Finite-difference time-domain solution of light scattering by an infinite dielectric column immersed in an absorbing medium

The two-dimensional (2-D) finite-difference time-domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions and extinction and absorption efficiencies, are derived by use of an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than approximately 0.1%. The errors in the scattering phase functions are typically smaller than approximately 4%. With the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.     

Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices

The finite-difference time-domain (FDTD) technique is examined for its suitability for studying light scattering by highly refractive dielectric particles. It is found that, for particles with large complex refractive indices, the FDTD solution of light scattering is sensitive to the numerical treatments associated with the particle boundaries. Herein, appropriate treatments of the particle boundaries and related electric fields in the frequency domain are introduced and examined to improve the accuracy of the FDTD solutions. As a result, it is shown that, for a large complex refractive index of 7.1499 + 2.914i for particles with size parameters smaller than 6, the errors in extinction and absorption efficiencies from the FDTD method are generally less than ~4%. The errors in the scattering phase function are less than ~5%. We conclude that the present FDTD scheme with appropriate boundary treatments can provide a reliable solution for light scattering by nonspherical particles with large complex refractive indices.     

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.     

Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium

The three-dimensional (3-D) finite-difference time-domain (FDTD) technique has been extended to simulate light scattering and absorption by nonspherical particles embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. When computing the single-scattering properties of a particle in an absorbing dielectric medium, we derive the single-scattering properties including scattering phase functions, extinction, and absorption efficiencies using a volume integration of the internal field. A Mie solution for light scattering and absorption by spherical particles in an absorbing medium is used to examine the accuracy of the 3-D UPML FDTD code. It is found that the errors in the extinction and absorption efficiencies from the 3-D UPML FDTD are less than approximately 2%. The errors in the scattering phase functions are typically less than approximately 5%. The errors in the asymmetry factors are less than approximately 0.1%. For light scattering by particles in free space, the accuracy of the 3-D UPML FDTD scheme is similar to a previous model [Appl. Opt. 38, 3141 (1999)].     

Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles

The pseudospectral time-domain (PSTD) method is a powerful approach for computing the single-scattering properties of arbitrarily shaped particles with small-to-moderate-sized parameters. In the PSTD method, the spatial derivative approximation based on the spectral method is more accurate than its counterpart based on the finite-difference technique. Additionally, the PSTD method can substantially diminish accumulated errors that increase with the spatial scale and temporal duration of simulation. We report on the application of the PSTD method to the scattering of light by nonspherical ice particles. The applicability of the PSTD method is validated against the Lorenz-Mie theory and the T-matrix method. The phase functions computed from the PSTD method and the Lorenz-Mie theory agree well for size parameters as large as 80. Furthermore, the PSTD code is also applied to the scattering of light by nonspherical ice crystals, namely, hollow hexagonal columns and aggregates, which are frequently observed in cirrus clouds. The phase functions computed from the PSTD method are compared with the counterparts computed from the finite-difference time-domain (FDTD) method for a size parameter of 20 and an incident wavelength of 3.7 microm. The comparisons show good agreement between the two methods.     

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.     

Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition

A three-dimensional finite-difference time-domain (FDTD) program has been developed to provide a numerical solution for light scattering by nonspherical dielectric particles. The perfectly matched layer (PML) absorbing boundary condition (ABC) is used to truncate the computational domain. As a result of using the PML ABC, the present FDTD program requires much less computer memory and CPU time than those that use traditional truncation techniques. For spheres with particle-size parameters as large as 40, the extinction and absorption efficiencies from the present FDTD program match the Mie results closely, with differences of less than ~1%. The difference in the scattering phase function is typically smaller than ~5%. The FDTD program has also been checked by use of the exact solution for light scattering by a pair of spheres in contact. Finally, applications of the PML FDTD to hexagonal particles and to spheres aggregated into tetrahedral structures are presented.     

Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium

The conventional Lorenz-Mie formalism is extended to the case for a coated sphere embedded in an absorbing medium. The apparent and inherent scattering cross sections of a particle, derived from the far field and near field, respectively, are different if the host medium is absorptive. The effect of absorption within the host medium on the phase-matrix elements associated with polarization depends on the dielectric properties of the scattering particle. For the specific cases of a soot particle coated with a water layer and an ice sphere containing an air bubble, the phase-matrix elements -P12/P11 and P33/P11 are unique if the shell is thin. The radiative transfer equation for a multidisperse particle system embedded within an absorbing medium is discussed. Conventional multiple-scattering computational algorithms can be applied if scaled apparent single-scattering properties are applied.     

Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols

We have examined the Maxwell-Garnett, inverted Maxwell-Garnett, and Bruggeman rules for evaluation of the mean permittivity involving partially empty cells at particle surface in conjunction with the finite-difference time-domain (FDTD) computation. Sensitivity studies show that the inverted Maxwell-Garnett rule is the most effective in reducing the staircasing effect. The discontinuity of permittivity at the interface of free space and the particle medium can be minimized by use of an effective permittivity at the cell edges determined by the average of the permittivity values associated with adjacent cells. The efficiency of the FDTD computational program is further improved by use of a perfectly matched layer absorbing boundary condition and the appropriate coding technique. The accuracy of the FDTD method is assessed on the basis of a comparison of the FDTD and the Mie calculations for ice spheres. This program is then applied to light scattering by convex and concave aerosol particles. Comparisons of the scattering phase function for these types of aerosol with those for spheres and spheroids show substantial differences in backscattering directions. Finally, we illustrate that the FDTD method is robust and flexible in computing the scattering properties of particles with complex morphological configurations.     

Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths

We investigate the errors associated with the use of circular cylinders as surrogates for hexagonal columns in computing the optical properties of pristine ice crystals at infrared (8-12-microm) wavelengths. The equivalent circular cylinders are specified in terms of volume (V), projected area (A), and volume-to-area ratio that are equal to those of the hexagonal columns. We use the T-matrix method to compute the optical properties of the equivalent circular cylinders. We apply the finite-difference time-domain method to compute the optical properties of hexagonal ice columns smaller than 40 microm. For hexagonal columns larger than 40 microm we employ an improved geometric optics method and a stretched scattering potential technique developed in previous studies to calculate the phase function and the extinction (or absorption) efficiency, respectively. The differences between the results for circular cylinders and hexagonal columns are of the order of a few percent. Thus it is quite reasonable to use a circular cylinder geometry as a surrogate for pristine hexagonal ice columns for scattering calculations at infrared (8-12-microm) wavelengths. Although the pristine ice crystals can be approximated as circular cylinders in scattering calculations at infrared wavelengths, it is shown that optical properties of individual aggregates cannot be well approximated by those of individual finite columns or cylinders.     

Scattering properties of needlelike and platelike ice spheroids with moderate size parameters

We use the current advanced version of the T-matrix method to compute the optical cross sections, the asymmetry parameter of the phase function, and the scattering matrix elements of ice spheroids with aspect ratios up to 20 and surface-equivalent-sphere size parameters up to 12. We demonstrate that platelike and needlelike particles with moderate size parameters possess unique scattering properties: their asymmetry parameters and phase functions are similar to those of surface-equivalent spheres, whereas all other elements of the scattering matrix are typical of particles much smaller than the wavelength (Rayleigh scatterers). This result may have important implications for optical particle sizing and remote sensing of the terrestrial and planetary atmospheres.     

Scattering matrix of infrared radiation by ice finite circular cylinders

Scattering matrix characteristics of polydisperse, randomly oriented, small ice crystals modeled by finite circular cylinders with various ratios of the length to diameter (L/D) ratio are calculated by use of the exact T-matrix approach, with emphasis on the thermal infrared spectral region that extends from the atmospheric short-wave IR window to the far-IR wavelengths to as large as 30 microm. The observed ice crystal size distribution and the well-known power-law distribution are considered. The results of the extensive calculations show that the characteristics of scattering matrix elements of small ice circular cylinders depend strongly on wavelengths and refractive indices, particle size distributions, and the L/D ratios. The applicability of the power-law distribution and particle shapes for light scattering calculations for small ice crystals is discussed. The effects of the effective variance of size distribution on light scattering characteristics are addressed. It seems from the behavior of scattering matrix elements of small ice crystals that the combination of 25 and 3.979 microm has some advantages and potential applications for remote sensing of cirrus and other ice clouds.     

Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles

Two grid configurations can be employed to implement the finite-difference time-domain (FDTD) technique in a Cartesian system. One configuration defines the electric and magnetic field components at the cell edges and cell-face centers, respectively, whereas the other reverses these definitions. These two grid configurations differ in terms of implication on the electromagnetic boundary conditions if the scatterer in the FDTD computation is a dielectric particle. The permittivity has an abrupt transition at the cell interface if the dielectric properties of two adjacent cells are not identical. Similarly, the discontinuity of permittivity is also observed at the edges of neighboring cells that are different in terms of their dielectric constants. We present two FDTD schemes for light scattering by dielectric particles to overcome the above-mentioned discontinuity on the basis of the electromagnetic boundary conditions for the two Cartesian grid configurations. We also present an empirical approach to accelerate the convergence of the discrete Fourier transform to obtain the field values in the frequency domain. As a new application of the FDTD method, we investigate the scattering properties of multibranched bullet-rosette ice crystals at both visible and thermal infrared wavelengths.     

Multiple-Scattering Influence on Extinction-and Backscatter-Coefficient Measurements with Raman and High-Spectral-Resolution Lidars

A formalism describing the influence of multiple scattering on cloud measurements with Raman and high-spectral-resolution lidars is presented. Model calculations including both particulate and molecular scattering processes are performed to describe the general effects of multiple scattering on both particulate and molecular lidar backscatter signals. It is found that, for typical measurement geometries of ground-based lidars, as many as five scattering orders contribute significantly to the backscattered light. The relative intensity of multiple-scattered light is generally larger in signals backscattered from molecules than in signals backscattered from particles. The multiple-scattering formalism is applied to measurements of water and ice clouds taken with a Raman lidar. Multiple-scattering errors of measured extinction coefficients are typically of the order of 50% at the bases of both water and ice clouds and decrease with increasing penetration depth to below 20%. In contrast, the multiple-scattering errors of backscatter coefficients are negligible in ice clouds and below 20% in water clouds.     

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.     

Application of the symplectic finite-difference time-domain method to light scattering by small particles

A three-dimensional fourth-order finite-difference time-domain (FDTD) program with a symplectic integrator scheme has been developed to solve the problem of light scattering by small particles. The symplectic scheme is nondissipative and requires no more storage than the conventional second-order FDTD scheme. The total-field and scattered-field technique is generalized to provide the incident wave source conditions in the symplectic FDTD (SFDTD) scheme. The perfectly matched layer absorbing boundary condition is employed to truncate the computational domain. Numerical examples demonstrate that the fourth-order SFDTD scheme substantially improves the precision of the near-field calculation. The major shortcoming of the fourth-order SFDTD scheme is that it requires more computer CPU time than a conventional second-order FDTD scheme if the same grid size is used. Thus, to make the SFDTD method efficient for practical applications, one needs to parallelize the corresponding computational code.     

Photographing impact of plasma-sprayed particles on rough substrates

Plasma-sprayed nickel and molybdenum particles (∼55 μm diameter) were photographed during spreading on silicon wafers that were patterned with micron-sized columns to make a textured rough surface. Impact on grit-blasted glass was also studied. The surfaces were maintained at either room temperature or at 350 °C. As the droplets approached the surface they were sensed by a photodetector and, after a known delay, a fast charge-coupled device (CCD) camera was triggered to capture time-integrated images of the spreading splat. A rapid two-color pyrometer was used to collect the thermal radiation from the spreading particles to record the evolution of their temperature and calculate splat cooling rates. It was found that micron-sized columns on the textured surfaces impeded fluid flow during spreading of splats, promoting splashing. When the column height was on the order of the splat thickness, increasing the space between each column increased the splat cooling rate as the columns penetrated into the liquid splat, providing a larger surface area for heat transfer. On the grit-blasted glass surfaces it was found that as the surface roughness increased, the maximum spread diameters of the molten droplets decreased, while the splat cooling rates increased. Impact on non-heated and heated roughened glass with similar roughness values produced splats with approximately the same maximum spread diameters, skewed morphologies, and cooling rates. On smooth glass, the splat morphologies were circular, with large maximum spread diameters and smaller cooling rates on non-heated smooth glass. An established model was used to estimate the splat-substrate thermal contact resistances. On highly roughened glass, the thermal contact resistance decreased as the glass roughness increased, suggesting that splat-substrate contact was improved as the molten metal penetrated the spaces between the large asperities.     

Light scattering from particles located under a rough dielectric layer

A discussion is presented of the effect of roughness on the detectability of subsurface particles by means of the light-scattering method. We have studied the scattering of light by calibrated spheres located under a slightly rough dielectric surface both experimentally and theoretically. In our experiments, the scattering from slightly rough layers with nonresonant particles was dominated by the roughness, and the scattering diagram did not bear any discernible indications of the spheres. However, at resonance, the subsurface particles manifested themselves by an increase in the total scattered intensity and by well-pronounced maxima in the angular dependence of both the scattering diagram and the backscattered intensity. Theoretical calculations show that the angular positions of the maxima in the scattering diagram are essentially determined by the interference of fields scattered by the particles and by the surface, whereas the contribution of the multiple interparticle scattering is negligible. By contrast, the oscillations in the angular dependence of the backscattered intensity are due primarily to the scattering between neighboring spheres.     

Attenuation of an electromagnetic wave by charged dust particles in a sandstorm

We calculate the light scattering properties of the partially charged dust particles with the Mie theory for electromagnetic waves with different frequencies, and the attenuation coefficients of an electromagnetic wave propagating in a sandstorm are also calculated. The results show that the electric charges distributed on the sand surface have a significant effect on the attenuation of the electromagnetic wave, especially for a frequency lower than 40 GHz, and attenuation coefficients increase with the magnitude of charges carried by the dust particles (expressed by the charge-to-mass ratio in this paper). For the higher frequency electromagnetic wave, such as visible light, the effect of charges carried by sand particles on its attenuation is very little, which can be ignored.     

Influence of crystal tilt on solar irradiance of cirrus clouds

The single and multiple scattering and absorption properties of hexagonal ice columns with different degrees of particle orientation are modeled in the solar spectral range by means of a ray-tracing single-scattering code and a Monte Carlo radiative-transfer code. The scattering properties are most sensitive to particle orientation for the solar zenith angles of 50 degrees (asymmetry parameter) and 90 degrees (single-scattering albedo). Provided that the ice columns are horizontally oriented, the usual assumption of random orientation leads to an overestimation (underestimation) of the reflected (transmitted) solar broadband radiation at high Sun elevation and to an underestimation (overestimation) at medium solar zenith angles. The orientation effect is more (less) pronounced in scattering and transmission (absorption) for smaller ice crystals.