(*^
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Rayleigh_intensity.m
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The intensity of Rayleigh scattering as a function of scattering angle and polarization
:[font = text; inactive; preserveAspect]
In order to match a clear blue sky, which has a maximum polarization of around 80%, these formulae include an additional isotropic, unpolarized fraction, a.
:[font = input; preserveAspect; startGroup]
a=0.0625
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0.0625
;[o]
0.0625
:[font = text; inactive; preserveAspect]
Number of plot points for 3D plots
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pp=10
:[font = output; output; inactive; preserveAspect; endGroup]
10
;[o]
10
:[font = subsection; inactive; preserveAspect; startGroup]
Definition of the angle of scattering, b, as a function of the 'altitude' angle in the 3D plot
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b[u_]:= u-Pi/2
:[font = subsection; inactive; preserveAspect; startGroup]
Scattered intensity
:[font = text; inactive; preserveAspect]
For the E-vector in the plane of the source, scatterer and observer,
:[font = input; preserveAspect]
scpar[u_]:= 0.5 Cos[b[u]]^2
:[font = input; preserveAspect; startGroup]
Plot[scpar[u], {u,-Pi/2,Pi/2}]
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Graphics["<<>>"]
;[o]
-Graphics-
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3D plots of the scattered intensity
:[font = text; inactive; preserveAspect]
For parallel polarization
:[font = input; preserveAspect; startGroup]
ParametricPlot3D[{
(a+scpar[u]) Sin[u], 
(a+scpar[u]) Cos[t] Cos[u], 
(a+scpar[u]) Sin[t] Cos[u]},
{t,-Pi/2,Pi/2}, {u,-Pi/2, Pi/2}, 
PlotPoints->pp,
Axes -> True, 
AxesStyle -> Thickness[0.01]]
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Graphics3D["<<>>"]
;[o]
-Graphics3D-
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For perpendicular polarization
:[font = input; preserveAspect; startGroup]
ParametricPlot3D[{
(a+1)/2 Sin[u], 
(a+1)/2 Cos[t] Cos[u], 
(a+1)/2 Sin[t] Cos[u]},
{t,-Pi/2,Pi/2}, {u,-Pi/2, Pi/2}, 
PlotPoints->pp,
Axes -> True, 
AxesStyle -> Thickness[0.01]]
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Graphics3D["<<>>"]
;[o]
-Graphics3D-
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The sum
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ParametricPlot3D[{
(0.5+a+scpar[u]) Sin[u], 
(0.5+a+scpar[u]) Cos[t] Cos[u], 
(0.5+a+scpar[u]) Sin[t] Cos[u]},
{t,-Pi/2,Pi/2}, {u,-Pi/2, Pi/2}, 
PlotPoints->pp,
Axes -> True, 
AxesStyle -> Thickness[0.01]]
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Graphics3D["<<>>"]
;[o]
-Graphics3D-
:[font = subsection; inactive; preserveAspect; startGroup]
Fractional polarization
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pray[u_]:= (0.5 - scpar[u])/(0.5 + 2a + scpar[u])
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Plot[pray[u], {u,-Pi/2, Pi/2}]
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Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
ParametricPlot3D[{
pray[u] Sin[u], 
pray[u] Cos[t] Cos[u], 
pray[u] Sin[t] Cos[u]},
{t,-Pi/2,Pi/2}, {u,-Pi/2, Pi/2}, 
PlotPoints->pp,
Axes -> True, 
AxesStyle -> Thickness[0.01]]
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Graphics3D["<<>>"]
;[o]
-Graphics3D-
^*)