1174 OPTICS LETTERS / Vol. 16, No. 15 / August 1, 1991
Frequency control using a complex effective
reflectivity in laterally coupled semiconductor laser arrays
Giora Griffel, William K. Marshall, Ilan Grav6, and Amnon Yariv
California
Institute of Technology, Pasadena, California 91125
Rashit Nabiev
P N. Lebedev Physical Institute, Moscow 117924, USSR
Received December 11, 1990
Frequency selectivity of a novel type of multielement, multisection laterally coupled semiconductor laser array
is studied using the roundtrip method. It is found that such a structure should lead to a strong frequency se
lectivity owing to a periodic dependency of the threshold gain on the frequency. A gainguided twocoupled
cavity device was fabricated.
prediction.
The experimental results show excellent agreement with the theoretical
It has been shown that laterally coupled
1 3
and verti
cally coupled
4
arrays of semiconductor lasers possess
a strong frequency selectivity, manifested by the
ability to control the longitudinalmode spectrum
and to tune the lasing wavelength. Roundtrip
analysis has been utilized in the past' to calculate
the laser threshold gain and lasing frequencies. In
this Letter we extend roundtrip analysis to allow for
multilongitudinal sections and varying mirror re
flectivities. The analysis is applied specifically to
the case of two laterally coupled elements, each
having two longitudinal electrodes, a long electrode
for providing the gain and a short electrode near one
facet used to control the phase and/or the magnitude
of the reflectivity for this channel. In this way we
create a complex, frequencydependent, and electri
cally controlled effective reflectivity for the entire
structure. This effective reflectivity is the key
feature of this device, which enables us to control its
lasing spectrum and to select a desired longitudinal
mode operation. We show that for proper selection
of the structural parameters in such a device, a
strong frequency selectivity is obtained.
In general, the steadystate condition for a self
reproducing mode field leads to
Trt U = U,
(1)
where U is a vector representing the optical field in
terms of the isolated channel mode amplitudes and
Trt is the roundtrip propagator matrix. This ma
trix is obtained by a chain multiplication of individ
ual matrices, each of which describes propagation
within a longitudinal section, boundary crossing of
two such sections, reflectivity at the end facet, or a
representation change from a supermode form of
field vector to a channel mode one and vice versa. It
turns out that for the general case of two laterally
coupled channels, each having a different (complex)
end reflectivity, such as shown schematically in
Fig. 1, the result of carrying the above matrix multi
plication is
Trt = (P 12)
2
[r(r
2
A
2
+ r
4
C
2
)
Lr
3
(r
2
A ± r
4
B)C
ri(r
2
A + r
4
B)C]
r
3
(r
2
C
2
+ r4B2)'
where A, B, and C are given by
A =_ P 1exp(icriL) + P2 exp(i2_2 L),
P2 P1i
B _ Pi exp(io
2
L) + 12 exp(iciL),
P2  i
C =_exp(ioiL)  exp(ic0r
2
L).
(3)
L is the length of the device and 1,2 are the (com
plex) propagation constants of the compound (super)
modes. These latter quantities are given by
0712 = Y +S,
(4)
where s _ (Ay ± kabkba)
1 2
, Ay (Ya  yb)/2,
and
Y = (ya + yb)/
2
; Ya,b and
kabba are the propagation
constants and the coupling coefficients of the iso
lated waveguides, respectively. The parameters Pl,2
in Eq. (2) are defined as
P1,2
2
= 1
± "Y /2.
(5)
The condition det(Trt  I) = 0, which follows from
Eq. (1), leads, using Eq. (2), to
(plp2)
2
[rjr
2
A
2
+ r3r
4
B
2
+ (rlr
4
+ r
2
r
3
)C
2
]
 r
1
r
2
r
3
r
4
exp[i2(o, + 0
2
)L] = 1. (6)
We now analyze the device depicted in Fig. 2.
Here each channel has two longitudinal electrodes, a
long one, which is forward biased to provide gain,
and a short endsection electrode for controlling the
phase and the magnitude of the reflectivity at the
channel facets. For this specific case r
1
= r4  r,
r
2
 eir exp(io
1
), and r
3

2
r exp(i0
2
). If we as
sume identical pumping conditions for the two gain
sections, we also have yi = Y2 , AY = 0, and kab =
01469592/91/15117403$5.00/0 © 1991 Optical Society of America
August 1, 1991 / Vol. 16, No. 15
/ OPTICS LETTERS 1175
r
r
3
r
4
Fig. 1. Schematic of the laterally coupled twochannel
[(a) and (b)] structure with different reflectivities at each
channel facet.
Fig. 2. Schematic of the proposed device. Each channel
has two longitudinal electrodes, a gain (long) electrode
and a phase/absorption (short) electrode.
kba  k,
hence 0
1
,
2
= 7 + k, p
1
= P2
= 1/\2, and
A = B = 2 exp(iyL)cos kL,
C = 2i exp(iyL)sin kL.
Equation (6) becomes
X2  2aX + 1 = 0,
where
X =_ V2 exp(i2yL) exp(ib),
a 2_ {[ exp(iAO)
+ \/ exp(iA4
 1 
e
_
7E1E2exp(iO) + V~l2exp(iO)
equally pumped laterally coupled strip quantumwell
lasers using the cosh
2
lateral distribution for the
modal gain. The result is
shown in Fig. 3. It is
seen that k is a complex number whose phase
is con
stantly increased with increasing gain
y. The mag
nitude of the coupling coefficient, although being
proportional to the gain, tends to acquire a constant
value owing to the increased confinement of the in
dividual waveguide mode profiles.
Equation (10) indicates that the parameter a has a
periodic dependency on the frequency through ¢,2
Therefore, the threshold gain, obtained by equating
the absolute values of each side of Eq. (11), is not
constant but instead has a periodic frequency depen
dence. The threshold condition, as a function of the
wavelength, for the case of a gainguided structure
with El = 62 = 1, L = 600
1
um, D
1
= 45 ,m, and
D2 = 55 ,um was calculated. The results are shown
in Fig. 4. The periodic behavior, with repetitive
minima, gives rise to a frequency selectivity in the
laser. Proper selection of the endsection lengths
and the biasing current should therefore lead to
singlelongitudinalmode operation.
In Fig. 5 we present experimental results of the
configuration described in Fig. 2, utilizing a gain
guided stripe laser structure as the channel ele
ments. The stripe widths were 4 ,m, and the
centertocenter separation was 9 ,um. The end
section electrodes were 215 and 195 Aim, and the
(7)
(8)
(9)
t)j
Cos2kL
sin2kL}.
(10)
Here AO = (02  Ol)/2 and k = (02 + 01)/2, where
01,2 = 2/31,2 AL
1
,
2
is the phase contributed
by each
of the short end sections. 81,2 is the propagation
constant at each of the endsection regions. The so
lution to Eq. (8) is
X = exp(i cos
1
a). (11)
For the simple case of two laterally coupled index
guided lasers with no end sections, Eq. (11) reduces
to X = exp(±i2kL), i.e., the threshold condition is
the same as that of the single channel except for a
longitudinal FabryPerot mode splitting that is due
to the appearance of lateral (super) modes. For the
case of gainguided channels, the parameter a (and
therefore qp) is a complex quantity, owing to el, e
2
, or
k being a complex number.
5
We have calculated the
coupling coefficient for the case of two identical,
07
6
15 5h 1
15 .10 5 0
5 10 15
Re [ k] [cm ]
Fig. 3. Calculated coupling coefficient k for the case of
gainguided coupledcavity channels with identical gain in
the two channels.
1 70
0
I
0
w)
I
I
165
1 60
155
150
1 45
140 
0.84
0.88
WAVELENGTH [1Lm]
Fig. 4. Threshold gain versus frequency for the case of
gainguided channels with El = E2 = 1.0 for the sym
metric (solid curve) and antisymmetric (dashed curve)
supermodes.
(a)
(b)
r2
1176
OPTICS
LETTERS
/ Vol.
16, No.
15 / August
1, 1991
1 .
0.
cE 0.
_ 0.
~0.
Z
8.
F
U)
z
wJ
Z
6
4
2
0
0.842
0.844 0.846
0.848
0.850
(
a )
1.0
II
I
I
).a (
0.6 
1.4
).2
).0
0.842 0.844
0.846 0.848
0.850
WAVELENGTH
[gm]
(b)
Fig. 5.
Lasing spectrum
of
the proposed
device
under
different
operating
conditions.
(a) The
endsection
elec
trodes
not
biased (contacts
opened),
(b)
the endsection
electrodes
shorted.
E
1
9
0
0
U)
C
M
M
WAVELENGTH
[gm]
Fig.
6. Calculated
threshold
gain for the
device
of Fig. 5
with end
sections
shorted
to
ground
(e1,
2
=
0.03).
total
length
of the
device
was 600
gm. The
laser
spectra
were
measured
for the
cases of
opencontact
and
shorted
(0V) endsection
electrodes
and
are
shown
in Figs.
5(a) and
5(b), respectively.
In the
case of open
contacts
the
end sections
are
bleached
by
a combined
influence
of
light penetrating
from
the longitudinally
coupled
gain
section
and leak
age current
arriving
from
the adjacent
electrode
in
the
parallel
channels.
In that
case the
end sections
will have
little
influence
on the
cavity loss,
and as
a
result
the
device will
function
as an
ordinary
later
ally
coupled
FabryPerot
stripe
laser,
exhibiting
a
multimode
spectrum
as
observed
in the
measured
spectrum
shown
in Fig. 5(a).
However,
if
the end
section
electrodes
are
shorted
instead,
photoinduced
carriers
are
swept away
from
the
junction.
There
fore
these
sections
are not
bleached,
and
they remain
lossy.
As a
result
their reflectivity
will
have a
dif
ferent
magnitude
and
a different
phase
from
those of
the
adjacent
elements.
The spectrum
observed
in
this case
is
shown in
Fig.
5(b). Here
a
strong
fre
quency
selectivity
is observed,
the
whole
spectrum
shifts
toward
a shorter
wavelength,
and three
iso
lated
longitudinal
modes are
observed.
The separa
tion
between
these
modes
is equal
to
the equivalent
of 27
longitudinal
modes.
A theoretical
calculation
of the
threshold
gain
as a function
of the
frequency
for
the specific
device
parameters,
with
shorted
end
sections
(e
1
,
2
= 0.03), is
shown
in Fig. 6.
The calcu
lation
predicts
minima
separation
of 28 longitudinal
modes
with
0.18nm
separation
between
adjacent
modes.
The
curve
of Fig.
6 represents
the
gains
that
would be
required
for the
structure
to
lase
within
the indicated
wavelength
range.
In
practice,
the
gain is
clamped
to the
value
at which
the
lowest
threshold
modes
lase,
i.e., 167
cm
1
. The
result
is
that only
modes
near the
periodic
minima
can
lase.
The
wavelength
separation
between
these preferred
modes
is
determined
by
the lengths
of
the sections
in
the device.
The 28longitudinalmode
separation
agrees
well
with
the observed
spectrum
of Fig.
5(b).
To
conclude,
we
have analyzed
multielectrode,
laterally
coupledcavity
semiconductor
laser struc
tures.
A detailed
roundtrip
analysis
showed
that
such devices
should
exhibit
strong
frequency
selec
tivity,
which
has been
confirmed
experimentally.
Using
this
approach,
one can
construct
singlelongi
tudinalmode
devices
with
a fairly
simple
fabrica
tion process.
Giora
Griffel
acknowledges
the
support
of
the
Bantrell
postdoctoral
Fellowship
at Caltech.
This
research
was
supported
by
the National
Science
Foundation,
the Office
of Naval
Research,
and
the
Army
Research
Office.
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1