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,
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:  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

#### some value

author name string: Jacopo Bertolotti

## File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current17:30, 24 February 2021 800 × 261 (2.93 MB)BertoUploaded own work with UploadWizard
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