Page 333 - Introduction to Information Optics

3 1 8 6. Interconnection with Optics

The boundary condition in Eq. (6.9) can be written as

D'(h) + (U h - P)D(h) = Q(h)S
(6.19)
D'(0) +(f/ 0 -P)D(0) -0.

Using Eqs. (6.17) and (6.19) we can find constant matrices A, B:

A = (W l + U h - P)~\W 2 + U h - P)B
B=[_(W 2 + U h-P)Qxp(W 2h)
w
U
l
- ( i + H - P}Z*P(WMW, + u - pr'(W + u - P)T Q s
h
2
h
h
(6.20)
The above equation gives an exact analytical solution in the case of a tilted
grating. In the case of a rectangular grating, W l and W 2 matrices are

To calculate exponential function in Eq. (6.20) we can factorize W v and W 2
matrices:

where O m is a diagonal matrix the elements of which are the eigen-values of
W m, and Z m is an orthogonal matrix whose columns are eigen-vectors of W m.
Using exponential function series and matrix factorization we can write:

00 k 1 k k
(W h\
°° (7~ W7 } h
n
vv
— \ m > V^ V-^m vv m^m) n _ y-
\ — — Z< m
Jc!
1
Thus, the exponential function in Eq. (6.20) is: exp(W mh) = Z m O mZ OT,
where O m is a diagonal matrix the elements of which are [Q] n = exp(Q, nh).
Coefficients R n and T n in Eqs. (6.3), (6.4) can be found from condition of E,
being continued at y — h and y = 0. Energy in the nth diifracted, reflected, and
transmitted orders can be found

E R = R*R ncos<x nl
E T= T*T ncosx n2
where * means complex conjugate.