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ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2021/section-b/b5/b/solution

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Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section B / B5 / b / Solution by

Codex  0 2026-10-18

The given spatial and carrier factors combine into retarded copies of the same random field:

E^{'} (r, t) = \frac{f ( t - t _{1} ) + f ( t - t _{2} )}{2} z, t_{j} = \frac{k _{j} \cdot r}{ω} .

On the

y

axis,

t_{1} = k y / (2 ω)

and

t_{2} = - k y / (2 ω)

. Expand the squared electric-field wave amplitude. Each self term contributes

I_{0} /2

, and the two cross terms together contribute the real part of the full-field correlation. Stationarity and

Γ (- τ) = Γ (τ)^{*}

give

I (y) = C ⟨ ∣ E^{'} ∣^{2} ⟩ = I_{0} + Re Γ (τ (y)), τ (y) = \frac{2 k y}{ω} .

k = ω / c

and

τ (y) = 2 y / c

. Reversing the delay convention replaces

τ

by

- τ

without changing this real-part result. This

I

is the electric-field light intensity convention of the question. For two nonparallel waves it should not automatically be identified with the magnitude of the net Poynting vector; part (d) gives that geometric conversion explicitly.

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