1 Fraunhofer Diffraction Wed. Nov. 20, 2002

2 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

3 Geometry of single slit R S P ’’    Have infinite screen with aperture A Radiation from source, S, arrives at aperture with amplitude Let the hemisphere (radius R) and screen with aperture comprise the surface (  ) enclosing P. Since R   E=0 on . Also, E = 0 on side of screen facing V. r’ r

4 Fresnel-Kirchoff Formula Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6)

5 Fresnel-Kirchoff Formula Now assume r, r’ >> ; then k/r >> 1/r 2 Then the second term in (7) drops out and we are left with, Fresnel Kirchoff diffraction formula

6 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(  )

7 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 = E A dA 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.

8 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 P S Hecht 10.2 Hecht 10.3

9 Fraunhofer vs. Fresnel Diffraction S P d’ d  ’’  h h’ r’ r

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

11 Fraunhofer diffraction limit Now, first term = path difference for plane waves ’’   sin  sin  ’ sin  ’≈ h’/d’ sin  ≈ h/d  sin  ’ +  sin  =  ( h’/d + h/d ) Second term = measure of curvature of wavefront Fraunhofer Diffraction 

12 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

13 Fraunhofer, Fresnel limits The near field, or Fresnel, limit is See of text

14 Fraunhofer diffraction Typical arrangement (or use laser as a source of plane waves) Plane waves in, plane waves out S f1f1 f2f2  screen

15 Fraunhofer diffraction 1. Obliquity factor Assume S on axis, so Assume  small ( < 30 o ), so 2. Assume uniform illumination over aperture r’ >>  so is constant over the aperture 3. Dimensions of aperture << r r will not vary much in denominator for calculation of amplitude at any point P consider r = constant in denominator

16 Fraunhofer diffraction Then the magnitude of the electric field at P is,

17 Single slit Fraunhofer diffraction y = b y dy P  roro r r = r o - ysin  dA = L dy where L   ( very long slit)

18 Single slit Fraunhofer diffraction Fraunhofer single slit diffraction pattern