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

be the electromagnetic power received by the detector downstream from the particle, and W0 the electromagnetic power received by the detector if the particle is removed. Evidently, W0 > W and we say that the presence of the particle has resulted in extinction of the incident beam. For a nonabsorbing medium, the electromagnetic power removed from the incident beam W0 −W is accounted for by absorption in the particle and scattering by the particle.

We now consider extinction from a computational point of view. In order to simplify the notations we will use the conventional expressions of the Poynting vectors and the electromagnetic powers in terms of the transformed fields introduced in Sect. 1.1 (we will omit the multiplicative factor 1/√ε0µ0). The time-averaged Poynting vector S can be written as [17]

S = 12 Re {E × H } = Sinc + Sscat + Sext ,

where E = Es + Ee and H = Hs + He are the total electric and magnetic fields,

Sinc = 12 Re {Ee × He }

is the Poynting vector associated with the external excitation,

Sscat = 12 Re {Es × Hs }

is the Poynting vector corresponding to the scattered field and

Sext = 12 Re {Ee × Hs + Es × He }

is the Poynting vector caused by the interaction between the scattered and

incident fields.

Taking into account the boundary conditions n×Ei = n×E and n×Hi = n × H on S, we express the time-averaged power absorbed by the particle as

With Sc being an auxiliary surface enclosing S (Fig. 1.11), we apply the Green second vector theorem to the vector fields E and E in the domain D bounded

50 1 Basic Theory of Electromagnetic Scattering

(1.82)

The divergence theorem applied to the excitation field in the domain Dc bounded by Sc gives

and

Re { · (Ee × He )} = 0 ,

yield

Winc = 0 .

Thus, Wext is the sum of the electromagnetic scattering power and the electromagnetic absorption power

Wext = Wscat + Wabs .

For a plane wave incidence, the extinction and scattering cross-sections are given by

the absorption cross-section is

Cabs = Cext − Cscat ≥ 0 ,

while the single-scattering albedo is

ω = Cscat ≤ 1 .

Cext

Essentially, Cscat and Cabs represent the electromagnetic powers removed from the incident wave as a result of scattering and absorption of the incident radiation, while Cext gives the total electromagnetic power removed from the incident wave by the combined e ect of scattering and absorption. The optical cross-sections have the dimension of area and depend on the direction and polarization state of the incident wave as well on the size, optical properties and orientation of the particle. The e ciencies (or e ciency factors) for extinction, scattering and absorption are defined as

where G is the particle cross-sectional area projected onto a plane perpendicular to the incident beam. In view of the definition of the normalized di erential scattering cross-section, we set G = πa2c , where ac is the area-equivalent-circle radius. From the point of view of geometrical optics we expect that the extinction e ciency of all particles would be identically equal to unity. In fact, there are many particles which can scatter and absorb more light than is geometrically incident upon them [17].

The scattering cross-section is the integral of the di erential scattering cross-section over the unit sphere. To prove this assertion, we express

Cscat as

52 1 Basic Theory of Electromagnetic Scattering

where Sc is a spherical surface situated at infinity and use the far-field representation

The scattering cross-section can be expressed in terms of the elements of the phase matrix and the Stokes parameters of the incident wave. Taking into account the expressions of Ie and Is, and using (1.76) we obtain

The phase function is related to the di erential scattering cross-section by the relation

and in view of (1.85) we see that p is dimensionless and normalized, i.e.,

1

4π Ω

p dΩ = 1 .

The mean direction of propagation of the scattered field is defined as

+ Z13 (er , ek ) Ue + Z14 (er , ek ) Ve] er dΩ (er ) .

The asymmetry parameter cos Θ is the dot product between the vector g and the incident direction ek [17, 169],