Scattering from a coated sphere

This section considers scattering of light by a coated sphere which consists of a spherical core surrounded by a concentric shell or mantle.

Fig. 1   Some geometric ray paths occurring within the coated sphere.
The ray paths are designated by (N, A, B) - see below for further explanation.

The ray paths shown in Fig. 1 can be grouped in terms of the number of internal reflections N, the number of chords in the shell A and the number of chords in the core B, thus giving the designations (N, A, B) as shown in Fig,. 1. Each of the ray paths is shown separately in Fig. 2 below, which includes diagrams of all ray paths for N ≤ 3 (i.e. involving up to 3 internal reflections).

(a) N = 0, A = 0, B = 0	(b) N = 0, A = 2, B = 1	(c) N = 1, A = 2, B = 0
(d) N = 1, A = 2, B = 2	(e) N = 1, A = 4, B = 2	(f) N = 2, A = 2, B = 3
(g) N = 2, A = 4, B = 1	(h) N = 2, A = 4, B = 3	(i) N = 2, A = 6, B = 3
(j) N = 3, A = 2, B = 4	(k) N = 3, A = 4, B = 0	(l) N = 3, A = 4, B = 2
(m) N = 3, A = 4, B = 4	(n) N = 3, A = 6, B = 2	(o) N = 3, A = 6, B = 2
(p) N = 3, A = 6, B = 4	(q) N = 3, A = 6, B = 4	(r) N = 3, A = 6, B = 4
(q) N = 3, A = 8, B = 4

The simplest ray path (0, 0, 0) in Fig. 2 (a) is reflected from the exterior of the spherical particle, whilst ray path (1, 2, 0) in Fig. 2 (b) involves reflection at the outside of the core. Other ray paths are much more complicated. Some diagrams in Fig. 2 show more than one path with the same (N, A, B) value. For example, the red and blue paths for (2, 4, 1) in Fig. 2 (g) have the same amplitude because each of these paths involves the same number of the different types of reflection and transmission. Furthermore, the paths are identical in terms of path length and, hence, the scattering contributions are in-phase. Such paths are known as "degenerate" paths: the (2, 4, 1), (2, 4, 3) and (3, 4, 2) paths each have a degeneracy factor of 2, whereas the (3, 4, 4) and (3, 6, 2) paths each have a degeneracy factor of 3. Note that two paths are sometimes shown in a single diagram, such as (g), (h), (l) and (n), whilst three paths are shown in (m). However, for reasons of clarity, the three paths for (3, 6, 2) are shown separately in two diagrams (n) and (o), whereas the three paths for (3, 6, 4) are shown in three individual diagrams (p), (q) and (r).

Another issue is that paths with value (N, A, B) have exactly the same path lengths as (N-2, A, B) - such as (3, 4, 2) and (1, 4, 2). Although the path lengths are identical, the amplitudes are different because they involve a different set of reflection and transmission coefficients. These are known as "repeated paths".

Much more information about the cataloguing of the various paths is available in "Understanding Light Scattering by a Coated Sphere. Part 1: Theoretical Considerations"³.

The following results examine the case of red light (λ = 0.65 μm = 650 nm) scattered by 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. 3     Intensity as a function of scattering angle θ calculated using the Aden-Kerker solution for scattering of red light (λ = 0.65 μm = 650 nm) by 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. 4     As Fig. 3 except that it isolates the Debye series contributions corresponding to specific ray trajectories (for perpendicular polarisation).

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     P. Laven and J. A. Lock, "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