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{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 }