IEEE
JOURNAL
OF
QUANTUM
ELECTRONICS,
VOL.
QE-23,
NO.
4,
APRIL
1987
395
Laterally Coupled-Cavity Semiconductor Lasers
Abstract-We analyze the threshold behavior of a pair of laterally
coupled semiconductor lasers of different lengths. The predictions in-
clude longitudinal mode selectivity leading to single longitudinal mode
operation with a periodicity determined by the length mismatch, and
ripples in the equipower curves in the current plane due to carrier-
induced index shifts. We present experimental measurements that con-
firm these predictions.
I. INTRODUCTION
T
HE desire for narrow linewidths in lasers used in fiber
optic transmission systems has made longitudinal
mode control the subject of study because single-mode
lasers are less noisy than their multimode counterparts.
The latter are plagued by partition noise resulting from
competition among the modes. Unfortunately, the most
common laser geometry-a longitudinally homogeneous
waveguide bounded by two flat mirrors-usually runs in
multiple longitudinal modes due to gain saturation and
high spontaneous emission.
More complicated laser structures have been proposed
that discriminate between longitudinal modes. They in-
clude distributed feedback lasers
[
11
and distributed Bragg
reflectors
[2],
which contain a corrugated grating with a
period of half the optical wavelength. DFB’s and DBR’s
suffer from difficulties in fabrication due to the need to
bury a very fine structure underneath the upper cladding
layers without introducing defects into the crystal struc-
ture. Another direction of research has encompassed cou-
pled-cavity lasers
[3],
[4]
in which multiple Fabry-Perot
resonators are coupled together. They are of particular in-
terest because they are relatively simple to fabricate and
offer the potential of FM operation
[SI,
linewidth reduc-
tion, and modulation speed enhancement
[6]-[8],
along
with single-mode operation.
To date, the most common geometry of coupled-cavity
lasers has been longitudinal, that is, the two lasers are
butted up against each other end to end. In this geometry,
the gap between the two lasers plays a crucial role in the
laser operation. For best gain selectivity, it must be a
.(small) integral number of half wavelengths [9],
[
101. Un-
fortunately, accurate control over the gap requires me-
chanical adjustment, which
is
undesirable in a system.
An alternative
is
to monolithically fabricate two lasers
Manuscript
received
July
28,
1986;
revised
November
21,
1986.
This
work
was
supported
by the
National
Science
Foundation
and the
Office
of
Naval
Research.
Pasadena,
CA
91125.
R.
J.
Lang
and
A.
Yariv
are
with
the
California Institute
of
Technology,
J.
Salzman
is
with
Bell
Communications
Research,
Red
Bank,
NJ
07701.
IEEE
Log
Number
8613068.
s102
p-GaAs
p-Goa6Al~.,As
GaAs
n-Gaa6Alo4As
n+-substrate
TCHED
MIRROR
Fig.
1.
Schematic
drawing
of
a
laterally
coupled-cavity
semiconductor
laser.
side by side
so
that the coupling occurs via the evanescent
fields of the individual lasers
[l
11
,
[
121.
If the lasers are
of different lengths, then the longitudinal spectra of the
two lasers differ, and we expect low thresholds only where
the longitudinal modes of the two lasers coincide. In this
paper, we present a theory of, and experimental measure-
ments on, a laterally coupled-cavity laser. In Section 11,
we outline the theory of operation and calculate some rep-
resentative threshold gain curves that illustrate the gain
discrimination. In Section 111, we present experimental
measurements on the device. In Section IV, we summa-
rize the important points of the paper.
11. THEORY
OF
OPERATION
The device under consideration is illustrated in Fig.
1.
It consists of two lasers of length
L1
(short) and
L2
(long),
characterized by propagation constants
PI
and P2, respec-
tively. We seek to understand the modes of such a struc-
ture and to calculate the threshold gains of each mode.
Our intuition suggests the following: each cavity is on
resonance when the optical path length seen by a field as
it completes a round trip of the cavity becomes an inte-
gral number of wavelengths. Only for a few select fre-
quencies will this condition be satisfied simultaneously in
both cavities. However, the situation
is
more complicated
than that. In the region of the laser where the two lasers
are side by side, a field cannot propagate in one cavity
alone, due to the coupling between the two cavities. The
appropriate description of the system is in terms of the
supermodes
[
121,
that
is,
the modes of the twin wave-
guide.
Any field at a fixed position in the cavity can be written
either as a sum of the supermodes of the cavity or as a
sum of the modes of the individual channels. Along the
laser length, there are portions of a single waveguide and
0018-9197/87/0400-0395$01
.OO
O
1987 IEEE
396
IEEE
JOURNAL
OF
QUANTUM ELECTRONICS.
VOL.
QE-23, NO.
4.
APRIL
1987
portions where the two waveguides are coupled,
so
we
need a means to switch from the supermode representa-
tion
(SM)
to a channel mode
(CM)
representation. We
first define the propagation constants of the isolated chan-
nels.
(1)
where the subscript
1,
2
refers to the channel,
w
is the
lasing frequency,
po
is the nonresonant refractive index,
y1
,2
is the gain in each channel, and
01
is the linewidth
enhancement factor relating changes in the real and imag-
inary index of refraction. When the two cavities are cou-
pled, standard coupled-mode theory gives a good approx-
imation to the propagation constants and fields by
assuming that the supermodes are composed of a linear
combination of the channel modes. We define coupling
coefficients as overlap integrals of the channel mode
fields:
K12
Ap?E](x) E2(X) dx,
S
S
(2)
K~~
=
ApiE2(x)
El(x)
dx
where
A pl,
is the perturbation in index seen by one chan-
nel mode due to the other channel. Then, if we make the
definitions
L
L
s
JK~~K~I
-t
AD2,
(3)
the propagation constants of the two supermodes are given
by
-
Ul.2
=
P
f
s.
(4
1
Any field that is represented by a linear sum of the chan-
nel mode fields can be written as a linear sum of the su-
permode fields as well. If we represent a field by a vector
A_
=
(
al
a2)
cM
where
al
and
a2
are the amplitudes of
the two channel modes; then the amplitudes
b,
and
b2
of
the two supermodes describing the same field can be writ-
ten as
The square matrix
v
-
is given by
with^:,^
=
(1
f
Ap/S)/2.
We point out that
-
vis un-
itary, that is,
v-'
=
yT.
Obviously,
2
=
yPTB.
We can also write the effects
of
any linear operatiocupon the fields as a square matrix that
is unique
within
a
given
representation.
For example, in
the channel mode representation, the field after an en-
counter with a mirror of reflectivity
ro
would be
1
Fig.
2.
Schematic representation
of
an idealized laterally coupled-cavity
laser.
However, to write the appropriate operator for a super-
mode vector, we must transform the operator to the new
representation. We accomplish this with the matrix
y.
For
an operator
T,
if we denote the channel representation of
this operator by
TCM
and the supermode representation by
ISM,
-
the two matrices are related by
TSM
-
=
EECME-'>
TCM
-
=
E-'TSME.
(8)
The need for switching between representations arises be-
cause the matrices for some operations (reflection, prop-
agation) assume a simpler form in one representation than
the other. Let us choose an arbitrary field in the super-
mode representation
ElSM
at
z
=
0
in Fig.
2,
and calculate
the matrix that propagates it through one round trip
of
the
resonator. We do this by composing a matrix for each
portion of the journey and appending it to the left side of
the initial matrix, the identity matrix [14],
[15].
We begin by propagating from
z
=
0
to
z
=
L.
That
matrix, in the supermode representation, is given by in-
spection. It is
since each supermode merely gains a phase factor. At
L,
we must switch over to a channel mode representation by
multiplying by a factor
y-'.
The field in channel
1
sees
a reflectivity
rl,
while tKe field in channel
2
propagates
further for a distance
D,
gets reflected by reflectivity
r2,
and then propagates back to
z
=
L.
This matrix can be
written as
Now we transfer back to the supermode representation by
multiplying by
y.
We propagate back to
z
=
0
with the
matrix
PsM,
andreflect
off
the left mirror. For the case
of
uniform-reflectivity on the left, the reflection matrix takes
the same form in either representation:
LANG
er
al.
:
LATERALLY
COUPLED-CAVITY SEMICONDUCTOR
LASERS
397
Fig.
3.
Threshold gain in the
(y,,
y2)
plane for an
LC2
laser consisting
of
two segments of lengths
200
and
240
pm
and intercavity coupling coef-
ficient
K
=
10
cm-’. Dashed lines indicate threshold gains of the indi-
vidual lasers in the absence
of
coupling.
So,
our round-trip matrix is given by
TSM
-
=
,SM,SM,,CM,
RL
P
VRR
I/-‘P
,SM-
(12)
The lasing condition that a field
BSM
reproduces itself ex-
actly after one round trip can be expressed as
ISMBSM
-
=
BSM.
(13)
We recognize this as an eigenvalue problem. The matrix
-
TSM
-
I
must be singular to have a nontrivial solution for
BSM.
-
If
we define
rleff
=
pf
r,
+
pi
r2e-2JP20
(
14a
)
rZeff
p$rl
+
pf
r2e-2JP2D
(
14b
1
rA
=
p1p2(
rl
-
r2e-2JP20),
(144
-
the associated secular equation can be written as
[
rleff
e-2JmL
-
11
[r2efe
11
-2jd
-
=
r;
e-2j
(01
+m)L
(15)
The roots to
(1
5) implicitly define the threshold gains
y
and lasing frequency
w
of the longitudinal modes because
the propagation constants and thus the supermode
propagation constants
(T~,~,
are functions of gain and fre-
quency.
In Fig.
3,
we have plotted the threshold gains for 11
adjacent longitudinal modes of a representative LC2 laser
consisting of two phase-matched
(
I
PI
-
Pz
l2
<<
I
K~K~
I
)
channels of lengths
200
and
240
pm (for this set of cavity
lengths, the longitudinal mode spectrum possesses
1
1-fold
periodicity). The mirror reflectivities of the two cavities
were taken to be
0.55
and
0.1,
respectively (the former
number is the dielectric reflectivity of
the
GaAs/air inter-
face; the latter reflects imperfections in the etched mirror
[
161). On the same graph, we have plotted the threshold
gains for the modes when the coupling disappears (inde-
pendent lasers or phase-mismatched channels). When the
channels are mismatched, the supermodes are localized
on one channel or the other; consequently, the longitudi-
nal modes of the resonator are just the longitudinal modes
of the individual cavities and are degenerate. Thus, the
horizontal and vertical dashed lines in Fig.
2
correspond
to
6
and
5
modes, respectively. Where the two lines cross,
all
11
modes are degenerate.
There are several features of interest to be gleaned from
this graph. The first is the broken degeneracy of the lon-
gitudinal modes, as seen by the spread curves in Fig.
3.
Fig.
3
shows a two-dimensional space of potential oper-
ating points; the only accessible region of the
yl
-
y2
plane for steady-state operation is the region below and
to the left
of
all of the curves in the graph (corresponding
to subthreshold operation) and the locus of sections of
threshold curves that makes up the boundary
of
that re-
gion (corresponding to laser operation). (The reasons for
this restriction are discussed in somewhat more detail in
[
171
.)
The gain differences between adjacent modes is re-
lated to the spacing between the first mode to lase (the
first line encountered as one moves out from the origin)
and subsequent modes. This spacing is shaded in Fig.
3.
The wider the shaded region is, the greater the gain sep-
aration at the adjacent operating point is. We see that the
greatest spacing and the greatest mode discrimination
arises when
y2
is large and
y1
is small or when we pump
the lossy laser hard. The modes become nearly degenerate
when both lasers are brought close to threshold. This mode
of operation is roughly analogous to the situation in axi-
ally coupled lasers when the “gap” is an odd number of
quarter wavelengths
[
181.
In our case, the equivalent pa-
rameter
(TA)
varies its phase with current.
So,
for exam-
ple, the “gap” could be adjusted by adding an indepen-
dent third contact to the additional section of laser that
acts as a tuning stub. Another feature to observe is that
the plot
of
71th
versus
72th
contains several ripples due to
the changes in optical path length with gain via the
a
pa-
rameter (taken to be
-
5
for the plots). Finally, we see
from the formulas that the distance that controls the pe-
riodicity of the structure is
D,
the difference in path length.
111.
EXPERIMENTAL MEASUREMENTS
The devices were fabricated upon GaAlAs double het-
erostructures grown by liquid phase epitaxy. Twin gain
stripes 4 pm wide with center-to-center separations of
9
pm were defined by proton implantation at 70 keV. CrAu
contacts were evaporated on the surface, and the mirror
of the shorter laser was etched using techniques similar to
those described in
[
161.
The devices were lapped down to
75-100 pm thickness and AuGe contacts were evaporated
on the bottom and annealed under
H2
at
380”
for
20 s.
The devices were then cleaved into varying lengths with
varying differences in cavity length.
One feature that became apparent immediately was that
nearly equal cavity lengths were better for getting single-
mode operation. As the model suggests, the difference in
cavity lengths determines the periodicity of the longitu-
dinal mode spectrum. The spectrum of a device with a
fairly long difference is shown in Fig.
4,
with a sinusoid
of period
c/2pD
superimposed over it. Also shown is the
spectrum of the two devices when operated indepen-
398
IEEE
JOURNAL
OF
QUANTUM
ELECTRONICS,
VOL.
QE-23,
NO.
4,
APRIL
1987
I
1
I
,
I
8500 8520 8540
9560
8580
8603
8620
Ah
(b)
Fig. 4. Longitudinal mode spectrum for a laser of length
L
=
450
pm,
path difference
D
=
60
pm.
(a) Spectrum when
lasers
are operated sep-
arately.
(b)
Spectrum of the composite structure with the periodicity
of
the gap superimposed.
IO0
'jOJl------
t
I,
(ma)
Fig.
6.
Threshold gains of a device with
D
=
3
pm.
I,
=
250
mA
I
A
A\
12.30
mA
8540
8560
E580
8600 8620 8640 8660
~(x)
Fig.
5.
Longitudinal mode spectrum
of
the device of Fig.
4
as a function
of
tuning current in cavity
2
(the long cavity).
dently. This shows another feature that is common to cou-
pled cavities, but that has not been adequately explained;
when two or more cavities are coupled together to reduce
the number of longitudinal modes, the gain curve (as in-
ferred from amplitudes of the longitudinal modes) appears
to shift to a longer wavelength. One possible explanation
is that the losses of coupled-cavity geometries that have
reported this phenomenon (see, for example,
[
191)
are
larger than in the uncoupled case (see Fig.
3),
and the
increased loss necessitates harder pumping and shifts the
gain curve. The shift in optical path length with carrier
density can be seen in Fig.
5
where cavity number
2
is
L
I
0
A
(a)+
Fig.
7.
Optical spectra for different currents for the device of Fig.
6
for
different currents (not to the same vertical scale). The device lased in the
same single longitudinal mode from threshold up to twice threshold.
pumped successively harder, thus increasing the carrier
density in the additional section of length
D
and shifting
the longitudinal modes.
Smaller differences in cavity length demonstrate other
phenomena. Fig.
6
shows the threshold currents (propor-
tional to the threshold gains) required by the two cavities
for a
D
=
3
k
1
pm device, illustrating the ripples from
interference. (For larger differences, the ripples are finer
and are beyond the measurement resolution
of
our sys-
tem.) This particular device lased in a single longitudinal
mode from threshold up to a current level of twice thresh-
old for asymmetric pumping (Fig.
7).
Yet another device
(D
=
10
pm) shows single-mode operation over limited
current ranges of about
20
percent
of
threshold, and shows
a mode hop between single modes; both modes are
on
the
long-wavelength side of the subthreshold gain curve.
As
before, the periodicity is controlled by the difference in
cavity lengths (Fig.
8).
IV.
CONCLUSIONS
In conclusion, we have presented a device capable
of
single-longitudinal mode operation that is easily fabri-
cated monolithically, a laterally coupled-cavity laser. We
note that for optoelectronic integration, it will be desira-
LANG
et
al.
:
LATERALLY COUPLED-CAVITY SEMICONDUCTOR LASERS
399
Fig.
8540 8560
8580
8600 8620 8640 8660
A
(A)
8.
Spectrum of a
D
=
10
pm device showing a mode hop and
periodicity of the path difference.
the
ble to etch all mirrors
of
a laser, and the scheme of etching
and one laser shorter than the other fits neatly within this
plan. Despite the operation of this laser under pulsed con-
ditions and the fact that the modes were gain guided
(which means that the coupling coefficients change some-
what with pump current), large regimes
of
single-mode
operation were obtained. We expect that more stable op-
eration (allowing larger current excursions) will result
from an index-guided structure where the coupling coef-
ficients are constant and the amount
of
spontaneous emis-
sion is less. These results indicate that a laterally coupled-
cavity laser designed for CW operation (e.g, a twin buried
heterostructure) may be suitable for
use
in single-mode
laser systems.
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spectral properties of
Robert
J.
Lang
(S’83-M’86) was born in Day-
ton, OH, on May 4, 1961, and was raised in At-
lanta, GA. In 1982, he received the
B.S.
degree
in electrical engineering from the California In-
stitute of Technology, Pasadena. He received the
M.S. degree, also in electrical engineering, from
Stanford University, Stanford, CA, in 1983.
In
1986, he received the Ph.D. degree in applied
physics from Caltech, for work
on
coupled-cavity
and unstable resonator semiconductor lasers.
Currently, he is investigating dynamic and
sinele-mode lasers at the Standard Elektrik Lorenz
..
Research Centre, StuttgG, West Germany. He has written over twenty
technical papers and two books.
~~~ ~