Anomalous Diffraction Approximation Research Articles

Light Scattering from Poly (Acrylate)/E7 Mixtures

A Light Scattering (LS) study of Poly(acrylate)/E7 mixtures is presented. The polarized component Ivv is measured as a function of the scattering angle for several compositions. The data are compared with the theoretical predictions of the scattering intensity in the case of isotropic and anisotropic spheres in the Rayleigh-Gans Approximation (RGA) and in the Anomalous Diffraction Approximation (ADA). Information on the size and optical symmetry of the droplets is obtained neglecting interparticle correlations and multiple scattering effects. Possible correlations between the present data and previous transmission measurements on these systems are discussed.

Polymer-Dispersed Liquid Crystal Droplets: Calculations of Light Scattering

The extinction efficiency factor and the angular distribution of radiation of polymer-dispersed liquid crystal (PDLC) droplets have been calculated by the discrete-dipole approximation (DDA) and compared with the data obtained by the commonly used Rayleigh-Gans approximation (RGA) and by the anomalous-diffraction approximation (ADA). The errors of calculations by the above-mentioned methods for droplets with homogeneous and droplets with radial director configuration are considered.

Light scattering from acrylate-based polymer dispersed liquid crystals: theoretical considerations and experimental examples

A light scattering (LS) study made using acrylate-based polymer dispersed liquid crystals (PDLCs) is presented. The polarized component IVV is measured for blends of a polyacrylate and the liquid crystal (LC) E7 at several compositions. Only the off-state configuration of the droplets with no external fields is considered here. These composites consist commonly of micron-sized nematic LC droplets dispersed in a solid polymer matrix. Theoretical expressions for the scattered intensities in the case of isotropic and anisotropic spherical droplets are given both in the Rayleigh-Gans approximation (RGA) and in the anomalous diffraction approximation (ADA). Series of VV and VH components of the scattering intensities are calculated using the models of Meeten, Stein and coworkers. The model calculations are compared with the light scattering data. This comparison enables us to extract information on the size and the shape of droplets assuming that the size distribution is uniform and that the scattering is due to single droplets, neglecting inter-particle correlation and multiple scattering effects. This paper demonstrates that the LS technique is a useful tool for studying the morphology of PDLC samples and estimating the average size of nematic droplets.

Single scattering by red blood cells

A highly diluted suspension of red blood cells (hematocrit 0.01) was illuminated with an Ar or a dye laser in the wavelength range of 458-660 nm. The extinction and the angle-resolved intensity of scattered light were measured and compared with the predictions of Mie theory, the Rayleigh-Gans approximation, and the anomalous diffraction approximation. Furthermore, empirical phase functions were fitted to the measurements. The measurements were in satisfactory agreement with the predictions of Mie theory. However, better agreement was found with the anomalous diffraction model. In the Rayleigh-Gans approximation, only small-angle scattering is described appropriately. The scattering phase function of erythrocytes may be represented by the Gegenbauer kernel phase function.

Polymer dispersed liquid crystal droplets: methods of calculation of optical characteristics

The extinction efficiency factor and the angular distribution of radiation for polymer dispersed liquid crystal (PDLC) droplets have been calculated in the new discrete-dipole approximation (DDA) and compared with the data obtained in the commonly used Rayleigh-Gans approximation (RGA) and in the anomalous-diffraction approximation (ADA). The errors of calculations by the above mentioned methods for droplets with a homogeneous and for droplets with a radial director configuration are considered. The results of calculations of the optical properties of droplets with a radial and with a bipolar director configuration in the DDA are presented. Based on the results obtained the potentialities of the above methods for describing the scattering by liquid crystal droplets are illustrated.

Diel patterns in optical properties of the chlorophyte Nannochloris sp.: Relating individual‐cell to bulk measurements

To investigate how cell growth and division affect the optical properties of phytoplankton, cultures of the chlorophyte Nannochloris were sampled over a diel cycle to measure cell size and concentration, light‐beam attenuation and absorption, flow cytometric forward light scattering and chlorophyll fluorescence, and carbon content. Refractive index was calculated using the anomalous diffraction approximation. At six different light levels, ranging from 60 to 1,500 µmol photons m−2 s−l, cell division was tightly phased to the light: dark cycle, occurring soon after dark. There were pronounced diel patterns, with minima near dawn and maxima near dusk, in cell size, cell‐specific beam attenuation and absorption, flow cytometric forward light scatter, and carbon content. The diel variations in attenuation cross section were primarily caused by changes in cell size due to growth and division, although there was some influence from changes in refractive index. Because the major constituents of many phytoplankton communities are small eukaryotic cells similar to Nannochloris in cell size and division pattern, the results presented here have important consequences for the interpretation of diel variations in optical properties observed in the ocean.

Anomalous diffraction approximation limits

It has been reported in a recent article [Liu, C., Jonas, P.R., Saunders, C.P.R., 1996. Accuracy of the anomalous diffraction approximation to light scattering by column-like ice crystals. Atmos. Res., 41, pp. 63–69] that the anomalous diffraction approximation (ADA) accuracy does not depend on particle refractive index, but instead is dependent on the particle size parameter. Since this is at odds with previous research, we thought these results warranted further discussion.

Validity of anomalous diffraction approximation in m− χ domain

In a recent paper, Liu et al. [Liu, C., Jonas, P.R., Saunders, C.P.R., 1996. Accuracy of the anomalous diffraction approximation to the light scattering by column-like ice crystals. Atmos. Res. 41, 63–69] reported that the anomalous diffraction approximation (ADA) accuracy is not sensitive to van de Hulst's condition | m−1|≪1, but is dependent on the size parameter χ. Videen and Chýlek [Videen, G., Chýlek, P., 1998. Anomalous diffraction approximation limits. Atmos. Res., this issue] pointed out that this result is at odds with previous research, and their results indicated that the accuracy of ADA is much dependent on the condition of | m−1|≪1. Some calculated results are presented here to provide further discussion of the ADA validity in the calculation of particle extinction and absorption efficiencies.

Particules diélectriques ou absorbantes: champ electromagnétique de surface et diffusion

The exact surface field for dielectric or absorbing spheres of various sizes is first compared to the fields obtained according to three approximations: Fraunhofer diffraction, (modified) anomalous diffraction and Fresnel's laws. In particular, it is shown that for large dielectric spheres, the characteristic reinforcements observed in the surface fields according to Fresnel correspond exactly to rainbow angles. The approximate far scattered fields, calculated using the vectorial Kirchhoff integral, are then studied. For the Fresnel and anomalous diffraction approximations, they are generally satisfactory when limited to small scattering angles and to large dimensions. In that case, the method can be fruitfully extended to non-spherical particles whose radii of curvature are not too small and especially to biological particles with low refractive index. An oblate spheroid and a surface of Cassini simulating a red cell at rest are used as examples.

Light transmission, linear dichroism and birefringence of nematic/polymer dispersions

A theoretical study of light transmission, linear dichroism and birefringence of partially ordered dispersions of bipolar nematic droplets in a polymer matrix is presented. The treatment rests on the single scattering approach to an ensemble of uncorrelated and noninteracting anisotropic particles. Theoretical evaluations of the extinction cross sections and phase functions are performed in the anomalous diffraction approximation. Four basic model systems are analysed: PDLC and NCAP films in an external electric field, and PDLC and NCAP films under uniaxial mechanical deformation. The calculated dependences of the generalized parameters characterizing the light transmission, dichroism and birefringence on the angle of incidence of the probing light, droplet size and shape, refractive indices of the LC and polymer, and parameters of the external field are presented graphically and discussed in detail.

Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm

A ray-by-ray integration (RBRI) method has been developed for the solution of light scattering by nonspherical dielectric particles. The principles of geometric optics are applied to solve the internal electric field within the scattering particles (near field) with the inclusion of complete phase and polarization configurations. The scattered field at the radiation zone (far field) and the extinction and absorption cross sections are obtained by integrating the near field along the propagation paths of geometric rays inside the scatterers by using a number of rigorous electromagnetic integral equations. In the computations of extinction cross section and single-scattering albedo, we demonstrate that the well-known anomalous diffraction approximation is a special case of the RBRI method when the scatterers are optically tenuous. The RBRI method is employed to compute the single-scattering properties of hexagonal ice crystals at visible and near-infrared wavelengths. Based on the reference results computed by the finite-difference time domain (FDTD) technique, we show that the RBRI method is more accurate than the conventional geometric ray-tracing technique and the anomalous diffraction approximation. The extinction efficiency and the single-scattering albedo computed by the RBRI method converge to the reference results when the size parameters along the ice crystal maximum dimension are larger than approximately 15. Substantial differences in terms of relative errors, in comparison with the FDTD solutions, are still noted in the phase function and polarization patterns computed by the RBRI method for size parameters of the order of 10.

A semi-analytic approach within the framework of the anomalous diffraction approximation for transverse magnetic scattering of light by an infinitely long cylinder at normal incidence

Near-forward scattering of light by an infinitely long homogeneous right circular cylinder has been considered using the anomalous diffraction approximation. The scattering function is shown to be expressible in terms of known functions by means of a series expansion. Simple formulae have been obtained for the specific cases of (i) near-forward scattering and (ii) a large-diameter cylinder. The validity of these formulae is examined numerically. The corresponding large-sphere formula is also examined to check that the analysis is not limited to particles of cylindrical geometry only.

Photometric immersion refractometry: a method for determining the refractive index of marine microbial particles from beam attenuation

Photometric immersion refractometry is a technique for determining the refractive index of particulate material. In this technique, the attenuation of light by a suspension of particles is measured as a function of the refractive index of the immersion medium. A minimum attenuation occurs at the refractive index of the medium equal to that of the particles. This technique can serve as a benchmark method for the refractive index determination because it is independent of assumptions invoked by other techniques, such as those based on the inversion of the spectral attenuation data. We present a rigorous model of the photometric immersion refractometry based on the anomalous diffraction approximation for the attenuation efficiency of particles. This model permits one to determine the average value of the real part of the refractive index of the particles, its variance, and the average imaginary part of the refractive index of the particles. In addition, the fourth moment of the particle size distribution can be determined if the concentration and shape of the particles are known. We analyze the sensitivity of this model to experimental errors and discuss the applicability of photometric immersion refractometry to marine microbial particles. We also present experimental results of this technique as applied to heterotrophic marine bacteria. The results indicate that the refractive index of these bacteria was narrowly distributed about the average value of 1.3886.

Angular Transmission of Polymer Dispersed Liquid Crystals Films

The angular transmission of Polymer Dispersed Liquid Crystals (PDLC) films has been experimentally investigated as a function of droplets dimension, refractive indices and thickness of sample. The experimental data have been interpreted in terms of the Anomalous Diffraction Approximation slightly modified in order to take into account extra-scattering phenomena generated by some of the peculiar PDLC characters. The dimension of the droplets and refractive index difference between polymer and liquid crystal are found to be the most important factors affecting the light transmission as well as extra scattering factors.

Refraction by spherical particles in the intermediate scattering region

Mie theory computations of the refraction efficiency for spherical particles are compared with predictions of the Rayleigh, Rayleigh-Gans and Anomalous Diffraction approximations. Attention is given to the intermediate region of particle size, where for extinction the Rayleigh-Gans and Anomalous Diffraction approximations have been shown to merge with each other and with Mie computations. In this region the Mie refraction efficiency is shown to have a first maximum which is not given by any approximation. The Rayleigh and Rayleigh-Gans refraction efficiencies are independent of particle size and approximate to the Mie results in the intermediate region, but not at larger particle sizes. Apart from the first maximum, the AD approximation is shown to closely predict the extrema and zeros of the refraction efficiency. In the intermediate region the Mie computations reveal a first maximum in the refraction efficiency which is not modelled by either of the RG or the AD approximations. Consequently a region where the Rayleigh-Gans and Anomalous Diffraction approximations merge with each other and with Mie computations is not found for refraction. The first maximum in refraction efficiency predicted by the Mie theory occurs when the optical wavelength inside a particle is about 20% greater than the particle's diameter. It is shown to result mainly from the contributions to refraction by the magnetic dipole and electric quadrupole.

A model of an aligned nematic droplet for small angle light scattering study

An aligned nematic droplet in the axial configuration is modeled as a hollow spheric scatterer for studying its small angle light scattering characters. When the thickness of the shell is reduced by an external field, calculations by anomalous diffraction approximation suggest that scattering intensity would decrease and scattering peak would drift to a smaller angle. Measured angular scattering intensities show good agreement with calculations. As the shell part originates from anchoring of the nematic molecules to the surrounding medium, angular scattering intensity can be associated with correlation length.

On the validity of soft particle approximations for the light scattering by a homogeneous dielectric sphere

Abstract The validity of various soft particle approximations has been examined for the scattering of light by a homogeneous dielectric sphere. A scalar analogue of the S-approximation has been presented. It is shown that this much simpler approximation could be nearly as good as the sophisticated S-approximation itself. Numerical results for the extinction and the scattered intensities are examined over a wide refracted index and size parameter domain. The anomalous diffraction approximation has also been included in numerical comparisons.

Accuracy of the Anomalous Diffraction approximation to light scattering by column-like ice crystals

The wavelength and the particle size effects on the accuracy of ADA (Anomalous Diffraction Approximation) method are discussed by comparing the scattering results between ADA method and Mie theory for spherical particles, which indicates that the ADA accuracy depends mainly on the particle size parameter, and is not sensitive to the condition of | m − 1 | ⪡ 1. The results comparison between ADA method and ray tracing method for hexagonal column-like ice crystal particles shows that within the visible wavelength range, results from both methods agree well, and within the infrared wavelength range, there is large discrepancy for co-albedo result due to different assumptions of the ray path inside the particle.

An approximation for spectral extinction of atmospheric aerosols

A closed-form approximation is derived for the frequency/wavelength dependence of the spectral extinction function for optical depths of columns of various categories of atmospheric aerosols. It adopts the Van de Hulst anomalous diffraction approximation for the Mie scattering efficiency factor, and the Deirmendjian modified γ-function form for the particle size distribution. Comparisons are made between the optical depths predicted by the approximation and from their numerical evaluation using the complete Mie function. Agreement is excellent over frequency segments of the ultraviolet, optical and infrared spectral regions for many modified γ-distributions. A method for extending the valid frequency range of the approximation is presented.

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.