Design of wide-core planar waveguides by an inverse scattering method. S. Lakshmanasamy*. Department of Electrical Engineering, University of Rhode Island, ...
April 15,1989 / Vol. 14, No. 8 / OPTICS LETTERS
411
Design of wide-core planar waveguides by an inverse scattering
method S. Lakshmanasamy* Department of Electrical Engineering, University of Rhode Island, Kingston,Rhode Island 02881
A. K. Jordan Space Science Division, Naval Research Laboratory,Washington, D.C. 20375 Received August 15, 1988; accepted January 27, 1989 An inverse scattering method is used to design single-mode planar optical waveguides.
Waveguides with wider
cores compared with those designed by direct scattering methods are obtained, and a numerical example is given.
Integrated-optics technology involves the synthesis of refractive-index profiles of planar thin-film waveguides. Especially important are single-mode waveguiding structures, which have applications in passive optical components and laser design. The design of planar optical waveguides has been studied by many authors and is usually based on the direct scattering analysis of the wave equation. In this Letter we present a method for synthesizing the refractive-index profile for a wide-core single-mode planar optical waveguide using inverse scattering theory. Inverse scattering theory is used to design an optical waveguide from its propagation characteristics, which have a specified functional form to fulfill system requirements. In the present case we choose a reflection coefficient that represents a single guided mode and a continuum of radiation modes in the waveguide. The propagation of light in an inhomogeneous planar optical waveguide for the case of TE modes is governed by the scalar wave equation" 2 d2 Egx, ko 0) -2
2 2( )+ [ko n2 ) -
2]Ey(
-
ko) xl]EG,
=
0,
width of the core region (Fig. 1). The normalized frequency V is defined as V2 = k02 L2 (n
2
(3)
-n22),
and the refractive-index profile n(x) is assumed to have the form n2(x)
=
{n 2[1
-
O< x < 1
g(X)]
x < 0, x > 1
(4)
where g(x) is a real piecewise continuous function be-
longing to L1(-o, A) such that max[g(x)] < 1, n1 is the maximum refractive index of the core, and n2 is the CLADDING
(1)
whereE5(x, ko)is the amplitude of the plane-polarized time-harmonic electromagnetic wave propagating in the z direction with a free-space wave number ko and a propagation constant j3, and n(x) is the refractive index whose variation is assumed to be a function of the x coordinate only. The fundamental mode in a planar optical waveguide is a TE mode, and, since our goal is
to design a single-mode waveguide, this discussion is restricted to the TE mode case. It is convenient to normalize the above equation by putting
x = x/L
t(k) exp (ikx)
r(k)
and E-(x, ,P~xko =.yE
ki )
I
x
(2)
where Eyo is maximum field amplitude and L is the 0146-9592/89/080411-03$2.00/0
0
1
Fig. 1. Physical model of an inhomogeneous planar optical waveguide for the inverse scattering application. © 1989 Optical Society of America
OPTICS LETTERS / Vol. 14, No. 8 / April 15,1989
412
refractive index of the cladding. When we make use of Eqs. (2)-(4), Eq. (1) can be rewritten as d2 f (x, k) + [k2
q(x)] t'(x, k) = 0,
-
(5)
function for the reflection coefficient: One pole on the positive imaginary axis of the k plane, h, = i, representing the single propagating mode, and two symmetric poles in the lower half of the k plane, k2 = 1/2(V - i) and k 3 = -1/2(V/3+ i), representing the radiating modes that occur in practical waveguides. The reflection coefficient for this case can be written
(6)
as
where k2 = L 2 (_-f 2 + k0 2n2 2 ) and
r(k) =q(x) =
(n1 2
n -
n22)
g(x) - V2 .
(7)
Equation (5) is a Schr6dinger-type equation with the spectral variable k and a potential function in one dimension, q(x). Since n(x) -n 2 as x - 1 and x • 0, q(x) - 0 as x - 1 and x < 0, the scattering parameters can be represented by meromorphic functions of the wave number 3 k. The model structure of the optical
which satisfies the normalization condition r(O) = -1 and the energy-conservation condition Ir(k)12< 1 for all real k; this reflection coefficient has been considered previously.4 The reflected transient function R(x + t) is R(x + t) =-l
waveguide can now be described in terms of the pole-
zero configurations of the reflection coefficient r(k). By considering the spectral properties of the differential Eq. (5), Gel'fand, Levitan, and Marchenko (GLM) have shown that the profile function q(x) is determined by the relation (8)
d K(x, x) = - q(x), 2
dx
which is a condition on the solution K(x, t) of the integral equation,3
(12)
k3 +i
+6iW exp[-
exp-(
-
(x + t) (1 +i)]
2 )(11- i3) + 3 exp(x+t). (13)
To solve Eq. (9) a differential
operator can be con-
structed such that the integral equation is converted into a differential equation for K(x, t) with its boundary conditions, which can easily be solved.4 If the operator p = d/dt is defined, an operator function f(p) can be obtained that has the property f(p)R(x + t) = 0. The appropriate operator f(p) for the case considered here is
R(x + t) + K(x, t) +J
K(x, S)R(Q+ t)dt = 0,
f(p) = (p + ikl)(p + ik2)(p + ik3) = p 3
t
