Page 330 - Introduction to Information Optics

6.3. Thin-Film Waveguide Couplers 3 I ."*

Below the groove region, in the area y > 0, the electric field can be written
for each corresponding m as:

E m 3 (y} — T m e x p( — ft m 2 y) (6.4)

where

a __ /prr ~
Pm2 ~~ V K 2 ~ ^1)12
2
Inside groove region 0 < y < /i, k (x, y) is periodic in the x direction and can
be represented by a Fourier series:

2
k ( x, y) = V C m(y) exp(27i/m.\7'A). (6.5)

Upon substituting Eqs. (6.2) and (6.5) into Eq. (6.1) and collecting all terms
with the same x-dependence, we can obtain an equation in the region
0 < v < h:

2
d E (v)
1
—f^T ~ <£E m2(y) + £ C m- q(y)E q2(y) = 0. (6.6)
°y q
Equation (6.6) can be written in a matrix form of

E' 2 = V(y)E 2, (6.7)

where E 2 is a column whose elements are E m2 and V(y) is a known square
matrix whose elements are defined by:

. , mentioned in manuscript
'
Art in box = - - (6 8)

The solution to Eq. (6.7) is subject to the boundary condition of E z and
dE z/dy being continuous at y = h and y = 0. Taking into account the bound-
ary conditions we get:

E' m(Q) + i0 m2JE m(0) - 0

E' m(h) - ifi mlE m(h)