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Interference and Diffraction

Interference and Diffraction. 2 . 2 . 0. 0. . . Destructive Interference. Constructive Interference. x. 2 . 0. . 0. path difference ( x )  phase difference (?). When x < . When x  . Two oscillators or two sources. To point P. . d. d sin .

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Interference and Diffraction

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  1. Interference and Diffraction 2 2 0 0   Destructive Interference Constructive Interference

  2. x 2 0  0 path difference (x)  phase difference (?) When x <  When x

  3. Two oscillators or two sources To point P  d dsin Phase difference when arriving at P is Original phase difference phase from path difference

  4. n equally spaced oscillators; equal amplitude; different in phase (different starting phase, different path length) E.g. C 2 Plane wave 1 n slits equally spaced A B a E D Huygen’s wavelets What is the result of the sum?

  5. Using complex amplitude to solve the problem 1 1

  6. n = 6 Q r T M O

  7. n = 6

  8. n = 6

  9. Path difference C 2 1 A B a …. …. E D Huygen’s wavelets Phase difference Phase difference …. …. 1st Constructive Interference: 2nd Constructive Interference: In general, constructive Interference:

  10. = I0

  11. Width: W10 < W5 ; Height:  n2; Integrated intensity?

  12. Fraunhofer Single Slit: 0 d/2 x x …..  d

  13. Interference between a pair of wavelets from the top and center of the slit } 0 d/2 C B  d next pair x = 0 and x = d/2; path difference BC = dsin/2.  phase difference = (2/)*dsin/2 = dsin/ Destructive interference: dsin/ = , 2, 3, …= n.  dsin = n.

  14. Resolution of single slit and circular aperture: The Rayleigh Criterion Well Resolved Barely Resolved http://www.kshitij-pmt.com/resolution-of-single-slit-and-circular-apertures

  15. Single Slit: Circular aperture: Minimum  1 Airy rings

  16. b The red ones: a The blue ones:  n sets of double slits

  17. What about n slits each with a slit width of d? d a 

  18. If d is very small, upper cap is more flat! a equals to a lot d!

  19. Diffraction Geometry 1a’, 2a’ X Y 1 1’ 1a 2’ 2 Plane normal 3 3’   2a X’ Y’ K P M N d   L S 2 Phase difference between different atoms interacted with X-ray. Atoms in neighboring plane: 2dsin

  20. Laue’s Equations: a  0 acos0 acos Integer Constructive interference: |acos-acos0| = h Similarly in the y direction: |bcos - bcos0| = k Similarly in the y direction: |ccos - ccos0| = l

  21. Reciprocal lattice and diffraction: S S0 OA= pa1+qa2+ra3 p, q, r:integers S-S0 O   m n -S Path difference: uA + Av = Om+On = S0OA + (-S) OA = (S0-S)OA. v u A b1, b2, b3: base vectors of G If  constructive interference

  22. Ewald Sphere k k k k |k| = |k| = |k| = 2/ k Diffraction codition k k k = G k k k 1/

  23. Reciprocal lattice k k k k O

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