# Analysis of resonant scatteringusing the Debye series

Resonant scattering of spherical particles is often referred to as "morphology-dependent resonances" (MDRs) or whispering-gallery modes. This topic has been studied by many authors (see the brief bibilography at the end of this page).

This page uses results of calculations using the Debye series to investigate the contributions made to these resonances by propagation paths of order p 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
• p = 3 corresponds to 2 internal reflections
• and so on ......
As a reference, Fig. 1 shows the results of Mie calculations (which include contributions from propagation paths for all integer values of p (i.e. from p = 0 through to p = ∞). Fig. 1   Mie calculations for scattering of red light (λ = 650 nm) by a spherical droplet as a function of size parameter x = 2 π r / λ of the droplet,
where r is the radius of the droplet and λ is the wavelength of the light.
Refractive index of the sphere = 1.33. Refractive index of the medium = 1. Scattering angle θ = 150°.

Fig. 2 below reproduces the Mie results from Fig. 1 (shown in red/brown) and compares them with the Debye series contributions (shown in blue) made by propagation paths for p = 0 through p = 20 (i.e. pmax = 20). If the Mie results exactly matched the Debye series results, none of the red/brown lines would be visible in Fig. 2 because they would have been overwritten by the blue lines. However, we can see that the match is far from perfect because most of the red/brown lines are still visible in Fig. 2. Although the blue lines in Fig. 2 reproduce the general trend of the red/brown lines, they entirely miss the maxima marked A-D in Fig. 1, thus indicating that the maxima are due to much higher values of p. The effects of using higher values of p are explored in Figs. 3 - 6 below. Fig. 2   As Fig. 1 but also showing the results of Debye series calculation for pmax = 20 (i.e. for p = 0 through p = 20). Fig. 3   As Fig. 1 but also showing the results of Debye series calculation for pmax = 100 (i.e. for p = 0 through p = 100). Fig. 4   As Fig. 1 but also showing the results of Debye series calculation for pmax = 200 (i.e. for p = 0 through p = 200). Fig. 5   As Fig. 1 but also showing the results of Debye series calculation for pmax = 500 (i.e. for p = 0 through p = 500). Fig. 6   As Fig. 1 but also showing the results of Debye series calculation for pmax = 1000 (i.e. for p = 0 through p = 1000).

The broader maxima (e.g. A and C in Fig. 1) can be explained by propagation paths of order p < 200 (see Fig. 4), but the very narrow maxima (e.g. B and D in Fig. 1) involve propagation path of order p = 1000 or even higher.

Figs. 7 and 8 below (which show only the Debye series results) indicate that the Debye series calculations are "almost" able to the reproduce the narrow maxima for p up to 500 and up to 1000 respectively, but obviously even higher values of p are needed to give an accurate reproduction of Fig. 1. Fig. 7   The results of Debye series calculation for pmax = 500 (i.e. for p = 0 through p = 500). Fig. 8   The results of Debye series calculation for pmax = 1000 (i.e. for p = 0 through p = 1000).

## Bibliography

• Petr Chýlek, "Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section," J. Opt. Soc. Am. 66, 285-287 (1976).
• B. R. Johnson, "Theory of morphology-dependent resonances: shape resonances and width formulas," J. Opt. Soc. Am. A 10, 343–352 (1993).
• G. Roll, T. Kaiser, S. Lange, and G. Schweiger, "Ray interpretation of multipole fields in spherical dielectric cavities," J. Opt. Soc. Am. A 15, 2879-2891 (1998).
• G. Roll and G. Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A 17, 1301-1311 (2000).

Page updated on 10 June 2010