Extended Boundary Condition Research Articles

Eliminating blowing-ups and evanescent waves when using the finite series technique in evaluating beam shape coefficients for some T-matrix approaches, with the example of Gaussian beams

When evaluating beam shape coefficients which encode the description of laser beams, for use in some T-matrix approaches such as generalized Lorenz-Mie theory or Extended Boundary Condition Method for structured beams, by using finite series, blowing-ups are observed. When numerical inaccuracies are ruled out, it has been firmly demonstrated that such blowing-ups correspond to genuine physical phenomena, namely they describe evanescent waves. We propose a method to eliminate these blowing-ups and the corresponding evanescent waves (at least most of them).

Rigorous justification of a localized approximation to encode off-axis Gaussian acoustical beams.

With the model of generalized Lorenz-Mie theory (GLMT) and the extended boundary condition method, describing the interaction between electromagnetic (EM) waves (more specifically laser beams) and particles, an acoustical GLMT-like framework can be used to deal with acoustical wave scattering. The incident acoustical wave may then be encoded in a set of beam shape coefficients (BSCs) similar to the ones used in EM scattering. Following a paper devoted to the evaluation of acoustical BSCs using a localized approximation in the case of on-axis Gaussian acoustical beams, the present paper deals with the evaluation of acoustical BSCs in the case of off-axis Gaussian acoustical beams.

An Experimental Method to Capture the Thermal Conductivity Coefficient of Fine-Grained Concretes during Transition from Liquid to Solid.

During the transition from liquid to solid, the thermal conductivity coefficient λ of concrete decreases. Although λ of hardened concrete is well investigated, there is limited research on the transition from liquid to solid and how it depends on hydration. Currently, only simplified qualitative approaches exist for the liquid state and the transient process. An experimental method is not available. For this purpose, a test rig is designed to experimentally capture the evolution of λ for fine-grain concretes during transition. The performance of the test setup is evaluated on a characteristic high-performance concrete (HPC). The results are compared to theoretical predictions from the literature. The developed test rig is mapped in a digital twin to investigate extended boundary conditions, such as different heat sources and temperatures of the experimental setup. It allows the experiment to be repeated and optimized for different setups with little effort. The test principle is as follows: A liquid concrete sample is heated through a controlled external source, while the transient temperature distribution over the height is measured with a fiber optic sensor. The thermal conductivity is derived from the heat flux induced and the temperature distribution over an evaluation length. Experiments show that λ in the liquid state is approximately 1.4 times greater than in the solid state and exponentially decreases for the transient process. Numerical results on the digital twin indicate that the robustness of the results increases with the temperature of the heat source. Moreover, the derivation in λ turns out to be strongly dependent on the evaluation length. A length of three times the maximum grain diameter is recommended.

A model for thermocapillary driven motion of liquid drops on an inclined plane

Thermocapillary drop migration is driven by surface tension gradient. On an inclined plane, gravity acts as an additional force that can have a promoting or resisting effect depending on the thermal gradient direction. The present 2-D model extends Brochard and Chebbi models for thermocapillary migration of small drops on horizontal plates by accounting for Marangoni and Poiseuille flow components, while including the effect of gravity, acting in the case of inclines. The present treatment is based on fundamentals of conservation of mass and momentum transport, and uses the lubrication approximation and extended boundary conditions at the receding and advancing contact lines. A model for the dynamics of migration is provided, and an analytical solution is derived for drop migration dynamics and the critical angle of inclination.

Light scattering by a periodically time-modulated object of arbitrary shape: the extended boundary condition method

A proper generalization of the extended boundary condition method to calculate the transition matrix, T, for electromagnetic scattering from a homogeneous and isotropic body of arbitrary shape, characterized by a periodically time-varying electric permittivity, is presented. The application of the method on a specific example of a spheroidal dielectric particle confirms that time modulation induces strong inelastic scattering, accompanied by energy transfer between the scatterer and the light field, when the difference of the incident wave frequency to a particle optical resonance matches an integer multiple of the modulation frequency. Moreover, it is shown that, for nonspherical scatterers, these effects can be selectively tuned by external means such as the polarization and the propagation direction of the incident light beam. The method is readily implementable in available dynamic multiple-scattering computer codes, and, because of its versatility and computational efficiency, it can offer new opportunities for studying more complex time-varying photonic structures.

Dynamics of Thermocapillary-Driven Motion of Liquid Drops.

The thermocapillary migration of a drop placed on a solid plate is examined. The Brochard model using the lubrication approximation provides both Marangoni and Poiseuille flow components. The present 2D model extends Brochard analysis and provides a solution for the dynamics of drop migration using extended boundary conditions at the advancing and receding contact lines to account for both Marangoni and Poiseuille flow components, derived approximate drop profiles, and conservation of mass. The model is analytical, and the results are presented in a dimensionless form. The effects of the temperature gradient, surface tension coefficient to surface tension ratio, liquid viscosity, and static advancing and receding contact angles on migration dynamics are analyzed.

Spatially Dispersive Metasurfaces—Part II: IE-GSTC-SD Field Solver With Extended GSTCs

An integral equation (IE) based field solver to compute the scattered fields from spatially dispersive metasurfaces is proposed and numerically confirmed using various examples involving physical unit cells. The work is a continuation of Part-1 [1], which proposed the basic methodology of representing spatially dispersive metasurface structure in the spatial frequency domain, k||. By representing the angular dependence of the surface susceptibilities in k|| as a ratio of two polynomials, the standard generalized sheet transition conditions (GSTCs) have been extended to include the spatial derivatives of both the difference and average fields around the metasurface. These extended boundary conditions are successfully integrated here into a standard IE-GSTC solver, which leads to the new IE-GSTC-SD simulation framework. The proposed IE-GSTC-SD platform is applied to various uniform metasurfaces, including a practical short conducting wire unit cell, as a representative practical example, for various cases of finite-sized flat and curvilinear surfaces. In all cases, computed field distributions are successfully validated, either against the semi-analytical Fourier decomposition method or the brute-force full-wave simulation of volumetric metasurfaces in the commercial Ansys FEM-HFSS simulator.

Optical coupled-mode theory for dielectric solids of revolution

We propose a single resonance coupled-mode approach to light scattering by dielectric solids of revolution. By using a biorthogonal decomposition of the $S$ matrix found with the extended boundary condition method we derived all parameters required for application of the temporal coupled-mode theory in a closed form. The proposed approach allows for constructing a frequency-dependent Fano response due to a single resonance after the full-wave solution has been found at a single incident frequency.

Model for the diffuse reflectance in spatial frequency domain imaging.

In spatial frequency domain imaging (SDFI), tissue is illuminated with sinusoidal intensity patterns at different spatial frequencies. For low spatial frequencies, the reflectance is diffuse and a model derived by Cuccia etal. (doi 10.1117/1.3088140) is commonly used to extract optical properties. An improved model resulting in more accurate optical property extraction could lead to improved diagnostic algorithms. To develop a model that improves optical property extraction for the diffuse reflectance in SFDI compared to the model of Cuccia etal. We derive two analytical models for the diffuse reflectance, starting from the theoretical radial reflectance for a pencil-beam illumination under the partial current boundary condition (PCBC) and the extended boundary condition (EBC). We compare both models and the model of Cuccia etal. to Monte Carlo simulations. The model based on the PCBC resulted in the lowest errors, improving median relative errors compared to the model of Cuccia etal. by 45% for the reflectance, 10% for the reduced scattering coefficient and 64% for the absorption coefficient. For the diffuse reflectance in SFDI, the model based on the PCBC provides more accurate results than the currently used model by Cuccia etal.

Dimension-variable invariant imbedding (DVIIM) T-matrix computational method for the light scattering simulation of atmospheric nonspherical particles.

The invariant imbedding (IIM) T-matrix method has shown great potential in light scattering field. However, the T-matrix need to be calculated through the matrix recurrence formula derived from the Helmholtz equation, thus its computational efficiency is much lower than Extended Boundary Condition Method (EBCM). To alleviate this problem, the Dimension-Variable Invariant Imbedding (DVIIM) T-matrix method is presented in this paper. Compared with the traditional IIM T-matrix model, the dimensions of the T-matrix and relevant matrices are gradually increasing as the iteration performed step by step, thus the unnecessary operations of large matrices can be avoided in early iterations. To optimally determine the dimension of these matrices in each iterative calculation, the spheroid-equivalent scheme (SES) is also proposed. The effectiveness of the DVIIM T-matrix method is validated from the modeling accuracy and calculation efficiency. The simulation results show that compared with traditional T-matrix method, its modeling efficiency can be improved notably, especially for the particles with large size and aspect ratio, where for the spheroid with a aspect ratio of 0.5, the computational time is cut down by 25%. Though the dimension of the T matrix is cut down in the early iterations, the computational precision of DVIIM T-matrix model is not decreased notably, and a good agreement is achieved between the calculation results of DVIIM T-matrix method, IIM T-matrix method and other well-validated models (like EBCM and DDACSAT), where the relative errors of the integral scattering parameters (e.g., extinction, absorption, scattering cross sections) are generally less than 1%.

Achieving Zero-Impact Emissions with a Gasoline Passenger Car

The Euro 7 legislation and the Zero-Impact Emissions concept aim at significantly improving air quality. Technologies that reduce pollutant emissions beyond current gasoline passenger cars have already been intensively investigated, but a holistic system layout considering extended boundary conditions is missing so far. This paper therefore develops technical solutions to achieve a Euro 7 scenario and Zero-Impact Emissions for a 2030+ vehicle. First, challenging test scenarios are identified to develop compliant vehicles. The scenarios cover extreme conditions in real-world driving, such as hot and cold ambient conditions, stop-and-go in rural areas or high speed and steep gradients on highways. Different technology options are discussed and selected for the investigations. An empirical–physical simulation model for the exhaust gas aftertreatment system is extended with new technologies, such as an electrical heater disc in front of the catalyst or a burner in the exhaust system. In addition to stoichiometric engine operation and increased catalyst volume, the results show that the expected Euro 7 regulations can be achieved in all extreme scenarios by combining additional exhaust gas heating with engine power limitation or pre-heating. Moreover, even Zero-Impact Emissions are achieved in most cases with the same technology options.

The influence of surface roughness to particle optical characteristics by a study of Chebyshev particles

In this work, Chebyshev particles are used to study the influence of surface roughness and nonsphericity to the optical characteristics of particles by exploiting the parameters of Chebyshev particles, namely the waviness parameter and the deformation parameter. This is analyzed by comparing the scattering, extinction, and absorption cross sections, asymmetry parameter, single-scattering albedo, phase function, and linear polarization degree of different types of particles, including spheres and several types of Chebyshev particles. To calculate the optical characteristics, Extended Boundary Condition Method is utilized. The results show that there are some deviations affected by nonsphericity or surface roughness of particles, especially in relatively large size parameter, real part of refractive index is larger than 1, and imaginary part of refractive index is equal to 0. By approximating the surface roughness of particles by Chebyshev particles, larger waviness and deformation parameters tend to give larger difference from ideal particles in optical characteristics, so the optical characteristics could be controlled by modifying these parameters.

EM Scattering by Core-Shell Gyroelectric-Isotropic and Isotropic-Gyroelectric BoRs Using the EBCM

We employ the extended boundary condition method (EBCM) and construct a solution for the problem of electromagnetic (EM) scattering by anisotropic core-shell bodies of revolution (BoRs). In particular, two different core-shell configurations are examined: the gyroelectric-isotropic and the isotropic-gyroelectric setup. To construct the solution, we employ two groups of integral representations (IRs)—one group for each configuration solved—in conjunction with the discrete eigenfunction (DE) expansion of the fields in terms of spherical vector wave functions (SVWFs) for the gyroelectric regions. We demonstrate the validity and the computational performance of the method by comparisons with the HFSS commercial software for various core-shell setups such as spheroidal, cylindrical, and combined spherical-cylindrical BoRs. We also employ ADDA, a particular version of the discrete dipole approximation (DDA) method, to trace the boundaries of validity of the EBCM. Finally, we present an application of the method to the study of magnetically-tunable spheroidal THz antennas. The method can be used in a variety of potential EM applications including microwaves, functional photonics structures, as well as nanoantenna engineering.

Scattering From Fractal Surfaces Based on Decomposition and Reconstruction Theorem

A decomposition and reconstruction theorem (DRT) is introduced to advance computation and provide physical understanding for the scattering from fractal surfaces (FPs). The profile of FP is decomposed into test profiles (TPs), and the scattering of FP is reconstructed using the scattering of TPs to enhance the computational efficiency. A new method that applies DRT on the extended boundary condition method combined with the truncated singular value decomposition technique (TEBCM) is presented and referred to as TEBCM-DRT. The method of TEBCM-DRT is employed to solve the scattering from realistic soil surfaces, and the results are validated against other scattering models. Moreover, the efficiency of TEBCM-DRT and its validity range are investigated. The result shows that TEBCM-DRT improves computational efficiency to <inline-formula> <tex-math notation="LaTeX">$1.95\times 10^{5}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$7.35\times 10^{2}$ </tex-math></inline-formula> times, respectively, compared to TEBCM and the conventional method of moments for a wide range of roughness. In addition, TEBCM-DRT indicates that the amplitude and direction of propagation of scattering modes are dependent on deterministic TPs. This relationship benefits for obtaining the accurate height of an arbitrary point on the profile from bistatic scattering coefficients.

Theory of perturbation of electric potential by a 3D object made of an anisotropic dielectric material

The extended boundary condition method (EBCM) was formulated for the perturbation of a source electric potential by a 3D object composed of a homogeneous anisotropic dielectric medium whose relative permittivity dyadic is positive definite. The formulation required the application of Green’s second identity to the exterior region to deduce the electrostatic counterpart of the Ewald–Oseen extinction theorem. The electric potential inside the object was represented using a basis obtained by implementing an affine bijective transformation of space to the Gauss equation for the electric field. The EBCM yields a transition matrix that depends on the geometry and the composition of the 3D object, but not on the source potential.

Morphology-dependent resonances in homogeneous and core-shell nonspherical particles

We formulate a perturbation theory for determining the frequencies of quasinormal modes in dielectric nonspherical core-shell particles. The theory is based on the extended boundary condition method (also known as the null-field method), which is used to represent the internal field in terms of a matrix $Q$. In this framework, the excitation of a morphology-dependent resonance corresponds to a vanishing eigenvalue of the $Q$ matrix, which can be determined using perturbation techniques to yield explicit formulas for the corresponding eigenfrequency. We specialize our results to the case of a core-shell spherical particle that has been deformed into a spheroid and provide a comparison between numerical simulations and the perturbation theory. Excellent agreement is found for the small perturbations considered here. Further, we investigate the eigenfrequencies of core-shell particles for an extensive range of parameters.

EBCM for Electromagnetic Modeling of Gyrotropic BoRs

We employ the extended boundary condition method (EBCM) that we properly extend so as to describe gyroelectric and gyromagnetic (i.e., gyrotropic) anisotropy and report on the electromagnetic (EM) complex resonances of magnetooptic (i.e., gyroelectric) bodies of revolution (BoRs), as well as on the complex magnetic plasmon resonances (MPRs) of ferrite (i.e., gyromagnetic) BoRs. The proposed extension is based on an alternative scheme for the expansion of the EM field inside a gyrotropic medium, namely, a discrete eigenfunction (DE) expansion in terms of spherical vector wave functions (SVWFs). This approach provides the transition matrix (namely, T-matrix) that allows not only for the direct computation of the scattered field from the incident one, but also for the determination of the complex resonances of open (i.e., situated in free space) gyrotropic BoR resonators. The EBCM is validated on two levels: first, by calculating the EM scattering from various BoRs, including anisotropic spheroids, cylinders, and rods, and comparing with HFSS commercial software; second, by computing the complex eigenfrequency spectrum of gyroelectric spheroidal resonators and comparing with a recently developed rigorous technique for the EM modeling of anisotropic spheroids.

Notes on Rayleigh's hypothesis and the extended boundary condition method.

Regarding wave scattering on a three-dimensional nonspherical obstacle, the Rayleigh hypothesis states that the scattered field can be expanded everywhere outside the obstacle using only outgoing eigensolutions of the underlying Helmholtz equation. However, the correctness of this assumption has not yet been finally clarified, although it is important for the near-field analysis of scattering processes and for multiple scattering. To circumvent this uncertainty, Waterman introduced the extended boundary condition to develop his T-matrix method. This approach leads to the restriction that, when modeling multiple scattering processes using this T-matrix, the smallest circumscribing spheres of the individual obstacles must not overlap. The purpose of this paper is to provide a justification of the correctness of Rayleigh's hypothesis and clarify its implications for modeling multiple scattering. We show that Waterman's T-matrix can in fact be used inside the critical region between the surface of the obstacle and its smallest circumscribing sphere to represent the near-field and that one does not necessarily have to exclude an overlap of these spheres in the multiple scattering modeling. The theoretical considerations in the first part of this paper are supplemented by a numerical study of a benchmark configuration for multiple scattering in the last part.

Accurate Simulation of Light Propagation in Complex Skin Tissues Using an Improved Tetrahedron-Based Monte Carlo Method

Understanding light transportation in skin tissues can help improve clinical efficacy in the laser treatment of dermatosis, such as port-wine stains (PWS). Patient-specific cross-bridge PWS vessels are structurally complicated and considerably influence laser energy deposition due to shading effects. The shading effect of PWS vessels is investigated using a tetrahedron-based Monte Carlo (MC) method with extended boundary condition (TMCE). In TMCE, body-fitted tetrahedra are generated in different tissues, and the precision of photon–surface interaction can be considerably improved via mesh refinement. Such improvement is difficult to achieve with the widely used voxel-based MC method. To fit the real physical boundary, the extended boundary condition is adapted by extending photon propagation to the semi-infinite tissue layers while restricting the statistics of photon propagation in the computational domain. Results indicate that the shading parameters, such as the cross angle, vessel distance, and geometric shadow (GS), of cross-bridge blood vessel pairs determine the peak characteristics of photon deposition in deep vessels by affecting the relative deposition of collimated and diffused light. Collimated light is shaded, attenuated, and partially transformed into diffused light due to the increase in vessel distance and GS of vessel pairs, resulting in difficulty in treating deep and shallow vessels with one laser pulse. The TMCE method can be used for the individualized and precise forecasting of laser energy deposition based on the morphology and embedding characteristics of vascular lesions.

Plane wave scattering by a three-dimensional chiral object immersed in a simple Lorentz-nonreciprocal medium with magnetoelectric gyrotropy.

Scattering by a three-dimensional object composed of a chiral medium (the interior medium) and immersed in a simple Lorentz-nonreciprocal medium with magnetoelectric gyrotropy (the exterior medium) was treated using the extended boundary condition method (EBCM). The exterior medium is quantified by εre, μre, and Γ, whereas the interior medium is quantified by εri, μri, and β. When irradiated by a plane wave, the differential scattering efficiency does not depend on the polarization state of the incident plane wave if the exterior medium is impedance-matched with the interior medium, regardless of the shape of the object, Γ, and β. Zero backscattering is possible if, in addition to impedance-matching condition, the object is rotationally symmetric about the propagation direction, and Γ is parallel to the propagation direction. Numerical results confirm these remarks for scattering by spheroids. On fixing εri, μri, εre, and μre, the effects of Γ and β on the total scattering efficiency were examined for a sphere. The total scattering efficiency does not depend on the polarization state of the incident plane wave, even when the exterior medium is not impedance-matched with the interior medium, and despite the presence of Γ and β. The total scattering efficiency when Γ is coparallel to the propagation direction can be either equal to, larger than, or smaller than when Γ is antiparallel or perpendicular to the propagation direction, depending on β and the electrical size of the sphere. It is found that parallel propagation of the incident plane wave with respect to Γ has a stronger influence than perpendicular propagation, regardless of β and the electrical size of the sphere. The effect of increasing/decreasing the magnitude of Γ can be envisioned only when its direction is parallel to the propagation direction.