1 Diffraction

2 Diffraction theory (10.4 Hecht) We will first develop a formalism that will describe the propagation of a wave – that is develop a mathematical description of Huygen’s principle We know this U Wavefront U U What is U here?

3 Diffraction theory Consider two well behaved functions U 1 ’, U 2 ’ that are solutions of the wave equation. Let U 1 ’ = U 1 e -i  t ; U 2 ’ = U 2 e -i  t Thus U 1, U 2 are the spatial part of the functions and since, we have,

4 Green’s theorem Consider the product U 1 grad U 2 = U 1  U 2 Using Gauss’ Theorem Where S = surface enclosing V Thus,

5 Green’s Theorem Now expand left hand side, Do the same for U 2  U 1 and subtract from (2), gives Green’s theorem

6 Green’s theorem Now for functions satisfying the wave equation (1), i.e. Consequently, since the LHS of (3) = 0

7 Green’s theorem applied to spherical wave propagation Let the disturbance at t=0 be, where r is measured from point P in V and U 1 = “Green’s function” Since there is a singularity at the point P, draw a small sphere  P, of radius , around P (with P at centre) Then integrate over  +  P, and take limit as   0

8 Spherical Wave propagation  PP  Thus (4) can be written,

9 Spherical Wave propagation In (5), an element of area on  P is defined in terms of solid angle and we have used Now consider first term on RHS of (5)

10 Kirchoff’s Integral Theorem Now U 2 = continuous function and thus the derivative is bounded (assume) Its maximum value in V = C Then since e ik   1 as  0 we have, The second term on the RHS of (5)

11 Kirchoff’s Integral Theorem Now as  0 U 2 (r)  U P (its value at P) and, Now designate the disturbance U as an electric field E

12 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

13 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

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

15 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

16 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(  )

17 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.

18 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

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

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

21 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 

22 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

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

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

25 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

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

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

28 Single slit Fraunhofer diffraction Fraunhofer single slit diffraction pattern