Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: a comparison of the T-matrix and the finite-difference time-domain methods

We calculated the scattering and absorption properties of randomly oriented hexagonal ice columns using T-matrix theory, employing analytic orientation averaging, and the finite-difference time-domain method, which uses a numerical procedure to simulate random orientation. The total optical properties calculated are the extinction efficiency, absorption efficiency, single-scattering albedo, and the asymmetry parameter. The optical properties are calculated at the wavelengths of 0.66, 8.5, and 12 mum, up to a size parameter of 20 at 0.66 mum and 15 at the two other wavelengths. The phase-matrix elements P11, P12, and P22 are also calculated and compared, up to a size parameter of 20 at 0.66 mum and 15 at 12.0 mum. The scattering and absorption solutions obtained from the two independent electromagnetic methods are compared and contrasted, as well as the central processing unit time and memory load for each size parameter. It is found that the total optical properties calculated by the two methods are well within 3% of each other for all three wavelengths and size parameters. In terms of the phase-matrix elements it is found that there are some differences between the T-matrix and the finite-difference time-domain methods appearing in all three elements. Differences between the two methods for the P11 element are seen particularly at scattering angles from approximately 120 degrees to 180 degrees ; and at the scattering angle of 180 degrees , relative differences are less than 16%. At scattering angles less than 100 degrees , agreement is generally within a few percent. Similar results are also found for the P12 and P22 elements of the phase matrix. The validity of approximating randomly oriented hexagonal ice columns by randomly oriented equal surface area circular cylinders is also investigated in terms of the linear depolarization ratio.     

Scattering of light by polydisperse, randomly oriented, finite circular cylinders

We use the T-matrix method, as described by Mishchenko [Appl. Opt. 32, 4652 (1993)], to compute rigorously light scattering by finite circular cylinders in random orientation. First we discuss numerical aspects of T -matrix computations specific for finite cylinders and present results of benchmark computations for a simple cylinder model. Then we report results of extensive computations for polydisperse, randomly oriented cylinders with a refractive index of 1.53 + 0.008i, diameter-to-length ratios of 1/2, 1/1.4, 1, 1.4, and 2, and effective size parameters ranging from 0 to 25. These computations parallel our recent study of light scattering by polydisperse, randomly oriented spheroids and are used to compare scattering properties of the two classes of simple convex particles. Despite the significant difference in shape between the two particle types (entirely smooth surface for spheroids and sharp rectangular edges for cylinders), the comparison shows rather small differences in the integral photometric characteristics (total optical cross sections, single-scattering albedo, and asymmetry parameter of the phase function) and the phase function. The general patterns of the other elements of the scattering matrix for cylinders and aspect-ratio-equivalent spheroids are also qualitatively similar, although noticeable quantitative differences can be found in some particular cases. In general, cylinders demonstrate much less shape dependence of the elements of the scattering matrix than do spheroids. Our computations show that, like spheroids and bispheres, cylinders with surface-equivalent radii smaller than a wavelength can strongly depolarize backscattered light, thus suggesting that backscattering depolarization for nonspherical particles cannot be universally explained by using only geometric-optics considerations.     

亚微米级扁椭球颗粒群的散射特性

采用T矩阵方法计算亚微米级扁椭球随机取向分布颗粒群的散射特性,研究消光截面、散射截面、吸光截面、单散射反照率、非对称因子以及散射矩阵元素与颗粒的大小、折射率、长短轴比之间的关系。结果表明,随颗粒粒径增大,消光截面、散射截面、吸光截面、非对称因子都单调增加,散射相函数F11的角分布曲线特征可以区分颗粒的大小;颗粒越偏离球形,颗粒对入射光的衰减效率越低,后向散射光强越强,在轴比不大时,前向50°内的F22/F11值可以区分颗粒的形状;折射率变化主要是对后向散射光的分布产生影响,实部、虚部的变化可分别通过F34/F11的角分布曲线、F12/F11的第一个峰值来体现。     

Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm

Natural particles such as ice crystals in cirrus clouds generally are not pristine but have additional microroughness on their surfaces. A two-dimensional finite-difference time-domain (FDTD) program with a perfectly matched layer absorbing boundary condition is developed to calculate the effect of surface roughness on light scattering by long ice columns. When we use a spatial cell size of 1/120 incident wavelength for ice circular cylinders with size parameters of 6 and 24 at wavelengths of 0.55 and 10.8 microm, respectively, the errors in the FDTD results in the extinction, scattering, and absorption efficiencies are smaller than approximately 0.5%. The errors in the FDTD results in the asymmetry factor are smaller than approximately 0.05%. The errors in the FDTD results in the phase-matrix elements are smaller than approximately 5%. By adding a pseudorandom change as great as 10% of the radius of a cylinder, we calculate the scattering properties of randomly oriented rough-surfaced ice columns. We conclude that, although the effect of small surface roughness on light scattering is negligible, the scattering phase-matrix elements change significantly for particles with large surface roughness. The roughness on the particle surface can make the conventional phase function smooth. The most significant effect of the surface roughness is the decay of polarization of the scattered light.     

Information content of the scattering matrix for spheroidal particles

The T-matrix method is shown to be an efficient and accurate procedure for calculating the scattering matrix for randomly oriented nonspherical particles. Calculated scattering matrix elements for spheroidal particles are identical to those obtained by the spheroidal harmonic approach. T-matrix calculations for a randomly oriented finite length cylinder agree well with microwave scattering measurements. Analysis of the information content of the angular variation of the matrix elements for a set of moderately sized absorbing spheroidal particles is presented. It is found that the Fourier spectrum of the phase function and a parameter related to the depolarization ratio contain particle size and shape information, respectively.     

Average path-length parameter of diffuse light in scattering media

We use Monte Carlo simulations to study in detail the propagation of light in a plane-parallel medium containing scattering particles. In particular, we compute the forward and backward average path-length parameters (FAPP and BAPP, respectively) of four-flux radiative transfer models as functions of the optical depth. Strong dependence on the single scattering albedo and phase function asymmetry is found for both quantities. In general the values of the FAPP decrease with increasing absorption, whereas the opposite occurs for the BAPP. A similar effect is produced when changing from isotropic phase functions to phase functions with a large asymmetry in the forward direction. We present analytical results for the asymptotic values of the FAPP and BAPP as functions of albedo for the particular case of isotropic scattering. Our results differ markedly from the predictions obtained recently with two multiple-scattering models by Vargas and Niklasson [J. Opt. Soc. Am. A 14, 2243 (1997); Appl. Opt. 36, 3735 (1997)]. The differences found point out the intrinsic limitations of these models.     

Modelling optical properties of soft tissue by fractal distribution of scatterers

Abstract

A knowledge of the local refractive index variations and size distribution of scatterers in biological tissue is required to understand the physical processes involved in light-tissue interaction. This paper describes a method for modelling the complicated soft tissue, based on the fractal approach, permitting numerical evaluation of the phase functions and four optical properties of tissue—scattering coefficient, reduced scattering coefficient, backscatter-ing coefficient, and anisotropy factor—by the use of the Mie scattering theory. A key assumption of the model is that refractive index variations caused by microscopic tissue elements can be treated as particles with size distribution according to the power law. The model parameters, such as refractive index, incident wavelength, and fractal dimension, that are likely to affect the predictions of optical properties are investigated. The results suggest that the fractal dimension used to describe how biological tissue can be approximated by particle distribution is highly dependent on how the continuous distribution is discretized. The optical properties of the tissue significantly depend on the refractive index of tissue, implying that the refractive index of the particles should be carefully chosen in the model in order accurately to predict the optical properties of the tissue concerned.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Effective medium theories for irregular fluffy structures: aggregation of small particles

The extinction efficiencies as well as the scattering properties of particles of different porosity are studied. Calculations are performed for porous pseudospheres with small size (Rayleigh) inclusions using the discrete dipole approximation. Five refractive indices of materials covering the range from 1.20+0.00i to 1.75+0.58i were selected. They correspond to biological particles, dirty ice, silicate, and amorphous carbon and soot in the visual part of the spectrum. We attempt to describe the optical properties of such particles using Lorenz-Mie theory and a refractive index found from some effective medium theory (EMT) assuming the particle is homogeneous. We refer to this as the effective model. It is found that the deviations are minimal when utilizing the EMT based on the Bruggeman mixing rule. Usually the deviations in the extinction factor do not exceed approximately 5% for particle porosity P = 0 - 0.9 and size parameters x(porous) = 2 pi r(s,porous)/lambda < or approximately = 25. The deviations are larger for scattering and absorption efficiencies and smaller for particle albedo and the asymmetry parameter. Our calculations made for spheroids confirm these conclusions. Preliminary consideration shows that the effective model represents the intensity and polarization of radiation scattered by fluffy aggregates quite well. Thus the effective models of spherical and nonspherical particles can be used to significantly simplify the computations of the optical properties of aggregates containing only Rayleigh inclusions.     

Light scattering by absorbing hexagonal ice crystals in cirrus clouds

An improved ray-optics theory for single scattering and polarization of hexagonal columns and plates randomly oriented in space has been developed by considering absorption and by using the Chebyshev solution for diffraction integrals. The vector-tracing method and statistics technique of random sampling are employed. The equivalent forms of Snell's law and Fresnel formulas for absorbing ice crystals are derived, and two equivalent optical constants, m' and m″, are obtained. Comparison is made of the computed results of our model and the Takano and Liou model for asymmetry factors, single-scattering albedos, and scattering phase matrix elements. Some characteristics of our model are discussed, and these analyses demonstrate that our ray-optics model is practical and much improved.     

COP: a data library of optical properties of hexagonal ice crystals

The data library of optical properties of hexagonal ice crystals for radiative modeling, Cirrus Optical Properties (COP), is introduced. It includes phase functions, asymmetry parameters, extinction cross sections, and single scattering albedos. Furthermore, lidar ratios and depolarization are given. The dependence of these parameters on wavelength, particle size, and shape is calculated, and different particle orientations are considered. In addition, a simple FORTRAN code is provided to calculate the corresponding properties of size distributions. Thus the data library is a very flexible tool for determining the optical parameters of ice clouds for climatological purposes and remote sensing. The data library and the FORTRAN code are distributed through electronic mail.     

Analytical light-scattering solution for Chebyshev particles

We develop a modification of the T-matrix method that allows for fast calculations of scattering properties of particles with irregular shapes. This modification uses the so-called Sh matrices, the elements of which depend on the shape of particles and do not depend on the particle size or optical constants; i.e., the introduction of Sh matrices makes possible the separation of these parameters within the T-matrix algorithm. For a given shape of a scattering object we calculate the Sh matrices only once and then can quickly calculate the T-matrix elements for a number of sizes and refractive indices. This, in particular, can provide rapid particle-size and refractive index averaging in a particle ensemble. This separation is useful for the derivation of an analytical light-scattering solution for Chebyshev particles.     

Modeling light scattering from Diesel soot particles

The Mie model is widely used to analyze light scattering from particulate aerosols. The Diesel particle scatterometer, for example, determines the size and optical properties of Diesel exhaust particles that are characterized by the measurement of three angle-dependent elements of the Mueller scattering matrix. These elements are then fitted by Mie calculations with a Levenburg-Marquardt optimization program. This approach has achieved good fits for most experimental data. However, in many cases, the predicted complex index of refraction was smaller than that for solid carbon. To understand this result and explain the experimental data, we present an assessment of the Mie model by use of a light-scattering model based on the coupled-dipole approximation. The results indicate that the Mie calculation can be used to determine the largest dimension of irregularly shaped particles at sizes characteristic of Diesel soot and, for particles of known refractive index, tables can be constructed to determine the average porosity of the particles from the predicted index of refraction.     

Dust optical properties retrieved from ground-based polarimetric measurements

We have systematically processed one year of sunphotometer measurements (recorded at five AERONET/PHOTONS sites in Africa) in order to assess mineral dust optical properties with the use of a new polarimetry-based algorithm. We consider the Cimel CE318 polarized sunphotometer version to obtain single-scattering albedo, scattering phase matrix elements F(11) and F(12) for dust aerosols selected with Angstr?m exponents ranging from -0.05 to 0.25. Retrieved F(11) and F(12) differ significantly from those of spherical particles. The degree of linear polarization -F(12)/F(11) for single scattering of atmospheric total column dust aerosols in the case of unpolarized incident light is systematically retrieved for the first time to our knowledge from sunphotometer measurements and shows consistency with previous laboratory characterizations of nonspherical particles.     

Mie scattering computation of spherical particles with very large size parameters using an improved program with variable speed and accuracy

The electromagnetic radiation scattering patterns were computed using an improved C program to study variations in the patterns with changes in the size distribution, size parameters and refractive index of small particles in a volume element. The particle size distributions considered were gamma, normal and lognormal. The program is stable for computation of the theoretical values of the non-zero elements of the scattering matrix, efficiency factors, single scattering albedo, radiation pressure and asymmetry parameter for particles ranging from very small to very large size parameters. One of the significant features of the program is that it incorporates two methods for the determination of the optimal number of terms required for the computation of Mie series with the added benefit of having the option of either going for computational speed or accuracy. After a comparison of the C program with other reported benchmark results, it has been found that the program is very accurate and reliable for electromagnetic scattering computations.     

Combined Henyey-Greenstein and Rayleigh phase function

The phase function is an important parameter that affects the distribution of scattered radiation. In Rayleigh scattering, a scatterer is approximated by a dipole, and its phase function is analytically related to the scattering angle. For the Henyey-Greenstein (HG) approximation, the phase function preserves only the correct asymmetry factor (i.e., the first moment), which is essentially important for anisotropic scattering. When the HG function is applied to small particles, it produces a significant error in radiance. In addition, the HG function is applied only for an intensity radiative transfer. We develop a combined HG and Rayleigh (HG-Rayleigh) phase function. The HG phase function plays the role of modulator extending the application of the Rayleigh phase function for small asymmetry scattering. The HG-Rayleigh phase function guarantees the correct asymmetry factor and is valid for a polarization radiative transfer. It approaches the Rayleigh phase function for small particles. Thus the HG-Rayleigh phase function has wider applications for both intensity and polarimetric radiative transfers. For microwave radiative transfer modeling in this study, the largest errors in the brightness temperature calculations for weak asymmetry scattering are generally below 0.02 K by using the HG-Rayleigh phase function. The errors can be much larger, in the 1-3 K range, if the Rayleigh and HG functions are applied separately.     

Laboratory measurements and T-matrix calculations of the scattering matrix of rutile particles in water

We present experimentally determined scattering matrix elements of birefringent rutile particles in water as a function of the scattering angle for a wavelength of 633 nm (in air). These elements are compared with the results of T-matrix calculations for prolate spheroids. For the diagonal matrix elements the results of the T-matrix calculations are in good agreement with those of the measurements. A good fit for the whole matrix, including the off-diagonal elements, is obtained when we compensate for the birefringence of the rutile particles by performing the computations for spheroids with a slightly larger length/width ratio than measured.     

Optical properties of atmospheric particles: complete parameter sets obtained through polar photometry and an improved inversion technique

Complete sets of optical parameters of dry particles sampled on a Nuclepore filter are derived through interpretation of photometric data with an improved inversion technique. The parameters are the volume-extinction and absorption coefficients, the single-scattering albedo, the asymmetry parameter of the volume scattering function, the apparent complex refractive index, and the apparent soot content. They may serve as input data for solar radiation-budget considerations. Results from preliminary measurements taken in Central Europe and Italy show an extreme variability of the optical parameters. Both large regional and temporal variabilities have been observed caused by the fluctuating midlatitude weather systems and human activities.     

Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation

We report the results of an extensive study of the scattering of light by size and size-shape distributions of randomly oriented prolate and oblate spheroids with the index of refraction 1.5 + 0.02i typical of some mineral terrestrial aerosols. The scattering calculations have been carried out with Waterman's T-matrix approach, as developed recently by Mishchenko [J. Opt. Soc. Am. A 8, 871 (1991); Appl. Opt. 32, 4562 (1993)]. Our main interest is in light scattering by polydisperse models of nonspherical particles because averaging over sizes provides more realistic modeling of natural ensembles of scattering particles and washes out the interference structure and ripple typical of monodisperse scattering patterns, thus enabling us to derive meaningful conclusions about the effects of particle nonsphericity on light scattering. Following Hansen and Travis [Space Sci. Rev. 16, 527 (1974)], we show that scattering properties of most physically plausible size distributions of randomly orientednonspherical part cles depend primarily on the effective equivalent-sphere radius and effective variance of the distribution, the actual shape of the distribution having a minor influence. To minimize the computational burden, we have adopted a computationally convenient power law distribution of particle equivalent-sphere radii n(r) α r(-3),r(1) ≤ r≤r(2). The effective variance of the size distribution is fixed at 0.1, and the effective size parameter continuously varies from 0 to 15. We present results of computer calculations for 24 prolate and oblate spheroidal shapes with aspect ratios from 1.1 to 2.2. The elements of the scattering matrix for the whole range of size parameters and scattering angles are displayed in the form of contour plots. Computational results are compared with analogous calculations for surface-equivalent spheres, and the effects of particle shape on light scattering are discussed in detail.     

How big should hexagonal ice crystals be to produce halos?

It has been hypothesized that the frequent lack of halos in observations of cirrus and contrails and laboratory measurements is caused by small ice crystal sizes that put the particles outside the geometrical optics domain of size parameters. We test this hypothesis by exploiting a strong similarity of ray tracing phase functions for finite hexagonal and circular ice cylinders and using T-matrix computations of electromagnetic scattering by circular cylinders with size parameters up to 180 in the visible. We conclude that well-defined halos should be observable for ice crystal size parameters of the order of 100 and larger and discuss remote-sensing implications of this result.