Lecture 10
 Aims:
Wave propagation.
 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.
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Fraunhofer diffraction
 Diffraction.
 Propagation of partly obstructed waves.
Apertures, obstructions etc...
 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)
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Young’s slits
 Fraunhofer conditions
 For us this means an incident plane wave and
observation at infinity.
 Two narrow apertures (2 point sources)
 Each slit is a source of secondary wavelets
 Full derivation (not in handout) is….
 p  A exp ikr1  t   A exp ikr2  t 
Applying “cos rule” to top triangle gives
2
d
2
2 d 
r1  R     2 R sin 
2
2
1/ 2
d


r1  R1  sin  
 R

d
d


 R 1 
sin    R  sin 
2
 2R

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2-slit diffraction
 Similarly for bottom ray
d
d
r2  R1 
sin    R  sin 
2
 2R

 Resultant is a superposition of 2 wavelets
kd
kd
i ( kR  sin  t )
i ( kR  sin  t )
2
2
 p  Ae
 Ae
kd
 i kd sin 
i
sin 
i ( kR t ) 
2
2
 Ae
e

e






 The term expi(kR-t)
will occur in all
expressions. We ignore
it - only relative phases
are important.

 p  A e ikd sin  / 2  Ae ikd sin  / 2

 2 A cos(kd sin  / 2)  2 A cos(kds / 2)
Where s = sin.
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cos-squared fringes
 We observe intensity
2
I   p  cos 2 kds / 2 
cos-squared fringes.
Spacing of maxima
 Spacing inversely proportional to separation of
the slits.
 Amplitude-phase diagrams.
Resultant
Slit 1
Slit 2
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Three slits
 Three slits, spacing d.


 p  A e ikds  ei 0  eikds  A1  2 cos kds
I  1  2 cos kds2
 Primary maxima separated by l/d, as before.
 One secondary maximum.
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N slits and diffraction gratings
 N slits, each separated by d.

 p  A ei 0  eikds  ei 2kds .......... .  ei ( N 1) kds

 A geometric progression, which sums to
 p  A(1  eiNkds ) /(1  eikds )
A
A


eikds / 2 e ikds / 2  eikds / 2 
eiNkds / 2 e iNkds / 2  eiNkds / 2
eiNkds / 2 sin( Nkds / 2)
eikds / 2 sin( kds / 2)
I  sin 2 ( Nkds / 2) / sin 2 (kds / 2)
Spacing, as before
N-2, secondary maxima
 Intensity in primary maxima a N2
 In the limit as N goes to infinity, primary maxima
become d-functions. A diffraction grating.
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Single broad slit
 Slit of width t. Incident plane wave.
 Summation of discrete sources
becomes an integral.
 p  e iky sin  dS
t/2
 p  A e ikysdy


t / 2


A
p 
e ikts / 2  eikts / 2
(iks)
2A

sin( kts / 2) At sinc( kts / 2)
ks
l/t
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Generalisation to any aperture
 Aperture function
 The amplitude distribution across an aperture
can take any form a(y). This is the aperture
function.
 The Fraunhofer diffraction pattern is

p 
a( y)e ikysdy

 putting ks=K gives a Fourier integral

p 
a( y)e iKydy

 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 width
Convolution of 2
d-functions with a
single broad slit.


FT(f*g) a FT(f).FT(g)
Cos fringes
sinc function
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