FERREIRA, LG

PUDENZI, MAA

Journal of the Optical Society of America

Repositorio da Producao Cientifica e Intelectual da Unicamp

Vol. 71, No. 11/November 1981/J. Opt. Soc. Am. 1377
Waveguiding in a dielectric medium varying slowly in one
transverse direction
Luiz G. Ferreira
Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 20516, Sao Paulo, Brazil
Marcio A. A. Pudensi
Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, Brazil
Received February 13, 1981
We solve the problem of wave propagation in a medium whose dielectric constant varies quickly in one transverse
direction and slowly in the other direction. This dielectric-constant profile is present in the waveguiding of dou-
ble-heterostructure-junction lasers. The advantage of our solution is in its generality because we do not make
other assumptions about the dielectric-constant function. From our solution we were able to correct the expres-
sion for the filling factor in TM modes that is seen in the literature.
1. INTaODUCTION AND RESULTS
We consider here the waveguiding of a medium whose com-
plex dielectric constant has a fast variation along the
transverse y direction and a slow variation along the
transverse x direction. This situation is found in double-
heterostructure (DH) injection lasers whose dielectric-con-
stant profiles have the slab structure sketched in Fig. 1.14
The active region has a larger dielectric constant e and thus
is able to guide light waves. Because of inhomogeneous in-
jection, a variation of e exists along the x direction, but this
variation is slow, with a typical length much longer than the
wavelength of the light.
Because of their importance to the theory of injection lasers,
the waveguiding properties of the slab structure have been
studied on many occasions.1-3 In all instances that we found
in the literature, the function e(x, y) was made to assume
special forms so as to make the solution of the vectorial wave
equation simple, and no mention was made of the z compo-
nent of the field and its dependence on x and y.'3 For this
reason we decided to undertake the present study, eliminating
all simplifying assumptions about the function E(x, y) and
retaining only the assumption that the variation along x is
much slower than that along y.
Our method of solution of the wave equation is perturbative.
We begin by solving the equation in a hypothetical medium
whose dielectric constant d(y) depends only on y. For a given
angular frequency w, we find the wave number k of a solution
with no variation along x:
aE
-=0.
ax
Then we consider the effect of the perturbation
E(x, y) - Z(y)
on-this solution. It turns out that this effect is a modulation
of the unperturbed solution by a function t(x). We postpone
to Section 2 many of the details of the solution, and here we
present just the results.
For the TE solution we find the following expression for the
electric field:
1 (X)X(°)(Y)
\E} i d{I(x) X(°)(Y)
ank dx XT)Jand for the TM solution:
exp(iwt - ikz), (1)
exu(icot - ikz). (2)Lix)Y(M)(y) , -
if(x)Z(0 )(y)
where X(°)(y), Y(O)(y), and Z(0 )(y) are the unperturbed so-
lutions of the wave equation for the medium e(y), that is,
d2X(0)(y) + W2 y)X(O)(y) = k 2X( 0 )(y)
dy2  c2
and
k42 - @(y)] Y(O)(Y)-ik Y = °,
-dY(f)(y) -d 2Z(0 )(y) w%2
dy dy 2  c2
The magnetic field follows from Eqs. (1) and (2):
c
H = i - X ) E.
co)
(3)
(4a)
(4b)
(5)
We notice that, because of the modulating function O(x), the
electric and magnetic fields are not exactly transverse in the
TE and TM solutions. Since the modulating function +(x)
is slowly varying, the longitudinal components of these fields
are small with respect to the transverse components.
The modulating function satisfies the following equa-
tion:
0030-3941/81/111377-04$00.50 © 1981 Optical Society of America
L. G. Ferreira and M. A. A. Pudensi
1378 J. Opt. Soc. Am./Vol. 71, No. 11/November 1981
and satisfies the following equation:
v(v -E) - vE - 2 EE = 0. (11)
EACTIVE CI / /1 x70
Fig. 1. Injection laser with slab waveguiding dielectric-constant
profile.
d244(X) co2 _
+ - AE(x)x) = (k2 -k2)4(X), (6)
dx 2  c2
where k is the wave number being looked for, k is the wave
number of the unperturbed solution, and AE(x) is an effective
dielectric constant of perturbation with the following defini-
tion: For the TE waves
In many previous works the effects of the divergence of E have
been neglected,6 but in what follows we make no such ap-
proximation.
The three components of E may be expanded in a complete
set of orthonormal functions (Hermite-Gaussian, for in-
stance):
X(x,y) = ajjh(x)Hj(y),
Y(x,y) = bjjhj WH (y)
Z(x, y) = cjjhj(x)Hj(y),
(12a)
(12b)
(12c)
AE(x) =
J X(0)(y)[e(x,y) - Z(y)JX(°)(y)dy
s- X y
and for the TM solutions
AE(X) =
S [e(x, y) - (y)] [Y(O)(y)+Y(O)(y) + Z(0)(y)+Z(0)(y)]dy
(O)+(O) dZ(0 )Gy) d
Y(°)(y)+Y(°)(y)dy- Y(-)(y)+ dZ) dy
X k Xdy
(8)
In Eq. (8), Y(O)(y)+ and Z(0)(y)+ are the solutions of Eqs. (4)
when we change h into -k.
The effective dielectric constants of perturbation given by
Eqs. (7) and (8) are the main results of our derivation. The
expression for Ae(x) for TE waves comes as no surprise and
leads trivially to the definition of the filling factor rTE that
is current in injection-laser literature.",2 If we use the di-
electric-constant profile of Streifer and Kapon3 in our Eqs.
(7) and (6), we arrive at their Eqs. (15) and (16). /In the case
of TM waves, Eq. (8) leads to the following filling factor:
rTM =
d3'd [Y(O)(Y)+Y()(y) + Z(0)(Y)+Z(0 )(y)]dy
f Y(°)(y)+Y( 0)(y)dy - 1 y(O)(y)+ dZ( )(dy
co k dy
(9)
which is different from the expression given by Hakki and
Paoli.5 In Fig. 2 we compare ITE and rTM for the dielec-
tric-constant profile sketched in Fig. 1. One notices that rTE
does not depend on the ratio (el - EO)/El and is always greater
than rTM-
2. MATHEMATICAL DERIVATION
Unperturbed Solution
The E field of the wave is written as
X(x,y) I
E = Y(x, y) X exp(iwt - ikz)
Z(xy)
where we sum in the repeated indices. We assume that the
expansions above converge rapidly if the functions hj (x) and
, (7) Hj(x) have conveniently chosen widths. For instance, in the
case of Hermite-Gaussian functions, we would choose the
width parameters according to the typical lengths of variation
of the fields in the x and y directions. Naturally, hj (x) will
have widths much larger than Hj(y). Then, by making scalar
products of Eq. (11) with the vectors
hi WXHI (y) O O
O , hi (x)HI (y) , O
O O hi (xi )HW
and by integrating in the xy plane, we obtain the following set
of equations:
1.00
0.90
0.80
C 0.70
0
° 0.60
L)
U 0.50
, 0.40
QL 0.30
0.20
0.10
0.00
0 0.8 1.6 2.4 3.2 4.0 4.8 5.6
p = d 0) f-T -,o
(10) Fig. 2. Filling factor r as function of frequency and thickness for
TE waves and TM waves and different values of the ratio Ae = (El -
co)/e1.
7
-d
d
L. G. Ferreira and M. A. A. Pudensi
L. G. Ferreira and M. A. A. Pudensi Vol. 71, No. 11/November 1981/J. Oct. Soc. Am. 1379
(p2 )Ij46j + h
2
1 Jb6ij - 2 1J -Pijpij
C2'
-PIJPij 45IJ(p 2)j + k
2b1 j81 - 2
kPjbij
kPebij
(I2
6 Ij(p 2 )uj + (P 2 )1J4 j8 - 2 EiijC2 te
where
Pij =-i hi x) d hj (x)dx, (14a)
dx
Pj= -if HI(y) -Hj(y)dy, (14b)
dy
EiIJj = rr hi (x)Hi(y)E(x, y)hj(x)Hj(y)dxdy, (15)
(P2 )14 = PIKPKJ, (16a)
K
(P2 ii = Z PikPkPi (16b)
k
In Eq. (13) we have a matrix multiplication followed by sums
in the repeated indices. Equation (13) can be solved to de-
termine k for any dielectric waveguide when w is given. In
our case, we begin the solution by considering a guide whose
dielectric constant ?(y) does not depend on x. For a wave with
no variation along x, we set
Pij = 0, (17a)
k k, (17b)
ailoEJ o ijbta, (17in
and from Eq. (13) we obtain
0 0
hPI~J
- aJ-2
C- _' E1J
2
(P2)I- 2 - Ej
cI
Perturbed TE Solution
We consider now the dielectric-constant perturbation that is
due to the difference
(i) E(X,y) - (y)
between the true dielectric constant E(x, y) and the approxi-
mative one e(y). Following this perturbation, there results
a difference
(ii) k2 - k 2
between the wave numbers, and the momentum matrix ele-
ments
(iii) Pij
cease to be null. Then we compare Eqs. (13) and (18) and use
the following formula of perturbation theory:
(1) Perturbations i, ii, and iii are taken to the second order
in the TE-TE matrix elements (upper-left-hand side).
(2) The perturbations are taken to the zeroth order in the
TM-TM matrix elements (lower-right-hand side).
(3) The perturbations are taken to the first order in the
TE-TM matrix elements (off-diagonal side).
X
We arrive at
(TE),jbij + (k 2 - A2)b1 4J51 - 2 (Eirjj-ZjJ6 -I c2
-PIJpii
k6,(JPij
= 0. (18)
-Pnmipg k61 4jpij
[TMlijbij
X/aij\
)(i
(21)
These equations have two solutions: The TE field corre-
sponds to the solution of the upper-left-hand submatrix, and
the TM field corresponds to the lower-right-hand submatrix.
We can verify readily that the upper-left-hand submatrix is
the matrix version of the differential equation [Eq. (3)] for
X(0 )(y) = a cj(W)Hj(y). (19)
J
Analogously, the lower-right-hand submatrix is the matrix
version of the coupled Eqs. (4), where
Y(O)(y) = E bj(0)Hj(y), (20a)
Z(o)(y) = Z cj(0 )Hj(y). (20b)
J
where (TE)1j and [TM]ij mean the submatrices in Eq. (18).
Then, by using the second and third rows of Eq. (21) to elim-
inate bgj and cyj, we obtaini(TE)1 jbi6 + (k 2 -2) 1,.ij--T (EiIrj - ZJ6ij)
- PikPkJ(-PIK Ji6IK)[TMIKr' (.{ )LJ ajj = 0. (22)
When taking the inverse of the matrix [TM], the reader must
not be misled by our compact notation: This matrix is not 2
X 2 but is much larger because of the indices K and L.
We look for a field whose x component is
X(x, y) = X 0)(y)Vf(x) = Eaj(O)Hj(y) F tjhj(x), (23)
J i
bjj
CYJ
x = 0, (13)
7
P2)1 J + 226aj -W2 ZIJ
C2
1380 J. Opt. Soc. Am./Vol. 71, No. 11/November 1981
namely, the unperturbed component X(°)(y) times a modu-
lating function +(x). Thus
When the formulas of perturbation theory are applied, Eq.
(13) becomes
-PIJpij
[TM]Iii
hbjjPij
ii +
jIJep2jij + (W -k2 )6ijij- 2 /fiiJ
I C2
[k - ipbj
(k - )Pjbij
_,,2 Af2 j
5lJ(p2 )ij-
ELIJ
ajj
I X bjj = 0.
J CYj
aji = aj(0 ) I, (24)
where the unperturbed coefficients aj(0) are normalized ac-
cording to
E aJl(0 )a( 0 ) = 1. (25)
Now, by inserting Eq. (24) into Eq. (22), multiplying by al(
0),
summing in I, and using the fact that
[TM]KLr [haL () = [- aJ(0)1 (26)
which follows from Eq. (18), we obtain
A ((k2 - fi2).ij - 2 fAc(x)hi(x)hj(x)dx
-J'hi(x) d - hj(x)dx]t; = 0, (27)
where Ae(x) has been defined by Eq. (7).
Equation (27) is equivalent to Eq. (6), where
(= E tjhj(x). (28)
Further, using Eqs. (26) and (21) we obtain
b jj = 0,
CjY = 1 ai(°)pjlt
(29a)
(29b)
from which follows the expression for the electric field given
by Eq. (1).
Perturbed TM Solution
We look for x -confined solutions of the form
Y(x,y) = i(x)Y( 0)(y) = E 1jbj(
0)hj(x)Hj(y), (30a)
Z(x,y) = (x)Z( 0)(y) = i Ejcj( 0)hj(x)Hj(y), (30b)
where the modulating function 4(x) has expansion coefficients
tj. Thus
bjj = (jbi(
0), (31a)
ciJ = tjcj(O). (31b)
The first line of this equation is used to solve for ajj:
a ji =-pjLl1(TE)JK'1[-PKLbL(
0 ) + kCKO0i, (33)
where we have used Eqs. (31). By inserting Eq. (33) into the
second and third lines of Eq. (32) and multiplying by the row
matrix
[bI(°)+cJ(°)+],
where b1 (°)+ and ci (0)+ are the solutions of the equations dual
to the lower-right-hand block of Eq. (18), namely,
(34)I [bi(0)'c/(M)] [TM]ij = O
we obtain a mathematical development similar to the one of
the perturbed TE solutions. The results are embodied by Eq.
(2), which gives the field, Eq. (6) for the modulating function,
and Eq. (8) for the effective dielectric constant of perturba-
tion.
3. SUMMARY
We have solved, in general terms, the problem of wave prop-
agation in a medium with poor confinement ability in one
transverse direction. The advantage of our solution is in its
generality. For TM modes we were able to correct the ex-
pression for the filling factor that is presented in the litera-
ture.
ACKNOWLEDGMENTS
This work is based on part of a doctoral thesis (MAAP). The
work was supported by Telebras and Fapesp, Brazil.
REFERENCES
1. T. L. Paoli,"Waveguiding in a stripe-geometry junction laser,"
IEEE J. Quantum Electron, QE-13, 662-668 (1977).
2. J. Buus, "Detailed field model for DH stripe lasers," Opt. Quantum
Electron. 10, 459-474 (1978).
3. W. Streifer and E. Kapon, "Application of the equivalent-index
method to DH diode lasers," Appl. Opt. 18, 3724-3725 (1979).
4. F. C. Prince, N. B. Patel, and D. J. Bull, "InP-In1_xGaxAsyPJ_,
embedded mesa stripe lasers," IEEE J. Quantum Electron. QE-16,
1034-1038 (1980).
5. B. W. Hakki and T. L. Paoli, "Gain spectra in GaAs double-het-
erostructure injection lasers," J. Appl. Phys. 46, 1299-1306
(1975).
6. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold,
New York, 1972).
(TE)Ijc3ij
-PIJpij
h63IJPij
(32)
L. G. Ferreira and M. A. A. Pudensi


WAVEGUIDING IN A DIELECTRIC MEDIUM VARYING SLOWLY IN ONE TRANSVERSE DIRECTION

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WAVEGUIDING IN A DIELECTRIC MEDIUM VARYING SLOWLY IN ONE TRANSVERSE DIRECTION

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