High Power Phased Array and Tailored Gain Semiconductor Lasers - CaltechTHESIS

First, the presence of a nonlinear interaction between the carriers and the op-
tical field in a conventional semiconductor laser with a uniform spatial gain profile
produces filaments,®° so-called because a photomicrograph of an operating device

exhibits small areas of enhanced optical intensity with a filamentary structure.

(A) FILAMENTATION (B) LATERAL MODES

Gain

ch
Indext{ JL |

near field
Optical intensity amplitude farfield intensity

. ‘|
° Gain * * | \
Aa * >
; / yo \

fe) x

Du

it
o°

FIGURE 1.4 Problems with broad area lasers (a) filament formation (b) higher order lateral modes.

Since the real part of the refractive index is related to its imaginary part (t.e., the
spatial gain) through the Kramers-Kroenig® relationship, and also due to the free
carrier plasma effect,?” an increase in the gain at any point within the waveguide
produces a decrease in the real part of the refractive index; this is referred to as the
antiguiding effect.°° Conversely, if a small localized “hot spot” of increased optical
intensity should develop as a result of a fluctuation within the waveguide, the

increased stimulated emission will deplete the gain,°9 thus creating an tncrease

—~10-

§1.2(a.i) The Filamentation Control Problem

in the local index of refraction. As illustrated in Figure 1.4a, this interaction
effectively forms a small waveguide 3 to 12um wide*® within the larger waveguide
defined by the entire broad area laser.

As a result of the translational invariance within a conventional uniform gain
broad area laser, the filaments become unstable and move about randomly. The
complicated motions and interactions of the many filaments in a conventional
uniform gain broad area laser are one cause of the poor beam quality characteristic
of these devices. If a broad area laser’s farfield pattern is to be improved, some
method of stabilizing the filaments must be found. In conventional semiconductor
lasers, this is usually achieved by making the laser’s width narrow enough, typically

less than at most ten to fifteen microns, so that only one filament can form.

The Lateral Mode Control Problem

The second problem that must be overcome in a broad area laser comes about
because the optical field must be guided in the horizontal (lateral) direction as
well as the vertical one. Since the width of a typical broad area laser is perhaps
50 to 100 times the lasing wavelength, the lateral waveguide in a broad area laser
will support many lateral optical modes. In a conventional broad area laser in
which the current injection throughout the device is untform, only the lateral
fundamental mode will have a predominantly single lobed farfield pattern. (82-6)
This is illustrated schematically in Figure 1.4b, which shows the waveguide model
of a dielectric waveguide that supports two guided modes. (The mode shapes in
a gain guided waveguide are very similar; see §2.6(a).) The lasing of the higher
order modes thus increases the width of the farfield pattern, possibly making
it multilobed. To make the laser’s farfield pattern single lobed and diffraction

limited (1.e., as narrow as possible), the fundamental mode must be the only lasing

-~li-

§1.2(a.li) The Lateral Mode Control Problem
mode. All other modes must be suppressed. This is not possible in conventional
broad area lasers due to the nearly uniform spatial gain profile. Therefore, the
conventional method of achieving single lobed farfield operation is to make the

laser narrow enough so that the waveguide supports only the fundamental mode,

making it the sole lasing mode.

(b) Single Element Stripe Geometry Lasers

We see that the twin problems of filamentation and lateral mode control may
be solved by the simple expedient of limiting the width of the laser, typically to
5um to 10um. One type of stripe geometry laser diode is illustrated in Figure 1.5a;
it is similar to the broad area semiconductor laser of Figure 1.3a except that the
surface resistivity of the entire wafer has been increased everywhere but along a

thin stripe by implanting high energy protons into the surface of the crystal.

(A) (B)

a H* implantation

50°
i L. 4 l ] 1 bn
+10 0 10
e239 current flow FARFIELD ANGLE 6
| be 10°-30 V4 (degrees)

FIGURE 1.5 (a) Proton implanted stripe geometry semiconductor laser (b) typical farfield pattern
for a strip geometry laser ~ 54m wide.

~12-
§1.2(b) Single Element Stripe Geometry Lasers
Thus current is injected only into the narrow stripe. The optical field is then
confined in the horizontal direction by the presence of the lossy unpumped GaAs
active region at the edges of the stripe.

Since unprotected GaAlAs laser mirrors cannot sustain an incident power den-
sity greater than about 5MW/ cm? without damage,*° limiting the width of the
laser stripe also limits the laser’s maximum power output to ~ 50mW and, as
shown in Figure 1.5b, also limits its minimum beamwidth to ~ 10°. New semicon-
ductor laser designs which achieve high power operation by increasing the laser’s
width must therefore solve both the filamentation and lateral mode control prob-
lems.

In this thesis, we are particularly interested in phased arrays of semiconductor
lasers that use single-element stripe geometry lasers as building blocks to form
the array. We therefore lay the groundwork for our discussion of arrays with a
discussion of some simple waveguide models that describe a semiconductor laser.
A waveguide in the lateral direction may be formed either by variations in the real
part of the index of refraction (real index guided) or by variations in the spatial gain
distribution (gatn guided). We consider several classes of laser structures based on
the strength of real index guiding which are candidates for use in phased arrays
in §2.2. Due to some fundamental and technological constraints, practical evanes-
cently coupled phased arrays are either gain guided or at best very weakly index
guided (§2.4). However, since such structures are notoriously difficult to analyze
analytically, we proceed by breaking the problem down into smaller, more man-
ageable parts by starting with simple waveguides and gradually working towards
more complicated structures.

In §2.6 we briefly review the properties of the simplest of all possible dielectric

waveguides, the symmetric three layer slab structure which forms the basic building

~13-

§1.2(b) Single Element Stripe Geometry Lasers

block we will use throughout this thesis. Although other, more accurate modes
for gain guided lasers have been introduced,°*:4! this simple “box” waveguide has
the great advantage that it may be used to display the essential physics without
undue mathematical and computational complexity. Many of the other types of
waveguides we will encounter have such complicated refractive index and gain
profiles that their analytical analysis becomes intractable. We therefore introduce
some very simple yet powerful numerical methods for finding the optical modes of
a nearly arbitrary waveguide in §2.7, and make extensive use of these methods in

Chapters 3 and 4.

(c) Phased Array Lasers

As noted in §1.1, one promising method of achieving high power semiconductor
laser operation is to place many lasers in close proximity so that their optical fields
overlap sufficiently to bring about “phase locking.” In a phase locked array, the
optical fields of each laser add coherently, thus potentially providing the dual
benefits of high power and narrow beamwidth operation.

Figure 1.6a&b shows the simplest example of a phased array semiconductor
laser, the untform array, which is formed by placing several identical lasers on
uniformly spaced centers. Increasing the number of lasing elements obviously will
increase the optical power output, and will also decrease the beam angle into
which it is emitted because of the inverse relationship between the width of the
nearfield and farfield patterns.4?:43 In such phase locked semiconductor laser arrays
the filamentation problem has been solved by confining the filaments within the

individual laser channels that comprise the array.

—~14-

§1.2(c) Phased Array Lasers

(A) (C)
d*9um
SS
FAR FIELD PAK
py UNLOCKED 10°
~ye
diffraction
limited
controlled
filaments
(B) (D) d* Gum
10 lasers

ANTI -PHASE

L I. L

AL 1
GaAs SUBSTRATE -20 -10 i} 10 20
FARFIELD ANGLE @ (degrees)

FIGURE 1.6 Uniform array of semiconductor lasers (a) proton implanted (b) air ridge (mesa
stripe). Problems with phased arrays (c) not phase locked (wide farfield pattern) (d) anti-phase
operation (twin lobed farfield pattern).

However, most uniform arrays suffer from a fundamental problem that makes
them unsuitable for many applications. Figure 1.6c&d illustrates the two diffi-
culties with the uniform array that motivated the work described in this thesis.
First, Figure 1.6c shows that, if the laser elements comprising the array are too far
apart, the lasers do not lock in phase and the optical fields add incoherently.44*°
The emitted beam then has the same width as that of an individual laser. This
implies that the laser elements must be more closely spaced. Second, if the spacing
between the elements is decreased, the lasers lock in phase (as evidenced by the
narrower beam and deep minima between them in Figure 1.6d), but the farfield
pattern is now twin lobed. These undesirable twin lobed farfield patterns have been
found in uniform arrays with a wide variety of single-element laser designs, and

thus appears to be a property of the uniform array structure itself and not of the

-15-

§1.2(c) Phased Array Lasers

individual lasers. Understanding the cause of this problem and demonstrating a
method of eliminating it form the core of this thesis.

Comparison of the uniform array’s experimental farfield pattern of Figure 1.6d
and the vy = 2 theoretical farfield pattern of Figure 1.4b suggests that the twin
lobed farfield pattern may be due to a high order lateral waveguide mode. We are
therefore led to consider waveguiding properties of arrays of box waveguides. One
of the very simplest possible arrays consists of two box waveguides placed suffi-
ciently close so that their evanescent fields overlap and the lasers phase-lock. The
detailed analytic study of these coupled waveguides forms the subject of Chapter 3.
Weakly coupled real index guided waveguides are studied with the help of coupled
mode theory in §3.1, while strongly coupled waveguides are considered in §3.2.
Coupled mode theory predicts that if the two individual waveguides are single
mode, then the composite system will support two “supermodes.”? In §3.2(a) we
find (and explain) the interesting result that when the spatial overlap between the
elemental fields becomes large, one of these supermodes disappears, leaving only
a new single mode system.

Central to the understanding of systems of coupled waveguides is the concept
of the phase matching wavelength(3-1) From the point of phased array semicon-
ductor laser design, the primary significance of the phase matching wavelength is
that at this wavelength equal power flows in each waveguide (t.e., the admizture
factor is unity). In §3.3 we show that not all coupled waveguides will have a phase
matching wavelength, and in §3.1(a) we present a simple method of designing a
two waveguide system around a predetermined phase matching wavelength (e.9.,

the peak in the spectral gain curve of GaAs).*%.47

~16—-

§1.2(c) Phased Array Lasers

Coupling between two waveguides is illustrated in Figure 1.7. Figure 1.7a
shows the nearfield and farfield patterns for two closely spaced identical real in-
dex waveguides —1.e¢., a two-element uniform array. We find that if we assume
that each of the individual waveguide channels supports only the fundamental
mode, the composite structure supports two supermodes which we label the v = 1
(++) and v = 2 (+—) supermodes, where (++) refers to the inphase addition of
the individual modes and (+—) refers to antiphase addition. Notice that if the
gain is concentrated in the core regions (as indicated by the hatched ovals), the

interchannel region between the waveguides is relatively lossy.

(A) IDENTICAL (B) NONIDENTICAL

Gain =~ Sain
eg
GD ae

lp CD
Index f J index |
™Loss

cs

N,
Oss

nearfield amplitude farfieid intensity near field farfieid

| |
+\ | - |
yA

ge

INTENSITY

ae

al

FIGURE 1.7 (a) Supermodes of two identical coupled waveguides. The lossy interchannel region
causes the high order (+—) supermode to lase, leading to a twin lobed farfield pattern. (b)
Supermodes of two nonidentical coupled waveguides. Greater gain in the left channel than in the
right one favors the fundamental (++) supermode, potentially leading to a single lobed farfield
pattern.

-~17-

§1.2(c) Phased Array Lasers

Since the twin lobed (+—) supermode has a null in the lossy interchannel
region, it will be relatively more concentrated in the high gain channel regions
than will the single lobed (++) mode; it therefore will be amplified more as it
travels down the guide (1.e., it has a higher modal gain) because it is attenuated
less by the lossy interchannel regions. The mode with the highest modal gain will
be the lasing mode at threshold, and thus near threshold, the farfield pattern will
be twin lobed. The above threshold behavior is very complicated, and must be
solved using the rate equations.4® This latter topic is beyond the scope of this
work.

We find it interesting to note that the very feature which makes good single
lasers with low threshold currents (t.e., placing the gain where the light intensity
is greatest) is the very cause of the undesirable twin lobed farfield patterns in a
uniform array.

Figure 1.7b shows the supermodes for two phase mismatched nontdentical
waveguides which have the same index step but different widths. As before, we
have a (++) and (+—) supermode with single and double lobed farfield patterns
respectively, but now we notice a difference between the (++) and (+—) super-
mode: the (++) supermode is more concentrated in the wider guide, while the
(+—) supermode is more concentrated in the narrower guide. (83-3) Thus, if it were
possible to have more gain in the wider laser than in the narrower (as indicated by
the differently sized hatched ovals), the (++) supermode should have the greatest
overlap with the gain distribution, and would then be the lasing mode — leading
to a single- lobed farfield pattern. |

Although the presence of gain has been indicated only schematically in Fig-
ure 1.7, we are able to examine its effects on the admixture factor for both weakly

and strongly coupled waveguides. For the weakly coupled case (§3.4), the major

—~18-

§1.2(c) Phased Array Lasers

effect is that if the gain mismatch between the two waveguides is sufficient, equal
power will never flow in each waveguide, even at the phase matching wavelength.
For the case of two strongly coupled gain induced waveguides (§3.5), we find the
surprising and unexpected result that while we might expect to find either one or
two supermodes for two coupled single mode waveguides, we actually find four.
We discuss this interesting result in terms of gain guided “leaky” modes in §3.5(b)
and the special nature of the complex coupling between two gain guided lasers in
§3.5(c).

In Chapter 4 we utilize the ideas of Chapter 3 in our quest to design and
fabricate a phase locked array with a single lobed farfield pattern by using a
nonuniform array structure based on the idea of Figure 1.7b. Chapter 4 tells how
the actual implementation of these ideas into a working device came to resemble
a rather exciting detective story which starts with the single clue of two phase
mismatched waveguides, and evolves as we work out its extension to more elements,
discover the limitations of the theoretical results of Chapter 3, modify our ideas,
and try again — and again... until we finally arrive at a working device. Along
the way, we will also discuss a variety of waveguides relevant to phased array lasers
so that the reader will emerge (we hope) with a good understanding of the lateral
mode control problem in evanescently coupled phased array semiconductor lasers.

We start our discussion of multi-element arrays by considering a uniform array
in §4.1 and real index guided nonuniform chirped arrays in §4.2.

Figure 1.8a shows the nearfield patterns for a five-element uniform array. Notice
that the envelope function (shown by the dashed line) for the vy = 1 (+ + +++)
supermode is identical to that of the vy = 5 (+ —+—-+) supermode, and as in the

case of the two-element laser, the two modes differ only in the lossy interchannel

~19-

§1.2(c) Phased Array Lasers

(A) (B) (C)
UNIFORM ARRAY CHIRPED ARRAY CHIRPED ARRAY
NEARFIELO NEARFIELD FARFIELD
pW TQ Dees uniform array

LJ ,u ud ——
wide —nerrow
taser — iaser

envelope
f hon--
uncho .

vel
fundamentai
supermode

5 + + + + + n°
Yn
2 y=5
a nhighest order
supermode
+ = + = + 0°
LATERAL DIMENSION LATERAL DIMENSION FARFIELD ANGLE

FIGURE 1.8 Supermodes of real index guided arrays (a) uniform array (b) chirped array (c)
farfield patterns for the chirped array. (The farfield pattern for the uniform array is very similar.)
region, thus implying that the twin lobed vy = 5 supermode will be the lasing mode.
This explains the twin lobed farfield patterns of the uniform array of Figure 1.6c.

Figure 1.8b shows the nearfield pattern for a nonuniform chirped array in
which the width of the laser waveguides decrease linearly from left to right. Now
we observe a great change in the envelope functions of the fundamental vy = 1 and
highest order antisymmetric vy = 5 supermodes: the fundamental supermode is
localized at one side of the laser, while the highest order antisymmetric supermode
is localized at the other. If it were possible to tailor the spatial gain profile to have

approximately the same shape as the fundamental supermode, that mode would

— 20 -

§1.2(c) Phased Array Lasers

become the lasing mode because it best utilizes the available gain, thus yielding
an array with the desired single lobed farfield pattern.

Prior to this work, there was no known way to easily tailor the spatial gain
profile within a semiconductor laser array without using complicated multilevel
metallizations.49 The invention of two different methods for easily achieving such
gain tailoring in §4.5(a) and §5.2 forms one of the major contributions of this
thesis.

Although we have described the concept of a nonuniform array in terms of a
real index guided structure, in §4.2(a) we show that a real index guided chirped
array suffers from some fundamental and technological limitations which make its
fabrication exceedingly difficult, if not impossible. Consequently, towards the end
of Chapter 4 we turn our attention to arrays of nonuniform gain guided lasers and
discover that gain tailoring may be easily achieved in a chirped array of proton
implanted lasers.(84-5(2)) We are thus led to the concept of the tailored gain chirped
array presented in Figure 1.9. However, as shown in Figure 1.9a, in order to achieve
the desired single lobed farfield pattern, it is necessary to make the interchannel
gain so large that the coupling between the array elements becomes so strong that
the distinction between an array and broad area laser becomes blurred .(84-5(6),4-6)
Figure 1.9b demonstrates a tailored gain chirped array with a 1.5° wide single
lobed diffraction limited beam; other devices were capable of high power (450mW
into 3}°) essentially single lobed operation. (§4-5(6))

We have therefore achieved our goal of demonstrating a semiconductor laser

array capable of single lobed high power operation.

~21-

§1.2(c) Phased Array Lasers

TAILORED GAIN CHIRPED ARRAY

(A) LOW INTERCHANNEL GAIN (B) HIGH INTERCHANNEL GAIN
Ht- implant
Sym f Ht implant 2. Sum /\
sams eo A em i a a = Sum
o.3umt SS S Ss SS iam
nt-GaAs n*-GaAs
5 |
2 |
3 |
: |
Ee h | |
rp) | |
ai
is i \ |

-20 -I0 0 10 20 30 40 50 60 -20 -I0 0 10 20 30 40 50 60
LATERAL DIMENSION (um)

i L

Lo
30 “26 =10 0 10 “30 “30 76 6 10 20 30

INTENSITY (arb units) —»

FARFIELD ANGLE @ (degrees)

FIGURE 1.9 Tailored gain chirped array with (a) low interchannel gain (b) high interchannel gain
showing diffraction limited single lobed farfield operation.

—~22-

§1.2(d) Tailored Gain Broad Area Lasers

(d) Tailored Gain Broad Area Lasers

It is interesting to note that although the schematic diagram of the strongly
coupled tailored gain phased “array” of Figure 1.9b superficially resembles an
array, examination of its gain profile via the spontaneous emission pattern below
threshold reveals that the effect of the array has been nearly, if not completely,
obliterated by current spreading in the upper cladding layer between the channels.
Therefore, in §4.6 we consider whether or not it is actually more proper to refer
to such a device as a tailored gain broad area laser rather than as an array.

However, it is possible that, despite the beneficial effects of the gain tailoring,
the improved performance was due to some residual effect of the array structure.
In Chapter 5 we propose, demonstrate, and analyze a very versatile innovation, re-
ferred to as the “halftone process,” for achieving nearly arbitrary two-dimensional
spatial gain profiles within an optoelectronic device such as a broad area semicon-
ductor laser and use it to fabricate an entirely new type of semiconductor laser, a
tailored gain broad area laser, in which all traces of the array structure have been
removed.(§5-2) This laser is illustrated in Figure 1.10a. The various sized black
dots represent areas on the surface of the laser where current is injected, while
the white regions represent insulating regions. (82-2(¢)) As may be seen from this
figure, the fractional surface coverage of the injecting contact decreases approxi-
mately linearly across the laser, and so, to a first approximation, does the injected
current density and hence spatial gain profile. This is confirmed by a plot of the
nearly linear, highly asymmetric spontaneous emission pattern below threshold in
Figure 1.10b. Figure 1.10c presents the single lobed farfield pattern of this laser
which was nearly 50um wide and emitted 200mW into 2.3°.

Chapter 5 also contains a discussion of the optical modes of a linear asym-

metric tailored gain waveguide. We introduce the method of Path Analysis29 for

—~ 23 -

§1.2(d) Tailored Gain Broad Area Lasers

p*GaAs
HALF TONE (A) Haiftone pattern
TAILORED GAIN Cr/Au
BROAD AREA LASER
Active
Region
(C)
Sym
AuGe/Au
2.3° 4 current flow
200 mw $y
; ¢-- minimum
hae conn maximum
3.2] th (B)
‘€
18° > O.7I ty
wo
Zz
tid
2.21 hh 5 i 1 i ! 1 r
—= (5 ie) 15 30 45 60
LATERAL DIMENSION (pm)
! —_ i a 1
-30 -15 ° 15 30

FARFIELD ANGLE @ (degrees)

FIGURE 1.10 Halftone tailored gain broad area laser (a) schematic plan view (b) spontaneous
emission below threshold showing asymmetric linear spatial gain profile (c) farfield pattern, showing
high power single lobed farfield operation.

—~24-

§1.2(d) Tailored Gain Broad Area Lasers

analyzing nonuniform gain induced waveguides which greatly eases the analysis of
these structures, and makes possible the calculation of many waveguide properties
using only simple algebraic and geometric arguments. ($2-6(c),5.4,5.8) We find that
the modes of asymmetric tailored gain broad area lasers have several very inter-
esting properties. First, the mode discrimination (t.e., difference in the modal
gains) between the fundamental and other higher order modes is much better than
it is in a uniform gain waveguide. (85-6) Second, it is well known that symmetric
waveguides and real index waveguides have higher order modes with nulls in the
nearfield patterns. (84-2) This is not true for asymmetric tailored gain waveguides.
The nearfield patterns of the modes of this latter structure all null-less. (85-19) py.
nally, the higher order modes of either symmetric or real index guided waveguides
have multilobed farfield patterns, (84-2) while all of the modes of a linear asymmet-
ric tailored gain waveguide have single lobed farfield patterns. (85-11) We show that
these unusual properties result from the complex nature of the electric field made
possible by gain guiding and the lack of left-right inversion symmetry in the asym-
metric tailored gain structures.(85-12) We then discuss the effect of these unusual
properties on device design, and briefly present some of the engineering tradeoffs
involved in designing tailored gain broad area lasers. (85-14)

In this work, we consider only asymmetric tailored gain waveguides for two rea-
sons. First, although higher powers may, in principle, be obtained from a symmet-
ric structure with twice the width of an asymmetric one, the available experimen-
tal evidence suggests that asymmetric structures show better single lobed farfield
operation at higher powers than do their symmetric counterparts.©°Second, asym-
metric gain induced waveguides have properties very different from other, more

commonly known, structures. We therefore concentrate on asymmetric structures,

~25-

§1.2(d) Tailored Gain Broad Area Lasers

and remark that the extension of this work to symmetric waveguides is straight-

forward.

§1.3 Applications and Future Extensions of This Work

Halftone tailored gain broad area lasers are very simple to fabricate, and are
thus well suited to large scale processing techniques. We anticipate that they may
find application wherever high power single lobe operation of a laser is desired,
provided that the lack of spectral purity associated with gain guided lasers is not

objectionable.

DETECTOR READ
——> SIGNAL
OUT

LASER
WRITE DIODE =
sicnai —{_}

IN
BEAM
COLLIMATING
LENS SPLITTER

FOCUSING
LENS

INFORMATION
ENCODED IN
PITS ON THE
DISC

FIGURE 1.11 Optical recording.

Figure 1.11 shows a possible application of asymmetric tailored gain broad
area lasers to optical data recording. A semiconductor laser and detector is used
to measure the amount of light reflected from microscopic pits on the surface of the

disc. In optical data recording, it is necessary to steer the laser beam to the correct

— 26 -

§1.8 Applications and Future Extensions of This Work

track on the optical disc. A closed loop servo mechanism is used to insure that
the optical beam will stay centered on the track even if it is not exactly concentric
with the axis of rotation. Conventionally, this is done using electromechanical
techniques. Figure 1.12 shows how the beam emission angle from an asymmetric
tailored gain broad area laser depends on the value of the spatial gain gradient.
However, it should be possible to utilize asymmetric tailored gain broad area lasers
to electronically steer the laser beam onto a track much more rapidly than with

conventional electromechanical methods.

(a) (b)

ar - -
- “ie BEM Yam a
> E
seseeT
{20pm i
Lateral dimension x canal !
fe) \ 2 3 4 5 6
FARFIELD ANGLE @
(degrees)

FIGURE 1.12 (a) Tailored gain broad area waveguides with varying gain gradients (b) farfield
patterns showing potential for beam steering.

Although electromechanical methods would probably still be required for gross
positioning of the beam, the ability to rapidly adjust the precise position of the
beam, combined with the high power output of a tailored gain broad area laser,
could potentially increase the optical bandwidth of the unit. Thus, we anticipate

that asymmetric tailored gain broad area lasers merit further development work.

~I27-

§1.4 Conclusion
§1.4 Conclusion

In conclusion, we have come full circle. We started by stating that a uniform
gain broad area laser was unsuitable for many applications due to the filamentation
and lateral mode control problems. The filamentation problem was solved by using
an array structure, while the lateral mode control problem was solved by intro-
ducing gain tailoring. However, we discovered that in order to make gain tailoring
work, it was necessary to increase the interchannel gain to the point where the
device resembled a broad area laser more than it did an array. We then demon-
strated a tailored gain broad area laser (!) which is capable of single lobed, high
power operation.

It is interesting to note that, although we began by assuming that an array
was necessary for high power operation, we have found that the introduction of
gain tailoring makes the array structure superfluous. In fact (as we will show
in §4.5(b)), it may be that since the modulation of the laser’s nearfield pattern
introduced by the array structure increases the power present in the sidelobes and
decreases the ability of the device to operate in a phase locked mode, it may well be
advantageous to consider other structures which resemble broad area lasers more
than they do arrays of individual lasers. In particular, it would be very interesting
to combine the present work with gain tailoring with new methods of tailoring
the real part of the refractive index to create tatlored indez tailored gain broad
area lasers that would potentially combine some of the benefits of both real index
and tailored gain structures. Furthermore, in one sense what we have done in this
thesis is essentially to redefine the maximum upper width of a semiconductor laser
from 10um — 154m to perhaps 60um — 100um. One might then speculate about

the possibility of creating arrays of broad area semiconductor lasers!

~ 28 -

§1.4 Conclusion

However, one crucial question remains unanswered: what is the role of fila-

mentation in a tailored gain broad area laser? Do filaments exist at all, and if

so, why don’t they degrade device performance? Unfortunately, the resolution of

these exceedingly interesting questions are beyond the scope of this thesis.

Finally, Figure 1.13 presents an overall flow chart for the work of this thesis

and for some possible future extensions.

§1.5 Conventions Used Throughout This Thesis; References

(1)

Here we summarize conventions used throughout this thesis.

The real part of a complex quantity q will be denoted by either ¢g or Re{q};
the imaginary part will be denoted by either g, by Sm{q}, or by a different
symbol. Since the symbol z is used to denote the lateral dimension along a
waveguide or laser, we will refer to the real axis of the complex z—plane as the
€ axis.

A plane wave moving in the +z direction is denoted by et (Bz—wt) | where @ is
the propagation constant and w is the free space radian frequency of the wave.
N.B.: some authors, [40] for example, use et(82z—-wt) to represent the same
wave. See §2.6 for a brief discussion of this difference.

When we make a waveguide model of a gain guided laser, we assume a loss
in the unpumped GaAs active region of 200cm~! (85-13) The peak gain within
the waveguide is set by the requirement that at threshold the power modal
gain y exactly equal the mirror losses, which we assume to be 40cm~! for a
device 250m long.>! Unless we are attempting to model an actual device, we
will ignore the effect of the antiguiding parameter b. This parameter relates

the decrease in the real part of the refractive index due to the presence of

— 29 —

§1.4 Conclusion

Chapter 2 Single Element Lasers
Chapter 3 Two Coupled Lasers
4 Uniform Arrays
Real Index Guided
Chapter 4 Chirped Array
400mW Tailored Gain
L 33 Chirped Array
7 ;
Halftone Process
2D Gain Tailoring
200mW Tailored Gain
i ~———__
Chapter 5 23 Broad Area Lasers
Device Analysis of
L Performance Tailored Gain Waveguide
v v
W . .
Future Work Filamentation Beam Steering Tailored Index & Gain

Broad Area Lasers

Arrays of
Broad Area Lasers

FIGURE 1.13 Flowchart for work described in this thesis.

~30-

§1.5 Conventions Used Throughout This Thesis; References

gain. When it is included, we use a value 6 = = 3.0.52 The e*4? convention

IE
Sys

described above implies that 6 > 0.
(4) Unless otherwise noted, the figures plot intensity nearfield and farfield patterns.
The phase plots for gain guided modes are in radian units. In order to simplify

the labeling of the axes, 3.419 1 2°" is used to represent 3.410, 3.411, 3.412, etc.

Throughout this work, a familiarity with the fundamentals of laser theory, and
of that of semiconductor lasers in particular, is assumed. An excellent elementary
treatment of both topics may be found in the book by Yariv,?? while more ad-
vanced treatments may be found in the comprehensive works by Yariv,°! Casey
and Panish,*9 Kressel and Butler,®? and Thompson.*4 An elementary knowledge
of waveguiding in dielectric media is also assumed. Tamir®? provides a good in-
troduction to waveguiding in real index media as well as to the field of integrated

optics. Marcuse®4:55

gives an advanced treatment of (mostly) real index guided
waveguides, with particular applications to optical fibers. A good introductory ar-
ticle on semiconductor lasers is that by Panish,°® and a reasonably complete review
of recent advances in phased arrays has been published by Botez and Ackley.
Finally, while each chapter in this thesis leads naturally into the next, the

extensive cross-referencing should make it possible to read each chapter independ-

ently of the others.

~31-—-

CHAPTER

TWO
Single-Element Stripe Geometry Lasers

‘Excellent!’ I cried. ‘Elementary,’ said he

—Sherlock Holmes, The Adventure of the Crooked Man
Sir Arthur Conan Doyle

§2.1 Introduction

Before considering arrays of semiconductor lasers, in §2.2 we review several
classes of single-element stripe geometry lasers which are potentially suitable for
use in phased arrays. These include strongly index guided structures such as
the buried heterostructure (§2.2(a)) and buried crescent lasers (§2.2(b)), buried
ridge (§2.2(c)) and air ridge (mesa stripe) lasers (§2.2(d)), and two types of gain
guided lasers, proton implanted and Schottky isolated (§2.2(e)). We find that
technological limitations make fabrication of strongly index guided lasers such
as the buried heterostructure unsuitable for use in evanescently coupled phased
arrays, so we next consider weakly index guided ridge structures. These latter
structures are really neither entirely gain nor index guided; we briefly examine
the interplay between gain and real index guiding in §2.3. Finally, we summarize
this information by discussing some design considerations for evanescently coupled
phased arrays in §2.4.

In §2.5 and §2.6 we summarize the properties of the simplest possible optical
model for a single-element laser, that of the symmetric three layer “box” waveguide

which forms the basic building block for all of our subsequent work. Finally, in

~32-

§2.1 Introduction

§2.7 we describe a powerful numerical method for finding the modes of a one-
dimensional waveguide with a nearly arbitrary index and gain profile; we will
make extensive use of a computer program based on this technique to check the
validity of the analytical results of Chapter 3 and also to analyze the complicated

array waveguides of Chapter 4.

§2.2 Semiconductor Lasers for Use in Phased Arrays

While almost all semiconductor lasers utilize a double heterostructure to
achieve carrier and optical confinement in the vertical direction(81-2) the variety
of methods of achieving carrier and optical confinement in the horizontal (lateral)
dimension is almost unlimited. We consider three broad classes of semiconduc-
tor lasers which are potentially suitable for use in phased arrays: material index
guided, effective index guided, and gain guided. The primary difference between
these types of lasers is the strength of the index of refraction difference between
the core and cladding regions which form the waveguide.

The index of refraction 7 of GaAs is 3.59, while that of Ga,_,Al,As is ap-
proximately given by! 7 = 3.590 — 0.710z + 0.91z”. Due to the possibility of
nonradiative recombination in Ga,_,AlzAs with z > 0.45,?the mole fraction of
aluminum is usually limited to z < 0.4. Thus, maximum feasible index step for
the GaAs/GaAlAs system is about An ~ 0.25.

Throughout this thesis, we will make waveguide models of both single-element
and multiple-element arrays of semiconductor lasers. In order to reduce the discus-
sion to the fundamental physical principles that apply to phased array lasers, we
will make use of the simplest possible model for a single-element waveguide, that

of the symmetric three layer slab waveguide shown schematically in Figure 1.4b.

— 33 -

§2.2 Semiconductor Lasers for Use in Phased Arrays

For obvious reasons, we will refer to this simple structure as a “box” waveguide;

extension of our work to more accurate waveguide models is straightforward. 34

(a) Buried Heterostructure Lasers

Material index guided lasers provide the largest index difference, and hence
the tightest confinement of the optical field. They make use of the differing ma-
terial properties of a GaAs/Ga,_,Al,As heterojunction with the mole fraction of
Al x = 0.2 to 0.4 in a manner entirely analogous to the double heterostructure
configuration in the vertical direction. A typical laser of this type is the buried

heterostructure”® illustrated in Figure 2.1.

fo diffused GaAlAs

p GaAlAs n GaAlAs
n GaAlAs GaAs active region

=— p GaAlAs

n* GaAs

FIGURE 2.1 Buried heterostructure laser.

The distinguishing feature of a buried heterostructure laser is that the GaAs
active layer is surrounded on all sides by GaAlAs. Figure 2.2a shows the lateral
waveguide model for a buried heterostructure laser 1.5m wide with z = 0.3, while

Figure 2.2b presents the intensity nearfield and farfield patterns of this waveguide

for the fundamental mode.

— 34 —-

§2.2(a) Buried Heterostructure Lasers

3.415 rT
3.385)

NE ARFIELD FARFIELD
1 L q qT | U q
7)
ii
i l 1 L 1 i AL L
-2 -| O | 2 -40 -20 0 20 40
LATERAL DIMENSION FARFIELD ANGLE @
(um) (degrees )

FIGURE 2.2 (a) Lateral waveguide model for a buried heterostructure laser with a cladding Al
mole fraction z = 0.3 and a width of 1.5m. (b) Nearfield and farfield for the fundamental mode.
Note the very wide farfield pattern.

Notice that as a result of the large index of refraction difference between GaAs
and GaAlAs the field is very well-confined. This, in conjunction with the large
energy band gap difference between GaAs and GaAlAs, provides excellent carrier
and optical confinement in both the horizontal and vertical directions, thus leading
to threshold currents as low as 15mA.’ However, if the laser is to have a single
lobed farfield pattern the widths of a buried heterostructure laser are limited to
about 1lum.® This limits the typical power output of buried heterostructure lasers
to a few milliwatts.

Strongly real index guided lasers such as the buried heterostructure have the
very desirable property that their spectral width is very much smaller than that
of a gain guided structure. This is of crucial importance in optical fiber appli-

cations, and especially so for heterodyne detection methods.? Therefore, it would

— 35 -
§2.2(a) Buried Heterostructure Lasers
be highly advantageous to be able to fabricate phased locked arrays of strongly
index guided lasers. Unfortunately, due to the techniques necessary to fabricate a
buried heterostructure laser, it is not currently possible to place two such lasers
closer than 2um— 3m apart. Figure 2.2b shows that as a result of the large index
step that tightly confines the field the overlap between the fields of adjacent lasers
will be very small, and thus an array of buried heterostructure lasers is not likely
to operate in a phased locked mode.!%!! It is possible to slightly decrease the re-
fractive index step by using smaller mole fractions z of aluminum in the cladding
layer. However, decreasing the mole fraction below z = 0.2, causes the threshold
current to rise dramatically because the carriers are no longer well-confined.!” We
therefore conclude that despite their great advantages of a low threshold current
and very pure spectral output, buried heterostructure lasers are unsuitable for use

in evanescently coupled phased arrays.

(b) Buried Crescent (Channeled Substrate) Lasers

Providing slightly less of an index step difference is the buried crescent (chan-
neled substrate or V groove) laser of Figure 2.3.!% This device takes advantage
of the differential growth rate of GaAs and GaAlAs inside and outside an etched
groove to create an active region that is thicker towards the center of the groove
than towards the edge, thus providing the good carrier confinement characteristic
of a double heterostructure.

It is possible to achieve some degree of control of the active layer thickness
between the elements, and thus control both the size of the lateral index variation
as well as the coupling between the lasers. Although the index step in these lasers
approaches that available in a buried heterostructure laser, the smooth variation

in the effective index profile provides improved mode discrimination between the

~ 36 -

§2.2(b) Buried Crescent (Channeled Substrate) Lasers

p* GaAs
pAlGaAs

nGaAs

AlGaAs

n GaAs
A cHaNWeL

FIGURE 2.3 Buried Crescent (Channeled Substrate) Laser.

fundamental and higher order modes in a single-element waveguide. These lasers
are attractive candidates for uniform phased arrays but not for nonuniform arrays
because it is not easy to obtain controllable growth using grooves of different
sizes.!4 We will therefore not consider such arrays in this work. We note, however,
that arrays of identical buried crescent lasers with variable spacing between the

array elements have shown promising results.}5

(c) Burted Ridge (Strip-Loaded) Lasers

Another method of confining the optical field in the horizontal direction makes
use of the “effective index” effect in a ridge guided structure.!® The refractive
index differences associated with effective index guided structures may vary widely
depending upon the particular details. Two such possible structures are shown
schematically in Figures 2.4 and 2.5. In each case, the optical mode tends to
concentrate in the high index region of the waveguide, which is indicated by the
dotted ovals in the figures. The first structure is referred to as a buried ridge or
sometimes as a strip-loaded waveguide, while the second is referred to as an air

ridge or mesa Stripe laser.

~37-

§2.2(c) Buried Ridge (Strip-Loaded) Lasers

Buried ridge structures are difficult to fabricate in the GaAs/GaAlAs sys-
tem because it is virtually impossible to obtain good regrowths over air exposed
Ga,_,Al,As with a mole fraction z of aluminum greater than 0.1.17 This fact, in
conjunction with the difficulty of controlling etching depths to a precision of better

than about 0.2m, limits the smallest obtainable index step to about An ~ 0.03.

Cr/Au SiQo
= NW p’-GaAs cap
Gdg7Alg 3As | {Caos Alo.iAs

active
region

Gdg7Alg As waveguide

FIGURE 2.4 Buried ridge laser.

The largest index step is limited to An = 0.1 by the requirement that the laser’s
farfield pattern be single lobed in the vertical direction as well. Nevertheless, de-
spite the difficulties with the regrowth process, buried ridge structures also provide
excellent candidates for phased array lasers, and in fact. have been used with some
success.18

We note that many laser waveguides, while technically not single lobed, do
lase with a single lobed farfield pattern. For example, a buried heterostructure
waveguide will support only a single mode if its width is less than about 0.5um,
yet many actual devices about lum wide show clean, single lobed farfield patterns,
suggesting that a better practical condition be that the vy = 3 mode be cut off.

The improved performance probably results from the fact that the vy = 2 mode

has a null at the high gain region in the center of the waveguide. (This is one

— 38 —

§2.2(c) Buried Ridge (Strip-Loaded) Lasers

instance in which our simplification of the actual waveguide to a box structure
is inappropriate.) This new criterion effectively doubles the useful width of the
waveguide, and places an upper limit of about 2um-— 3m on the width of a buried
ridge waveguide.

However, while this is a good design rule for an tsolated laser, it is not clear
that it holds for the elements in an array of lasers. For a given index step, as
the width of the laser increases the fields become more tightly confined, but it is
always true that the higher order modes with multilobed farfield patterns are less
well-confined than the single lobed fundamental. If the individual laser waveguides
support more than one mode, the coupling between the higher order modes in the
array will be greater than it will be for the fundamental mode, thus exacerbating
the tendency of the array to operate in an undesirable multilobed farfield pattern.
Therefore, it is probably advantageous to use single mode waveguides whenever
possible. This more stringent criterion would limit the width of the buried ridge

waveguide to about lum, with wider waveguides being marginally satisfactory.

(d) Air Ridge (Mesa Stripe) Lasers

An air ridge waveguide!%20

such as that of Figure 2.5 may be used to obtain
arbitrarily small index steps. These devices are sometimes referred to as mesa
stripe lasers because they are fabricated by etching a mesa into a four layer het-
erostructure which is similar to the three layer heterostructure described in §1.2
except for the addition of top ptGaAs cap layer. The metal to GaAs interface

forms a good ohmic contact, thus providing better injection into the laser than

does the metal to GaAlAs interface.

—39-

§2.2(d) Air Ridge (Mesa Stripe) Lasers

Schottky Barrier p'-GaAs cap
(blocking) Cr/Au p-contract
GaAlAs

SS rtng
region
GaAlAs 9

wavequide

FIGURE 2.5 Air Ridge (Mesa Stripe) Laser.

The maximum index step obtainable is limited by the precision with which
the etching action can be halted very close to the active region, typically about
~ 0.2um (although our experience indicates that it is very difficult to obtain uni-
form results when working to these tolerances; a better practical limit is probably
0.3um — 0.4um). This places an upper limit on the the refractive index step of
about An ~ 0.02; the lower limit is, of course, zero, and corresponds to no etching

at all.

(e) Proton Implanted and Schottky Isolated Gain Guided Lasers

A gain guided laser is usually considered to be a laser waveguide in which
there is no intentionally introduced real index guiding. Waveguiding is provided
solely by the gain distribution.24:22 There are a variety of types of gain guided
structures; we will make extensive use of both proton implanted”? and Schottky
isolated lasers*4 in Chapters 4 & 5.

Figure 2.6 presents a schematic diagram of such a gain guided proton im-

planted laser. The crystal damage caused by the implanted protons° creates high

— 40 —

§2.2(e) Proton Implanted and Schottky Isolated Gain Guided Lasers

resistivity regions in the upper cladding layer, thereby blocking injection every-
where except at the laser stripe that has been protected from the protons by a

thick (~ 34m) photoresist stripe.

H*- implant Cr/Au p-contact
( blocking)

a p* GaAs cap layer
ihhgkehihf Ltt. ae — ORLTUUT OUTED

LILLE \ KL
GaAlAs PEL Scniecte current
= + active

GaAlAs region

FIGURE 2.6 Proton implanted laser.

Typical implantion dosages are 5 x 10!6em-3. The implanation depth, and hence
depth of the insulating region, depends upon the proton energy. Proton implanted
lasers have been extensively studied, and offer several advantages over other types
of lasers from the point of view of phased array semiconducor laser design (see
§4.3). However, due to the thick photoresist pattern, it is difficult to achieve
feature sizes much smaller than 34m —5ym, and especially so for deeply implanted
devices.

A schematic diagram of another useful type of gain guided laser, a Schottky
isolated laser, is illustrated in Figure 2.7. It consists of a standard three layer
heterostructure with the 0.2um p*GaAs cap layer. After growth, photoresist
stripes are deposited on the surface of the wafer, and the thin pt GaAs cap layer
is etched away from the unprotected areas using a a noncritical wet chemical etch.

This type of laser is actually an “air ridge” laser with a zero etching depth. The

~ 41 —-

§2.2(e) Proton Implanted and Schottky Isolated Gain Guided Lasers

shallow etching depth does not affect the real refractive index profile, but does

control current injection into laser.

p*—GaAs cap (qoad injection)
Schottky Barrier C/A tact
(blocking) J. r/Au p-contac
GaAlAs AL inet current

GaAlAs

= active
region

FIGURE 2.7 Schottky isolated laser.

After removing the photoresist, a metal is deposited over the entire surface of
the device. The metal to p*GaAs interface (shown in black) forms an injecting
ohmic contact, while the metal to p Ga,_, Al, As interface (shown in white) forms
a Schottky blocking contact. Thus current is injected only into the region under
the ptGaAs stripe. The very thin cap layer allows the feature size to approach
the technological limit of about 2um.

It is important to note that in both structures, at any given point on the surface
of the wafer, current is either injected into the crystal or it isn’t; there is no simple
method for achieving partial injection and hence arbitrarily controlled variations
in the spatial gain profile. However, we will return to this point in §5.2 when we
demonstrate the halftone process for achieving nearly arbitrary two-dimenstonal
spatial gain profiles within a broad area semiconductor laser.

In an actual gain guided laser, the free carriers in the active region, and the

change in the electronic band edge due to the gain they introduce, leads to a

_ 42 ~
§2.2(e) Proton Implanted and Schottky Isolated Gain Guided Lasers

decrease in the real part of the refractive index within the core region; this is

known as the antiguiding effect.(81-2(¢4)) The ratio b = |An/An| is referred to

as the antiguiding parameter. Figure 2.8a shows the waveguide model for a gain

guided laser 64m wide. The gain in the core region is fixed by the requirement

that at threshold the modal gain of the fundamental mode be equal to the mirror

losses. (81-5)

+13.41500
tL
+13.41393
NEARFIELD FARFIELD
en ee T T 7
(B)
FE
LiJ
= —
oe ee | 1 !
-6 -3 0 3 6 -20 -I0 Oo 10 20
LATERAL DIMENSION FARFIELD ANGLE @
(ym) (degrees)

FIGURE 2.8 (a) Waveguide model showing the effect of antiguiding parameter b. (b) Nearfield
and farfield patterns for 6 = 0 (solid curve) and 6 = 3 (dashed curve).

Figure 2.8b shows the nearfield and farfield patterns for this mode with no
antiguiding (b = 0, solid line) and with an antiguiding parameter b = 3 (dashed

line) .“ Note that the nearfield patterns are very similar. Throughout this thesis

—~ 43 -

§2.2(e) Proton Implanted and Schottky Isolated Gain Guided Lasers

our waveguide models will therefore ignore the antiguiding parameter unless we
are attempting to model an actual device, (84-5(6),85-13)
Finally, we remark that most gain guided structures show single filament

operation (8!-2(2)) only if they are narrower than about 10um — 15um.*6

§2.3 Interplay Between Real Index and Gain Guiding

If the etching depth in an air ridge structure is too small, the effect of the real
index guiding becomes weaker than the effect of the gain guiding, and the laser
loses the advantages of a real index guided structure (low thresholds and spectral
purity). There is no clear dividing line separating a weakly real index guided laser
from a gain guided one. We therefore adopt the criterion that, to be considered
index guided, the size of the intentionally introduced refractive index step must be
approximately equal to the change in the :maginary part of the complex refractive
index in an otherwise equivalent purely gain guided laser.

The waveguide model for such a comparison is shown in Figure 2.9a. We write
the complex index of refraction step(82-5) as An = An+1 An with An = —AT/2kp
and AIT = Ig —TI¢. We assume a loss due to the unpumped GaAs active region
—T, ~ 200cm™!, and require that the peak gain Io inside the core region of the
waveguide be just large enough to give the fundamental mode a modal gain just
sufficient to balance the mirror losses of 40cm~1(§1.5)

A waveguide 6um wide a peak gain 9 ~ 50cm7! has a gain step AT = 250cm™!.
The peak gain Ip, and hence |n|, increases slightly as the guide becomes narrower

because the field extends farther into the lossy region. 250cm7!

corresponds to
a change in the magnitude of the imaginary part of the index of refraction of
An = AT/2kg = 0.0018. Figure 2.9b shows the superimposed intensity nearfield

and farfield patterns for the two equivalent (Am = |An| = 0.0018) waveguides

-— 44 —-

§2.3 Interplay Between Real Index and Gain Guiding

dashed curve solid curve
3.41500F +100 gain guided
. =|
(A) of saizest—l L|-a90 f Tem ) __ _ index quided
NEARFIELD FARFIELD
f rn re | ! i |! li i
>| 7] 4
rE
(ep) nn ~ ood
(B) Zz j |
Lil = | = —
a a -
— PF 7 <4 ]
je ee ee 1 a ee
-6 -3 0 3 6 -10 -6 -2 2 6 10
LATERAL DIMENSION _ FARFIELD ANGLE @
(zm) (degrees)

FIGURE 2.9 Waveguide model for gain guided (solid line) and equivalent real index guided (dashed
line) structure with An = An = 0.0018.

6um wide. The gain guided fields are indicated by the solid curves while the
index guided fields are indicated by the dashed curves. The gain guided nearfield
pattern is very slightly wider than the index guided nearfield, but because of the

§2.6(5)) the farfield patterns are virtually

phase front curvature due to gain guiding, |
identical. This indicates that using the criterion Aw = |An| as a dividing line to
distinguish a real index guided laser from a gain guided one is not an unreasonable
one.

In the preceding analysis, we have neglected the effect of the antiguiding factor
b which accounts for the depression in the index of refraction due to the free
carrier and band edge effects. (§1-2(4)) The antiguiding effect reduces the size of

the effective index step by about three times |f|.4 Therefore, we propose that

for an actual device to be considered as real index guided, it is necessary that

~ 45 -—

§2.3 Interplay Between Real Index and Gain Guiding

An > An +3|n| = 0.003 in order to offset the antiguiding effect. In an air ridge
waveguide, this corresponds to an upper cladding thickness of about 0.4um —
which is near the technoloical limit with which etching can be controlled. We
therefore conclude that almost all air ridge waveguides will be more or less gain
guided structures. We will refer to them as quasi-real index guided lasers.
Finally, we the presence of a metal film so close to the active layer will in-
troduce additional loss into the interchannel regions of an air ridge phased array,
thus promoting the tendency of the array to operate with a twin lobed farfield

pattern (§4-1) .

§2.4 Design Considerations for Evanescently Coupled Arrays

Figure 2.10 presents a graphical summary of the information presented in the
previous sections. We have plotted the waveguide width @ vs. the size of the
intentionally introduced real index step An. Lasers with widths narrower than
those indicated by the light dashed line support only the single fundamental mode,
while those wider than the heavy dashed line support more than two modes and
are hence probably unsuitable for use in phased array lasers. Lasers with widths
between the two lines support two modes and are therefore marginally suitable.

The vertical light dotted line indicates the approximate location of the regime
where An = An t.e., the dividing line between gain guided and real index guided
lasers if the antiguiding factor were to be ignored. The vertical heavy dotted line
accounts for the antiguiding factor of about three by indicating the approximate
location of the regime where An > An +3|n|. We consider lasers to the left of
the light dotted curve to be gain guided, those between the light and heavy dotted
curves to be quasi-real index guided, and those to the right of the heavy dotted

line to be real index guided.

~ 46 —

§2.4 Design Considerations for Evanescently Coupled Arrays

ao — Real Index Guided
| i274
_~ 11 Multiple : Ar
= lament: = ° Anw An:
4 aA eko
S =
An = An + 3) A

Co)
AS]
Oo
° AiR RIDGE
E= Ya
= i=—Y ‘ / BURIED RIDGE
| 4M mode /

2 : BURIED

| Technological ~ “Tae ble node /“HETEROSTRUCTURE

[4 Limit (stripe width) — Sex

single mode \
e) 0.0032 005 0.0! ae 05 .

Real Index Step An ——>

FIGURE 2.10 Design considerations for evanescently coupled phased array lasers. Ideally, an array
element should be real index guided and single mode. Current technological limitations make this
difficult, so we emphasize gain guided arrays in this work.

Finally, the upper limit on the width of a gain guided laser of about 10um
and the technological limit corresponding to the smallest practical feature size of
about 2um are indicated by the horizontal heavy solid lines.

This figure shows that the only candidates for truly real index guided arrays
are the buried heterostructure and buried ridge lasers, both of which will probably
support at least two modes. As discussed in §2.2(a), such waveguides are probably
unsuitable for use in evanescently coupled arrays requiring single lobed farfield
patterns. Almost all air ridge lasers are only weakly real index guided at best.
Given the current technological limit of about 24m feature size and the difficulty

of fabricating buried ridge waveguides, we conclude that the laser most suitable

~AT-—

§2.4 Design Considerations for Evanescently Coupled Arrays

for use in arrays will likely be either gain guided or at best partially gain and
partially index guided air ridge structures, and that even these will not be single
mode waveguides. For this reason, as well as others discussed in §4.2(a), most of
the experimental work of this thesis will be with gain guided lasers (also see §4.3).

We remark that these results indicate that many of the so-called “real index
guided” air ridge lasers described in the literature*”:?® are not truly real index
guided lasers as the term is commonly used. In particular, the on-axis farfield
pattern in Reference [28] is probably due to the symmetry of the waveguide, not
necessarily to the fact that they are real index guided. The effect of the gain guiding
is masked by the symmetry of the structure (see §4.2(a), especially Figure 4.5b,
and §5.12).

§2.5 The Helmholtz Equation

Having discussed some of the index of refraction profiles appropriate to semi-
conductor lasers suitable for use in phased arrays, we now summarize some of the
relevant properties of the optical modes of these waveguides.

The optical field E(r,t) inside any waveguide satisfies Maxwell’s wave

equation”?

2 2
n“(r) O°E _
3 5F =9 (2.5.1)

V7E —

where c is the speed of light in a vacuum, and n(r) is the index of refraction in

the medium. n(r) is, in general, a complex number*®

n(r) = nr) + 17(r)
(2.5.2)
= “n(r) — 10 (r)/2ko
and t = —1. The ordinary (real) index of refraction is denoted by n(r), while

['(r) = —2ko7(r) is the spatially dependent power gain experienced by an optical

-— 48 —

§2.5 The Helmholtz Equation

wave propagating through the point r. In the unpumped GaAs absorbing regions
of the waveguide, I(r) is a negative number.

In a semiconductor laser, E(r,t) is a complicated superposition of many
transverse, lateral, and longitudinal modes oscillating at several different fre-
quencies. We simplify the problem by considering only one oscillation frequency
(thus eliminating the longitudinal modes), and make the usual effective index
approximation!® (thereby eliminating the transverse modes). Furthermore, we
consider only TE waves*! traveling in the +z direction. After making these ap-

proximations, the electric field of a lateral mode may be written as
E(r,t) ~ XE(z)e(F2-%) = B = kon (2.5.3)

where E(x) is now a scalar electric field in the x direction, kg = 27/A is the
free-space wavevector, 2 = cko is the circular frequency of the wave.

Substituting (2.5.3) into (2.5.1) yields the scalar Helmholtz equation:*?

d2
zak t+ k2(n*(c) —n*)E=0. (2.5.4)

Solutions of this equation are referred to as modes of the waveguide. The effective
index of the mode is given by the constant 7; the propagation constant G in the z
direction is then given by 6 = kof. A very important quantity is the power modal
gain y = —2ko7 = — 28. The intensity of an optical wave which is an eigenmode
of Equation (2.5.4) will grow with z as e?”. In a laser, the lateral mode with the
highest modal gain will be the lasing mode at threshold.

Throughout this work, we will solve this equation for various refractive index
profiles n(x). In a real index guided waveguide, there is no gain or loss present, and
so the refractive index profile, the eigenmodes, and the electric fields are all real

quantities. In particular, we note that the phase ¢ of the electric field EF = |E| eb

— 49 -
§2.5 The Helmholtz Equation

is restricted to either 0 or 7. On the other hand, in a gain guided structure,
variations in the tmaginary part of the index of refraction determine the modal
properties more than do variations in the real part of the refractive index profile.
In this case, the eigenvalues are complex, the real part being the effective index
and the imaginary part being the modal gain as described by Equation (2.5.2). In
a gain guided waveguide, the phase ¢ of the electric field is no longer restricted to
either 0 or 7 and may take on any value.

We remark that the terms “real index guided” and “gain guided” refer to two
limiting cases in which variations in one part of the complex index of refraction
dominate the other. As we have seen in §2.3, an important class of lasers suitable
for use in phased arrays, the air ridge structures, may be considered to be either
gain or index guided. Such waveguides will play an important role at one point in

our work, and will be further discussed in §4.2(a).

§2.6 Symmetric Three Layer “Box” Waveguides

The simplest possible single-element waveguide is the symmetric three layer
slab “box” waveguide illustrated in Figure 2.11 and Figure 2.12, which has a

refractive index profile described by

Ne (cladding region) -co<24< —§
n(z) = ¢ no (core region) —§ <2< +§ (2.6.1)
Ne (cladding region) f <£2r<0

The quantities no and ne are, in general, complex. For waveguiding to occur, it
is necessary that in a real index guided laser ng > ne, while in a gain guided
laser [9 > Te, where [9 is related to the imaginary part of the complex index
of refraction through Equation (2.5.2). The properties of the real index guided

version of this waveguide have been extensively discussed in the literature,??—5

-50-

§2.6 Symmetric Three Layer “Box” Waveguides

while the gain guided version has been somewhat less well studied.2? In this work
we will introduce only selected aspects of the theory of these waveguides necessary
to aid our understanding of phased array semiconductor lasers.

In general, a waveguide will support many optical modes E”(z) where v is the

mode number. For the box waveguide of width @ illustrated in Figure 2.11, E”(z)

is given by
Beg(z+¢/2) -wo_ coskxz symmetric modes v = 1,3,5,... 2 2
E(z) = {se kx antisymmetric modes) vy = 2,4,6,... 2 St< 43
Be~9(2-4/2) f <2<0
(2.6.2)

where the normalization constants A and B are given by

E 2P

2 _ ke
n U/2+g-h B= Acos 2 (2.6.3)

with Zo = \/uo/€o the impedance of free space and P the power flowing in the

mode; this gives the conventional electromagnetic normalization®®
a ; W1L0 Zo
/ E;(z)E(j(z) dz = ae i i 2 koh (2.6.4)
— oo

with 6;; the Kronecker delta function. The lateral wavevector k inside the core

region and the evanescent wavevector g in the cladding region are given by

27
k= S(n6 — Pi?

Qn 5 ay 1/2 (2.6.5)
g= cw ~ ne)

where A is the free space optical wavelength. The eigenvalue 7 is determined by

the roots of the secular equation

_ } —k/g symmetric modes v = 1,3,5,...
tan(ke/2) = { —g/k antisymmetric modes) v = 2,4,6, eee (2.6.6)

~51-

§2.6 Symmetric Three Layer “Box” Waveguides

Equation (2.6.2) to (2.6.6) are valid for both real index guided and gain
guided box waveguides. The nearfield pattern is given by the magnitude of Equa-
tion (2.6.2), while the farfield pattern is given by a slightly modified Fourier Trans-
form of the amplitude nearfield pattern.37»38

Note that Equation (2.7-7) of Reference (37] uses the e~*9? convention, leading
to the (+7) transform of (2.7-28). Our use of the e+*#? convention in (2.5.3) implies
the use of a (—1) transform to obtain the farfield pattern. Furthermore, through

Equation (2.5.2), it also implies that the antiguiding factor 6 is greater than zero.

(a) Real Index Box Waveguides

For guided modes in a real index waveguide, ne < n < no. When 7 < n,-
the mode is cut off and is no longer guided. Figure 2.1la presents a typical real
index box waveguide. Figure 2.11b shows the intensity nearfield and corresponding
farfield patterns for the first five low order modes, while Figure 2.11c consists of a
plot of the effective index 7 for each of the eight guided modes. Notice that all the
higher order modes have nulls in their nearfield patterns, and that the intensity
of each maximum is the same. It is of particular interest to note that only the
fundamental mode has a single lobed farfield pattern; the farfield patterns of all

higher order modes are multilobed and symmetrical about 0° (85.12)

(b) Gain Guided Bor Waveguides

Figure 2.12 presents the corresponding plots for a gain guided box waveguide.
Since the refractive index is now a complex quantity, so also is the modal eigenvalue
n. As discussed in §2.5, the real part of the eigenvalue, denoted by 7 or Re{n}, is
the effective index for the mode, while the imaginary part, denoted by 7 or $m{n},

defines the power modal gain - = —2ko%. We therefore present the eigenmodes of

—-52-

§2.6 Symmetric Three Layer “Box” Waveguides

3.415
3,415 ‘Ora
+2

n¢

T T
Al. v
+ Ol

3.400

+h

NEARFIELD FARFIELD 3.410

aT}
+O +o

3.415

nN

=|(==|S|[-

3,400
LLL EL
ELLIE
U8 //)

SS

SS

=|=|=|-

‘a L 1 1 L i 1

“10-5 O 5 10 -30 -20 -iI0 0 i0 20 30
LATERAL DIMENSION FARFIELD ANGLE
(jem) (degrees }

FIGURE 2.11 Modes of the real index box waveguide (a) refractive index profile (b) nearfield and
farfield intensity patterns for the first five modes (c) mode structure. Note the nulls in the nearfield
patterns and the multilobed farfield patterns for all except the fundamental mode.

—53-

§2.6(b) Gain Guided Box Waveguides

(A) (C)
| und ~l i T i] i T i
40 cm ' BOL v=i@ _|
r = 2
Le —| ~200 em! = 3+
E +4
QO
(B) NEARFIELD FARFIELD — 5
~ 40 4
z +
oO
vel 2 :
s 30 4
w +
oO
a.
| 20h 4
vee 1 i i ! i i

3.414° 2 4 6 8
—Re {a}—

SEE

SN
Mc
im
Ih

a ee L_ oe
~-40 -20 0 20 40 -6-4 -2 0 2 4 6

LATERAL DIMENSION FARFIELD ANGLE
(pm) (degrees)

FIGURE 2.12 Modes of the gain guided box waveguide (no antiguiding) (a) gain profile (b)
nearfield and farfield intensity patterns for the first five modes (c) mode structure. Note the
slight differences from the modes of the real index box waveguide of Figure 2.11.

—~54-

§2.6(b) Gain Guided Box Waveguides

a waveguide with gain in a modal diagram which plots y vs. 7. The modal diagram
for the waveguide of Figure 2.12a is presented in Figure 2.12b. The modal gain of
a mode is approximately related to the overlap between the optical field and the
gain distribution within the waveguide. This implies that since the lower order
modes are more well-confined, they have higher modal gains than do the higher
order modes.

Another effect of the gain is to make the wavevectors k and g complex, and
to introduce curvature into the phase fronts of a mode, reflecting the fact that
power flows from the high gain region of a mode (inside the core region) towards a
low gain region (in the cladding region).°9 The phase front curvature (and hence
astigmatism) will therefore be less for the fundamental mode in a very wide wave-
guide than it will be in a narrow waveguide in which the fundamental has a modal
gain much less than the peak modal gain. Unfortunately, because of the poor
mode discrimination, the higher order modes of a wide laser will also lase; this
is one cause of the poorly characterized farfield patterns of conventional uniform
gain semiconductor lasers ($1-2(a-22)) (The effect of the complex wavevectors will
be further discussed in §3.5(c); also see §5.10).

Figure 2.12b presents the corresponding intensity nearfield and farfield pat-
terns for the first five modes of a gain guided box waveguide. Comparison of
nearfield patterns of the real index box waveguide of Figure 2.11 with that of the
gain guided box waveguide of Figure 2.12 shows that unlike the real index guided
case, the gain guided structure has deep minima in the nearfield patterns. The
only exact null occurs only at the center of the waveguide. Furthermore, the in-
tensity peaks in the nearfield pattern are no longer uniform but increase slightly

towards the edge of the waveguide.

—55-

These differences between real index and gain guided box waveguides may
be easily understood by introducing a new, very powerful method for analyzing
waveguide modes. In brief, the method consists of following the path of the ar-
gument of the optical eigenfunction (e.g., kx in the sine or cosine functions of
Equation (2.6.2)) throughout the complex plane; hence the name “Path Analy-
sis.” We will use this method to analyze the uniform box waveguide in this chapter;
however, the real power of Path Analysis will not become apparent until we begin
our study of the asymmetric linear tailored gain waveguide in Chapter 5, where
we will see that it allows analytical calculation of waveguide properties using some
simple geometric and algebraic arguments.

In general, a waveguide eigenmode must be described in terms of a linear com-
bination of the two linearly independent solutions of the second order differential
Helmholtz equation (2.5.4) inside the waveguide. In the case of the uniform box
waveguide, these are the sine and cosine functions. The boundary conditions then
determine the relative contribution of each of the two linearly independent solu-
tions to the optical field on each side of the interface. However, it is often possible
to simplify the analysis by eliminating one of the independent solutions by suitable
choice of the coordinate system. For example, if the origin is taken to be at the
center of the box waveguide, symmetry implies that each eigenmode has a definite
parity, either even or odd, corresponding to the cosine or sine functions, respec-
tively. Thus, the symmetry of the waveguide allows the eigenmode to be described
entirely in terms of the properties of either the cosine function (symmetric modes)
or sine function (antisymmetric modes).

Figure 2.13 illustrates the use of Path Analysis for the simple case of a real

dE (x

index box waveguide. The boundary conditions require (1) that E(x) and a

~ 56 —

§2.6(c) Path Analysis of Box Waveguides

be continuous at the edges +¢/2 of the waveguide, and (2) that the electric field

outside the waveguide be an evanescent exponential.

REAL INDEX GUIDED: k real

—AMPLITUDE—>

l N | L |
-2T1 -17 O 1 er

—€ = kx——

FIGURE 2.13 Path analysis for a wide real index guided box waveguide showing the path of the
argument kz of the sine or cosine function along the real axis for the first three modes.

For a well-confined mode, the argument kr must be approximately equal to a
zero of the cosine or sine function when |z| = ‘. This is shown by the endpoints
of the heavy solid horizontal line in Figure 2.13. This line plots the line £ that
the argument € = kz of the sine or cosine function follows along the real axis
for the first three modes in a wide real index waveguide with many well-confined

modes. For the fundamental mode, the line starts near € = —ké/2 ~ —1/2 and

—~57-

§2.6(c) Path Analysis of Box Waveguides

ends near € = +ké/2 = +/2. Equation (2.6.5) shows that the length of the line
£ increases with the mode number v. In a real index waveguide, k is real so that
L is restricted to the real axis.

Figure 2.13 also plots the sine and cosine functions using a solid line for the
first three modes, thus yielding the vy = 1,2,3, modes of Figure 2.11. Since the
sine and cosine function vary between 0 and +1 along the real axis, the minima
of the electric field will all be exact nulls, while the intensity maxima will be the

same for each peak across the guide, as shown in Figure 2.11b.

GAIN GUIDED: K COMPLEX

[_J< |sin(z)| <

oO
> a4
iT]
Poles

0 + 2x 30 4r Sm
—Re{z}—e

FIGURE 2.14 Path analysis for a wide gain guided box waveguide showing the path of the argument
ka of the sine or cosine function in the complex plane for the y = 11 mode. The values of the
complex sine function are indicated using level lines.

Now consider the case of a gatn guided box waveguide of Figure 2.12. The
eigenvalue 7 and wavevector k are now complex, and the path of the argument

£: z= kz of the sine or cosine function is no longer restricted to the real € axis. It

~ 58 -

§2.6(c) Path Analysis of Box Waveguides

now makes an angle @ = tan~! smi with the real axis, and will cross it only once

at the exact center of the waveguide when x = 0. We therefore need to consider
sin z or cosz as a function of the complex argument z = (k + ik) x. Figure 2.14
shows a plot of the level lines of |sin z| near the real axis along with the path £
for the vy = 11 mode of a very wide waveguide. Since the only zeros of the sine
and cosine functions occur along the real axis, we see that there will only be one
exact null in a gain guided box waveguide at z = 0. The exponential growth of the
sine and cosine functions away from the real axis suggests that the intensity of the
optical field will also grow exponentially away from the center of the waveguide.
Physically, this is due the incomplete reflection of the wave at the edge of the
waveguide, which in turn leads to net power flow away from the waveguide core
into the surrounding lossy media. This causes the phase fronts of the mode in the
core region to be curved (see §2.6(b) and 3.5(c.ii)). Mathematically, we can write

the electric field intensity inside the waveguide as

= |cos(k + ik)x ? (2.6.7)
= cos* kr + sinh? kz .
The nearfield pattern of a high order mode have an envelope function which grows
exponentially away from the center of the waveguide. These results are plotted in
Figure 2.15, which compares the high order y = 11 mode of a real index and gain
guided box waveguide, clearly illustrating the effect of the complex k vector and

its effect on the nearfield pattern.

—~ 59 —

§2.6(c) Path Analysis of Box Waveguides

Real Index Guided Gain Guided
(A) (B)

_ L__

y=l\ y =i

FIGURE 2.15 Comparison of the vy = 11 mode for (a) real index and (b) gain guided box waveguide
showing the effect of the complex k vector on the nearfield pattern.

Although the preceding discussion was directed towards waveguides which sup-
port many high order modes, we remark that the complex k vector also has an
effect on the width of the fundamental mode when compared with an equivalent
real index guided mode (see Figure 2.9). The evanescent wavevector g of Equa-
tion (2.6.5) is also complex. This has a very important effect on the modes of
coupled gain guided box waveguides (see §3.5(c.ii)), and may lead to enhanced

coupling between the elements in an array of gain guided lasers($3-5(¢.24))

§2.7 Numerical Solutions for Arbitrary Waveguides

The Helmholtz equation (2.5.4) describing optical wave propagation in a di-
electric media may be solved exactly over all space only for a few continuous index
profiles n(x) such as the quadratic and inverse cosh distributions.** As we saw
in §2.6, the effect of step discontinuities, such as those encountered by the stan-
dard double heterostructure, may be included by solving the Helmholtz equation
within piecewise continuous regions of space and then requiring that the electric

field and its derivative be continuous at the interface. This technique works well

—~60-

§2.7 Numerical Solutions for Arbitrary Waveguides

for the simple case of a double heterostructure because the eigenfunctions of the
free space Helmholtz equations are the sine and cosine functions, which are easy

to evaluate numerically.

—— exact

‘ eecv5e approx.

INDEX n(x)

LATERAL DIMENSION x

FIGURE 2.16 Numerical approximation for complicated waveguides showing an arbitrary index
profile (solid line) and its slab waveguide approximation (dashed line).

As illustrated in Figure 2.16 the same technique may be used to obtain the
approximate eigenvalues of waveguides with more complicated index profiles by
subdividing a wide waveguide with a continuously varying index of refraction into
many smaller contiguous slab waveguides, each having a constant index of refrac-
tion. The electric field inside each elemental slab waveguide may then be written
in terms of the sine and cosine functions (or, alternatively, as complex exponen-
tials). Matching the boundary conditions at each interface and requiring that
the field decay evanescently outside the waveguide yields an eigenvalue equation
which may be solved numerically. The net effect of this approach is to transform
the Helmholtz equation from a second order differential equation into an algebraic

equation whose roots give approximations for the eigenvalues.

-61-
§2.7 Numerical Solutions for Arbitrary Waveguides

We use a matrix propagation technique which has found application in quan-

tum mechanics?® and in the study of periodically stratified optical media*! and

extend its use to waveguides with arbitrary index and gain profiles. We write the

electric field at any point z inside the r** elemental slab waveguide as a superpo-

sition of traveling waves moving to the left and right:
E,(z) = (apetibr# + bre thr) etlkonz—Mt) 1,2,...7; (2.7.1)

ay and by are constant coefficients, which are, in general, complex. 1 is the angular

frequency of the optical wave, and ky is the spatial lateral wavevector in the r¢?

kp = koy/n2 — 2 (2.7.2)

where ko is the free space wavevector 2m ny is the (possibly complex) index of

slab which is defined by

refraction in the r‘” layer, and 7 is the effective index of the eigenmode. In
what follows, the z and t dependence is superfluous, and so we drop the second
exponential product in (2.7.1).

If we write the electric field in Equation (2.7.1) in vector form
; ik,
E,(z) = jeri ,e *] é| (2.7.3)

and match the boundary conditions at the interface between the r*? and rt? +1
layer. We can derive a relationship between the a and 6 coefficients on each side

of the interface in terms of a matrix equation

Or+1| = 7, | or 2.7.4
ict] =a [fe r
where the tnterface propagation matriz J is given by

= _1i krtt + ky ky44 —ky
an 2ky+t fae —kyp Kkpya thy | ° (2.7.5)

-~62—-

§2.7 Numerical Solutions for Arbitrary Waveguides

Similarly, after free space propagation a distance / through the layer, the elec-
tric field becomes

E,(x + 1) _ a,ettkr(z+l) + beter (z+)
_ al.etthrz 4 ble thr ;

The new coefficients a}. and b/. are given by the matrix equation

a a
where the free space propagation matriz F, is given by

tkyl

Given arbitrary az and by coefficients at the far left of a guide with n layers,
the corresponding ap and bp coefficients at the right of the guide may be found

by multiplication of the ¥ and I matrices:
ap | _ A B ay
|= [2 5 | (3 | (2.7.9)

where

M= E b | =InFnIn—tFn—1- heli (2.7.10)

and the layers are numbered from left to right.
For a real index waveguide, the fields must be evanescent in the cladding region,

so that ky and kp are imaginary. If we write
kr =t9LR (2.7.11)
the exponential functions then have real arguments gy, p:

tikaz —gz
jee | —_ | . (2.7.12)

~ 63 —

§2.7 Numerical Solutions for Arbitrary Waveguides

For a guided mode, the coefficients ay and bp to the exponentially growing terms

must be zero. This leads to the dispersion relationship
D=0. (2.7.13)

This result may also be derived by invoking the radiation condition that allows
only outward traveling waves to contribute to the field of a guided mode. Gain
guided waveguides are similar, but the condition (2.7.11) becomes Sm{kz} < 0.
The dispersion relation (2.7.13) remains unchanged, but the k, are complex.

As an example, consider the symmetric three layer slab waveguide

ny = Ne -—wo<2r<07
n(z) = {4 ng =n9 Oot<2rN3Z = Ne lt

where ne and ng are (possibly complex) constants. The evanescent wave vectors
in regions 1 and 3 are imaginary, kz = kp = tg and the propagation matrix is
simply M = I2F)J;. The eigenvalue condition becomes

(ig + k)2e—tke _ (—ig +4 k)ettke
4itkg

0 = Dthree layer — (2.7.15)

where k and g are given by Equation (2.6.5). Equation (2.7.15) reduces to

_ J -k/g positive root — symmetric modes)
tan(ke/2) = { —g/k tose square root —> antisymmetric modes)

(2.7.16)
which is the eigenvalue equation (2.6.6).

This method may be extended to slab waveguides with more layers, although
the analytic results rapidly become unwieldly. The multilayer slab approximation
to a continuous waveguide is a good one provided that k,l, is not too large. From
a numerical point of view the series of 2 x 2 matrix computations (2.7.10) are easy
to perform for any number of layers. D may be considered a function of n, with

the zeros to be found numerically. For real index waveguides, 7 is real, and the

—~ 64 —

§2.7 Numerical Solutions for Arbitrary Waveguides

zeros of D are easy to find. If, however, the effects of gain and/or loss are to be
considered, 7 becomes complex, and it is necessary to search a two-dimensional
parameter space Re{7} @ Sm{n}. The problem is further complicated because D
is not analytic due to the branch points arising from the square-root dependence
of k on 7 in Equation (2.7.2). Therefore, complex root finders which make use
of the analyticity of the function cannot be easily used. Furthermore, from the
point of view of semiconductor laser design, it is important to find all of the modes
with high modal gains, and especially the mode that has the highest modal gain
because that will be the lasing mode at threshold.

In waveguides in which gain is not a perturbation (e.g., in a gain guided laser),
the task of finding the zeros of D is considerably simplified by the equivalent
technique of minimizing |D|. We find that it is not unusual for V|D| to be very
large, often as much as 102°. As a result, automated root finders often miss roots.
This problem can be minimized by making use of a contour plot of |D|; roots may
be approximately located merely by the structure they introduce into the contour
lines.

For example, inspection of the contour plot of Figure 2.17 shows that there is
almost certainly a root within the area enclosed by box B, while there may be one
in the box marked C. Note that the minimum value of |D| in box B is 10%; the
gradients are so steep in this region that an automated root finder will probably
miss the root unless the program is given a good initial guess. Furthermore, it is
obvious that there are no roots to be found within box A; this is the most effective
way of insuring that all the high gain modes have been found.

Once the eigenvalues have been found, the electric field may be calculated by

setting (5%) to (°), multiplying by J, to get the a and 6 coefficients inside the first

~ 65 —

§2.7 Numerical Solutions for Arbitrary Waveguides

ate
W/.

— EFFECTIVE INDEX Re {n}——>

Im {y}/2k,—>
——

MODAL GAIN y

FIGURE 2.17 Contour plot of the level lines of the magnitude of the dispersion function of a
complicated waveguide showing several regions of particular interest.
layer, using (2.7.1) with x = O taken to be the left edge of the layer to get the
electric field within the layer, then using (2.7.4) to cross the boundary, ete.
Finally, we remark that, if one starts with an a priori index and gain distri-
bution, the method applied here is valid only for the unsaturated waveguide (t.e.,
only at threshold). However, an iterative scheme could easily be devised which
includes the effects of gain saturation via the rate equations. The effect of gain
saturation is to reduce the mode discrimination between the modes; therefore, an
accurate model requires that all the high gain modes be identified. This can be
easily done with the method described here.
A listing of the essential parts of the MODES and the CONTOUR plotting

computer programs used to calculate the dispersion function D, and to calculate

~ 66 —

§2.7 Numerical Solutions for Arbitrary Waveguides

the nearfield and farfield patterns of a mode are given in the Appendix. We have

made extensive use of these programs throughout this thesis.

~67-

CHAPTER

THREE
Two Coupled Lasers

Better two than one by himself,
since thus their work is really profitable.

—Ecclestiastes 4:9

In this Chapter, we consider in detail the very simplest possible array, that
of two coupled box waveguides. When two waveguides are placed close together
so that their optical fields overlap, an optical eigenmode of one of the individual
waveguides will not be an eigenmode of the composite two waveguide system. If
each waveguide supports only the fundamental mode, to an excellent degree of
approximation, the new eigenmode will be given by a linear combination of the
two individual waveguide modes. In a laser, the mode of the composite system
with the highest modal gain will be the lasing mode. We therefore need to find
an expression for this mode’s effective index, as well as its nearfield and farfield
pattern. |

We consider weakly coupled real index guided waveguides in §3.1, strongly
coupled waveguides in §3.2, show which classes of coupled waveguides have phase
matching wavelengths in §3.3, and give a means of designing two coupled wave-
guides with a given phase matching wavelength in §3.1(a). We then take up the
effect of gain on weakly coupled waveguides in §3.4. Finally, the complex character
of the wavevectors for the case of two strongly coupled gain induced waveguides
makes these systems especially interesting; they are studied both theoretically and

experimentally in §3.5.

— 68 —

§Two Coupled Lasers
§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

The solution of the Helmholtz equation with an index profile consisting of two
adjacent waveguides may be found by use of the coupled mode equations. These
equations and their derivation have been extensively discussed in the literature;
thus here we merely present an outline of their derivation with a view to under-
standing their applicability and limitations when applied to the understanding of

phased array semiconductor lasers.

(a) _ 2

| An,

Ne

(b) —~ P24, Ano
Ne

(c)
Ne

FIGURE 8.1 Spatial variation of the refractive index for two uncoupled waveguides (a) ny(zx) and
(b) n2(x), and (c) the two coupled waveguides n(z).

Following Yariv,!? we consider the case of the two planar waveguides illus-
trated in Figure 3.1. Refractive index distributions for the individual two guides
are given by n,(z) in Figure 3.la and n(x) in Figure 3.1b. The spatial refrac-
tive index variation for the combined two waveguide system, which is shown in

Figure 3.1c, is given by

n(x) = n3(z) + n3 (2) —n? (3.1.1)

— 69 ~

§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

where ne is the common cladding index external to all the core regions. We
will often use the term “channel region” interchangably with “core region” and
“interchannel region” to refer to the cladding layer between the waveguides.

We denote the eigenmodes of guide 1,2 as Ey, (zx) e*1,27 with both fields as-
sumed normalized by Equation (2.6.4). We also assume without loss of generality
that 62 > 61. For many purposes, the mode propagation constants B12 = kom,2
are more convienient to use for analytical calculations than are the mode effective
indices 71,9; we will use both formalisms interchangeably.

The eigenmodes of the combined guide are denoted by E(z,z). In the limit
of weak coupling, E may be approximated as a linear combination of the electric

fields in the two elemental waveguides:
E(z, 2) = 5 { A1(2) Ei (2)e*4 + Ap(z)Ea(z)e*} tee, (3.1.2)

where c. c. represents the complex conjugate. In the absence of coupling — that
is, if the distance between the waveguides were infinite — A(z) and Ao(z) will
not depend on z and will be independent of each other since each of the two
terms on the right-hand side of (3.1.2) satisfies the Helmholtz equation (2.5.4)
separately. When the guides are placed in close proximity, an eigenmode of the
composite system has the property that the shape of the mode (t.e., the ratio
|A2(z)|/|A1(z)| is independent of z). This implies that (3.1.2) may be rewritten
as (83-2)

E(z,z) = [Ey(x) + @ Eo(zx)] e?? . (3.1.3)

We desire to find the new mode propagation constant @ and the admixture factor
@ which relates the optical amplitude in one guide to that in the other.

If we define a column vector E(z) to represent the two terms in (3.1.2)

ee = eee | = (EG) a

~70-

§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

then the evolution of E(z) is described by the matrix equation

dE

— =1CE 1.5
de tC (3.1.5)
with
Py, K42

= 1.6
C | fe Bo. (3.1.6)

The coupling coefficients K12 are given by?

21
1k 9°
Ki = — 2 in (x) — ne | Ey(z)E2(z) dz . (3.1.7)
21 4Zo Jol 2

— . [Ho ;
where Zp = ~ is the impedance of free space. The new propagation constants
8 and the admixture factor g are given by the eigenvalues and eigenvectors of the

matrix (3.1.6).

The eigenvalues of (3.1.6) are given by the roots of the secular equation

Bi-B kip

Kot Bs B| = 9: (3.1.8)

Since the two basis vectors Ey 2(z) are assumed to be orthogonal, the diagonaliza-
tion of the perturbed matrix (3.1.6) also yields two eigenvectors, denoted E*(z),

which are known as the “supermodes”‘ of the composite waveguide. They may be

written as

E*(z) = oe eBee (3.1.9)

where
BX =B+tVA2+ x2 (3.1.10)

is the propagation constant of the (+) supermode and « = ,/K12K 1 is the mean
coupling constant. The average propagation constant B and phase mismatch pa-

rameter A are defined to be

B = 3(B1 + Ba)
(3.1.11)
A=

(81 — Be) .

-~71-

§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

Finally, the admixture factor o* is given in terms of the normalized phase

mismatch parameter 6

ot =6+V14+ 62

= +exp(+sinh~! 6)

Nn
Ill
al >

(3.1.12)

The special condition 6 = 0 (which corresponds to 3; = (2) is referred to as the
phase matching point. Obviously, identical waveguides will be phase matched at
all wavelengths, while nonidentical waveguides, having differing dispersion curves
(see Figure 3.3), will not be phase matched except at a possible phase matching
wavelength Xp. We note in passing that not all waveguides have a phase matching
wavelength; we will discuss this point further in §3.3.

The phase matching wavelength is a very important number which character-
izes a system of two coupled waveguides. Equation (3.1.12) shows that far away
from the phase matching wavelength 6 — oo and the supermodes are identical to

the modes of the isolated waveguides:
Et (z) — H P22 E~(z) — 3 Plz = (89 > 4) . (3.1.13)

As the two waveguides become more closely phase matched, the supermode
becomes more equally distributed between the two waveguides. At the phase
matching wavelength 6 = 0, the admixture factor is unity, and equal power flows

in each guide. The solutions E+(z) become
Et(z) — | bz — E-(z) 1 jee (3.1.14)

This is, of course, to be expected in the special case of two identical waveguides
where, by symmetry, |g| must be unity, and since all the quantities in (3.1.12) are
real, the phase of the fields can only be either 0 or 7. The inphase addition of the

two modes (corresponding to 8+) is usually referred to as the (++) supermode,

~72—

§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

while the out-of-phase combination (corresponding to @~) is referred to as the
(+—) supermode. Equation (3.1.14) also applies to two dissimilar waveguides at

the phase matching wavelength.

f \ ZX
e -
° LN LN
x INDIVIDUAL \
oO MODES
bk
= SUPER MODES
< ~~
(ep)
2 7
8 Ay f 2k LL an
Z ~~
= hase matchi
Ee eoint "
Os Np
ir \ No/
a \ a
L \
Ap
— Wavelength \ ——>

FIGURE 3.2 Schematic diagram of supermodes near the phase matching wavelength.

These results are shown schematically along with the dispersion curves for the
coupled waveguide system in Figure 3.2. The individual unperturbed dispersion
curves are indicated by the straight dashed lines, while the supermodes of the com-
posite waveguide are shown as the solid curves. The phase matching wavelength
is indicated by the intersection of the two dashed lines. Far away from the phase
matching wavelength, the dispersion curves for the two supermodes are asymptotic

to the dispersion curves for the uncoupled individual waveguide modes. Near the

~73-

§3.1 Coupled Mode Theory of Weakly Coupled Waveguides

phase matching wavelength at which the two individual waveguide modes cross,
the dispersion curves of the supermodes separate. The difference in the super-
mode propagation constants at the phase matching point is then simply twice the

coupling constant.

(a) Calculation of the Phase Matching Wavelength

From the point of view of optoelectronic device design utilizing two nonidenti-
cal waveguides, it is important to know the location of the phase matching wave-
length Ap. Given two arbitrary waveguides, the phase matching wavelength can
usually be found only by numerically solving the dispersion equation (2.6.6) for
each individual uncoupled waveguide separately. However, for the case of two
nearly identical coupled waveguides, it is possible to derive a general relationship
between the waveguide parameters so that two guides may be designed around a
predetermined phase matching wavelength (e.g., within the spectral gain curve of
GaAs).°

We start with the dispersion equation (2.6.6) for the fundamental mode of a
box waveguide, and write it as an explicit function of the parameters defining the

guide. Equation (2.6.6) is of the form

D(z2) = D(ne,no, A, 2,7) = ke/2tan ke/2 — g£/2 =0 (3.1.15)

20
k- <(n _ n?)1/2

x (3.1.16)
g = —(n? — n2)1/?

and the xo represent the five parameters n,,no,,0, which define a mode in
a box waveguide. Equation (3.1.15) is an implicit relationship among these five

parameters. If we expand the dispersion equation (3.1.15) for the second waveguide

—~74-
§3.1(a) Calculation of the Phase Matching Wavelength

about the unperturbed parameter values of the first guide using a first order Taylor

expansion, we can write

oD
Oz;

,6(2;) = 0 (3.1.17)

Ly=Tz;

D(z? + 52;) ~ D(x®) + )~

where z; represent the parameters corresponding to the first guide and 6(z,) rep-
resent the differences between the two waveguides. Since the dispersion equation
D(z?) = 0 is satisfied for the unperturbed parameter values, Equation (3.1.17)

establishes a linear relationship between the partials of D and changes in at most

four of the five parameters:
A6(ne) + B6(no) + C6(A) + E6(€)+ F6(n) =0. (3.1.18)

For example, if we allow changes only in the width @ of the guide and the core

index no, 6(£) and 6(ng) are related by

B ab

6(é) = ~ 34 (no) = jak 6(no) (3.1.19)
where ke/2
= —3—— > + tan(ké/2)
coe etl 2) (3.1.20)
°= Tek)?

and k and g are given by Equation (3.1.16). We remark that essentially the same
relationships may be obtained using first order perturbation theory;®” the present
method has the great advantage that it does not require a priori knowledge of
the electric field and power filling factors, and also that the method may be easily
extended to any waveguide for which the dispersion equation is known. It can
also be used to derive simple relationships between any other two or three of the

parameters in Equation (3.1.18).

—~75—

§3.1(a) Calculation of the Phase Matching Wavelength

We now consider a numerical example to help explore the limitations of the
coupled mode theory. If we start with a guide with the parameters ne = 3.550,
An = ng— ne = 0.050, 21 = 1.0um, and require the phase matching wavelength to
occur at 0.85um, we find that a waveguide with An = 0.051 will phase match to
the first waveguide at A, = 0.846975446um if the width of the second guide has a
width £2 = 0.943274347um. Equation (3.1.19) does not exactly predict the phase
matching wavelength due the effect of the Coulomb self-energy term (see §3.2) on
n and the effect of higher order terms in (3.1.17), both of which have been ignored;
the exact phase matching wavelength was found numerically. The dispersion curves
for these two guides at infinite separation is plotted in Figure 3.3 along with the
electric field amplitude at three different wavelengths. Note that the fields become
less well-confined at longer wavelengths, and so the coupling between two adjacent
waveguides a fixed distance apart will increase as the wavelength increases. Also,

there is only one phase matching wavelength; this point will be discused further

in §3.3.

(b) Comparison of the Coupled Mode Theory With the Exact Theory

We use the two waveguides of Figure 3.3 to illustrate the predictions of the
coupled mode solutions of §3.1 by plotting the exact numerical eigenmodes near the
phase matching wavelength (Figure 3.4) and far away from it (Figure 3.5). Close
to the phase matching wavelength the predictions of the coupled mode theory
appear to be correct: away from the exact phase matching wavelength, the optical
field is concentrated in either one or the other of the two guides, while at the phase

matching point the admixture factor is unity, as expected.

3.595

3.590

3.585

t 3.580

Index

— Effective

3.575

3.570

3.565

3.560

3.555

3.550

—~76-

§3.1(b) Comparison of the Coupled Mode Theory With the Exact Theory

GUIDE | An £(ym)
© |0.050 1.00

- §) {0.051 0.943274347

dp =.846975446 pum

| ! J l }

| a
O05 10 15 20 25 30 35 40
— Wavelength » (,.m)—>

FIGURE 8.3 Dispersion curves for two slightly different real index guided box waveguides. The
insets indicate the extent of the field amplitude relative to the guide width t at various wavelengths.

~77 —-

§3.1(b) Comparison of the Coupled Mode Theory With the Exact Theory

WAVE GUIDE

© ®)
£=1.00 -£#0,.94+

An=0.050_ An=0.05!
2b
FA2uem
15
‘O 10 \,
Xx 5
Qa
5 | Mn
£& 5 F

-10 fF |
-| 5 =
\p=0.846975446

“20 fn +3.58846179
a~b i i j L j j a i “yl i i —

-120 -80 -40 0 40 120

FIGURE 3.4 Exact solutions near the phase matching wavelength for the two waveguides of Fig-
ure 3.3 separated by 2um. Note that they compare well with the coupled mode theory predictions
of Figure 3.2.

However, Figure 3.5 shows that far away from the phase matching wavelength in
the direction of longer wavelength the admixture factor (which is predicted by the
coupled mode theory to decrease indefinitely) actually increases towards a limiting
value of unity! The reasons for this behavior will be discussed in the next section,

especially in §3.2(b).

—~ 78 -

§3.1(b) Comparison of the Coupled Mode Theory With the Exact Theory

WAVEGUIDE

eo © (R)
3.999 £={|.00 £=0.94+
An=0.050 An=0.05I
SLE

kK
2pm

3.580-F

3.5755

3.570F \

3.565

3.560fF

3.555-

] ] ! l r 1 |
05 10 15 20 25 30 35 40
— (pm) —>

FIGURE 8.5 The solid lines plot the exact solutions far away from the phase matching wavelength
for the waveguide of Figure 3.4. The dahsed line shows the dispersion curve of the individual
isolated waveguides of Figure 3.3. Note that the coupled mode theory prediction that power flows
in only one guide or the other when the waveguides are very phase mismatched is correct at short
wavelengths, but is incorrect at long wavelengths (or equivalently, large overlap between the fields).

~79-

§3.2 The Quantum Chemistry of Strongly Coupled Waveguides
§3.2 The Quantum Chemistry of Strongly Coupled Waveguides

Three assumptions have been made in the derivation of the coupled mode

equations:

(1) that the composite eigenmode can be expressed as a linear combination of the
individual eigenmodes (Equation (3.1.2)),

(2) an adiabatic approximation® which makes the coupled mode equations (Equa-
tion (3.1.5)) first order in z, and

(3) that the individual eigenmodes E}2(z) in (3.1.2) are orthogonal.

These assumptions restrict application of the coupled mode theory to weakly
coupled waveguides in which the optical fields do not overlap strongly. However,
as we pointed out in Chapter 1 (and discuss further in §4.6), there are a variety
of reasons for wanting to make strongly coupled arrays. We therefore take up the
study of two strongly coupled waveguides.

Assumption (1), that the composite eigenmode can be expressed as a linear
combination of the individual waveguide modes, ignores the fact that even in the
case of a single isolated waveguide, the guided modes are an incomplete basis set
because leaky and radiation modes have been ignored.? We will show in §3.2(c),
however, that for the symmetric single slab waveguides we are interested in assem-
bling into laser arrays, the contribution of the leaky and radiation modes can be
safely ignored except for the case of very strong coupling.

We will now construct a more precise theory of the coupling between two
waveguides that does not assume weak coupling (Assumption (2)) and explicitly
takes into account the nonorthogonality of the basis states EZ, 2(x) (Assumption
(3)). This theory is in many ways similar to the quantum mechanical approach

known as LCAO theory (for “Linear Combination of Atomic Orbitals”)!° that is

~ 80 -

83.2 The Quantum Chemistry of Strongly Coupled Waveguides

used to calculate the wavefunctions and eigenenergies of simple molecules such as
the H. 2 ion; hence the title of this section as “The Quantum Chemistry of Strongly
Coupled Waveguides.” (This problem has recently been treated by Hardy and
Streifer!! from a slightly different perspective.)

If we rewrite the Helmholtz equation (2.5.4) as

1 @ 2 2
(as + [—n*(z)] —[-n*] > E(z) =0 (3.2.1)
and compare it with the time independent quantum mechanical Schroedinger
Equation
A? @
—-—-—;+V(zr)-—E = 0 12.2
soa + V (2) - Ep v2) (3.2.2)
we see that the equations are mathematically identical if we make the correspon-
dence
1k 2(z) + V(z) 2 E (3.2.3)
=+e— —-n'(z rz) - . 2.
k2 2m 7

Although the equations are mathematically similar, there are some differences
between the quantum mechanical and electromagnetic theories. First, the time
dependent Schroedinger Equation is of first degree in the time variable, while
Maxwell’s equations are of second degree in both the time and spatial variables.
However, as we will show below, this difference is not important for the waveguides
of interest here. Secondly, there is no quantum mechanical analog of the optical
dispersion curve. Equation (3.2.3) shows the correspondence between the optical
wavelength and the quantum mechanical particle mass. Since the free space quan-
tum mechanical electronic mass is a fundamental constant of nature, the optical
analogy corresponds to considering only one wavelength. Finally, we note that
the effect of gain and/or loss may be conveniently included in the optical formula-
tion through a complex potential which has no analog in the quantum mechanical

formulation.

-81-
§3.2 The Quantum Chemistry of Strongly Coupled Waveguides

The optical problem of finding the effective indices and electric field for a single
real index guided dielectric slab waveguide is therefore essentially equivalent to the
quantum mechanical problem of finding the energy levels and wavefunctions for
a particle in a finite potential well. As remarked earlier, the problem of two
coupled real index guided dielectric waveguides is mathematically similar to the
quantum chemical problem of the Hydrogen molecule-ion. In particular, since
both the Helmholtz equation (3.2.1) and the Schroedinger equation (3.2.2) are
both Sturm-Liouville eigenvalue problems, we can adopt the powerful and elegant
Dirac notation of quantum mechanics to simplify the mathematical manipulations.

We therefore rewrite the Helmholtz equation (3.2.1) as an eigenvalue equation:
NIE >= B?|E> (3.2.4)

with ¥ being the “optical Hamiltonian”:

da
~ dx
n*(x) = nj(z) + n3(2) —n?

x + ken? (zx)

(3.2.5)
where n,(z), no(z), and ne are given by Figure 3.1.
We keep the LCAO approximation (Assumption (1)) by writing the electric

field E(x) (which is denoted in the Dirac notation by |E >) as a linear combination

of the electric fields £y(x) and E2(x) (or |1 > and |2 >, respectively):
|E >= 41 > +¢2|2 > (3.2.6)

where c 2 are constants (note that the coupled mode theory admixture factor

0 =c/c 1). The single guide field |1 > satisfies the Helmholtz equation

d2
My|1 >= lS + Kni(2)| jl >= B2|1 > (3.2.7)

~82-

83.2 The Quantum Chemistry of Strongly Coupled Waveguides

and similarly for ¥2|2 >. For |E > to be an eigenvector of ¥ it is necessary and

sufficient that

=6? fori=1,2