Diffraction
Fraunhofer and Fresnel Diffraction
Reports due this week
Two light sources have a frequency difference of 4000 GHz. Can
we resolve these two light sources?
A sodium atom has a doublet emission with a frequency difference
of 0.1 GHz. Can we resolve the fine structure?
A Fabry-Perot wave meter. Free spectral range: 3000 GHz.
Finesse 100,000.
Hecht, Pg 312, Fig. 7.34
The square pulse and its transform
Fourier Integrals and Fourier Transforms
0
sin( / 2)( )
/ 2
kLA k E L
kL
The sinc function.
Inverse relationship between widths
FT in single-slit Fraunhofer diffraction
0 0
1( ) ( )cos( ) ( )sin( )f x A k kx dk B k kx dk
( ) ( )cos( )A k f x kx dx
( ) ( )sin( )B k f x kx dx
aperture
incident wave
diffracted wave
Diffraction
Huygens-Fresnel Principle:Every unobstructed point of a wavefront acts
as a source of spherical secondary wavelets.
The field at any point is the superposition of all
of the secondary wavelets.
A
B
P Manifestation of the wave
nature of light – light “bends”
around corners.
When l ~ smallest dim
max AP BP AB
ABl Wave spreads in large angle
ABl Multitude of wavelets
emitted from the aperture
interfere
Analytical Approach
(of aperture)
ikr ikr
o o
S S
e eE E S E dS
r r
Individual Huygens wavelet generated from surface S
Superposition of Huygens wavelets at r.
Time dependence is same for all (same frequency)
Observer
(Detector)
Source
Co-ordinate
System
Eo
Analytical Approach - II
'z z d
The distance between source
plane and detector plane.
Criterion for Fraunhofer
Diffraction
Can ignore quadratic terms if
2 2
12
x yk
d
2wd
l
w is largest dim
of aperture
We assume the incoming wave is homogeneous
Fraunhofer Diffraction: Far field diffraction
1-D Fraunhofer Diffraction – Single Slit
Huygens’ Recipe
Fraunhofer limit
In general, related to
Fourier Transform
of aperture function
SINC sq.
function
Hecht page 452-469
Can determine width w
from minima location
Single Slit Diffraction Pattern
Smaller w, stronger diffraction
Fraunhofer Diffraction – Rectangular Aperture
( ' ')ik xx yy
o dE
E e dsd
Hecht page 452-469
Fraunhofer Diffraction – Rectangular Aperture
2
1, ( 1/ 2)
(0)
mi
i
In
I
Hecht page 452-469
Fresnel Diffraction
Occurs when w2/dl >> 1
Near Field Effect
Wavefront curvature
is important.
Fresnel Diffraction
rs rd
Fresnel Diffraction
Occurs when w2/dl >> 1
Near Field Effect
Wavefront curvature
is important.
(keep quadratic terms)
rsrd
Fresnel Diffraction
Occurs when w2/dl >> 1
Near Field Effect
Wavefront curvature
is important.
Either source or
Detector in near field
Fresnel Integrals
C
S
I=|P2P1|2
The Cornu Spiral
C
S
I=|P2P1|2
Once width is fixed,
the length of the
curve is fixed
Fresnel Diffraction
All the equations are derived for X’ = 0.
P
-w/2
w/2
u1
u2P
-w/2+d
w/2+d
u1
u2
P
The Cornu Spiral
Fraunhofer
Diffraction
on Cornu
Spiral
P
-0.5
0.5
u1
u2
u=0
Fraunhofer
Diffraction
on Cornu
Spiral
P
-0.5
0.5
u1 =0
u2 = 1
u=-0.5
Fraunhofer
Diffraction
on Cornu
Spiral
P
-0.5
0.5
u1 =0.5
u2 =1.5
u=-1
Detector is at the u=0.
Start with slit width =1. We continuously increase the slit width.
Detector is at the u=0.
Start with slit width =1. We continuously increase the slit width.
Detector is at the u=0.
Start with normalized slit width =1. We continuously increase the slit
width.
