Rayleigh-Gans-Debye Research Articles

The amplitude of light scattering by a hexagonal column with hollow ends in the Rayleigh-Gans-Debye approximation

The formulas for light scattering amplitude and phase function [or element of scattering matrixf11] of hexagonal column with concave ends in the Rayleigh-Gans-Debye (RGD) approximation are derived. Numerical results for the phase function of light scattering of hexagonal column with pyramidal hollow ends in the RGD approximation and in the method of Purcell-Pennypacker (or discrete dipole method, or coupled dipole method) are compared. The good agreement for hexagonal columns having small phase shifts has been found.

Rigorous analysis of the spectral sizing of single particles based on light scattering patterns

The sizing of a single particle based on the Fourier spectrum of its angle-resolved light-scattering pattern (LSP) is widely used in practice but has been lacking rigorous theoretical justification. In this paper we fill this gap, starting with systematic analysis based on the Rayleigh-Gans-Debye (RGD) approximation. We related the LSP spectrum with an integrated autocorrelation function (IAF), determined solely by the particle geometry. If the exact forward scattering is not included in the LSP angular range and the Hann window is used for the Fourier transform, the LSP spectrum vanishes everywhere except near the discontinuities of the IAF derivatives. Thus, the main spectral peak corresponds to the particle diameter, i.e. largest distance between two interior points. Next, we showed that the same qualitative conclusions hold in the Wentzel-Kramers-Brillouin (WKB) approximation and in the general case, when no approximations are employed. In the case of the WKB, we proposed a change of the LSP argument to simplify the description of the resulting peak.

A semi-empirical correction for the Rayleigh-Debye-Gans approximation for fractal aggregates based on phasor analysis: Application to soot particles

The Rayleigh Debye-Gans approximation for Fractal Aggregates (RDG-FA) is commonly used for the evaluation of the radiative properties of fractal aggregates of nanometer-scale nearly spherical particles as soot particles. The cost of its simplicity, however, is the precision of the aggregate cross sections when the refractive index deviates from unity and when the aggregate’s spheres, or monomers, are not sufficiently small compared to the wavelength. While correction factors have been highlighted before, their physical origin is not clear and no universal correction factors are proposed. The present study develops an approach based on phasor analysis of the aggregate’s internal electric field rigorously determined by the discrete dipole approximation. Aggregates representative of the Diffusion Limited Cluster Aggregation (DLCA) regime having a fractal dimension of Df=1.78 are considered as representative of a soot aggregate. The results reveal that correction factors to the RDG-FA for forward scattering (A) and the absorption cross section (h) are due to a competition between internal-field hot-spots caused by point contact between the spherical monomers and a decrease of the field amplitude as the field propagates through the aggregate. Both phenomena are neglected in the RDF-FA by definition. The absorption phenomenon explains the aggregate-size dependence of A and h. These effects are then studied as the aggregate size varies according to the number of monomers Nm ranging from 10 to 1000, as the monomer radius varies from Rm=5nm - 30nm, and as the wavelength varies from λ=266nm - 1064nm. Both constant and wavelength dependent refractive indices for organic, graphitic, and amorphous soot are considered. Finally, a semi-empirical model is proposed intended to correct the RDG-FA theory based on the analysis.

Генерация второй гармоники в поверхностном слое диэлектрической сфероидальной частицы. I. Аналитическое решение

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave in a thin optically nonlinear surface layer of a dielectric particle shaped as an ellipsoid of revolution is solved. The generalized Rayleigh–Gans–Debye approximation is used for an analytical description with taking into account the difference in refractive indices of the medium corresponding to the frequencies of the exciting and generated radiation. The limiting forms of functions are obtained, with the use of which the electric field strength of the generated radiation is expressed. The order of dependence of these functions on the linear dimensions is found, when the lengths of the semiaxes of the particle are small compared to the wavelength of the exciting radiation and their ratio remains constant. It was found that the power density of the generated radiation in this case is determined to a greater extent by the chiral components of the nonlinear dielectric susceptibility tensor and is proportional to the fourth power of the length of the semiaxis of the particle, if the shape of the spheroidal particle differs significantly from the spherical one. The solution of this problem, obtained by other authors, is supplemented for the possibility of applying to the description of generation in the surface layer of a dielectric particle not only in the form of a prolate, but also in the form of an oblate spheroid. Corrections of inaccuracies and misprints made in similar works by other authors are proposed. The relationships between the formulas used in these works are found, taking into account the corrections and the formulas used in this work.

Retrieving the refractive index of a sphere from the phase spectrum of its light-scattering profile

We studied the Fourier spectrum of the light-scattering profiles of single particles in the Rayleigh-Gans-Debye (RGD) and Wentzel–Kramers–Brillouin (WKB) approximations. In the case of a homogeneous sphere, we found the relationship between the key parameters of the spectrum (including its phase) and the sphere characteristics – both analytically and numerically in the framework of the approximations and the rigorous Lorentz–Mie theory, respectively. Based on these results, we have improved the existing spectral characterization method for spheres extending the applicability range to particles with a higher refractive index.

Effects of the Method of Preparation and Dispersion Media on the Optical Properties and Particle Sizes of Aqueous Dispersions of a Double-Chain Cationic Surfactant.

As inferred from visual observations and turbidity measurements, the average radius of the unilamellar vesicles formed in water from the cationic double-chain surfactant didodecyldimethylammonium bromide (DDAB) varies with the method of preparation, being ∼24 nm after sonication (SS method) and ∼74 nm after extrusion/ultrafiltration (SE method). The radii were larger when the vesicles were produced in 10 mM NaBr, ∼65 nm for the SS method and ∼280 nm for the SE method. The specific turbidity, or turbidity per unit path length divided by the surfactant weight fraction, w, of these vesicular dispersions increased with decreasing w until a constant value was reached at w*, which depends on the preparation method and the dispersion medium. The constant specific turbidities are indicative of single and independent scattering and were used to estimate vesicle radii by solving the specific turbidity equations derived for the Rayleigh-Debye-Gans (RDG) regime. Two turbidity equations were used, one accounting for absorbance errors due to some scattered light reaching the detector and another with no correction. Estimates of the average distances between the vesicles and their corresponding Debye lengths were obtained for evaluating the importance of intervesicle electrostatic interactions, which could lead to dependent scattering at higher weight fractions.

Crumpled few-layer graphene: Connection between morphology and optical properties

Comprehensive characterization of crumpled graphene materials is required to understand their morphologically dependent properties. This study connects the optical properties of crumpled few-layer graphene (FLG) aerosols and colloidal suspensions with particle morphology. The mass absorption cross-section (MAC) of crumpled FLG was measured in the aerosol phase and it is shown that such aerosols absorb visible light more efficiently than soot aerosols. The UV–Vis spectra of FLG suspensions in ethanol were correlated with the mean nanoparticle layer number and the lateral size. The measured optical properties were also simulated using discrete dipole approximation (DDA) applied to volume-reconstructed particles with refractive indices of graphene and graphite from literature. The measurements were more closely reproduced using the refractive index of graphite pellets rather than flat graphene on a substrate. Moreover, we show that a simpler Rayleigh–Debye–Gans (RDG) model can predict the wavelength dependence of the crumpled FLG absorption cross-section.

Tunable and flexible deep-ultraviolet bandpass filters based on micro- and nanoparticle/polydimethylsiloxane hybrid membranes

Abstract Deep-ultraviolet (DUV) bandpass filters play an important role in the fields of modern environmental disinfection and sterilization. A novel low-cost optical scattering material that can be used to manufacture flexible DUV bandpass filters is proposed. Considering both the refractive index matching condition and Rayleigh-Gans-Debye (RGD) scattering theory, the calcium fluoride (CaF2)-doped polydimethylsiloxane (PDMS) DUV bandpass filter shows over 90% transmittance near the peak wavelength of 272.5 nm. Strain measurements result in relatively stable peak wavelengths with stretching ratios of 12%. A temperature response of 1.5 nm °C−1 is measured experimentally. To simulate the DUV light filtering characteristics of the device, an improved random walk scattering model is developed based on a Monte Carlo numerical simulation. To verify the tunability of the transmittance spectra of the devices, a 10 nm blueshift of the peak wavelength is obtained by doping 50 vol% low-molecular-weight PDMS in the original PDMS matrix, and a 17 nm redshift is observed upon doping 1.4 wt% CdSe/ZnS colloidal quantum dots (CQDs). The realization of tunable and flexible DUV bandpass filters paves the way towards the development of environmental purification equipment and wearable photonic devices.

Assessing the limits of Rayleigh–Debye–Gans theory: Phasor analysis of a bisphere

Rayleigh–Debye–Gans (RDG) theory is a commonly used approximation for light scattering and absorption by nanoparticles in the visible spectrum. The aim of this study is to advance our understanding for the origin of empirically observed deviations between accurate calculations of forward scattering and the absorption cross-sections and those predicted by RDG. For this purpose, we investigate the internal electric field within bi-spherical nanoparticles in several cases. The fixed-size constituent spheres of the bispheres are either separated or overlapped and are investigated using the discrete dipole approximation (DDA). To study the internal electric fields, we apply a phasor approach, which provides a semi-graphical way to understand the deviation of the forward scattering (A) and that of the absorption cross section (h) with respect to the RDG. The phasor approach reveals the influence of the bisphere orientation, absorption function E(m), scattering function F(m), and wavelength. It is observed that RDG tends to overestimate the forward scattering, i.e., A<1, as well as absorption h<1 for larger size parameters, and thus, shorter wavelengths. At the opposite limit, a decreasing absorption function E(m) leads to RDG underestimating the true scattered intensity. An explanation for this effect is a competition between the non-uniformity of the internal electric field and the phase shift of the phasors, leading to different results than those predicted by RDG and the amount of dipoles scattering in-phase.

Генерация второй гармоники-суммарной частоты в тонком сферическом слое. I. Аналитическое решение

The problem of the simultaneous second–harmonic and sum–frequency generation by two coherent plane electromagnetic waves with elliptical polarizations and equal frequencies in a thin spherical layer is solved in the Rayleigh–Gans–Debye model. The second–order dielectric susceptibility tensor is chosen in a form containing four independent components (including one chiral component). Fundamental differences are shown between the problem statements as well as between solutions to the problem of second–order nonlinear generation by several coherent electromagnetic waves and the problem of sum–frequency generation in the limiting case of equality of the frequencies of the incident waves. Combinations of parameters are determined at which the spatial distribution of the radiation generated as a result of incidence of two plane waves coincides with the distribution of second–harmonic radiation generated by one plane wave. The limiting cases of the obtained solution are considered: small and large radii of the spherical particle. In these cases the contributions of the chiral and non-chiral anisotropy coefficients to the generated radiation are determined.

Leveraging Anisotropy for Coupled Optimization of Thermal Transport and Light Transmission in Micro‐Structured Materials for High‐Power Laser Applications

AbstractMaximum operating power densities in ceramic laser media scale with thermal conductivity k. This requires larger grain sizes in polycrystalline ceramics to reduce phonon scattering at grain boundaries. However, smaller grain sizes are preferred to minimize light scattering in the Rayleigh regime in polycrystals made from birefringent materials such as AlN and Al2O3, which are otherwise appealing for their high k. An optimization challenge arises from the opposite scaling laws governing the effects of grain sizes on k and light transmission. Here, this is tackled by introducing anisotropically microstructured materials (columnar/disk‐shaped grains) as the lasing media, and allowing orthogonal heat transfer and lasing directions. For columnar grains, larger grain sizes along the c‐axis help maintain high k for good heat dissipation, while preserving light transmission properties in the orthogonal lasing and pumping directions. Analytical models for the thermal conductivity in such structures are presented and verified using Monte‐Carlo ray‐tracing simulations. Similarly, an approximate Rayleigh–Gans–Debye model is used to predict light transmission and verified with exact simulations using FEM software. Finally, the tradeoff between thermal and optical phenomena is captured in a new anisotropic figure‐of‐merit tensor, which is optimized for the microstructure that maximizes lasing media performance in AlN and Al2O3 model systems.

Rayleigh and Rayleigh-Debye-Gans light scattering intensities and spetroturbidimetry of dispersions of unilamellar vesicles and multilamellar liposomes

Hypothesis: Since the volume fraction of the surfactant bilayer(s), of thickness db, in a vesicle and liposome is smaller than one, the dependences of the Rayleigh (R) scattering intensity and turbidity on the particle radius a are weaker than those for a homogeneous sphere, which are proportional to a3. The dependences of the Rayleigh-Debye-Gans (RDG) scattering intensity and turbidity on a are also weaker. Work done: The dependences of the effective relative refractive index on a, db, and dw (water layer thickness) were derived. The specific Rayleigh ratio Rθ** and the specific turbidity τ** for single and independent scattering were derived analytically for R and RDG scattering. Spectroturbidimetry data at 25°C for a cationic double-chain surfactant, didodecyldimethylammonium bromide (DDAB) were compared to the turbidity predictions. Findings: For R scattering, Rθ** and τ** are proportional to a2db for vesicles, and to a3dbdw+db for liposomes. For RDG and particle radii 20–1000 nm, τ** is proportional to an, where n is 2 to 0.4 for vesicles and 2 to 1.1 for liposomes. Turbidity data for DDAB vesicles are consistent with the RDG predictions, which are also used to estimate the vesicles’ sizes. RDG applies to liposomes < 800 nm and to much larger sizes for vesicles.

Absorption and scattering of light by soot aggregates with uniform and pendular ring coatings

The magnitude of radiative forcing by soot is strongly dependent on its morphology and mixing state, which continuously evolve during atmospheric aging. One of the major soot aging mechanisms is via interaction with condensable trace-gas chemicals, and our previous study has shown the formation of two distinct coating distributions on fractal soot aggregates, primarily depending on the radius of monomer spheres and supersaturation of the condensing vapor. The two coating distributions, spherical shells around monomers and pendular rings in junctions between monomers, differ profoundly in their ability to restructure soot aggregates. Here, we investigate the impact of these two distributions on the optical properties of soot, using Discrete Dipole Approximation (DDA) calculations, which are also compared against the Rayleigh Debye Gans (RDG) approximation and the core-shell model. For most optical parameters, the difference between the two coating distributions was minor, except for the asymmetry parameter, where it was significant (5–13%). We found that the placement of condensate in junctions significantly depressed the asymmetry parameter for most coating volumes, thus promoting isotropic scattering by soot. Compared to a uniform condensate distribution, the presence of coating between monomers always resulted in slightly larger absorption, scattering and single scattering albedo (SSA), with the difference reaching ~10% for scattering at high coating volumes. We found that the RDG approximation underestimated absorption, but was able to predict scattering to within ~3%. On the other hand, the core-shell model significantly overestimated absorption, scattering, and SSA, especially for thinly coated soot.

Scattering phase function of fractal aggregates of TiO2 particulate photocatalyst simulated with discrete dipole approximation

The aggregate structures of photocatalyst particles are critical for their light absorption and scattering properties and thus significantly impact the photocatalytic performance toward solar hydrogen production. Herein, taking P25 TiO2 nanoparticles as model photocatalysts, we calculate the optical properties of the aggregates of nanoparticles with the Discrete Dipole Approximation (DDA). A scattering phase function, which can reveal and predict the structural information of photocatalyst particles, is accordingly obtained. Specifically, it is found that the forward scattering is dominant when light is scattered on the particles, and this scattering mode becomes more intense with the increment of the particle size. Our results show that the scattering phase function is also in agreement with the Rayleigh-Gans-Debye (RGD) approximation.

Radiative properties of soot fractal superaggregates including backscattering and depolarization

The modeling of radiative properties of soot particles can be addressed by a number of methods. Among them, the canonical Rayleigh-Debye-Gans (RDG) theory and the Superposition T-Matrix Method (STMM) are particularly well adapted to the task. Here, both RDG and STMM are used to model the radiative properties of two types of fractal aggregates indicative of soot relevant to lidar sensing. One type, a canonical aggregate with a chain-like morphology, corresponds to low-sooting flames and is well characterized by a single fractal dimension. The other type, a superaggregate, exhibits multiple fractal dimensions and is seen in heavily sooting flames. Radiative properties such as the differential scattering cross-section, total scattering cross-sections, backscattering cross-sections, and linear depolarization-ratio (LDR) are calculated for a range of wavelengths from 300−1100 nm while including the wavelength dependence of the soot refractive index. Of particular interest is that depolarization by a superaggregate is found to be approximately ten times larger than the depolarization by a canonical chain-like aggregate and both exhibit a power-law with wavelength. Results for specific wavelengths of interest to lidar (355, 532, and 1064 nm) are tabulated for lidar applications. Additionally, RDG theory yields fairly accurate results for modeling the radiative properties, including lidar relevant parameters such as backscattering cross-sections, only for canonical soot fractal aggregates.

Refractive Indices of Biomass Burning Aerosols Obtained from African Biomass Fuels Using RDG Approximation

Biomass burning (BB) aerosols contribute to climate forcing, but much is still unknown about the extent of this forcing, owing partially to the high level of uncertainty regarding BB aerosol optical properties. A key optical parameter is the refractive index (RI), which influences the absorbing and scattering properties of aerosols. This quantity is not measured directly, but it is obtained by fitting the measured scattering cross section and extinction cross section to a theoretical model using the RI as a fitting parameter. We used the Rayleigh–Debye–Gans (RDG) approximation to retrieve the complex RI of freshly emitted BB aerosol from two fuels (eucalyptus and olive) from Africa in the spectral range of 500–580 nm. Experimental measurements were carried out using cavity ring-down spectroscopy to measure extinction over the range of wavelengths of 500–580 nm and nephelometry to measure scattering at three wavelengths of 450, 550, and 700 nm for size-selected BB aerosol particles. The fuels were combusted in a tube furnace at a temperature of 800 °C, which is representative of the flaming stage of burning. Filter samples were collected and imaged using tunneling electron microscopy to obtain information on the morphology and size of the particles, which was used in the RDG calculations. The mean radii of the monomers were 27.8 and 31.5 nm for the eucalyptus and the olive fuels, respectively. The components of the retrieved complex RI were in the range of 1.31 ≤ n ≤ 1.56 and 0.045 ≤ k ≤ 0.468. The real and complex parts of the RI increase with increasing particle mobility diameter. The real part of the RI is lower, and the imaginary part is higher than what was recommended in literature for black carbon generated by propane or field measurements from fires of mixed wood samples. Fuel dependent results from controlled laboratory experiments can be used in climate modeling efforts and to constrain field measurements from biomass burning.

Sensitive detection and estimation of particle non-sphericity from the complex Fourier spectrum of its light-scattering profile

We develop a fast method to estimate the non-sphericity of arbitrary-shaped particles from the complex Fourier spectrum of its light-scattering profile (LSP), measured with the scanning flow cytometer (SFC). We show that previously used amplitude spectrum is not sufficiently sensitive to the non-sphericity and extensively study the phase of the spectral peak for spheroids in the framework of the Rayleigh–Gans–Debye (RGD) approximation. Based on this analysis we construct a new spectral parameter P – the weighted deviation of the complex spectrum around the peak from that for an equivalent sphere determined by the previously published spectral characterization method for spheres (SCMS). We also propose a geometric indicator of non-sphericity η as the relative volume difference from that of the best-fit sphere. These two new parameters apply to particles of arbitrary shape and strongly correlate with each other for rigorously simulated LSPs for spheroids and biconcave disks in a wide range of sizes, refractive indices, and orientations. This correlation is the core of the new method, allowing one to provide both the estimate and the confidence range of η from the experimental value of P. The method is both sensitive and specific to small non-sphericity. For instance, the median error of estimated aspect ratio for simulated LSPs of spheroids is 0.024. We test the resulting algorithm on the real experimental measurements of milk fat globules and red blood cells (RBCs) during the spherization process. These results raise a question about the actual shape of a spherized RBC in the flow inside the SFC. The applicability domain of the method is determined mainly by that of the SCMS and includes biological objects with sizes larger than 7 wavelengths in the liquid host medium. Moreover, we briefly discuss the potential extension of the method to larger refractive indices.

Second-Harmonic Generation from a Thin Cylindrical Layer: II. Analysis of Solution

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave from a thin optically nonlinear layer on the surface of a cylindrical dielectric particle of finite dimensions placed in a dielectric medium has been solved analytically in the Rayleigh–Gans–Debye approximation. The result has been presented in the tensor and vector forms in the general case, in which the nonlinear dielectric susceptibility tensor has four independent components (one chiral component and three nonchiral components). It has been shown for the first time that the contribution of chiral components to the generation from the butt-end surfaces of a cylindrical particle differs in phase from the contribution of nonchiral components. It has also been revealed that, if the linear dimensions of a cylindrical particle (height and base radius) are small, the radiation determined by the chiral component of the second-order nonlinear dielectric susceptibility tensor makes a predominant contribution to the second-harmonic generation from a nonlinear cylindrical layer (butt-end and lateral surfaces).

Influence of porosity on transparency behavior of MgAl2O4 spinel, experiment vs Mie theory

The effect of porosity on the transmission behavior of MgAl2O4 was modeled using Mie and Rayleigh-Gans-Debye (RGD) single-scattering theories. The RGD theory showed that the porosity less than 0.01 percent (especially for pore radius less than 70 nm) had no remarkable effect on the transparency of this ceramic. While Mie theory showed that this limit is less than 0.1 percent (especially for pore radius less than 70 nm). At the wavelengths of 550 and 1100 nm respectively and for the porosity higher than 0.01 percent, the pore size had to be smaller than 70 and 90 nm in order to achieve the maximum transmission. To verify the simulation results, the transparent spinel ceramic was used which made by two techniques, gel-casting and cold isostatic pressing (CIP). The strength of the final gel-casted and CIPed sample was 15.32 and 2.47 MPa, respectively. After spark plasma sintering (SPS), the density of sintered sample was 98.8 and 99.8% respectively with a transmission of 74.1 and 86.8% at the wavelength of 1100 nm. It was found that RGD theory is not as suitable as Mie theory for simulating the transmission of spinel ceramics, because of the high difference between refractive indexes of medium (spinel) and pore. However at shorter wavelength, by considering the scattering coefficient relation of RGD theory, two theories agree well.

Light scattering from nanoparticle agglomerates

Accounting for the morphology of fractal-like particles is important for their in-situ characterization by optical diagnostics. Here, the formation of carbonaceous aerosol (soot) fractal-like particles by surface growth and agglomeration is investigated experimentally and numerically accounting for necking and multiple scattering between polydisperse primary particles (PPs) by Discrete Element Modeling (DEM) coupled with the Discrete Dipole Approximation (DDA). The DEM-derived number of constituent PPs, soot agglomerate effective density and differential scattering cross-sections for vertically-, Cv, and horizontally-polarized incident light, Ch, are in excellent agreement with those measured here in premixed ethylene flames. In contrast, using the Rayleigh-Debye-Gans (RDG) theory and neglecting necking (aggregation) and polydispersity of constituent PPs underestimates the above Cv and Ch up to 50 %. A DEM-derived scaling law for PP aggregation and polydispersity is developed. Using this law in the RDG theory, the soot scattering cross-sections are in agreement with those estimated by rigorous DDA. Thus, both DEM-DDA and revised RDG theory can be used to optimize optical diagnostics for soot or carbon black characterization and selective sensing by fire (smoke) detectors.