1 Waves 10

Lecture 10Lecture 10

Wave propagation.Wave propagation.

Aims:Aims:

Fraunhofer diffraction(waves in the “far field”).

Young’s double slits Three slits N slits and diffraction gratings A single broad slit General formula - Fourier transform.

2 Waves 10

Fraunhofer diffractionFraunhofer diffraction

Diffraction.Diffraction. Propagation of partly obstructed waves.

Apertures, obstructions etc...

Diffraction régimes.Diffraction régimes.

In the immediate vicinity of the obstruction: Large angles and no approximations

Full solution required. Intermediate distances (near field)

Small angles, spherical waves, Fresnel diffraction.

Large distances (far field) Small angles, and plane waves,

Fraunhofer diffraction. (More formal definitions will come in the Optics

course)

3 Waves 10

Young’s slitsYoung’s slits

Fraunhofer conditionsFraunhofer conditions For us this means an incident plane wave and

observation at infinity.

Two narrow apertures (2 point sources)Two narrow apertures (2 point sources) Each slit is a source of secondary wavelets Full derivation (not in handout) is….

Applying “cos rule” to top triangle gives

tkriAtkriAp 21 expexp

sin2

sin2

1

sin1

sin2

22

2/1

1

222

1

dR

Rd

R

Rd

Rr

dR

dRr

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2-slit diffraction2-slit diffraction

Similarly for bottom ray

Resultant is a superposition of 2 wavelets

The term expi(kR-t)will occur in allexpressions. We ignoreit - only relative phasesare important.

Where s = sin.

sin2

sin2

12d

RR

dRr

sin2

sin2)(

)sin2

()sin2

(

kdi

kditkRi

tkd

kRitkd

kRip

eeAe

AeAe

)2/cos(2)2/sincos(2

2/sin2/sin

kdsAkdA

AeeA ikdikdp

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cos-squared fringescos-squared fringes

We observe intensity

cos-squared fringes.

Spacing inversely proportional to separation of the slits.

Amplitude-phase diagrams.Amplitude-phase diagrams.

2/cos22kdsI p

Spacing of maximaSpacing of maxima

Slit 1Slit 1

Slit 2Slit 2

ResultantResultant

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Three slitsThree slits

Three slits, spacing Three slits, spacing dd..

Primary maxima separated by /d, as before. One secondary maximum.

2

0

cos21

cos21

kdsI

kdsAeeeA ikdsiikdsp

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N slits and diffraction gratingsN slits and diffraction gratings

NN slits, each separated by slits, each separated by dd..

A geometric progression, which sums to

Intensity in primary maxima N2

In the limit as N goes to infinity, primary maxima become -functions. A diffraction grating.

kdsNikdsiikdsip eeeeA )1(20 ...........

)2/sin(

)2/sin(

)1/()1(

2/

2/

2/2/2/

2/2/2/

kdse

NkdseA

eee

eeeA

eeA

ikds

iNkds

ikdsikdsikds

iNkdsiNkdsiNkds

ikdsiNkdsp

)2/(sin/)2/(sin 22 kdsNkdsI

Spacing, as beforeSpacing, as before

N-2, secondary maximaN-2, secondary maxima

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Single broad slitSingle broad slit

Slit of width Slit of width tt. . Incident plane wave.. Summation of discrete sources

becomes an integral.

/t/t

2/

2/

t

t

ikysp dyeA

dSe ikyp

sin

)2/sinc()2/sin(

2

)(2/2/

ktsAtktsksA

eeiksA iktsikts

p

9 Waves 10

Generalisation to any apertureGeneralisation to any aperture

Aperture functionAperture function The amplitude distribution across an aperture

can take any form a(y). This is the aperture function.

The Fraunhofer diffraction pattern is

putting ks=K gives a Fourier integral

The Fraunhofer diffraction pattern is the Fourier Transform of the aperture function.

Diffraction from complicated apertures can often be simplified using the convolution theorem.

Example: 2-slits of finite widthConvolution of 2-functions with asingle broad slit.

FT(f*g) FT(f).FT(g)Cos fringesCos fringes

sinc functionsinc function

dyeya iKyp )(

dyeya ikysp )(