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

where cos Θ = er · ek , and it is apparent that the asymmetry parameter is the average cosine of the scattering angle Θ. If the particle scatters more light toward the forward direction (Θ = 0), cos Θ is positive and cos Θ is negative if the scattering is directed more toward the backscattering direction (Θ = 180◦). If the scattering is symmetric about a scattering angle of 90◦,cos Θ vanishes.

1.4.5 Optical Theorem

The expression of extinction has been derived by integrating the Poynting vector over an auxiliary surface around the particle. This derivation emphasized the conservation of energy aspect of extinction: extinction is the combined e ect of absorption and scattering. A second derivation emphasizes the interference aspect of extinction: extinction is a result of the interference between the incident and forward scattered light [17]. Applying Green’s second vector theorem to the vector fields Es and Ee in the domain D bounded by S and Sc, we obtain

This result together with (1.82) and the identity Re{Ee ×Hs } = Re{Ee ×Hs} give

whence, using the explicit expressions for Ee and He, we derive

In the integral representation for the electric far-field pattern (cf. (1.69)) we set er = ek , take the dot product between Es∞(ek ) and Ee0, and obtain

54 1 Basic Theory of Electromagnetic Scattering

The above relation is a representation of the optical theorem, and since the extinction cross-section is in terms of the scattering amplitude in the forward direction, the optical theorem is also known as the extinction theorem or the forward scattering theorem. This fundamental relation can be used to compute the extinction cross-section when the imaginary part of the scattering amplitude in the forward direction is known accurately. In view of (1.88) and (1.74), and taking into account the explicit expressions of the elements of the extinction matrix we see that

1

Cext = Ie [K11 (ek ) Ie + K12 (ek ) Qe + K13 (ek ) Ue + K14 (ek ) Ve] . (1.89)

1.4.6 Reciprocity

The tensor scattering amplitude satisfies a useful symmetry property which is referred to as reciprocity. As a consequence, reciprocity relations for the amplitude, phase and extinction matrices can be derived. Reciprocity is a manifestation of the symmetry of the scattering process with respect to an inversion of time and holds for particles in arbitrary orientations [169]. In order to derive this property we use the following result: if E1, H1 and E2,

H2 are the total fields generated by the incident fields Ee1, He1 and Ee2,

He2, respectively, we have

where as before, Sc is an auxiliary surface enclosing S. Since E1 and E2 are source free in the domain bounded by S and Sc, the above equation follows immediately from Green second vector theorem. Further, applying Green’s second vector theorem to the internal fields Ei1 and Ei2 in the domain Di, and taking into account the boundary conditions n × Ei1,2 = n × E1,2 and n × Hi1,2 = n × H1,2 on S, yields

n · (E2 × H1 − E1 × H2) dS = 0 ,

S

whence

Sc

follows. We take the surface Sc as a large sphere of outward unit normal vector er , consider the limit when the radius R becomes infinite, and write the integrand in (1.90) as

E2 × H1 − E1 × H2

=Ee2 × He1 − Ee1 × He2 + Es2 × Hs1 − Es1 × Hs2

+Es2 × He1 − Ee1 × Hs2 + Ee2 × Hs1 − Es1 × He2 .

The Green second vector theorem applied to the incident fields Ee1 and Ee2 in any bounded domain shows that the vector plane wave terms do not contribute to the integral. Furthermore, using the far-field representation

and taking into account the transversality of the far-field patterns

Es∞2 × Hs∞1 − Es∞1 × Hs∞2 = 0 ,

we see that the integral over the scattered wave terms also vanishes. Thus, (1.90) implies that

the integrands in (1.91) contain the term exp(jksRek1,2 · er ). Since R is large, the stationary point method can be used to compute the integrals accordingly to the basic result

56 1 Basic Theory of Electromagnetic Scattering

where (θst, ϕst) is the stationary point of f , fθθ = ∂2f /∂θ2, fϕϕ = ∂2f /∂ϕ2 and fθϕ = ∂2f /∂θ∂ϕ. The integrals in (1.91) are then given by

er · (Es2 × He1 − Ee1 × Hs2) dS

Sc

= −4πj R εs Es∞2 (−ek1) · Ee01e−jksR ,

ks µs

er · (Es1 × He2 − Ee2 × Hs1) dS

Sc

= −4πj R εs Es∞1 (−ek2) · Ee02e−jksR ,

ks µs

and we deduce the reciprocity relation for the far-field pattern (Fig. 1.12):

Es∞2 (−ek1) · Ee01 = Es∞1 (−ek2) · Ee02 .

The above relation gives

Ee01 · A (−ek1, ek2) · Ee02 = Ee02 · A (−ek2, ek1) · Ee01

and since a · D · b = b · DT · a, and Ee01 and Ee02 are arbitrary transverse vectors, the following constraint on the tensor scattering amplitude:

A (−ek2, −ek1) = AT (ek1, ek2)

follows. This is the reciprocity relation for the tensor scattering amplitude which relates scattering from the direction −ek1 into −ek2 to scattering from ek2 to ek1. Taking into account the representation of the amplitude matrix elements in terms of the tensor scattering amplitude and the fact that for

Fig. 1.12. Illustration of the reciprocity relation

e k = −ek we have e β = eβ and e α = −eα, we obtain the reciprocity relation for the amplitude matrix:

If we choose ek1 = −ek2 = −ek , we obtain

Sϕβ (−ek , ek ) = −Sθα (−ek , ek ) ,

which is a representation of the backscattering theorem [169].

From the reciprocity relation for the amplitude matrix we easily derive the reciprocity relation for the phase and extinction matrices:

Z (−ek , −er ) = QZT (er , ek ) Q

and

K (−ek ) = QKT (ek ) Q ,

respectively, where Q = diag[1, 1, −1, 1].

The reciprocity relations can be used in practice for testing the results of theoretical computations and laboratory measurements. It should be remarked that reciprocity relations give also rise to symmetry relations for the dyadic Green functions [229].

1.5 Transition Matrix

The transition matrix relates the expansion coe cients of the incident and scattered fields. The existence of the transition matrix is “postulated” by the T -Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253, 256]), and for this reason, the null-field method has often been referred to as the T -matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] found a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-di erence technique.

In this section we review the general properties of the transition matrix such as unitarity and symmetry and discuss analytical procedures for averaging scattering characteristics over particle orientations. These procedures