FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
1
Fraunhofer Diffraction Pattern of Multiple Slits
Experimental Diagram
y0
y1
P1
x1
Po
r01
θ
Z
Aperture (N slits)
d = slit spacing (period)
w = slit width
φ
x0
FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
2
Intensity Distribution
Iq
=
If N
b
:=
Io
=
p
Sin@ D2
b
þþþþþþþþþþþþþþþþ
þþþþþ
b2
Iq
2, then
w Sin@qD
þþþþþþþþþþþþþþþþ
þþþþþþþþ
þþþþ
Sin@N D2
N 2 Sin@gD2
g
þþþþþþþþþþþþþþþþ
þþþþþþþþþþþ
;
l
=
g
Io
:=
Sin@ D2
b
þþþþþþþþþþþþþþþþ
þþþþþ
b2
p
d Sin@qD
Cos@gD2
þþþþþþþþþþþþþþþþ
þþþþþþþþþþþþ
;
l
amplitude@q_, n_, w_, d_ D := ao
p w Sin@qD
Sin@ þþþþþþþþ
þþþþþþþþþþþþ D
l
þþþþþþþþþþþþþþþþ
þþþþþþþþþþþþþþþþ
þþþþþþ
p w Sin@qD
l
þþþþþþþþþþþþþþþþ
þþþþ
n p d Sin@qD
Sin@ þþþþþþþþþþþþþþþþ
þþþþþþþ D
l
þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ
þþþþþþþþþþ
p d Sin@qD
n Sin@ þþþþþþþþ
þþþþþþþþþþþþ D
l
;
amplitude@0, n_, w_, d_ D := ao ; ao = 1;
intensity@q_, n_, w_, d_ D := amplitude @q, n, w, dD2 ;
Double Slit
Sin@bD2
PlotA9 þþþþþþþþþþþþþþþþ
þþþþþþ
b
2
ƒ. w “ 100 l, intensity@q, 2, 100 l, 275 lD=, 8q, - .02, .02<, PlotLabel ->
StyleForm @"Single and Double Slit Irradiance Distribution", FontSize “ 16D,
PlotRange “ All, AxesOrigin “ 80, 0<, Frame “ True, GridLines “ Automatic,
DefaultFont “ 8"Times-Roman", 10<, PlotStyle “
88RGBColor @0, 1, 0D, Thickness @.008D<, 8RGBColor @1, 0, 0D, Thickness @.008D<<,
PlotLegend “ 8"Single Slit", "Double Slit"<,
LegendPosition “ 8.3, - .2<, Background “ WhiteE;
Single and Double Slit Irradiance Distribution
1
0.8
Single Slit
0.6
Double Slit
0.4
0.2
0
- 0.02
- 0.01
0
0.01
0.02
FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
3
Multiple Slits
é slit width = 4 l, slit spacing = 100 l
n = 82, 3, 4, 5, 10<;
Do@Plot@intensity @q, n@@pDD, 4 l, 100 lD, 8q, - .02, .02<,
PlotLabel -> StyleForm @8n@@pDD, "Slits"<, FontSize “ 10D,
PlotRange “ All, AxesOrigin “ 80, 0<, Frame “ True,
GridLines “ Automatic, DefaultFont “ 8"Times-Roman", 10<,
PlotStyle “ 88RGBColor @1, 0, 0D, Thickness @.008D<<,
Background “ RGBColor @1, 1, 1DD, 8p, Length@nD<D;
82,
Slits<
1
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
0
83, Slits<
0.01
0.02
0
0.01
0.02
1
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
4
84,
Slits<
0
Slits<
0.01
0.02
85,
0
Slits<
0.01
0.02
810,
0
0.01
0.02
1
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
1
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
1
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
5
Plot3D@amplitude @q, 5, 4 l, 100 lD, 8q, - .02, .02<,
8f, - .02, .02<, PlotPoints -> 50, AxesLabel -> 8"q", "f", "amplitude"<,
DefaultFont “ 8"Times-Roman", 10<, PlotRange -> All, Background “ WhiteD;
1
0.02
amplitude0.5
0.01
0
- 0.02
0
- 0.01
q
f
- 0.01
0
0.01
0.02
- 0.02
Plot3D@intensity @q, 5, 4 l, 100 lD, 8q, - .02, .02<,
8f, - .02, .02<, PlotPoints -> 50, AxesLabel -> 8"q", "f", "intensity"<,
DefaultFont “ 8"Times-Roman", 10<, PlotRange -> All, Background “ WhiteD;
1
0.75
intensity 0.5
0.25
0
-0.02
0.02
0.01
0
- 0.01
q
- 0.01
0
0.01
0.02
- 0.02
f
FraunhoferDiffractionofMultipleSlits.nb
Optics 505 - James C. Wyant
6
é Varying the ratio of slit width (w) to slit spacing (d)
Plot@8intensity @q, n@@1DD, 4 l, 100 lD, intensity@q, n@@1DD, 40 l, 100 lD<,
8q, - .02, .02<, PlotLabel -> StyleForm @8n@@1DD, "Slits"<, FontSize “ 10D,
PlotRange “ All, AxesOrigin “ 80, 0<, Frame “ True, GridLines “ Automatic,
DefaultFont “ 8"Times-Roman", 10<, PlotStyle “ 88RGBColor@1, 0, 0D, [email protected]<,
8RGBColor @0, 1, 0D, Thickness @.008D, Dashing @8.04, .04<D<<,
Background “ RGBColor@1, 1, 1D, PlotLegend “ 8"w = 4 l", "w = 40 l"<,
LegendPosition “ 8.9, - .3<, LegendLabel -> "d= 100 l"D;
82,
Slits<
1
d= 100
l
0.8
0.6
0.4
0.2
0
- 0.02
- 0.01
0
0.01
0.02
w
=
4
w
=
40
l
l