A thin pure-phase object again gives a field to second order

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Fourier optics and image formation · Fourier optics and image formation

What an Instrument Can See, Ch Seeing what the transfer function forbids: phase contrast and super-resolution

Problem

A thin pure-phase object again gives a field 1+iϕ(x)−ϕ(x)2/21 + i\phi(x) - \phi(x)^2/2 to second order, with 11 the unscattered part and iϕ(x)−ϕ(x)2/2i\phi(x) - \phi(x)^2/2 the scattered part. Now defocus the imaging system slightly. As an idealised model of what defocus does, take it to add a single uniform relative phase ψ\psi between the scattered field and the unscattered field, leaving both amplitudes unchanged (a simplification of the true frequency-dependent defocus phase, adequate for this estimate). ((a)) Show that the image irradiance is I(x)=1−2ϕ(x)sin⁡ψ+ϕ(x)2(1−cos⁡ψ),I(x) = 1 - 2\phi(x)\sin\psi + \phi(x)^2(1-\cos\psi), and identify the value(s) of ψ\psi for which the term linear in ϕ\phi vanishes. ((b)) Take ϕ=0.080 rad\phi = 0.080\ \mathrm{rad} and a defocus phase ψ=π/6 rad\psi = \pi/6\ \mathrm{rad} (30∘30^\circ). Compute the irradiance contrast I−1I - 1 at the phase step.

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