Debye Series Research Articles

Inverse problem-solving method applied to the acoustic characterisation of a thin film via the Debye series method (DSM)

ABSTRACT The study of the ultrasonic acoustic response of materials is of increasing interest for the characterisation of high-frequency applications. In this work, some elastic properties (acoustic velocity and acoustic attenuation) and some physical properties (thickness and density), together with the phase shift of the transducer, are estimated via one method. Scanning acoustic microscopy was performed via an innovative and different method using a high-frequency (50 MHz) planar wave transducer. In addition, the Debye series model (DSM) was used to model the acoustic response of the bilayer structure of the glass substrate and the resin‒epoxy sample. The inverse problem of estimating the acoustic parameters that results in a DSM reflected echo that matches the experimental signal is solved by proposing an innovative step-by-step curve fitting algorithm. The algorithm for solving the inverse problem is then optimised via relative mean square error (RMSE) curves, resulting in a robust method for assessing the uniqueness of the multiparameter solution.

Online testing of extinction method of high-pressure natural gas oil mist based on light extinction

Abstract This paper proposes a theoretical model and scheme for accurately measuring droplet concentration in high-pressure natural gas environments, aiming for precise, online measurement of lubricating oil mist. The model integrates droplet behavior, light extinction properties of the medium, and pressure, using the light extinction method. It applies the Debye Series Expansion of Mie scattering to improve the light extinction coefficient for mediums such as lubricating oil particles. Differential Optical Absorption Spectroscopy is used to calculate the extinction index of natural gas at different pressures. The study innovates by replacing the traditional parallel light path with a non-parallel path that focuses the light beam, countering beam diameter expansion due to gas pressure. Empirical testing shows the method’s accuracy in online oil mist testing under 10 MPa pressure, with less than 5% error compared to offline methods. This work paves the way for future high-pressure gas particle monitoring, particularly in the oil and natural gas sector.

Distinct linear polarization of core-shell particles at near-backscattering directions.

The degree of linear polarization (-P12/P11) of scattered light by particles with a core-shell structure may display a distinct negative minimum at near-backscattering directions. However, the specific range of microphysical parameters within which this phenomenon occurs and the underlying physical mechanism are still unclear. Therefore, this study systematically investigated the impacts of particle size, shell-core ratio and refractive index on the negative minimum of -P12/P11 at near-backscattering angles for both coated spheres and coated super-spheroids. The findings reveal that the pronounced negative minimum at near-backscattering angles mostly appeared when the size parameter defined in terms of the mean radius was smaller than approximately 14.5 (e.g., the mean radius is smaller than approximately 2 μm at 0.865 µm wavelength) and the shell-core ratio was in a range of 1.4-1.9. The presence of weakly- and moderately-absorptive shells would lead to pronounced negative polarization at near backscattering directions. However, as the core absorption increased, the amplitude of negative minimum decreased and then stabilized. As for coated super-spheroids, the non-sphericity of the shell tended to suppress the negative polarization at near-backscattering directions. As a result, the pronounced negative minimum (<-0.4) mostly appeared when the aspect ratio and roundness of the shell were close to unity (the overall shape of the particle was nearly-spherical). However, the negative minimum of -P12/P11 showed little dependence on the shape of the core. Furthermore, the Debye series approach was employed to investigate the underlying mechanism of the negative minimum of -P12/P11 for coated spheres. The results demonstrated that the interference among the partial waves underwent one internal reflection on the shell-medium interface and, without internal reflection on the core-shell interface, led to the pronounced negative polarization at near-backscattering angles. When the core absorption was significant, the interference became negligible and the amplitude of the negative minimum was suppressed. This study enhances our understanding the scattering characteristic of coated particles and has implications in aerosol classification and polarized remote sensing.

Debye-series expansion of T-matrix for light scattering by non-spherical particles computed from Riccati-differential equations.

A new formulation of the Debye series based on the Riccati-differential equations was developed to compute electromagnetic wave scattering by non-spherical particles. In this formulation, the T-matrix was expanded in terms of the Debye series. The zeroth-order term, which corresponds to a combination of diffraction and external reflection, is given by unity minus the external reflection matrix. The higher-order terms are generated from the transmission matrix from the medium to the particle, the internal reflection matrix within the particle and the transmission matrix from the particle to the medium. We demonstrate that the aforementioned four reflection-transmission matrices satisfy the Riccati-differential equations, which can be numerically solved by the fourth-order Runge-Kutta method. The present algorithm can be applied to generalized convex non-spherical particles. The differential equations were analytically validated in the case of a homogeneous sphere. Representative results were given in the case of spheroids. The impacts of the Debye series with various orders on the optical properties of spheroids were revealed with significant details.

Modeling multiple scattering transient of an ultrashort laser pulse by spherical particles

The multiple scattering of an ultrashort laser pulse by a turbid dispersive medium (namely a cloud of bubbles in water) is investigated by means of Monte Carlo simulations. The theory of Gouesbet and Gréhan (2000) [14] is used to derive an energetic model of the scattering transient. It is shown that the spreading and extinction of the pulse can be decoupled from the transient of scattering, which allows to describe each phenomenon individually. The transient of scattering is modeled with the Lorenz-Mie Theory and thus is also valid for a relative refractive index lower than one, contrary to the Debye series expansion which does not converge close to the critical angle. This is made possible after the introduction of a new physical object, the Scattering Impulse Response Function (SIRF) which allows to detect the different modes of scattering transient, in time and direction. The present approach is more generic, as it enables to simulate clouds of air bubbles in water, which was not possible previously. Two different approaches are proposed within the Monte Carlo framework. The first is a pure Monte Carlo approach where the delay due to the scattering is randomly drawn at each event, while the second is based on the transport of the whole scattering signal. They are both embedded in the Monte Carlo code Scatter3D (Jenny et al., 2007) [21]. Both models produce equivalent trends and are validated against published numerical results. They are then applied to the multiple scattering of ultra short pulse by a cloud of bubble in water in the forward direction. The pulse spread due to the propagation in water is computed for a wide range of traveled distances and pulse durations, and the optimal pulse duration is given to minimize the pulse spread at a given distance. The main result is that the scattered photons exit the turbid medium earlier than the ballistic photons and produce a double peak related to the refraction in the bubble. This demonstrates the possibility to develop new diagnostics to characterize dynamic bubbly flows.

Analysis of Mie resonances using the Debye series.

Scattering of light from homogeneous spherical particles can exhibit sharp resonances as functions of particle size, wavelength, or refractive index. Such resonances, usually known as morphology-dependent resonances or whispering gallery modes, have been exhaustively studied using Mie theory. This paper demonstrates that the Debye series expansion provides a succinct and easily understood representation of these resonances: For example, the Debye coefficient Rn121 determines (a) the exact conditions for resonance, (b) the number of terms required to replicate the Mie result, and (c) the Q-factor of the resonance.

Debye series expansion for light scattering by a charged sphere.

Debye series expansion (DSE) is developed for electromagnetic (light) scattering by a charged sphere. By comparing our results with Mie theory (for a charged sphere) and with DSE (for a neutral sphere), we verify our theory numerically. DSE is employed for calculation of far-field intensity, and absorption and extinction efficiencies of charged spherical particles illuminated by a plane wave. The influences of various parameters (including surface charge, refractive index, size parameters, Debye mode, etc.) are studied. The rainbow produced by charged particles is analyzed. These results are of great significance in many fields including particle sizing, optical tweezers, etc.

Behavioral Modeling of Complex Magnetic Permeability With High-Order Debye Model and Equivalent Circuits

In this article, a systematic procedure to derive equivalent circuit networks accurately reproducing the frequency response of the input impedance of magnetic cores in a broad frequency range is presented. The proposed procedure foresees to represent the effective complex permeability spectra of a magnetic core (i.e., the permeability resulting from the superposition of intrinsic material properties and effects due to structural features of the core) by a high-order Debye series expansion, which is subsequently synthesized into suitable Foster and Cauer networks. Such networks can be implemented in any circuit simulator, and are particularly favorable for time-domain transient simulation since they can be easily combined with hysteresis models. Two nanocrystalline tape-wound cores and a commercial bulk current injection probe are used as test cases to prove the effectiveness of the proposed method both in terms of accuracy and ease of implementation.

Dislocated spots and triple splittings of natural rainbows generated by large drop distortions, oscillations, and tilts.

For an accurate modeling of natural rainbows, it is necessary to take into account the flattened shape of falling raindrops. Larger drops do also oscillate, and their axes exhibit tilt angles with respect to the vertical. In this paper, I will discuss two rare rainbow phenomena that are influenced by these effects: bright spots belonging to various rainbow orders, but appearing at remarkable angular distances from their traditional locations, as well as triple-split primary rainbows. While the former have not been observed in nature so far, the latter have been documented in a few photographs. This paper presents simulations based on natural drop size distributions using both a geometric optical model, as well as numerically calculated Möbius shifts applied to Debye series data.

Characterization of refractive index and size of a spherical drop by using Gaussian beam scattering in the secondary rainbow region

In the present study the structure of the secondary rainbow is examined for the case of a spherical drop illuminated by a Gaussian beam with its waist radius comparable to the drop radius. Computations are performed using the generalized Lorenz-Mie theory (GLMT) and Debye series. The influence of beam waist radius on the secondary rainbow pattern is investigated. The possibility of extracting size and refractive index from the secondary rainbow pattern is then evaluated by examining the achievable accuracy. The influence of incident position of the beam on extracting drop information is also investigated. For a Gaussian beam centered on the impact parameter associated with the Descartes ray, the absolute error of the refractive index is smaller than 6.8 × 10−4 and the value of relative error for drop radius is smaller than 4.0% for water drops with a radius between 50 and 200 μm.

Light scattering of a coated spherical particle with large absorption in Debye series expansion

Abstract Debye series expansion (DSE) of a coated sphere is reorganized by three numbers of (N, M, X), in which N is the total number of reflections, M is the number of reflections on the outer interface, and X is the number of chords in the core. With this reorganization, for a coated particle with highly absorbing core, DSE terms can be given by just one number of M, i.e., M = − 1 , 0 , 1 , 2 … , in which M = − 1 term represents the sum of diffraction and external reflection. We find that the overall scattered light is mainly composed of the first three terms, and the coating size that makes M = 0 and M = 1 terms greatly enhanced is proportional to the core size, demonstrating coating size has important effects on intensity distributions of the two terms.

Depolarization of nearly spherical particles: The Debye series approach

Backscattering depolarization of nonspherical particles plays a critical role in active Light Detection And Ranging (LiDAR) retrievals of cloud or aerosol parameters, as well as in particle characterization techniques. However, the interpretation of backscattering light from particles is a challenging research subject. This paper addresses the depolarization of nearly spherical particles by using the Debye series approach. Specifically, the T matrix is represented as an infinite sum of terms; the terms in the expansion are correspondingly associated with diffraction and reflection ($p=0$), and multiple transmissions ($pg0$) from the particle to the medium as waves undergo internal reflections. We found that the enhanced depolarization for optically soft particles stems from multiple transmissions. However, this is mostly from the transmission after one internal reflection ($p=2$), when the refractive index is larger than 1.3. Moreover, the interference among multiple transmissions was found to play an essential role in suppressing the depolarization ratio as the refractive index approaches unity. These findings have implications in interpreting the backscattering optical properties of atmospheric aerosols and hydrosols in water.

Co-polarized and cross-polarized scattering of an off-axis focused Gaussian beam by a spherical particle. 3. Diffraction, the Debye series, and time-domain scattering

Diffraction in the near-forward direction is examined for the localized model of an off-axis focused Gaussian beam scattered by a sphere when the beam is translated off-axis either in the scattering plane or perpendicular to it. The results are analyzed using the Fraunhofer diffraction approximation. We also obtain various Debye series contributions to the exact GLMT co-polarized VV and HH scattering amplitudes and cross-polarized VH and HV scattering amplitudes, and discuss their interpretation. Time-domain scattering is also considered for both the exact GLMT scattering amplitudes and our approximation to them, for which the sum over azimuthal modes is evaluated analytically. It is found that although the approximation accurately reproduces the overall magnitude of the scattered intensity, it is not as accurate in reproducing the phase of high-frequency oscillations in the intensity.

Primary rainbow of high refractive index particle ([formula omitted]) has refraction ripples

Primary rainbow (Debye series, p=2) of a common liquid droplet, i.e., water drop, has a smooth Airy rainbow structure, and the superimposed high frequency ripple structures are mostly generated by interference of refraction (p=2) and reflection (p=0). In this work, the primary rainbow (p=2) of a particle with high refractive index (1.547<n<2) is found to have ripples. This is because the primary rainbow transits from 2-rays rainbow to 3-rays rainbow. A third refraction light with a higher incident angle emerges at the same angle of the classical Airy rainbow, and its interference with the refractions around Descartes ray gives birth to the high frequency ripples. Characteristics of this refraction ripples, i.e., angular frequency, are investigated, and implications of this special ripple for particle measurement are also pointed out. This refraction ripple is not observed in other higher order rainbows.

Measurements of refractive index and size of a spherical drop from Gaussian beam scattering in the primary rainbow region

The primary rainbow observed when light is scattered by a spherical drop has been exploited in the past to measure drop size and relative refractive index. However, if higher spatial resolution is required in denser drop ensembles/sprays, and to avoid then multiple drops simultaneously appearing in the measurement volume, a highly focused beam is desirable, inevitably with a Gaussian intensity profile. The present study examines the primary rainbow pattern resulting when a Gaussian beam is scattered by a spherical drop and estimates the attainable accuracy when extracting size and refractive index. The scattering is computed using generalized Lorenz–Mie theory (GLMT) and Debye series decomposition of the Gaussian beam scattering. The results of these simulations show that the measurement accuracy is dependent on both the beam waist radius and the position of the drop in the beam waist.

Use of Debye's series to determine the optimal edge-effect terms for computing the extinction efficiencies of spheroids.

The extinction efficiencies of atmospheric particles are essential to determining radiation attenuation and thus are fundamentally related to atmospheric radiative transfer. The extinction efficiencies can also be used to retrieve particle sizes or refractive indices through particle characterization techniques. This study first uses the Debye series to improve the accuracy of high-frequency extinction formulae for spheroids in the context of Complex angular momentum theory by determining an optimal number of edge-effect terms. We show that the optimal edge-effect terms can be accurately obtained by comparing the results from the approximate formula with their counterparts computed from the invariant imbedding Debye series and T-matrix methods. An invariant imbedding T-matrix method is employed for particles with strong absorption, in which case the extinction efficiency is equivalent to two plus the edge-effect efficiency. For weakly absorptive or non-absorptive particles, the T-matrix results contain the interference between the diffraction and higher-order transmitted rays. Therefore, the Debye series was used to compute the edge-effect efficiency by separating the interference from the transmission on the extinction efficiency. We found that the optimal number strongly depends on the refractive index and is relatively insensitive to the particle geometry and size parameter. By building a table of optimal numbers of edge-effect terms, we developed an efficient and accurate extinction simulator that has been fully tested for randomly oriented spheroids with various aspect ratios and a wide range of refractive indices.

Electromagnetic scattering of a plane wave by a radially inhomogeneous sphere in the short wavelength limit

The formulas of ray theory and Airy theory for scattering of an electromagnetic plane wave by a radially inhomogeneous sphere are obtained from the generalization of exact Lorenz–Mie wave scattering theory using the WKB approximation to determine the radial portion of the partial wave scalar radiation potential, carrying out a Debye series expansion of it, approximating the sum over partial waves by an integral over an associated impact parameter, and approximately evaluating the integral using either the method of stationary phase or mapping it into one of the phase integrals that describe optical caustics. Various features of the results are commented on, and the procedure is extended to the merging of two rainbows in a given Debye series channel in the context of the longitudinal cusp caustic.

Debye series analysis of internal and near-surface fields for a homogeneous sphere illuminated by an axicon-generated vector Bessel beam

The interaction of an axicon-generated vector Bessel beam (AGVBB) with a homogeneous sphere is investigated in the framework of generalized Lorenz-Mie theory (GLMT). An analytical expression of beam shape coefficients (BSCs) is derived using angular spectrum decomposition method (ASDM), and the scattering coefficients are expanded using Debye series (DSE) in order to isolate the contribution of single scattering process. The internal and near-surface electric fields are numerically analyzed, and the effect of beam location, polarization, order of beam, half-cone angle, and scattering process (namely Debye mode p) are mainly discussed. Numerical results show that a curve formed by extreme peaks can be observed, and the electric fields can be locally enhanced after the interaction of AGVBBs with the particle. Internal and near-surface fields, especially its local enhancement, are very sensitive to the beam parameters, including polarization, order, half-cone angle, etc. The internal fields can also be enhanced by various scattering process (or Debye mode p). Such results have important applications in various fields, including particle sizing, optical tweezers, etc.

Optical coherence tomography images simulated with an analytical solution of Maxwell's equations for cylinder scattering.

An algorithm for the simulation of image formation in Fourier domain optical coherence tomography (OCT) for an infinitely long cylinder is presented. The analytical solution of Maxwell’s equations for light scattering by a single cylinder is employed for the case of perpendicular incidence to calculate OCT images. The A-scans and the time-resolved scattered intensities are compared to geometrical optics results calculated with a ray tracing approach. The reflection peaks, including the whispering gallery modes, are identified. Additionally, the Debye series expansion is employed to identify single peaks in the OCT A-scans. Furthermore, a Gaussian beam is implemented in order to simulate lateral scanning over the cylinder for two-dimensional B-scans. The fields are integrated over a certain angular range to simulate a detection aperture. In addition, the solution for light scattering by layered cylinders is employed and the various layers are identified in the resulting OCT image. Overall, the simulations in this work show that OCT images do not always display the real surface of investigated samples.

Transient Electromagnetic Scattering by a Radially Uniaxial Dielectric Sphere: The Generalized Debye and Mie Series Solutions

In this paper, a theoretical study is carried out to determine the scattering of a transient electromagnetics wave by a radially uniaxial dielectric sphere. This is achieved by inverse Laplace transformation of the frequency-domain scattering solution. To improve understanding of the scattering mechanism from a uniaxial dielectric sphere, two different frequency-domain solutions are employed. In the first approach, the impulse and step responses of a uniaxial dielectric sphere are evaluated by the Mie series solution. Following the high-frequency (HF) scattering solution of a large uniaxial sphere, the Mie series summation is split into high-frequency (HF) and low-frequency terms where the HF term is replaced by its asymptotic expression allowing a significant reduction in computation time of the numerical Bromwich integral. In the second approach, the generalized Debye series solution is introduced, and the generalized Mie series coefficients are replaced by their equivalent Debye series formulations. The results are then applied to evaluate the transient response of each individual Debye term allowing the identification of impulse returns in the transient response of a uniaxial sphere. The effect of variation in permittivity on the arrival time as well as amplitudes of each impulse return is studied, and the results are compared with those computed using the Mie series solution. The numerical results obtained from both methods are in complete agreement.