Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.4 EXERCISES

Determine the convolution offwith itself and, without further integration,

deduce its transform. Deduce that

∫∞

−∞

sin^2 ω

ω^2

dω=π,

∫∞

−∞

sin^4 ω

ω^4

dω=

2 π

3

.

13.8 Calculate the Fraunhofer spectrum produced by a diffraction grating, uniformly
illuminated by light of wavelength 2π/k, as follows. Consider a grating with 4N
equal strips each of widthaand alternately opaque and transparent. The aperture
function is then

f(y)=

{

A for (2n+1)a≤y≤(2n+2)a, −N≤n<N,

0 otherwise.

(a) Show, for diffraction at angleθto the normal to the grating, that the required

Fourier transform can be written

̃f(q)=(2π)−^1 /^2

∑N−^1

r=−N

exp(− 2 iarq)

∫ 2 a

a

Aexp(−iqu)du,

whereq=ksinθ.

(b) Evaluate the integral and sum to show that

̃f(q)=(2π)−^1 /^2 exp(−iqa/2)Asin(2qaN)

qcos(qa/2)

,

and hence that the intensity distributionI(θ) in the spectrum is proportional

to

sin^2 (2qaN)

q^2 cos^2 (qa/2)

.

(c) For large values ofN, the numerator in the above expression has very closely

spaced maxima and minima as a function ofθand effectively takes its mean

value, 1/2, giving a low-intensity background. Much more significant peaks

inI(θ) occur whenθ= 0 or the cosine term in the denominator vanishes.

Show that the corresponding values of| ̃f(q)|are

2 aNA

(2π)^1 /^2

and

4 aNA

(2π)^1 /^2 (2m+1)π

, withmintegral.

Note that the constructive interference makes the maxima inI(θ)∝N^2 , not

N. Of course, observable maxima only occur for 0≤θ≤π/2.

13.9 By finding the complex Fourierseriesfor its LHS show that either side of the
equation
∑∞

n=−∞

δ(t+nT)=

1

T

∑∞

n=−∞

e−^2 πnit/T

can represent a periodic train of impulses. By expressing the functionf(t+nX),

in whichXis a constant, in terms of the Fouriertransform ̃f(ω)off(t), show

that

∑∞

n=−∞

f(t+nX)=

√

2 π

X

∑∞

n=−∞

̃f

(

2 nπ

X

)

e^2 πnit/X.

This result is known as thePoisson summation formula.