Derive an expression for the ratio of the proportionality

Question

Field Distribution

The transverse distribution u_m(y) of the electric-field complex amplitude of a TE mode in a slab waveguide is given by:

a slab waveguide is given by:

            cos(2Π(sinθ_m/λ)y) → m = 0, 2, 4, 6 .....

um(y) ∝

              sin(2Π(sinθ_m/λ)y) → m = 1, 3, 5 .....

and

             exp(-γ_my) → y > d/2.

um(y) ∝

             exp(-γ_my) → y < -d/2.

Derive an expression for the ratio of the proportionality constants. Plot the distribution of the m = 0 TE mode for a slab waveguide with parameters n₁ = 1.48, n₂ = 1.46, d = 0.5 pm, and λ₀ = 0.85 μm, and determine its confinement factor (percentage of power in the slab).