Diffraction Overview
Diffraction is the bending and spreading of waves from
apertures or obstructions, including interference of the waves.
E ( x, y, z )  f ( E ( x ', y ', 0)
Diffraction for increasing screen distance
Looks like the
aperture with
fringes!
(Fresnel)
Aperture
200x100 l/2p
z screen
20 l/2p
z screen
100 l/2p
“Far field”
looks like
|FT|2 of
aperture!
(Fraunhoffer
diffraction)
z screen
500 l/2p
z screen
2500 l/2p
Diffraction Calculations
How could we solve
E ( x, y , z ) 
with no approximations?
2

E
 2 E   o o 2  0
t
plus boundary conditions.
…but there are easier
approximations!
f ( E ( x ', y ', 0)
http://www.falstad.com/ripple/
(download, extract and run the .jar file
with Java)
Huygens’ principle 1678
Every point on a wavefront acts like a “forward spherical”
scalar source.
Conceptual tool: gave Snell’s law,
finds diffraction maxes, mins
Fresnel’s update --- make it formal:
ei( kr t)
1  cos ( r , ˆz   2
r
Obeys a scalar wave equation
 2E
 E   o o 2  0
t
2
Hard to solve
vs
2 E ( r   k 2 E ( r   0
Helmholtz equation
ei( kr t )
r
Works when:

essentially single frequency

E doesn’t change significantly over a distance of l

Forget about polarization
(if we further required small l, we’d get the
Eikenol equation…then no diffraction)
Fresnel-Kirchoff diffraction formula
Fresnel’s diffraction model: add these Huygen waves from
the openings…it works pretty well!
ei( kr )
1  cos ( r , ˆz   2 dx ' dy '
E ( x, y, z   C  E ( x, y, 0 
R( x ', y ', z ')
aperture
Kirchhoff found the factor:
Put on firm math foundation with
Green’s theorem and Helmholtz equation
i meaning?
C
i
l
Fresnel approximation
i
ei( kr t)
1  cos ( r , ˆz  2 dx ' dy '
E ( x, y, z   
E ( x, y, 0 

l aperture
r
Becomes: (know how to do this step with small angle/binomial approx’s)
E ( x, y, z   
ikz
ie e
i
(
k 2 2
x y
2z
lz


E ( x, y, 0  e
i
(
k
x 2  y 2
2z

e
i
k
( xx yy
z
dxdy 
aperture
restrictions:
a (size of aperture) > l [scalar wave approx]
z of screen > a (but if get far enough, becomes simpler Fraunhofer)
x,y of screen <<z, so angles on screen are small
Diffraction for increasing z,
using Fresnel equations
Aperture
200x100 l/2p
z screen
20 l/2p
z screen
100 l/2p
Looks like the
aperture with
fringes!
(Fresnel
diffraction)
“Far field”
looks like
|FT|2 of
aperture!
(Fraunhoffer
diffraction)
z screen
500 l/2p
z screen
2500 l/2p
Fresnel diffraction for slit, increasing z
Babinet’s principle for complementary patterns
The diffraction features caused by an aperture are
also present in those of a “block”.
Complimentarity principle
The diffraction pattern for an aperture is similar (but not
identical) to the pattern for a “block” of the same shape
Eblock  Eincident  Eslit
Circular hole diffraction
D = 1 to 4 mm, screen 1 meter away, HeNe light
Fresnel diffr.: Center
alternates bright/dark
Complimentarity principle
Poisson’s spot in shadow of ball bearing
Center is always bright
Converging thin lens/mirror imaging review
Converging thin lens/mirror imaging review
Diverging thin lens/mirror imaging review
"The picture shows a spherical mirror,
resting on [my] left hand… Such a
globe reflection collects almost one's
whole surroundings in one diskshaped image. the whole room, four
walls, the floor, and the ceiling,
everything, albeit distorted, is
compressed into that one small circle.
Your own head, or more exactly the
point between your eyes, is the
absolute center. No matter how you
turn or twist yourself, you can't get out
of that central point. You are
immovably the focus, the unshakable
core, of your world." - M. C. Escher
Diverging thin lens/mirror imaging review