Developing Finite Element Methods for Maxwell's Equations in a Cole–Cole Dispersive Medium

In this paper, we propose a new time-domain finite element method for solving the time dependent Maxwell's equations coupled with the Lorentz metamaterial model. The Lorentz metamaterial Maxwell's equations are much more complicated than the standard Maxwell's equations in free space. Our fully discrete scheme uses edge elements to approximate the unknowns in space, and uses the leap-frog scheme in time discretization. Numerical stability and the optimal error estimate in the $$L^2$$L2 norm are proved for our proposed scheme. Extensive numerical results are presented to confirm the theoretical analysis and applications of our scheme to model many interesting phenomena happened when wave propagates in the Lorentz metamaterials. Examples include the convergence effect happened in the concave lenses formed by the negative refraction index metamatrials, and total reflection and total transmission observed in the zero index metamaterials.

Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media

Using the Hankel transform, we obtain, in their most general form, the spinor solutions of Maxwell's equations in free space for a light beam propagating along 0z with cylindrical symmetry around 0z. Then we compute a paraxial approximation of these solutions and we discuss the case of Gaussian beams. A comparison of the spinor solutions with the scalar ones supplied by the parabolic equation is also given.

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type. In the second part, we introduce the corresponding solution to the Maxwell equations in free space.

It is known that there is a close analogy, exhibited by the use of five-dimensional tensor notation, between the electromagnetic equations in free space and the first- and second-order equations of the quantum theory. In the present paper an analogy is traced between two tensors, one of which is related to the magnetic and electric moments of a doublet while the other represents the electric and magnetic field-strengths.

The aim of the present work is to introduce two monochromatic solutions to the scalar wave equation in free space, characterized by a caustic with a singularity of the hyperbolic umbilical type. The first solution, is a superposition of half-Mathieu beams, and the second one, is a superposition of parabolic beams. Since these solutions are determined by two particular complete integrals of the eikonal equation in free space, we compute their geometrical wavefronts, the caustic regions, and the corresponding Poynting vectors. Finally, we remark that, under certain conditions, these solutions describe three-dimensional accelerating beams in free space, propagating along semielliptical and parabolic paths, respectively.

We construct exact solutions to the paraxial wave equation in free space characterized by stable caustics. First, we show that any solution of the paraxial wave equation can be written as the superposition of plane waves determined by both the Hamilton-Jacobi and Laplace equations in free space. Then using the five elementary stable catastrophes, we construct solutions of the Hamilton-Jacobi and Laplace equations, and the corresponding exact solutions of the paraxial wave equation. Therefore, the evolution of the intensity patterns is governed by the paraxial wave equation and that of the corresponding caustic by the Hamilton-Jacobi equation.

Finite-element method for time-dependent Maxwell's equations based on an explicit-magnetic-field scheme

Numerical simulation of Bessel beams by FDTD employing the superposition principle

The first realistically photon-like Schrodinger solution of Maxwell's classical equations in dispersive media is presented. Classical modes of transverse electric or transverse magnetic fields with angular frequency ω propagating along an axis are shown to be able to be enveloped with counter-rotating helical modulations which have a different angular frequency Ω. These helical rotations, called distributed spin rotations, propagate at the group velocity. The formation of a completely closed packet of electromagnetic energy requires that the axial fields and transverse fields have a common axial length of envelope. This forces Ω to take quantized values in terms of ω with Ω related to the Schrodinger frequencies of a harmonic oscillator. The spin rotations permit flexible transverse confinement allowing for localization of the photon wave-packet over different spatial areas. It is argued that the energy of this packet is not related to its volume but depends on the quantized helical frequency Ω. Such photon-like packets possess classical phase and group velocities in keeping with experimental evidence. A single photon-like packet does not disperse in dispersive media. Incrementing or decrementing the rate of helical rotation promotes or demotes the packet energy in keeping with standard photon theory. The model offers explanations for self-interference and entanglement.

A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries

Subluminal, luminal and superluminal localized wave solutions to the paraxial pulsed beam equation in free space are determined. A clarification is also made to recent work on pulsed beams of arbitrary speed which are solutions of a narrowband temporal spectrum version of the forward pulsed beam equation.

A new solution of the paraxially approximated wave equation in free space is presented. It has the form of a Hermite-Gaussian function, where the argument of the Hermite polynomial is generally complex. The usual Gaussian beams where such argument is real and those recently described by Siegman where it is complex but coinciding with that of the Gaussian function are particular cases of the general solution presented here. The application of these generalized Gaussian beams to the analysis of the beam emerging from a complex graded-index medium is discussed in detail.

As standard microelectronic technology approaches fundamental limitations in speed and power consumption, novel computing strategies are strongly needed. Analogue optical computing enables the processing of large amounts of data at a negligible energy cost and high speeds. Based on these principles, ultrathin optical metasurfaces have been recently explored to process large images in real time, in particular for edge detection. By incorporating feedback, it has also recently been shown that metamaterials can be tailored to solve complex mathematical problems in the analogue domain, although these efforts have so far been limited to guided-wave systems and bulky set-ups. Here, we present an ultrathin Si metasurface-based platform for analogue computing that is able to solve Fredholm integral equations of the second kind using free-space visible radiation. A Si-based metagrating was inverse-designed to implement the scattering matrix synthesizing a prescribed kernel corresponding to the mathematical problem of interest. Next, a semitransparent mirror was incorporated into the sample to provide adequate feedback and thus perform the required Neumann series, solving the corresponding equation in the analogue domain at the speed of light. Visible wavelength operation enables a highly compact, ultrathin device that can be interrogated from free space, implying high processing speeds and the possibility of on-chip integration.

It is rigorously showed that any solution of the homogeneous Helmholtz equation in free space is band-limited to a circle of radius k in the plane z=0 and evanescent waves in the angular spectrum representation are really absent. If the part of spectrum that gives rise to evanescent waves is not equal to zero then specified field distribution is not exactly the solution of the wave equation.