April 1984
/ Vol.
10, No.
4 / OPTICS
LETTERS
125
Supermode
analysis
of
phase-locked
arrays
of
semiconductor
lasers
E.
Kapon
California
Institute
of Technology,
Pasadena,
California
91125
J. Katz
Jet Propulsion
Laboratory,
4800 Oak
Grove
Drive,
Pasadena,
California
91109
A. Yariv
California
Institute
of Technology,
Pasadena,
California
91125
Received
December
23, 1983;
accepted
February 3,
1984
The
optical
characteristics
of
phase-locked
semiconductor
laser
arrays
are
formulated
in terms
of the
array
super-
modes,
which are
the
eigenmodes
of
the composite-array
waveguide,
by using
coupled-mode
theory.
These
super-
modes
are
employed
to
calculate
the near
fields,
the
far fields,
and
the difference
in the longitudinal-mode
oscilla-
tion wavelengths
of
the array.
It
is shown
that
the broadening
in the
far-field
beam
divergence,
as well
as the
broadening
of
each of
the longitudinal
modes
that were
observed
in phase-locked
arrays,
may
arise
from
the excita-
tion
of an increasing
number
of
supermodes
at increasing
pumping
levels.
Phase
locking
of several
diode
lasers
that
are integrat-
ed in
parallel
provides
a
useful
means
for
obtaining
high-power
injection
lasers
having
low
beam
diver-
gence.
1
-
7
Moreover,
it
was
recently
demonstrated
that
phase-locked
arrays
incorporating
separate
laser
con-
tacts
7
also
exhibit
a remarkable
degree
of
longitudi-
nal-mode
selectivity
as
well
as
output-wavelength
tunability.
8
Many
of
the
observed
characteristics
of
these
useful
devices,
however,
are yet
not
fully
under-
stood.
The only
attempts
to explain
the
optical
prop-
erties
of phase-locked
arrays
have
been
limited,
to date,
to
evaluating
the
array
far field,
assuming
that
it con-
sists
of identical
radiators,
2
and
to deriving
the
phase
relationship
between
adjacent
emitters.
9
In this
Letter,
we present
an
optical
model
of phase-
locked
semiconductor
laser
arrays.
This
model
yields
the
optical
characteristics
of the
array
in
terms
of
its
supermodes,
i.e.,
the
eigenmodes
of the
composite-array
waveguide.
For some
special,
yet important,
cases
we
calculate
analytically
the near
fields,
the far
fields,
and
the
propagation
constants
of these
supermodes.
Consider
an
array of
N coupled
lasers,
as
shown
in
Fig. 1.
Each individual
laser waveguide,
when isolated
from
its
neighbors,
is presumed
to support
a single,
TE-like,
spatial
mode.
This
mode
is
described
by its
electric
field
61 (x, y)
exp(iflz),
I
= 1, 2,.
.. N,
where
At
is the
complex
propagation
constant.
The
total electric
field of
the array
is
N
Ey (x,
y IZ)
= F,
&j1(x,
y)Aj
(z) exp(i~jz),
1=4
dE/dz =
iAE,
(2)
where
E is a
vector whose
elements
are El
_= Al exp(ilz)
and
the only
nonvanishing
elements
of
the matrix
A are
ALjl
= $j, with
I = 1,
2,... N,
and A1
1
,j+i
= K',t+i,
Atl+ i
= KI+i
l, with I =
1, 2,. . .
N - 1.
The definition
of the
coupling
coefficients
Kij
is the same
as in
the case
of a
pair of
coupled
waveguides.
1
1
The
array
supermodes
are,
by definition,
the
eigen-
solutions
of Eq.
(2), i.e.,
those vectors
that
satisfy
EV(Z)
= EI(O)
exp(iavz),
(3)
a,
being the
propagation
constant
of the
supermode
E".
Substitution
of Eq.
(3) into
Eq. (2)
gives
(A. -,J)ER
= 0,
(4)
where
I is the unit
matrix.
A
solution
of Eq. (4)
yields the
N supermodes
that are
supported
by an
array of
N single-mode
lasers.
The
eigenvectors
EB, v
= 1, 2, ....
N, can
be used
in Eq. (1)
to evaluate
the
near field
of each
supermode;
each
such
mode
say,
B', describes
a phase-locked
combination
of
the
individual
laser
modes with
amplitudes
E
1
". Gen-
z
(1)
where
the
z
dependence
of
Al(z)
is
due
to
the interac-
tion
among
the
array
elements.
Assuming
only
the
nearest-neighbor
coupling,
the
coupled-mode
equa-
tions
10
for
the
N-channel
array
can
be written
in
the
form
K12
K23
KN-1,
go ~~~~~~~~~~~~W
151
I
I
0 N
IN
g1
02
0,-I
A
Fig. 1. Schematic
illustration
of an N-channel
laser
array.
The y
axis is in the
p-n-junction
plane.
0146-9592/84/040125-03$2.00/0
©
1984, Optical
Society
of America
126
OPTICS
LETTERS /
Vol. 10, No.
4 / April 1984
Table 1. Propagation
Constants
of the Supermodes
of
Identical-Channel Arrays
Number
of
Propagation
Channels N
Constants oa,
2
4 iK12
3 4; K13
4 i(K142/2)
4 [(K
14
2
/2)
2
- K122K34
2
11/
2
11/2
5 f3;f J3 (K12/2)
i [(K
15
2
/2)
2
- (K
1
2
2
K
3
4
2
+ K12
2
K45
2
+ K
23
2
K
45
2
)]
1
/2J1/2
K
11
2
K12
2
+
K2
3
2
+ ... +
KI-1,1
2
and where
E(8) is the far-field
pattern of each individual
array element,
P
111
_ ElP/ElV
are the
admixture
factors,
S
is the center-to-center
separation
of adjacent
lasers,
0
is the angle
in the
junction
plane, and
ko = 2-r/Xo,
X0
being the free-space
wavelength.
Generally, the eigenvalues
of a given N-channel array
can be
found by solving
Eq. (4) numerically.
It
is useful,
however,
to consider
special cases
that allow
for analytic
solution
and provide some
physical insight. The
sim-
plest case is
that of an array
with identical
channels,
Il
= $2 =**
AN
- with uniform
coupling,
i.e., Kij
=
K. In
this case, the
solution
of Eq. (4) is
.,3 2
/-3;
VI
V
A
An
\/2v
3V
A
EzY =sin 1N+)'
U=
+ 2K COS (N7 V
_ +3K
Note
that the splitting in
the propagation
constants of
the
supermodes
is proportional
to the coupling
coeffi-
cient
K. When N >> 1, these propagation
constants
form a quasi-continuum
in the range:
- 2K < of < $
+
2K, and the
maximum wavelength
splitting
in the FP
modes
becomes
[see Eq.
(5)]
A+ K
Axos = (4/r)KLAXOFP
Since usually KL S 1, the excitation
of several super-
modes would
result in an effective
broadening
of each
of
the longitudinal modes
of the laser
array. Such a
broadening of the
FP modes with
increasing pump
current was indeed observed experimentally.
2
In the case of similar
channels ($A =
i) but nonuni-
3- d3K
Fig. 2.
Schematic illustration of the supermodes in a
five-
channel array of equal
waveguides (fj = /) and
uniform
coupling (Kij = K). The numbers beside the near-field
lobes
indicate the relative magnitude of the
field amplitude. The
expressions beside each field
pattern are the corresponding
supermode eigenvalues.
erally, the arrayn near field will consist
of a superposition
of the
near fields of a number of supermodes.
The
different
propagation constants o-p of the different
su-
permodes may give
rise to a group of Fabry-Perot
(FP)
resonances associated
with a given longitudinal mode
of the laser cavity. The
wavelength separation AXOS
between two such
modes, which is usually smaller than
the FP mode spacing AXoFP, is
given by
Ax
0
S =
(AWrL/iX0oFP,
Cl)
z
z
Lii
>-
-J
(5)
where
AO is the difference in the supermode propaga-
tion constants and L is the
laser-cavity length.
The supermode near fields
can be readily employed
to evaluate the far-field radiation pattern
of each su-
permode. In the case of arrays with
similar individual
near
fields 61 = 6, the far-field intensity
pattern in the
junction plane
(y-z plane in Fig. 1) is given
by
Pv(0) = E(6)GV(O),
with
G"(0)
-- | A, P1 exp(ikoS sin
0)
(6a)
Fig. 3. The grating function G for
the supermodes of Fig. 2.
(6b)
The dashed
curve corresponds
to an
array of five
identical
radiators.
(7a)
(++O--)
(+O-O+)
(+-O+ -)
= 1, 2,... N.
(7b)
(8)
*.1 ~
~ ~
1
1+++++ .
) I
F,. -X
I -
'
-- !.
1,I
(++O--)
I
AA:
VVVX
p- Y - 1 _! ' ea -- ' -- 1-
: +0-0+)
l LAX
(+-O+-)
koS-sine
(radians)
April 1984 / Vol. 10, No.
4 / OPTICS LETTERS 127
3 43
KC
2 3
K
34
,/-3/2 r372
AAAAL\iLA
K
2 3
K
3 4
K(
12
K
4 5
K
2 3
K
3 4
I
K1
2
K
4 5
2
Fig. 4. Effect
of variation of the coupling
coefficients on the
near field
of the (+++++) supermode
for K23 = K34.
form coupling
(KiJ # -lm
for ij F# im), Eq. (4)
was solved
analytically
for N < 5. The eigenvalues
for this case are
summarized in
Table 1. Note that in this case the ad-
mixture factors
P11 depend only on
the ratios of the
coupling coefficients
of adjacent pairs
of channels. In
what follows,
we illustrate in more
detail the supermode
features of an array
of five identical lasers.
Figure 2
shows schematically the near-field
patterns
of the supermodes
in such an array
when the coupling
is uniform. Note that the
relative excitation of the
channels
is different
in different supermodes.
In
par-
ticular, some of the supermodes
are characterized by
unexcited channels (which
is indicated by a 0 in the
notation
of Fig. 2). These peculiar forms
of the su-
permode
near fields have
an important
effect on the
value of the saturated-gain
coefficient in each channel
for a given current combination through the
array lasers
and a
given total output
intensity.1
2
The
modal gain
of a given supermode
depends
on the phase
relationship
between the fields
in adjacent channels, which deter-
mines the supermode intensity in the regions
between
the pumped laser stripes. For example,
in the case of
more-or-less equal
currents, which are injected
mainly
below each laser
stripe contact, it is expected that the
(+-+-+)
mode would have the lowest threshold since
the
unpumped regions correspond
to a small
modal
intensity.
Figure 3 shows the grating
function G [Eq. (6b)] for
the supermodes
of Fig. 2. The actual far-field intensity
patterns are obtained
by superimposing upon these
curves the envelope function E(Q). In practical
arrays
(e.g., GaAs arrays with S - 10
Am and 4-,4-m laser
stripes) one can concentrate
on the region IkoS sin 01
' 27r,
outside which E(0) is practically zero.
For
comparison, we also show the grating function
for (five)
identical radiators (dashed curve, Fig. 3),
which was
used in Ref. 2. Note that
the main lobes in the far-field
patterns of the
supermodes that are characterized by
unexcited
channels are displaced with respect to those
of the (+++++) and (+-+-+) ones. Thus
it is clear
that, when several supermodes
are excited, each will
contribute to an effective
angular divergence in the far
field,
as indicated by
the arrows in Fig. 3.
Thus angular
divergencies that can be almost as much
as four times
the diffraction-limited width are
expected. This may
explain the wide
beam divergencies that are observed
with most arrays."
3
'
4
'
6
For the GaAs five-element
array
of Ref. 1 with X0 = 0.8
gm and S = 8 Am, we find that
the
effective angular divergence
(FWHP)
of the
(+-+-+), (+-O+-),
and (+0-0+) supermodes is 1.2',
2.60, and 3.8°, respectively. The experimental
results'
show a broadening
of the main lobe
in the far field
from
1.90 at I = Ith to 3.80
at I = 2.1 X Ith, which was ac-
companied by the appearance
of a structure in this main
lobe, in qualitative agreement
with our prediction.
Finally, we briefly discuss the
effect of varying the
coupling coefficients Kij.
Figure 4 shows the near fields
of the (+++++) supermode for
K23 = K34 and for two
values
of K23/K12
= K34/K45. Variations
in Kij
can be ac-
complished by
fabricating the array with different
spacing of the laser stripes or by controlling
the coupling
by using additional contacts intermediate
to the laser
stripes.'
3
Decreasing the coupling
of the two outermost
lasers results
in further decrease
in their excitation
in
the (+++++) supermode. A stronger coupling
of the
outermost lasers yields a more uniform
excitation in the
near field. This illustrates the
potential use of the
coupling coefficients
in tailoring the near fields of a
phase-locked array.
In conclusion, we have presented an optical
model for
phase-locked semiconductor
laser arrays that is based
on the array supermodes.
The description of the array
optical field in
terms of these supermodes, which are
derived
by coupled-mode theory, is intermediate
be-
tween
treating the array as a single, giant waveguide
and
viewing it as a group
of coupled waveguides. This
model uses the eigenmodes of the
total, composite-array
waveguide while maintaining
and using the information
on the coupling
between the array elements.
The research described in this
Letter was performed
jointly
by the Applied Physics Department,
California
Institute
of Technology, under contracts
with the U.S.
Office of Naval Research and
the National Science
Foundation, and the Jet
Propulsion Laboratory under
contracts with
the National Aeronautics and Space
Administration. E. Kapon acknowledges
the support
of a Weizmann Postdoctoral
Fellowship.
References
1. D. R. Scifres,
R. D. Burnham, and W. Streifer, Appl. Phys.
Lett. 33, 1015 (1978).
2. D.
R. Scifres, W. Streifer, and R. D. Burnham, IEEE
J.
Quantum Electron. QE-15, 917
(1979).
3. D. R. Scifres, C. Lindstrom, R. D. Burnham,
W. Streifer,
and T. L. Paoli, Electron. Lett.
19, 169 (1983).
4. D. E. Ackley and R. W. H. Engleman,
Appl. Phys. Lett.
39,27 (1981).
5. W. T. Tsang, R. A. Logan,
and R. P. Salathe, Appl. Phys.
Lett. 34, 162 (1979).
6. D. Botez and J. C. Connolly,
Appl. Phys. Lett. 43, 1096
(1983).
7. J. Katz, E. Kapon,
C. Lindsey, S. Margalit, U. Shreter, and
A. Yariv, Appl. Phys. Lett. 43, 521
(1983).
8. E. Kapon, J. Katz, S. Margalit, and A. Yariv,
Appl. Phys.
Lett. 44, 157 (1984).
9. K. Otsuka, Electron.
Lett. 19, 723 (1983).
10. A. Yariv, IEEE J. Quantum
Electron. QE-9, 919
(1973).
11. A. Yariv, Introduction
to Optical Electronics, 2nd ed.
(Holt, Rinehart
& Winston, New York, 1976), Chap. 13.
12. J.
Katz, E. Kapon, S. Margalit, and A. Yariv (unpub-
lished).
13. E. Kapon, J. Katz, C.
Lindsey, S. Margalit, and A. Yariv,
Appl.
Phys. Lett. 43, 421 (1983).