Rigorous Integral Equation Research Articles

Scattering integral equation formulation for intravascular inclusion biosensing

A dielectric waveguide, inserted into blood vessels, supports its basic mode that is being scattered by a near-field intravascular inclusion. A rigorous integral equation formulation is performed and the electromagnetic response from that inhomogeneity is semi-analytically evaluated. The detectability of the formation, based on spatial distribution of the recorded signal, is estimated by considering various inclusion sizes, locations and textural contrasts. The proposed technique, with its variants and generalizations, provides a generic versatile toolbox to efficiently model biosensor layouts involved in healthcare monitoring and disease screening.

On an electrostatic micropump with a rigorous mathematical model

The novel electrostatic micropump model for applications such as in biomedical drug delivery is presented. The geometrical arrangement of the coupling rigid electrodes lets us exploit our mathematically rigorous boundary integral equation formulation and its solution. Thus, the charge densities involving the fringe effects on the plates are obtained by means of analytical regularization method (ARM) constructed for annular strips earlier. The efficiency of the constructed method is demonstrated with respect to the direct integral equation solvers implemented via the entire domain Galerkin method and point matching. The main physical characteristics of the suggested system and their deviation from that of infinitely large plane approximation are discussed.

Magnetic switching of Kerker scattering in spherical microresonators

Abstract Magneto-optical materials have become a key tool in functional nanophotonics, mainly due to their ability to offer active tuning between two different operational states in subwavelength structures. In the long-wavelength limit, such states may be considered as the directional forward- and back-scattering operations, due to the interplay between magnetic and electric dipolar modes, which act as equivalent Huygens sources. In this work, on the basis of full-wave electrodynamic calculations based on a rigorous volume integral equation (VIE) method, we demonstrate the feasibility of obtaining magnetically-tunable directionality inversion in spherical microresonators (THz antennas) coated by magneto-optical materials. In particular, our analysis reveals that when a high-index dielectric is coated with a magneto-optical material, we can switch the back-scattering of the whole particle to forward-scattering simply by turning off/on an external magnetic field bias. The validity of our calculations is confirmed by reproducing the above two-state operation, predicted by the VIE, with full-wave finite-element commercial software. Our results are of interest for the design of state-of-the-art active metasurfaces and metalenses, as well as for functional nanophotonic structures, and scattering and nanoantennas engineering.

L-Kuramoto–Sivashinsky SPDEs vs. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws

We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto–Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by the space–time white noise in one-to-three dimensional spaces. Both classes were introduced—with Brownian-time-type kernel formulations—by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in d=1,2,3. We show that both classes are (1/2)− Hölder continuously differentiable in space when d=1, and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by the space–time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order <1/2 and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba's earlier works on (1) and (2). Similar regularity results hold for the Allen–Cahn nonlinear members of (1) and (2) on compacts via change of measure.

Anomalous reflection of visible light by all-dielectric gradient metasurfaces

Plane wave scattering by a planar metasurface composed of two periodically alternating rectangular dielectric rods is considered. A rigorous integral equation methodology is employed for the analysis and the accurate determination of the reflected and transmitted fields. Systematic optimizations with respect to the configuration’s parameters are performed, which reveal that it is possible to obtain significantly enhanced anomalous reflection (with simultaneously suppressed ordinary reflection predicted by Snell’s law) with power varying from 92% to almost 100% of the input one, depending on the color of the incident light. It is shown that these reflection properties are supported by metasurfaces easily realizable with specific low-loss dielectric materials. In this way, several all-dielectric optimal designs are reported that can be used in numerous applications demanding anomalous reflection in the visible range.

Efficient integral equation modeling of scattering by a gradient dielectric metasurface

Plane-wave scattering by a gradient dielectric metasurface, composed of a dielectric slab on which has been etched an infinitely periodic grating, is analyzed by means of rigorous semi-analytical integral equation techniques based on the Method of Moments with entire domain basis functions. The electric field integral equation is considered with unknown function the electric field on the grating and is then solved by applying an entire domain Galerkin’s technique. The proposed methodology is characterized by high numerical stability and controllable accuracy; the obtained solution is analytic with the sole approximation of the final truncation of the expansion functions sets. The operation of the metasurface as a narrow-band reflection frequency filter is investigated. Moreover, it is also shown that under certain conditions anomalous reflection and transmission can be generated even for a metasurface composed of conventional materials. The numerical results obtained provide design guidelines, which may pave the way for more systematic investigations of gradient dielectric metasurfaces assisting the control and manipulation of electromagnetic waves.

Isotope effects of trapped electron modes in the presence of impurities in tokamak plasmas

The trapped electron modes (TEMs) are numerically investigated in toroidal magnetized hydrogen, deuterium and tritium plasmas, taking into account the effects of impurity ions such as carbon, oxygen, helium, tungsten and others with positive and negative density gradients with the rigorous integral eigenmode equation. The effects of impurity ions on TEMs are investigated in detail. It is shown that impurity ions have substantially-destabilizing (stabilizing) effects on TEMs in isotope plasmas for (), opposite to the case of ion temperature gradient (ITG) driven modes. Detailed analyses of the isotope mass dependence for TEM turbulences in hydrogenic isotope plasmas with and without impurities are performed. The relations between the maximum growth rate of the TEMs with respect to the poloidal wave number and the ion mass number are given in the presence of the impurity ions. The results demonstrate that the maximum growth rates scale as in pure hydrogenic plasmas. The scale depends on the sign of its density gradient and charge number when there is a second species of (impurity) ions. When impurity ions have density profiles peaking inwardly (i.e. ), the scaling also depends on ITG parameter . The maximum growth rates scale as for the case without ITG () or the ITG parameter is positive () but the impurity ion charge number is low (). However, when and the impurity ion charge number is moderate (), the scaling law is found as . Here, Z is impurity ion charge number, and the effective mass number, , with and being the mass numbers of the hydrogenic and impurity ions, respectively, and being the charge concentration of impurity ions. In addition, with regard to the case of , the maximum growth rate scaling is . The possible relations of the results with experimental observations are discussed.

Radiative properties of diffractively-coupled optical nano-antennas with helical geometry.

In this paper, using the rigorous Surface Integral Equation (SIE) method, we study light scattering by Au nano-helices with geometrical dimensions comparable to the wavelength of visible light and we demonstrate that they behave as highly directional nano-antennas with largely controllable radiation and polarization characteristics in the optical regime. In particular, we systematically investigate the radiation properties of helical nano-antennas with realistic Au dispersion parameters in the visible spectral range, and we establish general design rules that enable the engineering of directional scattering with elliptical or circular polarization. Given the realistic material and geometric parameters used in this work, our findings provide novel opportunities for the engineering of chiral sensors, filters, and components for nano-scale antennas with unprecedented beam forming and polarization capabilities.

Rectangular dipoles in the discrete dipole approximation

We performed a comprehensive analysis of the extension of the discrete dipole approximation (DDA) to a rectangular cuboid lattice of dipoles. The theoretical analysis of two different approaches, based either on the point–dipole interaction or on the integration of Green׳s tensor (IGT), was performed starting with the rigorous integral equation for the electric field. We showed that the expressions for polarizability and interaction terms must strictly conform to each other, which resolves the existing controversy in the literature. Moreover, there are large differences between the spectra of the interaction matrix in the static limit for those DDA formulations. In particular, the point–dipole formulation leads to unphysical edges of the spectrum that deteriorate the convergence of the iterative solver with increasing refractive index. This severely limits the applicability of point–dipole DDA formulations with rectangular dipoles in contrast to the case of cubic dipoles. We implemented both above formulations in the open-source code ADDA and illustrated their performance on a number of test cases. In particular, we considered a graphene sheet, with thickness much smaller than the wavelength. The use of rectangular dipoles (with IGT) resulted in up to 100-times decrease of both simulation time and memory requirements, keeping the satisfactory accuracy. Similar improvements are expected for any strongly oblate or prolate particles in which the smallest dimension is much smaller than the wavelength.

Fano-resonant aluminum and gold nanostructures created with a tunable, up-scalable process

An up-scalable approach for creating Fano-resonant nanostructures on large surfaces at visible wavelengths is demonstrated. The use of processes suitable for high throughput fabrication and the choice of aluminum as a cost-efficient plasmonic material ensure that the presented insights are valuable even in consideration of typical industrial constraints. In particular, wafer-scale fabrication and the process compatibility with roll-to-roll embossing are demonstrated. It is shown that through adjustment of readily accessible evaporation parameters, the shape and position of the optical resonance can be tuned within a spectral band of more than 70 nm. The experimental data are complemented with rigorous coupled wave analysis and surface integral equation simulations. Calculated electric fields as well as surface charges shed light onto the physics behind the present resonances. In particular, a surface plasmon polariton is found to couple to a localized plasmonic mode with a hexapolar charge distribution, leading to a Fano-like resonance. Further understanding of the interactions at hand is gained by considering both aluminum and gold nanostructures.

Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function

A rigorous surface integral equation approach is proposed to study the spontaneous emission of a quantum emitter embedded in a multilayered plasmonic structure with the presence of arbitrarily shaped metallic nanoscatterers. With the aid of the Fermi's golden rule, the spontaneous emission of the emitter can be calculated from the local density of states, which can be further expressed by the imaginary part of the dyadic Green's function of the whole electromagnetic system. To obtain this Green's function numerically, a surface integral equation is established taking into account the scattering from the metallic nanoscatterers. Particularly, the modeling of the planar multilayered structure is simplified by applying the layered medium Green's function to reduce the computational domain and hence the memory requirement. Regarding the evaluation of Sommerfeld integrals in the layered medium Green's function, the discrete complex image method is adopted to accelerate the evaluation process. This work offers an accurate and efficient simulation tool for analyzing complex multilayered plasmonic system, which is commonly encountered in the design of optical elements and devices.

Axial soil–pile interaction in a transversely isotropic half-space

A rigorous integral equation formulation is presented for the axisymmetric load-transfer analysis of a thin-walled pile embedded in a transversely isotropic half-space under axial load. By virtue of a set of ring-load Green’s functions for the pile and one for the half-space, the problem is shown to be reducible to a pair of Fredholm integral equations. Through a mathematical analysis of an auxiliary pair of Cauchy integral equations, the inherent singularities of the contact stress distributions are rendered explicit. With a direct incorporation of the singular nature of the resultant load transfers, the numerical solution of the integral equations is shown to be possible by an interpolation of regular functions only. Typical results for various material and geometrical conditions are presented, including a comparison with past classical solutions, to illustrate the effects of transverse anisotropy on the load-transfer process.

Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities

The scattering of a surface plasmon polariton (SPP) by a rectangular dielectric channel discontinuity is analyzed through a rigorous magnetic field integral equation method. The scattering phenomenon is formulated by means of the magnetic-type scalar integral equation, which is subsequently treated through an entire-domain Galerkin method of moments (MoM), based on a Fourier-series plane wave expansion of the magnetic field inside the discontinuity. The use of Green's function Fourier transform allows all integrations over the area and along the boundary of the discontinuity to be performed analytically, resulting in a MoM matrix with entries that are expressed as spectral integrals of closed-form expressions. Complex analysis techniques, such as Cauchy's residue theorem and the saddle-point method, are applied to obtain the amplitudes of the transmitted and reflected SPP modes and the radiated field pattern. Through numerical results, we examine the wavelength selectivity of transmission and reflection against the channel dimensions as well as the sensitivity to changes in the refractive index of the discontinuity, which is useful for sensing applications.

High Scalability FMM-FFT Electromagnetic Solver for Supercomputer Systems

An efficient parallel implementation of the Fast Multipole Method (FMM) combined with the fast Fourier transform (FFT) is presented in this paper. The good scaling properties of the FMM-FFT, combined with a smart parallelization strategy, has been shown to be very effective when using large parallel supercomputers. A challenging problem with more than 150 million unknowns has been solved, demonstrating that the proposed implementation of the FMM-FFT constitutes a real alternative to the more frequently used multilevel approaches, such as the Multilevel FMM (MLFMA). Even more importantly, we have achieved a high efficiency with 1,024 parallel processors, which indeed constitutes one of the better scalability results ever reached for a rigorous integral-equation electromagnetic solver.

Magnetic field integral equation analysis of interaction between a surface plasmon polariton and a circular dielectric cavity embedded in the metal

A rigorous integral equation (IE) analysis of the interaction between a surface plasmon polariton (SPP) and a circular dielectric cavity embedded in a metal half-space is presented. The device is addressed as the plasmonic counterpart of the established integrated optics filter comprising a whispering gallery (WG) resonator coupled to a waveguide. The mathematical formulation is that of a transverse magnetic scattering problem. Using a magnetic-type Green's function of the two-layer medium with boundary conditions that cancel the line integral contributions along the interface, an IE for the magnetic field inside the cavity is obtained. The IE is treated through an entire-domain method of moments (MoM) with cylindrical-harmonic basis functions. The entries of the MoM matrix are determined analytically by utilizing the inverse Fourier transform of Green's function and the Jacobi-Anger formula for interchanging between plane and cylindrical waves. Complex analysis techniques are applied to determine the transmitted, reflected, and radiated field quantities in series forms. The numerical results show that the scattered SPPs' spectra exhibit pronounced wavelength selectivity that is related to the excitation of WG-like cavity modes. It seems feasible to exploit the device as a bandstop or reflective filter or even as an efficient radiating element. In addition, the dependence of transmission on the cavity refractive index endows this structure with a sensing functionality.

Feasibility of simplified integral equation modeling of low-frequency marine CSEM with a resistive target

We have assessed the accuracy of a simplified integral equation (SIE) modeling approach for marine controlled-source electromagnetics (CSEM) with low applied frequencies and a resistive target. The most computationally intensive part of rigorous integral equation (IE) modeling is the computation of the anomalous electric field within the target itself. This leads to a matrix problem with a dense coefficient matrix. It is well known that, in general, the presence of many grid cells creates a computational disadvantage for dense-matrix methods compared to sparse-matrix methods. The SIE approach replaces the dense-matrix part of rigorous IE modeling by sparse-matrix calculations based on an approximation of Maxwell’s equations. The approximation is justified theoretically if a certain dimensionless parameter [Formula: see text] is small. As opposed to approximations of the Born type, the validity of the SIE approach does not rely on the anomalous field to be small in comparison with the background field in the target region. We have calculated [Formula: see text] for a range of parameter values typical for marine CSEM, and compared the SIE approach numerically to the rigorous IE method and to the quasi-linear (QL) and quasi-analytic (QA) approximate solutions. It is found that the SIE approach is very accurate for small [Formula: see text], corresponding to frequencies in the lower range of those typical for marine CSEM for petroleum exploration. In addition, the SIE approach is found to be significantly more accurate than the QL and QA approximations for small [Formula: see text].

Scattering by a Grating Slab Waveguide With Regular Plane Regions Grooves: Integral Equation Modeling

A rigorous integral equation formulation in conjunction with Green's function theory is used to analyze the scattering phenomena by a periodic dielectric grating slab waveguide. Such a structure is mainly utilized to obtain a frequency selective surface, finding applications both in the microwave and the optical region. The grating is modeled as a periodic layer with grooves regular plane regions. The advantages of the developed method with respect to already established ones are discussed. The dependence of the waveguide's reflectance and transmission from the grooves characteristics is investigated.

Diffraction of plane waves incident on a grated dielectric slab: An entire domain integral equation analysis

Diffraction of electromagnetic waves by periodic grating waveguides is investigated by using a rigorous integral equation method, which combines semianalytical techniques and the Method of Moments with entire domain basis functions. The electric field integral equation is employed with unknown function the electric field on the grooves. This equation is subsequently solved by applying an entire domain Galerkin's technique. The proposed analysis provides high numerical stability and controllable accuracy. All the involved computations are analytically carried out, leading to an analytic solution with the sole approximation of the final truncation of the expansion functions sets. The computed results exhibit superior accuracy and numerical efficiency compared with those already derived by applying different methods. The effect of the incident field's and grating's characteristics on the diffraction process as well as the grating structure's efficient operation as a narrow band reflection filter are thoroughly investigated. The numerical results obtained provide design guidelines, which may be exploited appropriately in the development of millimeter and optical waveguide structures.

Efficient Analysis and Design of Low-Loss Whispering-Gallery-Mode Coupled Resonator Optical Waveguide Bends

Waveguides that are composed of electromagnetically coupled optical microcavities [coupled resonator optical waveguides (CROWs)] can be used for light guiding, slowing, and storage. In this paper, we present a 2-D analysis of finite-size straight and curved CROW sections based on a rigorous Muller boundary integral equation method. We study the mechanisms of the coupling of whispering-gallery (WG) mode and guiding light around bends in CROWs composed of both identical and size- mismatched microdisk resonators. Our accurate analysis reveals the differences in WG modes coupling in the vicinity of bends in CROWs composed of optically large and wavelength-scale micro- cavities. We propose and discuss possible ways to design low-loss CROW bends and to reduce bend losses. These include selecting specific bend angles depending on the azimuthal order of the WG mode and tuning the radius of the microdisk positioned at the CROW bend.

Rigorous 3D inversion of marine CSEM data based on the integral equation method

Marine controlled-source electromagnetic (MCSEM) surveys have become an important part of offshore petroleum exploration. However, due to enormous computational difficulties with full 3D inversion, practical interpretation of MCSEM data is still a very challenging problem. We present a new approach to 3D inversion of MCSEM data based on rigorous integral-equation (IE) forward modeling and a new IE representation of the sensitivity (Fréchet derivative matrix) of observed data to variations in sea-bottom conductivity. We develop a new form of the quasi-analytical approximation for models with variable background conductivity (QAVB) and apply this form for more efficient Fréchet derivative calculations. This approach requires just one forward modeling on every iteration of the regularized gradient-type inversion algorithm, which speeds up the computations significantly. We also use a regularized focusing inversion method, which provides a sharp boundary image of the petroleum reservoir. The methodology is tested on a 3D inversion of the synthetic EM data representing a typical MCSEM survey conducted for offshore petroleum exploration.