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Comparison_Sommerfeld-Fresnel-Fraunhofer.gif (549 × 225 pixels, file size: 1.86 MB, MIME type: image/gif, looped, 201 frames)

Summary

Description
English: Comparison between the (scalar) diffraction pattern at different distances from a 4λ-wide slit between the Rayleigh-Sommerfeld equation, the paraxial approximation (Fresnel), and the far field approximation (Fraunhofer).
Date
Source https://twitter.com/j_bertolotti/status/1199661806538633216
Author Jacopo Bertolotti
Permission
(Reusing this file)
https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

\[Lambda] = 0.5; k0 = (2 \[Pi])/\[Lambda];
fU = 1/(2 \[Pi]) Integrate[E^(I kx x), {x, -1, 1}];
fres = E^(I k0 z)/(I \[Lambda] z)  Sqrt[z/2] Integrate[E^(-I k0/(2 z) (x2 - x1)^2), {x1, -1, 1}];
fraun = Sqrt[z/2] E^(I k0 z)/(I \[Lambda] z) E^(-I k0/(2 z) x2^2) Integrate[E^(I k0/z (x2 x1)), {x1, -1, 1}];
p1 = Table[
   somm = Table[{x, If[z == 0, HeavisidePi[x/2], Abs[NIntegrate[fU E^(I Sqrt[k0^2 - kx^2] z) E^(I kx x), {kx, -2 k0, 2 k0}]]^2]}, {x, -10 \[Lambda], 10 \[Lambda], \[Lambda]/10}];
   Legended[Show[
     Plot[If[z == 0, HeavisidePi[x2/2], Abs[fres]^2], {x2, -10 \[Lambda], 10 \[Lambda]}, PlotRange -> All, PlotStyle -> {Thick, Orange}],
     Plot[If[z == 0,  10 Sinc[100 x2]^2, Abs[fraun]^2], {x2, -10 \[Lambda], 10 \[Lambda]}, PlotRange -> All, PlotStyle -> {Thick, Cyan}],
     ListPlot[somm, PlotRange -> All, Joined -> True, PlotStyle -> {Thick, Purple}], PlotRange -> {0, 2}, Axes -> False, 
     Epilog -> {Text[Style[StringForm["z=``\[Lambda]", z/\[Lambda]], Medium, Bold], {3, 1.5}]}
     ], LineLegend[{Purple, Orange, Cyan}, {"Rayleigh-Sommerfeld", "Fresnel (paraxial)\napproximation", "Fraunhofer (far field)\napproximation"}] ]
   , {z, 0 \[Lambda], 20 \[Lambda], \[Lambda]/10}];
ListAnimate[p1]

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

Comparison between the (scalar) diffraction pattern at different distances from a 4λ-wide slit between the Rayleigh-Sommerfeld equation, the paraxial approximation (Fresnel), and the far field approximation (Fraunhofer).

27 November 2019

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1,949,435 byte

225 pixel

549 pixel

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