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

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

and

d

dx [xzn(x)] = xzn−1(x) − nzn(x), (A.8)

where zn stands for any spherical function. The Wronskian relation for the spherical Bessel and Neumann functions is

as x → ∞.

Spherical Bessel functions can be expressed in terms of trigonometric func-

256 A Spherical Functions

and

The spherical Bessel functions are computed by downward recursion. This recursion begins with two successive functions of small values and produces functions proportional to the Bessel functions rather than the actual Bessel functions. The constant of proportionality between the two sets of functions is obtained from the function of order zero j0. Alternatively, the spherical Bessel functions can be computed with an algorithm involving the auxiliary function χn [169]

χn(x) = jn−1(x) .

In this case, the functions χn are calculated with the downward recurrence relation

1

χn(x) = 2nx+1 − χn+1(x) ,

the constant of proportionality is obtained from the function of order one, χ1(x) = 1/x − cot x, and the spherical Bessel functions are computed with the upward recursion

jn(x) = χn(x)jn−1(x)

starting at j1. The spherical Neumann functions are calculated by upward recursion starting with the functions of order zero and one, y0 and y1, respectively.

A.2 Legendre Functions

With the substitution x = cos θ, the associated Legendre equation transforms to

This equation is characterized by regular singularities at the points x = ±1 and at infinity. For m = 0, there are two linearly independent solutions to the Legendre di erential equation and these solutions can be expressed as power series about the origin x = 0. In general, these series do not converge for

x = ±1, but if n is a positive integer, one of the series breaks o after a finite number of terms and has a finite value at the poles. These polynomial solutions are called Legendre polynomials and are denoted by Pn(x). For m = 0, the solutions to (A.13) which are finite at the poles x = ±1 are the associated Legendre functions. If m and n are integers, the associated Legendre functions are defined as

The angular functions πm and τ m are related to the associated Legendre

n n functions by the relations