DiffractionNew1.html

Plotting Diffraction Patterns

> restart: with(plots):

Fresnel-Kirchoff Formula - Fresnel Approximation

> F:=(ax,cx,ay,cy,k,d)->-I/lambda*exp(I*k*d)*exp(I*(k/2/d*(x^2+y^2)))*int(int(exp(I*k/2/d*(u^2+v^2))*exp(I*k/d*(x*u+y*v)),u=ax..cx),v = ay..cy);

proc (ax, cx, ay, cy, k, d) options operator, arrow; `+`(`-`(`/`(`*`(`+`(I), `*`(exp(`*`(I, `*`(k, `*`(d)))), `*`(exp(`/`(`*`(`*`(`/`(1, 2), `*`(I)), `*`(k, `*`(`+`(`*`(`^`(x, 2)), `*`(`^`(y, 2)))))),... (1)

Fresnel-Kirchoff Formula - Fraunhofer Approximation

> G:=(ax,cx,ay,cy,k,d)->-I/lambda*exp(I*k*d)*exp(I*(k/2/d*(x^2+y^2)))*int(int(exp(I*k/d*(x*u+y*v)),u=ax..cx),v = ay..cy);

Narrow Slit Intensities

Fresnel Approximation

> lambda:=5: M:=10: L:=1: EE:=F(-L/2,L/2,-M/2,M/2,2*Pi/lambda,10):

> Intensity1:=abs(EE):

> plot(subs(y=0.001,Intensity1),x=-50*lambda..50*lambda);

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Fraunhofer Approximation

> EE:=G(-L/2,L/2,-M/2,M/2,2*Pi/lambda,10):

> Intensity2:=abs(EE):

> plot(subs(y=0.001,Intensity2),x=-50*lambda..50*lambda);

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Comparison of Approximations

> plot({subs(y=0.001,Intensity1),subs(y=0.001,Intensity2)},x=-50*lambda..50*lambda,color=[black,blue]);

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Fraunhofer Condition: Need d >> k/2 (aperture area). d above is 10. But

> 2*Pi/lambda/2*L*M;

(3)

Special Functions

Sinc Function - Single Slit

> sinc:=x->sin(x)/x:

> plot(sinc(x)^2,x=-10..10);

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Bessel Functions

> plot(BesselJ(1,x),x=0..30);

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Zeros of Bessel Functions

> BesselJZeros(0.,1..5);

(4)

> BesselJZeros(1.,1..5);

(5)

Jinc Function

> Jinc:=x->(2*BesselJ(1,x)/x): plot(Jinc(x),x=0..30);

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Fresnel Integrals and the Cornu Spiral

> plot([FresnelC(x),FresnelS(x),x=-10..10]);

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Simple Diffraction Patterns

Narrow Slit:

> densityplot(sqrt(sqrt(sinc(x)^2)),x=-20..20,y=-5..5,axes=frame,style=PATCHNOGRID,grid=[50,50],color=red,scaling=constrained);

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Rectangular Aperture:

> densityplot(sqrt(sqrt(sinc(x)^2*sinc(2*y)^2)),x=-20..20,y=-20..20,axes=frame,style=PATCHNOGRID,grid=[100,100],color=red,scaling=constrained);

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Circular Aperture:

> densityplot(sqrt(sqrt(Jinc((sqrt(x^2+y^2)))^2)),x=-10..10,y=-10..10,axes=frame,style=PATCHNOGRID,grid=[70,70],color=red,scaling=constrained);

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Resolving Peaks

> c:=3.8: plot({Jinc(x)^2+Jinc(x-c)^2,Jinc(x)^2,Jinc(x-c)^2},x=-30..30,color=[red,blue,black]);

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Plot

> densityplot(sqrt(sqrt(Jinc(sqrt((x+c/2)^2+y^2))^2+Jinc(sqrt((x-c/2)^2+y^2))^2)),x=-10..10,y=-10..10,axes=frame,style=PATCHNOGRID,grid=[70,70],color=red,scaling=constrained);

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Plot

Double Slit - width b, separation a with a>b.

> beta:=a/2*sin(x): alpha:=b/2*sin(x):

> Intensity:=(a,b)->4*cos(a/2*sin(x))^2*sinc(b/2*sin(x))^2:

> plot(Intensity(20,5),x=-5..5);

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> densityplot(sqrt(sqrt(Intensity(20,5))),x=-5..5,y=-1..1,axes=frame,style=PATCHNOGRID,grid=[50,50],color=red,scaling=constrained);

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> lambda:=5: a:=20: b:=19: L:=1: EE:=G(-L/2,L/2,(a-b)/2,(a+b)/2,2*Pi/lambda,10)+G(-L/2,L/2,-(a+b)/2,-(a-b)/2,2*Pi/lambda,10):

> Intensity:=abs(EE):

> plot(subs(x=0.001,Intensity),y=-5..5,numpoints=200);

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> densityplot(sqrt(sqrt(subs(x=0.001,Intensity))),y=-5..5,x=-1..1,axes=frame,style=PATCHNOGRID,grid=[50,50],color=red);

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N Slits

> beta:=a/2*sin(x): alpha:=b/2*sin(x):

> Intensity:=(a,b)->sin(a*N/2*sin(x))^2*sinc(b/2*sin(x))^2/sin(a/2*sin(x))^2;

proc (a, b) options operator, arrow; `/`(`*`(`^`(sin(`+`(`*`(`/`(1, 2), `*`(a, `*`(N, `*`(sin(x))))))), 2), `*`(`^`(sinc(`+`(`*`(`/`(1, 2), `*`(b, `*`(sin(x)))))), 2))), `*`(`^`(sin(`+`(`*`(`/`(1, 2),... (6)

> N:=2:plot(Intensity(20,19),x=-5..5,numpoints=200);

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> densityplot(sqrt(sqrt(Intensity(20,19))),x=-5..5,y=-1..1,axes=frame,style=PATCHNOGRID,grid=[50,50],color=red);

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> N:=4:plot(Intensity(20,19),x=-5..5,numpoints=200);

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