File:Light scattering - Averaging.gif
Summary
| Description |
English: Coherent light incident on a scattering medium will produce a volume speckle inside the material. But if you start averaging over more and more realizations of the disorder, the average doesn't look like speckle anymore.
The real part of the electric field (right panel) inside the scattering medium averages to zero, but not immediately. There is a region inside where the amplitude decays exponentially, showing that not all the light is scattered immediately (Lamber-Beer law). The intensity of the electric field (left panel) can not average to zero, as it is a positive quantity, but it gets smoother and smoother. To a good approximation the average light intensity in a scattering medium can actually be described using the diffusion equation! |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1413159977854849029 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
| GIF development |
Mathematica 12.0 code
\[Lambda]0 = 1.; k0 =
N[(2 \[Pi])/\[Lambda]0]; (*The wavelength in vacuum is set to 1, so all lengths are now in units of wavelengths*)
\[Delta] = \[Lambda]0/10; \[CapitalDelta] = 50*\[Lambda]0; (*Parameters for the grid*)
ReMapC[x_] := RGBColor[(2 x - 1) UnitStep[x - 0.5], 0, (1 - 2 x) UnitStep[0.5 - x]];
\[Sigma] = 10 \[Lambda]0;
d = \[Lambda]0/2; (*typical scale of the absorbing layer*)
imn = Table[
Chop[5 (E^-((x + \[CapitalDelta]/2)/d) + E^((x - \[CapitalDelta]/2)/d) + E^-((y + \[CapitalDelta]/2)/d) + E^((y - \[CapitalDelta]/2)/d))], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}]; (*Imaginary part of the refractive index (used to emulate absorbing boundaries)*)
dim = Dimensions[imn][[1]];
L = -1/\[Delta]^2*KirchhoffMatrix[GridGraph[{dim, dim}]]; (*Discretized Laplacian*)
optimizedpoint = Round[0.7 dim];
sourcelist = Table[x0, {x0, -\[CapitalDelta]/2 + 2, \[CapitalDelta]/2 - 2, 1}];
sourcedim = Dimensions[sourcelist][[1]];
ren0 = 2;
\[Phi] = Table[
ren = ren0*Clip[Total[Table[RotateRight[DiskMatrix[5, dim], {RandomInteger[{-Round[dim/2], Round[dim/2] }], RandomInteger[{ -Round[dim/8], Round[dim/2] - 10}]}], {200}]], {0, 1}] + 1;
n = ren + I imn;
phases = Table[
\[Phi]in = Table[E^(-((x - sourcelist[[j]])^2/(2 (\[Lambda]0/2)^2))) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))), {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];(*Discretized source*)
b = -(Flatten[n]^2 - 1) k0^2 Flatten[\[Phi]in]; (*Right-hand side of the equation we want to solve*)
M = L + DiagonalMatrix[SparseArray[Flatten[n]^2 k0^2]]; (*Operator on the left-hand side of the equation we want to solve*)
\[Phi] = Partition[LinearSolve[M, b], dim]; (*Solve the linear system*)
Arg[\[Phi][[Round[dim/2] , optimizedpoint ]] ]
, {j, 1, sourcedim, 1}];
\[Phi]in = Total@Table[Table[E^(-((x - sourcelist[[j]])^2/(2 (\[Lambda]0/2)^2))) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))) E^(-I phases[[j]]), {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}], {j, 1, sourcedim}];(*Discretized source*)
b = -(Flatten[n]^2 - 1) k0^2 Flatten[\[Phi]in]; (*Right-hand side of the equation we want to solve*)
M = L + DiagonalMatrix[SparseArray[Flatten[n]^2 k0^2]]; (*Operator on the left-hand side of the equation we want to solve*)
Partition[LinearSolve[M, b], dim] (*Solve the linear system*)
, {50}];
\[Phi]0 = Table[ren =
ren0*Clip[Total[Table[RotateRight[DiskMatrix[5, dim], {RandomInteger[{-Round[dim/2], Round[dim/2] }], RandomInteger[{ -Round[dim/8], Round[dim/2] - 10}]}], {250}]], {0, 1}] + 1;
n = ren + I imn;
\[Phi]in = Total@Table[Table[E^(-((x - sourcelist[[j]])^2/(2 (\[Lambda]0/2)^2))) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))) , {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}], {j, 1, sourcedim}];(*Discretized source*)
b = -(Flatten[n]^2 - 1) k0^2 Flatten[\[Phi]in]; (*Right-hand side of the equation we want to solve*)
M = L + DiagonalMatrix[SparseArray[Flatten[n]^2 k0^2]]; (*Operator on the left-hand side of the equation we want to solve*)
Partition[LinearSolve[M, b], dim] (*Solve the linear system*)
, {50}];
frames = Table[GraphicsRow[{
ArrayPlot[
Transpose[(Mean[Abs[#]^2 & /@ \[Phi]0[[1 ;; j]] ]/Max[avg\[Phi]0abs[[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]]])][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors", DataReversed -> True, Frame -> False, PlotRange -> {0, 1}, LabelStyle -> {Black, Bold}, ColorFunctionScaling -> False],
ArrayPlot[
Transpose[(Mean[Re[#] & /@ \[Phi]0[[1 ;; j]] ]/Sqrt[Max[avg\[Phi]0abs[[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]]]])][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> ReMapC, DataReversed -> True, Frame -> False, PlotRange -> {-1, 1}, LabelStyle -> {Black, Bold}, ColorFunctionScaling -> True]
}]
, {j, 1, 50}];
ListAnimate[frames]
Licensing
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
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