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Diffraction

Diffraction. Monday Nov. 18, 2002. 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. Wavefront U. What is U here?. We know this. Diffraction theory.

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Diffraction

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  1. Diffraction Monday Nov. 18, 2002

  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 Wavefront U What is U here? We know this

  3. Diffraction theory • Consider two well behaved functions U1’, U2’ that are solutions of the wave equation. • Let U1’ = U1e-it ; U2’ = U2e-it • Thus U1 , U2 are the spatial part of the functions and since, • we have,

  4. Green’s theorem • Consider the product U1 grad U2 = U1U2 • Using Gauss’ Theorem • Where S = surface enclosing V • Thus,

  5. Green’s Theorem • Now expand left hand side, • Do the same for U2U1 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 U1 = “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   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 U2 = continuous function and thus the derivative is bounded (assume) • Its maximum value in V = C • Then since eik  1 as 0 we have, • The second term on the RHS of (5)

  11. Kirchoff’s Integral Theorem Now as 0 U2(r)  UP (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 Have infinite screen with aperture A Let the hemisphere (radius R) and screen with aperture comprise the surface () enclosing P.  P S r r’ ’  Radiation from source, S, arrives at aperture with amplitude Since R  E=0 on . R  Also, E = 0 on side of screen facing V.

  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/r2 • 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 = 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.

  18. 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 S P Hecht 10.2 Hecht 10.3

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

  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 sin’ sin • 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 

  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 10.1.2 of text

  24. Fraunhofer diffraction • Typical arrangement (or use laser as a source of plane waves) • Plane waves in, plane waves out screen S  f1 f2

  25. 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

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

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

  28. Single slit Fraunhofer diffraction Fraunhofer single slit diffraction pattern

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