Coated sphere: impulse response

Although the previous web page noted that there are many potential geometric ray paths through a coated sphere, we need to determine the relative importance of individual paths in terms of scattering.

For a coated sphere, the scattered intensity can be calculated using the solution provided by Aden-Kerker.¹ which is comparable to Mie theory for a homogeneous sphere. This web site contains many examples of the use of the Debye series to identify the scattering mechanisms for scattering by a homogeneous sphere. The Debye series decomposition can also be applied to the Aden-Kerker solution for coated spheres ². The MiePlot computer program now permits calculations for scattering from coated spheres with real values of refractive index - using the algorithms developed by J. A. Lock.

All of the graphs shown on this page assume a coated sphere with a spherical core with refractive index m₁ = 1.5, surrounded by a shell with refractive index m₂ = 1.3333 immersed in a medium of refractive index m₃ = 1. The radius of the core a₁₂ = 7.5 μm and the shell has a thickness of 2.5 μm, giving a total radius of a₂₃ = 10 μm.

Fig. 1   Time domain results for Mie scattering by a coated sphere

Fig. 2   Time domain results for Mie scattering by an homogeneous sphere with refractive index n = 1.3333 immersed in a medium of refractive index n = 1

Comparing Figs. 1 and 2 reveals that scattering from a coated sphere is, as you might expect, more complicated than scattering from a homogeneous sphere. Nevertheless, there are significant simlarities between the two diagrams.

Figs. 3-18 below show the time domain results obtained using the Debye series for a coated sphere for all ray paths involving up to 3 internal reflections (N ≤ 3). Two diagrams, (a) and (b), are shown for each term of the Debye series. In the diagrams marked (a), the false-color intensity scale is normalised to the maximum intensity in Fig. 1 - thus the colours can be directly compared with those in Fig. 1. In the diagrams marked (b), the false color scales have been optimised to reveal more of the scattering than in the corresponding (a) diagrams.

Fig. 3   Time domain results for the Debye (0, 0, 0) term (external reflection & diffraction)

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The pattern in Fig. 7 at θ ≈ 170° corresponds to a rainbow caused by one internal reflection. Note that the straight line at t > 200 fs is due to surface waves.

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1    A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
2     J. A. Lock, J. M. Jamison, and C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677-4690 (1994). Free download
3     J. A. Lock and P. Laven, "Understanding Light Scattering by a Coated Sphere. Part 1: Theoretical Considerations," Journal of Optical Society of America A, Vol. 29, Issue 8, pp. 1489-1497 (2012). Free download
4     J. A. Lock and P. Laven, "Understanding Light Scattering by a Coated Sphere. Part 2: Time domain analysis," Journal of Optical Society of America A, Vol. 29, Issue 8, pp. 1498-1507 (2012). Free download