Babinet’s complimentarity principle
The diffraction pattern for an aperture is similar (but not
identical) to the pattern for a “block” of the same shape.
Comes from rewriting integral limits.
E( x, y, z)block  E( x, y, z)incident  E( x, y, z)slit
Babinet’s principle for complementary patterns
E( x, y, z)both  E( x, y, z)square  E( x, y, z)circ
I both  E( x, y, z)square  E( x, y, z)circ
2
Atmospheric “corona” caused by
diffraction around uniform droplets or
ice particles
Review of Fresnel approximation
E  x, y, z   
ieikz e
i

k 2 2
x y
2z
z


E  x, y, 0  e
i

k
x 2  y 2
2z

e
i
k
 xx yy
z
dxdy 
aperture
Approximation made: z > aperture size, interference pattern size
For I on screen at z, we can ignore the factors in front
I  x, y, d  

E  x, y, 0  e

k
i
x 2  y 2
2z

e
k
 i  xx  yy  
z
aperture
(We only need the factors in front if we interfere
the diffracted light with another source)
2
dxdy
Next approx as z gets bigger
(or aperture smaller)
Which goes to zero faster as z gets big (and why)?
A.
B.

k
2
2


x y
2z

k
xx  yy 

z
Fraunhofer approximation:
e
i

k
x 2  y 2
2z

e
i
k
xx  yy  

z

Rough boundary between Fresnel and
Fraunhofer diffraction:
Or: for big z, the aperture-screen length
2
aperture size
differences between any 2 points on the
z
aperture becomes less than /2… (the

center can’t become dark)!


Diffraction for increasing screen distance
Looks like the
aperture with
fringes!
(Fresnel)
Aperture
200x100 /2p
z screen
20 /2p
z screen
100 /2p
“Far field”
looks like
|FT|2 of
aperture!
Fraunhoffer
diffraction)
z screen
500 /2p
z screen
2500 /2p
Fraunhofer angular form
EFraun  x, y, d   ( phase factor) 
E  x, y, 0  e

i
k
 xx yy
z
dxdy 
aperture


EFraun  x , y  ( phase factor) 

E  x, y, 0  e

 ik  x x  y y 
aperture


EFraun  x , y  ( phase factor)  FT E( x ', y ')


I Fraun  x , y  FT E( x ', y ')
2
  
dx dy
Rectangular slit


EFraun  x , y 

aperture
E  x, y, 0  e

 ik  x x  y y 
  
dx dy
Square aperture plots

Square aperture
Zooming in…
The square diffraction pattern is shown on the left. Which pattern
on the right belongs to the rectangular aperture shown? A) top B)
bottom
Fraunhofer diffraction of a very tall slit, uniformly
illuminated


EFraun  x , y 

aperture
E  x, y, 0  e

 ik  x x  y y 
  
dx dy
Single slit Fraunhoffer diffraction
In reality just need to illuminate slit by beam >> slit width:
Angular scale of diffraction
How big should we expect the pattern to be on the screen?
What is the average change of angle of a photon?
What should it depend on?
a
Can use photon momentum uncertainty
 

a
Two interference views of Fraunhoffer diffraction
1. Huygen’s wavelets in the aperture weighted by Eaperture[x’,y’]
interfere to give Escreen[x,y]
2. Reverse time: Plane waves coming from screen at angle x,y
weighted by Escreen[x,y] interfere to give Eaperture[x’,y’] !
a
Can laser beams propagate without spreading?
Use angular scale of diffraction
Diffraction of a “Gaussian aperture”
A laser beam has a Gaussian profile of 2mm width, and
a constant phase across the beam. How wide will it be
after propagating 100m?
Fresnel vs Fraunhofer
Figures in
order were
made by
increasing
something.
Which could it
be?
a) a
b 
c) z
Review of approximations, regimes
z > aperture size > 
ei( kr t )
1  cos  r , ˆz   2 dx ' dy '
E  x, y, z    E  x, y, 0 
r
aperture
z > aperture size, z > interference pattern size:
E  x, y, z  Fresnel  eikz e
i

k 2 2
x y
2z


E  x, y , 0  e
i

k
x 2  y 2
2z

e
i
k
 xx yy
z
dxdy 
aperture
z > (aperture size)2 /  :
EFraun  x, y, d   ( phase factor) 

E  x, y, 0  e
i
k
 xx yy
z
dxdy
aperture


EFraun  x , y  ( phase factor) 

aperture
 ik  x x  y y  


E  x , y ,0 e
dxdy
Atmospheric (pilot’s) “Glory”
Light scattered back from droplets
Complicated mixture of diffraction and refraction: For a
recent discussion (2012) of “glory” physics see
http://www.scientificamerican.com/article.cfm?id=the-science-of-the-glory&page=1