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|Fig. 1 Three ray paths through a sphere which result in a scattering angle of 141° for wavelength λ = 0.65 µm and a refractive index of 1.33257|
Fig. 1 shows that there are two paths of order p = 2 (ray A and ray B) and one path of order p = 0 (ray C) which result in scattering at angle of 141°. Interference between ray A and ray B causes the supernumerary arcs of the primary rainbow because the two rays have almost identical amplitudes:
Fig. 2 Primary rainbow: Debye series calculation of scattering by a water drop of radius r = 100 µm for wavelength λ = 0.65 µm (perpendicular polarisation)
Fig. 2 shows that the p = 2 rays are responsible for the smooth maxima and minima of the primary rainbow and its supernumeraries. However, Mie calculations also show a high frequency ripple structure. The red curve in Fig. 10 represents the vector sum of the p = 0 and p = 2 contributions, indicating that the ripples are caused by interference between the p = 0 rays (external reflections from the surface of the sphere) and the p = 2 rays.
Note that the intensity at the rainbow angle predicted by geometrical optics (in this case 137.9°) is about 50% of the maximum intensity (at about 138.8°) predicted by Mie and Debye methods.
MiePlot offers the option of using the Debye series.
Page updated on 13 October 2002
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