Extinction Theorem Research Articles

A note on Feynmanʼs calculation of reflection amplitudes for radiation striking a glass surface

In this paper we present a detailed calculation of reflection amplitudes for s- and p-polarized radiation striking a glass surface, closely following the derivation found in the Feynman Lectures on Physics, vol I. The basic idea underlying Feynmanʼs exposition is the extinction theorem, which is used here in a very unique, Feynmanesque way. The calculation is carried out both for the case of radiation coming from the air and from the glass. We also show that the same reasonings are useful to discuss the internal Brewsterʼs law.

A Calderón Preconditioner for the Electric Field Integral Equation With Layered Medium Green's Function

A Calderón preconditioner is developed for the analysis of electromagnetic scattering of perfect electrically conducting (PEC) objects embedded in a layered medium. The electric field integral equation (EFIE) is formulated with the kernel of layered medium Green's function to account for the effects from the multilayered background. The Calderón projector is derived based on the general source-field relationship and the extinction theorem for inhomogeneous environment in electromagnetic theory. The Calderón identities can be naturally deduced based on this projector, which is then leveraged to precondition the EFIE with layered kernel. An alternative implementation is then proposed to make the implementation of the preconditioner as efficient as the one in free space. Different numerical examples are designed to show the performance of the preconditioner, where the objects are located in different positions with respect to the layered medium, or different types of excitation are adopted. It is shown that the proposed effective and robust preconditioner makes the EFIE system converge rapidly in all cases, independent of the discretization density.

VECTOR POTENTIAL ELECTROMAGNETICS WITH GENERALIZED GAUGE FOR INHOMOGENEOUS MEDIA: FORMULATION (Invited Paper)

The mixed vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The present rapid development in quantum optics applications calls for electromagnetic solutions that straddle both the quantum and classical physics regimes. The vector potential formulation using A and ' (or A-' formulation) is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetics problems in inhomogenous media that can be extended to the anistropic case. An advantage of the resulting equations is their absence of catastrophic breakdown at low-frequencies. Hence, the usual difierential equation solvers can be used to solve them over a wide range of scales and bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell's equations. Also, the classical Green's theorem can be extended to such a formulation, resulting in an extinction theorem and a surface equivalence principle similar to the classical case. Moreover, surface integral equation formulations can be derived for piecewise homogeneous scatterers. Furthermore, the integral equations neither exhibit the low-frequency catastrophe nor the frequency imbalance observed in the classical formulation using E-H flelds. The matrix representation of the integral equation for a PEC (perfect electric conductor) scatterer is given.

Single-Source Equivalence Principle Algorithm for the Analysis of Complex Structures

In this letter, a novel single-source equivalence principle algorithm is presented. In the traditional equivalence principle algorithm (EPA), double sources, which are equivalence electric and magnetic currents, are constructed on the equivalence surface to replace the currents of the enclosed scatterers. According to the extinction theorem, electric current is related to magnetic current. Therefore, only electric or magnetic current, called single-source, is the final unknown on the equivalence surface. The accuracy of two kinds of single-source EPA is investigated. Multilevel Fast Multipole Algorithm (MLFMA) is used to accelerate the couplings between equivalence surfaces. Numerical results demonstrate the accuracy and efficiency of the proposed method.

Accuracy Improvement of the Second-Kind Integral Equations for Generally Shaped Objects

In computational electromagnetics, second-kind integral equations are usually considered less accurate than their first-kind counterparts. The loss of the numerical accuracy is mainly due to the discretization error of the identity operators involved in second-kind IEs. In the previous studies, it was shown that by using the Buffa-Christiansen (BC) functions as testing functions, such a discretization error can be suppressed significantly, and the numerical accuracy of the second-kind IEs in the far-field calculation for spherical objects can be improved dramatically. In this paper, this technique is generalized for generally shaped objects in both perfect electric conductor and dielectric cases by using the BC functions as the testing functions, and by handling the near-singularities in the evaluation of the system matrix elements carefully. The extinction theorem is applied for accurate evaluation of the numerical errors in the calculation of scattering problems for generally shaped objects. Several examples are given to demonstrate the performance of this technique, and several important conclusive remarks are drawn.

Method of Image Green's Function in Grating Theory: Reflection Extinction Theorem

Method of Image Green's Function in Grating Theory: Reflection Extinction Theorem

Wave Propagation in Parallel Plate Metallic Waveguide With Finite Conductivity and Three Dimensional Roughness

We study the effects of random roughness on wave propagation in a parallel plate metallic waveguide with finite conductivity. The rough surface is three dimensional (3D) with roughness heights varying in both horizontal directions. Integral equations are obtained from the extinction theorem formulated with layered medium Green's function. The second order small perturbation method is then applied to solve the integral equations. A closed form expression for the coherent wave is derived, which is expressed in terms of a three-fold Sommerfeld type integral due to the waveguide structure. Approximate methods are applied to calculate the Sommerfeld integral. The results based on the approximations are shown to agree with that of direct integration. The coherent wave enhancement factors of absorption are computed. Results of the present 3D roughness case are compared with the previous results of two dimensional (2D) roughness. The results for waveguides are also compared with the results obtained for a plane wave incident on a metal surface with 3D roughness. Results are illustrated for 3D roughness with a variety of power spectra. It is shown that enhancement factors of 3D roughness are larger than for 2D roughness. We also show that enhancement factors for a waveguide are larger than that of the plane wave case. Comparisons are also made between the theoretical results and measurements.

A New Green's Function Formulation for Modeling Homogeneous Objects in Layered Medium

A new Green's function formulation is developed systematically for modeling general homogeneous (dielectric or magnetic) objects in a layered medium. The dyadic form of the Green's function is first derived based on the pilot vector potential approach. The matrix representation in the moment method implementation is then derived by applying integration by parts and vector identities. The line integral issue in the matrix representation is investigated, based on the continuity property of the propagation factor and the consistency of the primary term and the secondary term. The extinction theorem is then revisited in the inhomogeneous background and a surface integral equation for general homogeneous objects is set up. Different from the popular mixed potential integral equation formulation, this method avoids the artificial definition of scalar potential. The singularity of the matrix representation of the Green's function can be made as weak as possible. Several numerical results are demonstrated to validate the formulation developed in this paper. Finally, the duality principle of the layered medium Green's function is discussed in the appendix to make the formulation succinct.

Second-order small perturbation method for transmission from dielectric rough surfaces

Although the small perturbation method (SPM) for rough surface scattering has been studied extensively in problems in optics, remote sensing and propagation, there are fewer studies on rough surface transmission by the SPM. In this paper, from Huygens’ principle and the extinction theorem, the SPM is used to derive the transmitted field to the second order, and expressions for the bidirectional transmission coefficient and the total surface transmittance to the second order are developed for the random rough surface. The refined expressions can be applied to the situations where the transmission characteristics of a random rough surface need to be more accurately calculated. For example, to calculate the brightness temperature of stratified rough media by the incoherent method, we have to know the bidirectional transmission coefficient or transmittance of random rough surface as accurately as possible. The accuracy of the presented expressions is verified through the conservation of energy. It is shown that the transmission characteristics calculated by SPM to the first order violate conservation of energy, whereas solutions to the second order conform to energy conservation much better. This is particularly important for the calculation of transmittance or emissivity.

Effective medium theory for random media composed of two-layered spheres

We study the effective index of random media composed of two-layered spheres by using the energy-density coherent potential approximation method. As expected from the Ewald-Oseen extinction theorem, in the long-wavelength limit, the optical properties of a random medium composed of two-layered spheres are identical to those of a random medium composed of the corresponding pure spheres, while in the Mie-scattering region, the single-scattering resonances lead to an overall shift of the effective refractive index with the modified volume fraction.

Critical Multitype Branching Systems: Extinction Results

We consider a critical branching particle system in $\mathbb{R}^d$, composed of individuals of a finite number of types $i\in\{1,\ldots,K\}$. Each individual of type i moves independently according to a symmetric $\alpha_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more complex dynamics by assuming in case (2) that the most mobile particle type corresponds to a finite-mean lifetime: in this case, local extinction of the population is determined by an interaction of the parameters (offspring variability, mobility, longevity) of the long-living type and those of the most mobile type. The proofs are based on a precise analysis of the occupation times of a related Markov renewal process, which is of independent interest.

Integral Equation Method for Electromagnetic Wave Propagation in Stratified Anisotropic Dielectric-Magnetic Materials

We investigate the propagation of electromagnetic waves in stratified anisotropic dielectric-magnetic materials using the integral equation method (IEM). Based on the superposition principle, we use Hertz vector formulations of radiated fields to study the interaction of wave with matter. We derive in a new way the dispersion relation, Snell's law and reflection/transmission coefficients by self-consistent analyses. Moreover, we find two new forms of the generalized extinction theorem. Applying the IEM, we investigate the wave propagation through a slab and disclose the underlying physics, which are further verified by numerical simulations. The results lead to a unified framework of the IEM for the propagation of wave incident either from a medium or vacuum in stratified dielectric-magnetic materials.

Linking quasi-normal and natural modes of an open cavity

The present paper proposes a comparison between the extinction theorem and the Sturm–Liouville theory approaches for calculating the electromagnetic (e.m.) field inside an optical cavity. We discuss for the first time to the best of our knowledge, in the framework of classical electrodynamics, a simple link between the quasi normal modes (QNMs) and the natural modes (NMs) for one-dimensional (1D), two-sided, open cavities. The QNM eigenfrequencies and eigenfunctions are calculated for a linear Fabry–Pérot (FP) cavity. The first-order Born approximation is applied to the same cavity in order to compare the first-order Born approximated and the actual QNM eigenfunctions of the cavity. We demonstrate that the first-order Born approximation for an FP cavity introduces symmetry breaking: in fact, each Born approximated QNM eigenfunction produces values below or above the actual QNM eigenfunction value on the terminal surfaces of the same cavity. Consequently, the two error-functions for an approximated QNM are not equal in proximity to the two terminal surfaces of the cavity.

Coupled Integral Equations for Microwave Induced Elastic Wave in Elastic Media

The interaction of waves with materials could include multiple physics (multiphysics), depending on the properties of waves and materials. For microwave illumination on elastic media, the electromagnetic (EM) and mechanic processes exist simultaneously and are coupled together. To exactly describe the involved process with interaction of microwave and induced elastic wave, the coupled Maxwell's equations and elastic wave equations should be solved. Although such coupled equations have been derived earlier but they are in partial differential equation (PDE) form. The solutions for the equations are then only based on finite difference method (FDM) or finite element method (FEM) in addition to analytical approaches. In this work, we first derive coupled integral equations for governing the process from its PDE counterpart. The derivation is based on the Huygens' equivalence principle and extinction theorem by recognizing that the excitation of elastic wave is EM force and the induced elastic wave will be a new excitation source to affect the EM in return. The coupled integral equations is solved using Nyström method in first time and some basic results from numerical examples for microwave illumination on piezoelectric materials are presented.

Estimation of scattering object characteristics for image reconstruction using a nonzero background.

Two methods are described to estimate the boundary of a 2-D penetrable object and the average sound speed in the object. One method is for circular objects centered in the coordinate system of the scattering observation. This method uses an orthogonal function expansion for the scattering. The other method is for noncircular, essentially convex objects. This method uses cross correlation to obtain time differences that determine a family of parabolas whose envelope is the boundary of the object. A curve-fitting method and a phase-based method are described to estimate and correct the offset of an uncentered radial or elliptical object. A method based on the extinction theorem is described to estimate absorption in the object. The methods are applied to calculated scattering from a circular object with an offset and to measured scattering from an offset noncircular object. The results show that the estimated boundaries, sound speeds, and absorption slopes agree very well with independently measured or true values when the assumptions of the methods are reasonably satisfied.

Polar decomposition of Mueller matrices for 2D-structured surfaces

In this research, the Polar Decomposition (PD) has been applied to the Mueller matrices (MMs) of the light scattered by linear ribs of rectangular profile on a flat substrate. Although photo-lithographic technique produces a silicon surface, metallic character is acquired by sputtering with gold some of the samples. With a dual rotating compensator polarimeter the MMs are obtained by Fourier Transform analysis. The samples have been numerically modeled by using both FDTD and Extinction Theorem (ET) and MMs have been computed from the results. The scattering depends strongly on the geometry and composition of the ribs, and this sensi- tivity is noticed for instance in M11 element. But information offered by PD parameters is shown to be more apprehensible, like the substrate-induced depolarization or the retardance associated to the rib width.

RECONSTRUCTION AND VISUALIZATION OF EQUIVALENT CURRENTS ON A RADOME USING AN INTEGRAL REPRESENTATION FORMULATION

In this paper an inverse source problem is investigated. The measurement set-up is a re∞ector antenna covered by a radome. Equivalent currents are reconstructed on a surface shaped as the radome in order to diagnose the radome's interaction with the radiated fleld. To tackle this inverse source problem an analysis of a full-wave integral representation, with the equivalent currents as unknowns, is used. The extinction theorem and its associated integral equation ensure that the reconstructed currents represent sources within the radome. The axially symmetric experimental set-up reduces the computational complexity of the problem. The resulting linear system is inverted by using a singular value decomposition. We visualize how the presence of the radome alters the components of the equivalent currents. The method enables us to determine the phase shift of the fleld due to the transmission of the radome, i.e., the IPD (insertion phase delay). Also, disturbances due to defects, not observable in the measured near fleld, are localized in the equivalent currents.

SINTONIZACIÓN DE RESONANCIAS ELECTROMAGNÉTICAS EN NANO CILINDROS RECUBIERTOS: UN ESTUDIO PARAMÉTRICO TUNING OF ELECTROMAGNETIC RESONANCES IN COATED NANOCYLINDERS: A PARAMETRICAL STUDY

In this paper a study on electromagnetic scattering of coated particles is showed and a rigorous formalism based on the Extinction Theorem, for a 2D geometry, is implemented. Through this integral Method results for the electromagnetic scattering of systems, which consist in two coaxial circular shaped cylinders are performed for two fundamental modes of polarization: s and p mode. The particles are filled with homogeneous linear and isotropic media, these being able to have magnetic properties in the whole classic electromagnetic range. Particularly in nanoparticles, results of optical cross sections vs. wavelength of the illumination are showed. A characterization of eigenmodes (plasmons when the particles have metal) is performed.

Equations for the description of wave scattering by multi-valued random surfaces

We consider a scattering theory for multi-valued rough surfaces which cannot be described by the conventional equation of the type z = ζ(x,y). Both Dirichlet and Neumann problems are analyzed. Starting with Green's theorem we obtain a representation of the scattered field, the surface integral equation, and the extinction theorem for such surfaces. In contrast to conventional theory, these equations contain three random functions x = x(u 1,u 2), y = y(u 1,u 2), and z = z(u 1,u 2), where u 1 and u 2 are the parameters describing the surface. We introduce two scattering amplitudes S ± for describing the scattered wave above and below the surface. The extinction theorem, if formulated in terms of S −, allows us to determine S − for an arbitrary multi-valued surface and after this it becomes possible to derive a simple integral equation for surface sources. Knowledge of the surface sources allows us to find S + by integration.

Effective optical parameters of ordered nanocomposites

The analysis of composite structures consisting of metal (gold) nanoparticles in a nonconducting matrix (for simplicity, vacuum) is based on the theory of integral equations. The validity of such an approach is proven by an excat electrodynamic calculation using the finite-element method. The expressions for optical fields inside and outside the system are derived and investigated. It is demonstrated that the ordering of particles leads to a substantial decrease in the effective absorption coefficient due to the mutual polarization effect, which is significant for the creation of metal-dielectric nanocomposites with exotic optical properties (negative, giant, or ultralow refractive index; a refractive index that is close to the refractive index of vacuum; etc.). As in the case of conventional crystals, the extinction theorem is valid for the nanocrystal structures under study.