The Amplitude Scattering Matrix

At sufficiently large distances from tlje origin (kr 1), in the far-field region, the scattered electric field E5 is approximately transverse (er E5 — 0) and has the asymptotic form (see, e.g., Jackson, 1975, p. 748) 

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We again omit the multiplicative factor k/2cojii. The relation between incident and scattered Stokes parameters follows from the amplitude scattering matrix (3.12) ... 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

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 ... 

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. 

In Section 4.4 we showed that the off-diagonal elements of the amplitude scattering matrix (3.12) are zero for a nonactive sphere. If the sphere is optically active, however, the matrix elements are... 

In problems involving optically active particles it is usually more convenient to use the amplitude scattering matrix in the circular polarization representation. The transformation from linearly to circularly polarized electric field components is... 

In Chapter 3 we derived a general expression for the amplitude scattering matrix for an arbitrary particle. An unstated assumption underlying that derivation is that the particle is confined within a bounded region, a condition that is not satisfied by an infinite cylinder. Nevertheless, we can express the field scattered by such a cylinder in a concise form by resolving the incident and scattered fields into components parallel and perpendicular to planes determined by the cylinder axis (ez) and the appropriate wave normals (see Fig. 8.3). That is, we write the incident field... 

The amplitude scattering matrix for a normally illuminated cylinder is diagonal ... 

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

For spheres sufficiently small that Rayleigh theory (Chapter 5) is applicable, or for arbitrarily shaped particles that satisfy the requirements of the Rayleigh-Gans approximation (Chapter 6), incident light with electric field components parallel and perpendicular to the scattering plane may be scattered with different amplitudes however, there is no phase shift between the two components. Hence, the amplitude scattering matrix has the form... 

The off-diagonal elements of the amplitude scattering matrix for an optically active sphere (Section 8.3)... 

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... 

To calculate the amplitude scattering matrix elements, the incident and the scattered fields are decomposed into the components parallel and perpendicular to the scattering plane as shown in Figure 4. 

From Eqs. (36-38), we can derive the amplitude scattering matrix as follows ... 

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