As discussed in Chapter 5, scattering in the far zone is unique in that the scattered field always evolves into a simple outgoing spherical wave, irrespective of the physical nature of the scatterer. It should therefore be important, as well as instructive, to analyze how the general concepts introduced in Chapters 7–12 apply to this simplest type of electromagnetic scattering. Indeed, electromagnetic scattering in the far zone of an object was previously described in terms of the incident and scattered fields related by the amplitude scattering matrix, that is, in terms of typically unobservable quantities. The aim of this chapter is to describe FFS in terms of quantities directly measurable with a WCR and/or directly quantifying the electromagnetic energy budget of a inite volume element enclosing the scattering object.

We will begin by considering the simplest case of monochromatic scattering by a fixed object imbedded in a lossless homogeneous medium and then generalize the results by allowing the incident field to be a polychromatic parallel beam and the object to change randomly in time. The main results of this chapter will be straightforward mathematical corollaries of the total field in the far zone being a superposition of plane and spherical wavefronts. Unlike the more complex case of near-field scattering by a large multi-particle group, the relative mathematical simplicity of FFS will allow us to bypass the computation of the PST and work directly with the Poynting vector and the Stokes parameters.

In Chapters 13 and 15 we have introduced a number of important far-field characteristics and discussed their general mathematical properties. In Chapter 14, we have seen that the same quantities enter the FOSA for a small group of randomly and sparsely distributed particles observed from a sufficiently remote location. In Chapter 19, we will see that the same far-field characteristics enter the RTE for a very large random group of sparsely distributed particles observed from a point located in the near zone of the group. Given the ubiquity and great practical importance of these quantities, it would be essential to discuss their typical qualitative and quantitative traits. We will do this mostly with the help of the numerically exact LMT, TMM, and STMM described in the preceding chapter.

We have seen in Chapter 14 that, in the framework of the FOSA, the actual particles forming a random N-particle group are effectively replaced by N statistically identical copies of the virtual FOSA particle. We will see later that the same happens in the RTT. As a consequence, the far-field characteristics serve a triple purpose of representing: (i) the whole object in the framework of the FFS formalism, (ii) the virtual FOSA particle, and (iii) the virtual RTT particle. In this chapter we will not make a distinction between these three scenarios and will use the term “particle” in application to the whole FFS object, the virtual FOSA particle, or the virtual RTT particle.

We have seen in the preceding chapter that a WCR directly measures the absolute value of the time-averaged Poynting vector (i.e., the intensity) for the superposition of plane or near-plane wavefronts filtered out by the {objective lens, diaphragm} combination, by relaying it onto the sensitive surface of the photodetector. Since all these wavefronts propagate in essentially the same direction, i.e., along the optical axis of the instrument, they can be thought of as forming a parallel beam that can be characterized by all four Stokes parameters rather than only by the first one, i.e., the intensity (Sections 9.2 and 9.3). It turns out that by inserting special optical elements between the relay lens and the detector, it is possible to modify this beam in such a way that the new first Stokes parameter of the beam reaching the photodetector contains information about the second, third, or fourth Stokes parameters of the original beam. This is usually done by using so-called polarizers and retarders, and typically involves a succession of several measurements to fully characterize the four-component Stokes column vector.

The following discussion will be based on the assumption that the WCR faces a monochromatic plane wave. However, the additivity of the Stokes parameters derived in Sections 9.2 and 9.3 allows for a straightforward generalization of the results to polychromatic light assuming that, according to Section 11.5, the range of angular frequencies involved is sufficiently narrow.

The main advantage of the microphysical approach to radiative transfer is that it establishes a direct link between the MMEs and the RTE via a sequence of nine unambiguously defined and physically realizable assumptions and approximations summarized in Section 19.13. By keeping assumptions 1, 2, and 4–8 from Section 19.13, but relaxing approximations 3 and 9, one can extend the micro-physical approach and establish a similar direct link between the MMEs and the effect of WL of electromagnetic waves by a sparse DRM. Specifically, one can supplement the computation of the ladder component of the dyadic correlation function with the computation of the so-called “cyclical” component. The latter is caused by pairs of multi-particle sequences exemplified by Plates 18.1e and 18.1h. As we have seen in Section 18.4, the sum of the ladder and cyclical components can be expected to provide a better representation of optical observables at certain points located in the far zone of the particulate medium.

It is important to recognize that WL is not an independent physical phenomenon. It is implicitly contained in the exact solution of the MMEs (Section 18.4) but “falls through the cracks” when one resorts to the ladder approximation in order to simplify the computation. Therefore, one may characterize WL as the difference between the exact solution of the MMEs for a sparse DRM and the ladder approximation, although this characterization may still not be fully accurate since it neglects the existence of multi-particle sequences that go through a particle more than once.

In order to use the results of Sections 19.10—19.12 in various practical applications, one needs efficient techniques for solving the RTE in either the integral or the integro-differential form. Unfortunately, like many other integral and integro-differential equations, the RTE is difficult to solve analytically or numerically. In order to facilitate the solution, it is customary to make several simplifying assumptions. The most typical of them, which will be used throughout this chapter, are the assumptions that the particulate medium:

1. • is plane parallel;

2. • has an infinite horizontal extent; and

3. • is illuminated from above by a plane electromagnetic wave or a parallel polychromatic beam with quasi-monochromatic components.

These assumptions mean that all statistically averaged optical properties of the medium and all observable characteristics of the radiation field may vary only in the vertical direction and are independent of the horizontal coordinates. Taken together, these assumptions specify the so-called standard one-dimensional problem of the RTT and provide a model relevant to a great variety of applications in diverse fields of science and engineering.

To simplify the standard problem even further, we will also assume that the particulate medium is populated by statistically isotropic and mirror-symmetric random particles and use the extinction matrix given by Eq. (15.42) and the phase matrix given by Eqs. (15.20) and (15.21).

In this chapter we will derive several general equations describing the radiation field in the particular case of plane-parallel scattering geometry.

In accordance with the preceding discussion, the theoretical foundation for describing electromagnetic scattering by particles and particle groups in this book is provided by classical macroscopic electromagnetics. This chapter is intended to summarize basic concepts and equations of electromagnetic theory that will be used extensively in the remainder of the book and introduce the necessary notation.

We start by formulating the primordial set of time-domain MMEs, constitutive relations, and boundary conditions. This is followed by a general analysis of time-harmonic fields and the frequency-domain MMEs. Finally, we discuss energy conservation in the framework of the frequency-domain macroscopic electromagnetics.

The macroscopic Maxwell equations and constitutive relations

As already mentioned, the basic laws of macroscopic electromagnetics are adopted in this textbook essentially as axioms describing the spatial distribution and temporal behavior of the electromagnetic field and its interaction with matter. A thorough justification of this approach can be found in the textbook by Roth-well and Cloud (2009). The microphysical derivation of the MMEs from more fundamental physical principles and the range of their validity are discussed by de Groot (1969), Robinson (1973), Akhiezer and Peletminskii (1981), Suttorp (1989), and Jackson (1998).

The formalism described in Chapters 4 and 5 applies equally to a scatterer in the form of a single body and to a fixed multi-particle group. However, when the scattering object is a cluster consisting of touching and/or separated distinct components, then it is often convenient to use a modified formalism in which the total scattered field is explicitly represented as a vector superposition of the partial fields contributed by the cluster components. This approach is based on the system of integral so-called Foldy equations (FEs) which follow directly from the MMEs, automatically incorporate all boundary conditions and the radiation condition at infinity, and rigorously describe the scattered electric field at any point in space. In this chapter, we will derive both the exact form of the FEs and an approximate far-field version. The latter applies to a group of widely separated particles and offers significant simplifications essential for the development of microphysical theories of radiative transfer and WL.

Vector form of the Foldy equations

Consider electromagnetic scattering by a fixed group of N finite particles collectively occupying the interior region VINT, according to Eq. (4.1). As before, we assume that the particles are imbedded in an infinite, homogeneous, linear, isotropic, and nonabsorbing medium.

Although the far-field formulas derived in Chapter 13 for a stochastic scattering object are attractively simple, they have a rather limited range of applicability. Indeed, one of the criteria of the far zone, Eq. (5.14), becomes quite challenging for an object significantly greater than the wavelength (Problems 14.1 and 14.2) and often prohibits direct use of the far-field approximation. It turns out, however, that many aspects of the far-field formalism can be preserved if the stochastic scattering object belongs to a particular morphological type. Specfically, in this chapter we will assume that the object can be defined as a group of N distinct particles separated from each other and distributed throughout an imaginary volume element V. Accordingly, the starting point of the analysis of electromagnetic scattering by this object will be the FEs derived in Section 6.1 rather than the VIE. In addition, we will assume that:

1. • the total number of particles forming the object is sufficiently small and the average distance between them is sufficiently large that in the framework of the FEs (Section 6.1), each particle can be assumed to be “excited” only by the incident field;

2. • the N-particle object is observed from a distance much greater than any linear dimension of the volume element V;

3. • although the observation point is allowed to be in the near zone of the entire object, it is remote enough to be in the far zone of any of the N distinct particles forming the object; and

4. • the N particles are moving randomly and independently of each other throughout the volume element V.

The discussion in the preceding chapter was limited to the far field of a random particulate volume comprising a moderate number of particles. However, one is often interested in the near field in order to compute the energy budget of a volume element of a DRM rather than of the entire DRM. Furthermore, in the majority of actual applications the detector of light is located in the near zone of the entire DRM, including the cases of being inside the particulate volume. Although numerically exact solvers of the MMEs such as the MSM (Section 16.1.3) can be used to compute the near field of a DRM (see, e.g., Mackowski and Mishchenko 2013), the applicability of this direct approach is still limited in terms of the number of constituent particles and the overall size of the particulate volume relative to the wavelength.

This implies that the near-field solution of the MMEs for stochastic multi-particle objects such as those shown in Plates 1.1b—1.1f has to be based on a number of simplifying assumptions such as ergodicity, the sparsity and statistical uniformity of the particles' spatial distribution, and the asymptotic limit N → ∞, where N is the number of particles in a DRM. The main objective of the following analysis is to show that this methodology can indeed be used to derive analytically a set of closed-form equations that can be solved numerically with relative ease.

As we have discussed in Chapter 1, the presence of an object with a refractive index different from that of the surrounding host medium changes the electromagnetic field that would otherwise exist in an unbounded homogeneous space. The difference between the total field in the presence of the object and the total field that would exist in the absence of the object can be thought of as the field scattered by the object. In other words, the total electromagnetic field in the presence of the object is mathematically represented as the vector sum of the incident and scattered fields.

The spatial distribution of the scattered field depends on specific characteristics of the incident field as well as on such properties of the scatterer as its size relative to the wavelength and its morphology, composition, and orientation. Therefore, in practice one usually must solve the scattering problem anew every time some or all of these input parameters change. It is appropriate, however, to consider first the general mathematical description of electromagnetic scattering without making any detailed assumptions about the scattering object, except that it is composed of linear, isotropic, and nonmagnetic materials. Hence the goal of this chapter is to establish a basic theoretical framework underlying specific problems discussed in the following chapters. Consistent with this objective and with the discussion in Section 1.1, we will assume that the scattering object is stationary and will consider only monochromatic fields whose dependence on time is completely described by the time-harmonic factor exp(—iωt).