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"Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 214 "Ti mes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 207 "" 0 "" {TEXT 206 57 "Feynman's sum over pat hs method for Fraunhofer scattering" }}{PARA 0 "" 0 "" {TEXT 215 0 "" }}{PARA 0 "" 0 "" {TEXT 215 411 "The sum over paths method is used to \+ produce the single-slit diffraction pattern. The idea is that photons \+ can take all kinds of possible paths which connect the source with a p oint on the screen. We can illustrate this idea by considering straigh t-line paths between the source and a point y(i) somewhere in the aper ture. We subdivide the aperture in equally spaced intervals according \+ to the following scheme:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "N:=40; # the nu mber of paths" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "a:=10; # the slit width in wavelengths" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "y_i:=i->evalf(a/2-(i-1/2)*a/N); # a subdivision of th e aperture" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"iG6\"F%6$I)operator GF%I&arrowGF%F%-I&evalfG%*protectedG6#,&I\"aGF%#\"\"\"\"\"#*(,&9$F0#! \"\"F1F0F0F.F0I\"NGF%F6F6F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y_i(1),y_i(N/2),y_i(N/2+1),y_i(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"++++v[!\"*$\"++++]7!#5$!++++]7F($!++++v[F%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 215 366 "We have chosen N paths through t he aperture at equidistant heights. We choose a distance between sourc e and aperture (called X), we use this same distance between aperture \+ and screen, and use the variable Y to denote how far away from the cen ter position we are on the screen. All these distances are measured in wavelength (this is of the order of microns or less)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "X:=500; # the source-screen and scr een-aperture distance" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$+&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Y:='Y';" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"YG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 90 "The dis tance travelled by a photon along one of these paths is given by two c ontributions:" }}{PARA 0 "" 0 "" {TEXT 215 37 "source to aperture: sqr t(X^2 + y_i^2)" }}{PARA 0 "" 0 "" {TEXT 215 41 "aperture to screen: sq rt(X^2 + (y_i-Y)^2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "d_i: =(i,Y)->sqrt(X^2+y_i(i)^2)+sqrt(X^2+(y_i(i)-Y)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"iG6\"I\"YGF%F%6$I)operatorGF%I&arrowGF%F%,&-I%sqr tGF%6#,&*$I\"XGF%\"\"#\"\"\"*$-I$y_iGF%6#9$F1F2F2-F,6#,&F/F2*$,&F4F29% !\"\"F1F2F2F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t_i:=i ->d_i(i)/c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"iG6\"F%6$I)operato rGF%I&arrowGF%F%*&-I$d_iGF%6#9$\"\"\"I\"cGF%!\"\"F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=c/lambda;" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I\"cG6\"\"\"\"I'lambdaGF$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 215 540 "Feynman's idea to explain the phase: think of the ph oton travelling along the straight-line path with a clock whose handle shows how much time has passed. The photon oscillates with a frequenc y which depends on the wavelength (color) of the light. All we need to know is the orientation of the clock handle at the end of the path, w hen it hits the screen. Technically this determines the phase (in phas or language we determine whether the wavelet that travelled along this path arrives with a crest or a through, i.e., with what amplitude)." }}{PARA 0 "" 0 "" {TEXT 215 0 "" }}{PARA 0 "" 0 "" {TEXT 215 49 "To de termine the phase: we want exp(I*2*Pi*f*t_i)" }}{PARA 0 "" 0 "" {TEXT 215 139 "We realize that c drops out, and that we need a distance divi ded by wavelength, i.e., the distance measured in multiples of the wav elength " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\1673\", italic = \"false\", mathvariant = \"normal\"));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q)λF'/%'italicGQ& falseF'/%,mathvariantGQ'normalF'" }{TEXT 215 1 "." }}{PARA 0 "" 0 "" {TEXT 215 0 "" }}{PARA 0 "" 0 "" {TEXT 215 322 "Then we add up all the phases, i.e., we add the clock handles in the sense of vector additio n. It is an addition of complex numbers of equal modulus (magnitude). \+ This gives us the probability amplitude with which the wave arrives. T he squared modulus (magnitude) of this number gives us the light inten sity at the point Y." }}{PARA 0 "" 0 "" {TEXT 215 0 "" }}{PARA 0 "" 0 "" {TEXT 215 85 "Let us illustrate how this vector addition works usin g little arrows (clock handles):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{MPLTEXT 1 0 17 " with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Y:=70;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Lis:=[]: v0:=[0,0]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "fo r i from 1 to N do: w1:=evalf(exp(2*Pi*I*d_i(i,Y))); v1:=[Re(w1),Im(w1 )]; Lis:=[op(Lis),[arrow(v0,v0+v1,.2,.4,.1,color=red)]]: v0:=v0+v1; od : Lis:=[op(Lis),[arrow([0,0],v0,.2,.4,.1,color=blue)]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(seq(Lis[j],j=1..nops(Lis)),scaling=con strained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#q" }}{PARA 13 "" 1 "" {TEXT 216 0 "" }{GLPLOT2D 400 300 300 {PLOTDATA 2 "6_p-%)POLYGONSG6&7& 7$$\"+X\\W)***!#6$!+%4#[j!\"*7%7$$!+MW]BL!#5$\"+Ymy_=!\"*7$$!+3aXA:!#5$\"+)=4`)>! 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