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

1.4.3 Phase and Extinction Matrices

As in optics the electric and magnetics fields cannot directly be measured because of their high frequency oscillations, other measurable quantities describing the change of the polarization state upon scattering have to be defined. The transformation of the polarization characteristic of the incident light into that of the scattered light is given by the phase matrix. The coherency phase matrix Zc relates the coherency vectors of the incident and scattered fields

1

J s (rer ) = r2 Zc (er , ek ) J e ,

where the coherency vector of the incident field J e is given by (1.19) and the coherency vector of the scattered field J s is defined as

The phase matrix Z describes the transformation of the Stokes vector of the incident field into that of the scattered field

r2

and we have

Z (er , ek ) = DZc (er , ek ) D−1 ,

where the transformation matrix D is given by (1.21), the Stokes vector of the incident field Ie is given by (1.20) and the Stokes vector of the scattered field Is is defined as

46 1 Basic Theory of Electromagnetic Scattering

The above phase matrix is also known as the pure phase matrix, because its elements follow directly from the corresponding amplitude matrix that transforms the two electric field components [100]. The phase matrix of a particle in a fixed orientation may contain sixteen nonvanishing elements. Because only phase di erences occur in the expressions of Zij , i, j = 1, 2, 3, 4, the phase matrix elements are essentially determined by no more than seven real numbers: the four moduli |Spq | and the three di erences in phase between the Spq , where p = θ, ϕ and q = β, α. Consequently, only seven phase matrix elements are independent and there are nine linear relations among the sixteen elements. These linear dependent relations show that a pure phase matrix has a certain internal structure. Several linear and quadratic inequalities for the phase matrix elements have been reported by exploiting the internal structure of the pure phase matrix, and the most important inequalities are Z11 ≥ 0 and |Zij | ≤ Z11 for i, j = 1, 2, 3, 4 [102–104]. In principle, all scalar and matrix properties of pure phase matrices can be used for theoretical purposes or to test whether an experimentally or numerically determined matrix can be a pure phase matrix.

Equation (1.76) shows that electromagnetic scattering produces light with polarization characteristics di erent from those of the incident light. If the incident beam is unpolarized, Ie = [Ie, 0, 0, 0]T , the Stokes vector of the scattered field has at least one nonvanishing component other than intensity, Is = [Z11Ie, Z21Ie, Z31Ie, Z41Ie]T . When the incident beam is linearly polarized, Ie = [Ie, Qe, Ue, 0]T , the scattered light may become elliptically polarized since Vs may be nonzero. However, if the incident beam is fully polarized (Pe = 1), then the scattered light is also fully polarized (Ps = 1) [104].

As mentioned before, a scattering particle can change the state of polarization of the incident beam after it passes the particle. This phenomenon is called dichroism and is a consequence of the di erent values of attenuation rates for di erent polarization components of the incident light. A complete description of the extinction process requires the introduction of the so-called extinction matrix. In order to derive the expression of the extinction matrix we consider the case of the forward-scattering direction, er = ek , and define the coherency vector of the total field E = Es + Ee by

Eα (rek ) Eα (rek )

Using the decomposition

Ep(r)Eq (r) = Ee0,pEe0,q + Ee0,pejke·r Es,q (r)

+Es,p(r)Ee0,q e−jke·r + Es,p(r)Es,q (r) ,

where p and q stand for β and α, we approximate the integral of the generic term EpEq in the far-field region and over a small solid angle ∆Ω around the

Di

Ds

Fig. 1.10. Elementary surface ∆S in the far-field region

direction ek by

Ep(r)Eq (r)r2 dΩ (er ) ≈ Ep (rek ) Eq (rek ) ∆S ,

∆Ω

where ∆S = r2∆Ω (Fig. 1.10). On the other hand, using the far-field representation for the scattered field

and the asymptotic expression of the plane wave exp(jke · r) (cf. (B.7)) we approximate the integrals of each term composing EpEq as follows:

and the above relation gives

J (rek ) ∆S ≈ J e∆S − Kc (ek ) J e ,

48 1 Basic Theory of Electromagnetic Scattering

where the coherency extinction matrix Kc is defined as

The elements of the extinction matrix have the dimension of area and only seven components are independent. Equation (1.78) is an interpretation of the so-called optical theorem which will be discussed in the next section. This relation shows that the particle changes not only the total electromagnetic power received by a detector in the forward scattering direction, but also its state of polarization.

1.4.4 Extinction, Scattering and Absorption Cross-Sections

Scattering and absorption of light changes the characteristics of the incident beam after it passes the particle. Let us assume that the particle is placed in a beam of electromagnetic radiation and a detector located in the far-field region measures the radiation in the forward scattering direction (er = ek ). Let W