Wave equations One dimensional wave equation Maxwells wave

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

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

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

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

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

Electromagnetic waves

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

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

Random and Coherent source

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

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

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

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

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

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?

SUPERPOSITION OF WAVES y Waves of (nearly) the same k vector, same frequency: P Young’s double-slit experiment S 1 S 2 Young’double-slit as a spectrometer: to each l corresponds an angle q,

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

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)

Waves mixing: not necessarily optical waves Example: AOM

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

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 q A k. AP = k. R - kyy AP = R – y sin q R P q O x y’ L x’ z

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

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 radiating points 1 point 1 line as an ensemble of rays Uniform distribution (k in all directions) Delta in one direction Uniform in the other