Previous page: Mie scattering 
To understand scattering from a sphere, it is useful to define various types of ray paths, as shown in Fig. 3 where:

In 1908, Peter Debye solved the problem of electromagnetic scattering from a cylinder and his method can also be adapted to scattering from a sphere. The Debye series is equivalent to the Mie series, but the Debye series has the great advantage that it can be used to isolate the contributions made by rays of order p  as shown in Fig. 4 below for scattering from a water drop of radius r = 100 µm.
Fig. 4 reveals the intricacy of the scattering processes  as well as the potential of the Debye series to improve our understanding of the mechanisms causing atmospheric optical effects. Note that this and subsequent graphs on this page show the scattered intensity for perpendicular polarization only.
It is interesting to compare Figs. 4 and 5 which use the Debye series to identify the ray paths contributing to scattering from, respectively, r = 100 µm and r = 10 µm water drops. The primary and secondary rainbows due to the p = 2 and p = 3 contributions respectively are clearly separated for r = 100 µm, whereas they overlap for r = 10 µm. On the other hand, it can be seen that the corona (θ < 10°) for r = 10 µm is predominantly due to p = 0 rays. Similarly, the glory (θ > 170°) for r = 10 µm is due to a combination of p = 2 rays and higher order rays.
Figs. 6 and 7, shown below, are polar plots corresponding to the rectangular plots in Figs. 4 and 5 respectively. The intensity scales for the polar plots are logarithmic  with each division representing a 10:1 change in intensity. These plots demonstrate that the forward scatteringlobe (&theta near 0°) due to the p = 0 ray is much stronger than other scattering mechanisms: the beamwidth of this lobe is about + 1° for r = 10 µm and even less for r = 100 µm.
Despite its similarity to geometrical optics, it must be emphasized that the Debye series is NOT an approximation: the sum of the Debye series terms from p = 0 to p = ∞ gives exactly the same result as the Mie calculation. In practice, higher order terms can often be ignored, as indicated by Figs. 7 and 8 below which compare the results of Mie calculations with Debye series calculations for p = 0 through p = 7 and for p = 0 through p = 12 respectively. In each case, these graphs show the Mie results in red with the Debye results overplotted in blue. If the Mie and Debye series results were identical, there would be no hint of red in these graphs.
MiePlot offers the option of using the Debye series.
Page updated on 16 November 2021
Previous page: Mie scattering 
