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.     

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)].     

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.     

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.     

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.     

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.     

Scattering of light by bispheres with touching and separated components

We use the T-matrix method as described by Mishchenko and Mackowski [Opt. Lett. 19, 1604 (1994)] to compute light scattering by bispheres in fixed and random orientations extensively. For all our computations the index of refraction is fixed at a value 1.5 + 0.005i, which is close to the refractive index of mineral tropospheric aerosols and was used in previous extensive studies of light scattering by spheroids and Chebyshev particles. For monodisperse bispheres with touching components in a fixed orientation, electromagnetic interactions between the constituent spheres result in a considerably more complicated interference structure in the scattering patterns than that for single monodisperse spheres. However, this increased structure is largely washed out by orientational averaging and results in scattering patterns for randomly oriented bispheres that are close to those for single spheres with size equal to the size of the bisphere components. Unlike other nonspherical particles such as cubes and spheroids, randomly oriented bispheres do not exhibit pronounced enhancement of side scattering and reduction of backscattering and positive polarization at side-scattering angles. Thus the dominant feature of light scattering by randomly oriented bispheres is the single scattering from the component spheres, whereas the effects of cooperative scattering and concavity of the bisphere shape play a minor role. The only distinct manifestations of nonsphericity and cooperative scattering effects for randomly oriented bispheres are the departure of the ratio F(22)/F(11) of the elements of the scattering matrix from unity, the inequality of the ratios F(33)/F(11) and F(44)/F(11), and nonzero linear and circular backscattering depolarization ratios. Our computations for randomly oriented bispheres with separated wavelengthsized components show that the component spheres become essentially independent scatterers at as small a distance between their centers as 4 times their radii.     

Formulation and validation of Berenger's PML absorbing boundary forthe FDTD simulation of acoustic scattering

Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic (EM) waves is derived to absorb 2-D and 3-D acoustic waves in finite difference time domain (FDTD) simulation of acoustic wave propagation and scattering. A PML medium suitable for acoustic waves is constructed. Plane wave propagation in the PML medium is solved for both 2-D and 3-D cases and explicit FDTD boundary conditions are derived. The equations show that a matched PML medium is a perfect simulation of free space in that a plane wave does not change its direction of propagation or its speed when it propagates from free space into a matched PML medium. FDTD simulation of a pulsed point source propagating in two dimensions is carried out to test the performance of the PML boundary for acoustic waves. Results show that an eight layer PML boundary condition reduces the reflected error 40 dB over Mur's second order boundary condition     

Parallel implementation of the biorthogonal multiresolution time-domain method

The three-dimensional biorthogonal multiresolution time-domain (Bi-MRTD) method is presented for both free-space and half-space scattering problems. The perfectly matched layer (PML) is used as an absorbing boundary condition. It has been shown that improved numerical-dispersion properties can be obtained with the use of smooth, compactly supported wavelet functions as the basis, whereas we employ the Cohen-Daubechies-Fouveau (CDF) biorthogonal wavelets. When a CDF-wavelet expansion is used, the spatial-sampling rate can be reduced considerably compared with that of the conventional finite-difference time-domain (FDTD) method, implying that larger targets can be simulated without sacrificing accuracy. We implement the Bi-MRTD on a cluster of allocated-memory machines, using the message-passing interface (MPI), such that very large targets can be modeled. Numerical results are compared with analytical ones and with those obtained by use of the traditional FDTD method.     

Geometrical-optics approximation of forward scattering by gradient-index spheres

By means of geometrical optics we present an approximation method for acceleration of the computation of the scattering intensity distribution within a forward angular range (0-60 degrees ) for gradient-index spheres illuminated by a plane wave. The incident angle of reflected light is determined by the scattering angle, thus improving the approximation accuracy. The scattering angle and the optical path length are numerically integrated by a general-purpose integrator. With some special index models, the scattering angle and the optical path length can be expressed by a unique function and the calculation is faster. This method is proved effective for transparent particles with size parameters greater than 50. It fails to give good approximation results at scattering angles whose refractive rays are in the backward direction. For different index models, the geometrical-optics approximation is effective only for forward angles, typically those less than 60 degrees or when the refractive-index difference of a particle is less than a certain value.     

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.     

Spherical particle size determination by analytical inversion of the UV-visible-NIR extinction spectrum

An analytic inversion method, based on the anomalous diffraction approximation for nonabsorbing spherical particles, was developed to retrieve the size distribution from the optical turbidity or extinction spectrum. This method makes use of a differential Fourier cosine transform approach and provides a simple and fast inversion by means of fast Fourier transform and the Savitzky-Golay filter. The applicability of this algorithm was tested on the extinction data generated by the Mie solution. The effects of noise, modality, band limits, and data set size were analyzed by comparison with simulated data. This method can be used to reconstruct the original monomodal and bimodal distributions from 10% noise-corrupted data. The peak position and ratio of peak heights can be recovered with 10% or less deviation. The experiments with latex spheres showed that the inversion result from this method compares favorably with that from the dynamic light scattering measurement.     

Standardization of Light Scattering Measurement in Conjunction With Immunochemical Analysis

Light scattering methods for the physical analysis of synthetic and biological polymers necessitates the use of scattering standards and absolute light scattering measurements. Standardization has not been employed when light scattering has been used to monitor immunochemical reactions using a kinetic or thermodynamic mode.The concentration of a specific protein present in a complex matrix such as urine, serum or cerebrospinal fluid, is measured by reacting the protein of interest with its specific antibody and then measuring the excess light scattering of the solution produced by the formation of antigen antibody complexes. The lack of established light scattering standards in the area of immunochemical measurements makes instrumental quality control difficult and has hindered direct comparison of data among investigators. Both solid and liquid light scattering standards would be necessary to encompass the wide range of instrumentation currently in use. Several solid standards which have been used in the past include reflecting diffusers such as vitrolite, magnesium carbonate crystals with a ground surface, magnesium oxide coatings on magnesium carbonate crystal, casein paint on vitrolite, and solid opal glass transmitting diffusers such as flashed opal glass and solid opal glass. These standards, while applicable to manual light scattering photometers, are not suitable for recently developed automated instrumentation. Liquid standards in the form of Ludox^®, solutions of polystyrene, suspensions of small diameter latex spheres and even pure organic solvents could be used more easily with the continuous flow and discrete automated analyzers. The introduction of instrumental standards at this level of analysis would result in improved overall quality control and facilitate data and method comparison between laboratories.     

A New Closed-Form Solution to Light Scattering by Spherical Nanoshells

Light or electromagnetic wave scattered by a single sphere or a coated sphere has been considered as a classic Mie theory. There have been some further extensions that were made further based on the Mie theory. Recently, a closed-form analytical model of the scattering cross section of a single nanoshell has been considered. The present paper is documented further, based on the work in 2006 by Alam and Massoud, to derive another different closed-form solution to the problem of light scattered by the nanoshells using polynomials of up to order 6. Validation is made by comparing the present closed-form solution to the exact Mie scattering solution and also to the other closed-form solution by Alam and Massoud. This study is found to be, however, more generalized and also more accurate for the coated spheres of either tiny/small or medium sizes than that of Alam and Massoud. Therefore, the derived formulas can be used for accurately characterizing both surface plasmon resonances of nanoparticles (of small sizes) or nanoantenna near-field properties (of medium sizes comparable with half wavelength).     

A general form of perfectly matched layers for three-dimensional problems of acoustic scattering in lossless and lossy fluid media

The concept of perfectly matched layer (PML) has been proven very effective in absorbing electromagnetic waves in lossless media. An extension of this method and a complete three-dimensional (3-D) scheme properly suited for finite-difference, time-domain (FDTD) modeling of acoustic propagation and scattering in unbounded problems are presented in this paper. This generalized PML is constructed in such a way that it performs significant absorption of traveling waves in acoustics for both lossless and lossy media. Theoretically, no reflections occur when propagating waves encounter the lossy medium-PML interface, no matter what the angle of incidence is, introducing at the same time the possibility for further wave attenuation via the stretched coordinates idea. Numerical results support the suggested PML theory as well as reveal the proper modifications, which lead to the achievement of the optimum absorbing-boundary condition.     

Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations

We present an approach based on three-dimensional Monte Carlo radiative transfer simulations for estimating scattering error in measurements of light absorption by aquatic particles with a typical laboratory double-beam spectrophotometer. The scattering error is calculated by combining the weighting function describing the angular distribution of photon losses that are due to scattering on suspended particles with the volume scattering function of particles. We applied this method to absorption measurements made on marine phytoplankton, a diatom Thalassiosira pseudonana and a cyanobacterium Synechococcus. Assuming that the scattering phase function is described by the Henyey-Greenstein formula, we determined the backscatter probability of phytoplankton, which yields the best correction for scattering error at a light wavelength of 750 nm, where true absorption is null. The backscattering ratio estimated for both phytoplankton species is significantly higher than previously reported data based on Mie-scattering calculations for homogeneous spheres. Depending on the type of particles, the corrected absorption spectra obtained with our method may be similar or significantly different from spectra obtained with the null-point correction based on wavelength-independent scattering error.     

Implementing the near- to far-field transformation in the finite-difference time-domain method

When the finite-difference time-domain (FDTD) method is applied to light-scattering computations, the far fields can be obtained by means of integrating the near fields either over the volume bounded by the particle's surface or on a regular surface encompassing the scatterer. For light scattering by a sphere, the accurate near-field components on the FDTD-staggered meshes can be computed from the rigorous Lorenz-Mie theory. We investigate the errors associated with these near- to far-field transform methods for a canonical scattering problem associated with spheres. For a scatterer with a small refractive index, the surface-integral approach is more accurate than its volume counterpart for computation of the phase functions and extinction efficiencies; however, the volume-integral approach is more accurate for computation of other scattering matrix elements, such as P12, P32, and P43, especially for backscattering. If a large refractive index is involved, the results computed from the volume-integration method become less accurate, whereas the surface method still retains the same order of accuracy as in the situation for the small refractive index.     

Accuracy in multiangle light scattering measurements for molar mass and radius estimations. Model calculations and experiments

Multiangle light scattering (MALS) is a well-established technique used to determine the size of macromolecules and particles. In this study, different extrapolation procedures used in MALS were investigated with regard to accuracy and robustness in the obtained molar mass and rms radius. Three different mathematical transformations of the light scattering function referred to as the Debye, Zimm, and Berry methods for constructing the Debye plot were investigated for two idealized polymer shapes, homogeneous spheres and random coils, with radii from 25 to 250 nm. The effect of the angular interval used for the extrapolation was investigated, as was the robustness of the different transformations toward errors in the measured light scattering intensity at low angles. For an rms radius less than 50 nm, the relative error in molar mass due to extrapolation was less than 1% independent of the method used. For larger radii, the error increased and the extrapolation procedure became more critical. For random coil polymers, the Berry method was superior in terms of accuracy and robustness. For spheres, the Debye method was superior. The Zimm method was inferior to the others. The different extrapolation methods were evaluated and compared on experimental data from a size exclusion chromatography-MALS analysis of an ultrahigh molar mass poly(ethylene oxide) (PEO). The PEO data qualitatively verified the calculations and stressed the importance of optimizing the extrapolation procedure after careful evaluation of the experimental data. A discussion of how to detect erroneous data in an experimental Debye plot is given.     

Effect of particles on ultraviolet light penetration in natural and engineered systems

The effect of light scattering on measurement of UV absorbance and penetration of germicidal UVC irradiance in a UV reactor were studied. Using a standard spectrophotometer, absorbance measurements exhibited significant error when particles that scatter light were present but could be corrected by integrating sphere spectroscopy. Particles from water treatment plants and wastewater effluents exhibited less scattering (20%-30%) compared with particles such as clay (50%) and alumina (95%-100%). The distribution of light intensity in a UV reactor for a scattering suspension was determined using a spherical chemical actinometry method. Highly scattering alumina particles increased the fluence rate in the reactor near the UV lamp, whereas clay particles and absorbing organic matter reduced the fluence rate. A radiative transfer fluence rate model reasonably predicted the fluence rate of absorbing media and highly scattering suspensions in the UV reactor.     

Air bubble-induced light-scattering effect on image quality in 193 nm immersion lithography

As an emerging technique, water immersion lithography, offers the capability of reducing critical dimensions by increasing the numerical aperture that is due to the higher refractive indices of immersion liquids than that of air. However, in the process of forming a water fluid layer between the resist and the lens surfaces, air bubbles are often created because of the high surface tension of water. The presence of air bubbles in the immersion layer will degrade the image quality because of the inhomogeneity-induced light scattering in the optical path. Analysis by geometrical optics indicates that the total reflection of light causes the enhancement of scattering in the region in which the scattering angle is less than the critical scattering angle, which is 92 deg at 193 nm. Based on Mie theory, numerical evaluation of scattering that is due to air bubbles, polystyrene spheres, and poly(methyl methacrylate) spheres was conducted for TE, TM, or unpolarized incident light. Comparison of the scattering patterns shows that the polystyrene spheres and air bubbles resemble each other with respect to scattering properties. In this paper, polystyrene spheres are used to mimic air bubbles in studies of lithographic imaging of bubbles in immersion water. In an interferometric lithography system, the distance beyond which bubbles will not print can be estimated by direct counting of defect sites.