Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 1 / ii

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- Solution ii

Solution

Let

M (x; z_{1}, z_{2}) = ⟨ E (x, z_{1}) E (x, z_{2})⟩ .

Applying the parabolic wave equation to each factor gives

2 ik M_{x} + (\partial_{z_{1}}^{2} + \partial_{z_{2}}^{2}) M = 0.

If

C (s) = ⟨ w (z) w (z + s)⟩

, Gaussian averaging at the screen gives

M (0; z_{1}, z_{2}) = exp {- k^{2} ξ^{2} [σ^{2} + C (z_{1} - z_{2})]} .

Stationarity makes this a function only of

s = z_{1} - z_{2}

. Since

\partial_{z_{1}}^{2} + \partial_{z_{2}}^{2} = 2 \partial_{s}^{2}

on such functions,

M_{x} = \frac{i}{k} M_{ss} .

M_{0} (q)

for the Fourier transform of the screen value, the solution at arbitrary range is

M (x; s) = \frac{1}{2 π} \int_{R} M_{0} (q) e^{i q s - i q^{2} x / k} d q .

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