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Fraunhofer Diffraction. Wed. Nov. 20, 2002. Kirchoff integral theorem. This gives the value of disturbance at P in terms of values on surface  enclosing P. It represents the basic equation of scalar diffraction theory. Geometry of single slit. Have infinite screen with aperture A.

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Fraunhofer Diffraction

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fraunhofer diffraction

Fraunhofer Diffraction

Wed. Nov. 20, 2002

kirchoff integral theorem
Kirchoff integral theorem

This gives the value of disturbance at P in terms of values on surface  enclosing P.

It represents the basic equation of scalar diffraction theory

geometry of single slit
Geometry of single slit

Have infinite screen with aperture A

Let the hemisphere (radius R) and screen with aperture comprise the surface () enclosing P.






Radiation from source, S, arrives at aperture with amplitude

Since R 

E=0 on .


Also, E = 0 on side of screen facing V.

fresnel kirchoff formula
Fresnel-Kirchoff Formula
  • Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6)
fresnel kirchoff formula5
Fresnel-Kirchoff Formula
  • Now assume r, r’ >>  ; then k/r >> 1/r2
  • Then the second term in (7) drops out and we are left with,

Fresnel Kirchoff diffraction formula

obliquity factor
Obliquity factor
  • Since we usually have ’ = - or n.r’=-1, the obliquity factor F() = ½ [1+cos ]
  • Also in most applications we will also assume that cos   1 ; and F() = 1
  • For now however, keep F()
huygen s principle
Huygen’s principle
  • Amplitude at aperture due to source S is,
  • Now suppose each element of area dA gives rise to a spherical wavelet with amplitude dE = EAdA
  • Then at P,
  • Then equation (6) says that the total disturbance at P is just proportional to the sum of all the wavelets weighted by the obliquity factor F()
  • This is just a mathematical statement of Huygen’s principle.
fraunhofer vs fresnel diffraction
In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)

If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction

Fraunhofer vs. Fresnel diffraction



Hecht 10.2

Hecht 10.3

fraunhofer vs fresnel diffraction9
Fraunhofer vs. Fresnel Diffraction










fraunhofer vs fresnel diffraction10
Fraunhofer Vs. Fresnel Diffraction

Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it 

fraunhofer diffraction limit
Fraunhofer diffraction limit



  • Now, first term = path difference for plane waves


sin’≈ h’/d’

sin ≈ h/d

sin’ + sin =  ( h’/d + h/d )

Second term = measure of curvature of wavefront

Fraunhofer Diffraction 

fraunhofer diffraction limit12
Fraunhofer diffraction limit
  • If aperture is a square -  X 
  • The same relation holds in azimuthal plane and 2 ~ measure of the area of the aperture
  • Then we have the Fraunhofer diffraction if,

Fraunhofer or far field limit

fraunhofer fresnel limits
Fraunhofer, Fresnel limits
  • The near field, or Fresnel, limit is
  • See 10.1.2 of text
fraunhofer diffraction14
Fraunhofer diffraction
  • Typical arrangement (or use laser as a source of plane waves)
  • Plane waves in, plane waves out





fraunhofer diffraction15
Fraunhofer diffraction
  • Obliquity factor

Assume S on axis, so

Assume  small ( < 30o), so

  • Assume uniform illumination over aperture

r’ >>  so is constant over the aperture

  • Dimensions of aperture << r

r will not vary much in denominator for calculation of amplitude at any point P

consider r = constant in denominator

fraunhofer diffraction16
Fraunhofer diffraction
  • Then the magnitude of the electric field at P is,
single slit fraunhofer diffraction
Single slit Fraunhofer diffraction


y = b





r = ro - ysin

dA = L dy

where L   ( very long slit)

single slit fraunhofer diffraction18
Single slit Fraunhofer diffraction

Fraunhofer single slit diffraction pattern