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

It should be mentioned that for notation simplification we omit to indicate the dependency of T on the matrix indices.

Using the rotation transformation rule for the transition matrix (cf. (1.116)), we obtain

Taking into account the definition of the Wigner D-functions (cf. (B.34)) and integrating over αp and γp, yields

we use the symmetry relations (cf. (B.39) and (B.41))

given by (B.47), and the orthogonality property of the d-functions (cf. (B.43)).

78 1 Basic Theory of Electromagnetic Scattering

We obtain

I = (−1)m+m +m+m (−1)n+n1+n+n1 δm1−m,m1−mδm1−m ,m1−m

where Cm+m1u are the Clebsch–Gordan coe cients defined in Appendix B,

mn,m1n1

and

umin = max (|n − n1| , |n − n1| , |m1 − m| , |m1 − m| , |m − m1| , |m − m1|) ,

umax = min (n + n1, n + n1) .

Further, using the symmetry properties of the Clebsch–Gordan coe cients (cf. (B.48) and (B.51)) we arrive at

The orientation-averaged quantities Spq (er , ek )Sp1q1 (er , ek ) can be computed from the set of equations (1.127)–(1.132).

For an incident wave propagating along the Z-axis, the augmented vector of spherical harmonics vq (ez ) can be computed by using (1.121). Choosing the XZ-plane as the scattering plane, i.e., setting ϕ = 0, we see that the matrices V pp1 (er ) involve only the normalized angular functions πn|m|(θ) and τn|m|(θ). The resulting orientation-averaged scattering matrix can be computed at a set of polar angles θ and polynomial interpolation can be used to evaluate the orientation-averaged scattering matrix at any polar angle θ.

For macroscopically isotropic media, the orientation-averaged scattering matrix has sixteen nonzero elements (cf. (1.113)) but only ten of them are independent. For macroscopically isotropic and mirror-symmetric media, the orientation-averaged scattering matrix has a block-diagonal structure (cf. (1.114)), so that only eight elements are nonzero and only six of them are independent. In this case we determine the six quantities |Sθβ (θ)|2 ,

Other scattering characteristics as for instance the orientation-averaged scattering cross-section and the orientation-averaged mean direction of propagation of the scattered field can be expressed in terms of the elements of the orientation-averaged scattering matrix. To derive these expressions we consider the scattering plane characterized by the azimuth angle ϕ as shown in Fig. 1.15. In the scattering plane, the Stokes vector of the scattered wave is given by Is(rer ) = (1/r2) F (θ) Ie, whence, using the transformation rule of the Stokes vector under coordinate rotation Ie = L(ϕ)Ie, we obtain

1

Is (rer ) = r2 F (θ) L (ϕ) Ie .

Further, taking into account the expression of the Stokes rotation matrix L (cf. (1.23)) we derive

Is (er ) = F11 (θ) Ie + [ F12(θ) cos 2ϕ + F13(θ) sin 2ϕ] Qe

− [ F12(θ) sin 2ϕ − F13(θ) cos 2ϕ] Ue + F14(θ) Ve .

Integrating over ϕ, we find that the orientation-averaged scattering crosssection and the orientation-averaged mean direction of propagation of the scattered field are given by

80 1 Basic Theory of Electromagnetic Scattering

Fig. 1.15. Incident and scattering directions ek and er . The incident direction is along the Z-axis and the scattering matrix relates the Stokes vectors of the incident and scattered fields specified relative to the scattering plane characterized by the azimuth angle ϕ

and

respectively. Because the incident wave propagates along the Z-axis, the nonzero component of g is the orientation-averaged asymmetry parametercos Θ . In practical computer simulations, we use the decomposition

1

Cscat = Ie ( Cscat I Ie + Cscat V Ve) ,

and compute the quantities Cscat I and Cscat V by using (1.124) and the

relation

π

Cscat V = 2π F14(θ) sin θ dθ , (1.133)

0

respectively. These quantities do not depend on the polarization state of the incident wave and can be used to compute the orientation-averaged scattering cross-section for any incident polarization. For the asymmetry parameter we proceed analogously; we use the decomposition