Fraunhofer Diffraction
Geometrical Optics:
Bill Mageors
…light can’t turn a corner.
I
Fraunhofer Diffraction
Physical Optics:
Francesco Maria Grimaldi
…actually, it can.
I
Huygens-Fresnel Principle
Every point on a wavefront may be regarded as a secondary
source of spherical wavelets.
The propagated
wave follows the
periphery of the
wavelets.
Huygens, just add the
wavelets considering
interference!
(That’s a sweet ascot by
the way.)
Augustin-Jean Fresnel
Far-Field Diffraction
(a.k.a. Fraunhofer Diffraction)
Far enough that
source illumination is
a plane wave.
Far enough that
diffracted wavelets are
plane wave.
Single Slit Fraunhofer Diffraction
s sinq
ds
s
b
r
P
Get the amplitudes right!
spherical wavelets:
field
field amplitude
 dEo  j  kr t 
dEP  
e
 r 
As ds goes to zero, dEo
must go to zero:
dEo  E L ds
field amplitude
per unit width
 E ds 
dEP   L e j  kr t 
 r 
Get the phases right! (just like Young’s Double Slit)
 E L ds  j  kr t ks sin q 
dE P  
e
 r 
Integrate over the slit:
b 2
b 2
EL j  kr t 
j  ks sin q 
dE

e
e
ds
P


r
b 2
b 2
j  ks sin q   b 2
EL j  kr t  e
EP 
e
r
jk sin q
b 2
EL j  kr t  1
e j kb sin q 2   e j kb sinq 2  
EP 
e
r
jk sin q
  12 kb sin q
new variable:
EL j  kr t  b
e j  e j 
EP 
e
r
2 j
EL b sin  j  kr t 
EP 
e
r 
Calculate irradiance:
I
 oc
2
EP
2
field at r
 o c  E L b  sin 2 
I


2  r  2
2
sin x
sinc x 
x
“sinc”
“sampling function”
“sine cardinal”
an irradiance
I  I o sinc 2 
1
  m
0.8
0.6
Minima at:
m  1,  2,...
m  bsin q
0.4
0.2
Maxima at:
0
  tan 
-8
-4
0
4
8
(graph it!)
Beam spreading