A new decomposition of exact solutions to the scalar wave equation into bidirectional, forward and backward, traveling plane wave solutions is described. The resulting representation is a natural basis for synthesizing pulse solutions that can be tailored to give directed energy transfer in space. The development of known free‐space solutions, such as the focus wave modes, the electromagnetic directed energy pulse trains, the spinor splash pulses, and the Bessel beams, in terms of this decomposition will be given. The efficacy of this representation in geometries with boundaries, such as a propagation in a circular waveguide, will also be demonstrated.

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A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

Localized Light Waves: Paraxial and Exact Solutions of the Wave Equation (a Review)

The first article by Turunen and Friberg deals with a class of fields with propagation-invariant properties such as the optical intensity distribution. Coherent and partially coherent stationary and pulsed solutions are treated in view of scalar and electromagnetic approaches. Approximations of ideal propagation-invariant fields and methods for their generation are discussed. Finally, some application areas are covered.

Propagation-Invariant Optical Fields

The Lorentz transformations of propagation-invariant localized waves (also known as nondispersive or nondiffracting or undistorted progressive waves) are studied in the frequency-momentum space For supports of wave functions in this space rules of transformation are derived which allow one to group all localized waves into distinct classes: subluminal, luminal, and superluminal localized waves It is shown that for each class there is an inertial frame in which any given localized wave takes a particularly simple form In other words, any localized wave is nothing but a relativistically aberrated and Doppler shifted version of a simple ``seed'' wave Also discussed are the relations of the physical (subluminal) Lorentz tranformation to other mathematical tranformations used in the literature on localized waves, as well as physical interpretation of the substantial changes that localized waves undergo if observed and generated in different inertial frames

Generation and classification of localized waves by Lorentz transformations in Fourier space.

Pulsed beams (PB’s) are time-dependent wave fields that are confined in beamlike fashion in transverse planes perpendicular to the propagation axis, whereas confinement along the axis is due to temporal windowing. Because they have these properties, pulsed beams are useful wave objects for generating and synthesizing highly focused transient fields. The PB problem is addressed here within the context of fundamental Green’s-function propagators for the time-dependent field equations. In a departure from known results in the frequency domain, by which beam solutions can be generated from point-source solutions by displacing the source coordinate location into a complex coordinate space, the complex extension is applied here as well to the source initiation time. This procedure converts the conventional causal impulsive-source Green’s-function propagator into a noncausal PB propagator, which must be defined as an analytic signal because, owing to causality, the analytic continuation into the complex domain cannot be performed by direct substitution. This being done, PB’s can be manipulated as conventional Green’s functions. Some previous results obtained by similar methods are viewed here from a sharper perspective, and new results, both analytical and numerical, are presented that grant basic insight into the PB behavior, including the ability to excite these fields by finite-causal-aperture-source distributions. Besides the basic (analytic Green’s-function) PB, examples include PB’s with frequency spectra of special interest. Particular attention is paid to the PB synthesis of focus-wave modes, which are source-free solutions of the time-dependent wave equation, and to the compact PB formulation of wave fields synthesized by focus-wave-mode spectral superposition.

Complex-source pulsed-beam fields

A general formulation is given for electromagnetic pulses which remain localized in a multidimensional space, and which propagate at the speed of light without dispersing (focus wave modes). It is shown that such modes necessarily have infinite electromagnetic energy in the source‐free, three‐dimensional space. Finite‐energy focus wave modes cannot exist without sources. A set of complete focus wave modes with Hermite–Gaussian transverse variation is derived. The relation of focus wave modes to the solutions of the paraxial wave equation is established.

A. Sezginer

Papers

A general formulation of focus wave modes