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

26 1 Basic Theory of Electromagnetic Scattering

where λ = 1/ε and λz = 1/εz . Equation (1.35) is the dispersion relation for ordinary waves, while (1.36) is the dispersion relation for extraordinary waves. For f = 0 it follows that Dβ(1) = 0 and Dα(2) = 0, and further that Dα(1) = Dα

and Dβ(2) = −Dβ . The electric displacement is then given by

00

(1.37) where k1(β, α) = k1ek (β, α), k2(β, α) = k2(β)ek (β, α), and for notation simplification, the dependence of the spherical unit vectors eα and eβ on the spherical angles β and α is omitted. For εxy = ε, the integral representations for the electric and magnetic fields become

respectively. For isotropic media, the only nonzero λ functions are λββ and λαα, and we have λββ = λαα = λ. The two waves degenerate into one (ordinary) wave, i.e., k1 = k2 = k, and the dispersion relation is

k2 = k02εµ .

Next we proceed to derive series representations for the electric and magnetic fields propagating in uniaxial anisotropic media. On the unit sphere, the tangential vector function Dα(β, α)eα −Dβ (β, α)eβ can be expanded in terms of the vector spherical harmonics mmn and nmn as follows:

Because the system of vector spherical harmonics is orthogonal and complete in L2(Ω), the series representation (1.40) is valid for any tangential vector field. Taking into account the expressions of the vector spherical harmonics (cf. (B.8) and (B.9)) we deduce that the expansions of Dβ and Dα are given by

respectively. Inserting the above expansions into (1.38) and (1.39), yields the series representations

∞n

28 1 Basic Theory of Electromagnetic Scattering

In (1.38)–(1.39), the electromagnetic fields are expressed in terms of the unknown scalar functions Dα and Dβ , while in (1.41) and (1.42), the electromagnetic fields are expressed in terms of the unknown expansion coe cients cmn and dmn. These unknowns will be determined from the boundary condi-

tions for each specific scattering problem. The vector functions Xe,h and Y e,h

mn mn

can be regarded as a generalization of the regular vector spherical wave functions M 1mn and N 1mn. For isotropic media, we have ελββ = 1, λkβ = 0 and k1 = k2 = k, and we see that both systems of vector functions are equivalent:

As a result, we obtain the familiar expansions of the electromagnetic fields in terms of vector spherical wave functions of the interior wave equation:

H(r) = −j µ n=1 m=−n cmnN mn (kr) + dmnM mn(kr) .

Although the derivation of Xemn,h and Y emn,h di ers from that of Kiselev et al. [119], the resulting systems of vector functions are identical except for a multiplicative constant. Accordingly to Kiselev et al. [119], this system of vector functions will be referred to as the system of vector quasi-spherical wave functions. In (1.43)–(1.46) the integration over α can be analytically performed by using the relations

ek = sin β cos αex + sin β sin αey + cos βez ,

eβ = cos β cos αex + cos β sin αey − sin βez ,

eα = − sin αex + cos αey ,

and the standard integrals

− jm−1ej(m−1)ϕJm−1 (x) ,