Extended Boundary Condition Method Research Articles

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.

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.

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%.

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.

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.

Analytical Jacobians of single scattering optical properties using the invariant imbedding T-matrix method

Integrated and differential optical properties of a single particle, such as the scattering, absorption, and extinction cross sections, single scattering albedo, asymmetry factor, and scattering phase matrix, are derived from electromagnetic scattering theory. This process depends on microphysical inputs which include particle shape, refractive index, aspect ratio, and size parameter. In this work, we use the invariant imbedding T-matrix method (IITM) to derive analytic expressions for Jacobians of these optical properties with respect to the input parameters. These IITM-derived Jacobians for spheroids, cylinders, and hexagonal prisms are validated by comparison with results calculated with the extended boundary condition method (EBCM) and further validated using finite-difference estimates. We examine the dependencies of these Jacobians as functions of the input microphysical parameters, focusing again on spheroids, cylinders, and hexagonal prisms.

Solution of the Problem of Diffraction by a Circular Inhomogeneous Cylinder Partially Covered by a Reflective Screen

Based on the method of extended boundary conditions, an algorithm for the numerical solution of the two-dimensional problem of diffraction by a dielectric cylinder with wave number depending on the radial coordinate, which is partially covered by a perfectly reflective screen. The method is tested for the case of a homogeneous medium inside a dielectric cylinder. The accuracy of the fulfillment of the optical theorem is checked for various dependences of the wave number on the radial coordinate and the angular dependences of the scattering diagram are plotted for different opening angles of the reflective screen.

Application of the Method of Extended Boundary Conditions to the Solution of the Problem of Wave Diffraction by Magnetodielectric Scatterers Having a Compound Geometry

A technique allowing to model scattering characteristics for bodies of arbitrary geometries is suggested on the basis of the method of extended boundary conditions. The scattering characteristics include those ones, which are averaged over orientation angles. The 2D problem of diffraction of a plane wave by dielectric bodies having a complicated geometry of the cut and, in particular, by bodies similar to fractals is considered. The numerical algorithms of the diffraction problem solution on the basis of the systems of integral equations of the first and second kinds are compared. The correctness of the method is confirmed with the help of the verification of the optical theorem fulfillment for various bodies and by comparing with the calculation results obtained by the modified method of discrete sources.

Efficient design of the realization scheme of the invariant imbedding (IIM) T-matrix light scattering model for atmospheric nonspherical particles

The invariant imbedding T-matrix method has been recognized as one of the rigorous computational tools to simulate the light scattering by atmospheric nonspherical particles after the efficient implementation by L. Bi and P. Yang. Compared with the T-matrix model based on the Extended Boundary Condition Method (“EBCM”), it is suitable for the light scattering simulation of particles with any shapes. However, since the T-matrix must be updated in each iterative process, its computational efficiency is relatively lower than EBCM, especially for the particles with large sizes and large aspect ratios. To this end, an efficient realization scheme of the IIM T-matrix method is proposed in this paper. The principle and the structure of our IIM T-matrix code is firstly introduced, and then the symmetric properties of the U-matrix (an important matrix in the IIM T-matrix method) is derived; to simplify the iteration of T-matrix calculation, the implicit matrix inversion scheme is also proposed. To validate the effectiveness of these improvements, the computational efficiency and accuracy is systematically investigated for the particles with different shapes and sizes. The simulation results indicate that the computational time is reduced notably after the model is improved by the symmetry of U-matrix and the implicit matrix inversion scheme, where for the spheroid with (a, b) = (2.5 µm, 3.0 µm) at λ=0.5 µm (the particle is treated as complex geometries without any symmetry), the computational efficiency is improved by nearly 40%. Excellent agreement is also achieved between the scattering parameters obtained by the IIM T-matrix method and the well-test scattering models (such as BECM, DDASCAT, MRTD and IGOA, etc.), indicating that the IIM T-matrix method can simulate the light scattering by nonspherical particles effectively.

Modified finite series technique for the evaluation of beam shape coefficients in the T-matrix methods for structured beams with application to Bessel beams

The finite series technique is a rigorous mathematical formulation to exactly calculate the beam shape coefficients gn,TMm and gn,TEm which describe an arbitrary structured beam within the framework of the T-matrix methods for structured beams, e.g. generalized Lorenz–Mie theory or Extended Boundary Condition Method. It allows for fast computation without having recourse to time-consuming quadratures with double or triple integrals. Here, a simpler version of the finite series – the Modified Finite Series – is presented in which gn,TMm and gn,TEm are obtained by using an arbitrary choice of the spherical angle θ in the development of the finite series technique. To illustrate it, a linearly polarized vector Bessel beam is described within this new formalism. We expect that the present method will be an important alternative tool in the field of light scattering and light-matter interaction within the optical domain.

Transmission and reflection characteristics of a multi-layered wall with doubly periodic interfaces

In this paper, a semi-analytical technique to derive scattering coefficients of a three-dimensional multi-layered wall with doubly periodic interfaces is presented. First, the proposed method uses the periodicity of interfaces in the surface integral equation formulation to confine the integration interval to just one period. It also simplifies the formulation by using the extended boundary condition method (EBCM) and the spectral formulation of the background Green’s function. Then, each of the scattering coefficient matrices of interfaces and layers is treated as two-port networks, and the scattering matrix of the whole structure is derived using the scattering transfer matrix method. In a limiting case, the proposed method is validated by comparing the results to a two-dimensional semi-analytical approach that is available in the literature. Then for a single layer wall, the effects of side interfaces’ roughness, layer’s thickness, and dielectric constant on reflection and transmission coefficients are analyzed.

On an infinite number of quadratures to evaluate beam shape coefficients in generalized Lorenz-Mie theory and the extended boundary condition method for structured EM beams

When dealing with light scattering theories such as the T-matrix methods for structured laser beams, e.g. Generalized Lorenz-Mie Theory (GLMT) or the Extended Boundary Condition Method (EBCM), EM fields are expanded over a set of Vector Spherical Wave Functions (VSWFs) involving spherical Bessel functions, with expansion coefficients expressed in terms of Beam Shape Coefficients (BSCs). Although spherical Bessel functions are orthogonal over the range (−∞,+∞), the GLMT may be expressed using a non-orthogonal set of spherical Bessel functions defined over (0,+∞), allowing one to generate an infinite number of quadratures for evaluating the BSCs. This paper points out the difference between orthogonal and non-orthogonal spherical Bessel functions, establishes the infinite number of quadratures and discusses its properties.

Design and Validation of the Invariant Imbedded T-Matrix Scattering Model for Atmospheric Particles with Arbitrary Shapes

Light scattering by non-spherical particles is an important factor influencing atmospheric radiative transfer. To accurately simulate the scattering properties of non-spherical particles, the Invariant Imbedded T-matrix method (IIM T-Matrix) is developed by combining the Lorenz–Mie theory and invariant imbedding technique. In this model, the non-spherical particle is regarded as an inhomogeneous sphere and discretized into multiple spherical layers in the spherical coordinate system. The T-matrix of the inscribed sphere is firstly calculated by the Lorenz–Mie theory, and then taking it as the initial value, the T-matrix is updated layer by layer by using the invariant imbedding technique. To improve the computational efficiency, the model is further parallelized by the OpenMP technique. To verify the simulation accuracy of the IIM T-Matrix method, the results of the model are compared with those of the EBCM (Extended Boundary Condition Method) T-Matrix method, DDA (Discrete Dipole Approximation) and MRTD (Multi-Resolution Time Domain). The results show that the scattering phase matrix simulated by the IIM T-Matrix method closely agrees with that of the well-tested models, indicating that the IIM T-Matrix method is a powerful tool for the light scattering simulation of non-spherical particles. Since the IIM T-Matrix method is derived from the volume integral equation, compared to the T-Matrix method which is based on surface integral principles (i.e., “EBCM” or the “null field method”), it can be applied to the scattering calculations of particle with arbitrary shapes and inhomogeneous compositions, which can greatly expand the application scope of the T-Matrix method.

Sufficient Conditions for Zero Backscattering by a Uniaxial Dielectric-Magnetic Scatterer Endowed With Magnetoelectric Gyrotropy

As vector wavefunctions are available to represent incident and scattered fields in an isotropic dielectric-magnetic medium endowed with magnetoelectric gyrotropy, a transition matrix can be conceptualized to relate the scattered field coefficients to the incident field coefficients for scattering by an arbitrary scatterer composed of a linear medium. The elements of the transition matrix must satisfy certain conditions for zero backscattering. For a scatterer composed of a uniaxial dielectric-magnetic medium endowed with magnetoelectric gyrotropy, the extended boundary condition method (EBCM) can be formulated to determine the transition matrix. The numerical results obtained thereby lead to the formulation of a sufficient set of three zero-backscattering conditions: (i) the scatterer is a body of revolution with the incident plane wave propagating along the axis of revolution; (ii) the impedances of both mediums are identical; and (iii) the magnetoelectric-gyrotropy vectors of both mediums are aligned along the axis of revolution, whether or not both magnetoelectric-gyrotropy vectors are co-parallel.