For the wave equation with three spatial variables, we give a solution that has a finite energy integral but does not tend to a spherical wave at infinity.

A solution of the wave equation with three spatial variables is given, which has a finite energy integral, but does not tend to a spherical wave at infinity.

A new class of localized solutions of paraxial parabolic equation is introduced. Each solution is a product of some Gaussian-type localized axisymmetric function (different from the fundamental mode) and an amplitude factor. The latter can be expressed via an arbitrary solution of the Helmholtz equation on an auxiliary two-sheet complex surface. The class under consideration contains well known and novel solutions, including those describing optical vortices of various orders. Keywords: parabolic equation, quadratic beams, Gauss, Helmholtz, Bessel.

A brief overview of the current state of the theory of unidirectional pulses is given.

A theoretical description of a class of unidirectional axisymmetric localized pulses, is given. The equivalence of their representations in the form of relatively undistorted quasi-spherical waves, in the form of Fourier-Bessel integrals and in the form of a superposition of plane waves with wave vectors having positive projections on a given direction is established.

The paper is aimed at constructing exact solutions of Maxwell's equations for homogeneous media, convenient for modeling ultrashort pulses of various shapes. An analytical description of a family of simple closed-form few-cycle electromagnetic pulses that are free of backward propagating components and have finite energy is presented. The mathematical framework rests on using, as a component of Hertz's potential, a certain axisymmetric exact solution of the linear wave equation, which is studied here in detail. Depending on the choice of free parameters in this solution and on polarization of the potential, the resulting electromagnetic pulses can be pancake-like, ball-like, needle-like, and doughnut-like. Expressions for spectra of the electric field components of the pulses are obtained. Based on the derived formulas, typical examples of pulses with different types of localization and their spectra are calculated and plotted. (c) 2024 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.

A brief review of the authors' latest work in the field of extremely short electromagnetic pulses, including unipolar pulses, is presented. Фамилии авторов на английском: Rosanov N.N., Arkhipov M.V., Arkhipov R.M., Plachenov A.B., Tumakov D.A. Благодарностей грантам нет.

Simple closed-form analytical expressions for tilted astigmatic wave beams and wavepackets that are exact solutions of the wave equation are constructed. They are obtained through two different but equivalent derivations, one is based on a complex shift in the Bateman-type solutions, the other employs their Lorentz transformation. Analytic expressions for the propagation invariants: energy, momentum and orbital angular momentum of tilted waveobjects with general astigmatism are presented.

We analyze the possibility of existence of electromagnetic-field pulses with a non-zero electric area (time integral of the electric field) within the framework of the Maxwell equations. It is demonstrated that in the absence of charges and currents in vacuum, the non-zero electric area of a pulse would lead to its infinite energy. A pulse with the non-zero area is shown to form already in the case of uniform and rectilinear motion of the charge. The conditions for formation of such pulses by a localized charge system are found, and an example of such a system is presented. The asymptotics of the electric area far from the system of charges is obtained.

A brief review of the authors’ latest work in the field of extremely short electromagnetic pulses, including unipolar pulses, is presented.

Simple formulas expressing the total energy of solutions of 3D wave equation, and for electro- and elastodynamics equations via their asymptotics at large time and distance, are obtained. Several examples are considered.

A simple solution of the wave equation with two spatial variables having a similarity to the well known splash mode solutions is presented. It depends on two free parameters and describes a localized few-cycle pulse having finite energy. The behavior of the solution is studied both in the vicinity of the focus and at large times and distances. It was found that its real part at all points and at all instants of time is strictly positive. Integral representations for the solutions are derived and an expression for its total energy is found.

A new class of localized solutions of paraxial parabolic equation is introduced. Each solution is a product of some Gaussian-type localized axisymmetric function (different from the fundamental mode) and an amplitude factor. The latter can be expressed via an arbitrary solution of the Helmholtz equation on an auxiliary two-sheet complex surface. The class under consideration contains well known and novel solutions, including those describing optical vortices of various orders.

We obtain a family of simple closed-form exact solutions of the three-dimensional wave equation that are free of backward components. The solutions have no singularities and possess finite energy. They describe single-cycle and subcycle pulses. Depending on two parameters, the family yields pulses with different types of localization, in particular, focused pancakes, balls, and needles.

In this work, we establish the relationship existing between astigmatic Gaussian beams and Bessel-modulated Gaussian beams with quadratic radial dependence. We introduce an alternative representation of quadratic Bessel-Gaussian beams. This result permits us to expand the astigmatic Gaussian beams in a Fourier series with respect to the polar azimuthal angle with coefficients that are expressible in terms of Bessel-modulated Gaussian beams with quadratic radial dependence. The analysis presented also includes an analytic derivation of the power azimuthal spectral distribution associated with the decomposition in orbital angular momentum eigenmodes and a comprehensive characterization of the properties of this distribution as the beam parameters are varied.

We introduce a simple exact solution of the wave equation in free space. It describes unidirectional finite-energy single-cycle and sub-cycle pulses. Depending on the ratio of its free parameters, the solution can yield, in particular, pancake, ball, and needle pulses.

We introduce a simple exact solution of the wave equation in free space. It describes unidirectional finite-energy single-cycle and sub-cycle pulses. Depending on the ratio of its free parameters, the solution can yield, in particular, pancake, ball and needle pulses.

We address propagation of paraxial Gaussian-type beam modes in longitudinally inhomogeneous axisymmetric lens-like media, with a specific case of a transition of waveguide into antiwaveguide. Higher-order modes are described in a unified manner using a so-called secondary parabolic equation. Analyzing a linear transition from waveguide to antiwaveguide where beams are described in terms of Airy functions, we conclude that the width of a localized Gaussian-type beam in inhomogeneous media can dramatically grow. This fact should be taken into account in numerical procedures employing summation of Gaussian beams. Copyright (C) EPLA, 2019

We have considered a new family of localized solutions of a parabolic (paraxial wave) equation that generalizes the well-known Bessel–Gaussian beams and includes asymmetric and noncoaxial Bessel–Gaussian beams as subfamilies. The Fourier spectra of these solutions have been found, and their expansions into nonshifted Bessel–Gaussian beams have been obtained.

We propose a new class of localized solutions of the paraxial wave equation. They have a form of a product of a Gaussian term and an amplitude which contains only elementary coordinate functions. Solutions are obtained by summing of the quadratic Bessel-Gauss beams with odd indices. Due to the configuration of the obtained solutions, we named them quadratic cosine-Gauss beams.