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Luneburg_Lens.gif (360 × 408 pixels, file size: 4 MB, MIME type: image/gif, looped, 53 frames, 5.3 s)

Summary

Description
English: A Luneburg lens has a gradually varying refractive index, such that a point source on its edge will be converted into a plane wave (and vice-versa, a plane wave will be focussed on its edge).
Date
Source https://twitter.com/j_bertolotti/status/1394635354209230849
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

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]0/20; \[CapitalDelta] = 40*\[Lambda]0; (*Parameters for the grid*)

ReMapC[x_] := RGBColor[(2 x - 1) UnitStep[x - 0.5], 0, (1 - 2 x) UnitStep[0.5 - x]];
R = \[CapitalDelta]/3;
ren = Table[
   If[x^2 + y^2 <= R^2, Sqrt[2 - ((x^2 + y^2)/R^2)], 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
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[ren][[1]];
L = -1/\[Delta]^2*KirchhoffMatrix[GridGraph[{dim, dim}]]; (*Discretized Laplacian*)
n = ren + I imn;

sinstep[t_] := 20 - (5/6 \[CapitalDelta]) Sin[\[Pi]/2 t]^2;
frames1 = Table[
  \[Phi]in = Table[E^(-((x)^2 + (y + sinstep[t])^2)/(2 (\[Lambda]0/5)^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]s = Partition[LinearSolve[M, b], dim]; (*Solve the linear system*)
  MatrixPlot[Transpose[(Re[(\[Phi]in + \[Phi]s)]/Max[Abs@Re[\[Phi]in + \[Phi]s][[(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}, PlotLabel -> "Luneburg Lens: n=\!\(\*SqrtBox[\(2 - \*FractionBox[SuperscriptBox[\(r\), \(2\)], SuperscriptBox[\(R\), \(2\)]]\)]\)", LabelStyle -> {Black, Bold}, Epilog -> {White, Circle[{Round[dim/2 - (4 d)/\[Delta]], Round[dim/2 - (4 d)/\[Delta]]}, dim/3 - (1 d)/\[Delta]]}](*Plot everything*)
  , {t, 0, 1, 1/20}];

sinstep[t_] := 20 - (5/6 \[CapitalDelta]) Sin[\[Pi]/2 t]^2;
frames2 = Table[
  \[Phi]in = Table[E^(-((x - \[CapitalDelta]/3 Cos[\[Theta]])^2 + (y - \[CapitalDelta]/3 Sin[\[Theta]])^2)/(2 (\[Lambda]0/5)^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]s = Partition[LinearSolve[M, b], dim]; (*Solve the linear system*)
  MatrixPlot[Transpose[(Re[(\[Phi]in + \[Phi]s)]/Max[Abs@Re[\[Phi]in + \[Phi]s][[(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}, PlotLabel -> "Luneburg Lens: n=\!\(\*SqrtBox[\(2 - \*FractionBox[SuperscriptBox[\(r\), \(2\)], SuperscriptBox[\(R\), \(2\)]]\)]\)", LabelStyle -> {Black, Bold}, Epilog -> {White, Circle[{Round[dim/2 - (4 d)/\[Delta]], Round[dim/2 - (4 d)/\[Delta]]}, dim/3 - (1 d)/\[Delta]]}](*Plot everything*)
  , {\[Theta], \[Pi]/2, 3/2 \[Pi], \[Pi]/20}];

sinstep[t_] := 1/3 \[CapitalDelta] + (\[CapitalDelta]/2 - \[CapitalDelta]/3) Sin[\[Pi]/2 t]^2;
frames3 = Table[
  \[Phi]in = Table[E^(-((x)^2 + (y + sinstep[t])^2)/(2 (\[Lambda]0/5)^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]s = Partition[LinearSolve[M, b], dim]; (*Solve the linear system*)
  MatrixPlot[ Transpose[(Re[(\[Phi]in + \[Phi]s)]/Max[Abs@Re[\[Phi]in + \[Phi]s][[(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}, PlotLabel -> "Luneburg Lens: n=\!\(\*SqrtBox[\(2 - \*FractionBox[SuperscriptBox[\(r\), \(2\)], SuperscriptBox[\(R\), \(2\)]]\)]\)", LabelStyle -> {Black, Bold}, Epilog -> {White, Circle[{Round[dim/2 - (4 d)/\[Delta]], Round[dim/2 - (4 d)/\[Delta]]}, dim/3 - (1 d)/\[Delta]]}](*Plot everything*)
  , {t, 0, 1, 1/10}];

ListAnimate[Join[frames1, frames2, frames3]]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

Luneburg lens illuminated with a point source at varying positions.

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18 May 2021

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current08:05, 19 May 2021Thumbnail for version as of 08:05, 19 May 2021360 × 408 (4 MB)BertoUploaded own work with UploadWizard

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