Rayleigh Expansion Research Articles

Analytical model of the enhanced light transmission through subwavelength metal slits: Green’s function formalism versus Rayleigh’s expansion

We present an analytical model of the resonantly enhanced transmission of light through a subwavelength nm-size slit in a thick metal film. The simple formulae for the transmitted electromagnetic fields and the transmission coefficient are derived by using the narrow-slit approximation and the Green's function formalism for the solution of Maxwell's equations. The resonance wavelengths are in agreement with the semi-analytical model [Y. Takakura, Phys. Rev. Lett. 86, 5601 (2001)], which solves the wave equations by using the Rayleigh field expansion. Our formulae, however, show great resonant enhancement of a transmitted wave, while the Rayleigh expansion model predicts attenuation. The difference is attributed to the near-field subwavelength diffraction, which is not considered by the Rayleigh-like expansion models.

Ring-Shaped Functions and Wigner 6j-Symbols

We obtain an explicit expression for the ring-shaped matrix relating the ring-shaped functions corresponding to different values of an axiality parameter and find the relation between this matrix and the Wigner 6j-symbols. We investigate the motion of a quantum particle in a ring-shaped model with a zero “bare” potential and find bases factored in spherical and cylindrical coordinates for this model. We derive a formula generalizing the Rayleigh expansion for a plane wave in terms of spherical waves in a ring-shaped model.

Theoretical method for the study of plasmon generation in hybrid multilayer-optical fiber structures

A theoretical method is presented for the determination of the behavior of devices based on the deposition of multilayer structures on polished optical fibers. Plasmon generation in metallic layers is modeled. The method is based on the Rayleigh expansion of the electric fields and permits us to determine their distribution over the whole structure by an application of boundary conditions. Once the distribution is known, the power transmitted by the fiber can be computed as a function of the geometrical and refractive parameters of the device. The method is versatile and can be used as a theoretical tool for the design of devices of that type used for many different purposes. We present real experimental results obtained with an operative sensor that agree with the theoretical predictions of our technique and prove its suitability.

Acoustic Propagation in Dispersions in the Long Wavelength Limit

The problem of scattering of ultrasound by particles in the long wavelength limit has a well-established solution in terms of Rayleigh expansions of the scattered fields. However, this solution is ill-conditioned numerically, and recent work has attempted to identify an alternative method. The scattered fields have been expressed as a perturbation expansion in the parameter Ka (the wavenumber multiplied by the particle radius), which is small in the long wavelength region. In the work reported here the problem has been formulated so as to be valid for all values of the thermal wavelength, which varies in order of magnitude, from much smaller to much larger than the particle size in the long wavelength region. Thus the present solution overlaps the limiting solutions for very small thermal wavelength (geometric theory) and very large thermal wavelength (low frequency) previously reported. Close agreement is demonstrated with the established Rayleigh expansion solution.

Overlap integrals with respect to quantum numbers over Slater‐type orbitals via the Fourier‐transform method

AbstractA unified analytical treatment of overlap integrals over Slater‐type orbitals, using the Fourier transform method, is presented. Our approach leads to considerable simplification of the derivation of analytical representations for the overlap integrals over Slater type orbitals. In this approach, the representations of overlap integrals with both the same and different screening parameters have been obtained with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. Analytical expressions obtained by this method have been expressed in terms of Gaunt coefficients, which are the product of spherical harmonics, Gegenbauer, and binomial coefficients. These overlap integrals have also been calculated with respect to the differences in n1 − l1 and n2 − l2 numbers that are even or odd. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

An inverse problem in periodic diffractive optics: global uniqueness with a singlewavenumber

We consider the problem of recovering a perfectly reflecting two-dimensionaldiffraction grating from the knowledge of one wavenumber, one incident directionand the total field measured above the grating. We prove a global uniquenessresult within the class of polygonal grating profiles. The proof relies on theanalyticity of solutions to the Helmholtz equation and the Rayleigh expansion ofthe scattered field.

A remark on the critical speed for vortex nucleation in the nonlinear Schrödinger equation

The critical speed for the nucleation of quantized vortices in the nonlinear Schrödinger equation (NLS) for a flow around a disk in two spatial dimensions is discussed in this paper. This problem is closely related to a compressible flow around a disk. The flow is computed via a Janzen–Rayleigh expansion for low Mach number. The calculation leads to an estimate for the critical Mach number M c=0.36969(7)…

The improved point matching method for triangular metal gratings

In this paper, we present the “point matching method” (PMM) adapted for the calculation of Smith–Purcell (SP) radiation parameters in H-polarization, generated by relativistic electron beams (REB) passing close to a triangular metal grating. A truncated expression H d y, n ( N; β, ω) was used in order to find the coefficients H d y, n ( β, ω) in the infinite Rayleigh expansion of the diffracted field of a moving electron. A linear system of 2 N+1 equations was solved, increasing the number N of harmonics until convergence solutions were achieved. When N→∞ then lim N→∞H d y,n(N;β,ω)=H d y,n(β,ω) and the diffracted field can be calculated. SP radiation factors were calculated using the modified PMM for electron energies in 1–30 MeV domain and metal gratings with period length in 1–10 mm range.

Development of Computational Aeroacoustics Equations for Subsonic Flows Using a Mach Number Expansion Approach

Computational aeroacoustics equations are developed using a Janzen–Rayleigh expansion of the compressible flow equations. Separate expansions are applied to an inner region characterized to lowest order by an incompressible flow field and an outer region characterized by propagating acoustic waves. Several perturbation equation sets are developed in the inner and outer regions by truncating the expanded equations using different orders in the perturbation variable, ε, where ε2 is proportional to the square of the Mach number characterizing the flow. Composite equation sets are constructed by matching the equations governing the inner and outer regions. The highest-order perturbation continuity and momentum equations include an infinite series in ε2 and are shown to be identical to the equations used in the expansion about incompressible flow approach. As such, the perturbation analysis is used to interpret the physical meaning of the perturbation variables and to highlight the assumptions inherent in this approach. Differences between numerical solutions obtained with the composite equation sets are evaluated for two unsteady flow problems. The lowest-order perturbation equation set is shown to yield adequate acoustic predictions for low Mach number flows. This equation set is considerably simpler to implement into a numerical solver and reduces the required CPU time relative to the highest-order equation set.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary–value problem for the Helmholtz equation in a non–locally perturbed half–plane. This problem models time–harmonic electromagnetic scattering by a one–dimensional infinite rough perfectly conducting surface; the same problem arises in acoustic scattering by a sound–soft surface. Chandler–Wilde and Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler–Wilde and Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary–value problem for the scattered field has a unique solution.

Mode Truncation Effects in an Explicit Mode Elastic Characterization Method

The second author and colleagues developed an elastic identification method based on model data obtained experimentally from free transverse vibration of impacted freely supported plates. Application of the method to thin, square isotropic metal and composite plates is described in this paper. For these modes, some additively and subtractively coupled modes arising from diagonal symmetry of geometry and boundary conditions (thus termed diagonal modes) are important. The focus of this paper is on how the number of modes used to represent the vibration affects accuracy. One-, three-, and six-term optimized Rayleigh displacement expressions are used for plate vibration, including the diagonal modes. Sensitivity analysis of the frequencies to each elastic constant is included. Results show that the optimal three-term approximation gives results comparable to the normal 36-term Rayleigh expansion, while the optimal six-term compares favorably with the 64-term Rayleigh expansion.

Regularized Waterman and Rayleigh methods: extension to two-dimensional gratings

The origin of the instabilities of the Waterman method was studied previously and an improvement in the method was developed for one-dimensional gratings and s polarization [J. Opt. Soc. Am.15, 1566 (1998)]. Later, the same kind of regularization was used to improve Rayleigh’s expansion method. We show that the same well-adapted regularization process can be generalized to two-dimensional (2D) gratings. Numerical implementations show that the convergence domain of the Waterman method is extended by a factor of ∼40% in the range of groove depth. In the same way, the convergence domain of the Rayleigh expansion method is extended by a factor of ∼35% for 2D sinusoidal gratings. As a consequence, the new versions of Waterman and Rayleigh methods become simple and efficient tools for use in investigating the properties of 2D gratings that have ratios of groove depth to period up to unity.

Scattering of Electromagnetic Waves by Rough Interfaces and Inhomogeneous Layers

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations.

Acoustic Scattering by an Inhomogeneous Layer on a Rigid Plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Model with two roughness levels for diffraction gratings: the generalized Rayleigh expansion

A model with two roughness levels for the diffraction of a plane wave by a metallic grating with periodic imperfections is presented. The grating surface is the sum of a reference profile and a perturbation profile. First, the diffraction by the reference grating is treated. At this stage the Chandezon method is used. This method leads to the resolution of eigenvalue systems. Each eigensolution defines an elementary wave function that characterizes a propagating or an evanescent wave. Second, the periodic errors are taken into account and a Rayleigh hypothesis is expressed: Everywhere in space the diffracted fields can be written as a linear combination of reference wave functions. The boundary conditions on the perturbed grating allow the diffraction amplitudes to be determined and therefore lead to the energetic magnitudes (efficiencies). The domain of analytical validity of this hypothesis is not defined. In fact, this method is considered to be an approximation. The proposed numerical study leads to some utilization rules. With a plane as the reference surface, the electromagnetic fields are given by classical Rayleigh expansions. Here the reference profile is a grating, hence the term generalized Rayleigh expansion.

Long Series in Mechanics: Janzen–Rayleigh Expansion for a Circle

We reconsider the transonic controversy – whether a given airfoil can exhibit a continuous range of shock-free supersonic flow. We employ the circle as our ‘airfoil’, and examine the Janzen–Rayleigh expansion in even powers of the free-stream Mach number, which we previously carried to 29 terms. In addition to expressing doubt about our previous conclusions, we introduce a promising modification of the graphical ratio test of Domb and Sykes.

Low-frequency scattering of coated spherical obstacles

This work deals with the scattering of a plane harmonic elastic wave by a penetrable spherical scatterer with a concentric spherical penetrable inclusion. We evaluate the zeroth and first-order approximations of the Rayleigh expansion of the displacement fields. The major line of applications belongs to the science of the particulate composite material. So, as an application of the method, a typical particulate composite material is examined and the behaviour of the scattering cross section with respect to the elastic properties of the medium is presented.

Diffraction from corrugated gratings made with biaxial crystals: Rayleigh methods

We apply the Rayleigh hypothesis to study the diffraction of electromagnetic waves at corrugated gratings made of biaxial crystals with arbitrary orientations of their optic axes. The method is valid for gratings made of shallow grooves of any shape, with arbitrary orientation of the plane of incidence with respect to the main section of the cylindrical corrugation (conical mounting). We solve the dispersion equation to find the propagation constants and the polarization of the plane waves involved in Rayleigh expansions. These expansions are used to impose Maxwell boundary conditions at the grating interface. This leads to a linear system of equations with the amplitudes of the diffracted fields as unknowns, which is solved numerically. As examples of application of this procedure, we calculate the co- and cross-polarized components in the specularly diffracted order when the periodically corrugated biaxial crystal is illuminated from an isotropic medium by s and p polarized plane waves, and we study the excitation of surface plasmons for incidence from the biaxial crystal onto a metal surface.

Multiple multipole program computation of periodic structures

The three-dimensional multiple multipole program (MMP) code based on the generalized multipole technique is outlined for readers who are not familiar with its concepts. This code was originally designed for computational electromagnetics. Rayleigh expansions and periodic boundary conditions are two new features that make MMP computations of arbitrary periodic structures efficient and that at the same time allow us to take advantage of the benefits of other MMP features, including surface impedance boundary conditions and a variety of available basis functions for modeling the electromagnetic field. The application of three-dimensional MMP to a simple grating of highly conducting wires with rectangular cross sections illustrates the high accuracy and the fast convergence of the method as well as the use of surface impedance boundary conditions. A more complicated biperiodic array of helical antennas demonstrates the application of thin-wire expansions in conjunction with regular MMP expansions. This model can be considered a simulation of a thin, anisotropic chiral slab with interesting characteristics.

An Expansion for Use in the Simulation and Analysis of NMR Data

Abstract The time-domain NMR signal that results from chemical shifts or first-order quadrupolar effects in materials with either a few well-defined sites or a distribution in orientation or coupling constants is given by an integral of exp[ikt(3 cos2θ − 1)] or exp[ikt(3 cos2θ − 1 + η sin2θ cos 2 φ)] times a weighting probability for orientation and/or coupling constants. Data analysis requires either execution of angular averages, as for a powder sample, or determination of the weighting function, as for an oriented sample. Development of expansions of the above exponential forms in spherical harmonics for use in such data analysis is initiated. This development is analogous to the Ray leigh expansion of exp (ikr cos θ) in spherical harmonics that is ubiquitous throughout scattering theory. Applications to analysis of a 2D NMR experiment, characterization of anisotropic materials, and determination of the distribution of coupling constants are discussed as examples of the reduction of computation time, by use of analytic, rather than the currently used numerical, execution of angular averages. It is also expected that the availability of algebraic forms provides a useful alternative perspective, complementary to numerical calculations, offering a potential for direct extraction of motional correlation functions from NMR data and estimation of orientation probabilities. A transformation needed for application of the expansion to magic-angle spinning of rigid-lattice samples is developed.