Amr M. Shaarawi

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.

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 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.

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.

on the evanescent fields and the causality of the focus wave modes

The diverging and converging field components of the source‐free focus wave modes are studied within the framework of both the Whittaker and Weyl plane wave expansions. It is shown that, in the Weyl picture, the evanescent fields associated with the diverging and converging components of the focus wave mode solution cancel each other identically. The source‐free focus wave modes are, hence, composed solely of backward and forward propagating components of the Whittaker type. It will also be shown that no evanescent fields are associated with the causal excitation of an aperture by an initial focus wave mode field. The diverging field, however, is composed solely of causal components that propagate away from the aperture. With a specific choice of parameters, the field generated by the aperture is a very good approximation to the source‐free solution. Under the same conditions, the original focus wave mode solution is composed predominantly of causal forward propagating fields.

Comparison of two localized wave fields generated from dynamic apertures

The performances of two localized pulses that have lately attracted attention are compared. The first localized field, the focus wave modes, was recently shown to be causally launchable from flat apertures characterized by having time-dependent radii. It is demonstrated that the second localized field, the X waves, can be also be generated from finite-time dynamic sources. A comparison of the decay patterns of the two wave fields is carried out. It is emphasized that such carrier-free pulses with ultrawide bandwidths can perform differently, even if they are excited by use of apertures with the same dimensions and with initial illumination wave fields of comparable frequency bandwidths. The propagation characteristics of such localized pulsed fields in the near- and far-field regions are studied. Based on an understanding of the spectral depletion of such pulses, a method is suggested to slow the decay rate of the pulses in the far-field region.

Focused X-shaped pulses.

The space-time focusing of a (continuous) succession of localized X-shaped pulses is obtained by suitably integrating over their speed, i.e., over their axicon angle, thus generalizing a previous (discrete) approach. New superluminal wave pulses are first constructed and then tailored so that they become temporally focused at a chosen spatial point, where the wave field can reach very high intensities for a short time. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics.The space-time focusing of a (continuous) succession of localized X-shaped pulses is obtained by suitably integrating over their speed, i.e., over their axicon angle, thus generalizing a previous (discrete) approach. First, new Superluminal wave pulses are constructed, and then tailored in such a wave to get them temporally focused at a chosen spatial point, where the wavefield can reach for a short time very high intensities. Results of this kind may find applications in many fields, besides electromagnetism and optics, including acoustics, gravitation, and elementary particle physics. PACS nos.: 41.20.Jb; 03.50.De; 03.30.+p; 84.40.Az; 42.82.Et; 83.50.Vr; 62.30.+d; 43.60.+d; 91.30.Fn; 04.30.Nk; 42.25.Bs; 46.40.Cd; 52.35.Lv. Keywords: Localized solutions to Maxwell equations; Superluminal waves; Bessel beams; Limited-diffraction pulses; Finite-energy waves; Electromagnetic wavelets; X-shaped waves; Electromagnetism; Microwaves; Optics; Special relativity; Localized acoustic waves; Seismic waves; Mechanical waves; Elementary particle physics; Gravitational waves