A transparency with transmittance t(x) = 12 [1 + 0.4 (2 90x) + 0.2 (2 220x) ] ( in…

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

A transparency with transmittance

t(x) = \tfrac12\left[1 + 0.4\cos(2\pi\cdot 90x) + 0.2\cos(2\pi\cdot 220x)\right]

(

x

in millimetres, the two spatial frequencies in cycles per millimetre) sits against the front focal plane of a lens of focal length

f = 250\ \mathrm{mm}

, illuminated at

\lambda = 532\ \mathrm{nm}

. ((a)) Find the positions

x_1

and

x_2

of the

+1

orders belonging to the

90\ \mathrm{mm^{-1}}

and

220\ \mathrm{mm^{-1}}

components in the back focal plane. ((b)) Find the ratio of each of these two orders' amplitude to that of the zeroth order.