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

with εrt being the complex relative permittivity. Both the conductivity and the susceptibility contribute to the imaginary part of the permittivity, Im{εt} = ε0Im{χe} + Re{σ/ω}, and a complex value for εt means that the medium is absorbing. Usually, Im{χe} is associated with the “bound” charge current density and Re{σ/ω} with the “free” charge current density, and absorption is determined by the sum of these two quantities. Note that for a free-source medium, σ = 0 and εrt = εr = 1 + χe. The simplest solution to Maxwell’s equations in source-free media is the vector plane wave solution. The behavior of a vector plane wave in an isotropic medium is characterized by the dispersion relation

k = ω√εµ ,

which relates the wave number k to the properties of the medium and to the angular frequency ω of the wave. The dimensionless quantity

m = c√εµ

is the refractive index of the medium, where c = 1/√ε0µ0 is the speed of light in vacuum, and if k0 = ω√ε0µ0 is the wave number in free space, we see that

where ε and µ are the permittivity and permeability tensors, respectively. In our analysis we will consider electrically anisotropic media for which the permittivity is a tensor and the permeability is a scalar. Except for amorphous materials and crystals with cubic symmetry, the permittivity is always a tensor, and in general, the permittivity tensor of a crystal is symmetric. Since there exists a coordinate transformation that transforms a symmetric matrix into a diagonal matrix, we can take this coordinate system as reference frame and we have

This reference frame is called the principal coordinate system and the three coordinate axes are known as the principal axes of the crystal. If εx = εy = εz ,

81 Basic Theory of Electromagnetic Scattering

the medium is biaxial, and if ε = εx = εy and ε = εz , the medium is uniaxial. Orthorhombic, monoclinic and triclinic crystals are biaxial, while tetragonal, hexagonal and rhombohedral crystals are uniaxial. For uniaxial crystals, the principal axis that exhibits the anisotropy is called the optic axis. The crystal is positive uniaxial if εz > ε and negative uniaxial if εz < ε.

In our analysis, we will investigate the electromagnetic response of isotropic, chiral media exposed to arbitrary external excitations. The lack of geometric symmetry between a particle and its mirror image is referred to as chirality or optical activity. A chiral medium is characterized by either a leftor a right-handedness in its microstructure, and as a result, leftand right-hand circularly polarized fields propagate through it with di ering phase velocities. For a source-free, isotropic, chiral medium, the constitutive relations read as

D = εE + βε × E ,

B = µH + βµ × H ,

where the real number β is known as the chirality parameter. The Maxwell equations can be written compactly in matrix form as

where

1βk2 jωµ

K = 1 − β2k2 −jωε βk2

and k = ω√εµ.

Without loss of generality and so as to simplify our notations we make the following transformations:

1 1

E → √ε0 E, H → √µ0 H ,

D → √ε0D, B → √µ0B .

As a result, the Maxwell equations for a free-source medium become more “symmetric”:

× E = jk0B ,

× H = −jk0D ,

· D = 0 ,

the constitutive relations are given by (1.6) and (1.7) with ε and µ being the relative permittivity and permeability, respectively, the wave number is k = k0√εµ, and the K matrix in (1.9) takes the form

10 1 Basic Theory of Electromagnetic Scattering

In the frequency domain, a vector plane wave propagating in a medium with constant wave number ks = k0√εsµs is given by

where k0 is the wave number in free space, ke is the wave vector, ke = ksek , Ee0 is the complex amplitude vector,

Ee0 = Ee0,β eβ + Ee0,αeα ,

and Ee0,β and Ee0,α are the complex amplitudes in the β- and α-direction, respectively. An equivalent representation for Ee0 is

where epol is the complex polarization unit vector, |epol| = 1, and

1

epol = |Ee0| (Ee0,β eβ + Ee0,αeα) .

Inserting (1.13) into (1.12), gives the representation

Ee(r) = |Ee0| epolejke·r , epol · ek = 0 ,

and obviously, |Ee(r)| = |Ee0|.

There are three ways of describing the polarization state of vector plane waves.

1. Setting

where aβ and aα are the real non-negative amplitudes, and δβ and δα are the real phases, we characterize the polarization state of a vector plane wave by aβ , aα and the phase di erence ∆δ = δβ − δα.

2. Taking into account the representation of a vector plane wave in the time domain

Ee(r, t) = Re Ee(r)e−jωt = Re Ee0ej(ke·r−ωt) ,

where Ee(r, t) is the real electric vector, we deduce that (cf. (1.14))

Ee,β (r, t) = aβ cos (δβ + ke · r − ωt) ,

Ee,α(r, t) = aα cos (δα + ke · r − ωt) ,

where

Ee(r, t) = Ee,β (r, t)eβ + Ee,α(r, t)eα .