Duration-bandwidth_product.gif(800 × 261 pixels, file size: 2.93 MB, MIME type: image/gif, looped, 305 frames, 31 s)

Summary

Description
English: How short you can make a pulse depends on many frequencies you are using (bandwidth). But he "duration-bandwidth product" depends only on the shape of your power spectrum. Interestingly, the question "which power spectrum will result in the shortest pulse" depends A LOT on how you decide to measure how wide things are. In particular using the standard deviation or the full-width half-maximum, give very different numbers in many cases.
Date 19 February 2021
Source https://twitter.com/j_bertolotti/status/1362719916449742848
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 12.0 code

labels = {"f(\[Omega])\[Proportional] \[CapitalPi](\[Omega])",    "f(\[Omega])\[Proportional] \[CapitalLambda](\[Omega])",    "f(\[Omega])\[Proportional] \!\(\*SuperscriptBox[\(e\), \\(-\*SuperscriptBox[\(\[Omega]\), \(2\)]\)]\)", "f(\[Omega])\[Proportional] \!\(\*SuperscriptBox[\(sech\), \\(2\)]\)(\[Omega])", "f(\[Omega])\[Proportional] \!\(\*SuperscriptBox[\(e\), \(-\(\(|\)\\(\[Omega]\)\(|\)\)\)]\)"};
frames = Table[
   Table[
    GraphicsRow[{
      Plot[(1 - \[Tau]) f[[j]]^2 + \[Tau] f[[Mod[j + 1, 5, 1]]]^2, {\[Omega], -5, 5}, PlotRange -> {-0.1, 1.1}, Exclusions -> None, PlotStyle -> Black, Axes -> False, FrameLabel -> {{None, None}, {"\[Omega]", "Power spectrum"}}, Frame -> True, FrameStyle -> Directive[White, FontColor -> Black], LabelStyle -> {FontSize -> 14, Bold}, FrameTicks -> None, Epilog -> {Opacity[1 - \[Tau]], Text[Style[labels[[j]], Black, Bold], {3, 0.8}],          Opacity[\[Tau]], Text[Style[labels[[Mod[j + 1, 5, 1]]], Black, Bold], {3, 0.8}]}
       ]
      ,
      Plot[(1 - \[Tau]) p[[j]]*Cos[\[Omega]0 t] + \[Tau] p[[Mod[j + 1, 5, 1]]]*Cos[\[Omega]0 t], {t, -20, 20}, PlotStyle -> Black, PlotRange -> {-1, 1}, Axes -> False, FrameLabel -> {{None, None}, {"t", "Pulse"}}, Frame -> True, FrameStyle -> Directive[White, FontColor -> Black], LabelStyle -> {FontSize -> 14, Bold}, FrameTicks -> None]
      ,
      Graphics[{Text[Style["Duration-bandwidth product", Black, Bold, FontSize -> 9], {0, 0.8}],
        Text[Style["\!\(\*SubscriptBox[\(\[Sigma]\), \(\[Omega]\)]\) \!\(\\*SubscriptBox[\(\[Sigma]\), \(t\)]\) = ", Black, Bold, FontSize -> 10], {0.08, 0.3}], Opacity[1 - \[Tau]], 
        Text[Style[StringForm["``", NumberForm[\[Sigma]\[Omega][[j]]*\[Sigma]t[[j]] // N, {3, 2}]], Black, Bold, FontSize -> 10], {0.5, 0.32}],  Opacity[\[Tau]], 
        Text[Style[StringForm["``", NumberForm[\[Sigma]\[Omega][[Mod[j + 1, 5, 1]]]*\[Sigma]t[[Mod[j + 1, 5, 1]]] // N, {3, 2}]], Black, Bold, FontSize -> 10], {0.5, 0.32}], Opacity[1],
        Text[Style["\!\(\*SubscriptBox[\(FWHM\), \(\[Omega]\)]\) \\!\(\*SubscriptBox[\(FWHM\), \(t\)]\) = ", Black, Bold, FontSize -> 10], {0, -0.1}], Opacity[1 - \[Tau]], 
        Text[Style[StringForm["``", NumberForm[fwhm\[Omega][[j]]*fwhmt[[j]], {3, 2}]], Black, Bold, FontSize -> 10], {0.85, -0.08}], Opacity[\[Tau]], 
        Text[Style[StringForm["``", NumberForm[fwhm\[Omega][[Mod[j + 1, 5, 1]]]*fwhmt[[Mod[j + 1, 5, 1]]], {3, 2}]], Black, Bold, FontSize -> 10], {0.85, -0.08}]
        }, PlotRange -> {{-1, 1}, {-1, 1}}]
      }]
    , {\[Tau]1, 0, 1, 0.02}]
   , {j, 1, 5}];
ListAnimate[Join[Flatten@Table[{Table[frames[[j, 1]], {10}], frames[[j]]}, {j, 1, 5}] ]]

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

How short you can make a pulse depends on many frequencies you are using (bandwidth). But he "duration-bandwidth product" depends only on the shape of your power spectrum.

Items portrayed in this file

depicts

spectral density

creator

some value

Wikimedia username: Berto
author name string: Jacopo Bertolotti

copyright status

copyrighted, dedicated to the public domain by copyright holder

copyright license

CC0

inception

19 February 2021

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Date/TimeThumbnailDimensionsUserComment
current17:30, 24 February 2021Thumbnail for version as of 17:30, 24 February 2021800 × 261 (2.93 MB)BertoUploaded own work with UploadWizard
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