A pure phase specimen

PHYSICS · Subjective · 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, weakly perturbing an on-axis coherent field so that

U_O(x) = A\big(1+i\phi(x)\big)

with

|\phi(x)|\ll1

and mean irradiance

I_O = A^2

, is recorded holographically against an off-axis plane-wave reference

U_R(x) = B\,e^{i2\pi\nu_R x}

of irradiance

I_R = B^2

, derived from the same laser so that the two beams are mutually coherent. The hologram records

I(x) = |U_R(x) + U_O(x)|^2

. (a) Show that structured illumination alone, using a real fringe pattern in place of the reference beam and without recording an off-axis interference term, would still give no first-order contrast for this object, as in the previous problem. (b) Expand

I(x)

to first order in

\phi(x)

and identify the term that carries phase information at first order, explaining why the reference beam breaks the quadrature symmetry that hides

\phi

in ordinary imaging. (c) Given

I_R = 4

(arbitrary irradiance units) and

I_O = 1

, with

\phi(x)

locally of amplitude

\phi_0 = 0.05

rad, compute the fractional modulation of the interference fringe carrier due to

\phi_0

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