Chapter 11: Fraunhofer Diffraction
Diffraction is…
-
a consequence of the wave nature of light
-
an interference effect
-
any deviation from geometrical optics
resulting from obstruction of the wavefront
Diffraction is…
interference on the edge
…on the edge of sea
…on the edge of night
…on the edge of dawn
…in the skies
…in the heavens
…on the edge of the shadows
…on the edge of the shadows
With and without diffraction
The double-slit experiment
interference explains the fringes
-narrow slits or tiny holes
-separation is the key parameter
-calculate optical path difference D
diffraction shows how the size/shape of the slits
determines the details of the fringe pattern
Josepf von Fraunhofer (1787-1826)
Fraunhofer diffraction
-
far-field
-
plane wavefronts at aperture and obserservation
-
moving the screen changes size but not shape of
diffraction pattern
Next week: Fresnel (near-field) diffraction
Diffraction from a single slit
slit  rectangular aperture, length >> width
Diffraction from a single slit
plane waves in

- consider superposition of segments of the wavefront arriving at point P
- note optical path length differences D
Huygens’ principle
every point on a wavefront may be regarded as a secondary source of wavelets
curved
wavefront:
planar
wavefront:
c Dt
Not any more!!
obstructed
wavefront:
In geometrical optics, this
region should be dark
(rectilinear propagation).
Ignore the peripheral and
back propagating parts!
Diffraction from a single slit
for each interval ds:
 E L ds
dEP  
 r
 i ( kr   t )
e

Let r = r0 for wave from center of slit (s=0).
Then:
 E L ds
d E P  
 r0  D
 i ( k  r0  D   t )
e


where D is the difference in path length.
-negligible in amplitude factor
-important in phase factor
EL (field strength) constant for each ds
Get total electric field at P by integrating over
width of the slit
Diffraction from a single slit
After integrating:
EP 
E L b sin 

r0
e
i ( kr 0   t )
where b is the slit width
and  
2
E0
Irradiance:
  0c
I 
 2
1
 2  0c  E Lb

E0 
2  r0

 sin 2 

2



I0
I  I 0 sinc
2
2
 
kb sin 
Recall the sinc function
1 for  = 0
sinc  
sin 

zeroes occur when sin = 0
i.e. when  
1
kb sin   m 
2
where m = ±1, ±2, ...
Recall the sinc function
sinc  
sin 

maxima/minima when
d  sin 

d  
 cos  sin 
 cos   sin 
 


0
2




 
sin 
cos 
 tan 
Diffraction from a single slit
I  I 0 sinc
2
 
Central maximum:
image of slit
angular width
D 
2
b
hence as slit narrows, central maximum spreads
Beam spreading
angular spread of central maximum independent of distance
Aperture dimensions determine pattern
Aperture dimensions determine pattern
I  I 0 sinc
2
   sinc 2  
where  
kb
 
ka
sin 
2
2
sin 
Aperture shape determines pattern
Irradiance for a circular aperture
 2 J 1 
I  I 0 
 

2


J1(): 1st order Bessel function
Friedrich Bessel
(1784 – 1846)
where  
1
kD sin 
2
and D is the diameter
Irradiance for a circular aperture
Central maximum: Airy disk
circle of light; “image” of aperture
angular radius D 

1/ 2
1 . 22 
D
hence as aperture closes, disk grows
How else can we obstruct a wavefront?
Any obstacle that produces local amplitude/phase
variations create patterns in transmitted light
Diffractive optical elements (DOEs)
Diffractive optical elements (DOEs)
Phase plates
change the spatial profile of the light
Demo
Resolution
Sharpness of images limited by diffraction
Inevitable blur restricts resolution
Resolution
measured from a ground-based telescope, 1978
Charon
Pluto
Resolution
measured from the Hubble Space Telescope, 2005
http://apod.nasa.gov/apod/ap060624.html
Rayleigh’s criterion
for just-resolvable images
 D  min

1 . 22 
D
where D is the diameter
of the lens
Imaging system (microscope)
x min  f D  min
 1 . 22  
 f

 D 
- where D is the diameter and f is the focal length of the lens
- numerical aperture D/f (typical value 1.2)
x min  
Test it yourself!
visual acuity
Test it yourself!
Double-slit diffraction
considering the slit width and separation
Double-slit diffraction
 
I  4 I 0 sinc
2
1
2
   cos 2 
 
1
2
single-slit
diffraction
double-slit
interference
kb sin 
ka sin 
Double-slit diffraction
I  4 I 0 sinc
2
   cos 2 
Double-slit diffraction
Double-slit diffraction
 sin    
2
 cos 
 4 I 0 
 

2
IP
single slit
two beam
diffraction interference
Multiple-slit diffraction
IP
 sin    

 I 0 
 

single slit
diffraction
2
 sin  N   


 sin(  ) 
2
multiple beam
interference
Importance of spatial coherence
If the spatial coherence length is less than
the slit separation, then the relative phase
of the light transmitted through each slit
will vary randomly, washing out the finescale fringes, and a one-slit pattern will be
observed.
Max
Fraunhofer diffraction patterns
Good spatial
coherence
Poor spatial
coherence
The double slit and quantum mechanics
Imagine using a beam so weak that only one photon passes through the
screen at a time. In this case, the photon would seem to pass through only
one slit at a time, yielding a one-slit pattern.
Which pattern occurs?
Possible Fraunhofer diffraction patterns
Each photon
passes
through only
one slit
Each photon
passes
through
both slits
The double slit and quantum mechanics
Dimming the incident light:
Each individual
photon goes
through both
slits!
How can a particle go through both slits?
“Nobody knows, and it’s best if you try not to
think about it.”
Richard Feynman
Exercises
You are encouraged to solve
all problems in the textbook
(Pedrotti3).
The following may be covered
in the werkcollege on
12 October 2011:
Chapter 11:
1, 3, 4, 10, 12, 13, 22, 27