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

cos σ1 = eα · eα = −sin β cos θ − cos β sin θ cos (ϕ − α) , sin Θ

cos σ2 = eϕ · eϕ = cos β sin θ − sin β cos θ cos (ϕ − α) .

For an ensemble of randomly positioned particles, the waves scattered by di erent particles are random in phase, and the Stokes parameters of these incoherent waves add up. Therefore, the scattering matrix for the ensemble is the sum of the scattering matrices of the individual particles:

F = N F ,

where N is number of particles and F denotes the ensemble-average scattering matrix per particle. Similar relations hold for the extinction matrix and optical cross-sections. Because the particles are identical, the ensembleaverage of a scattering quantity X is the orientation-averaged quantity

e cient analytical techniques for computing X . These methods work much faster than the standard approaches based on the numerical averaging of results computed for many discrete orientations of the particle. We begin with the derivation of the rotation transformation rule for the transition matrix and then compute the orientation-averaged transition matrix, optical crosssections and extinction matrix. An analytical procedure for computing the orientation-averaged scattering matrix will conclude our analysis.

Rotation Transformation of the Transition Matrix

To derive the rotation transformation rule for the transition matrix we assume that the orientation of the particle coordinate system Oxyz with respect to the global coordinate system OXY Z is specified by the Euler angles αp, βp and γp.

In the particle coordinate system, the expansions of the incident and scattered field are given by

∞n

Ee (r, θ, ϕ) = amnM 1mn (ksr, θ, ϕ) + bmnN 1mn (ksr, θ, ϕ) ,

n=1 m=−n

∞n

Es (r, θ, ϕ) = fmnM 3mn (ksr, θ, ϕ) + gmnN 3mn (ksr, θ, ϕ) ,

n=1 m=−n

70 1 Basic Theory of Electromagnetic Scattering

while in the global coordinate system, these expansions take the form

Assuming the T -matrix equations s = T e and s = T e, our task is to express the transition matrix in the global coordinate system T in terms of the transition matrix in the particle coordinate system T . Defining the “augmented” vectors of spherical wave functions in each coordinate system

and using the rotation addition theorem for vector spherical wave functions, we obtain

nn1

for i, j = 1, 2.

Because the elements of the amplitude matrix can be expressed in terms of the elements of the transition matrix, the above relation can be used to express the elements of the amplitude matrix as functions of the particle orientation angles αp, βp and γp. The properties of the Wigner D-functions can then be used to compute the integrals over the particle orientation angles.

Orientation-Averaged Transition Matrix

The elements of the orientation-averaged transition matrix with respect to the global coordinate system are given by

Using the definition of the Wigner D-functions (cf. (B.34)), the symmetry relation of the Wigner d-functions dnm m(−βp) = dnmm (βp), and integrating over αp and γp, yields

where ∆mm is given by (B.36). Taking into account the identities: ∆m m = ∆mm and (∆mm )2 = 1, and using the orthogonality property of the d- functions (cf. (B.43)), we obtain [168, 169]

'(

The above relation provides a simple analytical expression for the orientationaveraged transition matrix in terms of the transition matrix in the particle coordinate system. The orientation-averaged T ij matrices are diagonal and their elements do not depend on the azimuthal indices m and m1.

Orientation-Averaged Extinction and Scattering Cross-Sections

In view of the optical theorem, the orientation-averaged extinction crosssection is (cf. (1.88) with |Ee0| = 1)

Cext = 4kπ Im )epol · Es∞ (ez )* . s