Poggio-Miller-Chang-Harrington-Wu-Tsai Research Articles

Advanced Numerical and Experimental Analysis of Ultra-Miniature Surface Resonators

Many scientific and technological applications make use of strong microwave fields. These are often realized in conjunction with microwave resonators that have small geometric features in which such fields are generated. For example, in magnetic resonance, large microwave and RF magnetic fields make it possible to achieve fast control over the measured electron or nuclear spins in the sample and to detect them with high sensitivity. The numerical analysis of resonators with small geometric features can pose a significant challenge. This paper describes a general method of analysis and characterization of surface microresonators in the context of electron spin resonance (ESR) spectroscopy and spin-based quantum technology. Our analysis is based on the Electric Field Integral Equation (EFIE) and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. In particular, we focus on a class of resonator configurations that possesses extremely small subwavelength features, which normally would require an ultra-fine mesh. We present several efficient techniques to numerically model, solve, and analyze these types of configurations for both normal and superconducting structures. The validation of these techniques is established both numerically and experimentally by the S11 parameters as well as the provision of direct mapping of the resonator’s microwave magnetic field component using a unique electron spin resonance micro-imaging method.

Single-Source SIE for 2-D Arbitrarily Connected Penetrable and PEC Objects With Nonconformal Meshes

We proposed a simple and efficient modular single-source surface integral equation (SS-SIE) formulation for electromagnetic analysis of arbitrarily connected penetrable and perfectly electrical conductor (PEC) objects in two-dimensional space. In this formulation, a modular equivalent model for each penetrable object consisting of the composite structure is first independently constructed through replacing it by the background medium, no matter whether it is surrounded by the background medium, other media, or partially connected objects, and enforcing an equivalent electric current density on the boundary to remain fields in the exterior region unchanged. Then, by combining all the modular models and any possible PEC objects together, an equivalent model for the composite structure can be derived. The troublesome junction handling techniques are not needed and non-conformal meshes are intrinsically supported. The proposed SS-SIE formulation is simple to implement, efficient, and flexible, which shows significant performance improvement in terms of CPU time compared with the original SS-SIE formulation and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Several numerical examples including the coated dielectric cuboid, the large lossy objects, the planar layered dielectric structure, and the partially connected dielectric and PEC structure are carried out to validate its accuracy, efficiency and robustness.

An advanced two-scale model of EM backscattering from rough surfaces

Based on the random and composite rough surface theory, an advanced two-scale model (ATSM) is proposed to calculate the electromagnetic scattering from the rough surface. The large-scale rough surface part is calculated by Kirchhoff Approximation Method (KAM). Compared with the previous two-scale method, the small-scale roughness is calculated by the Advanced Integral Equation Method (AIEM), which can improve the calculation accuracy and applicability range in terms of the roughness and frequency. By calculating the rough surface backscattering, it can be proved that the proposed two-scale method agrees well with the KAM results for near-vertical incidence and is identical to the AIEM results for large-angle incidence. It should be noted that the proposed method can make up for the drawback that KA decays too fast for swept incidence. The proposed ATSM is also compared with the Shooting and Bouncing Ray (SBR) method and the Poggio-MillerChang-Harrington-Wu-Tsai (PMCHWT) method to verify the accuracy.

Vector Single-Source Surface Integral Equation for TE Scattering From Cylindrical Multilayered Objects

A single-source surface integral equation (SS-SIE) for transverse electric (TE) scattering from cylindrical multilayered objects is proposed in this paper. By incorporating the differential surface admittance operator (DSAO) and recursively applying the surface equivalence theorem from innermost to outermost boundaries, an equivalent model with only electric current density on the outermost boundary can be obtained. In addition, an integration approach is proposed, where the small argument expansion of the Hankel function is used to evaluate the singular and nearly singular integrals. Compared with other SIEs, such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation, the computational expenditure is reduced for multilayered structures because only a single source is needed on the outermost boundary. As shown in the numerical results, the proposed method generates only 19% of unknowns, uses 26% of memory, and requires 29% of the CPU time of the PMCHWT formulation.

Towards a unified approach to electromagnetic analysis of objects embedded in multilayers

According to the surface equivalence theorem [1], any enclosed surface with electric and magnetic current densities enforced on it can generate exactly the same fields as the original problem. The problem is how to construct the current sources. In this paper, an efficient and accurate unified approach with an equivalent current density enforced on the outermost boundary is proposed to solve transverse magnetic (TM) scattering problems by objects embedded in multilayers. In the proposed approach, an equivalent current density is derived after the surface equivalence theorem is recursively applied on each boundary from innermost to outermost interfaces. Then, the objects are replaced by the background medium and the equivalent electric current density only on the outermost boundary is derived. The scattering problems by objects embedded in multilayers can be solved with the electric field integral equation (EFIE). Compared with the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and other dual source formulations, the proposed approach shows significant benefits: only the surface electric current density instead of both the electric and magnetic current densities is required to model the complex objects in the equivalent problem. Furthermore, the electric current density is only enforced on the outermost boundary of objects. Therefore, the overall count of unknowns can be significantly reduced. At last, several numerical experiments are performed to validate its accuracy and efficiency. Although overhead is required to construct the intermediate matrices, the overall performance improvement is still significant as numerical results are shown. Therefore, it shows great potential useful in the practical engineering applications.

Eddy Current Modeling in Multiply Connected Regions via a Full-Wave Solver Based on the Quasi-Helmholtz Projectors

Eddy currents are central to several industrial applications and there is a strong need for their efficient modeling. Existing eddy current solution strategies are based on a quasi-static approximation of Maxwell’s equations for lossy conducting objects and thus their applicability is restricted to low frequencies. On the other hand, available full-wave solvers such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation become highly ill-conditioned and inaccurate in eddy current settings. This work presents a new well-conditioned and stable full-wave formulation which encompasses the simulation of eddy currents. Our method is built upon the PMCHWT equation and thus remains valid over the entire frequency range. Moreover, our scheme is also compatible with structures containing holes and handles (multiply connected geometries). The effectiveness of quasi-Helmholtz projectors is leveraged to obtain a versatile solver, which is computationally efficient and allows for a seamless transition between low and high frequencies. The stability and accuracy of the new method are demonstrated both theoretically and through numerical experiments on canonical and realistic structures.

Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

We consider the simulation of electromagnetic scattering by single and multiple isotropic homogeneous dielectric particles using boundary integral equations. Galerkin discretizations of the classical Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) boundary integral equation formulation provide accurate solutions for complex particle geometries, but are well-known to lead to ill-conditioned linear systems. In this paper we carry out an experimental investigation into the performance of Calderón preconditioning techniques for single and multiple absorbing obstacles, which involve a squaring of the PMCHWT operator to produce a well-conditioned second-kind formulation. For single-particle scattering configurations we find that Calderón preconditioning is actually often outperformed by simple “mass-matrix” preconditioning, i.e. working with the strong form of the discretized PMCHWT operator. In the case of scattering by multiple particles we find that a significant saving in computational cost can be obtained by performing block-diagonal Calderón preconditioning in which only the self-interaction blocks are preconditioned. Using the boundary element software library Bempp (www.bempp.com) the numerical performance of the different methods is compared for a range of wavenumbers, particle geometries and complex refractive indices relevant to the scattering of light by atmospheric ice crystals.

Electromagnetic Field Analysis of an Electric Dipole Antenna Based on a Surface Integral Equation in Multilayered Dissipative Media

In this paper, a novel method based on the Poggio–Miller–Chang-Harrington–Wu–Tsai (PMCHWT) integral equation is presented to study the electromagnetic fields excited by vertical or horizontal electric dipoles in the presence of a layered region which consists of K-layered dissipative media and the air above. To transform the continuous integral equation into a block tridiagonal matrix with the feature of convenient solution, the Rao–Wilton–Glisson (RWG) functions are introduced as expansion and testing functions. The electromagnetic fields excited by an electric dipole are calculated and compared with the available results, where the electric dipole antenna is buried in the non-planar air–sea–seabed, air–rock–earth–mine, and multilayered sphere structures. The analysis and computations demonstrate that the method exhibits high accuracy and solving performance in the near field propagation region.

On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

We present a hierarchical basis preconditioning strategy for the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries.To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the Electric Field Integral Equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severly ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach.

Fast Solution of Electromagnetic Scattering From Homogeneous Dielectric Objects with Multiple-Traces EF/MFIE Method

In this letter, a novel multiple-traces electric field/magnetic field integral equation (MT-EF/MFIE) method is developed for solving electromagnetic (EM) scattering from dielectric objects. Different from the traditional Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) method, the nonconformal MT-EF/MFIE method can discretize the surface with two pairs of nonconformal meshes (exterior and interior). Meanwhile, the transmission conditions are enforced to ensure the continuity of currents, on the interface of the exterior domain and interior domain. Compared to the traditional PMCHWT method, this MT-EF/MFIE method leads to a better conditioned matrix and obviously interior memory and time consumption.

Calderón Preconditioned PMCHWT Equations for Analyzing Penetrable Objects in Layered Medium

We study Calderón preconditioners for analyzing electromagnetic scattering by penetrable objects in a layered medium. To account for the scattering effects of the multilayered background, the layered medium Green's function is adopted in the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) method. However, similar to the free-space case, the spectrum of the resulting equation is undesirable. This leads to a slow convergence of an iterative solver, especially when the geometry is densely meshed. To improve the convergence, a highly effective preconditioner is proposed. Different from its free-space counterpart, the preconditioning operator is constructed based on the Calderón identities for inhomogeneous medium. To reduce the relatively high construction cost of the preconditioning operator, several alternative simplified schemes are proposed and analyzed. Finally, the performances of different preconditioners are examined and compared carefully through different numerical examples. It is shown that the convergence of the PMCHWT system in a layered medium can be significantly improved by using the proposed Calderón preconditioners.

Monostatic scattering from two-dimensional two-layer rough surfaces using hybrid 3DMLUV-ACA method

The electromagnetic scattering from two-dimensional two-layer dielectric random rough surfaces is analyzed in this work based on Monte Carlo (MC) simulations. The random rough surface is characterized with Gaussian statistics for surface correlation length and for surface root mean square (RMS) height. The Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) surface integral equation is adopted and implemented. A rigorous three-dimensional (3-D) multilevel UV method combined with adaptive cross approximation (ACA) is proposed to solve for the unknown electric and magnetic surface currents. By calculating monostaitc statistical radar cross section (RCS) of the layered structure, the influence of the rough surface RMS height, the correlation length, the medium permittivity and the layer thickness on the scattering characteristic is investigated.

Surface-enhanced Raman scattering enhancement factor distribution for nanoparticles of arbitrary shapes using surface integral equation method

Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) surface integral equation method is applied for the first time to accurately estimate the surface-enhanced Raman scattering (SERS) enhancement factor distribution for arbitrary nanoparticles and nano-aggregates. It is the first time in literature that the distributions of SERS enhancement factors of nanoparticles of a large variety are reported. It is shown that not every SERS substrate exhibits a long-tail distribution as a dimer consisting of two spheres in close proximity. Generic methods are proposed to evaluate the performance of nanoparticles on SERS substrates. A cumulative distribution is proposed to examine the contributions of hot and warm spots around the nanoparticles. It is used to identify the importance of warm spots on a SERS substrate. A parameter q is proposed to describe the likelihood of a randomly positioned molecule that can be activated. This study provides guidance and insights for the optimization of SERS substrate fabrication techniques.

Electromagnetic Scattering From General Bi-Isotropic Objects Using Time-Domain Integral Equations Combined With PMCHWT Formulations

Electromagnetic scattering by general bi-isotropic objects is calculated by using the time-domain integral equations which are incorporated with the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulations. By introducing a pair of equivalent electric and magnetic sources, electromagnetic fields inside a homogeneous bi-isotropic region can be represented by these sources over its boundary after applying fields splitting method. A series of coupled surface integral equations are obtained after imposing boundary conditions. These equations are then solved numerically using the method of moment (MoM) which involves separate spatial and temporal testing procedures. The Rao-Wilton-Glisson (RWG) functions are used as the spatial expansion and testing functions, and the weighted Laguerre functions are derived as the temporal basis and testing functions. Numerical results such as transient currents, far scattered fields, and normalized radar cross sections are presented and compared with analytical results as well as MoM-based frequency-domain analysis, and good agreements are observed.

Generalized method of moments: a framework for analyzing scattering from homogeneous dielectric bodies

In this paper, we present a novel framework for discretizing integral equations--specifically, those used for analyzing scattering from dielectric bodies. The candidate integral equations chosen for the analysis are the well-known Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and the Müller equations. Discrete solutions to these equations are typically obtained by representing the spatial variation of the currents using the Rao-Wilton-Glisson (RWG) basis functions or their higher order equivalents. In this paper, we propose a framework for defining basis functions that departs significantly from those of RWG functions in that approximation functions can be chosen independent of continuity constraints. We will show that using this framework together with a quasi-Helmholtz type representation has a number of benefits. Namely, (i) it shows excellent convergence, (ii) it permits inclusion of different orders of polynomials or different functions as basis functions without imposition of additional constraints, (iii) the method can easily handle nonconformal meshes, and (iv) the method is well conditioned at all frequencies. These features will be demonstrated via a number of numerical experiments.

A Comparative Study of Calderón Preconditioners for PMCHWT Equations

The Calderón identities are used to precondition the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equations for wave scattering by dielectric objects. Based on the Calderón identities, several versions of preconditioners are presented and studied. The memory requirements and computational costs of different preconditioners are analyzed and discussed. The convergence properties of the iterative solutions and the solution accuracy of the Calderón preconditioned PMCHWT equations are also investigated and compared theoretically and numerically at different frequencies. With the help of the Calderón preconditioners, the convergence rate of the iterative solutions of the PMCHWT equations is significantly improved. Several numerical examples are given to show the performance of the Calderón preconditioners and to draw some conclusions.

Time Domain Integral Equation Analysis of Scattering From Composite Bodies via Exact Evaluation of Radiation Fields

A novel marching on in time and Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)-based time domain integral equation solver for analyzing transient scattering from composite multiregion structures comprising homogeneous penetrable volumes and perfectly conducting surfaces is presented. To render the marching on in time procedure stable, observer and source integrals are approximated via multipoint collocation and evaluated in closed form. The proposed scheme avoids the costly evaluation of five dimensional integrals required in full space-time Galerkin schemes, and proves to be stable for all targets investigated, including coated perfect electrically conducting surfaces and multiregion penetrable volumes.

Surface Integral Equation Solutions for Modeling 3-D Uniaxial Media Using Closed-Form Dyadic Green's Functions

In this paper, a method of moments (MoM) implementation of surface integral equations (SIEs) is presented for uniaxial dielectric materials using closed-form dyadic Green's functions. The standard Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) SIE is adapted for homogeneous uniaxial materials. A challenge in the implementation was the evaluation of the self impedance matrix elements and singularities due to the distinguished axis of the uniaxial media. For validation, bistatic scattering patterns of uniaxial targets are computed and compared with the well-known finite-element boundary integral (FE-BI) method solutions. A parametric study of the radiation by dipoles placed within high-contrast uniaxial layers is also presented for various distinguished axis orientations.

SOLVING MIXED DIELECTRIC/CONDUCTING SCATTERING PROBLEM USING ADAPTIVE INTEGRAL METHOD

This paper presents the adaptive integral method (AIM) utilized to solve scattering problem of mixed dielectric/conducting objects. The scattering problem is formulated using the Poggio-Miller- Chang-Harrington-Wu-Tsai (PMCHWT) formulation and the electric field integral equation approach for the dielectric and conducting bodies, respectively. The integral equations solved using these approaches can eliminate the interior resonance of dielectric bodies and produce accurate results. The method of moments (MoM) is applied to discretize the integral equations and the resultant matrix system is solved by an iterative solver. The AIM is used then to reduce the memory requirement for storage and to speed up the matrix-vector multiplication in the iterative solver. Numerical results are finally presented to demonstrate the accuracy and efficiency of the technique.