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.     

Concept of the equiphase sphere for light scattering by nonspherical dielectric particles

We introduce the concept of the equiphase sphere for light scattering by nonspherical dielectric particles. This concept facilitates the derivation of a simple analytical expression for the total scattering cross section of such particles. We tested this concept for spheroidal particles and obtained a bound on the minor-to-major axis ratio for the valid application of this technique. We show that this technique yields results that agree well with the rigorous numerical solution of Maxwell's equations obtained with the finite-difference time-domain method. The new technique has the potential to be extended to the study of light scattering by arbitrarily shaped convex dielectric particles.     

Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals

A new geometric-optics model has been developed for the calculation of the single-scattering and polarization properties for arbitrarily oriented hexagonal ice crystals. The model uses the ray-tracing technique to solve the near field on the ice crystal surface, which is then transformed to the far field on the basis of the electromagnetic equivalence theorem. From comparisons with the results computed by the finite-difference time domain method, we show that the novel geometric-optics method can be applied to the computation of the extinction cross section and single-scattering albedo for ice crystals with size parameters along the minimum dimension as small as ~6. Overall agreement has also been obtained for the phase function when size parameters along the minimum dimension are larger than ~20. We demonstrate that the present model converges to the conventional ray-tracing method for large size parameters and produces single-scattering results close to those computed by the finite-difference time domain method for size parameters along the minimum dimension smaller than ~20. The present geometric-optics method can therefore bridge the gap between the conventional ray-tracing and the exact numerical methods that are applicable to large and small size parameters, respectively.     

Equiphase-sphere approximation for analysis of light scattering by arbitrarily shaped nonspherical particles

We extend the previously proposed concept of equiphase sphere (EPS) to analyze light-scattering properties of arbitrarily shaped particles. Our analyses based on the Wentzel-Kramers-Brillouin technique and numerical studies based on the finite-difference time-domain method demonstrate that a wide range of irregularly shaped particles can be approximated as their equivalent equiphase ellipsoids to determine their total scattering cross-section (TSCS) spectra. As a result, a simple expression given by the EPS approximation can be used to calculate the TSCS spectra of these particles. We find that the accuracy of the EPS approximation is influenced by both the magnitude and the geometric scale of the surface perturbation of the particle, and we derive validity conditions of the EPS approximation to guide the practical application of this method.     

Computation of light scattering by axisymmetric nonspherical particles and comparison with experimental results

A laboratory prototype of a novel experimental apparatus for the analysis of spherical and axisymmetric nonspherical particles in liquid suspensions has been developed. This apparatus determines shape, volume, and refractive index, and this is the main difference of this apparatus from commercially available particle analyzers. Characterization is based on the scattering of a monochromatic laser beam by particles [which can be inorganic, organic, or biological (such as red blood cells and bacteria)] and on the strong relation between the light-scattering pattern and the morphology and the volume, shape, and refractive index of the particles. To keep things relatively simple, first we focus attention on axisymmetrical particles, in which case hydrodynamic alignment can be used to simplify signal gathering and processing. Fast and reliable characterization is achieved by comparison of certain properly selected characteristics of the scattered-light pattern with the corresponding theoretical values, which are readily derived from theoretical data and are stored in a look-up table. The data in this table were generated with a powerful boundary-element method, which can solve the direct scattering problem for virtually arbitrary shapes. A specially developed fast pattern-recognition technique makes possible the on-line characterization of axisymmetric particles. Successful results with red blood cells and bacteria are presented.     

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.     

Size-shape determination of nonspherical particles in suspension by means of full and depolarized static light scattering

Full and depolarized static light-scattering (LS) experiments have been carried out to characterize the size and shape of colloidal suspensions. Results have been compared with theoretical predictions following the extended-boundary-condition method (T-matrix) formalism for scattering by nonspherical particles. Theory-to-experiment data fitting has yielded size-shape data that compare well with electron-microscopy determinations. Depolarized light-scattering has been found to be an especially useful tool to use to find the correct geometrical parameters of the suspended particles. Size (though not shape) is also correctly fitted through full LS experiments.     

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)].     

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.     

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.     

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.     

Application of dynamic light scattering to the study of small marine particles

Small particles ranging from approximately 0.1 μm to several micrometers in size, which include detrital material, bacteria, and other planktonic microorganisms, make a significant contribution to light scattering in the upper ocean. The scattering properties of these particles are strongly dependent on their size, which is difficult to measure in the submicrometer range with commonly used electronic resistive counters and microscopic techniques. We examined the size of small marine particles by application of the dynamic light scattering (DLS) method. In this method the time-dependent autocorrelation function of scattered intensity by particles undergoing Brownian motion provides information about the size of particles. The samples were collected in clear oceanic waters off the coast of Southern California. The mean hydrodynamic diameter of particles, determined from the DLS measurements at a scattering angle of 45°, was 0.54μ m. This indicates that the major contribution to scattering at this angle comes rom submicrometer particles. We also described an inverse method for estimating the general slope of the size distribution of small marine particles from the mean hydrodynamic diameter. This method is based on calculations of the size distribution weighted by distribution from Mie theory and assumes that a power-law approximation represents the actual particle scattered intensity. These calculations suggested that particulate assemblage in our seawater samples was best characterized by a differential size distribution with a slope of -4.35. This estimation was supported by independent measurements of particle size distribution and the spectral beam attenuation coefficient taken from the same samples as those used for the DLS measurements. We also demonstrated that multiangle DLS measurements may be used to determine the representative value of the refractive index of particles.     

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.     

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.     

Applying a mapped pseudospectral time-domain method in simulating diffractive optical elements

A new technique for the analysis of two-dimensional diffractive optical elements, by use of the pseudospectral time-domain (PSTD) method, is presented. In particular, the method uses a nonuniform (NU) grid and a mapping technique to obtain very accurate spatial derivatives in an efficient manner. To this end, we present the formulation of the PSTD method by using a NU grid and compare its application to the analysis with that of the finite-difference time-domain (FDTD) method. Using only a fraction of the memory and a fraction of the computation time used by FDTD, the mapped PSTD was able to obtain very close results to FDTD.     

Light scattering from nonspherical airborne particles: experimental and theoretical comparisons

A laser light-scattering instrument has been designed to permit an investigation of the spatial intensity distribution of light scattered by individual airborne particles constrained within a laminar flow, with a view to providing a means of classifying the particles in terms of their shape and size. Ultimately, a means of detecting small concentrations of potentially hazardous particles, such as asbestos fiber, is sought. The instrument captures data relating to the spatial distribution of light scattered from individual particles in flow. As part of an investigation to optimize orientation control over particles within the sample airstream, the instrument has been challenged with nonspherical particles of defined shape and size, and a simple theoretical treatment based on the Rayleigh-Gans formalism has been used to model the spatial intensity distribution of light scattered from these particle types and hence derive particle orientation data. Both experimental and theoretical scattering data arepresented, showing good agreement for all particle types examined.     

Model for optical forward scattering by nonspherical raindrops

We describe a numerical model for the interaction of light with large raindrops using realistic nonspherical drop shapes. We apply geometrical optics and a Monte Carlo technique to perform ray traces through the drops. We solve the problem of diffraction independently by approximating the drops with area-equivalent ellipsoids. Scattering patterns are obtained for different polarizations of the incident light. They exhibit varying degrees of asymmetry and depolarization that can be linked to the distortion and thus the size of the drops. The model is extended to give a simplified long-path integration.