Introduction to the Debye series

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

• p = 0 corresponds to external reflection plus diffraction
• p = 1 corresponds to direct transmission through the sphere
• p = 2 corresponds to 1 internal reflection (primary rainbow)
• p = 3 corresponds to 2 internal reflections (secondary rainbow
• and so on ......

Fig. 3     Definition of types of ray path scattered from a sphere

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    Use of the Debye series to identify the contributions made by different ray paths to scattering of 0.65 µm wavelength light
by a water drop of radius r = 100 µm (showing only perpendicular polarization)

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.

Fig. 5    Use of the Debye series to identify the contributions made by different ray paths to scattering of 0.65 µm wavelength light
by a water drop of radius r = 10 µm (showing only perpendicular polarization)

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.

Fig. 6   Polar plot showing the use of the Debye series for scattering of 0.65 µm wavelength light by a water drop of radius r = 100 µm

Fig. 7   Polar plot showing the use of the Debye series for scattering of 0.65 µm wavelength light by a water drop of radius r = 10 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.

Fig. 8   Comparison of the results of Mie calculations (in red) with the results of Debye series (in blue) for p = 0 through p = 7 for scattering of 0.65 µm wavelength light by a water drop of radius r = 100 µm



Fig. 9   Comparison of the results of Mie calculations (in red) with the results of Debye series (in blue) for p = 0 through p = 12 for scattering of 0.65 µm wavelength light by a water drop of radius r = 100 µm

MiePlot offers the option of using the Debye series.

Page updated on 16 November 2021