A pure phase specimen has amplitude transmittance with

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 pure phase specimen has amplitude transmittance

t(x) = e^{i\phi(x)}

with

|\phi(x)| \ll 1

. It is illuminated coherently by a structured pattern

U_{\mathrm{ill}}(x) = 1 + \cos(2\pi\nu_i x)

of the kind used in structured illumination, with

\nu_i

chosen at

0.8

times the incoherent cutoff of the objective. The transmitted field is

U(x) = U_{\mathrm{ill}}(x)\,t(x)

, and this field passes, unattenuated at every spatial frequency present, through an aberration-free lens acting as a Fourier transformer (with the usual transfer function of an imaging system) onto a camera that records

I(x) = |U(x)|^2

. ((a)) Expand

U(x)

to first order in

\phi(x)

and show that the terms proportional to

\phi(x)

I(x)

vanish, for any real illumination profile including this structured one. ((b)) For a phase amplitude

\phi_0 = 0.06

rad, keep the next term,

e^{i\phi}\approx1+i\phi-\phi^2/2

, and find the phase-dependent irradiance contrast to second order. Compare it with the value for uniform illumination at the same

\phi_0

. ((c)) In one sentence, explain why the resolution gain of structured illumination does not rescue a pure phase object.

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