Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation

We show that the use of a matrix inversion scheme based on a special lower triangular-upper triangular factorization rather than on the standard Gaussian elimination significantly improves the numerical stability of T-matrix computations for nonabsorbing and weakly absorbing nonspherical particles. As a result, the maximum convergent size parameter for particles with small or zero absorption can increase by a factor of several and can exceed 100. We describe an improved scheme for evaluating Clebsch-Gordon coefficients with large quantum numbers, which allowed us to extend the analytical orientational averaging method developed by Mishchenko [J. Opt. Soc. Am. A 8, 871 (1991)] to larger size parameters. Comparisons of T-matrix and geometrical optics computations for large, randomly oriented spheroids and finite circular cylinders show that the applicability range of the ray-tracing approximation depends on the imaginary part of the refractive index and is different for different elements of the scattering matrix.     

Applicability of regular particle shapes in light scattering calculations for atmospheric ice particles

We ascertain the usefulness of simple ice particle geometries for modeling the intensity distribution of light scattering by atmospheric ice particles. To this end, similarities and differences in light scattering by axis-equivalent, regular and distorted hexagonal cylindric, ellipsoidal, and circular cylindric ice particles are reported. All the results pertain to particles with sizes much larger than a wavelength and are based on a geometrical optics approximation. At a nonabsorbing wavelength of 0.55 μm, ellipsoids (circular cylinders) have a much (slightly) larger asymmetry parameter g than regular hexagonal cylinders. However, our computations show that only random distortion of the crystal shape leads to a closer agreement with g values as small as 0.7 as derived from some remote-sensing data analysis. This may suggest that scattering by regular particle shapes is not necessarily representative of real atmospheric ice crystals at nonabsorbing wavelengths. On the other hand, if real ice particles happen to be hexagonal, they may be approximated by circular cylinders at absorbing wavelengths.     

Comparison of electromagnetic theory and various approximations for computing the absorption efficiency and single-scattering albedo of hexagonal columns

The applicability of various approximations for computing the absorption efficiency and single-scattering albedo of a randomly oriented hexagonal column is tested versus electromagnetic theory. To calculate the absorption efficiency and single-scattering albedo of the hexagonal column from electromagnetic theory we used a generalization to the separation-of-variables method, which enables continuous calculation of optical properties up to size parameters of 86. We found that the asymptotic absorption efficiency is independent of particle shape, and that, as the size parameter increases, the hexagonal column tends to its asymptotic absorption value more quickly than Mie theory. The asymptotic absorption limit of the hexagonal column is calculated accurately (to within 1%) and rapidly by use of the complex-angular-momentum approximation, indicating that this approximation could be used to calculate the absorption limit of nonspherical particles. The equal-volume sphere best approximates the hexagonal column single-scattering albedo at a strongly absorbing wavelength (e.g., 11.9 mum for an ice particle). However, in the resonance region (e.g., 80 mum for an ice particle) Mie theory fails to approximate the single-scattering albedo of the hexagonal column, but as the size parameter exceeds 10 the error in the sphere approximation reduces to within 2%. At 80-mum wavelength there is a characteristic ripple structure superimposed on the hexagonal column absorption efficiency solutions between size parameters from approximately 1 to 4. The ripple structure is indicative of surface-wave interference and is similar to the sphere but less pronounced on the hexagonal column. We investigated the applicability of ray tracing for calculating the single-scattering albedo at absorbing wavelengths relevant to remote sensing of ice particles in the atmosphere and found it to be within 4% for size parameters between 3 and 42 at 3.7-mum wavelength. At mid-infrared wavelengths (e.g., 8.5 and 11.9 mum) ray tracing is within 5% of electromagnetic theory for size parameters exceeding 10. We also tested the Bryant and Latimer absorption approximation to anomalous diffraction theory by using the separation-of-variables method.     

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)].     

Case study of modeled aerosol optical properties during the SAFARI 2000 campaign

We present modeled aerosol optical properties (single scattering albedo, asymmetry parameter, and lidar ratio) in two layers with different aerosol loadings and particle sizes, observed during the Southern African Regional Science Initiative 2,000 (SAFARI 2,000) campaign. The optical properties were calculated from aerosol size distributions retrieved from aerosol layer optical thickness spectra, measured using the NASA Ames airborne tracking 14-channel sunphotometer (AATS-14) and the refractive index based on the available information on aerosol chemical composition. The study focuses on sensitivity of modeled optical properties in the 0.3-1.5 microm wavelength range to assumptions regarding the mixing scenario. We considered two models for the mixture of absorbing and nonabsorbing aerosol components commonly used to model optical properties of biomass burning aerosol: a layered sphere with absorbing core and nonabsorbing shell and the Maxwell-Garnett effective medium model. In addition, comparisons of modeled optical properties with the measurements are discussed. We also estimated the radiative effect of the difference in aerosol absorption implied by the large difference between the single scattering albedo values (approximately 0.1 at midvisible wavelengths) obtained from different measurement methods for the case with a high amount of biomass burning particles. For that purpose, the volume fraction of black carbon was varied to obtain a range of single scattering albedo values (0.81-0.91 at lambda=0.50 microm). The difference in absorption resulted in a significant difference in the instantaneous radiative forcing at the surface and the top of the atmosphere (TOA) and can result in a change of the sign of the aerosol forcing at TOA from negative to positive.     

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。     

Inverse relationships for reflection diagnostics of uniaxially anisotropic nanoscale films on isotropic materials

The possibilities of determining the parameters of uniaxially anisotropic ultrathin nonabsorbing dielectric films on absorbing or transparent isotropic substrates by surface differential reflectance measurements are analyzed. The analysis is based on analytical reflection formulas obtained in the framework of a long-wavelength approximation. It is shown that, in the case of transparent substrates, it is always possible to determine the thickness of a uniaxially ultrathin film and its four parameters of anisotropy (optical constants n(o) and n(e) and angles θ and φ) simultaneously. However, for such films on absorbing substrates, it is possible to decouple the thickness and optical constants by differential reflectance measurements only if θ≠0. The accuracy of the obtained analytic formulas for determining the parameters of ultrathin films is estimated by computer simulations where the reflection problem was solved numerically on the basis of the rigorous electromagnetic theory for anisotropic layered systems.     

Analysis of the enhancement of backscattering by nonspherical particles with flat surfaces

We examine backscattering by analyzing large nonspherical particles with flat surfaces for which where the size is much larger than the wavelength, using ray optics and diffraction theory. We show that the backscattering cross section for rectangles can be 1 order of magnitude larger than that for spheres with same geometrical cross sections, depending on the orientation of the particles. Then we show that there is a difficulty in estimating the backscattering cross section for hexagonal columns with the available solutions but that it is possible to estimate the integration of the differential scattering cross section over small solid angles in backward directions. The integral values for hexagonal columns are found to be more than 1 order of magnitude larger than that for spheres with the same volume. As an application, the use of power from hexagonal columns for lidar observations is analyzed. Unlike for spherical particles with their dependence on Z(-2) (where Z is the distance between the particle and the detector), for nonspherical particles such dependence varies with the particles' nonsphericity, such as shape and orientation: Z(0) for a hexagonal plate randomly oriented in the horizontal plane; Z(-1) for a hexagonal column randomly oriented in the horizontal plane.     

On the integral characteristics of single scattering on nonspherical particles

Expressions are obtained for the integral characteristics of single scattering of polarized radiation on particles of arbitrary shape. Polarized radiation is described by Stokes parameters. The two known scattering characteristics are examined—the full scattering cross section and the scattering matrix normalization constant. A dimensionless scattering integral is analyzed that takes into account possible scattering of incident radiation in all directions and determines the two considered integral characteristics. The integral is expressed via the scattering matrix elements and Stokes parameters of incident radiation. In the case of a nonspherical particle, the matrix elements depend on the direction of radiation incident on the particle. In this connection, the total scattering is affected by the structure of the incident beam. The practically important cases of particle illumination by parallel and convergent beams are considered. Expressions are obtained for the integral characteristics, averaged over the directions of incident radiation. Simple relations between the two scattering characteristics under different particle illumination are derived.     

Optical characterization of nonabsorbing and weakly absorbing thin films with the wavelengths related to extrema in spectral reflectances

In this contribution a new efficient modification of a method that enables us to perform the optical characterization of nonabsorbing and weakly absorbing thin films without using the absolute values of the reflectances measured is presented. Namely, this modification is based on determining the values of the wavelengths corresponding to touching the spectral dependences of the reflectances of the studied films measured for several angles of incidence with the envelopes of maxima and minima of these spectral dependences. By means of combining the explicit formulas containing the wavelengths mentioned and the suitable iteration procedure one can evaluate the values of the thicknesses and spectral dependences of the refractive indices of the films analyzed in reliable and precise ways.     

Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination

A previously developed [Appl. Opt. 34, 5542 (1995)] theoretical procedure for the calculation of the internal and the near-surface electromagnetic fields for nonabsorbing spheroidal particles with arbitrary monochromatic illumination has been generalized to permit solutions for absorbing (i.e., complex relative index of refraction) spheroidal particles. Calculations are presented that demonstrate the general effects of absorption on the internal and near-surface electromagnetic-field distributions for the particular case of a plane wave that is incident upon a 2:1-axis-ratio oblate spheroidal particle.     

Determination of the aerosol size distribution by analytic inversion of the extinction spectrum in the complex anomalous diffraction approximation

A new derivation is presented for the analytical inversion of aerosol spectral extinction data to size distributions. It is based on the complex analytic extension of the anomalous diffraction approximation (ADA). We derive inverse formulas that are applicable to homogeneous nonabsorbing and absorbing spherical particles. Our method simplifies, generalizes, and unifies a number of results obtained previously in the literature. In particular, we clarify the connection between the ADA transform and the Fourier and Laplace transforms. Also, the effect of the particle refractive-index dispersion on the inversion is examined. It is shown that, when Lorentz's model is used for this dispersion, the continuous ADA inverse transform is mathematically well posed, whereas with a constant refractive index it is ill posed. Further, a condition is given, in terms of Lorentz parameters, for which the continuous inverse operator does not amplify the error.     

Absorption and scattering properties of dense ensembles of nonspherical particles

The purpose of this work is to show that an appropriate multiple T-matrix formalism can be useful in performing qualitative studies of the optical properties of colloidal systems composed of nonspherical objects (despite limitations concerning nonspherical particle packing densities). In this work we have calculated the configuration averages of scattering and absorption cross sections of different clusters of dielectric particles. These clusters are characterized by their refraction index, particle shape, and filling fraction. Computations were performed with the recursive centered T-matrix algorithm (RCTMA), a previously established method for solving the multiple scattering equation of light from finite clusters of isotropic dielectric objects. Comparison of the average optical cross sections between the different systems highlights variations in the scattering and absorption properties due to the electromagnetic interactions, and we demonstrate that the magnitudes of these quantities are clearly modulated by the shape of the primary particles.     

Evanescent field in surface plasmon resonance and surface plasmon field-enhanced fluorescence spectroscopies

The highly sensitive nature of surface plasmon resonance (SPR) spectroscopy and surface plasmon field-enhanced fluorescence spectroscopy (SPFS) are governed by the strong surface plasmon resonance-generated evanescent field at the metal/dielectric interface. The greatest evanescent field amplitude at the interface and the maximum attenuation of the reflectance are observed when a nonabsorbing dielectric is employed. An absorbing dielectric decreases the evanescent field enhancement at the interface. The SPR curve of an absorbing dielectric is characterized by a greater reflectance minimum and a broader curve, as compared to those of the nonabsorbing dielectric with the same refractive index. For a weakly absorbing dielectric, such as nanometer-thick surface-confined fluorophores, the absorption is too small to induce a significant change in the SPR curve. However, the presence of a minute amount of the fluorophore can be detected by the highly sensitive SPFS. The angle with the maximum fluorescence intensity of an SPFS curve is always smaller than the resonance angle of the corresponding SPR curve. This discrepancy is due to the differences of evanescent field distributions and their decay characteristics within the metal film and the dielectric medium. The fluorescence intensity in an SPFS curve can be expressed in terms of the evanescent field amplitude. Excellent correlations between the experimentally measured fluorescence intensities and the evanescent field amplitudes are observed.     

Direct calculation of the optical constants for a thin film using a midpoint envelope

An algebraic method to calculate the optical constants for a weakly absorbing thin film from the spectrum of normal reflectance is described. The calculation of the refractive index of the thin film is simplified by constructing a midpoint envelope through the reflection spectrum. If a portion of the spectrum includes a region where the film is nonabsorbing, the results can be used to calculate an algebraic solution for the refractive index and the absorption coefficient of the thin film throughout the entire spectrum. The method is used to determine the constants for a coating of alumina on a glass substrate. The results are compared to the calculation from the extrema of the spectrum.     

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.     

Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution

The link between the spectral shape of the beam attenuation spectrum and the shape of the particle size distribution (PSD) of oceanic particles is revisited to evaluate the extent to which one can be predicted from the other. Assuming a hyperbolic (power-law) PSD, N(D) ? D(-xi), past studies have found for an infinite distribution of nonabsorbing spheres with a constant index of refraction that the attenuation spectrum is hyperbolic and that the attenuation spectral slope gamma is related to the PSD slope xi by xi = gamma + 3. Here we add a correction to this model because of the finite size of the biggest particle in the population. This inversion model is given by xi = gamma + 3 - 0.5 exp(-6gamma). In most oceanic observations xi > 3, and the deviation between these two models is negligible. To test the robustness of this inversion, we perturbed its assumptions by allowing for populations of particles that are nonspherical, or absorbing, or with an index of refraction that changes with wavelength. We found the model to provide a good fit for the range of parameters most often encountered in the ocean. In addition, we found that the particulate attenuation spectrum, c(p)(lambda), is well described by a hyperbolic relation to the wavelength c(p) ? lambda(-gamma) throughout the range of the investigated parameters, even when the inversion model does not apply. This implies that knowledge of the particulate attenuation at two visible wavelengths could provide, to a high degree of accuracy, the particulate attenuation at other wavelengths in the visible spectrum.     

Complex ray tracing in uniaxial absorbing media

We theoretically analyze waves propagating between an isotropic nonabsorbing medium and a uniaxial absorbing medium in the general case where the incidence plane does not coincide with the principal section of the uniaxial crystal. Expressions for the reflection and transmission coefficients are derived by a combination of complex ray tracing and the 4 x 4 matrix method. The work presented here is useful for the design of optical systems that incorporate the use of birefringent absorbing components.