Scattering matrix element

The physical interpretation of the scattering matrix elements is best understood in tenns of its square modulus... 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Thus, if we have in hand the scattering coefficients an and bn, we can determine all the measurable quantities associated with scattering and absorption, such as cross sections and scattering matrix elements. 

It is easy to show from (5.4) that for sufficiently small frequencies the forward scattering matrix element is given by... 

The scattering matrix elements StJ corresponding to (5.47) can be obtained from (3.16). However, of possibly greater interest than the most general scattering matrix is that for a collection of identical, but randomly oriented, anisotropic dipoles this scattering matrix is proportional to where %... 

If the amplitude scattering matrix elements (5.4) for a homogeneous, isotropic sphere of radius a are divided by the volume v, the resulting quotients approach finite limits as the sphere radius tends to zero ... 

Because of condition (6.2) it is customary (but not necessary) to write the scattering matrix elements (6.1) as... 

In Chapter 8 we shall derive the field scattered by an infinite cylinder of arbitrary radius and refractive index we shall also consider scattering by a finite cylinder in the diffraction theory approximation. Although the finite cylinder scattering problem is not exactly soluble, we can obtain analytical expressions for the amplitude scattering matrix elements in the Rayleigh-Gan s approximation. 

The (4 X 4) scattering matrix elements (3.16) for an optically active sphere satisfy the following six relations ... 

A computer program for calculating the scattering coefficients (8.38) and the corresponding cross sections and scattering matrix elements is described in Appendix C all the examples in this section were obtained with this program. 

The amplitude scattering matrix elements correct to terms of order x2 are... 

Equality holds for a single sphere or a collection of identical spheres inequality holds if they are distributed in size or composition. This inequality was used by Hunt and Huffman (1973), for example, as an indicator of dispersion in suspensions of spherical particles. It was pointed out by Fry and Kattawar (1981) that the inequalities they derived are useful consistency checks on measurements of all 16 scattering matrix elements. 

The simplest, and probably most obvious, way to measure scattering matrix elements is with a conventional nephelometer (Fig. 13.5) and various optical elements fore and aft of the scattering medium. Recall that we introduced Stokes parameters in Section 2.11 by way of a series of conceptual measurements of differences between irradiances with different polarizers in the beam. Although we did not specify the origin of the beam, it could be light scattered in any direction. Combinations of scattering matrix elements can therefore be extracted from these kinds of measurements. There are now, however, two beams—incident and scattered—and many possible pairs of optical elements these are discussed below. 

Table 13.1 Combinations of Scattering Matrix Elements That Result from Measurements with a Polarizer Ps Forward of the Scattering Medium and an Analyzer A s aft"...

The possible outcomes of measurements—combinations of scattering matrix elements—listed in Table 13.1 follow from multiplication of three matrices those representing the polarizer, the scattering medium, and the analyzer. If U is an element in the optical train, then the measured irradiance depends on only two matrix elements. In general, however, there are four elements in a combination, so that four measurements are required to obtain one matrix element. 

Few measurements or calculations of all 16 scattering matrix elements have been reported. There are only four nonzero independent elements for spherical particles and six for a collection of randomly oriented particles with mirror symmetry (Section 13.6). It is sometimes worth the effort, however, to determine if the expected equalities and zeros really occur. If they do not, this may signal interesting properties such as deviations from sphericity, unexpected asymmetry, or partial alignment some examples are given in this section. But we begin with spherical particles. 

One of the few sets of measurements of all scattering matrix elements for nonspherical particles was made by Holland and Gagne (1970), who used various combinations of polarizers and retarders (see Section 13.7). They studied quartz (sand) particles with a fairly broad range of sizes. To investigate further the effects of nonsphericity on all matrix elements Perry et al. (1978),... 

Scattering matrix elements off the block diagonal are zero for mirror-symmetric collections of randomly oriented particles, as they are for spheres. 

For incident unpolarized light to be (partially) circularly polarized upon scattering by a collection of particles, the scattering matrix element S4l must not be zero. It was shown in Section 13.6 that the scattering matrix for a collection (with mirror symmetry) of randomly oriented particles has the form... 

As a check on the amplitude scattering matrix elements, we compute Qcxt in BHMIE from the optical theorem (4.76), whereas Qsca is computed from the series (4.61). POL, the degree of polarization, must vanish for scattering angles of 0 and 180°, as must 34- Also, the 4x4 scattering matrix elements must satisfy... 

As in the previous programs, series for scattering matrix elements and efficiencies are truncated after NSTOP terms, where NSTOP = x + 4x1/3 + 2. Gn(mx) is computed by (C.l) beginning with CNMX, successive lower-order logarithmic derivatives CNMXGx are computed by downward recurrence. Provided that NMX is sufficiently greater than NSTOP and mx, Gp for... 

SUBROUTINE BHCYL CALCULATES AMPLITUDE SCATTERING MATRIX ELEMENTS AND EFFICIENCIES FOR EXTINCTION AND SCATTERING FOR A GIVEN SIZE PARAMETER AND RELATIVE REFRACTIVE INDEX THE INCIDENT LIGHT IS NORMAL TO THE CYLINDER AXIS PAR .ELECTRIC FIELD PARALLEL TO CYLINDER AXIS PERIELECTRIC FIELD PERPENDICULAR TO CYLINDER AXIS sc sc sc scsc scsc sc sc sc sc ... 

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