For a basic three-layer planar waveguide such as that shown in Fig. 3.1, show that the phase constant h in Region 2, and the extinction coefficients q and p in Region 1 and 3, respectively, are given...

For a basic three-layer planar waveguide such as that shown in Fig. 3.1, show

that the phase constant h in Region 2, and the extinction coefficients q and p

in Region 1 and 3, respectively, are given by

q = (β2 − n2

1 k2)

1/2

p = (β2 − n2

3 k2)

1/2

h = (n2

2 k2 − β2)

1/2,

where β is the propagation constant in the z direction and k ≡ ω/c.