Ioannis M. Besieris

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

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.

A note on an accelerating finite energy Airy beam

A recently derived Airy beam solution to the (1+1)D paraxial equation is shown to obey two salient properties characterizing arbitrary finite energy solutions associated with second-order diffraction; the centroid of the beam is a linear function of the range and its variance varies quadratically in range. Some insight is provided regarding the local acceleration dynamics of the beam. It is shown, specifically, that the interpretation of this beam as accelerating, i.e., one characterized by a nonlinear lateral shift, depends significantly on the parameter a entering into the solution.

Aperture realizations of exact solutions to homogeneous-wave equations

Several new classes of localized solutions to the homogeneous scalar wave and Maxwell’s equations have been reported recently. Theoretical and experimental results have now clearly demonstrated that remarkably good approximations to these acoustic and electromagnetic localized-wave solutions can be achieved over extended near-field regions with finite-sized, independently addressable, pulse-driven arrays. We demonstrate that only the forward-propagating (causal) components of any homogeneous solution of the scalar-wave equation are actually recovered from either an infinite- or a finite-sized aperture in an open region. The backward-propagating (acausal) components result in an evanescent-wave superposition that plays no significant role in the radiation process. The exact, complete solution can be achieved only from specifying its values and its derivatives on the boundary of any closed region. By using those localized-wave solutions whose forward-propagating components have been optimized over the associated backward-propagating terms, one can recover the desirable properties of the localized-wave solutions over the extended near-field regions of a finite-sized, independently addressable, pulse-driven array. These results are illustrated with an extreme exampl—one dealing with the original solution, which is superluminal, and its finite aperture approximation, a slingshot pulse.

Two Fundamental Representations of Localized Pulse Solutions to the Scalar Wave Equation

A study is undertaken of two fundamental representations suitable for the derivation of localized pulse (LW) solutions to the scalar wave equation. The first one uses superpositions over products of plane waves moving in opposite directions along the characteristic variables z - ct and Z + ct. This bidirectional representation, introduced in an earlier publication, has proved instrumental in advancing our understanding of Focus Wave Mode (FWM)-like pulses. The second representation, based on the Lorentz invariance of the scalar wave equation, uses products of plane waves propagating along the subluminal and superluminal boost variables. This representation is suitable for the derivation of X-wave-type solutions. Subluminal and superluminal Lorentz transformations are used to derive closed-form LW solutions to the scalar wave equation by boosting known solutions of other equations, e.g., the 2-D scalar wave equation, the Helmholtz equation and Laplaces equation. Several of these LW solutions are deduced in this manner and their properties are discussed. Of particular interest is the derivation of a novel finite energy LW solution, named the Modified Focus X-Wave pulse. It is characterized by low sidelobe levels, a desirable property for applications, e.g., in pulse echo techniques used in medical imaging.

A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations

A systematic approach to the derivation of exact nondispersive packet solutions to equations modeling relativistic massive particles is introduced. It is based on a novel bidirectional representation used to synthesize localized Brittingham‐like solutions to the wave and Maxwell’s equations. The theory is applied first to the Klein–Gordon equation; the resulting nondispersive solutions can be used as de Broglie wave packets representing localized massive scalar particles. The resemblance of such solutions to previously reported nondispersive wave packets is discussed and certain subtle aspects of the latter, especially those arising in connection to the correct choice of dispersion relationships and the definition of group velocity, are clarified. The results obtained for the Klein–Gordon equation are also used to provide nondispersive solutions to the Dirac equation which models spin 1/2 massive fermions.A systematic approach to the derivation of exact nondispersive packet solutions to equations modeling relativistic massive particles is introduced. It is based on a novel bidirectional representation used to synthesize localized Brittingham‐like solutions to the wave and Maxwell’s equations. The theory is applied first to the Klein–Gordon equation; the resulting nondispersive solutions can be used as de Broglie wave packets representing localized massive scalar particles. The resemblance of such solutions to previously reported nondispersive wave packets is discussed and certain subtle aspects of the latter, especially those arising in connection to the correct choice of dispersion relationships and the definition of group velocity, are clarified. The results obtained for the Klein–Gordon equation are also used to provide nondispersive solutions to the Dirac equation which models spin 1/2 massive fermions.

Localized wave representations of acoustic and electromagnetic radiation

Novel space-time solutions to general classes of wave equations and their properties are reviewed. The localized-wave (LW) solutions, in contrast to their continuous-wave (CW) or monochromatic counterparts, exhibit enhanced localization and energy fluence characteristics. This has led to the analysis and construction of pulse-driven, independently addressable arrays to investigate nonstandard methods of wave energy transmission based upon these LW solutions. Such arrays allow different signals to be transmitted from different locations in the array, thus allowing shading of the spectral features as well as the amplitudes seen by the array. It has been shown experimentally that the beams transmitted by these LW pulse-driven arrays outperform conventional CW driven arrays. >

Localized energy pulse trains launched from an open, semi-infinite, circular waveguide

A new decomposition of exact solutions to the scalar wave equation into bidirectional, backward and forward traveling plane waves is described. These elementary blocks constitute a natural basis for synthesizing Brittinghamlike solutions. Examples of such solutions, besides Brittingham’s original modes, are Ziolkowski’s electromagnetic directed energy pulse trains (EDEPTs) and Hillion’s spinor modes. A common feature of these solutions is the incorporation of certain parameters that can be tuned in order to achieve slow energy decay patterns. The aforementioned decomposition is used first to solve an initial boundary‐value problem involving an infinite waveguide. This is followed by considering a semi‐infinite waveguide excited by a localized initial pulse whose shape is related directly to parameters similar to those arising in Ziolkowski’s EDEPT solutions. The far fields outside the semi‐infinite waveguide are computed using Kirchhoff’s integral formula with a time‐retarded Green’s function. The resulti...

Generation of approximate focus-wave-mode pulses from wide-band dynamic Gaussian apertures

It is demonstrated that an approximation to the focus-wave-mode field can be generated from a dynamic Gaussian aperture. A source of this type is characterized by the time variation of its effective radius. The performance of such an aperture is studied in detail; it is demonstrated that the dynamic aperture shows a great enhancement over the corresponding static one. The types of source investigated provide an efficient scheme to launch narrow Gaussian pulses from extended apertures.

Stochastic wave‐kinetic theory in the Liouville approximation

The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic, random media is investigated on the basis of a stochastic, collisionless, Liouville‐type equation governing the temporal evolution of a phase‐space Wigner distribution density function. Within the framework of the first‐order smoothing approximation, a general diffusion–convolution‐type kinetic or transport equation is derived for the mean phase‐space distribution function containing generalized (nonloral, with memory) diffusion, friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long‐time, Markovian, diffusion approximation, a general set of Fokker–Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.

Nondispersive accelerating wave packets

Motivated by the work of Berry and Balazs and Greenberger on the 1‐D Schrodinger equation, we have investigated a class of nonspreading solutions to the 3‐D Schrodinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy‐type solutions to the 3‐D Schrodinger equation has led us also to analogous solutions for the 3‐D scalar wave equation.