Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

is a wavefunction and the distribution of the wave intensity in time is given by

|f|^2 (also a Gaussian). Similarly, the intensity distribution in frequency is given

by| ̃f|^2. These two distributions have respective root mean square deviations of

τ/

√

2and1/(

√

2 τ), giving, after incorporation of the above relations,

∆E∆t=
/2and∆p∆x=
/ 2.

The factors of 1/2 that appear are specific to the Gaussian form, but any

distributionf(t) produces for the product ∆E∆ta quantityλ
in whichλis

strictly positive (in fact, the Gaussian value of 1/2 is the minimum possible).

13.1.2 Fraunhofer diffraction

We take our final example of the Fourier transform from the field of optics. The

pattern of transmitted light produced by a partially opaque (or phase-changing)

object upon which a coherent beam of radiation falls is called adiffraction pattern

and, in particular, when the cross-section of the object is small compared with

the distance at which the light is observed the pattern is known as aFraunhofer

diffraction pattern.

We will consider only the case in which the light is monochromatic with

wavelengthλ. The direction of the incident beam of light can then be described

by thewave vectork; the magnitude of this vector is given by thewave number

k=2π/λof the light. The essential quantity in a Fraunhofer diffraction pattern

is the dependence of the observed amplitude (and hence intensity) on the angleθ

between the viewing directionk′and the directionkof the incident beam. This

is entirely determined by the spatial distribution of the amplitude and phase of

the light at the object, the transmitted intensity in a particular directionk′being

determined by the corresponding Fourier component of this spatial distribution.

As an example, we take as an object a simple two-dimensional screen of width

2 Yon which light of wave numberkis incident normally; see figure 13.2. We

suppose that at the position (0,y) the amplitude of the transmitted light isf(y)

per unit length in they-direction (f(y) may be complex). The functionf(y)is

called anaperture function. Both the screen and beam are assumed infinite in the

z-direction.

Denoting the unit vectors in thex-andy- directions byiandjrespectively,

the total light amplitude at a positionr 0 =x 0 i+y 0 j, withx 0 >0, will be the

superposition of all the (Huyghens’) wavelets originating from the various parts

of the screen. For larger 0 (=|r 0 |), these can be treated as plane waves to give§

A(r 0 )=

∫Y

−Y

f(y) exp[ik′·(r 0 −yj)]

|r 0 −yj|

dy. (13.8)

§This is the approach first used by Fresnel. For simplicity we have omitted from the integral a

multiplicative inclination factor that depends on angleθand decreases asθincreases.