Identify the limits of geometric optics ray tracing by numerically solving the vector Kirchhoff integral.

The properties of a pencil of light as defined approximately in the geometric optics ray tracing method are investigated. The vector Kirchhoff integral is utilized to accurately compute the electromagnetic near field in and around the pencil of light with various beam base sizes, shapes, propagation directions and medium refractive indices. If a pencil of light has geometric mean cross section size of the order p times the wavelength, it can propagate independently to a distance p2 times the wavelength, where most of the beam energy diffuses out of the beam region. This is consistent with a statement that van de Hulst made in a classical text on light scattering. The electromagnetic near fields in the pencil of light are not uniform, have complicated patterns within short distances from the beam base, and the fields tend to converge to Fraunhofer diffraction fields far away from the base.

Spectrally dependent linear depolarization and lidar ratios for nonspherical smoke aerosols.

We use the numerically exact T-matrix method to model light scattering and absorption by aged smoke aerosols at lidar wavelengths ranging from 355 to 1064 nm assuming the aerosols to be smooth spheroids or Chebyshev particles. We show that the unique spectral dependence of the linear depolarization ratio (LDR) and extinction-to-backscatter ratio (or lidar ratio, LR) measured recently for stratospheric Canadian wildfire smoke can be reproduced by a range of model morphologies, a range of spectrally dependent particle refractive indices, and a range of particle sizes. For these particles, the imaginary part of the refractive index is always less than (or close to) 0.035, and the corresponding real part always falls in the range [1.35, 1.65]. The measured spectral LDRs and LRs could be produced by nearly-spherical oblate spheroids or Chebyshev particles whose shapes resemble those of oblate spheroids. Their volume-equivalent effective radii should be large enough (r eff = 0.3 µm or greater) to produce the observed enhanced LDRs. Our study demonstrates the usefulness of triple-wavelength LDR measurements as providing additional size information for a more definitive characterization of the particle morphology and composition. Non-zero LDR values indicate the presence of nonspherical aerosols and are highly sensitive to particle shapes and sizes. On the other hand, the LR is a strong function of absorption and is very responsive to changes in the particle refractive index.

Retrieval of volcanic and man-made stratospheric aerosols from orbital polarimetric measurements.

Stratospheric aerosols that are caused by a major volcanic eruption can serve as a valuable test of global climate models, as well as severely complicate tropospheric-aerosol monitoring from space. In either case, it is highly desirable to have accurate global information on the optical thickness, size, and composition of volcanic aerosols. We report sensitivity study results, which reveal the implications of making precise multi-angle photopolarimetric measurements in a 1.378-µm spectral channel residing within a strong water-vapor absorption band. We demonstrate that, under favorable conditions, such measurements would enable near-perfect retrievals of the optical thickness, effective radius, and refractive index of stratospheric aerosols. Besides enabling accurate retrievals of volcanic aerosols, such measurements can also be used to monitor man-made particulates injected in the stratosphere for geoengineering purposes.

Additivity of integral optical cross sections for a fixed tenuous multi-particle group.

We use the volume integral equation formulation of frequency-domain electromagnetic scattering to settle the issue of additivity of the extinction, scattering, and absorption cross sections of a fixed tenuous group of particles. We show that all the integral optical cross sections of the group can be obtained by summing up the corresponding individual-particle cross sections, provided that the single-scattering approximation applies.

Modeling study of scattering and absorption properties of tar-ball aggregates.

Atmospheric tar balls (TBs) form an important class of atmospheric brown carbon (BrC) particulates. The morphology of the individual TBs is typically described as amorphous and nearly spherical. However, several studies reported observations of TBs aggregated with other aerosols or agglomerations consisting of up to tens of individual TBs. We use the superposition $T$T-matrix method to compute the scattering matrix elements and optical cross sections for a variety of TB aggregates, each of which is composed of a number of monomers whose sizes follow a lognormal distribution. The results for a TB aggregate can differ fundamentally from those calculated for two simplified models commonly used in climate modeling; viz., the external mixture of TBs and the respective volume-equivalent sphere model. Clustering of individual TBs into an aggregate can either enhance or weaken absorption depending on the wavelength, the monomer size, and how absorptive the BrC material is. In the case of strongly absorptive BrC, aggregation results in enhanced absorption only at 1064 nm, while at 355 and 532 nm TB aggregates become less effective absorbers relative to the corresponding external mixtures. The effect of aggregation is always to increase the single-scattering albedo and asymmetry parameter, sometimes more than tenfold. The significant scattering-matrix differences between a TB aggregate, the "equivalent" external mixture, and the volume-equivalent sphere model demonstrate the failure of the conventional Lorenz-Mie theory to represent the scattering properties of morphologically complex BrC aerosols. We show that TB aggregates can help explain exceptionally strong and spectrally dependent lidar depolarization ratios reported in several recent studies.

Multiple scattering of polarized light by particles in an absorbing medium.

We study multiple scattering of light by particles embedded in an absorbing host medium using a recently developed single-scattering and vector radiative-transfer methodology directly based on the Maxwell equations. The first-principles results are compared with those rendered by the conventional heuristic approach according to which the single-scattering properties of particles can be computed by assuming that the host medium is nonabsorbing. Our analysis shows that the conventional approach yields very accurate results in the case of aerosol and cloud particles suspended in an absorbing gaseous atmosphere. In the case of air bubbles in water, the traditional approach can cause large relative errors in reflectance, but only when strong absorption in the host medium makes the resulting reflectance very small. The corresponding polarization errors are substantially smaller.

Electromagnetic scattering by discrete random media. IV: Coherent backscattering.

The problem of backscattering of light by a discrete random medium illuminated by an obliquely incident plane electromagnetic wave is considered. The analysis is performed in a linear-polarization basis and includes (i) a complete derivation of the cross reflection matrix for a layer with densely and sparsely distributed particles, (ii) the design of an approximate method for computing the ladder and cross reflection matrices in the case of a semi-infinite medium with a sparse distribution of particles, (iii) the derivation of the relations between the elements of the ladder and cross reflection matrices in the exact backscattering direction for dense and sparse media, and (iv) the development of practical algorithms for solving the underlying integral equations by the method of Picard iterations and the discrete ordinate method. Simulation results for particles with large size parameters are also presented.

An overview of methods for deriving the radiative transfer theory from the Maxwell equations. II: Approach based on the Dyson and Bethe-Salpeter equations.

In this paper, the vector radiative transfer equation is derived by means of the vector integral Foldy equations describing the electromagnetic scattering by a group of particles. By Assuming that in a discrete random medium the positions of the particles are statistically independent and by applying the Twersky approximation to the order-of-scattering expansion of the total field, we derive the Dyson equation for the coherent field and the ladder approximated Bethe-Salpeter equation for the dyadic correlation function. Then, under the far-field assumption for sparsely distributed particles, the Dyson equation is reduced to the Foldy integral equation for the coherent field, while the iterated solution of the Bethe-Salpeter equation ultimately yields the vector radiative transfer equation.

Electromagnetic scattering by discrete random media. II: The coherent field.

The computation of the coherent field in the case of a plane electromagnetic wave obliquely incident on a discrete random layer with non-scattering boundaries is addressed. For dense media, the analysis is based on a special-form solution for the conditional configuration-averaged exciting field coefficients, and is restricted to the computation of the so-called zeroth-order fields without a special treatment of the boundary regions. In this setting, we calculate the coherent fields reflected and transmitted by the layer, and the coherent field inside the layer. We found that these fields are analytically equivalent to plane electromagnetic waves, and investigated the fulfillment of the boundary conditions for the electric fields at the layer interfaces. The results are then particularized to the cases of normal incidence and a semi-infinite discrete random medium. For sparsely distributed particles, we present a self-consistent derivation of the coherent field and discuss the Twersky and Foldy approximations.

Radiative transfer in a discrete random medium adjacent to a half-space with a rough interface.

For a macroscopically plane-parallel discrete random medium, the boundary conditions for the specific coherency dyadic at a rough interface are derived. The derivation is based on a modification of the Twersky approximation for a scattering system consisting of a group of particles and the rough surface, and reduces to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave propagating in a discrete random medium with non-scattering boundaries. In a matrix-form setting, the boundary conditions for the specific coherency dyadic imply the boundary conditions for specific intensity column vectors which in turn, yield the expressions for the reflection and transmission matrices. The derived expressions are shown to be identical to those obtained by applying a phenomenological approach based on a facet model to the solution of the scattering problem for a rough surface illuminated by a plane electromagnetic wave.

Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial.

Although free space cannot generate electromagnetic waves, the majority of existing accounts of frequency-domain electromagnetic scattering by particles and particle groups are based on the postulate of existence of an impressed incident field, usually in the form of a plane wave. In this tutorial we discuss how to account for the actual existence of impressed source currents rather than impressed incident fields. Specifically, we outline a self-consistent theoretical formalism describing electromagnetic scattering by an arbitrary finite object in the presence of arbitrarily distributed impressed currents, some of which can be far removed from the object and some can reside in its vicinity, including inside the object. To make the resulting formalism applicable to a wide range of scattering-object morphologies, we use the framework of the volume integral equation formulation of electromagnetic scattering, couple it with the notion of the transition operator, and exploit the fundamental symmetry property of this operator. Among novel results, this tutorial includes a streamlined proof of fundamental symmetry (reciprocity) relations, a simplified derivation of the Foldy equations, and an explicit analytical expression for the transition operator of a multi-component scattering object.

Far-field Lorenz-Mie scattering in an absorbing host medium. II: Improved stability of the numerical algorithm.

A recently developed FORTRAN program computing far-field optical observables for spherical particles in an absorbing medium has exhibited numerical instability arising when the product of the particle vacuum size parameter and the imaginary part of the refractive index of the host becomes sufficiently large. We offer a simple analytical explanation of this instability and propose a compact numerical algorithm for the stable computation of Lorenz-Mie coefficients based on an upward recursion formula for spherical Hankel functions of a complex argument. Extensive tests confirm an excellent accuracy of this algorithm approaching machine precision. The improved public-domain FORTRAN program is available at https://www.giss.nasa.gov/staff/mmishchenko/Lorenz-Mie.html.

Retrieval of microphysical characteristics of particles in atmospheres of distant comets from ground-based polarimetry.

We summarize unique aperture data on the degree of linear polarization observed for distant comets C/2010 S1, C/2010 R1, C/2011 KP36, C/2012 J1, C/2013 V4, and C/2014 A4 with heliocentric distances exceeding 3 AU. Observations have been carried out at the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences (Nizhnij Arkhyz, Russia) during the period from 2011 to 2016. The measured negative polarization proves to be significantly larger in absolute value than what is typically observed for comets close to the Sun. We compare the new observational data with the results of numerical modeling performed with the T-matrix and superposition T-matrix methods. In our computer simulations, we assume the cometary coma to be an optically thin cloud containing particles in the form of spheroids, fractal aggregates composed of spherical monomers, and mixtures of spheroids and aggregate particles. We obtain a good semi-quantitative agreement between all polarimetric data for the observed distant comets and the results of numerical modeling for the following models of the cometary dust: (i) a mixture of submicrometer water-ice oblate spheroids with aggregates composed of submicrometer silicate monomers; and (ii) a mixture of submicrometer water-ice oblate spheroids and aggregates consisting of both silicate and organic monomers. The michrophysical parameters of these models are presented and discussed.

Information content of bistatic lidar observations of aerosols from space.

We present, for the first time, a quantitative retrieval error-propagation study for a bistatic high spectral resolution lidar (HSRL) system intended for detailed quasi-global monitoring of aerosol properties from space. Our results demonstrate that supplementing a conventional monostatic HSRL with an additional receiver flown in formation at a scattering angle close to 165° dramatically increases the information content of the measurements and allows for a sufficiently accurate characterization of tropospheric aerosols. We conclude that a bistatic HSRL system would far exceed the capabilities of currently flown or planned orbital instruments in monitoring global aerosol effects on the environment and on the Earth's climate. We also demonstrate how the commonly used a priori "regularization" methodology can artificially reduce the propagated uncertainties and can thereby be misleading as to the real retrieval capabilities of a measurement system.

Extinction by a homogeneous spherical particle in an absorbing medium.

We use a recent computer implementation of the first-principles theory of electromagnetic scattering to compute far-field extinction by a spherical particle embedded in an absorbing unbounded host. Our results show that the suppressing effect of increasing absorption inside the host medium on the ripple structure of the extinction efficiency factor as a function of the size parameter is similar to the well-known effect of increasing absorption inside a particle embedded in a nonabsorbing host. However, the accompanying effects on the interference structure of the extinction efficiency curves are diametrically opposite. As a result, sufficiently large absorption inside the host medium can cause negative particulate extinction. We offer a simple physical explanation of the phenomenon of negative extinction consistent with the interpretation of the interference structure as being the result of interference of the field transmitted by the particle and the diffracted field due to an incomplete wavefront resulting from the blockage of the incident plane wave by the particle's geometrical projection.

On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles.

Although the model of randomly oriented nonspherical particles has been used in a great variety of applications of far-field electromagnetic scattering, it has never been defined in strict mathematical terms. In this Letter, we use the formalism of Euler rigid-body rotations to clarify the concept of statistically random particle orientations and derive its immediate corollaries in the form of the most general mathematical properties of the orientation-averaged extinction and scattering matrices. Our results serve to provide a rigorous mathematical foundation for numerous publications in which the notion of randomly oriented particles and its light-scattering implications have been considered intuitively obvious.

On Babinet's principle and diffraction associated with an arbitrary particle.

Babinet's principle is widely used to compute the diffraction by a particle. However, the diffraction by a 3-D object is not totally the same as that simulated with Babinet's principle. This Letter uses a surface integral equation to exactly formulate the diffraction by an arbitrary particle and illustrate the condition for the applicability of Babinet's principle. The present results may serve to close the debate on the diffraction formalism.

Electromagnetic scattering and emission by a fixed multi-particle object in local thermal equilibrium: General formalism.

The majority of previous studies of the interaction of individual particles and multi-particle groups with electromagnetic field have focused on either elastic scattering in the presence of an external field or self-emission of electromagnetic radiation. In this paper we apply semi-classical fluctuational electrodynamics to address the ubiquitous scenario wherein a fixed particle or a fixed multi-particle group is exposed to an external quasi-polychromatic electromagnetic field as well as thermally emits its own electromagnetic radiation. We summarize the main relevant axioms of fluctuational electrodynamics, formulate in maximally rigorous mathematical terms the general scattering-emission problem for a fixed object, and derive such fundamental corollaries as the scattering-emission volume integral equation, the Lippmann-Schwinger equation for the dyadic transition operator, the multi-particle scattering-emission equations, and the far-field limit. We show that in the framework of fluctuational electrodynamics, the computation of the self-emitted component of the total field is completely separated from that of the elastically scattered field. The same is true of the computation of the emitted and elastically scattered components of quadratic/bilinear forms in the total electromagnetic field. These results pave the way to the practical computation of relevant optical observables.

Electromagnetic scattering by spheroidal volumes of discrete random medium.

We use the superposition T-matrix method to compare the far-field scattering matrices generated by spheroidal and spherical volumes of discrete random medium having the same volume and populated by identical spherical particles. Our results fully confirm the robustness of the previously identified coherent and diffuse scattering regimes and associated optical phenomena exhibited by spherical particulate volumes and support their explanation in terms of the interference phenomenon coupled with the order-of-scattering expansion of the far-field Foldy equations. We also show that increasing nonsphericity of particulate volumes causes discernible (albeit less pronounced) optical effects in forward and backscattering directions and explain them in terms of the same interference/multiple-scattering phenomenon.

First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media.

A discrete random medium is an object in the form of a finite volume of a vacuum or a homogeneous material medium filled with quasi-randomly and quasi-uniformly distributed discrete macroscopic impurities called small particles. Such objects are ubiquitous in natural and artificial environments. They are often characterized by analyzing theoretically the results of laboratory, in situ, or remote-sensing measurements of the scattering of light and other electromagnetic radiation. Electromagnetic scattering and absorption by particles can also affect the energy budget of a discrete random medium and hence various ambient physical and chemical processes. In either case electromagnetic scattering must be modeled in terms of appropriate optical observables, i.e., quadratic or bilinear forms in the field that quantify the reading of a relevant optical instrument or the electromagnetic energy budget. It is generally believed that time-harmonic Maxwell's equations can accurately describe elastic electromagnetic scattering by macroscopic particulate media that change in time much more slowly than the incident electromagnetic field. However, direct solutions of these equations for discrete random media had been impracticable until quite recently. This has led to a widespread use of various phenomenological approaches in situations when their very applicability can be questioned. Recently, however, a new branch of physical optics has emerged wherein electromagnetic scattering by discrete and discretely heterogeneous random media is modeled directly by using analytical or numerically exact computer solutions of the Maxwell equations. Therefore, the main objective of this Report is to formulate the general theoretical framework of electromagnetic scattering by discrete random media rooted in the Maxwell-Lorentz electromagnetics and discuss its immediate analytical and numerical consequences. Starting from the microscopic Maxwell-Lorentz equations, we trace the development of the first-principles formalism enabling accurate calculations of monochromatic and quasi-monochromatic scattering by static and randomly varying multiparticle groups. We illustrate how this general framework can be coupled with state-of-the-art computer solvers of the Maxwell equations and applied to direct modeling of electromagnetic scattering by representative random multi-particle groups with arbitrary packing densities. This first-principles modeling yields general physical insights unavailable with phenomenological approaches. We discuss how the first-order-scattering approximation, the radiative transfer theory, and the theory of weak localization of electromagnetic waves can be derived as immediate corollaries of the Maxwell equations for very specific and well-defined kinds of particulate medium. These recent developments confirm the mesoscopic origin of the radiative transfer, weak localization, and effective-medium regimes and help evaluate the numerical accuracy of widely used approximate modeling methodologies.