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:

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

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

and

u_m(Y) ∝ (exp (-y_my) → y > d /2.

             exp (y_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).