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

Fig. 1.7. General orientations of the particle and beam coordinate systems

where

m =−n m =−n

and the Wigner D-functions Dmn m are given by (B.34).

Remark. Another representation of a Gaussian beam is the integral representation over plane waves [48, 116]. This can be obtained by using Fourier analysis and by replacing the pseudo-vector potential of a nth Davis beam by an equivalent vector potential (satisfying the wave equation), so that both vector fields have the same values in a plane z = const.

1.3 Internal Field

To solve the scattering problem in the framework of the null-field method it is necessary to approximate the internal field by a suitable system of vector functions. For isotropic particles, regular vector spherical wave functions of the interior wave equation are used for internal field approximations. In this section we derive new systems of vector functions for anisotropic and chiral particles by representing the electromagnetic fields (propagating in anisotropic

22 1 Basic Theory of Electromagnetic Scattering

and chiral media) as integrals over plane waves. For each plane wave, we solve the Maxwell equations and derive the dispersion relation following the treatment of Kong [122]. The dispersion relation which relates the amplitude of the wave vector k to the properties of the medium enable us to reduce the three-dimensional integrals to two-dimensional integrals over the unit sphere. The integral representations are then transformed into series representations by expanding appropriate tangential vector functions in vector spherical harmonics. The new basis functions are the vector quasi-spherical wave functions (for anisotropic media) and the vector spherical wave functions of leftand right-handed type (for chiral media).

1.3.1 Anisotropic Media

Maxwell equations describing electromagnetic wave propagation in a sourcefree, electrically anisotropic medium are given by (1.10), while the constitutive relations are given by (1.7) with the scalar permeability µ in place of the permeability tensor µ. In the principal coordinate system, the first constitutive relation can be written as

E = λD ,

where the impermittivity tensor λ is given by

and λx = 1/εx, λy = 1/εy and λz = 1/εz .

The electromagnetic fields can be expressed as integrals over plane waves by considering the inverse Fourier transform (excepting the factor 1/(2π)3):

where A stands for E, D, H and B, and A stands for the Fourier transforms E, D, H and B. Using the identities

× A(r) = j k × A (k) ejk·r dV (k) ,

· A(r) = j k · A (k) ejk·r dV (k) ,

we see that the Maxwell equations for the Fourier transforms take the forms

k × E = k0B , k × H = −k0D , k · D = 0 , k · B = 0 ,

24 1 Basic Theory of Electromagnetic Scattering

The last two equations in (1.31) and the first two equations in (1.33) yield a homogeneous system of equations for Dβ and Dα

Requiring nontrivial solutions we set the determinant equal to zero and obtain two values for the wave number k2,

The above relations are the dispersion relations for the extraordinary waves, which are the permissible characteristic waves in anisotropic media. For an extraordinary wave, the magnitude of the wave vector depends on the direction of propagation, while for an ordinary wave, k is independent of β and α. Straightforward calculations show that for real values of λx, λy and λz , λββ λαα > λ2βα and as a result λ1 > 0 and λ2 > 0. The two characteristic

where

f = −λ∆βαλ

and

Next, we define the tangential fields

vα = f eβ + eα ,

vβ = −eβ + f eα ,

and note that the vectors vα and vβ are orthogonal to each other, vα ·vβ = 0, and vα = −ek × vβ and vβ = ek × vα. Taking into account that k1 (ek ) = k1 (ek ) ek and k2 (ek ) = k2 (ek ) ek , we find the following integral representation for the electric displacement:

D(r) = Dα (ek ) vα (ek ) ejk1(ek )·r + Dβ (ek ) vβ (ek ) ejk2(ek )·r dΩ (ek ) ,

Ω

with Ω being the unit sphere. The result, the integral representation for the electric field is

E(r) =

1

εxy

×Dα (ek ) weα (ek ) ejk1(ek )·r − Dβ (ek ) weβ (ek ) ejk2(ek )·r dΩ (ek ) ,

Ω

where

1

εxy = 2 (εx + εy ) ,

and

weα = εxy [(λkβ f + λkα) ek + λ1vα] , weβ = εxy [(λkβ − λkαf ) ek − λ2vβ ] ,

while for the magnetic field, we have

+ Dβ (ek ) whβ (ek ) ejk2(ek )·r dΩ (ek ) ,

where

whα = − εxy λ1vβ ,

whβ = εxy λ2vα .

For uniaxial anisotropic media we set λ = λx = λy , derive the relations

and use the identities λ1 = λαα and λ2 = λββ , to obtain