File:Spherical Lens.gif

Spherical_Lens.gif ‎(543 × 543 pixels, file size: 6.8 MB, MIME type: image/gif, looped, 92 frames, 9.2 s)

Summary

Description
English: Visualization of light through a spherical lens, as a function of the radii of curvature of the two facets. Notice that we are far from the "thin lens" approximation.
Date
Source https://twitter.com/j_bertolotti/status/1392058658520027138
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]0/15; \[CapitalDelta] = 40*\[Lambda]0; (*Parameters for the grid*)
\[Sigma] = 10 \[Lambda]0; (*width of the gaussian beam*)

sourcef[x_, y_] := E^(-(x^2/(2 \[Sigma]^2))) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))) E^(I k0 y);
\[Phi]in = Table[Chop[sourcef[x, y]], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/ 2, \[Delta]}]; (*Discretized source*)
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[\[Phi]in][[1]];
L = -1/\[Delta]^2*KirchhoffMatrix[GridGraph[{dim, dim}]]; (*Discretized Laplacian*)

ycenter = Map[y0 /. # &, FullSimplify[Solve[(x1)^2 + (y1 - y0)^2 == r^2, {y0}]][[All, 1, All]]  ];

surface2[x_] := Evaluate[Evaluate[((Sqrt[r^2 - (x)^2] + y0) /. {y0 -> ycenter[[1]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 100 \[CapitalDelta]} ] ];
surface1[x_] := Evaluate[((-Sqrt[r^2 - (x)^2] + y0 - 1) /. {y0 -> ycenter[[2]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 100 \[CapitalDelta]}];
frames1 = Table[
  ren = Table[ If[y < Re@Evaluate[surface2[x]] && y > Re@surface1[x], n0, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
  n = ren + I imn;
  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*)
  
  ImageAdd[
   ArrayPlot[ Transpose[(Abs[\[Phi]in + \[Phi]s]/Max[Abs[\[Phi]in + \[Phi]s]])^2][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors" , DataReversed -> True, Frame -> False, PlotRange -> {0, 1}],
   ArrayPlot[Transpose@Re[(n - 1)/5] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
   ](*Plot everything*)
  , {n0, 1, 2.5, 0.24}];

surface2[x_] := Evaluate[Evaluate[((Sqrt[r^2 - (x)^2] + y0) /. {y0 -> ycenter[[1]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> (c^2 + (c \[CapitalDelta])/2 + (5 \[CapitalDelta]^2)/16)/(
        2 (c + \[CapitalDelta]/4))} ] ];
surface1[x_] := Evaluate[((-Sqrt[r^2 - (x)^2] + y0 - 1) /. {y0 -> ycenter[[2]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 100 \[CapitalDelta]}];
frames2 = Table[
  ren = Table[ If[y < Re@Evaluate[surface2[x]] && y > Re@surface1[x], 2.5, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
  n = ren + I imn;
  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*)
  
  ImageAdd[
   ArrayPlot[
    Transpose[(Abs[\[Phi]in + \[Phi]s]/Max[Abs[\[Phi]in + \[Phi]s]])^2][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors" , DataReversed -> True, 
    Frame -> False, PlotRange -> {0, 1}],ArrayPlot[Transpose@Re[(n - 1)/5] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
   ](*Plot everything*)
  , {c, -(\[CapitalDelta]/4) + 0.01, 0, \[CapitalDelta]/(20*3)}];

surface2[x_] := Evaluate[Evaluate[((Sqrt[r^2 - (x)^2] + y0) /. {y0 -> ycenter[[1]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 5/8 \[CapitalDelta]} ] ];
surface1[x_] := Evaluate[((-Sqrt[r^2 - (x)^2] + y0 - 1) /. {y0 -> ycenter[[2]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> (c^2 + (c \[CapitalDelta])/2 + (5 \[CapitalDelta]^2)/16)/(2 (c + \[CapitalDelta]/4))}];
frames3 = Table[
  ren = Table[If[y < Re@Evaluate[surface2[x]] && y > Re@surface1[x], 2.5, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
  n = ren + I imn;
  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*)
  
  ImageAdd[
   ArrayPlot[Transpose[(Abs[\[Phi]in + \[Phi]s]/Max[Abs[\[Phi]in + \[Phi]s]])^2][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors" , DataReversed -> True, Frame -> False, PlotRange -> {0, 1}],
   ArrayPlot[Transpose@Re[(n - 1)/5] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
   ](*Plot everything*)
  , {c, -(\[CapitalDelta]/4) + 0.01, -\[CapitalDelta]/10, \[CapitalDelta]/(20*3)}];

surface2[x_] := Evaluate[Evaluate[((Sqrt[r^2 - (x)^2] + y0) /. {y0 -> ycenter[[1]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> (c^2 + (c \[CapitalDelta])/2 + (5 \[CapitalDelta]^2)/16)/(2 (c + \[CapitalDelta]/4))} ] ];
surface1[x_] := Evaluate[((-Sqrt[r^2 - (x)^2] + y0 - 1) /. {y0 -> ycenter[[2]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 109/120 \[CapitalDelta]}];
frames4 = Table[
  ren = Table[If[y < Re@Evaluate[surface2[x]] && y > Re@surface1[x], 2.5, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
  n = ren + I imn;
  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*)
  
  ImageAdd[
   ArrayPlot[Transpose[(Abs[\[Phi]in + \[Phi]s]/Max[Abs[\[Phi]in + \[Phi]s]])^2][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors" , DataReversed -> True, Frame -> False, PlotRange -> {0, 1}], ArrayPlot[Transpose@Re[(n - 1)/5] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
   ](*Plot everything*)
  , {c, 0, -(\[CapitalDelta]/4) + 0.01, -(\[CapitalDelta]/(20*3))}];

surface2[x_] := Evaluate[Evaluate[((Sqrt[r^2 - (x)^2] + y0) /. {y0 -> ycenter[[1]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> 100 \[CapitalDelta]} ] ];
surface1[x_] := Evaluate[((-Sqrt[r^2 - (x)^2] + y0 - 1) /. {y0 -> ycenter[[2]]}) /. {y1 -> -(\[CapitalDelta]/4), x1 -> \[CapitalDelta]/2, r -> (c^2 + (c \[CapitalDelta])/2 + (5 \[CapitalDelta]^2)/16)/(2 (c + \[CapitalDelta]/4))}];
frames5 = Table[
  ren = Table[If[y < Re@Evaluate[surface2[x]] && y > Re@surface1[x], 2.5, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
  n = ren + I imn;
  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*)
  
  ImageAdd[
   ArrayPlot[Transpose[(Abs[\[Phi]in + \[Phi]s]/Max[Abs[\[Phi]in + \[Phi]s]])^2][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], ColorFunction -> "AvocadoColors" , DataReversed -> True, Frame -> False, PlotRange -> {0, 1}], ArrayPlot[Transpose@Re[(n - 1)/5] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
   ](*Plot everything*)
  , {c, -\[CapitalDelta]/10, -(\[CapitalDelta]/4) + 0.01, -(\[CapitalDelta]/(20*3))}];

ListAnimate[ Flatten[ Join[Table[frames1[[1]], {5}], frames1, Table[frames2[[1]], {5}], frames2, Table[frames3[[1]], {5}], frames3, Table[frames4[[1]], {5}], frames4, Table[frames5[[1]], {5}], frames5, Table[frames1[[-1]], {5}], Reverse@frames1]] ]

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

Visualization of light through a spherical lens

Items portrayed in this file

depicts

optics

spherical lens

Gaussian beam

creator

some value

Wikimedia username: Berto
author name string: Jacopo Bertolotti
URL: https://commons.wikimedia.org/wiki/user:Berto

copyright status

copyrighted, dedicated to the public domain by copyright holder

copyright license

Creative Commons CC0 License

inception

11 May 2021

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