Page 332 - Introduction to Information Optics

6.3. Thin-Film Waveguide Couplers 51 /

Matrix V(y\ in the case of a tilted grating, can be written as follows:

V(y) =Q(y)V RQ*(y\ (6.12)

where Q is diagonal matrix the elements of which are [<2L = exp(yjw), n =
1,2,,.,, and the sign * means complex conjugation. Now, Eq. (6.7) can be
rewritten as

E"(y) = Q(y)V RQ(y)E(y). (6.13)

We can introduce a new matrix variable:

D(y) = Q(y)E(y). (6.14)

Derivatives of the Q matrix can be found:

2
Q'(y) = PQ(y), Q"(y) = P Q(y), (6.15)

where P is a constant diagonal matrix,

[/*]» = niy.

Upon substituting Eqs. (6.13), (6.14) into (6.12) we have an equation

2
/)" + 2PD' + (P - V R)D = 0. (6.16)

This equation is a second-order differential equation with constant coeffi-
cients, and its general solution is

D = QXp(W ty)A + e\p(W 2y)B, (6.17)

where exp is the exponential function from matrix (computation of this
function will be discussed below), A, B are constant matrices that are deter-
mined from boundary condition. In Eq. (6.17) matrix multiplication is non-
commutative; therefore, order of its fractions is important W t, W 2 matrices are
roots of a second-order matrix equation:

2 2
W + 2PW + (P - V R) = 0 (6.18)