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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 "" 0 278 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 "" 0 279 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 "" 0 280 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 "" 0 281 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 "" 0 282 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 " " 0 283 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 "" 0 284 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 "" 0 285 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 "" 0 286 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 "" 0 287 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 "" 0 288 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 "" 0 289 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 "" 0 290 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;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "d:=100 /4;" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 53 "The amplitude as a funct ion 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*sin(theta));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(Am(theta)^2,theta=-Pi/50..Pi/50,numpoints=500 );" }}}{EXCHG {PARA 265 "" 0 "" {TEXT 261 11 "Exercise 1:" }}{PARA 266 "" 0 "" {TEXT -1 39 "Vary the length parameters (wavelength " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 21 " and slit separati on " }{TEXT 260 1 "d" }{TEXT -1 66 ") and observe the change in the in tesity pattern as a function of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG " }{TEXT -1 30 " which is measured in radians." }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 54 "We can look at t he intensity pattern at larger angles:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(Am(theta)^2,theta=Pi/4-Pi/50..Pi/4+Pi/50,numpoints=500); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 87 "The nonlinearity of the expr ession in theta becomes more apparent at even larger theta:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot(Am(theta)^2,theta=3*Pi/8-Pi/25..3*Pi /8+Pi/25,numpoints=500);" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 262 39 "Di ffraction gratings and Ronchi rulings" }}{PARA 269 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 160 "To observe diffraction patterns f or gratings and Ronchi rulings we calculate the Fourier transform of t he respective aperture functions. The distance parameter " }{TEXT 263 1 "d" }{TEXT -1 289 " plays the role of the spacing between the slits. For a diffraction grating the aperture function is smooth: a cos-squa red behaviour of the transmissivity as a function of separation y acro ss the aperture is produced (using holography in modern times) and the periodicity is controlled by " }{TEXT 264 1 "d" }{TEXT -1 102 ". Foll owing eqns. (9.10-9.11) from the reference we calculate the amplitude \+ first for a 'single' 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 per fectly transmitting part up to the perfectly blocking parts on both si des (the zeroes of the cosine) at |y|=d/2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d:='d';" }}}{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));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Af(x);" }}}{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 compared 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);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 271 70 "It is useful to look at a logarithmic representation o f the amplitude:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(subs(d=10, log10(Af(x)^2)),x=-Pi/2..Pi/2,numpoints=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 65 "A log-plot can also be produced directly using the pl ots-package:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "logplot(subs(d=10,Af(x)^2),x =-Pi/2..Pi/2,numpoints=500);" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 30 "Nothing dramatic happened at |" }{TEXT 273 1 "x" }{TEXT -1 4 "|=1/" } {TEXT 274 1 "d" }{TEXT -1 88 ". Now we add more slits gradually by ext ending the range of integration. First just one:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Af1:=x->evalc(int(cos(Pi*y/d)^2*exp(I*2*Pi*x*y),y=-3* d/2..3*d/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(subs( d=10,Af1(x)^2),x=-Pi/8..Pi/8,numpoints=500);" }}}{EXCHG {PARA 275 "" 0 "" {TEXT -1 47 "Observe how a strong pair of peaks appears at |" } {TEXT 275 1 "x" }{TEXT -1 4 "|=1/" }{TEXT 276 1 "d" }{TEXT -1 26 ". Th is is shown below for " }{TEXT 277 1 "d" }{TEXT -1 4 "=20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(subs(d=20,log10(Af1(x)^2)),x=-0.3..0.3 ,-3..3,numpoints=1000);" }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 58 "We a re ready to calculate 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));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 95 "Observe that the intensi ty 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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(subs(d=10,Afn(x,19)^2),x=-Pi/8..Pi/8,numpoints=500);" }}} {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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Afn(x,9); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Afn(x,19);" }}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 87 "Now we set up the calculation for the R onchi ruling (a set of equidistant black lines)." }}{PARA 279 "" 0 "" {TEXT -1 10 "We choose " }{TEXT 282 1 "d" }{TEXT -1 52 " as the spacin g of a line and a gap (the aperture). " }}{PARA 280 "" 0 "" {TEXT -1 51 "First the single slit. The aperture has a width of " }{TEXT 283 1 "d" }{TEXT -1 40 "/2 and is chosen 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));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "plot(subs(d=10,Bf(x)^2),x=-Pi/2..Pi/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 64 "We found the intensity pattern for a sin gle slit (cf. page 134):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Bf(x);" }}}{EXCHG {PARA 281 "" 0 "" {TEXT -1 133 "To combine more steps we sim ply add more sections of the integral. The calculation is straightforw ard and can be done easily by hand!" }}{PARA 282 "" 0 "" {TEXT -1 62 " A graph of the aperture function (showing the transmissivity):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Step:=(a,b)->Heaviside(b-y)*Heavisi de(y-a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot(subs(d=1,S tep(-5*d/4,-3*d/4)+Step(-d/4,d/4)+Step(3*d/4,5*d/4)),y=-7/4..7/4);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 286 58 "A straightforward implementation of the required integral:" }}{PARA 0 "> " 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))*e xp(I*2*Pi*x*y),y=-5*d/4..5*d/4));" }}}{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 "assu me(d>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Bf1(x);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 288 78 "We can also define a new function that performs 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(e xp(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)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Bf1(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "res:=evalc(subs(d=10,Bf1(x)^2));" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(log10(res),x=-Pi/4..P i/4,numpoints=500);" }}}{EXCHG {PARA 283 "" 0 "" {TEXT -1 50 "Observe \+ the dominant peaks: strong side peaks at |" }{TEXT 291 1 "x" }{TEXT -1 4 "|=1/" }{TEXT 292 1 "d" }{TEXT -1 26 ", further strong ones at | " }{TEXT 293 1 "x" }{TEXT -1 4 "|=3/" }{TEXT 294 1 "d" }{TEXT -1 3 ",5 /" }{TEXT 295 1 "d" }{TEXT -1 145 ", etc. These are high Fourier comp onents that result from the analysis of a step function. The change of the dominant side peak location from 2/" }{TEXT 289 1 "d" }{TEXT -1 6 " to 1/" }{TEXT 290 1 "d" }{TEXT -1 99 " compared to the cosine-squa red transmission profile of the aperture is just a result of defining: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "Bfn:=(x,n)->evalc(int(exp(I*2 *Pi*x*y),y=-d/4..d/4)+sum(int(exp(I*2*Pi*x*y),y=-(4*i+1)*d/4..-(4*i-1) *d/4),i=1..n)+sum(int(exp(I*2*Pi*x*y),y=(4*i-1)*d/4..(4*i+1)*d/4),i=1. .n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Bfn(x,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(Bfn(x,1)-Bf1(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Bfn(x,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#Bfn(x,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#Bfn(x,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "res:=evalc(subs(d=10,Bfn(x,2)^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(log10(res),x=-0.6..0.6,-3..3,numpoin ts=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "#n=3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(log10(evalc(subs(d=10,Bfn(x,3)^2))),x=-0.6. .0.6,-3..3,numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot(log10(evalc(subs(d=10,Bfn(x,5)^2))),x=0..1.6,-3..4,numpoints= 500);" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 297 11 "Exercise 1:" }}{PARA 285 "" 0 "" {TEXT -1 0 "" }}{PARA 286 "" 0 "" {TEXT -1 44 "Explore the pattern for different values of " }{TEXT 296 1 "d" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 287 "" 0 "" {TEXT -1 306 "The observed pattern for a Ronchi ruling shows spots of weak inte nsity interspersed in the above pattern. We investigate whether they a re possibly caused by an asymmetry of the aperture function, i.e, that the spacing of the gaps (perfect transmission) is not identical to th e thickness of the black lines." }}{PARA 288 "" 0 "" {TEXT -1 36 "We i ntroduce 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)+su m(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));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Cfn(x,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(log10(evalc(subs(d=10,C fn(x,2)^2))),x=-0.8..0.8,-3..3,numpoints=500);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "plot(log10(evalc(subs(d=10,Cfn(x,5)^2))),x=0.. 1,-2..4,numpoints=500);" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 207 "Thu s, it is evident that small imperfections in the pattern (thickness of lines vs. gaps, but perfectly reproduced over 11 rulings in the above calculation) have a dramatic effect on the diffraction pattern." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 292 "" 0 "" {TEXT -1 341 "One sh ould keep in mind that illumination of Ronchi rulings and diffraction \+ gratings by laser beams represents a the use of a grating with a finit e number of aperture slits. The laser spot covers a finite area on the grating, which for a fine grating includes a large number of 'lines', but for a Ronchi 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 ruli ngs with different aperture functions ('duty cycles' of the square-wav e 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 }