Phase and Extinction Matrices

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

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 differences 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 S pq and the three differences in phase between the Spq, where p = 6, f and q = f3, a. 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 Zn 0 and Zij Z fori,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 different from those of the incident light. If the incident beam is unpolarized, 7e = ]/e, 0,0,0], the Stokes vector of the scattered field has at least one nonvanishing component other than intensity, 7g =. 21- 6)- 4i7e]. When the incident beam is linearly 

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 different values of attenuation rates for different 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, = 6fc, and define the coherency vector of the total field E = Eg + E. by 

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. 

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Our presentation is focused on the analysis of the scattered field in the far-held region. We begin with a basic representation theorem for electromagnetic scattering and then introduce the primary quantities which dehne the single-scattering law the far-held patterns and the amplitude matrix. Because the measurement of the ampUtude matrix is a comphcated experimental problem, we characterize the scattering process by other measurable quantities as for instance the optical cross-sections and the phase and extinction matrices. 

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 H and E2 H2 are the total fields generated by the incident fields E i, ffei and E 2, He2, respectively, we have... 

Prom the reciprocity relation for the amplitude matrix we easily derive the reciprocity relation for the phase and extinction matrices ... 

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