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ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2017/2/b/solution

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Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 2 / b / Solution by

Codex  0 2026-10-25

Apply the first lens, propagate, and apply the second lens. Omitting only common prefactors and uniform phases,

E_{out} (x) \propto \int d x^{'} exp [- \frac{ik x ^{2}}{2 f} + \frac{ik ( x - x ^{'} ) ^{2}}{2 L} - \frac{ik x ^{'2}}{2 f}] E_{in} (x^{'}) .

The coefficients of both

x^{2}

and

x^{'2}

vanish if and only if

f = L

. The remaining cross term is

- ik x x^{'} / L

, so

f = L, E_{out} (x) \propto \int d x^{'} e^{- i (k x / L) x^{'}} E_{in} (x^{'}) = E_{in} (\frac{k x}{L}) .

The Fourier convention here is

E (q) = \int e^{- i q x} E (x) d x

. Finite clear apertures would truncate the field and limit the spatial-frequency range; the ideal relation assumes the paraxial field fits within both lenses.

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