ϕ = X (x) sin (nπ y / h_{0})

gives

X^{''} + k_{n}^{2} X = 0, k_{n}^{2} = k_{0}^{2} - \frac{n ^{2} π ^{2}}{h _{0}^{2}} .

Hence

X = A_{n} e^{i k_{n} x} + B_{n} e^{- i k_{n} x}

, with imaginary

k_{n}

representing an evanescent mode.

For

X = ϵ x

and varying width,

k_{n} (X)^{2} = k_{0}^{2} - \frac{n ^{2} π ^{2}}{h ( X ) ^{2}}, Θ_{n} = \frac{1}{ϵ} \int_{0}^{X} k_{n} (ξ) d ξ .

Projection of the next-order equation onto

sin (nπ y / h)

, whose squared norm is proportional to

h

, gives

2 k_{n} A_{n}^{'} + (k_{n}^{'} + k_{n} \frac{h ^{'}}{h}) A_{n} = 0, 2 k_{n} B_{n}^{'} + (k_{n}^{'} + k_{n} \frac{h ^{'}}{h}) B_{n} = 0.

A_{n} (X) = A_{n} (0) [\frac{k _{n} ( 0 ) h ( 0 )}{k _{n} ( X ) h ( X )}]^{1/2}, B_{n} (X) = B_{n} (0) [\frac{k _{n} ( 0 ) h ( 0 )}{k _{n} ( X ) h ( X )}]^{1/2} .

This applies for

n \geq 1

k_{n} = 0

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