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.     

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.     

Apparent optical properties of spherical particles in absorbing medium

An apparent absorption efficiency for spherical particles in absorbing medium is introduced to take into account the non-exponential absorption of the near-field scattered radiation in the absorbing medium. The apparent extinction, which is the summation of the apparent scattering efficiency following previous studies and the apparent absorption efficiency, is the same as the actual extinction. These apparent optical properties are suited to radiative transfer equations.     

漫反射各向异性散射介质内的温度热流场

本文采用射线踪迹结合节点分析法和谱带模型,研究了漫反射不透明边界下吸收、发射、各向异性散射介质内的热辐射传递过程。考虑介质辐射能的入射和散射方向,导出漫反射、不透明边界、各向异性散射介质的辐射传递系数。在辐射平衡的情况下,考察了表面发射率和散射反照率对介质内辐射热流和温度场的影响。研究表明,介质不透明边界处存在温度跃迁现象,而且,内界面发射率越大,相应界面温度跃迁越小。     

An inverse radiative transfer problem of simultaneously estimating profiles of temperature and radiative parameters from boundary intensity and temperature measurements

A parallel-plane space filled with absorbing, emitting, isotropically scattering, gray medium is studied in this paper. The boundary intensity and boundary temperature profiles are calculated for the inverse analysis. For the simultaneous estimation of temperature, absorption and scattering coefficient profiles in the medium, the sum of residuals of boundary intensity and temperature after being weighted by a balance factor is minimized through using a Newton-type iteration algorithm and the least-squares method. To avoid over-updating for the parameters, the relative updating magnitude during the iteration process is constrained not to be >0.5. It is shown that the boundary intensity measurement alone is not enough to estimate simultaneously the temperature (source) and the radiative properties (both absorption and scattering coefficients) when the measurement data contain sensitive random errors. The boundary temperature measurement can serve as a necessary supplementation to the boundary intensity to make this kind of inverse radiative transfer problem resolvable. It was shown that a compensation relationship between absorption and scattering coefficients makes it difficult to fix them accurately. Parabolic profiles for the three parameters are used to validate the estimation method. When the optical thickness approaches 4.0, the results for the radiative properties are not acceptable, although the result for temperature profile is reasonable. This means the method needs further improvements.     

Mie scattering efficiency of a large spherical particle embedded in an absorbing medium

The Mie scattering calculations are usually performed for non-absorbing spherical particles embedded in a non-absorbing medium. We consider a case of an absorbing sphere placed in an absorbing medium. We find, by numerical calculation for large size parameter of the order of 10⁴, that the scattering efficiency of a spherical particle in an absorbing medium approaches the reflectance of a plane surface at perpendicular incidence.     

Influence of the vertical profile of Saharan dust on the visible direct radiative forcing

The vertical profile of Saharan dust in the atmosphere is generally characterized by a large aerosol concentration in the mid troposphere, differently from the climatological distribution of other types of particles, that show a peak at the surface and a rapid decrease with height. Saharan dust is also characterized by particles of relatively large size of irregular shape, and variable values of the single scattering albedo (the ratio between radiation scattering and extinction). The dust's peculiar vertical distribution is expected to produce an effect on the calculation of the direct aerosol radiative forcing at the surface and at the top of the atmosphere. This effect is investigated by comparing estimates of aerosol direct visible radiative forcing at the surface and at the top of the atmosphere for dust vertical profiles measured in the Mediterranean, and for the climatological profile. The radiative forcing is estimated by means of an accurate radiative transfer model, and for the ocean surface. The sensitivity of the results on the solar zenith angle, aerosol optical depth, and aerosol absorption is also investigated. The aerosol radiative forcing at the surface shows a very small dependency on the aerosol vertical profile. At the top of the atmosphere, the radiative forcing is weakly dependent on the vertical profile (up to 10% variation on the daily average forcing) for low absorbing particles; conversely, it shows a strong dependency (the daily radiative forcing may vary up to 100%) for absorbing particles. The top of the atmosphere visible radiative forcing efficiency produced by dust having single scattering albedo <0.7 is higher by 4 W m⁻² when the observed vertical profile instead of the standard profile is used in the calculations (i.e. it produces a lower cooling). For values of the single scattering albedo around 0.67, the sign of the forcing depends on the vertical profile. The influence of the vertical distribution on the radiative forcing is largest at small values of the solar zenith angle, and at short wavelengths.     

Control volume finite element method for radiation

In this paper a new methodology is presented by the authors for the numerical treatment of radiative heat transfer in emitting, absorbing and scattering media. This methodology is based on the utilisation of Control Volume Finite Element Method (CVFEM) and the use, for the first time, of matrix formulation of the discretized Radiative Transfer Equation (RTE). The advantages of the proposed methodology is to avoid problems that confronted when previous techniques are used to predict radiative heat transfer, essentially, in complex geometries and when there is scattering and/or non-black boundaries surfaces. Besides, the new formulation of the discretized RTE presented in this paper makes it possible to solve the algebraic system by direct or iterative numerical methods. The theoretical background of CVFEM and matrix formulation is presented in the text. The proposed technique is applied to different test problems, and the results compared favourably against other published works. Moreover this paper discusses in detail the effects of some radiative parameters, such as optical thickness and walls emissivities on the spatial evolution of the radiant heat flux. The numerical simulation of radiative heat transfer for different cases using the algorithm proposed in this work has shown that the developed computer procedure needs an accurate CPU time and is exempt of any numerical oscillations.     

Coupled radiative and conductive heat transfer in a non-grey absorbing and emitting semitransparent media under collimated radiation

This paper deals with heat transfer in non-grey semitransparent two-dimensional sample. Considering an homogeneous purely absorbing medium, we calculated the temperature field and heat fluxes of a material irradiated under a specific direction. Coupled radiative and conductive heat transfer were considered. The radiative heat transfer equation (RTE) was solved using a S₈ quadrature and a discrete ordinate method. Reflection and absorption coefficients of the medium were calculated with the silica optical properties. The conduction inside the medium was linked to the RTE through the energy conservation. Validation of the model and two original cases are also presented.     

Transient coupled heat transfer in a rectangular medium with black surfaces

The main objective of this paper is to extend to two-dimensional (2-D) medium the ray tracing-node analyzing method, which has already been successfully used to solve one-dimensional (1-D) problem of coupled heat transfer in a semitransparent medium. For simplicity, an infinitely long rectangular semitransparent medium with four black opaque surfaces is chosen as our studying object. A control volume method in the implicit scheme is adopted for discretizing the partial transient energy equation. In combination with spectral band model, the radiative heat source term is calculated using the radiative transfer coefficients (RTCs), which are deduced by the ray tracing method. The Partankar's linearization method is used to linearize the radiative source term and the opaque boundary condition, and the linearized equations are solved by the ADI method. Effects of absorption coefficient, refractive index and conductivity on transient cooling process in the 2-D gray rectangular medium are investigated under the condition that the radiation and convection processes cool one side of the rectangular medium while heat the remaining three sides.     

Numerically exact computer simulations of light scattering by densely packed, random particulate media

Direct computer simulations of electromagnetic scattering by discrete random media have become an active area of research. In this progress review, we summarize and analyze our main results obtained by means of numerically exact computer solutions of the macroscopic Maxwell equations. We consider finite scattering volumes with size parameters in the range [20] and [59], composed of varying numbers of randomly distributed particles with different refractive indices. The main objective of our analysis is to examine whether all backscattering effects predicted by the low-density theory of coherent backscattering (CB) also take place in the case of densely packed media. Based on our extensive numerical data we arrive at the following conclusions: (i) all backscattering effects predicted by the asymptotic theory of CB can also take place in the case of densely packed media; (ii) in the case of very large particle packing density, scattering characteristics of discrete random media can exhibit behavior not predicted by the low-density theories of CB and radiative transfer; (iii) increasing the absorptivity of the constituent particles can either enhance or suppress typical manifestations of CB depending on the particle packing density and the real part of the refractive index. Our numerical data strongly suggest that spectacular backscattering effects identified in laboratory experiments and observed for a class of high-albedo Solar System objects are caused by CB.     

Effects of polarization on radiation absorption in one-dimensional semitransparent slab exposed to collimated irradiation

The internal distribution of radiative absorption in one-dimensional semitransparent slab exposed to collimated irradiation is determined by the ray tracing method, and the detailed computation formulae for the distribution of radiative absorption are deduced. It is found that the ray will be polarized by specular reflection of slab boundaries even if the collimated irradiation is unpolarized, and the extent of polarization increases with the internal reflection times. The effects of polarization on the radiative absorption are analyzed and the radiative absorption distributions are compared with those obtained from the case in which the effects of polarization are omitted. The results show that the large differences between reflectances of perpendicular and parallel polarized components are the main causes resulting in errors of radiative absorption distribution. The polarization of incident beam has significant influences on the radiative absorption. Even for an unpolarized beam irradiated near Brewster's angle, omitting the effects of polarization will result in large errors, and the errors increase with the refractive index.     

Comparison of the radiative transfer equations for constant and spatially varying refraction

The radiative transfer equation for scattering media with constant refraction index (RTE) and the radiative transfer equation for scattering media with spatially varying refraction index (RTEvri) are compared by using the principle of conservation of energy. It is shown that the RTEvri, not only accounts for the spatial variations of refraction index, but also contains a term that accounts for the divergence of the rays. The latter term is missing in the RTE. A corrected RTE is proposed.     

掺杂对随机取向团簇粒子辐射特性的影响

利用Bruggeman有效介质理论得到了占有不同体积份额杂质的团簇粒子的等效复折射率.采用离散偶极子近似方法对包含有不同化学成分的随机取向团簇粒子的散射相函数、消光、吸收、散射效率因子、单次散射反照率以及不对称因子等辐射特性参量进行了数值计算,深入探讨了掺杂量对随机取向团簇粒子辐射特性的影响.研究表明,掺杂对随机取向团簇粒子的辐射特性产生显著的影响,并且此影响随着粒子尺度参量的变化而变化.这一工作对研究多种化学成分组成的混合气溶胶的辐射及气候效应具有重要科学价值.     

Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium

Light scattering by particles is often used to determine velocities or concentrations of particles in gaseous or liquid streams. Within the Lorenz-Mie theory, light scattering is well understood both for a single compact spherical particle and a single multilayered particle in a non-absorbing surrounding medium. However, in some cases of practical importance the Lorenz-Mie theory in its present form may fail to describe the scattering because the host medium is absorbing (e.g. water droplets in oil). In this case, a new treatment of the scattering theory is required. In previous work, solutions were obtained in the far-field of the scattering sphere. In this paper, a rigorous solution is derived from the calculation of the total absorption rate of the particle in the host medium, which is valid for all distances from the surface of the encapsulated particle. It is shown that it is necessary to consider finite sizes R of the integrating sphere when dealing with absorbing host media. Cross-sections are defined which are characteristic quantities not only for the particle, depending on the size of a conceptual sphere around the scatterer and the imaginary part of the refractive index of the host medium. The results obtained are discussed for the case of non-absorbing host media and in the far-field approximation. Some numerical examples are given which are also related to experimental results.     

Composites of resonant dielectric rods: A test of their behavior as metamaterial refractive elements

We report numerical experiments of optical wave propagation in composites of high refractive index dielectric rods at frequencies where their first electric and magnetic Mie resonances are excited. The arrays of these particles have been extensively studied and proposed as non-absorbing and isotropic metamaterials. We show that negative refraction, observed in such ordered particle arrays, is due to diffraction and that an effective medium theory (EMT) yields constitutive parameters that do not reproduce the observations in these composites, whose transmission also depends on the sample shape. This is further confirmed by disordering the arrays, a case in which large transmission losses appear due to extinction by resonant scattering from the particles. Therefore, these composites, although having little absorption, give rise to large extinction due to scattering and do not constitute an improvement, as low loss refractive elements, upon all previously designed highly absorbing metamaterials.     

Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index

The curved ray tracing method (CRT) is extended to radiative transfer in the linear-anisotropic scattering medium with graded index from non-scattering medium. In this paper, the CRT is presented to solve one-dimensional radiative transfer in the linear-anisotropic scattering gray medium with a linear refractive index and two black boundaries. The predicted temperature distributions and radiative heat flux at radiative equilibrium are determined by the proposed method, and numerical results are compared with the data in references. The results show that the CRT has a good accuracy for radiative transfer in the linear-anisotropic scattering medium with graded index and the dimensionless emissive power and dimensionless radiative heat flux depend on the dimensionless refractive index gradient. It can also be seen that the dimensionless refractive index gradient has important effects on the temperature discontinuity at the boundaries.     

A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer

The Sensitivity Function-based Conjugate Gradient Method (SFCGM) is described. This method is used to solve the inverse problems of function estimation, such as the local maps of absorption and scattering coefficients, as applied to optical tomography for biomedical imaging. A highly scattering, absorbing, non-reflecting, non-emitting medium is considered here and simultaneous reconstructions of absorption and scattering coefficients inside the test medium are achieved with the proposed optimization technique, by using the exit intensity measured at boundary surfaces. The forward problem is solved with a discrete-ordinates finite-difference method on the framework of the frequency-domain full equation of radiative transfer. The modulation frequency is set to 600 MHz and the frequency data, obtained with the source modulation, is used as the input data. The inversion results demonstrate that the SFCGM can retrieve simultaneously the spatial distributions of optical properties inside the medium within a reasonable accuracy, by significantly reducing a cross-talk between inter-parameters. It is also observed that the closer-to-detector objects are better retrieved.     

A theoretical study of radiation between two concentric spheres using a modified discrete ordinates method associated with Legendre transform

A modified discrete ordinates method (DOM) is used in spherical participating media. The radiative intensity is broken up into two components. One component is traced back to the enclosure's source. It is called direct intensity. The other component is rather traced back to the contribution of the medium itself. It is called diffuse intensity. Thus, the radiative transfer equation (RTE) is transformed into two simultaneous equations: a direct RTE and a diffuse RTE. The direct RTE is solved analytically. The diffuse RTE is solved numerically using the DOM. The streaming angular derivative term appearing in spherical geometry is modeled by making use of the Finite Legendre Transform. We study a pure radiation transfer problem between two concentric spheres. The medium is assumed to be gray and isotropically scattering. The limiting spheres are considered to be opaque, gray, diffusely emitting and diffusely reflecting with uniform emissivity over each surface. The obtained results are compared with available cases reported in the literature. In particular, relative importance of the direct radiation in optically thin media is studied.