Chapter 12 defined the Fresnel number for an aperture of half-width observed a…

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

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

Problem

Chapter 12 defined the Fresnel number

N_F = a^2/(\lambda z)

for an aperture of half-width

a

observed a distance

z

away, as the single number that decides whether the Fraunhofer (far-field) transform can be trusted at that distance. This chapter's transfer function

G(\nu_x,\nu_y) = \exp[ikz\sqrt{1-\lambda^2\nu^2}]

is exact at every distance, and reduces to the Fresnel form where

\lambda\nu\ll1

and the dropped quartic phase

\pi z\lambda^3\nu^4/4

stays well below a radian. A grating of half-width

a = 5.0\ \mathrm{mm}

and pitch

\Lambda = 20\ \mu\mathrm{m}

is lit at

\lambda = 633\ \mathrm{nm}

and observed at

z = 30\ \mathrm{mm}

. ((a)) Compute the Fresnel number

N_F

for this aperture and distance, and state whether the far-field Fraunhofer transform is a trustworthy description of the pattern at this

z

. ((b)) Compute the grating's spatial frequency

\nu_0 = 1/\Lambda

and the cutoff

1/\lambda

, confirm that

\lambda\nu_0 \ll 1

, and evaluate the dropped quartic phase

\pi z\lambda^3\nu_0^4/4

at this

z

, to decide whether the small-angle step that turns this chapter's exact

G

into Chapter 12's Fresnel transfer function is justified for this grating. ((c)) In one or two sentences, say why

N_F

and the condition for the Fresnel transfer function answer two different questions, even though both involve

\lambda

and

z

Related theory

Interference and diffraction

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