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1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 76 "Fourier Optics: a study o f diffraction patterns in the focal plane of a lens" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 130 "This worksheet deals with the generation of diffraction patterns produced by various apert ures illuminated by monochromatic light." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 86 "Following Smith-Thomson: Opti cs (2nd ed.), chapter 9, we use as a convenient variable " }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "x = sin(theta)/lambda;" "6#/%\"xG*&-%$sinG6 #%&thetaG\"\"\"%'lambdaG!\"\"" }{TEXT 257 1 " " }{TEXT -1 9 " (where \+ " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 30 " is the diffracti on angle and " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 17 " t he wavelength)." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 " " {TEXT -1 115 "For illustration purposes we begin with the Fraunhofer diffraction pattern for a pair of narrow slits displaced by " }{TEXT 258 1 "d" }{TEXT -1 84 " (Young's expt.). We use dimensionless quantit ies, and should use the displacements " }{TEXT 259 1 "d" }{TEXT -1 38 " which are larger than the wavelength " }{XPPEDIT 18 0 "lambda;" "6#% 'lambdaG" }{TEXT -1 206 ". We can think of the length unit as being mi crons, in which case a typical (yellow) wavelength equals 1/2. The ape rture width (or slit separation for Young's experiment) should be at l east several microns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "lambda:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG#\"\" \"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "d:=100/4;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"#D" }}}{EXCHG {PARA 264 "" 0 " " {TEXT -1 53 "The amplitude as a function of the diffraction angle " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 12 " in radians:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Am:=theta->cos(Pi*d/lambda*s in(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AmGR6#%&thetaG6\"6$% )operatorG%&arrowGF(-%$cosG6#*&*(%#PiG\"\"\"%\"dGF2-%$sinG6#9$F2F2%'la mbdaG!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(Am (theta)^2,theta=-Pi/50..Pi/50,numpoints=500);" }}{PARA 13 "" 1 "" {GLPLOT2D 682 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7f[m7$$!35++i%H&=$G'! 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larger angles: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(Am(theta)^2,theta=Pi/4-Pi/ 50..Pi/4+Pi/50,numpoints=500);" }}{PARA 13 "" 1 "" {GLPLOT2D 695 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7`jl7$$\"3q****p)*)y)F*7$$\"3xtNb];uSsF*$\"3YfSu-\"y\"*f )F*7$$\"31`F?b7AVsF*$\"3!efXpIb1R)F*7$$\"31X!yem&yXsF*$\"3-[W4VC#H;)F* 7$$\"3cydX_=M[sF*$\"3B,z\"[e?X#zF*7$$\"35Zia/7(4D(F*$\"3;me2'>P$owF*7$ $\"3g=()HRrG`sF*$\"3/)3!=KFEMuF*7$$\"3CYBILV*eD(F*$\"3m%p#3WK@irF*7$$ \"3'o/&pKA^esF*$\"3mq#egJY4)oF*7$$\"31v[%Q1N5E(F*$\"3<_Qw#[6Kg'F*7$$\" 39i/AOgKjsF*$\"3#[4b0T_hM'F*7$$\"3+HE'RA]gE(F*$\"3*G=dpIVb.'F*7$$\"3mP DJozNosF*$\"3#\\B*e%[5\"pdF*7$$\"3!HR4OUU5F(F*$\"3#*4`\"y3pkX&F*7$$\"3 ]a0T@'=MF(F*$\"3Of&yKgR#y^F*7$$\"3m?b+(oDgF(F*$\"3%z3#f`.Ws[F*7$$\"33O #p*H#3&ysF*$\"31`)GU^R[4H/'G(F*$\"3(HW=!zJd7PF*7$$\"3q `Gxyzq)G(F*$\"3$GC\\'=p]8MF*7$$\"3M,]UHz-\"H(F*$\"35\"3!G;@AeJF*7$$\"3 YYkoRN`$H(F*$\"3un/6I(F*$\"3ZF%*>y/ZA@F*7$$\"3w (Q=@@DQI(F*$\"3MJqhDE6o=F*7$$\"3)Hfi%Q(piI(F*$\"31oxN8'3/l\"F*7$$\"3=J 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***H'[4w.8F*$\"3CPS/&poHp&F`p-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F^itF]i t-%+AXESLABELSG6$Q&theta6\"Q!6\"-%%VIEWG6$;$\"+RNV_5!\"*$\"+^4w.8F\\jt %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 268 "" 0 "" {TEXT 262 39 "Diffraction grating s and Ronchi rulings" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 270 " " 0 "" {TEXT -1 160 "To observe diffraction patterns for gratings and \+ Ronchi rulings we calculate the Fourier transform of the respective ap erture functions. The distance parameter " }{TEXT 263 1 "d" }{TEXT -1 289 " plays the role of the spacing between the slits. For a diffracti on grating the aperture function is smooth: a cos-squared behaviour of the transmissivity as a function of separation y across the aperture \+ is produced (using holography in modern times) and the periodicity is \+ controlled by " }{TEXT 264 1 "d" }{TEXT -1 102 ". Following eqns. (9.1 0-9.11) from the reference we calculate the amplitude first for a 'sin gle' slit." }}{PARA 271 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 225 "We set up the cos-squared profile such that the distance d contains the full width of a slit that includes the perfectly trans mitting part up to the perfectly blocking parts on both sides (the zer oes of the cosine) at |y|=d/2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d:= 'd';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Af:=x->evalc(int(cos(Pi*y/d)^2*exp(I*2*Pi*x*y ),y=-d/2..d/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AfGR6#%\"xG6\"6 $%)operatorG%&arrowGF(-%&evalcG6#-%$intG6$*&)-%$cosG6#*&*&%#PiG\"\"\"% \"yGF:F:%\"dG!\"\"\"\"#F:-%$expG6#**^#F>F:F9F:9$F:F;F:F:/F;;,$F<#F=F>, $F<#F:F>F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Af(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#*(%\"dG\"\"\"%#PiGF*%\"xG F*F**(F+F*,&*&)F,\"\"#F*)F)F1F*F*F*!\"\"F*F,F*F3#F3F1" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 37 "Note the behaviour of the result at |" }{TEXT 267 1 "x" }{TEXT -1 39 "|=1/d: a 0/0 expression results as for \+ " }{TEXT 268 1 "x" }{TEXT -1 18 "=0, however the 1/" }{TEXT 269 1 "x" }{TEXT -1 61 " factor yields some suppression of the amplitude compare d to " }{TEXT 270 1 "x" }{TEXT -1 3 "=0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(subs(d=10,Af(x)^2),x=-Pi/2..Pi/2);" }}{PARA 13 " " 1 "" {GLPLOT2D 692 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7ep7$$!3+++lBj zq:!#<$\"3EqoCxN!>2\"!#C7$$!3WNzQW&=B]\"F*$\"3#R;s!45Ix6!#E7$$!3.42![R OFW\"F*$\"3AcVzQ*Q3p#F-7$$!3?xt3O+tv8F*$\"3.D=ML?Z,=F-7$$!3xxIt:&z#38F *$\"37(oOdL-3Q$!#D7$$!3\")zz#fd\\6C\"F*$\"3Wc@y&\\k8\\'F-7$$!3W:#)\\G: \"*y6F*$\"3\"f\\-tdWy/\"F*$\"3*)pFDwD7S>!#B7$$!3EWfpKW&Q\")*!#=$\"3;)QL:f^C#))F-7$$!3(fe 1Vw'\\I\"*Ffn$\"3)z%38'p$=:rF-7$$!333*H!e\\fG&)Ffn$\"3br<3p1f7nFX7$$!3 5`u@%3'*4&yFfn$\"3!f&49')GHvAFX7$$!3%ewEV#\\hqrFfn$\"35P$>***[ta]FX7$$ !3SI*3_gT\\^'Ffn$\"3ko+oY$!#A7$$!3B!RBoTF&>fFfn$\"3OW+%*HA#H!RFX7$ 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45.000000 45.000000 0 0 " Curve 1" }}}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 58 "We are ready to cal culate the result for an aperture with " }{TEXT 278 1 "n" }{TEXT -1 18 " cycles ('slits'):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Afn:=(x,n )->evalc(int(cos(Pi*y/d)^2*exp(I*2*Pi*x*y),y=-n*d/2..n*d/2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AfnGR6$%\"xG%\"nG6\"6$%)operatorG%& arrowGF)-%&evalcG6#-%$intG6$*&)-%$cosG6#*&*&%#PiG\"\"\"%\"yGF;F;%\"dG! \"\"\"\"#F;-%$expG6#**^#F?F;F:F;9$F;FF?,$FI#F; F?F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 95 "Observe that the int ensity ratio between the central and side peaks is close to the value \+ of 4:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(subs(d=4,Afn(x,19)^2) ,x=-Pi/8..Pi/8,numpoints=500);" }}{PARA 13 "" 1 "" {GLPLOT2D 692 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7a^m7$$!3@++943*p#R!#=$\"3!)[mR= 7$$!3E(e7p?B0\"RF*$\"3=0L]B_!*)p%F-7$$!3_0jEM[>'*QF*$\"3^X(eLyvu@(F-7$ $!3kZym!3\"3!)QF*$\"3elHY!=`kY)F-7$$!3!*>2l90'Q'QF*$\"3ZH9t\\4-:tF-7$$ 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3zZE$QF*$\"3u8uOh['Rj#F-7$$\"3eh@F9q6[QF*$\"3nf*3t%[a16F-7$$\"3y%f)fV& )fjQF*$\"3:+gQAI2V!)F-7$$\"3G6CaB()QzQF*$\"3sthvA`Wl`F-7$$\"3O\\jv\\g7 '*QF*$\"3m8//_i]$p&F87$$\"3zN!>*QDZ6RF*$\"3d'*3YB1UDdF-7$$\"3@++943*p# RF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F\\[wF[[w-%+AXESLABELSG6$Q\"x6 \"Q!6\"-%%VIEWG6$;$!+=3*p#R!#5$\"+=3*p#RFj[w%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 277 "" 0 "" {TEXT -1 28 "The special role played by |" }{TEXT 280 1 "x " }{TEXT -1 5 "|= 1/" }{TEXT 281 1 "d" }{TEXT -1 39 " is evident from \+ the analytical result:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Afn(x,3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#,$*(%\"dG\"\"\"%#PiGF +%\"xGF+\"\"$F+*(F,F+,&*&)F-\"\"#F+)F*F3F+F+F+!\"\"F+F-F+F5#F5F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Afn(x,9);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&-%$sinG6#,$*(%\"dG\"\"\"%#PiGF+%\"xGF+\"\"*F+*(F, F+,&*&)F-\"\"#F+)F*F3F+F+F+!\"\"F+F-F+F5#F5F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Afn(x,19);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&-%$sinG6#,$*(%\"dG\"\"\"%#PiGF+%\"xGF+\"#>F+*(F,F+,&*&)F-\"\"#F+)F*F 3F+F+F+!\"\"F+F-F+F5#F5F3" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 87 "No w we set up the calculation for the Ronchi ruling (a set of equidistan t black lines)." }}{PARA 279 "" 0 "" {TEXT -1 10 "We choose " }{TEXT 282 1 "d" }{TEXT -1 52 " as the spacing of a line and a gap (the apert ure). " }}{PARA 280 "" 0 "" {TEXT -1 51 "First the single slit. The ap erture has a width of " }{TEXT 283 1 "d" }{TEXT -1 40 "/2 and is chose n to be symmetric around " }{TEXT 284 1 "y" }{TEXT -1 3 "=0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Bf:=x->evalc(int(exp(I*2*Pi*x*y),y=-d/4.. d/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BfGR6#%\"xG6\"6$%)operato rG%&arrowGF(-%&evalcG6#-%$intG6$-%$expG6#**^#\"\"#\"\"\"%#PiGF89$F8%\" yGF8/F;;,$%\"dG#!\"\"\"\"%,$F?#F8FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "plot(subs(d=10,Bf(x)^2),x=-Pi/2..Pi/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 685 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7]s7$$!3++ +lBjzq:!#<$\"3!>-fV;W=0)!#?7$$!3WNzQW&=B]\"F*$\"3CV!pG*HK$[%!#>7$$!3.4 2![ROFW\"F*$\"3?J%*Qg4G$)=F37$$!3?xt3O+tv8F*$\"3f\")*eopp3T(F-7$$!3xxI t:&z#38F*$\"3E1g+D'H,#eF37$$!3\")zz#fd\\6C\"F*$\"3s?>i\"*)>gQ#F37$$!3W :#)\\G:\"*y6F*$\"3it>w\">P7r(F-7$$!3,%4``jnW6\"F*$\"3e#z44n\\Nu(F37$$! 3)yYu)o'>y/\"F*$\"3]1yi0^R)H%F37$$!3EWfpKW&Q\")*!#=$\"37aO0I^&3u)F-7$$ !3(fe1Vw'\\I\"*FZ$\"31wi:`7-l6FZ7$$!333*H!e\\fG&)FZ$\"3upsB(\\[(*e(F37 $$!3g!oB6_&z*=)FZ$\"3&*fPlx%[LI\"F37$$!35`u@%3'*4&yFZ$\"32Y0`oN!=%))F- 7$$!3\"*3@F/b!3^(FZ$\"3m#3\"4Yjjv')F37$$!3%ewEV#\\hqrFZ$\"3&o#Q](3sB$= FZ7$$!3)pIZXf'p1qFZ$\"39Y+Nn-gj?FZ7$$!36[ywk#yF%oFZ$\"3\"**>F?^uX.#FZ7 $$!3E*Q))\\$*f)ymFZ$\"3*epR5.33u\"FZ7$$!3SI*3_gT\\^'FZ$\"3WLdT$*)z&\\7 FZ7$$!3(3;;5^Ms@'FZ$\"3VvYeNF\\NHF37$$!3B!RBoTF&>fFZ$\"3MK-\\!\\bcf%F- 7$$!3YX:)p)[_lbFZ$\"3^c0P%GB7I\"FZ7$$!3!=qRrNA:@&FZ$\"3^9Ga()GjLLFZ7$$ !3#z`d%o!z:1&FZ$\"3U_XB))f%z\"RFZ7$$!38v`xzdj6\\FZ$\"35]6'4;)e>TFZ7$$! 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The calculation is \+ straightforward and can be done easily by hand!" }}{PARA 282 "" 0 "" {TEXT -1 62 "A graph of the aperture function (showing the transmissiv ity):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Step:=(a,b)->Heaviside(b-y )*Heaviside(y-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%StepGR6$%\"aG% \"bG6\"6$%)operatorG%&arrowGF)*&-%*HeavisideG6#,&9%\"\"\"%\"yG!\"\"F3- F/6#,&F4F39$F5F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "p lot(subs(d=1,Step(-5*d/4,-3*d/4)+Step(-d/4,d/4)+Step(3*d/4,5*d/4)),y=- 7/4..7/4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-% 'CURVESG6$7_r7$$!3+++++++]eaH\"F*F+7$$!3GLek3#4vF \"F*F+7$$!3#******\\rM&o7F*F+7$$!3cmTN@-cf7F*F+7$$!3))\\7`uH2b7F*F+7$$ !3?L$3xs&e]7F*F+7$$!3wuoH/@M[7F*$\"\"\"F,7$$!3a;a)3[)4Y7F*FW7$$!3IeRZd [&QC\"F*FW7$$!3&)*\\iSB6;C\"F*FW7$$!3ALeR(\\&[/7F*FW7$$!3em\"H2wft;\"F *FW7$$!35+D\"GTYL4\"F*FW7$$!37LL3(e9s,\"F*FW7$$!3-mmTNjd,&*!#=FW7$$!3Y )***\\iQnY()F[pFW7$$!31*****\\xswO)F[pFW7$$!3m****\\(or'))zF[pFW7$$!3d 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\"3!4+vo%zsnKF[pF+7$$\"3#>+](=%[Vj$F[pF+7$$\"3!G+vVt'zVVF[pF+7$$\"36** *\\78=:8&F[pF+7$$\"3@kmmT3KReF[pF+7$$\"3K/+]780&f'F[pF+7$$\"3-oTgxLXPp F[pF+7$$\"3uJ$3FWb)zsF[pF+7$$\"3M!HdqTRMP(F[pF+7$$\"3/]iS\"RBqY(F[pF+7 $$\"3o*[$*\\Q>/\\(F[pF+7$$\"3XI2ey`\"Q^(F[pFW7$$\"35qz;s8@PvF[pFW7$$\" 3w4_vltggvF[pFW7$$\"30*oHHN*R2wF[pFW7$$\"3MoT5S8>awF[pFW7$$\"3m'3-))Gf 8%yF[pFW7$$\"3'\\++vBF&G!)F[pFW7$$\"3ON3FW\"[2Q)F[pFW7$$\"3xl;/^!pHt)F [pFW7$$\"3?,](=s8$p%*F[pFW7$$\"3umm\"H_A*=5F*FW7$$\"3%)*\\Pfe!H%4\"F*F W7$$\"3#RLL$))*yo;\"F*FW7$$\"3sLe9^]*R?\"F*FW7$$\"3^L$eR666C\"F*FW7$$ \"3w!R]\"f;TV7F*FW7$$\"3+[CM/ArX7F*FW7$$\"3E0X`\\F,[7F*FW7$$\"3sils%H8 .D\"F*F+7$$\"3Wx16&Q9\\D\"F*F+7$$\"3#>z%\\va^f7F*F+7$$\"39@IEcwro7F*F+ 7$$\"3M]7.P)>zF\"F*F+7$$\"3u3xc)>CjH\"F*F+7$$\"3b[8F*F+7$$\"3Y++]Z`P#Q\"F*F+7$$\"3#pm\"z*>1*f9F*F+7$$\"3[LLL=2DH :F*F+7$$\"33+vVQk=.;F*F+7$$\"3I+DccB&Rn\"F*F+7$$\"3+++++++] " 0 "" {MPLTEXT 1 0 107 "Bf1:=x->evalc(int((Step(-5*d/4,-3* d/4)+Step(-d/4,d/4)+Step(3*d/4,5*d/4))*exp(I*2*Pi*x*y),y=-5*d/4..5*d/4 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Bf1GR6#%\"xG6\"6$%)operatorG %&arrowGF(-%&evalcG6#-%$intG6$*&,(-%%StepG6$,$%\"dG#!\"&\"\"%,$F8#!\"$ F;\"\"\"-F56$,$F8#!\"\"F;,$F8#F?F;F?-F56$,$F8#\"\"$F;,$F8#\"\"&F;F?F?- %$expG6#**^#\"\"#F?%#PiGF?9$F?%\"yGF?F?/FW;F7FLF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 81 "This integral is not worked out by Maple unless the following assumption is made:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "assume(d>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Bf1(x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,(-%$sinG6#,$*(% #PiG\"\"\"%\"xGF,%#d|irGF,#\"\"$\"\"#!\"\"-F'6#,$F*#F,F1F,-F'6#,$F*#\" \"&F1F,F,*&F+F,F-F,F2F,*&,&F&#F2F1*&F6F,F7F,F,F,*&F+F,F-F,F2F,*&,&F&F6 *&#F,F1F,F7F,F2F,*&F+F,F-F,F2F,*&^#F,F,,&*&,&-%$cosGF(F?*&F6F,-FMF8F,F ,F,*&F+F,F-F,F2F,*&,&FOF?*&F6F,FLF,F,F,*&F+F,F-F,F2F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 288 78 "We can also define a new function that p erforms the sum of required integrals." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "Bf1:=x->evalc(int(exp(I*2*Pi*x*y),y=-d/4..d/4)+int(exp(I*2*Pi *x*y),y=-5*d/4..-3*d/4)+int(exp(I*2*Pi*x*y),y=3*d/4..5*d/4));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Bf1GR6#%\"xG6\"6$%)operatorG%&arrow GF(-%&evalcG6#,(-%$intG6$-%$expG6#**^#\"\"#\"\"\"%#PiGF99$F9%\"yGF9/F< ;,$%\"dG#!\"\"\"\"%,$F@#F9FCF9-F16$F3/F<;,$F@#!\"&FC,$F@#!\"$FCF9-F16$ F3/F<;,$F@#\"\"$FC,$F@#\"\"&FCF9F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Bf1(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$sinG 6#,$*(%#PiG\"\"\"%\"xGF+%#d|irGF+#F+\"\"#F+*&F*F+F,F+!\"\"F+*&*&F/F+,& -F&6#,$F)#\"\"$F/#F1F/*&F.F+-F&6#,$F)#\"\"&F/F+F+F+F+*&F*F+F,F+F1F+*&^ #F+F+,&*&,&-%$cosGF6F:*&F.F+-FHF=F+F+F+*&F*F+F,F+F1F+*&,&FJF:*&F.F+FGF +F+F+*&F*F+F,F+F1F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " res:=evalc(subs(d=10,Bf1(x)^2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% $resG,.*&*$)-%$sinG6#,$*&%#PiG\"\"\"%\"xGF/\"\"&\"\"#F/F/*&)F.F2F/)F0F 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We investigat e whether they are possibly caused by an asymmetry of the aperture fun ction, i.e, that the spacing of the gaps (perfect transmission) is not identical to the thickness of the black lines." }}{PARA 288 "" 0 "" {TEXT -1 36 "We introduce slightly thicker lines:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 210 "Cfn:=(x,n)->evalc(int(exp(I*2*Pi*x*y),y=-d/4+d/128 ..d/4-d/128)+sum(int(exp(I*2*Pi*x*y),y=-(4*i+1)*d/4+d/128..-(4*i-1)*d/ 4-d/128),i=1..n)+sum(int(exp(I*2*Pi*x*y),y=(4*i-1)*d/4+d/128..(4*i+1)* d/4-d/128),i=1..n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$CfnGR6$%\"x G%\"nG6\"6$%)operatorG%&arrowGF)-%&evalcG6#,(-%$intG6$-%$expG6#**^#\" \"#\"\"\"%#PiGF:9$F:%\"yGF:/F=;,$%\"dG#!#J\"$G\",$FA#\"#JFDF:-%$sumG6$ -F26$F4/F=;,&*&,&%\"iG\"\"%F:F:F:FAF:#!\"\"FS*&#F:FDF:FAF:F:,&*&,&FRFS F:FUF:FAF:FT*&#F:FDF:FAF:FU/FR;F:9%F:-FI6$-F26$F4/F=;,&FY#F:FS*&FWF:FA F:F:,&FPFao*&#F:FDF:FAF:FUFgnF:F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Cfn(x,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&-%$si nG6#,$*(%#PiG\"\"\"%\"xGF+%#d|irGF+#\"#J\"#kF+*&F*F+F,F+!\"\"F+*&*&\" \"#F+,&-F&6#,$F)#\"#(*F0#F2F5*&#F+F5F+-F&6#,$F)#\"$f\"F0F+F+F+F+*&F*F+ F,F+F2F+*&^#F+F+,&*&,&-%$cosGF8F<*&F>F+-FKF@F+F+F+*&F*F+F,F+F2F+*&,&FM F<*&F>F+FJF+F+F+*&F*F+F,F+F2F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(log10(evalc(subs(d=10,Cfn(x,2)^2))),x=-0.8..0.8, -3..3,numpoints=500);" }}{PARA 13 "" 1 "" {GLPLOT2D 693 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7fcp7$$!3U+++++++!)!#=$!3wK&R`*QgoBF*7$$!3; `h2QiA$)zF*$!3Ju#3*4aSI>F*7$$!3y/B:wCXmzF*$!3%=Z(f+8\"Gs\"F*7$$!3%H=\" f:x&=&zF*$!3X4Z%oPaPt\"F*7$$!32h+.bHEPzF*$!3!y!z!=eP:$>F*7$$!3\\@2O?' \\3#zF*$!35*[3l.;]R#F*7$$!3,$Q\"p&GOW!zF*$!3C]Leg[DXJF*7$$!3Yg(z)>?Rry F*$!3g@u-:EfwdF*7$$!3!ed(y8%[\\&yF*$!3EUc(z&z>)*yF*7$$!3:\"R&p2[]QyF*$ !3+HW#*>ss&4\"!#<7$$!3hR)>*z%fK#yF*$!3Gq)3'QM*>a\"FZ7$$!3=*GW@:9!3yF*$ !3o=5y2)[q[#FZ7$$!3hA3TX4/1yF*$!38W1S=b#ft#FZ7$$!3/ctnQx1/yF*$!3/:`YZn *H3$FZ7$$!3?A1JN63.yF*$!3Kx%Hn,RhK$FZ7$$!3O))Q%>`%4-yF*$!3[nmm`QHjOFZ7 $$!3jbrdGz5,yF*$!3QjU?,8S=UFZ7$$!3z@/@D87+yF*$!3gcD]N'H:9'FZ7$$!3%zoV= sM\"*z(F*$!3'[1v@FxrV%FZ7$$!3?bpZ=\"[\")z(F*$!3%H`*Gb'p$yPFZ7$$!3O@-6: :;(z(F*$!3&>d-Bm!e4MFZ7$$!3_([V<\"\\<'z(F*$!33#zf*z$zE:$FZ7$$!3'4-5]q, Uz(F*$!3w]')yr9$fz#FZ7$$!3QalF)\\GAz(F*$!310$[J<(HYDFZ7$$!3)eoU8nNVy(F *$!31'R#R/5ne>FZ7$$!3]=)3W%GWwxF*$!3(*=kF[0()G;FZ7$$!3_rby_z`,\"FZ7$$!3^7k?j&=vs(F*$!3!*RDegv[F*)F*7$$!3gasit`C6xF*$!3_$4oB=D DP)F*7$$!3#Hard)e(Gq(F*$!38M3*3k&35$)F*7$$!3MKe\"zR1Xp(F*$!3iq&\\ZXCFZ7$$!3or`o#QI'\\vF*$!3O;^Q\\qeX6FZ7$$!3y(R)>*fX]`(F*$!3D c5f%>(*)>)*F*7$$!3xA9r:3Y?vF*$!3^#pT)>*G0\"))F*7$$!3U'=$fc%*y6vF*$!3e' )y)Ri&*H #*)F*7$$!3U*p\\[E#RcuF*$!3]v_o#[Dy+\"FZ7$$!3I\">&fF1B?FZ7$$!3\"*pV=s`32uF*$!33Wy%)RoNQEF Z7$$!3C0uF*$!3CsWO\"*)zY!HFZ7$$!3ck;$e`.LS(F*$!3%4%HYC\"4ZH$FZ7 $$!3*=Jbwh79S(F*$!3AUf]u\\vHSFZ7$$!3Af*y%*p@&*R(F*$!3A(QG:%*ox'\\FZ7$$ !3)Gy!RSid)R(F*$!3'oxA;Zm!>SFZ7$$!3a1EI\"yIwR(F*$!3+XcMYfbvNFZ7$$!3?IW @A`o'R(F*$!3s^O$*Htw#G$FZ7$$!3'QDEJ')RdR(F*$!3Y&ps,&QujIFZ7$$!3=,*\\\\ %*[QR(F*$!3LaWm^buUFFZ7$$!3SZNxE!e>R(F*$!3xy/mrD?3DFZ7$$!3%eg-8$4OvtF* $!337SD2(*\\I:FZ7$$!3;j;$e$QwetF*$!3m%Ha#)))fh4\"FZ7$$!3&R9sgZfHM(F*$! 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The laser spot covers a finite area on the grating, whic h for a fine grating includes a large number of 'lines', but for a Ron chi ruling can represent a finite number." }}{PARA 293 "" 0 "" {TEXT -1 1 " " }}{PARA 290 "" 0 "" {TEXT 298 11 "Exercise 2:" }}{PARA 291 " " 0 "" {TEXT -1 116 "Explore diffraction patterns for rulings with dif ferent aperture functions ('duty cycles' of the square-wave train)." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "55" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }