File:Bessel Beam - Self healing.ogg

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Original file(Ogg Theora video file, length 3.4 s, 968 × 531 pixels, 6.21 Mbps, file size: 2.52 MB)

Captions

Captions

Comparison between a Gaussian and a Bessel beam when an obstacle is inserted in the path.

Summary[edit]

Description
English: Comparison between a Gaussian and a Bessel beam when an obstacle is inserted in the path. Bessel beams are said to be "self-healing" because, if you put a small obstacle in their course, the result will still look very much like a Bessel beam.
Date
Source https://twitter.com/j_bertolotti/status/1535222926227668992
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 13.0 code[edit]

\[Lambda]0 = 0.25; 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] = 60*\[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] = 20 \[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*)

f = 10;
sourcef1[x_, y_] :=(*\[ExponentialE]^(-(x^2/(2 \[Sigma]^2)))*) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))) E^(I k0 y) E^(-I k0/(2 f) x^2);
\[Phi]in1 = Table[Chop[sourcef1[x, y]], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}]; 
f2 = 2;
sourcef2[x_, y_] :=(*\[ExponentialE]^(-(x^2/(2 \[Sigma]^2)))*) E^(-((y + \[CapitalDelta]/2)^2/(2 (\[Lambda]0/2)^2))) E^(I k0 y) E^(-I k0/(2 f2) Abs[x]);
\[Phi]in2 = Table[Chop[sourcef2[x, y]], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}]; (*Discretized source*)
frames = Table[
   ren = Table[ If[-(\[CapitalDelta]/50) < x + shift < \[CapitalDelta]/50 && -\[CapitalDelta]/400 < y + \[CapitalDelta]/4 < \[CapitalDelta]/400, 2 I, 1], {x, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}, {y, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[Delta]}];
   n = ren + I imn;
   M = L + DiagonalMatrix[SparseArray[Flatten[n]^2 k0^2]]; (*Operator on the left-hand side of the equation we want to solve*)
   b = -(Flatten[n]^2 - 1) k0^2 Flatten[\[Phi]in1]; (*Right-hand side of the equation we want to solve*)
   \[Phi] = Partition[LinearSolve[M, b], dim]; (*Solve the linear system*)
   b2 = -(Flatten[n]^2 - 1) k0^2 Flatten[\[Phi]in2]; (*Right-hand side of the equation we want to solve*)
   \[Phi]2 = Partition[LinearSolve[M, b2], dim]; (*Solve the linear system*)
  
    Grid[{{
      Show[ImageAdd[
        ArrayPlot[
         Transpose[(Re@\[Phi]/Max[Abs@\[Phi][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]]])][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], DataReversed -> True, Frame -> False, PlotRange -> {-0.5, 0.5}, LabelStyle -> {Black, Bold}, ColorFunctionScaling -> True, ColorFunction -> ReMapC, ClippingStyle -> {Blue, Red}],
        ArrayPlot[Transpose[Im@ren/1] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
        ], PlotLabel -> "Gaussian beam", LabelStyle -> {Black, Bold, FontSize -> 26}, ImageSize -> Medium]
      ,
      Show[ImageAdd[
       ArrayPlot[
        Transpose[(Re@\[Phi]2/Max[Abs@\[Phi]2[[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]]])][[(4 d)/\[Delta] ;; (-4 d)/\[Delta], (4 d)/\[Delta] ;; (-4 d)/\[Delta]]], DataReversed -> True, Frame -> False, PlotRange -> {-0.5, 0.5}, LabelStyle -> {Black, Bold}, ColorFunctionScaling -> True, ColorFunction -> ReMapC, ClippingStyle -> {Blue, Red}]
        ,
        ArrayPlot[Transpose[Im@ren/1] , DataReversed -> True , ColorFunctionScaling -> False, ColorFunction -> GrayLevel, Frame -> False]
        ], PlotLabel -> "Bessel beam", LabelStyle -> {Black, Bold, FontSize -> 26}, ImageSize -> Medium]
      }}]
   , {shift, -\[CapitalDelta]/2, \[CapitalDelta]/2, \[CapitalDelta]/50}];
ListAnimate[frames]

Licensing[edit]

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.

File history

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Date/TimeThumbnailDimensionsUserComment
current09:46, 14 June 20223.4 s, 968 × 531 (2.52 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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VP9 480P 821 kbps Completed 09:46, 14 June 2022 6.0 s
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