :J '"0 Q) "' :J c (Jl _.J <1> c X 8/T] = 0.9 +- "- '+0 8/T] = 0.7 CJI ""'-.. 0" a;.,= o.o 2 3 4 5 6 7 8 9 10 N = Number of Periods in Ref lector Imagi nary pa r t of t he index of t he act ive region required -208power loss into the substrate per unit lenoth, divided by the t otal R(x) ei(Kx)/2 eiBz + S(x) e-i(Kx)/2 e i f\z , 0 < X < L (So e -ik qx + Ro e ik qx ) ei ~ z (2.38) -t < X " O (So e ik CJX CJ + Ro e -ik t ) k (x+t ) < X < -t Hx = +i _L E = ~ E (2.39) (2.40) (2.4 1) Sz - 8 IE I l2 Re ( Ey H*) (2.42) 2w~ Im(E .jL_ E*) r) X (2.43) Using{2.36) through (2.43), it is straightforward to show (n -o )K 4n 2 wp (2.44) (2.45) -2090 IEY/ -t e2Vn2 _c; 2 L { 2''n2 _()2 + _v.:___ sin(2k t ) 0 cos (2k gx t 9 ) } (2.46) cos(2k qx t g ) (2.47) a = [ (1 + l Jn2-o2 ~0 + i- ) t This result simplifies somewhat f or the case o (2.48) n gx 2n 2+ 4n2 + 1 + k ax for C\ 0, 0 . ( 2 . 49) The key steps involved in calculating a are outlined in Figure while in Figure ber of periods. 2.9 we have a plot of a as a function of the num- Equation(2.49) was used for this plot and the device parameters are those of the actual laser tested, and are given in the · KX k0 217' Substrate J~ Sz(x)dx 3) a= Sx(x=L) Fig. 2.8 Calculation of a where S is the Poynting vector 2) Use coupled mode equations to solve for R(x) and S(x) throughout structure 9x . {3z + S(x)e-'2e' L=NA 2rr . {3z E y =R(x)e'Te' . KX ... ractive reg10n ...... -211- 100 Fabry- Perot Loss for Q) '+'+- Q) 10 :J Q) II .I I 10 II N =Number of Periods in Reflector Attenuation Coefficient, a , Due to Losses into -212Fabrication and Testing section. Figure 2.10 is a comparison of the effects of other Al concentrations with that of the actual device for All other parameter s such as quide thickness and Al concentration in the superstrate are the same for the four curve~. Diffcrentidling equa- tion (2 . 15) we obtain - - [n k M = 6 ~ 8 - B ~] + ~ [n 2 k - B JB ] (2.50) If we evaluate o around the point S = 0 or, in other words, Ak k - k(O) k~O) = k0 value for which o - 0 A~O) = A0 value for which o = 0. -213- E 100 +- Q) ..._ 10 +- ::J Q) ++- II 10 N = Number of Periods tn Reflector -214- where 6A 0 is qiven in ~. We thus find Figure 2.11 is a plot of the increase of the loss constant a as The structure drawn in Figure 2.12 does, in fact, represent the device which was later te s ted. This can be shown as follows: ~ e:ow ngrngi 1Ey l2 polarization= e: 0 (n~ - l)EY Py ngr + in g1 91 << n gr In order to find the power generated per unit length of active -t g e: w n n . 2kqx ( 2.51) \\ 100 200 300 I I Dependence of loss constant on lasing frequency 6>... in A N = Number of periods -400 -300 - 200 - 100 Fig. 2. 11 ~1~0 1.1 II 1- (/') (/') 8~ E§ 1.2 ~ c c ( /')+- +- +- (,() 1.31- II N=3 400 JN=5 1.4,----,---,---r---r--,--.--~-----.,r-~----. 01 _, no 14 t9 - I I ng II nl ""1 "--/ n1=3.42 ng=3.59 n0 =3.32 t2 =2200A t1 =3900A tg =500A Fig. 2.12 Laser structure used in analysis Field Distribution of Transverse Bragg Reflector Laser 0:: Q) lo.. Q) c: "'0 Q) )( II II 0'\ --' -217- where o= 0 has been used for simplicity. If we equate (2.51) to the power flow into the substrate, Sx(x=L) = 4~~ , we find -K 2w2 ~£ n (t o g sin(2k (2.52) gx 9 ) If we now use [sin(2k gx t g)]/2k gx = kax /(k 2gr+ k2ax ) which is aresult of eigenvalue equation (2.26),as well ask gr ng , we (2.53} k 0 x 1\ = n In order for the coupled mode equations to be valid, k1X "' k2X "' kQX , p = reflection coefficient. (2.23) and (2.24) we find CL -2nL) _ 2 -2nL 17-372 kgx e -2nL (2.55) -218- 2.4 Conclusion reflector laser has been presented. This method was based on the require- ment that maximum confinement of light be provided for the radiation The imaginary component of the index of re- fraction of the active region was then calculated, assumin9 steady state It was found that a device with 4 to 5 periods would reduce losses into the substrate to the -219CHAPTER 2 REFERENCES 1. Shyh Wang, IEEE J. Quant. Electr. QE-10, 413 (1974). 2. K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, H. W. Yeh , 3. H. C. Casey, Jr., S. Somekh, and M. I1egems, App1. Phys. Lett.~. 4. F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. 5. H. Koge1nik and C. V. Shank, Appl. Phys. Lett.~. 152 (1971). 6. C. V. Shank, J. E. Bjorkho1m, and H. Kogelnik, Appl. Phys. Lett . .l§_, 7. P. Yeh, Optical Waves in Layered Media, Caltech Ph.D. The sis , 1978. Lett.~. -220Chapter 3 Introduction transverse Bragq reflector laser. The device, which was grown using liquid phase epitaxy, was designed so as to support only a Bragg reflector type mode. A thin active layer and asymmetric structure makes con- ventional guiding impossible. The laser was then tested and the longi- tudinal mode spectrum and transverse mode profile measured. Fabrication and Testing shown in Figures 2.12 and 3.1. It was grown by Willie Ng and P. C. Chen using liquid phase epitaxy, and consists of a GaAs substrate, followed tion, at A0 = 8900 A we have -22 1- Al. 4 mode no (active region) t, A1.2s tg nl A1. 1 n2 A1.2s nl Al . r .•• .••• •• A1.2s Go As . ·:·:·:·: -:-:-:::.:: :::.: -: ·. . :::-: :· ·: ··:·: .·: Fin. 3 ·1 ·=·=·=··: :·. ·::·:·:=: ·: ;::·: ·:·.::: :·. .·: :;:·:. ·: Transverse Bragg Reflector Laser -222- k0 = 7.06 X 1o6;m 3.371 kax = 3.96 x 10 ;m ( 3. 1 ) kgx = 8 . 72 x 10 /m 3. 2 1 is a plot of refractive index of A1xGa 1 _xAs at l. 38 eV and is very useful in the design of the laser structures. thin layer of Au-Zn was evaporated on the p+ side of the device to form The substrate side was then coated with Au-Ge and the sample was annealed in flowing hydrogen at 400° C for approximately 5 minutes, The sample was then cleaved into lasers with dimensions of 250 ~ x 500 ~ on the average. As the Al concentration in GaAs is increased, the bandgap increases and the Thus light generated inside the GaAs active layer is strongly absorbed in the MBE structure, but is not absorbed in Since the majority of the radiation is outside of the active gain region it is impossible for a device to lase if the material out- - 223- 3.6 llr"T"~...-.-.....-.--t""".....--r--. IC: xw 3.5 3.0 "AiAs= h11 =1.38 ev 0 .1 0.2 0 .3 0.4 0 .5 0 .6 0.7 MOLE FRACTION AJAs, x 0.8 -224side the pumped region contains highly absorbent GaAs. Figure 3.3 (courtesy of Al Cho of Bell Laboratories) is an SEM of the MBE sample The great precision in MBE growth is evident from Figure 3.3. Because of the problems with absorption, pure GaAs (high in- dex) must be used in the active region, while lower index AlxGa 1 _xAs It is well known that for an asymmetric dielectric guide, a cut-off thickness exists for guiding. That i s , for guiding layers of thickness below a certain cut-off tc' no propagation is possible. For symmetric struct ure s, the guiding layer can be arbitrari ly thin and guiding is still possible. For the asymmetric laser design shown in Figure 2.12 the cut-off thickness is approxi0 mately 900 A at the lasing frequency. Thus guiding in the conventional sense is not possible and any modes must be of the Bragg l~aveguide Figure 3.5 is a plot of light output as a function of current, indicating a threshold of about 4 amps or 5 ka/cm . It is evident that appro xi mate ly t-=n l ongi tudi na 1 modes are lasing. The roode spacing is approximately 2.8 A. and agrees well with what is expected for mode spacing in a Fabry-Perot cavity 225 Fig.3.3 SEM of a Molecular Beam Epitaxy structure. -226- Fig.3.4 SEM of a Liquid Phase Epitaxy structure . -227- :::l 0.. : ::l ....c. 0'1 -_j Q) 0:: Current in Amps Relative light output as a function of current -228- 8630 8650 8670 8690 Wavelength in A Light output as a function of wavelength 8710 -229- s.e = ( 3. 2) p = integer (3.3) an a~ff %- 0.25, neff= 3.37, A0 % 8670, and .e = 350 ~ (the measured length), we obtain 6A = 3.0 A. This is in agreement thus verifying that the measured modes are longi- tudinal roodes. This is shown in Fiqure 3.7 , where three oscillations under a decaying envelope are apparent, thus indicating Bragg Waveguiding. It is expected that each peak of the intensity profile decreases by a fac2 j. ( ( ~~:) in Fiqure 3.7. The D.C. voltage which is used to rotate the mirrors is measured by the x-input of an x-y recorder, while the output of the The sche- 01 .r::. Q) ·(/) Distance in Lateral Direction (normal to junction plane) Fig . 3.7 Mode Profile- -Light intensity distribution inside Bragg laser Fig . 3 . 8 Integrator Box Car Supply D.C. Power Photomultiplier -·-Slit Galvanometer Experimental Apparatus for Measuring Mode Profile X- Y Recorder y-lnput x-Input Go As Injection Lens ..... -2323.3 Conclusion sions for a , the loss constant, as well as the imaqinary part of the It appears that very little light leaks into the substrate compared with Fabry-Perot losses The sample was designed so as to support only a Bragq type mode, and transverse mode profile measurements have confirmed this. -233CHAPTER 3 REFERENCES Casey , Sell and Panish, Appl. Phys. Lett. 24, 63 (1974).
the dielectric discontinuity is reflected with the amplitude
nL coseL - nR coseR
nL cos8L + nR cos8R
subscripts refer to regions on the left and right side of the di scontinuity.
with
tl ( n cos e) = __;ll::..:n-'------=--
equation (l-B.23) and
~n
dx
Thus for the continuous case being considered, an amplitude--~--~~
n(x) cos 2e
is reflected from a slab of thickness dx.
amplitude of light reflected per unit length.
amplitude incident
the wave-vector k(kx) as being equal to 1r/ A0 , and have considered only
the case of the incident electric field being parallel to the slab layers
(TE modes). That is, the device i s a "quarter wave stack" (k; k 2 n ~ - s2 ) .
Referring to Figure 1-8.1, we seek "' n A 1Tcos o and the expression in
0 0
equation (1-8.24) reduces to (k n1 A )/41T which is in agreement with
equation (1-8.17) for
{3 2 = k~
(l-8.25)
comes
equation (l - B.l 5) be2n0
gives
( 1-B .2 7)
decomposed into a series of sinusoidal variations and the amplitude
of the first harmonic is iTT
A final important case to consider, after the previous cases of
slab structures, is that of waveguiding in a thin film with a periodic
perturbation on a surface, leaking to the transfer of power from one
mode to another (Figs. l-8.2c,d). Note that now, in order to follow
convention the coordinate system has been changed with the periodicity
in the z direction.
sinusoidal slab reflector
{Note coordinate system
used for slab structure)
discontinuous slab
ref lector
dielectric waveguide with
square wave perturbation
{Note coordinate system
used for waveguide structure}
dielectric waveguide
with sinusoidal surface
per turbot ion
equations are appl icab l e .
between a guide of index n 2 and a superstrate of index n1 , the coupling
cons tant for contradirectional coupling i s given by
2 2 2
n m
ln l = 3l A
A/'--'a~--,--,---,-;:;2n ( 2 _ n 2)1/2
n2
2 - n 2) J
11
4n 2
a = height of square wave perturbation
A = wavelength of light
In a slightly different analysis Yen
- nl ) k +
4 K
where teff is the effective width of the guide
Ym
m subscript denotes the ~mode
t defined through B = ~
Flanders 3
lnl = 4 B teff
the same result as for slab structures; again because the square well
can be Fourier decomposed into sinusoidal functions with the amplitude
of the first Fourier harmonic being i7T
lnsinusoidall
Ins quare
arbitrary corrugation, including blazed qratings.
We now proceed to give the solution to the coupled mode equation,
equation (1-8.19). The boundary conditions on equation (l-8.19)for a
stratified structure extending from x = 0 to x = L are
R'(O)
S'(L)
o sinh(TL) + i T cosh(TL)
T ::: jK2- 02
of the guide, it is customary to define the direction along the length
of the guide as the z axis (see Fig.l-8.2). In this case S'(x) -+ S'(z)
and R'(x)-+ R'(z) in equation(l-8.34) and (l-8.35).
Figure l-B.3 2 is a plot of the incident field R(z) = Ei (z) and reflected field S(z) = Er(z) for KL = 1.0 and KL = 4.0 indicating the
effect of increasing coupling.
Except for special cases, such as ~(x) being quadratic in x( 5) or
solved exactly.
nonlinear Riccati equation 7 which can easily be solved by numerical
1Ei(O)I
2.)
The x-derivative of this is
dp = 1'( -28 + ax
d
numerical results are presented el sewhere in this thesis.
This appendix has merely been an outline of some fea tures of
the coupled mode formalism.
references.
(1)
New Yo rk , 1975 ) , p . 51 5 .
Lett. 24, 194 (1974).
}l, 867 (1975).
1359 (1976).
(13)
J. Opt. Soc. Am.).
Quantum Electron. }l, 296 (1977).
BROAD-BAND FILTERS
2.1
In this chapter the properties of a dielectric waveguide with a
iB z
guide; that is, a mode which propagates with a z dependence e 0 will
interact with the grating and be converted into a mode which propagates
-i 8 z
as e
Figure 1.4 of Chapter 1 illustrates this effect. The chirp
This tends also to decrease the amount of light reflected at any one
frequency, thus allowing a simple analysis based on direct integration
of the coupled mode equations and the use of the method of stationary
phase.
The fabrication and testing of several broad-band filters is
covered and the experimental results are found to be in excellent aoreement with theory.
2.2
Following coupled mode theory, we assume the field in the guide can
modes under consideration, and A(O) is the grating period at z = 0.
only interaction is between a particular forward and reverse traveling
mode and that these do not couple to the other modes.
These amplitudes are related by the usual coupled mode equations
which were presented in Appendix 1-B {eq. 1-8.5).
dz
(2. 3b)
11 is
2TT
o is the phase mismatch and o
traveling waves.
the reflection may be small.
dS'
* +2i 6z -iyz
crz=-ne
dS 1
. 2
e-lyz
condition S (L) = 0
n(u + ......2.)
u e1yu
du
extend the limits of integration from minus infinity to positive infinity.
The result is
= 0 otherwisewhere zB = y- 1s the point where the Bragg condition is satisfied.
case of gain (a < 0).
If our reflection is not small, equations (2.2) and (2. 3a) can be
combined to give second order differential equations for R and S.
The solutions of these equations are the parabolic cylinder functions,
for RandS and using the asymptotic expansions for the parabolic
cylinder functions we get
7TK
5K
far from the grating edges. That 1s,
when-< zB < L -5K
--. 3 If 1osses
Next we will take a specific case corresponding to typical re4
flection grating made by the author. For small grating depth
na
K =
2Weff N
ns = index of substrate
nf = index of film
so
N = ko
t = guide thickness
a = grating depth
qo = (So2- nc2 ko2)1 /2
p = (S 2 _ n 2 k 2)1/2
If we take the following values
nf -- 1. 540
ns = 1. 510
N = 1. 524
t =
8 x l o7 m- 1
reflectivity= .21. This agrees favorably with the experimental results
for which the measured reflectivity was 18%.
Thus far we have assumed a linear chirp in ~tz)
make thi s linear approximation 5
s i n2cp
sin2cp
period formed by the interference of two plane waves with '.'Javelength >incident on a surface at angles e1 and o2 is given by sln e~+ s in
For (z - zf)
(2 . 12)
For a general A(z), the coupled mode equations can be written.
· .... ~ \
gratings . (After rPfPrPnrP 13.)
Lens
Con t ainer
dz
dZ
Again, for large chirps the reflection may be small.
2n
-- -;:-y--,......./\{zg) - 28o = 0
(2.14)
d21¥(z8)
dz 2
- /..Xf
/..X f
(zB
dz 2
y 1 [ 1 - (-'IT- - s i n ) ]
the forward traveling mode to backward traveling mode, the liqht had to
be coupled into the end with smaller gratinq period.
we couple to the substrate radiation modes.
= Bo + kn s
J\(zs)
to z8 it will still interfere with the desired reflection.
Most of the light is reflected in a region of width - -about
point.
Combining equations (2.19) and (2.20) we get
1 ~
th e re fl ectlon
lS zs - ZB > 2 v7-Y
a problem in cases where N ~ ns.
expand K(u + z8 ) in equation (2.6) about u = 0, and integrate term by
term.
00
equation (2.19) usually reduces to (2.7) for large chirps.
Glass waveguides were fabricated by sputter deposition of a
and spun-coated at 3600 rpm on the waveguide, resulting in a layer
These are reported in Appendix 2- 2 and have been published 6
The chirped grating was recorded in the photoresist film by
exposing the resist to the interference fringes of a plane wave and
a cylindrically focused beam.
use a prism containing a Xylene solution as shown in Figure 2.1. Since
the index of refraction of the Xylene i s approximately 1 .5, the interfering light which produced the grating had an effective wavelenqth
of
The angle between the plane incident wave and the bisector of
ten seconds in AZ-303A, diluted 6-1 in deionized water.
The grating pattern was then transferred into the glass film by ionbeam etching
higher energies of 2 keV gave good results.
For convenience in later testing, only single mode waveguides
were used and their thicknesses were measured with a Sloan-Dektak.
The refractive index of the substrate was measured by the Brewster
angle method and the index of refraction of the film was determined
10-12
by the prism coupler method.
2.4
Figure 2.3 is a schematic of the filter evaluation setup.
index prism into the waveguide, is reflected contradirectionally back
through the prism, reflected by a beam-splitter and measured by a detector.
light into the end of the grating with smaller period.
attenuation before being reflected.
Dye Laser
Coupler
Beam
the factor e 2azB(A) to obtain the intrinsic filter reflectivity.
Table 2.1 summarizes the properties of three tested filters.
are in good agreement with theoretical predictions (within 10%) ,
while the filter bandwidths are in excellent agreement with the
values for which the devices were designed.
one uniform-period filter shown in detail in the inset.
The author wishes to thank C. S. Hong for much of the experimental
work as well as for his guidance during the fabrication process.
2.5 Conclusion:
been fabricated and a simple theory based on a direct integration of
the coupled mode equations gives results that are consistent with
experiment.
strongly on the value of the coupling constant at the Bragg point
and weakly on the derivatives of this constant at the Bragg point.
Losses have been included in some of the study and it is found
that they lead to a response that decays as e -2azB. This correspon ds to
th" attenuation due to the round tri~ distance to the Bragg point.
25oA
depth (h)
th1ckness (t)
refraction at X= 5950~a
period
response
0~
<.>
Q)
Q)
represents filter l, and dot represents filter 2. Details
of the filter with uniform period are given in the inset.
(After reference 14.)
new round trip distance, z , larger than z for gain and less than z8
for losses.
2.
(1975).
Phys. Lett. 24, 194 (1974).
Quantum Electron. QE-13, 296 (1977).
]2_, 1633 (1977).
Huqhes Aircraft Co., 1972; paper
presented at the Kodak Microelectronics Seminar, San Diego, Cal.
Dec. 11-1 2, 1972.)
App. Phy. Lett. ll• 276 (1977).
We start with the coupled mode equations
dR
. 2
dZ- i 6R = - nSe 1 yz
After making the substitution
R' = Re-i oz
we find
dS'-_ -n *R' e (2i 6z - i yz 2)
dz
equation for the parabolic cylinder functions
z 8 = Braqq point = ~
n = iK
In equation (2-A.8) WK ,]J (z) is the Whittaker function and G and H are
constants of integration to be determined by the boundary conditions. After
substituting equation (2-A.8) into equation (2-A.5), s• is found to be
-K,l.l (-iyx )- 2(l.l + K + -2)( l.l- K- -2) •
(2-A.9) and using the asymptotic expansion for WK;v' we find
R I ( 0) = P(x) R 1 (x) + Q* (x) S 1 (x)
e Y (yx )
1T
- K ei
together with equations (2-A. 12) and (2-A. 13) we finally derive the expressions
RI ( L)
expressions for the parabolic cylinder functions, the followin~ must
hold
5K < z
cannot be near the edge of the grating region.
contained in equation (2-A. 15).
values of K /y . The power reflected at the far right side (z = L) is
zero from the boundary conditions, while the reflectivity at z = 0, which
is the most important single parameter of the reflector, is given by
equation (2-A. 15) if equation (2-A. 16) is satisfied.
certain asymptotic limits, the plots were made by directly solving equations (2.2) and (2.3a) of the main body of this chapter numerically.
increases as K increases, it is a 1so necessary to increase the 1ength
of the gratinq asK increases if a steady state non-oscillatory solution
near z = 0 is desired.
(Z =U
reflectivity.
LL
0::: 0.8
-6
(Z =Ol
120.0
= 35.88
(Z = Ll
reflectivity.
(Z = Ol
reflectivity .
(Z=O)
(Z =L)
0'1
LINEARITY AND ENHANCED SENSITIVITY OF
THE SHIPLEY AZ-13508 PHOTORESIST
Current work in integrated optics requires the fabrication of
relief grating structures on photoresist and the subsequent chemical
or ion beam etching through the photoresist. 1 If the period of the
grating is to be less than .4~ the Shipley AZ-l350C photoresist is
commonly used.
have been examined in detail 2 • 3 and it was found that the photoresist
Bartolini 4 • 5 and others 6 have shown that
photoresist removes the nonlinearity and improves the sensitivity by
a factor of two or three.
second.
sensitivity and linearity.
Bartolini
6d = T[r 1 - 6r exp(-cE)]
characteristic of the photoresist, r 1 is the rate of etching of exposed
molecules, and r 2 is the rate of etching of unexposed ones and Ar = (r 1 -r2 ).
If the term cE is much less than l, equation (2-8. l) can be linearized as
follows:
and for some experiments the back surface was painted black with
3M Nextel 101-ClO velvet coating.
rpm for 30 seconds. The samples were baked
rate of the unexposed resist as a function of development time, for
various solutions of AZ-303A developer with distilled water.
baked under vacuum at 100°C for 30 min.
(2-B.l). The 4:1 solution (4 parts distilled water, 1 part AZ-303A developer) gave unacceptably high etch rat~ and the 8:1 gave low and nonlinear ones.
AZ-303A
(_)
Fig. 2-B.l
ratios of AZ-303A developer. The slope of the Hurves determines r 2 , which for the 6:1 dilution is 35A0 ± 5A.
per sec.
For comparison, the experiment was repeated using the Shipley
MF-312 developer.
semiconductor substrates.
is 5~ ±1~ per sec, which
is a much lower etch rate than the one for AZ-303 developer. If resist
recommended dilution of 1:1.
linearity is unimportant, this may indeed be a better choice.
To demonstrate the thickness change as a function of exposure time,
the .4416 ~ m line of a He-Cd laser was used to illuminate half of the
sample.
AZ-303A developer 6:1 dilution.
of the photoresist in the important ranqe .1 ~to .2 ~m is linear.
the same range of exposures.
and then tested.
development time An mi~utes for MF-312 developer . The 1:1 dilution
results in r 2 = 51\ ± 1 .
...........,
__,
• MF- 3 I 2 ( I : I )
DEVELOPMENT TIME= 10 sec
E in mw/cm2. The circles represent the AZ-303A developer and the squares represent the MF - 312 developer .
(f)
......__
the AZ-303A developer in making high efficiency gratings, the following experiment was performed.
intensity distribution produced by the interference pattern of t~to colThe wavelength used was .4579 ~m. the angle
between the beams 94.5°, and the intensity per beam was .60 mw/cm .
limated Ar+ laser beams.
vacuum, and the efficiency of the gratings was measured.
for two different development times.
.31 ~m. and the peak to trough height was .28 ~m.
In conclusion, it has been found that the use of AZ-303A developer
with the AZ-13508 photoresist results in an unexposed etch rate of
the photoresist in the .05 to .2 ~m range. Gratings with constant and
variable period 9 have been made using this method, and they have been
transferred to glass usinq ion beam etchinq techniques.
sec development times in AZ- 303A deve loper . Initial resist thic kness was 31 ~m .
50
60
1.
Opt. 1£, 455 (1973}.
s. Austin, F. T. Stone, Appl. Opt. .!_?_. 2126 (1976).
M. s. Htoo, Photogr. Sci. Eng. g, 169 ( 1968).
3.1
In the last chapter we dealt with the topic of chirped gratings
3000~, light will be coupled out of the guide and into the air and the
substrate.
focused to a line parallel to the quide and normal to the propaaation
direction, with different wavelengths focusing in different positions.
Thus long grating regions can be used to efficiently couple out weak signals and focus them to a line where they can easily be detected.
In this chapter we will consider in greater detail the des ign
considerations involved in making chirped gratings.
of grating region and recording geometry are established.
is found and compared with experiment.
3.2
The grating is fabricated similarly to the broad band f il t er
recording plate is located at the x ~ 0 plane, the angle of incidence
of the plane wave is 8/2, and the angle subtended by the colli mated beam
and the bisector of the converging beam angle is e.
width d, and the focus is located at point P(xf,zf).
Simple geometrical calculations relate the focal line coordinates
with f, L, d·and 8,namely 1
zf
¢ = tan -1 ( d2f )
is the convergence half angle.
the angles 8 and ¢.
The electric field in the recording plane (x ~ O) is given by the
sum of the reference wave and converging one and is given by:
E(x=O,z) = Ae-i kzsin( 8/2)
amplitudes of the plane and converging wave respectively.
chirped gratings.
to EE*
, and that A= a,then
t = 8[1 + cos{kzsin(0/2) + k/(z-zf) 2 + xf 2}]
where S is a proportionality constant.
(3.5)
( z-zf ) 2<<
In the pal~axla
xf
Xf
/\(z) =
It is seen from Eqs. (3.1), (3.2) and (3.5) that the period variation
angle, >-the wavelength of illumination, and L the lengt h of the grating.
The dependence of the period variation on F is illustrated by
3.2 and 3.3. In Fig. 3.2 the angle
length of l em.
A= .I.J 579 f-Lm
0.0
0.2
0.4
Fig. 3.2
lens ( F ~ f/d).
of
0 9CY'
>- = .11579 pm
~. o
0.0
to .28 f.Jm, again over a distance of l em.
F =1.33
A.= .IJ ~79 f-Lm
...--...
o_
the bisector of the converging wave). The F number of the lens
is 1.33 and the illumination v-1avelength is .4579 lJin.
cording distance is kept constant at 1 em .
>- - .LlS-f9;1m
L in em
o._
0.0
QLl
Fig. 3.5
tot a 1 amount of chirp is
angle
over
Thi~
extends over a distance of l em.
.5)Jm.
The linearity of the period variation as a function of grating length
L is shown in Fig. 3. 5.
3.3 Waveguide Coupling
Chirped grating etched onto a dielectric waveguide results in a
simultaneous output coupling and focusing to a point P(xA.,zA.) which
will vary as a function of the modes supported by the wa·teg ui de and the
wavelength of the guided modes.
Consider the geometry described by Fig. 3.6.
e-iSz, where s = kn cos e . When the wave reaches the perturbation
-ikzz
the radiated mode will have a z dependence given by e
. At po1nt
z=O, kz is given by
waveguide of index n 1 .
wavelength .
( 3. 9)
for light incident on a periodic medium.
k (z=L)
3 z
kz(L)
zA = .,...t-an--=e,_L-_-t.,...a_n__,..e- =
(3.11)
function of wavelength and period variation is illustrated by Fig. 3.7.
Taking n
It can be seen from this figure that (a) the larger the period variation
the closer to the waveguide the locus of the focal points will be,
(b) the smaller the period variation the larger the separation between
the different wavelengths and the larger the distance of the lo cus of the
focal points from the waveguide, and (c) if tl1e average period of
the c~irped grating is increased the focus will shift towa rds greater
values of z.
• .5500flm
• .5145 fLm
-t .'1880 f.L.Ill
.L1579prn
... ··• ···
(_)
- - .3LJ - .29fLm
······· .38 - .34f-Lm
Fig. 3.7 LocJs
A(O) is the lon gest period and A(l em) is the shortest.
(After reference 1.)
In the previous section the characteristics of the chirped gratings
by a chirped grating.
To analyze this problem we expand the electric field of the
perturbed waveguide in terms of the guided modes, the substrate modes
and the air modes. This work is essentially an extension of Marcuse's
work( 3 ) for which the symmetric case was treated. In our case the waveguide is not symmetric,
A closed form solution for the ·power radiated into air by a
chirped grating is presented and the solution is illustrated with
examples of gratings where the amount of chirp and the wavelength of
the guided radiation is varied.
Consider the geometry and notation as presented in Fig. 3.Ra.
Using the results from Appendix 1-A, we have for the TE guided modes
Fig. 3.8a
3.8b
Dielectric waveguide with a chirped grating e tched on the top
surface.
substrate and air, k is the wavenumber in air and B describes the
z dependence
Furthermore,
o)
tan Kd = K(y+
K -yo
carried by the mode 1 namely:
4K 2w).1 0 P
00
0 -oo
w is the radian frequency and p 0 is the magnetic permeability of
vacuum.
These guided modes occur for kn 2 ~ lsi ~ kn 1 .
0 < IBI
want to calculate the power radiated by the waveguide into the air.
Appendix 1-A gives the electric field as:
(3.22)
(sinod + F.cosod)sinp(x+d)]
6 = (n32k2-82) l/2
[(cosa d- Fisinod) 2 + -a 2 (sinod + F .cosod) 2 + (l +~F. 2 ).;:_T
It is instructive to compare the solution for a confined, guided
mode with that of the above continuum modes.
tions (E
and ~) at x=O and x=-d give
oX
Maxwell •s equations are second order and the field must be found in three
regions, we have to determine six constants using five conditions.
This cannot be done for the continuum case, and thus we have one
degree of freedom indicated by the
coefficients, with the well-known result being an eigenvalue equation
given by equation (3. 18).
Since the F.1 is arbitrary, it is convenient to choose two fixed
values F1 and F2 .
modes are orthogonal to one another
+ 2(p/!i){o -r/)(o -l'l ) · cos2od +(p 2 / 6 2 )(i-A 2 ) 2] 112 }
J ~Y (p)~Y (p' )dx
Next we expand an arbitrary T E electric field for the
perturbed waveguide in terms of the discrete guided modes and the
continuum of both substrate and air modes:
k ( n22- n32) 1/2
discrete
2 2)1/2
oddk(n2-n3
equation (3.18).
of F2 and F1 (Eq. 3.27). f s are the substrate modes which have not
been presented explicitly since they will not affect thi s cal cul ation.
It is to be noted that the above expansion for the total el ectric field
EY i s possible since the set of eigenfunctions is complete.
of the choice of Fi.
The limits of inteoration in equation (3.28) were found from
equation (3.25) and the requirement that for guided modes kn 2 < B < kn 1 ,
for substrate modes kn
To determine the value of h( p ,z), we substitute equation (3.28)
into the Helmholtz wave equation, multiply by ~~a. inteqrate over x,
and using the orthogonality relations, get a differential equation f or
h ( p, z).
a h
. ah
- 2 - 21Saz- = H(p,z)
equation.
where
kn2
+ J
constant from that of a uniform waveguide.
Note that for .6n 2 ~o, H(p,z)=O, and we have
e-iBz, we get the sum of a forward traveling mode with z dependence
e- i Bz and a backward traveling mode with z dependence e+i Bz (time
dependence e+"lw t ).
words, we use Cn(O) ~ con' where o0 n is the Kronecker delta, instead
of Cn(z) and set g(p,z) = h(p,z) = 0 in Ea. (3.30), resulting in
( 3. 31 )
-00
its ideal shape is on the top surface of the quide as shown in Fiq. 3.8b.
By taking a sha 11 ow grati nq and thus setting x = 0 in the above equation
we get
Equation (3.29) can then be divided into parts as follows:
h+ = Q- ~
H(p,s)ds
electric field arises from the product of h( p,z) · f.>a(p ,z). The z
dependence off a (p,z) 1s
e -i Bz . If we consider the z dependence of
the product, then
(z dependence)
forward traveling radiating mode and the term in brackets with the
negative traveling one.
The power radiated into air is given by
t,p
into air
even
odd
of the air mode radiated into the air. The remainder is, of course,
radiated into the substrate.
h+ (p.z=O) = 0
f H(p,s)ds
h+ = - 2TB
2 2
k (n - n )
h (p,L) =
cj>+ AC
4iw~P
3 cj>
dz
air we use equations (3.20), (3.21) and (3.22)
1 +a
1 +v. + w. +-- F.
= ~ [j
!2+
cp+
air per unit B
'+'
62 1
(3.50)
an arbitrary perturbation on the top surface.
For the particular case of the chirped grating with a transmission function given by equation (3.4), f(z) can be written as
method of stationary phase results in
(3 . 52)
if
" 0
(c)
l
= 4Y
of 8 for which the guide radiates.
8 - Po
- i\TZT
2rr
az
B - 80 +
conditions given in equations (3.52a,b,c). Since we have taken 8 as an
inherently positive quantity, the case of backward scattering must be considered separately and the equation governing it is
To ill.ustrate equation (3.48) we present Figure 3.9. The guide is 1.0 em
long, its thickness is .6425pm and tt1c index ot refract1on n
n3 = 1.0.
The calculation for the fundamental mode gave
x l0 7 (m- 1 ) corresponding to a wave number of 9.78 x 10 6 m- 1 .
as a function of 8
the limit of no chirp we expect the familiar delta function.
theoretically possible range (kz = 0 to k).
from 10 to 15 percent of the incident mode power.
The sharp vertical boundaries to the curves are easily interpreted as follows.
, =
Using Bo = 1.505 X 10 7/m and A = .35~ we find a'
-3.0 X 10 /m.
Similarly for A = .4, 8' = -.66 x 106;m.
.2 1o / 1 p rn
2 . 2 ~> 1o '1 11m
3 .:-s to .'1 1Lm
4 .3~> 1o A f Lm
<('
L = 1.0 ern
6 G0
(d egrees)
function of B or e, where e is the angle of scattering with
respect to the z-axis (O=TI- 8 0 , see Fi9. 3.6) for various
chirps. The area under each curve represents the total
power radiated into air for a qiven chirp. (After reference
1.)
It is important to note that in the actual calculation the waveguide is divided into approximately one hundred sections.
This new value of P is used as input power for the next section, and so
on.
Fig. 3.10 .
has an index of refraction n2 = 1.51 and air n = 1.0. The guide is
1.1 em long and again the perturbation on the top surface is given by equation (3.51). In this case a is 5ooR, and the period varies from .295 to
.33 ~m.
S ranges.
In addition, we have calculated the fractional power (per unit
B) radiated into air and substrate, and found that as predicted by the
theory 5 , it is twice as large as the one radiated into air.
/l5 /9;Lill
.51 1J5;'-rn
LJJ
nl = 1.~()~
5
n3 '-'1.0
LL
functi on of B for a chirp of
wave l engths .
reference 1 .)
The dielectric wavequides were made, as described in the previous
5.5 ion beam etching machine.
with a Sloan Dektak, was 1 .35~.
Chirped gratings were fabricated on the surface of the waveguide
as follows: a layer of undiluted Shipley AZ1350B photoresist was spin
coated at 6000 rpm on the waveguide.
a converging beam.
resulting in gratings with periorls varyinq from 0.29 to 0.33 ~m over a
distance of 1.2 em.
Typically the laser beam intensity was 0.6 mw/cm 2 (in each leg)
and the exposure time used was 60 sec.
The photoresist was next post baked under vacuum for 30 min and
the waveguide was ion beam etched through the photoresist, at ion current
density 0.1 mA/cm 2 and accelerating voltage of 1800V, for 30 min. The
sample was kept at an angle of 30° with respect to the ion beam.
about 500~.
into the waveguide using a prism coupler.
argon laser.
An output prism coupler was added at the end of the corrugated
region.
3.6
In this work ·the properties of chirped grating output couplers
As one important application of this device, the focusing effect in a
waveguide incorporating chirped grating was demonstrated.
light about 6 em away from the waveguide.
about 1-2 em when the wavelength was changed from 4579~ to 5145~.
waveguide.
'-----~o~~
6 .~--------------;~~~ o
5145.&
o 4880A
corrugated s tructurc used .
The solid dots represent the focus of the prominent lines of
the Ar+ laser.
amount of lioht radiated out of the guide at various anqles.
Quant um Elect . OE-13, 296 (1977).
New York, 1974).
R· 494 ( 1976).
STATISTICAL ANALYSIS OF BRAGG REFLECTORS
In previous chapters we considered the effects of introducing a
predetermined and controlled a periodicity in an opti ca 1 structure.
aperiodicity in a multilayer reflector.
properties of an ensemble.
These expressions are then compared with values obtained using a co~p uter
routine whicll "builds" a reflector with the desired parameters and a
value, and then calculates the reflection.
4.1
ideal periodic multilayered media. 1 •2 Amon9 the many uses of such structures are coatings for both high reflection and antireflection.
in stratified media composed of isotropic or cubic materials 6 • 7 .
In practice, however, it is not possible to fabricate perfect structures, and to date the standard deviation in layer thicknesses of commer-
greater when measured mechanically 8 •9 . Great precision in layer thickness
can be achieved by using new techniques such as molecular-beam epitaxy,
but these techniques are also costlier than the standard electron beam
evaporation.
Although there is ample literature on periodic structures, the study
of aperiodic structures has been rather limitedlO,ll •12 . The primary effect
of a slight aperiodicity is to decrease the amplitude and broaden the
width of the reflectivity spectrum .
than ideal reflectance from a mirror.
What follows is simply an analysis of one of the imperfections, namely a
statistical fluctuation in layer thickness about some predetermined mean.
The case of a low reflectivity structure is easily handled using
the undepleted incident wave approximation.
Yariv and Hong 2 to predict the expectation value of p and IPI 2 as a function of a as well as p(IPI 2 ), the probability of manufacturing a sample
of given reflection.
We start by calculating the reflectivity of a mirror with N cells in
p=rl[l+e
+ ...
2i[k 1 (a 1+a 2 )+k 2 (b 1+b 2 )]
+e
_x
in a layer of material 2.
the reflection from a single layer.
ap is the thickness of the layer of
in the pth cell (see Figure 4.1).
We denote the random deviation of the layers' thickness by parameters up, vp defined by
a = a(o) + u
P P distribution and standard deviations oa and ob
--'
This can easily be shown by expanding eiG in a Taylor series and averaging
term by term.
A typical term in (4.1) is
2ip[k
= e
2x
21. p [k 1 a (o) + k2x b(o)] -2[k 1 xpa a +k 2x pab ]
2 + k2
k2a 2 =
- lx a a
2x a b
1 - e
_ e
2k 2 2
2x a b (l _ e2iNkJ\ e-2Nk a )}
case N = 25 and r 1 = 1.96 x 10- 3 . We have taken k~x a~ = k~x a! =
k2a 2 and p = • 1 for a = 0 at the center of the band qap, as well
2x
1x
value of a has the effect of broadenina the response as well as lowering
center of the band gap, indicating the broadening and lowering of the response
curve for the case of 25 cells.
0 .31 4
0.0
0.942
.57
0 .628
.25""'
variable o aives the chirp of the mirror.
statistical variation illustrated in Fiqure 4.2.
indicating the increasing sensitivity of
a perfect reflector.
It is interesting to compare this expression to the well-known DebyeWaller factor for X-ray diffraction from a crystal at a finite temperature
for which 13
a = standard deviation in atomic position due to lattice
0 .6
ko
N= 10
give a 10% reflectance for a perfect reflector.
the correction factor in (4.4a) does depend on N.
The difference between ( 4. 4a) and ( 4. 5) can be reconciled.
precisely, but for which the surface of each layer is not perfect but is
rough and uneven, then the reflection from the entire structure i s reduced
by a term similar to (4.5), that is, independent of N. This can be seen
as follows. The reflection from a rough surface is given 14 by
~ is the wavelength of linht reflected, 1 is the standard deviation of
from each surface of a Braqq reflector is reduced by the factor exp[- ~ a~.J.
Thus for a structure of many layers, we have
complicated expression:
2 2
e-2Nk a
2 2
2 2 2 [1 - (N+l)e2k J N
( 1 _ e2k a )
2 2
a [l-Ne2ka(N-1)
2 2
-2(N-l)k
+Ne2N+1)ka] + N+2e
222
(l _ /k a )
+ (N-1)e2Nk a] - 2 cos(2k
-2EN-l)k 2a 2
lxaa)
222
.-1
( 1 _ e2k a )
same conditions as Figure 4.4. Although equation (4.7) is quite complicated
it reduces to [ ( 2N+ 1) r ] z for ka-+0 and to ( 2N+ 1 )( r 21 ) as ka ~ ""
This
is to be expected, since as a ~ 0, the reflections from each dielectric
interface are correlated and thus the amplitudes add.
intensities from the 2N+1 interfaces add.
N= 10
Fig. 4.5 Average intensity reflectivity in low reflectance li mit as a function of cell standard deviation and the number of cells. Note how, as ko becomes large, the asymptotic reflectance
becomes .Ol / (2N+l).
In many cases of practical interest, we deal with high reflecti-
prob 1em can be overcome by using the coupled mode theor) 5 ' 16 and the effects
of a random statistical variation in layer thickness can be included by using
a perturbation scheme similar to Streifer et al. 17 , but carried to a higher
order.
Consider a periodic structure which extends from z = -L/2 to z = L/2.
A wave propagating in the z direction, R(z)eiBz, with time dependence e-i wt
will generate a contradirectional wave S(z) e-isz.
period. Although the dielectric constant of a periodic slab guide does not
vary sinusoidally, we can decompose the index variation into its Fourier components and allow coherent interaction with the propagating wave and the first
Fourier harmonic of the structure.
The coupled mode equations at the Bragg condition S = n/ A are
In equations (4.9) and (4.10)
amplitude reflected/unit length l
amplitude incident
R(- 2 L) = 1
S(2
I R(- 2) = S(- 2)
(4.13)
Next we expand R in a power series in c
(4. 14)
we get
Rl
(4. 11) and (4. 12) are solved in Appendix 4-A where it is shown
Ro = cosh KL
n = 1 ,2 , ...
a different reflection since ccp '(z) is a random variable for each
structure.
w~
(4.19a)
(4 .19b)
Although other forms can be considered, the final result will not depend
on the exact form because of subsequent approximations (Appendix 4-B).
Also (4. 19b) can be shown to be the unique autocorrelation funct i on for
the case of a slab reflector (Appendix 4-D).
The quantity .t is a correlation length and
standard deviation of the random variable.
Using the results of Appendices 4-B and 4-C, we arrive at the
L > = tanh 2 KL
(4. 23)
L > =tanh 2 KL- E
K e
(2KL- 1)
and, a(o), b(O), and a 2 of the slab
In order to apply the results of the last section which assumed
a sinusoidal variation of the index to the case of multilayered mirrors with abrupt index discontinuities, we establish the connection
between the parameters used in characterizing these systems .
We start by defining the local period through the relationship
xczr
dz =
Next we take the auto-correlation function of 6/\.(z) to be
From equation (4.29) we see immediately that
( 4.30)
reflector is given by
A )
Ao
from its ideal thickness of~
Comparing equations (4.32) and (4. 29)
we see that the correlation length ~ is equal to the slab thickness
A0 /2. In order to find the connection between ;2 and E (and thus s
through equation (4.30), we compare either equation (4. 22) to equa t ion
(4.5), or alternatively, (4.23) to (4.8)(in limit of large N).
either case for the equations to agree, we must take
(4.34)
appear in equations (4.20) throuoh (4.25).
Also by comparing equation (4.5) to (4.22) or equation (4.8) to
(4.23) for large N, we again find K = (2Nr 1 )/L.
After using equation (4.34) and t
a n
A3
c4
0 K 1
1 S
4 - T6 4
or
tanh 2x
while for large x, G(x) ~ 2e- 2x(2- l).
If equ~tions (4.9) and (4. 10) are used directly and a series expansion is used in both R and S, we can avoid havinq to use equation (4. 13)
which involves ~ · and instead work only with the random variable
~.
R'
( 4.38)
\>Jhere
u. + -w n
v.
2 . 1
1=
v.1 are given in equation (4.2).
We define the random variable x = iK(eie:¢_1), assume it is small
and expand R and S in a series, with the n!b. term being of order x"'
_J
no , Sn(L) = 0. After substituting
(4.42) and (4.43) into ( 4. 39) and (4.40) and collecting terms of order
we find
R'
0 = iKS
( 4A4)
s• = -i KR
n-1
n.
= cosh
Using the boundary conditions on s , integrating by parts to get rid
of the derivative of x* and using equations (4.49) and (4.50) we find
pp* =
( 4. 54)
Or2 = - K tanh KL
cosh 2KL
2 2
2 2
2 2
-I\
[ 2
'I' =
-2 nl a a
-'l'x
(1 - e
)dz=c3
<6r 2>
ro
accurate as we shall see in the next section.
4.5
The analytical results of the last section are compared with cal-
stratified media.
distribution about predetermined thicknesses a 0 and b0 of the n1 and
n2 layers respectively. The same relative s tandard deviation was used
for all layers , i.e. cra/a(o) = ab/b(o). The reflectivity was ca lc ulated
using the matrix multiplication method.
Samples of 50 cells each were prepared this way.
50 cells.
b(o) = nl J\
n1+n 2
incidence.
less than Rp =
as o
standard deviation of the computed quantity
o
- - where N is the number of structures tested.
The results of the computer experiment are presented in Table l.
For comparison, results are also given for the two theories.
results are plotted in Fiaure 4.7.
.02
In Figure 4.8 are the results for a structure with 10 unit cells and
various values of or.
small values of or.
Finally Figures 4.9 through 4.14 illustrate the probability distribution function for various reflections. The points were determined by
the computer routine, while the solid line represents the theoretical
prediction which is described in the next section.
Figures 4.9, 4.10 and 4.11 give the probability distribution for 10, 25,
and 50 cells with relative standard deviation of 2%. Notice how the distribution is broader for 25 cells than for 10 or 50 cells.
and plot the probability distributions for relative standard deviations
of 5%, 7.5% and 10%.
The computer routine used in this analysis is presented in
Appendix 4-E.
-.26420
8.07XI0- 3
---· 5.49X 10- 3
6.53 xlo- 3
--
.._
. 3
1. 12 ±.01 XI0-
-+---
the close agreement between the second order ~ · expression and the experi ment.
- -
---- o I st order theory with cp
r Computer Experiment
various values of or.
25.0
Rp=.266795
PEAK ( R=.2655) =19.07 °/o
OF THE SAMPLES
dR =5.0E -04
27.0
Rp=.794689
PEAK ( R=.7920) = 16.13 °/o
OF THE SAMPLES
dR = I.OE -03
98.0
Rp=.986913
PEAK ( R=.9862) = 19.87 °/o
OF THE SAMPLES
dR =2.0E -04
98.8
6.25
"'0
~~
a~ 0 o£??:
Rp=.794689
PEAK ( R =. 7800) =23.67 °/o
OF THE SAMPLES
dR = I.OE -02
Rp=.794689
PEAK ( R=. 7500) =10.40 °/o
OF THE SAMPLES
dR =I.OE -02
~~
Rp=.794689
PEAK ( R=. 7300) = 5.87 °/o
OF THE SAMPLES
dR =I.OE -02
0 0
0o
60.0
As seen from Figures 4.9-4.14
in distribution, or the most likely value of reflectivity.
does not answer the following important question.
80%, a substantial number of reflectors reflect less than 60%.
Based on Figures 4.9-4.14 we will fit these data to the function
p(r) = r {q-1)
reflection from a perfect structure, and the parameters C and q are
determined from the average value and standard deviation of the distribution function.
It is easily shown that the parameters C and q are related to
the average and standard deviation of p(r) through
(r
q = 3 +
-
important point should be made.
been retained.
Nevertheless the result s are in good agreeme nt with experiment
as well as the more accurate second order res ults and because of its
simplicity expression (4.55) will be used to compute
Equation (4.55) gives us
(4.66)
keeping only the lowest order term in a , we find
or the interesting result
-
15
r
line and agrees well with the computer results.
Finally from equation (4.70) we find that the peak of the function
p(r) occurs at the point.
( 4.71)
rp-
As a final example we consider a multilayered dielectric mirror
HL(LH), 4 Air (S =Substrate, H =High index material, L = low index
material) and is depicted in Figure 4.15.
(for a perfect structure) of .986.
Twelve such devices were built and tested and the followin g reflections were measured:
.946
.972
.966
.954
.964
.973
.974
. 974
.986
.972
. 980
.974
(4.72)
<<
//
Fig. 4,15
calculations do not consider any systematic errors.
to changing temperatures and growing conditions.
Using equation (4.59) with the following variable values
1J! =
the experimental value of /
agreement with (4.73).
The effect on reflectivity of a statistical variation in the
coupled mode equations.
agreed well with the experiment.
found to be consistent.
In this appendix we solve
RI I
n-1
R(-
and
The procedure for determining R0 is straightforward and the
cosh[K(h - z)]
- cosh(K.L)
(4-A.7)
A homogeneous solution to (4-A.2) is given by
where Bn is a constant to be determined.
-1
-L/2
( 4-A. 2).
-L/2
determined through the boundary condition R~ (f) = 0.
is given by equation (4. 17).
We wish to solve for the expectation value of p (-~) ;
given that <4> 1(z) ¢ 1(z + z ) > ; f (1 and using equations
(4-8.1)
After differentiation we find from equation (4.17)
Equation (4.18) also gives
L ; --i
R I ( -71)
(4-8.3)
We now use equation (4. 17) to express Rl in terms of R and 4> 1.
2 > ; -K
c2 L/2J
K(2L - n )]d ~ dn
From equation (4.19)
This is a quite reasonable approximation and is good whenever Kt
J o(x) f(x) dx = f(O)
2 2
equation (4.20) .
is calculated in a fashion
The expectation value of p(-2)
p*(2)
L) = - ie
P ( --
+ E2
2 l) p*(-l
2 L) > = __
o 2
expectation value of R2 has been calculated in Appendix 4-B. Thus
we need only determine
c4
sinh[K( 2L - n)] sinh[K( 2L - s )J cosh[K( 2L - n )] cosh[K(2using {4-8.5) this reduces to
c.
32 4 - B K)
(4.21).
In this appendix we solve Rt(z 0 ) = <~t(z) ~t(z + z 0 ) >
Equation (4-0. l) can be interpreted as
From reference 18 we have
ao
Pn
otherwise
ao
If we now combine (4-0.2)
/\.o/2
Computer program used as a check of the analytical expressions
9 - . Ja rr - 7 H 9 : ':> 1 : f) 4
C PI ()f<:: r'(R>liR (llt~:n~ll
c HAS THI· flf'J tnN m I· I II) l INh fiN ffil .. Yf·fo't'l'< (li'J\1 <.:l'tol 1!11:
C ( t-· R U U R A MM~ I I H Y I ' • A l ~ M1 lN >
l.l 'I MF: NS TON h:F ( ~r)()() > • 1111 ( ,A; > • ·-:t; r I <'c'o ·t ·,
n 1M f N s 1 n N 1' A~; ( 1 > • f\1' s ( 1 > , H N' c 4 ' • '.1.· A 1• , 1. '
TI 'f MI N S T 0 N Y ( I 0 0 0 ·t ) , X r I '·' (\ i) ·1 > ., t I · c 'I '• • Y I · ; : ' 1
T11 M~· N s T n N t·U'l x < -' ' • fU1 y < ·' ) • , ",· t· , : , •, > • '., 11 ,.~ c t • ' •• ·, "1 · • " •
Rl- AI N1 • N : 1
f'q ll JHI F F' ~\ f l ' I ~ : I ( l N N f'l Mf • r: I • '( N N • I \1-'
f ClMMUN I Nf'lM('O I /NA MI·
COMMDNIXHF'/Xfll'l : 1. 1:
Tl A T A N t , N '. 1 / . ~ , t. • ~ , 4 I
\1 A T A R A s I '
998
1-ClRMAT<1 X •A 4l
t. ··, 1
wf\ J n <~'i • '~ • , o >
9 :"i 0
f IH\ MA T < I X Y ' l Y N I· W' >
AI' I' I f' T 6<' • Nf 1.<1
1> 0
F rm MA 1 <1 : • • o '
T F ( N f· w• f 0 • () ) I i I l l I I I ::,
A(' C f f·' T "·, 0 • I· I • N ~ i M I · • I '\ Y
50
F" 0 R MA 1
X N N ~- ( N l I N ~:> > * :>
T rf <; T = I
'>LBI ~· • 21
STT 1..=•• . : 1 '1.
SSCI
• lj
ST 1 f'K :: . ;> 1
f~ FA ll ( 1 • 1 0 1 ) N I ; v <:)I I • f\ t~·
·1. 0 l
f " 0 R MA T ( .I ~ • t >
RF' ::: ( ( t . -X NN**Nt >/ ( 1 . t XNN:tt:I'JI' >I *:!< :1
., I. I (\ • ,', I
F () R MA ·y ( 1 X • ' H II ' • I :•) • ,~ y • ~X • ' N 1 .,. ' " I ·.~; " •. '< • ' I\ I '
REA 11 < :1 , 1 0 0 >
100
Fllf''MA I ( ·t 01 I./,>
CON T I' NU E
4H
FfH;M()f(FlO •• ~/~lO •. ~/~·.IO.:~/II\l,:) / 1 I~> -~-' ()H•
J F < [I r V • 1.. F • 0 • ) (:; T U ~:·
OF
MINX =KX
XMAX=KX
YMAX:· nMAX
RA·::(),
[I()
X N <:; ·:·· N ~;; Mf·'
h:A::::h'f', /X N~;
WF< J l F ( :::; ~ -~ ~3) Sli v Nf.' v 1\l ·
WRI1F(~•10~>RA,N~MP
::> ()
CDN T I Nllf·
ACCF F'T 4<;, li I
FORMATCF10.3)
nn r.:, ·1 ::=1 .. ·u>oo·1
s R < r > == o •
{I il .:> I :::: 'I • N !~ MF'
K :::: II .1 \J »: h: I ( I > t I , ::=;
~; F< ( I\ > =:: S f\:
CClN I I'Nl.IF
h:l·'h ::c()
l F ( <:; h' ( K ) • I T • I·\ I·'" ) ( j ( ) l I l ' ) ::>
RPI\::::C)f\' ( K)
KF'K c:l·;
1:< K I ·' I\ ·:" ( KP K -· I ) /
700
XMN==h:M J N* 1 00,
XMX-.. h:MAX* t 00.
YMN .... O.
YMX::::<.;MAX
NCC ==NC
!:iJ'il-'h' ··SD* ·1 00.
FNf~Ul'IF ( l 0 • !l~,~-'j • Hl lf') !:\t iPI~
F () 1:< MA T ( 1 !-\ • JI • ::: 1 ~ 1.. 4 < 'I • 'i,, 1 'o
FNCOlil ( .~H v J 1 0 ~ f'I'AI\) 1:<1' I·' I\ • XF'I'F n F.: Mf'l r ( ' P FA K c F<::: ' • 1- ? • 1. , , ' "" • 1 1, • •: • • ·(
ENC('IOF ( 1 ::'i, 700, <.;Nl' > N1 I
F. !W MA r < ' :lf: 0 F C' F I I ~-; : ' , I ..~ >
Fh'A:::I:i•A
I 0 ..'
FUR M A T ( ' ~.: ~' "- ' • I 'l . f> • 3 X • ' h' .. ' • I I , h '
E N C U Die < :1 1 • 7 (\ ::>• S F< 1.. ~:; ) )I h' I· ! :
IC' () F< MA 1 ( ' I IF~ . ' y 1 1·:· 1:. 8 • l )
F 0 F~ MA I < ' F< F S 01 U T J UN:..: X . • I 4 l
R A )( 'I :::: ( h' A .... f~ M I N ) / ( 1:\ M (.~ X I ' 1'1 I N ) I I ., • + • ) : I
FW X :::: ( ~\ ~·· -- F\ M I N ) I ( I·' Mt"l x .. I< M I N ' I '-, • t , ' ' •
( ' () I 1.. I· h.: A U F·
CAt I I A B F 1. ( 0 • • 0 , • X MN • X M X • I ..., , • 4 • h'l·:·1 I I I ' I 1 \ 1 I t Y ( ·x. • ' • I '"! • r\ •
F'FRMX::::Y MX/NSMF'
C A l.. L. I () l
' . tH'll 'I I ' . ' ,. I <' • I •
F' R M:::: P f· 1·\ MX II f V
C A I L. I A H F I. < l !':, , • 0 • ~ Y MN • I·' h: ~1 • ·1 0 • • 4 ,, ' F' ( I\ ) ·• ... 4 ·• ·1 )
[I 0 3 ·1 ,,. M:l N X ' MA X X
YY :::S f;: < I )
IF
CONTINIIF
FORMAT<316)
CAL L X Y PL.. T ( K 1 , X , Y , X M 1 N , X MA X • 0 , • Y MA X • lt l1 • (\ • I q Y )
RAX< J )::::RA
R A X <2 ) :. I ·A X <·t )
F\AY(l):;;:(),
RAY< 2 >::: YMAX l!'.'i.
CALL.. XYF'L. OT <:) , F;:AX, r;:AY • f\M IN, RMAX • 0, • YMt'lX • I'll!,. 0)
XP<2) ::=XF'< 1 >
YP < 2) ::::YMAX/~:'i. *4.
YF'<:I.)::::(),
L.Afl:::: .... l
CA U
!1 Y !:; !:; Y M <1:;: F' X , ·1 • , • 1 , F;: P S , ::> , 9 0 • l
CAI..I. HYS~iYM
10 , 0 , \
CAl. L. !:;ys!:IYM < :> • 7. • • :l'"i • c;NC v I'•. n. )
C A L I. <:;y S !; Y M <::.> • • 6 • / ~i • • .I .:> • SF\ A f·' • ::> 4 • 0 , )
CAll 'IYS!:;YM<::.>. •6.~ •• l ' 1 vF·' I AK· ~8·f),)
C A 1... L.. !:; Y !:; S Y M ( ? • • .~ • :.> ~'i • • I ·:> • ':i I\ f.: ,:; • :1 ·1 • 0 , '
C A L I. X Y F'l () l ( :.> v X F' • Y F' • I \ M I' N • h: MA X • 0 • ,, Y M() X • 1111 .. I ('! H )
CflNl T NIH
GOTCl :1 :'.'i I
FNfl
In this appendix we find an expression for
l.
in the main body of this paper.
L L
0 0
We now examine the second, third and fourth terms on the right
side of (4-F.3).
_E <
e 2
dz
function as
'¥( , )
- -4z'¥ - 1\.
z -z
= e
A e
term we find
0 0
dz dz'
power series in ~ . retaining terms up to order ~~we arrive at the
desired result
l.
4.
~.
7.
(to be published).
Mountain View, California.
Electron.
Appl. Phys. Lett.
Surfaces, (MacMillan, N.Y., 1963), p. 81.
18.
(McGraw-Hill, N.Y., 1963), p. 75.
ADDITIONAL USES OF APERIODIC STRUCTURES
5.1 Introduction
In this chapter we will study some of the additional properties
and uses of aperiodic devices in optics.
filter is first discussed.
Next a perturbation solution to the coupled mode equations is presented, which is used to study active devices such as DFB lasers and the
effects of tapering the quidinq layer or chirpinq the feedback corruaations.
Finally, the topics of pulse compression and optical coatings to
reduce electric field intensities are discussed.
5.2 Effects of a Tapered Coupling Coefficient
In his 1976 paper Kogelnik 1 uses the coupled mode equations in the
form of a Riccat i equation (see Appendix 1-B) to obtain numerical results
for reflection from chirped gratings and tapered gratin9s .
has been modified so as to be consistent with that used in previous
chapters. It was shown in Appendix 2-A that the reflectivity for frequencies that reflect from orating reqions sufficiently far from the grating
1--
broad-band filter. (After reference l.)
Figure 5.1 for small
of the reflectivity spectrum of a chirped slab reflector.
A further interesting effect observed by Kogelnik which has not
been covered in this thesis is that of a tapered 9rating; that is, one
in which the period is constant, but the depth of the corruaation varies.
Figure 5.2 is a plot of the results. The form of the coupling constant
was taken as
( 5. 1)
corruqation that is shallower near the center of the qratinq than near
the ed9es, while a neqative T value represents the opposite.
ing region is located between z = 0 and z = L.
are strongly dependent on the T value.
shallow qrating, then a deeper orating, and finally a shallow orating.
Thus, while a chirped qratinq increases the filter bandwidth, a tapered
qrating serves to suppress the side lobes. Cross, t1atsuhara, and Hill
also investioated this effect usinq various taperino functions such as
the Hamminq function, raised-cosine or Hannina function, the Blackman
function, and the Kaiser function.
_J
l.U
pattern with side lobes.
edges, then considerable side-lobe suppression will result.
A further, novel application of this concept was investigated by
..,
placed between two polarizers.
characteristic contained side lobes as in Figure 2.
adjacent elements were rotated.
the filter by more than a and elements near the edge of the filter by
less than a, the side lobes were qreatly reduced.
If we integrate equation (l-B.9) of Appendix 1-B for the case of
low reflectivity (R'(z) = const = 1) and ¢= 0 (no chirp), we find
00
the couplinq constant K(z).
Gaussian, and thus IS'(O)I will not contain side lobes.
Since the far field pattern resulting from light incident on an aperture is the Fourier transform of the near field pattern, it is also
apparent why a "soft" aperture eliminates side lobes.
5.3 Perturbation Solutions to Aperiodic Bragg Reflectors
Although numerical methods must be employed for solutions to problems with the most general chirped or tapered gratinqs, it is possible
to use perturbation techniques for cases of small variations.
technique.
which
After combining equations (1-8.5) of Appendix 1-B, we find
to
~ = ~ + ~ (z),
correction due to nonzero¢', Kl and ~l values.
It is then straightforward to show 4 that the reflectance i s given
by
p ( 0) =
2 = 2 +
R0 (z) =
variations in coupling strength and periodicity. 4 The variations are taken
as
values (shorter wavelenqths).
()toL = - 1.0
''
''
0.8
-6.0
Filter regponse for sin1ultaneous linear variations in
coupling strength and periodicity. (After referen ce 4.)
these regions interact more strongly with the incident beam. This occurs
in regions where z is greater and the light travels a greater distance,
resulting in increased gain.
In Figures 5.4-5.6 we have plotted the contours ofequal reflectivity
(0)1 2
= E~{O)
) as a function of various chirp fac1
. l
2Tl
2 TI
8£
and frequency is determined by the poles of the contour plots. 5 It is
increased.
Thus, for example, a contour of reflectivity equal to 600 (contour F)
must be drawn closer to the pole as the factor E. is increased.
It should be noted that the gain condition for laser oscillation
is changed either as a re s ult of grating aperiodicity or, alternatively, if
the grating i s uniform, by a tapering in guide thickness.
. 6
0000
600.0
t')O
IE.i6f.,
os o funct1on of SL ond aL for KL = 0.4
E = - .02 .
3.0
IOOOOO.C
I.C
10.0
00.0
600.0
000.0
3000.0
4000.0
8000.0
150000
Tre ref ,ec•,on gou, ~
vacuum.
The perturbation analysis presented above is too crude to give
results which show how the gain condition for laser oscillation changes
as the guide thickness is tapered or
chirped.
determined.
In Figures 5.7 and 5.8 the gain required for oscillation of the first
four laser modes is plotted for KL = .4 anrl KL = 2.0.
required for higher order modes drops slightly as an aperiodicity is
introduced.
Figures 5.7 and 5.8 represent a change of 1% in the grating period •.
5.4
A chirped gratinq or zone plate can also be used to compress a
pulse of liqht is incident on the reflector it may be compressed.
be delayed upon reflection, since it is reflected from the far riqht
side of the reflector, while the trailinq low frequency lioht is
'v . v
., u11•s ~f 2rr ._ ·-· -~ ----
'o: ' 8€
00
Q)
(.)
s::
Q)
-o
(.)
Q)
:::3
Q)
1.!)
single mode fiber may be partially compressed again by this device.
Figure 5.10 is reproduced from Kock's article. 7 A short pulse of
light has been directed at the entire zone plate and upon diffraction
is focused to a point.
short incident pulse is broadened (Fig. 5.10b). This process can also be
reversed (Fig. 5.10c); a rapidly rotating plane mirror (marked horn in
the figure) first illuminates the morP. off-axis (more distant) portions
of the zone plate; later, through its rotational motion, it illuminates
the nearer portions.
made to arrive at this focal area at approximately the same time by making the rotational motion of the mirror match the travel times of the
various portions of the outgoing pulse.
must be used, rather than a "single frequency" pulse.
5.5 Use of Aperiodic Dielectric Mirrors to Reduce the Electric
Field Intensit,Y
High electric fields within dielectric mirrors reflecting intense
pulsed laser radiation can damage the mirror. 8 This damaae is a result
of absorption and the consequent heatina of the dielectric layer; the
exact mechanism is not understood at this time.
(a) All portions of a grating having a zone plate outline direct reflected singlefrequency waves toward tts focal area. (b) More distant reflections return later, resulting
in a stretched echo. (c) If, however, the areas are illuminated at different times by a
rotating hom pulse, pulse compression results.
After ref ere• e 7 . .
Typically, the damage occurs in only one of the two dielectric
materials being used in the mirror.
within the critical layer, the resistance to laser damage will be increased. Apfel 11 has suggested coating a conventional quarter-wave
stack with additional layers of varying thickness, and he has described
a method of design so as to minimize the liqht intensity inside the
easily damaged dielectric material (at the expense of the stronqer material). In Figure 5.11 the intensity distribution within a conventional
quarter-wave stack is shown; it consists of layers of index n2 and n .
easily damaged layer (of index n2 ) which must be protected.
done by adding two additional layers to the stack.
while the second added layer of index n2 is slightly thinner than a
quarter-wave thickness.
distribution is shown in Figure 5.12. 11 It is evident that the maxin1um
field has been reduced in the critical layer of index n 2 .
is a maximum and that the peak field intensity within the added n 2 layer
equals the peak field in the next high index layer.
I I
5.6
In this chapter additional properties of aperiodic structures were
Next, in order to study effects of varying the thickness of the active
layer in DFB lasers, a perturbation solution to the coupled mode equations was presented.
The possibility of coating a quarter-wave reflector with adrlitional
layers of varying thicknesses for the purpose of reducing field intensity
in the reflector was reviewed.
1 35 9 ( 19 76) .
New York, 1975).
Phys. Lett. ~. 651 ( 19 74).
of Electric Field Strength in Laser Damage of Dielectric Multilayers
(U.S. Government Printing Office, ~Jashington, D.C., 1976), NBS
Special Publication #462, p. 301.
(U.S. Government Printing Office, Washington, D.C., 1976),
NBS Special Publication #435, p. 248.
1974), NBS Special Publication #414 , p. 39.
THE TRANSVERSE BRAGG REFLECTOR U\SER
INTRODUCTION
It was shown in Appendix 1-A that the guidinq of liqht can occur in
a guide whose index of refraction is laraer than the surrounding material.
Since the medium surrounding the guidinq layer is of lower index, the
electric field is evanescent in this reoion and the lioht is confined to
the high index region.
reqion is a multilayered reflector medium.
In a conventional guide, the light is confined as a result of total
internal reflection with the adjacent lower index material.
interfaces and thus the guiding layer can be of arbitrary index.
In 1970, E. A. Ash 1 originally suggested the possibility of replacing the conventionally used low index substrate with a layered medium.
Four years later A. J. Fox 2 presented a plane-wave theory for this device
which he called the integrated optics grating guide.
that of the adjacent layer. Subsequent to this, Yariv, Yeh and Honq 3 • 4 • 5
ana lyzed in detail, by using a Bloch wave formulation, the problem of
electromaqnetic propaqation in layered media.
region.
Recently this guiding has been demonstrated 6 in structures grown by
Molecular Beam Epitaxy (MBE), a technique of crystal growth which allows
successful demonstration of guiding in a passive structure immediately
suggested the possibility of growing injection lasers either by MBE or
LPE (Liquid Phase Epitaxy) which would contain a periodic layered medium
adjacent to the active layer.
Although a thick active layer may support several transverse modes
(normal to the junction plane),the Braqg reflector would not reflect,
and thus confine, all of these modes to the same extent.
a means to discriminate a9ainst certain poorly confined and thus high
loss modes.
with arbitrarily low losses in a layer of air surrounded by Bragg reflectors.*
to surrounding material) guide, but the losses are usually quite hioh
and the loss constant increases as the third power of the reciprocal
thickness of the inner layer.
1.
International Microwave Symp., Newport Beach, CA., May 1970.
Part 3, pp. 157-191 (Pergamon Press, 1975).
THEORY OF BRAGG WAVEr,UIDES
2. 1
In this chapter design considerations and the loss constant of a
used as the reflector region.
of refraction of the active region required for steady s.tate conditions.
Finally the loss constant a is calculated.
2.2
structures have been used to provide feedback and lonaitudinal mode
5 6
selectivity since the work of Koqelnik and co-workers in 1971 . •
A grating is used to provide the feedback, rather than reflection
from the cleaved ends of a Fabry-Perot cavity.
mode selectivity (transverse) is new.
Figures 2.1 and 2.2 show a typical Bragg waveguide and field di s tribution for the case ng
structure is depicted in Figure 2.3. Figure 2.4 indicates how the field
structure changes for the case ng > n1 .
nl
nl
nl
; ~. . "'~%"::.,...'~ . ~~,
. ~,:-·
~ ~ j
~-:,..,.-;,i""'.-./0-,.,,
,.. ~~.---~,~~}·~"/,~~ ...
(slab) waveguide for the case ng < n1. (After reference 7. )
_Q
Jlllll
I, I:,
.'//,'/'//
i/J ;//1
waveguide with air as the guiding channel. (After reference 7. )
.B
II
II
II
II
II
II
traveling toward the right are:
hiah index on riqht
aE (x =interface, type B)
ax
~-------------------- =
the case ng > n1
Ey (x) = cos kgxx
kaxx
EY ( x) = A e
- 00
jn~
"o
jn;ff
"o
aE
Matching the boundary conditions on EY and~ at x = -t g , we arrive at
the following eigenvalue equation
tan(k gx t g )
If we assume that
for maximum confinement. (Field incident from left.)
solve for the thicknesses of the layers t 1 and t 2 in the reflector region.
chosen, it is a simple matter to calculate t 1 and t 2
t.
k.
2n
(2.8)
- oo a:
cot(k gx t g)
since the lasers tested were of this design.
2.3
As is evident from Figure 2.4, the electric field is not evanescent
k· = 271'
kgx- AQ V ng - neff
tan(kg t) = -kx
gx
at x = - t
produced to compensate for these substrate losses.
We start with the wave equation and the usual assumed form of the
solution for coupled mode calculations:
w = radian frequency of light
n2 = l 2 2
K =
R(x)
right)
Note the importance of the proper sign and phase of on' 2 .
on' 2 = +on2 sin KX would be the proper choice only if n < n 1 , and in
neither case would on cos KX be correct with a coordinate system
chosen for which x = 0 is the guide-reflector interface.
find
N = number of periods
The solutions to ( 2. 13) and ( 2. 14) are
S(x) = sinh[ ~n 2 - o2 {x-L)]
R(x) = --s1n
n- o {x-L)]
(2. 18)
(2.19)
(So e
eka x( x + t ) · a
9 el ~Z
gx ~+Roe
gx 9)
- oo < X <
ik
(2 . 22)
medium is quite high and thus E: << 1.
we find
oth order
qr g
gr
gx g
Next we equate terms of order E: and arrive at
gr g ax
sin k t
1 -
)n2- o2
n -o2 L
kI
91
2[t
,...
2[ t
qr
k2 + k2
gr
ax
2 = n2 k2 - e2 it is easy to show that in order to osci1Since kgx
g o
k . < 0
k2 n
91
o gr
91
91
k . is given in equation (2.31).
g1
and o = 0, we find
ax
plotted for various conditions.
Next we calculate the loss constant a which is defined by
a= (dl/dz)/1, where I is the light intensity.
- co
-....
Q)
L-
'"0
-X
'"0
<1>
-'"0
'+-
II
1Cf 6 ~~--~~~--~~~~------~
Fi g. 2. 7
for steady state condition as a function of the number of
peri ods i n t he refl ecto r l ayer
power flow in the z direction.
From the previous section we have
(2.36)
(2.37)
CJ
qx q e ax
q ei Bz ,
- m
w~ az
w~
dx = - - . . - - 2t
gx <1
2k gx
- 00
Finally, after combining (2.35), (2.44) through (2.47), and using
the eigenvalue equation tan(k t ) = k /k
to simplify the result, we
gx g
ax gx
arrive at the final result for the loss constant a
ax ?
Ke -2nl
6{t g +
2n
2.8,
=A
I) Fmd {3 = >-o neff from tan k9 x t = k
Typical Diode Laser
Fig. 2.9
Substrate Resulting from Finite
Number of Periods
which the reflector layers were Al _25 Ga_ 75 As and Al _10 Ga_ 90 As.
It is well known that if klx and k2 x differ substantially, the
coupled mode equations still qive surprisinqly accurate results, despite
the fact that they were derived under the assumption klx ~ k2x. Nevertheless, because of the larqe differences often encountered in klx and k2 x'
a modification must be made in expression (2. 15).
2L\k 0
~K.
)I(
near the frequency for which o ~ 0, we have
Using the above equations, together with A~O ) = 89ooa, and quide
parameters as oiven in the section on laser fabrication, we find
..._
Q)
Fig. 2.10 Dependence of loss constant a on laser structure
It is easily verified that the major effect of o occurs in the
exponent of equation (2.48).
0 << 11
the lasing wavelength is varied about the central wavelength for which
the structure provides maximum confinement.
Of course the power flow into the substrate is exactly balanced
by the power generated in the guide due to the complex index of refraction.
Power generated by complex nq per unit volume = l2 Re( ~*E
y y
where
region, we must integrate along the width of the guide.
Power generated/unit length of guide = e: 0 w narng1
sin(2k t )
gr g1 e2nl [tg +
ax g ]
in reflector
0 (.)
n2= 3.52
II
II
II
II
e -2nL
2kgx
kgx' ngr
find that (2.52) and (2.34) are in agreement, provided
or
thus (2.53) is just an expression of the Bragq condition.
Fox 3 obtains a simple expression for the loss coefficient o. of a
symmetric Braqg guide as shown in Fiqure 2.3.
(2.54)
where
Using the coupled mode theory, o = 0, p = S0 /R 0 , and equations
logep = loqe ( 1 - 2e
- - e
In this chapter a simple method for the desiqn of a transverse Braqa
within the active region.
oscillation and the loss constant a determined.
point where they were comparable to losses due to light coupled out the
cleaved ends of the laser cavity.
App1. Phys. Lett.~. 145 (1975).
142 (1975).
45 (1975).
395 (1971).
FABRICATION AND EXPERIMENTAL RESULTS
3.1
This chapter describes the fabrication and testing of the first
3.2
The desiqn of the first transverse Braqq reflector waveouide is
by nine layers of Al. 10 Ga. 90 Ar-Al . 25 Ga. 75 Ar, the active region of pure
GaAs, a superstrate of Al . 4Ga. 6Ar, and a GaAs cap. The Al . 4Ga . 6Ar is p+
doped with Ge to a concentration of 10 18 cm 3 . The substrate is n+ doped
with Sn to a concentration of 10 18;cm3 and the entire region is n
doped to a concentration of lo 17 ;cm3 . The index of refraction profile
is shown in Figure 2.12 along with the expected field distribution.
Importan t des ign parameters are indicated in this figure and in addi0
profile
} Unit cell
repeated
9 times
tl
n2
.•
Al.1
·:·: .::
.;:··
and Intensity Profile
neff
kl X = 4.07 x 1o /m
k2x = 7.15 x 1o /m
Figure
a contact, after which the laser was lapped down to a thickness of about
100 ~·
or until a color change was observed.
Before growing the sample by LPE, several lasers were grown using
MBE, but because of restrictions on the MBE apparatus only samples with
pure GaAs (rather than Al. 1Ga. 9As) and A1 . 25 Ga. 75As could be grown.
index of refraction drops.
the LPE structure which contains only layers with Al outside the guiding
region.
3.4
IC: 3.2
2 .971
T = 297°K
Fig. 3. 2 Refractive index as a function of Al concentration in
Al Ga
As. (After reference 1 .)
X 1 -X
containing pure GaAs in the reflector region, while Figure 3.4 is an
SEM of the LPE growth.
It should be noted that although the guiding of light in a guiding region of index lower than that of the surrounding material i s
possible with Bragg waveguiding, it was not demonstrated in the laser
tested.
is used in the other regions.
type with field profile as shown in Figure 2.1 2.
The samples were pulsed at a rate of 140 hz with a pulse width
of 10 ).lSec.
Figure 3.6 is a plot of light intensity as a function of wave1ength.
Q)
Fig. 3.5
Fig. 3.6
after differentiating equation 3.2 we find
For A0
with Figure 3.6,
A final important measurement is that of the transverse mode profile .
tor of
)from that of the adjacent peak. This is verified
The apparatus for measuring the mode profi l e consists of a x43
microscope objective which images the near field onto a galvanometer
mirror which reflects the 1i ght through a 30 ~ slit and into a photomultiplier.
photomultiplier is measured by they-input of the plotter.
matic of the apparatus is shown in Figure 3.8.
·_j
Mirror
Loser
An analysis of Brago waveguidinq has been presented and expres-
index of the guiding layer have been derived.
out the ends of the cavity for structures with more than half a dozen
or so periods.
A structure was grown by liquid phase epitaxy and successfully
tested.
1.