Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

Size Parameter

Fig. 3.79. E ective loss tangent versus size parameter computed with the EFMED routine and the Matlab program QCAMIE

and the quasi-crystalline approximation with coherent potential [118],

Figure 3.80 shows plots of Im{Ks/k0} versus concentration for spherical particles of radius a = 3.977 ·10−3 µm and relative refractive index (with respect to the ambient medium) mr = 1.194. The wave number in free space is k0 = 10, while the refractive index of the ambient medium is m = 1.33.

In Fig. 3.81 we plot Im{Ks/k0} versus concentration for a = 1.047 · 10−2 µm, mr = 1.789 and m = 1.0. Computed results using the T -matrix method agree with the quasi-crystalline approximation. It should be observed that both the quasi-crystalline approximation and the quasi-crystalline approximation with coherent potential do predict maximum wave attenuation at a certain concentration.

In Figs. 3.82 and 3.83, we show calculations of Re{Ks} and Im{Ks} as functions of the size parameter x = ks max{a, b} for oblate and prolate spheroids. The spheroids are assumed to be oriented with their axis of symmetry along the Z-axis or to be randomly oriented. Since the incident wave is also assumed to propagate along the Z-axis, the medium is not anisotropic and is characterized by a single wave number Ks. The axial ratios are a/b = 0.66 for oblate spheroids and a/b = 1.5 for prolate spheroids. As before, the fractional

A

Spherical Functions

In this appendix we recall the basic properties of the solutions to the Bessel and Legendre di erential equations and discuss some computational aspects. Properties of spherical Bessel and Hankel functions and (associated) Legendre functions can be found in [1, 40, 215, 238].

In a source-free isotropic medium the electric and magnetic fields satisfy the equations

× × X − k2X = 0,

· X = 0,

where X stands for E or H. Since

× × X = −∆X + ( · X) ,

we see that X satisfies the vector wave equation

∆X + k2X = 0

and we deduce that each component of X satisfies the scalar wave equation or the Helmholtz equation

is separable, so that upon replacing

u(r) = f1(r)Y (θ, ϕ) ,

254 A Spherical Functions

where (r, θ, ϕ) are the spherical coordinates of the position vector r, we obtain

The above equations are known as the Bessel di erential equation and the di erential equation for the spherical harmonics. Further, setting

The di erential equation (A.3) is known as the associated Legendre equation, while the solution to the di erential equation (A.4) is f3(ϕ) = exp(j mϕ).

A.1 Spherical Bessel Functions

With the substitution x = kr, the spherical Bessel di erential equation can be written in the standard form

x2f (x) + 2xf (x) + x2 − n(n + 1) f (x) = 0 .

For n = 0, 1, . . . , the functions

where 1 · 3 · · · (2n + 2p + 1) = (2n + 2p + 1)!!, are solutions to the spherical Bessel di erential equation (the first coe cient in the series (A.5) has to be set equal to one). The functions jn and yn are called the spherical Bessel and Neumann functions of order n, respectively, and the linear combinations

h(1n ,2) = jn ± jyn

are known as the spherical Hankel functions of the first and second kind. From the series representations we see that