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|Fig. 1 Two rays emerging at a scattering angle of 138.9° (near the primary rainbow at 137.9°) for refractive index n = 1.33257,
As rainbows are caused by minima or maxima in the variation of scattering angle as a function of angle of incidence, there are often circumstances in which 2 light rays emerge from the sphere at the same scattering angle, as shown in Fig. 1. For example, the red and blue lines in Fig. 1 correspond to rays with angles of incidence 52.3° and 66°, both of which generate rays emerging at 138.9° - which is just 1° away from the primary rainbow at 137.9°.
In the era of Descartes and Newton, the intensities of these two rays
would simply be added together - as was done to generate the graphs on
the previous page using the ray tracing method. In 1801, Young endorsed
the wave theory of light and, crucially, suggested that the supernumerary
arcs near the primary rainbow were the result of interference between waves
which had followed separate paths through the raindrops. As
the lengths of the paths taken by the two rays in Fig. 1 are different,
the scattered intensity depends on the difference in phase of the two rays.
If the phase difference is 0° or a multiple of 360°, there will
be constructive interference (i.e. the resultant is the sum of the amplitudes
of the two rays). If the phase difference is an odd multiple of 180°,
there will be destructive interference (i.e. the resultant is the difference
between the amplitudes of the two rays). For a given wavelength of
light and a given radius r of the scattering sphere, the result is a series
of maxima and minima as a function of scattering angle - as shown by the
blue line in Fig. 2 below.
|Fig. 2 Comparison of ray tracing calculations with Airy theory for scattering of 0.65 µm wavelength light from a water drop of radius r = 100 µm (p = 2, n = 1.33257, perpendicular polarisation)
Young's method certainly provides a simple explanation for supernumerary
arcs, but its predictions of the positions of the supernumerary arcs do
not agree with Airy theory. Fig. 2 is equivalent to the graph published
in 1838 by Airy in his article "On the intensity of light in the neighbourhood
of a caustic". In Airy's version of this graph, he described the
blue line as the "Imperfect theory of interference". This term "imperfect"
is undoubtedly valid because Young's method also wrongly predicts infinite
intensity at the geometric rainbow angle.
|Fig. 3 As Fig. 2, except that the Young method has been modified to take account of a phase difference of 90° between rays scattered either side of the rainbow angle
In practice, Young made a simple mistake - as explained by van de Hulst
on page 243 of his book "Light scattering by small particles".
Young was unaware that light suffers a phase change of 90° whenever
a ray passes through a focal line. When Young's method is modified
to correct this error, the amplitudes and angles of the supernumeraries
agree quite well with the Airy theory - as shown in Fig. 3 above.
|Fig. 4 Comparison of the modified version of Young's method with Airy theory and the Debye series for r = 100 µm for the primary rainbow (p = 2, n = 1.33257, perpendicular polarisation)
As Airy theory is only an approximation, it is important to compare
the modified version of Young's method with the rigorous solution provided
by the Debye series - as shown in Fig. 4 above. The surprise is that
the modified version of Young's method gives results which are essentially
identical to the Debye series for scattering angles above 139.5°.
Nevertheless, the correction of the phase error in Young's method has not
solved the remaining problem of infinite intensity at the rainbow angle.
|Fig. 5 Comparison of ray tracing calculations and Airy theory for r = 100 µm and the secondary rainbow (p = 3, n = 1.33257, perpendicular polarisation)
Fig. 5 shows the results of similar calculations for the secondary rainbow
for r = 100 µm. In this case, there are significant differences
between the modified version of Young's method and Airy theory.
|Fig. 6 Comparison of ray tracing calculations with calculations using Airy theory and the Debye series for r = 100 µm for the secondary rainbow (p = 3, n = 1.33257, perpendicular polarisation)
Fig. 6 shows that, using the Debye series as the reference, the modified
version of Young's method seems to be more accurate than Airy theory for
scattering angles below 127°. Perhaps, it is time to rehabilitate
Young's method - or, at least, the modified version of Young's method!
MiePlot offers the option of calculations based on ray tracing, using the methods developed by Descartes/Newton and Young.
Page updated on 11 May 2003
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