A sinusoidal amplitude grating

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

What an Instrument Can See, Ch The lens as a Fourier transformer

Problem

t(x) = \tfrac12[1 + m\cos(2\pi\nu_0 x)]

, with

m = 0.80

and

\nu_0 = 50

lines per millimetre, is the object of a 4f system built from two lenses of focal length

f = 200\ \mathrm{mm}

, illuminated at

\lambda = 633\ \mathrm{nm}

. ((a)) Find the distance from the axis, in the shared Fourier plane, of the

-\nu_0

order that a small opaque strip must cover if it is to block that order alone, leaving the zero order and the

+\nu_0

order untouched. ((b)) Write the field just after this mask, and show that the image-plane irradiance it produces is

I(x) = \left(\tfrac14 + \tfrac{m^2}{16}\right) + \tfrac{m}{4}\cos(2\pi\nu_0 x)

. ((c)) Compute the fringe contrast

(I_{\max}-I_{\min})/(I_{\max}+I_{\min})

of this masked image and compare it with the contrast of the unfiltered image,

I_0(x) \propto t(x)^2

. State whether blocking one sideband raised or lowered the contrast, and why that is not simply "half the light removed".