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

72 1 Basic Theory of Electromagnetic Scattering

Considering the expansion of the far-field pattern in the global coordinate system (cf. (1.96)), taking the average and using the expression of the orientationaveraged transition matrix (cf. (1.118) and (1.119)), gives

where the summation over the index m involves the values −1 and 1. In the next chapter we will show that for axisymmetric particles, T−ijmn,−mn =

−Tmn,mnij and T0ijn,0n = 0 for i = j, while for particles with a plane of symmetry, Tmn,mnij = 0 for i = j. Thus, for macroscopically isotropic and mirrorsymmetric media, (1.119) gives t12n = t21n = 0. Further, using the expressions of the incident field coe cients (cf. (1.26))

and the special values of the vector spherical harmonics in the forward direction

The above relation shows that the orientation-averaged extinction crosssection for macroscopically isotropic and mirror-symmetric media is determined by the diagonal elements of the transition matrix in the particle coordinate system. The same result can be established if we consider an ensemble of randomly oriented particles (with t12n = 0 and t21n = 0) illuminated by a linearly polarized plane wave (with real polarization vector epol).

For an arbitrary excitation, the scattering cross-section can be expressed in the global coordinate system as

Since

T (αp, βp, γp) = RT (−γp, −βp, −αp) T RT (αp, βp, γp) ,

T † (αp, βp, γp) = R (αp, βp, γp) T †R (−γp, −βp, −αp) ,

and in view of (B.54) and (B.55),

R (−γp, −βp, −αp) = RT (−γp, −βp, −αp) −1 ,

R (αp, βp, γp) = RT (−γp, −βp, −αp) ,

we see that

T † (αp, βp, γp) T (αp, βp, γp) = RT (−γp, −βp, −αp) T †T RT (αp, βp, γp) .

The above equation is similar to (1.115), and taking the average, we obtain

74 1 Basic Theory of Electromagnetic Scattering

The orientation-averaged scattering cross-section then becomes

where as before, the summation over the index m involves the values −1 and

1. For macroscopically isotropic and mirror-symmetric media, t12 = t21 = 0,

n n

and using (1.120) and (1.121), we obtain [120, 162]

Thus, the orientation-averaged scattering cross-section for macroscopically isotropic and mirror-symmetric media is proportional to the sum of the squares of the absolute values of the transition matrix in the particle coordinate system. The same result holds true for an ensemble of randomly oriented particles illuminated by a linearly polarized plane wave.

Despite the derivation of simple analytical formulas, the above analysis shows that the orientation-averaged extinction and scattering cross-sections for macroscopically isotropic and mirror-symmetric media do not depend on the polarization state of the incident wave. The orientation-averaged extinction and scattering cross-sections are invariant with respect to rotations and translations of the coordinate system and using these properties, Mishchenko et al. [169] have derived several invariants of the transition matrix.

Orientation-Averaged Extinction Matrix

To compute the orientation-averaged extinction matrix it is necessary to evaluate the orientation-averaged quantities Spq (ez , ez ) . Taking into account the expressions of the elements of the amplitude matrix (cf. (1.97)), the equation of the orientation-averaged transition matrix (cf. (1.118) and (1.119)) and the expressions of the vector spherical harmonics in the forward direction (cf. (1.121)), we obtain

∞

Sθβ (ez , ez ) = Sϕα (ez , ez ) = −2kj s (2n + 1) t11n + t22n ,

n=1

∞

Sθα (ez , ez ) = − Sϕβ (ez , ez ) = −21ks (2n + 1) t12n + t21n .

n=1

Inserting these expansions into the equations specifying the elements of the extinction matrix (cf. (1.79)), we see that the nonzero matrix elements are

In terms of the elements of the extinction matrix, the orientation-averaged extinction cross-section is (cf. (1.89))

1

Cext = Ie [ K11 Ie + K14 Ve] ,

while for macroscopically isotropic and mirror-symmetric media, the identities t12n = t21n = 0, imply

K14 = K41 = K23 = K32 = 0 .

In this specific case, the orientation-averaged extinction matrix becomes diagonal with diagonal elements being equal to the orientation-averaged extinction cross-section per particle, K = Cext I.

Orientation-Averaged Scattering Matrix

By definition, the orientation-averaged scattering matrix is the orientationaveraged phase matrix with β = 0 and α = ϕ = 0. In the present analysis we consider the calculation of the general orientation-averaged phase matrixZ(er , ek ; αp, βp, γp) without taking into account the specific choice of the incident and scattering directions. We give guidelines for computing the quantities of interest, but we do not derive a final formula for the average phase matrix.

According to the definition of the phase matrix we see that the orientation-

averaged quantities Spq (er , ek )Sp1q1 (er , ek ) , with p, p1 = θ, ϕ and q, q1 = β, α, need to be computed. In view of (1.99), we have

76 1 Basic Theory of Electromagnetic Scattering

where, as before, T stands for the transition matrix in the global coordinate system. Defining the matrices

V pp1 (er ) = vp1 (er ) vTp (er )

A11pp1 = 'T 11†V 11pp1 T 11 + T 11†V 12pp1 T 21 + T 21†V 21pp1 T 11 + T 21†V 22pp1 T 21( , A12pp1 = 'T 11†V 11pp1 T 12 + T 11†V 12pp1 T 22 + T 21†V 21pp1 T 12 + T 21†V 22pp1 T 22( , A21pp1 = 'T 12†V 11pp1 T 11 + T 12†V 12pp1 T 21 + T 22†V 21pp1 T 11 + T 22†V 22pp1 T 21( , A22pp1 = 'T 12†V 11pp1 T 12 + T 12†V 12pp1 T 22 + T 22†V 21pp1 T 12 + T 22†V 22pp1 T 22( .

(1.128) It is apparent that each matrix product in the above equations is of the form

where the permissive values of the index pairs (i, j), (k, l) and (u, v) follow