Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

−Y

Y

y

k x

k′

0

θ

Figure 13.2 Diffraction grating of width 2Ywith light of wavelength 2π/k

being diffracted through an angleθ.

The factor exp[ik′·(r 0 −yj)] represents the phase change undergone by the light

in travelling from the pointyjon the screen to the pointr 0 , and the denominator

represents the reduction in amplitude with distance. (Recall that the system is

infinite in thez-direction and so the ‘spreading’ is effectively in two dimensions

only.)

If the medium is the same on both sides of the screen thenk′=kcosθi+ksinθj,

and ifr 0 Ythen expression (13.8) can be approximated by

A(r 0 )=

exp(ik′·r 0 )

r 0

∫∞

−∞

f(y) exp(−ikysinθ)dy. (13.9)

We have used thatf(y)=0for|y|>Yto extend the integral to infinite limits.

The intensity in the directionθis then given by

I(θ)=|A|^2 =

2 π

r 02

| ̃f(q)|^2 , (13.10)

whereq=ksinθ.

EvaluateI(θ)for an aperture consisting of two long slits each of width 2 bwhose centres

are separated by a distance 2 a,a>b; the slits are illuminated by light of wavelengthλ.

The aperture function is plotted in figure 13.3. We first need to find ̃f(q):

̃f(q)=√^1

2 π

∫−a+b

−a−b

e−iqxdx+

1

√

2 π

∫a+b

a−b

e−iqxdx

=

1

√

2 π

[

−

e−iqx

iq

]−a+b

−a−b

+

1

√

2 π

[

−

e−iqx

iq

]a+b

a−b

=

− 1

iq

√

2 π

[

e−iq(−a+b)−e−iq(−a−b)+e−iq(a+b)−e−iq(a−b)

]

.