Problem 1:

In a wave guide switch, the propagation constants b₁ and b₂ are changed by an applied voltage (linear electro-optic effect).  In a coupler in which kd = p/2, full transfer occurs from guide 1 to guide 2.  What is the value of  |b₁ - b₂| so that no transfer occurs?

Problem 2:

Assume that the equations of mode coupling between two identical optical wave guides are 

dx₁/dz = ib₀x₁ - ikx₂   and   dx₂/dz = ib₀x₂ - ikx₁,

where k is real and positive.

(a)  Find the propagation constants of the coupled system and the two modes corresponding to these propagation constants.

(b)  Assume the two wave guides extend from z = 0 to z = d.  Perfect reflectors are placed at z = 0 and z = d.  The reflected waves satisfy the same equations,

dy₁/dz = ib₀y₁ - iky₂   and   dy₂/dz = ib₀y₂ - iky₁,

with y₁ = -x₁,  y₂ = - x₂ at z = 0 and z = d.

If b = w/v, what are the resonance frequencies of the structure?