Spinor solutions to Maxwell's equations in free space

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.

In this paper, we consider the time-dependent Maxwell's equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell's equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

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

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.

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.

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.

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.

Numerical simulation of Bessel beams by FDTD employing the superposition principle

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.

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.

Spatiotemporal behavior of ultrashort pulses in the far field

It is well known that the reliability of radar measurements to characterize targets is dependent upon the signal-to-noise ratio of echoes. The power of received signal becomes weak as range increases as described by radar equation in free space. For weather radar, the echoes can suffer extra power loss due to propagation in rain media. A set of system parameters need to be determined in radar design to assure the received signal in sufficient quality at given range. However, rain attenuation is dependent on rain intensity and path length. Therefore attenuation statistics can predict the power loss due to precipitation. This is especially important for radar designers to leverage the transmit power and maximum range. The higher the frequency the greater the amount of power attenuation the signal will suffer as it travels along its path mainly through precipitation as well as other attenuating medium. Conventionally long range weather radar operates at S- band where the attenuation is almost negligible and the application of X-band radars to weather sensing has seen limited application. X-band radars typically operate at frequencies in the neighborhood of 8-12 GHz with respective wavelengths between 3.75 cm to 2.5 cm, and S-band radars operate in the range of 2-4 GHz with wavelengths between 15 cm and 7.5 cm respectively. However the use of X-band radars in comparison to S-band radars has some advantages particularly in the physical size of the radar where the width of the dish decreases as the frequency of the radar increases. Therefore, the costs of building and maintaining a smaller radar can outweigh the costs of larger radars. This can then lead to the usage of multiple radars in place of a single large radar. Further more, the size of an X-band radar can lends itself to portability. So with the size advantage the statistics can give insight into X-band radar design.

A simple formalism is found for the measurement of wave fields that satisfy the Helmholtz equation in free space. This formalism turns out to be analogous to the well-known theory of measurements for quantum-mechanical wave functions: A measurement corresponds to the squared magnitude of the inner product (in a suitable Hilbert space) of the wave field and a field that is associated with the detector. The measurement can also be expressed as an overlap in phase space of a special form of the Wigner function that is tailored for Helmholtz wave fields.