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Wave equations

Wave equations. Maxwell’s wave equation, second order. One dimensional wave equation Complex numbers and trigonometric functions Three dimensional wave equation Spherical wave. Maxwell’s second order wave equation. forward. backward. SOLUTIONS?.

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Wave equations

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  1. Wave equations Maxwell’s wave equation, second order • One dimensional wave equation • Complex numbers and trigonometric functions • Three dimensional wave equation • Spherical wave

  2. Maxwell’s second order wave equation forward backward SOLUTIONS? Principle of superposition: all solutions can be written as a linear combination of (complex numbers) Substitute in wave equation to find the wave velocity

  3. Complex number Im Z = a + i b a = Re (z) b = Im (z) Z* = a - i b b z q Re a Add and subtract exponentials and trigonometric functions Plane waves, spherical waves

  4. Plane waves Solution of: Substitute: Surfaces of constant phase perpendicular to the vector k Relation between k and l At t = cst, the field repeats itself after l

  5. Spherical wave Wave equation: Write in spherical coordinates, assuming spherical symmetry: Conservation of energy

  6. Electromagnetic waves

  7. SUPERPOSITION OF WAVES Waves of the same k vector,same frequency Waves of the same k vector, different frequencies Beat note Creation of an arbitrary Group velocity Waves mixing (AOM) Waves of different k vector, same frequency Counter-propagating waves Co-propagating, random phase Intersecting waves

  8. Waves of the same k vector,same frequency Constructive and destructive y Superposition is just like adding two vectors , x

  9. Random and Coherent source

  10. Waves of the same k vector, different frequencies Two sine waves traveling in the same direction

  11. Waves of the same k vector, different frequencies Two sine waves traveling in opposite directions “standing wave”

  12. Waves of the same k vector, same frequency Energy conservation A beam splitter is an element with a complex reflection coefficient Energy conservation: If the energy is lost by destructive interference, it has to reappear somewhere else by constructive interference and a complex transmission coefficient

  13. Waves of the same k vector, same frequency Energy conservation Mach Zehnder: Michelson

  14. Waves of the same k vector, same frequency Energy conservation Antiresonant ring

  15. SUPERPOSITION OF WAVES Waves of (nearly) the same k vector, same frequency: Fresnel biprism Given the angle, what is the fringe spacing? How to determine the angle of the biprism?

  16. SUPERPOSITION OF WAVES y Waves of (nearly) the same k vector, same frequency: Young’s double-slit experiment P S1 S2 Young’double-slit as a spectrometer: to each lcorresponds an angle q,

  17. Young’s double slit Shape of the interferences: How far do the interference fringes extend? Fringe visibility Transverse coherence of a beam

  18. SUPERPOSITION OF WAVES Waves of the same k vector, same frequency Waves of different k vector, same frequency Counter-propagating waves Co-propagating, random phase Intersecting waves Waves of the same k vector direction, different frequencies Beat note Creation of an arbitrary Group velocity Waves mixing (AOM)

  19. Waves mixing: not necessarily optical waves Example: AOM

  20. The plan… Fraunhofer and Fourier – what is the connection? Fraunhofer and Fourier – what is the physical meaning? Fraunhofer and Correlations, applications to optical filtering of images Fraunhofer and Fourier – Application to gratings

  21. Fraunhofer and Fourier – what is the connection? Given: Field in plane z=0 e(x,y) Find: Field in plane z=L e(x’,y’,z) y y’ kAP = kR - kyy q AP = R – y sin q P A R q z O L x’ x

  22. Case 3: lens f Case 1: L finite Case 2: L infinite kx = (k/L)x’ kx = (k/f)x’ kx = kqx ky = (k/L)y’ ky = (k/f)y’ ky = kqy e(x,y) P P e(x,y) e(x,y) k k y ky y’ k A ky qy A ky f A L

  23. Fraunhofer and Fourier – what is the physical meaning? A curve can be defined… as an ensemble of points as an ensemble of lines An electric field distribution can be defined… as an ensemble of rays as an ensemble of radiating points Uniform distribution (k in all directions) 1 point Delta in one direction Uniform in the other 1 line

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