File:Pulse broadening due to absorption.gif

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Pulse_broadening_due_to_absorption.gif ‎(600 × 214 pixels, file size: 1.79 MB, MIME type: image/gif, looped, 193 frames, 19 s)

Captions

Captions

A Gaussian pulse is as short as you can make it. If you cut away part of its spectrum (e.g. absorbing it), the pulse will get longer.

Summary[edit]

Description
English: A Gaussian pulse is as short as you can make it. If you cut away part of its spectrum (e.g. absorbing it), the pulse will get longer.
Date
Source https://twitter.com/j_bertolotti/status/1357341710573969413
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.0 code[edit]

f[x_] := E^-x^2 E^(I k1 x);
ff = InverseFourierTransform[f[x], x, k];
aff = ff*(1 - a*E^(-((k - k2)^2/(2 \[Sigma]^2))));
faff = FullSimplify[FourierTransform[aff, k, x]];
sinstep[t_] := Sin[\[Pi]/2 t]^2;
\[Sigma]1 = 0.3; k3 = 16; a1 = b;
frame1 = Table[
   norm = NIntegrate[Evaluate[Abs[faff] /. {k1 -> 15, k2 -> k3, \[Sigma] -> \[Sigma]1, a -> sinstep[t]}], {x, -\[Infinity], \[Infinity]}];
   stdev = \[Sqrt]NIntegrate[Evaluate[x^2 Abs[faff/norm] /. {k1 -> 15, k2 -> k3, \[Sigma] -> \[Sigma]1, a -> sinstep[t]}], {x, -\[Infinity], \[Infinity]}];
   GraphicsRow[{
     Show[
      Plot[aff^2 /. {k1 -> 15, k2 -> k3, \[Sigma] -> \[Sigma]1, a -> 0}, {k, 9, 21}, PlotStyle -> {Gray}, PlotRange -> {-0.01, 0.6}, PlotPoints -> 100, Axes -> False, Frame -> True, FrameLabel -> {"\[Omega]", "Spectrum"}, FrameTicks -> None, LabelStyle -> {Black, Bold}]
      ,
      Plot[aff^2 /. {k1 -> 15, k2 -> k3, \[Sigma] -> \[Sigma]1, a -> sinstep[t]}, {k, 9, 21}, PlotStyle -> {Black}]
      ]
     ,
     Show[
      Plot[Re[f[x]] /. {k1 -> 15}, {x, -5, 5}, PlotRange -> All, PlotStyle -> {Gray}, Axes -> False, Frame -> True, FrameTicks -> None, FrameLabel -> {{"Field", None}, {"t", Style[StringForm["\[Sigma]=`` (arbitrary units)", NumberForm[stdev, {3, 2}]], Black, Bold]}}, LabelStyle -> {Black, Bold}]
      ,
      Plot[Re[faff] /. {k1 -> 15, k2 -> k3, \[Sigma] -> \[Sigma]1, a -> sinstep[t]}, {x, -5, 5}, PlotRange -> All, PlotStyle -> {Black}]
      ]
     }]
   , {t, 0, 1, 0.025}];
\[Sigma]1 = 0.3; a1 = 1;
\[CurlyPhi] = \[Phi] /. Solve[15 + 1.5*Cos[0 + \[Phi]] == 16, \[Phi]][[2]];
frame2 = Table[
   norm = NIntegrate[Evaluate[Abs[faff] /. {k1 -> 15, k2 -> (15 + 1.5*Cos[2 \[Pi] sinstep[t] + \[CurlyPhi]]), \[Sigma] -> \[Sigma]1, a -> a1}], {x, -\[Infinity], \[Infinity]}];
   stdev = \[Sqrt]NIntegrate[Evaluate[x^2 Abs[faff/norm] /. {k1 -> 15, k2 -> (15 + 1.5*Cos[2 \[Pi] sinstep[t] + \[CurlyPhi]]), \[Sigma] -> \[Sigma]1, a -> a1}], {x, -\[Infinity], \[Infinity]}];
   GraphicsRow[{
     Show[
      Plot[aff^2 /. {k1 -> 15, k2 -> 16, \[Sigma] -> \[Sigma]1, a -> 0}, {k, 9, 21}, PlotStyle -> {Gray}, PlotRange -> {-0.01, 0.6}, PlotPoints -> 100, Axes -> False, Frame -> True, FrameLabel -> {"\[Omega]", "Spectrum"}, FrameTicks -> None, LabelStyle -> {Black, Bold}]
      ,
      Plot[aff^2 /. {k1 -> 15, k2 -> (15 + 1.5*Cos[2 \[Pi] sinstep[t] + \[CurlyPhi]]), \[Sigma] -> \[Sigma]1, a -> a1}, {k, 9, 21}, PlotStyle -> {Black}, PlotRange -> {-0.01, 0.6}]
      ]
     ,
     Show[
      Plot[Re[f[x]] /. {k1 -> 15}, {x, -5, 5}, PlotRange -> All, PlotStyle -> {Gray}, Axes -> False, Frame -> True, FrameTicks -> None, FrameLabel -> {{"Field", None}, {"t", Style[StringForm["\[Sigma]=`` (arbitrary units)", NumberForm[stdev, {3, 2}]], Black, Bold]}}, LabelStyle -> {Black, Bold}]
      ,
      Plot[Re[faff] /. {k1 -> 15, k2 -> (15 + 1.5*Cos[2 \[Pi] sinstep[t] + \[CurlyPhi]]), \[Sigma] -> \[Sigma]1, a -> a1}, {x, -5, 5}, PlotRange -> All, PlotStyle -> {Black}]
      ]
     }]
   , {t, 0, 1, 0.01}];
\[Sigma]1 =.; a1 =.; k3 =.
\[Sigma]1 =.; k3 =.
ListAnimate[Join[Table[frame1[[1]], {10}], frame1, frame2, Reverse[frame1]]]

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.

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Date/TimeThumbnailDimensionsUserComment
current14:47, 5 February 2021Thumbnail for version as of 14:47, 5 February 2021600 × 214 (1.79 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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