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

We next seek to find a series representation for the total electric field. For this purpose, we use the decomposition

where the vector spherical wave functions M 2mn and N 2mn have the same expressions as the vector spherical wave functions M 3mn and N 3mn, but with the spherical Hankel functions of the second kind h(2)n in place of the spherical

Hankel functions of the first kind h(1)n . It should be remarked that for real arguments x, h(2)n (x) = [h(1)n (x)] . In the far-field region

as r → ∞, and we see that M 2mn and N 2mn behave as incoming transverse vector spherical waves.

The expansion of the incident field then becomes

+ 12 amnM 2mn + bmnN 2mn , n,m

follows. Setting

cmn = 2fmn + amn ,

dmn = 2gmn + bmn ,

and using the T -matrix equation, yields

The coe cients amn and bmn are determined by the incoming field. Since in the far-field region M 2mn and N 2mn become incoming vector spherical waves, we see that the S matrix determines how an incoming vector spherical wave is scattered into the same one.

In the far-field region, the total electric field

can be expressed as a superposition of outgoing and incoming transverse spherical waves

The orthogonality relations of the vector spherical harmonics on the unit sphere give

and since the incident field is arbitrarily, (1.107) and (1.108) implies that [217, 228, 256]

The above relation is the unitary condition for nonabsorbing particles. In terms of the transition matrix, this condition is

T †T = −12 T + T † ,

or explicitly

For absorbing particles, the integral in (1.107) is negative. Consequently, the equality in (1.110) transforms into an inequality which is equivalent to the contractivity of the S matrix [169]. Taking the trace of (1.110), Mishchenko et al. [169] derived an equality (inequality) between the T -matrix elements of an axisymmetric particle provided that the z-axis of the particle coordinate system is directed along the axis of symmetry.

To obtain the symmetry relation we proceed as in the derivation of the reciprocity relation for the tensor scattering amplitude, i.e., we consider the electromagnetic fields Eu, Hu generated by the incident fields Eeu, Heu, with u = 1, 2. The starting point is the integral (cf. (1.90))

er · (E2 × H1 − E1 × H2) dS = 0

Sc

66 1 Basic Theory of Electromagnetic Scattering

and the identities

mmn (π − θ, π + ϕ) = (−1)nmmn (θ, ϕ) , nmn (π − θ, π + ϕ) = (−1)n+1nmn (θ, ϕ) ,

and

m−mn (θ, ϕ) = mmn (θ, ϕ) , n−mn (θ, ϕ) = nmn (θ, ϕ) .

Additional properties of the transition matrix for particles with specific symmetries will be discussed in the next chapter. The “exact” infinite transition matrix satisfies the unitarity and symmetry conditions (1.110) and (1.112), respectively. However, in practical computer calculations, the truncated transition matrix may not satisfies these conditions and we can test the unitarity and symmetry conditions to get a rough idea regarding the convergence to be expected in the solution computation.

Remark. In the above analysis, the incident field is a vector plane wave whose source is situated at infinity. Other incident fields than vector plane waves can be considered, but we shall assume that the source of the incident field lies outside the circumscribing sphere Sc. In this case, the incident field is regular everywhere inside the circumscribing sphere, and both expansions (1.93) and (1.94) are valid on Sc. The T -matrix equation holds true at finite distances from the particle (not only in the far-field region), and therefore, the transition matrix is also known as the “nonasymptotic vector- spherical-wave transition matrix”. The properties of the transition matrix (unitarity and symmetry) can also be established by considering the energy flow through a finite sphere Sc [238]. If we now let the source of the incident field recede to infinity, we can let the surface Sc follow, i.e., we can consider the case of an arbitrary large sphere and this brings us to the precedent analysis.

1.5.3 Randomly Oriented Particles

In the following analysis we consider scattering by an ensemble of randomly oriented, identical particles. Random particle orientation means that the orientation distribution of the particles is uniform. As a consequence of random particle orientation, the scattering medium is macroscopically isotropic, i.e., the scattering characteristics are independent of the incident and scattering directions ek and er , and depend only on the angle between the unit vectors ek and er . For this type of scattering problem, it is convenient to direct the Z-axis of the global coordinate system along the incident direction and to choose the XZ-plane as the scattering plane (Fig. 1.13).

Z

θ

Fig. 1.13. The Z-axis of the global coordinate system is along the incident direction and the XZ-plane is the scattering plane

General Considerations

The phase matrix of a volume element containing randomly oriented particles can be written as

Z (er , ek ) = Z (θ, ϕ = 0, β = 0, α = 0) ,

where, in general, θ and ϕ are the polar angles of the scattering direction er , and β and α are the polar angles of the incident direction ek . The phase matrix Z(θ, 0, 0, 0) is known as the scattering matrix F and relates the Stokes parameters of the incident and scattered fields defined with respect to the scattering plane. Taking into account that for an incident direction (β, α), the backscattering direction is (π − β, α + π), the complete definition of the scattering matrix is [169]

The scattering matrix of a volume element containing randomly oriented particles has the following structure:

F14(θ) F24(θ) −F34(θ) F44(θ)

If each particle has a plane of symmetry or, equivalently, the particles and their mirror-symmetric particles are present in equal numbers, the scattering