Assignment 3, solutions:

Problem 3:

For m pairs (n₁, n₂) of quarter-wave layers on a substrate with index n we have for a TE wave at the input plane

Z(z_i) = [(m₀/e₀)^1/2](n cosq)^-1[(n₁ cosq₁)/(n₂ cosq₂)]^2m.

Z is continuous at the air-dielectric interface.

The reflection coefficient is therefore given by

G = (Z(z_i) - Z₀^air)/(Z(z_i) + Z₀^air).

For normal incidence  Z₀^air = [(m₀/e₀)^1/2] and Z(z_i) = [(m₀/e₀)^1/2]n^-1[n₁/n₂]^2m.

Therefore we have 

G = ([n₁/n₂]^2m - n)/([n₁/n₂]^2m + n),

or

(n₁/n₂)^m = [n(1 + G)/(1 - G)]^1/2,

m ln(n₁/n₂) = ln([n(1 + G)/(1 - G)]^1/.).

With n₁ = 2,5, n₂ = 1.5, n = 1.5, and  G = 0.995^1/2 this yield m = 6.94.

The minimum number of layers therefore is 7.