Page 224 - Introduction to Information Optics

4.2. All-Optical Switches 209

have a coherent interaction and periodically exchange power. Assume these
waveguides are single mode. The fields in the core of the waveguide can be
written as

J/(
£ fc(x, y, z) = B(z)e h(x, v)<r »% '
where f$ a and ^ are the propagation constants in the two waveguides. The
coupling origins from the polarization perturbation form the presence of the
evanescent field of the adjacent waveguide. For different propagation constants
in A and B, we have coupled mode equations for field amplitudes A and B [7]:

(4J2>
— = -JKA
az

where K is the coupling coefficient which depends on the refractive indices
«,, « 2, geometry of the waveguide, and separation s. With input to waveguide
A only; i.e., ,4(0) = 1 and B(0) = 0, the solution to Eq. (4.12) is
/( io
A(z) = [cos(0z) — y'A/(20) sin(0z)]e~ ' ~~ A)z
A}z
i(li
5{z) = [—j(K/g) sin(gz)]e~ "~ .
2
2
2
Here # = K + A and the phase mismatch A = (/i a — f} b)/2.
In terms of normalized power P a and P h in the two waveguides,
2
2
P h = (K /0 ) sin V)

4.2.3.1. Self-switching
Self-switching occurs when the waveguides are made of nonlinear materials
with refractive index n = n 0 + n 2I. The coupled equations, however, become
more complicated, and are given as

dA n .
—r- — —JKD — ip aA + y\A\ A
az
(4.15)
2
_ = -j KA - ifi hB + y\B\ B.

Here y is the coefficient, depending on the nonlinearity of the material and the