Rayleigh Expansion Research Articles

Closed-form analytical model for plasmonic metasurfaces consisting of graphene disks

In this paper, a closed-form analytical model is proposed for the fundamental plasmon mode of a metasurface of graphene disks. First, we derive an equivalent admittance for the graphene metasurface at the normal incidence of a plane wave using the Rayleigh expansion. This admittance is then expressed as a series R-L-C circuit obtained by analyzing the integral equation that governs the surface current on the graphene disks. Finally, by using the equivalence substitution method, the equivalent circuit for this structure is fully analyzed. The accuracy of the model is verified by comparison with the results of full-wave simulations. Examples of applications are provided to predict tunable transmission resonances and design broadband absorbers, thus demonstrating their convenience for practical device design.

Nucleation of a Vapor Phase and Vapor Front Dynamics Due to Boiling-Up on a Solid Surface

The effect of temperature and pressure on the nucleation of the vapor phase and the velocity of the vapor front in the initial stage of activated boiling-up of n-pentane on the surface of a quartz fiber was studied. Using a developed approach combining the “pump-probe” and laser Doppler velocimetry methods, this velocity was tracked in the course of sequential change in the degree of superheating with respect to the liquid–vapor equilibrium line. The studied interval according to the degree of superheating was 40–100 °C (at atmospheric pressure). In order to spatiotemporally localize the process, the activation of boiling-up at the end of the light guide was applied using a short nanosecond laser pulse. A spatial locality of measurements was achieved in units of micrometers, along with a time localization at the level of nanoseconds. An increase in temperature at a given pressure was found to lead to an increase in the speed of the transition process with a coefficient of about 0.2 m/s per degree, while an increase in pressure at a given temperature leads to a decrease in the transition process speed with a coefficient of 25.8 m/s per megapascal. The advancement of the vapor front velocity measurements to sub-microsecond intervals from the first signs of boiling-up did not confirm the existence of a Rayleigh expansion stage with a constant velocity.

Reverse time migration for imaging periodic obstacles with electromagnetic plane wave

Abstract We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles.

Uniqueness in Inverse Diffraction Grating Problems With Infinitely Many Plane Waves at a Fixed Frequency

This paper is concerned with uniqueness of solution of the inverse diffraction by problems by a periodic curve with Dirichlet boundary condition in two dimensions. It is proved that the periodic curve can be uniquely determined by the near-field measurement data corresponding to infinitely many incident plane waves with distinct directions at a fixed frequency. Our proof is based on Schiffer’s idea which consists of two ingredients: (i) the total fields for incident plane waves with distinct directions are linearly independent, and (ii) for a fixed wave number there exist only finitely many linearly independent Dirichlet eigenfunctions in a bounded domain or in a closed waveguide under additional assumptions on the waveguide boundary. Based on the Rayleigh expansion, we prove that the phased near-field data can be uniquely determined by the phaseless near-field data in a bounded domain, with the exception of a finite set of incident angles. Such a phase retrieval result leads to a new uniqueness result for the inverse grating diffraction problem with phaseless near-field data at a fixed frequency. Since the incident direction determines the quasi-periodicity of the boundary value problem, our inverse issues are different from the existing results of [F. Hettlich and A. Kirsch, Inverse Problems, 13 (1997), pp. 351–361], where fixed-direction plane waves at multiple frequencies were considered.

Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

On Rayleigh expansion for nonlinear long water waves

We consider strongly nonlinear long waves on the surface of a homogeneous fluid layer. By modifying the formulation for the high-order spectral (HOS) method for waves in water of finite depth, we present a higher-order nonlinear system for the surface elevation and the velocity potential on the free surface to describe the two-dimensional evolution of large amplitude long waves. It is shown that the resulting system preserves the Hamiltonian structure of the Euler equations and can be transformed to the strongly nonlinear long-wave model for the depth-averaged velocity. Due to truncation of the linear dispersion relation for water waves, both the system for the surface velocity potential and that for the depth-averaged velocity are ill-posed when the order of approximation is odd and even, respectively. To avoid this ill-posedness, fully dispersive models are also proposed. Under the same order approximation, the long-wave model is found more effective for numeral studies of large amplitude long waves than the finite-depth model.

A meshless numerical method for time harmonic quasi-periodic scattering problem

This paper documents a numerical method for time-harmonic wave scattering by an array of periodic penetrable obstacles. The fundamental solutions combining Rayleigh expansion is used to approximate the scattering field. The algorithm does not need to mesh or truncate the domain, it could obtain high accuracy by adding the number of basis functions. Some examples are given to illustrate the effectiveness of the approach.

Fourier transform of hydrogen-type atomic orbitals

Hydrogen-type atomic orbitals (HTOs) are an important type of exponential-type orbital. These orbitals have some mathematical properties and they are used usually in the theoretical atomic and molecular investigations as special functions to figure out analytical expressions. The Fourier transform method is a great way to convert basis functions into the momentum space, because their Fourier transforms are easier to use in mathematical calculations. In this paper, we obtain new and useful mathematical representations for the Fourier transform of HTOs related with Gegenbauer polynomials and hypergeometric functions, by using recurrence relations of Laguerre polynomials, Rayleigh expansion and some properties of normalized HTOs.

Spherical Harmonics Expansion of Fundamental Solutions and Their Derivatives for Homogeneous Elliptic Operators

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

General analytical solution for the electromagnetic grating diffraction problem.

Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.

Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions

A unified method to evaluate the fundamental solutions for generally anisotropic multi-field materials is presented. Based on the relation between the Rayleigh expansion and the three-dimensional Fourier representation of a homogenous partial differential operator, the proposed technique allows to obtain the fundamental solutions and their derivatives up to the desired order as convergent series of spherical harmonics. For a given material, the coefficients of the series are computed only once, and the derivatives of the fundamental solutions are obtained without any term-by-term differentiation, making the proposed approach attractive for boundary integral formulations and efficient for numerical implementation. Useful general relationships for the computation of derivatives of various order of the fundamental solutions are presented. Furthermore, no particular treatment is needed for mathematically degenerate cases. The fundamental solutions of the Laplace equation and isotropic elastic solids are exactly retrieved as special cases. Numerical results are presented to demonstrate the accuracy of the approach for isotropic elastic, generally anisotropic elastic, transversely isotropic and generally anisotropic piezo-electric and magneto-electro-elastic materials.

Impact of anisotropy on the interaction of an atom with a one-dimensional nano-grating

We study the interaction of an atom with a one-dimensional nano-grating within the framework of macroscopic QED, with special emphasis on possible anisotropic contributions. To this end, we first derive the scattering Green’s tensor of the grating by means of a Rayleigh expansion and discuss its symmetry properties and asymptotes. We then determine the Casimir–Polder potential of an atom with the grating. In particular, we find that strong anisotropy can lead to a repulsive Casimir–Polder potential in the normal direction.

Hydrodynamic characteristics of offshore and pile breakwaters

A combined method is presented to study the wave interaction with an offshore or pile (cylinder) breakwater in finite water depth. The influences of the fluid viscosity, the thickness of the structure and evanescent waves on the hydrodynamic behaviours are included in the model. Rayleigh expansion is used to describe the reflected and transmitted waves. The accuracy of the present formulas is verified by a comparison with existing results. The wave reflection and transmission coefficients predicted by the present model agree with the theoretical results and experimental data in open literature. A ratio of the grating constant, 2b (the distance between adjacent gaps), to the wavelength, L, equal to one is a critical value. When 2b/L>1, the influence of the thickness of the breakwater on the hydrodynamic characteristics can be ignored, whereas if 2b/L<1, the increase of the thickness can lead to wave resonance in these gaps and to dramatically enhanced wave transmission. The diffraction evanescent waves can cause a phase shift and tune the resonance condition. The fluid viscosity can repress but not completely eliminate the enhanced wave transmission. Phenomenon of the enhanced wave transmission at certain special frequencies induced by the structure thickness should be given attention.

A Fast and Stabilized Method for Gratings

The widely used Rayleigh expansion is not generally bounded and causes the instability and some limitations in numerical process. We extend the radiation condition for diffraction grating problems and stress that the radiation field excited by a planar periodic current source satisfies the generalized radiation condition; hence, it can be used as a modified Rayleigh expansion to formulate an enhanced Rayleigh method. The proposed method can considerably extend the domain of validity. Invoking the internal regularization mechanism, it can work with success when amplitude/period ratio equals 0.5 or higher for a sinusoidal metal grating and $s$ polarization with normal incidence. At the same time, the simplicity, efficiency, and ability to explicitly depict the interactions among diffracted orders are as good as, if not better than in the Rayleigh method.

Curvilinear coordinate method as an initial value problem: application to gratings.

We propose a C-method approach for the efficient calculation of the scattering matrix of a grating. We consider two horizontal planes above and below the grating and define a coordinate system such that the grating surface and both horizontal planes correspond to surface coordinates. Inside the area A delimited by the horizontal planes, the covariant formalism of Maxwell's equations leads to an initial value problem that is solved for independent initial conditions satisfying the boundary conditions. Outside the area A, the fields are represented by Rayleigh expansions. The scattering matrix is obtained by using continuity relations between different components of fields on the horizontal planes.

Casimir energies of periodic dielectric gratings

Reflection of electromagnetic waves from a periodic grating can be described in terms of a discrete coupled multichannel scattering problem. By modeling the grating as a space- and frequency-dependent dielectric, it is possible to use a variable phase method, applied to a generalized Helmholtz equation incorporating both transverse and longitudinal modes, to efficiently compute the scattering $S$ matrix. The projection onto transverse modes of this result, evaluated for imaginary wave vector, provides the information necessary for a Casimir energy calculation. This approach is of particular interest for gratings with deep corrugations, which can limit the applicability of techniques based on the Rayleigh expansion. We demonstrate the method by calculating the Casimir interaction energy between sinusoidal grating profiles as a function of separation and lateral displacement.

ELECTROMAGNETIC WAVE SCATTERING FROM ROUGH LAYERED INTERFACES: ANALYSIS WITH THE SMALL PERTURBATION METHOD AND THE SMALL SLOPE APPROXIMATION

We propose a theoretical study on the electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces by using the small perturbation method and the small slope approximation. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. They can be correlated or not. The electromagnetic fleld in each medium is represented by a Rayleigh expansion and a perturbation method is used for solving the boundary value problem and determining the flrst-order scattering amplitudes by recurrence relations. The scattering amplitude under the flrst-order small slope approximation are deduced from results derived from the flrst-order small perturbation method. Comparison between these two analytical models and a numerical method based on the combination of scattering matrices is presented. The study of electromagnetic wave scattering from rough layered interfaces has many applications in remote sensing, communication techniques, civil engineering, geophysics and optics. Several models give the average scattered fleld and the average intensity. Analytical methods are based on physical approximations and give closed-form formulae for the flrst- and second-order moments of the scattered fleld. Exact methods estimate the average scattered fleld and the average intensity from the results over many realizations of rough layered interfaces. In this paper, we propose a theoretical study on the electromagnetic wave scattering from layered structures with an arbitrary number of rough interfaces by using two analytical models: the flrst-order small perturbation method (SPM) and the flrst-order small slope approximation (SSA). Elson was one of the flrst authors to develop a vector theory of scattering from a stratifled medium. This vector theory allows the angular distribution of scattered light to be determined and can be used with correlated or uncorrelated surface roughness (1,2). The SPM has been used for the study of light scattering from multilayer optical coatings (1{5) and many authors have also implemented a perturbative theory for analyzing remote sensing problems (6{12). The small slope approximation (SSA1) has an extended domain of applicability (13{15) which includes the domain of the small-perturbation method that is only valid for surfaces with small roughness (16) and the domain of the Kirchhofi approximation that is applicable to surfaces with long correlation length (17,18). In the present paper, the structure under consideration is a stack of several rough one-dimensional interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fleld in each region is represented by a continuous spectrum of plane waves, the amplitudes of which are found by matching the boundary conditions

Full wave solution for hydrodynamic behaviors of pile breakwater

Rayleigh expansion is used to study the water-wave interaction with a row of pile breakwater in finite water depth. Evanescent waves, the wave energy dissipated on the fluid resistance and the thickness of the breakwater are totally included in the model. The formulae of wave reflection and transmission coefficients are obtained. The accuracy of the present model is verified by a comparison with existing results. It is found that the predicted wave reflection and transmission coefficients for the zero order are all highly consistent with the experimental data (Hagiwara, 1984; Isaacson et al., 1998) and plane wave solutions (Zhu, 2011). The losses of the wave energy for the fluid passing through slits play an important role, which removes the phenomena of enhanced wave transmission.

Scattering by Anisotropic Rough Layered 2D Interfaces

We propose a statistical study of the scattering of an incident plane wave by a stack of two two-dimensional rough interfaces. The interfaces are characterized by Gaussian height distributions with zero mean values and Gaussian correlation functions. The electromagnetic fields are represented by Rayleigh expansions, and a perturbation method is used for solving the boundary value problem and determining the first-order scattering amplitudes. For slightly rough interfaces with a finite extension, we show that the modulus of the co- and cross-polarized scattering amplitudes follows a Hoyt law and that the phase is not uniformly distributed. For interfaces with an infinite extension, the modulus follows a Rayleigh law and the phase is uniformly distributed. We show that these results are true for correlated or uncorrelated interfaces and for isotropic or anisotropic interfaces.

A least-squares non-polynomial finite element method for solving the polygonal-line grating problem

This paper is concerned with the diffraction by a polygonal-line grating. Using the Fourier–Bessel functions that capture the singularities at corners, and representing the diffraction field towards infinity by a truncation of the Rayleigh expansion, we propose a least-squares finite element method for solving the problem. We analyze the convergence and give an error estimate for the method. Numerical experiments are also presented to show the effectiveness of our method.