Electromagnetic scattering by a morphologically complex object: Fundamental concepts and common misconceptions

Following Keller (Proc Symp Appl Math 1962;13:227-46), we classify all theoretical treatments of electromagnetic scattering by a morphologically complex object into first-principle (or “honest” in Keller's terminology) and phenomenological (or “dishonest”) categories. This helps us identify, analyze, and dispel several profound misconceptions widespread in the discipline of electromagnetic scattering by solitary particles and discrete random media. Our goal is not to call for a complete renunciation of phenomenological approaches but rather to encourage a critical and careful evaluation of their actual origin, virtues, and limitations. In other words, we do not intend to deter creative thinking in terms of phenomenological short-cuts, but we do want to raise awareness when we stray (often for practical reasons) from the fundamentals. The main results and conclusions are illustrated by numerically-exact data based on direct numerical solutions of the macroscopic Maxwell equations.     

Incoherent and coherent backscattering of light by a layer of densely packed random medium

The problem of light scattering by a layer of densely packed discrete random medium is considered. The theory of light scattering by systems of nonspherical particles is applied to derive equations corresponding to incoherent (diffuse) and interference parts of radiation reflected from the medium. A solution of the system of linear equations describing light scattering by a system of particles is represented by iteration. It is shown that the symmetry properties of the T-matrices and of the translation coefficients for the vector Helmholtz harmonics lead to the reciprocity relation for an arbitrary iteration. This relation is applied to consider the backscattering enhancement phenomenon. Equations expressing the incoherent and interference parts of reflected light from statistically homogeneous and isotropic plane-parallel layer of medium are given. In the exact backscattering direction the relation between incoherent and interference parts is identical to that of sparse media.     

Numerical simulations of single and multiple scattering by fractal ice clusters

We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4≤R≤10 cm and fractal dimensions D_f=2.5 and 3.We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D_f=2.5, but the differences can increase somewhat for D_f=3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.     

Scattering of electromagnetic waves by ensembles of particles and discrete random media

Current problems of the theory of multiple scattering of electromagnetic waves by discrete random media are reviewed, with an emphasis on densely packed media. All equations presented are based on the rigorous theory of electromagnetic scattering by an arbitrary system of non-spherical particles. The main relations are derived in the circular-polarization basis. By applying methods of statistical electromagnetics to a discrete random medium in the form of a plane-parallel layer, we transform these relations into equations describing the average (coherent) field and equations for the sums of ladder and cyclical diagrams in the framework of the quasi-crystalline approximation. The equation for the average field yields analytical expressions for the generalized Lorentz-Lorenz law and the generalized Ewald-Oseen extinction theorem, which are traditionally used for the calculation of the effective refractive index. By assuming that the particles are in the far-field zones of each other, we transform all equations asymptotically into the well-known equations for sparse media. Specifically, the equation for the sum of the ladder diagrams is reduced to the classical vector radiative transfer equation. We present a simple approximate solution of the equation describing the weak localization (WL) effect (i.e., the sum of cyclical diagrams) and validate it by using experimental and numerically exact theoretical data. Examples of the characteristics of WL as functions of the physical properties of a particulate medium are given. The applicability of the interference concept of WL to densely packed media is discussed using results of numerically exact computer solutions of the macroscopic Maxwell equations for large ensembles of spherical particles. These results show that theoretical predictions for spare media composed of non-absorbing or weakly absorbing particles are reasonably accurate if the particle packing density is less than ∼30%. However, a further increase of the packing density and/or absorption may cause optical effects not predicted by the low-density theory and caused by near-field effects. The origin of the near-filed effects is discussed in detail.     

Maxwell's equations, radiative transfer, and coherent backscattering: A general perspective

This tutorial paper provides a general overview of the hierarchy of problems involving electromagnetic scattering by particles and clarifies the place of the radiative transfer theory and the theory of coherent backscattering in the context of classical electromagnetics. The self-consistent microphysical approach to radiative transfer is compared with the traditional phenomenological treatment.     

Coherent backscattering effects for discrete random media: Numerical and theoretical results

Manifestation of the backscattering enhancement phenomenon in the reflection matrix elements of the coherent component of scattered radiation is considered. The dependence of the coherent backscattering effects on the microphysical properties of the medium scatterers are investigated. It is shown that random media of fractal-like clusters exhibit brightness and polarization opposition effects, which are like those observed for some atmosphereless Solar system bodies. Conditions for a bimodal angle dependence in the degree of linear polarization are discussed and the manifestation of the enhanced backscattering phenomenon in the intensity of scattered radiation is studied.     

Boson peak, flickering noise, backscattering processes and radiative transfer in random media

The method of matrix Green’s functions in the classical theory of electromagnetic waves is stated. This method allows to obtain a closed equation system in the presence of the random media for the calculation both coherent, and incoherent (fluctuating) components of radiation. The density and heterogeneity of scattering media can be arbitrary. The coherent channel is calculated independently. The fluctuating radiation distribution in the medium is developed initially by an interference pattern generated by the coherent channel. The limitations of the processes speed are absent. The theory embraces such phenomena as the boson peak, flickering noise, memory effect, backscattering processes and also conventional radiative transfer equation and Fresnel’s formulae.     

Directional radiometry and radiative transfer: A new paradigm

Measurements with directional radiometers and calculations based on the radiative transfer equation (RTE) have been at the very heart of weather and climate modeling and terrestrial remote sensing. The quantification of the energy budget of the Earth's climate system requires exquisite measurements and computations of the incoming and outgoing electromagnetic energy, while global characterization of climate system's components relies heavily on theoretical inversions of observational data obtained with various passive and active instruments. The same basic problems involving electromagnetic energy transport and its use for diagnostic and characterization purposes are encountered in numerous other areas of science, biomedicine, and engineering. Yet both the discipline of directional radiometry and the radiative transfer theory (RTT) have traditionally been based on phenomenological concepts many of which turn out to be profound misconceptions. Contrary to the widespread belief, a collimated radiometer does not, in general, measure the flow of electromagnetic energy along its optical axis, while the specific intensity does not quantify the amount of electromagnetic energy transported in a given direction.The recently developed microphysical approach to radiative transfer and directional radiometry is explicitly based on the Maxwell equations and clarifies the physical nature of measurements with collimated radiometers and the actual content of the RTE. It reveals that the specific intensity has no fundamental physical meaning besides being a mathematical solution of the RTE, while the RTE itself is nothing more than an intermediate auxiliary equation. Only under special circumstances detailed in this review can the solution of the RTE be used to compute the time-averaged local Poynting vector as well as be measured by a collimated radiometer. These firmly established facts make the combination of the RTE and a collimated radiometer useful in a well-defined range of applications. However, outside the domain of validity of the RTT the practical usefulness of measurements with collimated radiometers remains uncertain, while the theoretical modeling of these measurements and the solution of the energy-budget problem require a more sophisticated approach than solving the RTE.     

Light scattering by aggregates of varying porosity and the opposition phenomena observed in the low-albedo particulate media

The rule that the opposition phenomena in brightness and linear polarization observed in many regolith surfaces usually accompany each other is violated in the cases of very dark asteroids and particulate samples: practically no nonlinear surge of brightness to opposition is observed while the branch of negative polarization at small phase angles exists. To explain this fact, we model the light scattering by particulate media with ensembles of spherical particles (with size comparable to the wavelength) of varying packing density and refractive index. The superposition T-matrix method is used. The increase in the absorption and/or packing density diminishes the amplitude of the brightness opposition peak, and its profile becomes wider. The influence on the branch of negative polarization is more complex and depends on the relation between the size parameters of the constituents, the refractive index, and the porosity. However, the feature common to all considered cases is that the negative branch changes its shape and the polarization minimum moves to the inversion point. This behavior radically differs from that observed in nonabsorbing ensembles of particles and reflects the fact that the efficiency of the coherent backscattering, which mainly determines these characteristics in nonabsorbing ensembles (to the packing density of about 30%), decreases. Moreover, since the angular profiles are not simply damped, but the polarization minimum changes its angular position, we may conclude that the near-field interaction of the constituents becomes important: the shielding of particles by each other eliminates many constituents from the scattering and the near-field effects promote the negative polarization and smooth the backscattering brightness surge. Due to this, when the packing density exceeds 10-20%, the opposition phenomena in absorbing ensembles are caused not only by the coherent backscattering, and situations, when the opposition brightness surge is practically suppressed, but the negative branch of polarization still survives, are possible. This may explain the fact that the dark regolith surfaces show no brightness opposition effect, but produce the branch of negative polarization with the minimum shifted from opposition.     

Coherent backscattering of light by a layer of discrete random medium

We consider the problem of backscattering of light by a layer of discrete random medium illuminated by an obliquely incident plane electromagnetic wave. The multiply scattered reflected radiation is assumed to consist of incoherent and coherent parts, the coherent part being caused by the interference of multiply scattered waves. Formulas describing the characteristics of the reflected radiation are derived assuming that the scattering particles are spherical. The formula for the incoherent contribution reproduces the standard vector radiative transfer equation. The interference contribution is expressed in terms of a system of Fredholm integral equations with kernels containing Bessel functions. The special case of the backscattering direction is considered in detail. It is shown that the angular width of the backscattering interference peak depends on the polar angle of the incident wave and on the azimuth angle of the reflection direction.     

Coherent backscattering: Conceptions and misconceptions (reply to comments by Bruce W. Hapke and Robert M. Nelson)

Although the note by Hapke and Nelson has virtually no relevance to our original publication, it contains a number of statements that are misleading and/or wrong. We, therefore, use this opportunity to dispel several profound misconceptions that continue to hinder the progress in remote sensing of planetary surfaces.     

Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation

This paper continues a systematic theoretical analysis of electromagnetic scattering by a group of arbitrarily sized, shaped, and oriented particles embedded in an absorbing, homogeneous, isotropic, and unbounded medium. The previously developed microphysical approach is used to derive the generalized form of the radiative transfer equation (RTE) applicable to a large group of sparsely, randomly, and uniformly distributed particles. The derivation of the RTE directly from the macroscopic Maxwell equations yields unambiguous and definitive analytical expressions for the participating quantities and thereby fully resolves the lasting controversy caused by the conflicting outcomes of several phenomenological approaches.     

Solution of the equation of transfer for coherent anisotropic scattering by double interval spherical harmonic method

The double interval spherical harmonic method introduced effectively by Wilson and Sen has already been used by Ghosh and Karanjai to solve the equation of radiative transfer in coherent isotropic scattering atmosphere, originally developed by Woolley and Stibbs. The same method has been successfully used in this paper to solve the equation of transfer for coherent anisotropic scattering.     

Radiative properties of densely packed spheres in semitransparent media: A new geometric optics approach

This contribution presents a new Ray-tracing method for calculating effective radiative properties of densely packed spheres in non-absorbing or semitransparent host medium. The method is restricted to the geometric optic objects and neglects the wave effects. The effective radiative properties such as the absorption and scattering coefficients, and phase function are retrieved from the calculation of mean-free paths of scattering and absorption, and the angular scattering probability of radiation propagating in the dispersed medium. The model accounts for the two geometric effects called here as non-point scattering and ray transportation effects. The successful comparison of the current model with data of radiative properties and transmittances of particle beds in a non-absorbing medium reported in the literature confirm its suitability. It is shown that: (i) for opaque or absorbing particles (not systematically opaque), the non-point scattering is the dominant geometric effects whereas both non-point scattering and ray transportation effects occur for weakly absorbing and transparent particles. In the later cases, these two geometric effects oppose and may cancel out. This may explain why the Independent scattering theory works well for packed of quasi-transparent particles; (ii) the non-point scattering and ray transportation effects can be captured through the scattering and absorption coefficients while using the classical form of phase function. This enables using the standard radiative transfer equation (RTE); (iii) the surrounding medium absorption can be accounted for without any homogenization rule. It contributes to increasing the effective absorption coefficient of the composite medium as expected but, at the same time, it reduces the particle extinction; and (iv) the current transfer calculation predicts remarkably the results of direct Monte Carlo (MC) simulation. This study tends therefore to confirm that the RTE can be applied to densely packed media by using effective radiative properties.     

T-matrix method and its applications to electromagnetic scattering by particles: A current perspective

This note serves as a short introduction to the reprint of our article “T-matrix computations of light scattering by nonspherical particles: a review” (JQSRT 1996; 55:535-75). We first discuss the motivation for writing that article and explain its historical context. This is followed by a short overview of more recent developments.     

Radiation force caused by scattering, absorption, and emission of light by nonspherical particles

General formulas for computing the radiation force exerted on arbitrarily oriented and arbitrarily shaped nonspherical particles due to scattering, absorption, and emission of electromagnetic radiation are derived. For randomly oriented particles with a plane of symmetry, the formula for the average radiation force caused by the particle response to external illumination reduces to the standard Debye formula derived from the Lorenz–Mie theory, whereas the average radiation force caused by emission vanishes.     

Temperature response in participating media with anisotropic scattering caused by pulsed lasers

It is not by isotropic scattering but by anisotropic scattering that radiant energy is redistributed in some materials containing real particles, fibers, or impurities. In some instances, great difference can be caused in transient thermal behavior between isotropic scattering and anisotropic scattering media. Ray tracing method combined with Hottel's zonal method is introduced to deduce thermal radiative source term for various optical boundary conditions induced by collimated incidence passing through translucent boundary. Temperature response caused by laser pulse at non-incident side of participating and anisotropic scattering media is examined. We investigate effects of scattering albedo, scattering phase function, initial temperature of media and thickness of media on temperature response. Results obtained for anisotropic scattering media are compared with those for isotropic scattering one and show that anisotropic scattering must be considered in the simulating measurement of thermophysical properties by the laser flash method for some materials with big scattering albedo which behave anisotropically, or big error will be introduced; forward scattering can increase excess temperature and backward scattering can decrease it at non-incident side of the considered sample irradiated by laser pulse.     

Comparison between experimental and 2-D numerical studies of multiple scattering in Inconel600® by means of array probes

Ultrasonic non-destructive testing of polycrystalline structures can be disturbed by scattering at grain boundaries. Understanding and modeling this so-called “structural noise” is crucial for characterization as well as detection purposes. Structural noise can be considered as a fingerprint of the material under investigation, since it contains information about its microstructure. The interpretation of experimental data necessitates an accurate comprehension of complex phenomena that occur in multiple scattering media and thus robust scattering models. In particular, numerical models can offer the opportunity to realize parametrical studies on controlled microstructures. However, the ability of the model to simulate wave propagation in complex media must be validated. In that perspective, the main objective of the present work is to evaluate the ability of the finite-element code ATHENA 2D to reproduce typical features of multiple wave scattering in the context of ultrasonic non-destructive evaluation, with an array of sources and receivers. Experiments were carried out with a 64-element array, around 2 MHz. The sample was a mock-up of Inconel600® exhibiting a coarse grain structure with a known grain size distribution. The numerical model of this microstructure is based on Voronoi diagrams. Two physical parameters were used to compare numerical and experimental data: the coherent backscattering peak, and the singular value distribution of the array response matrix. Though the simulations are 2-D, a good agreement was found between simulated and experimental data.