Finite-energy spatiotemporally localized Airy wavepackets.

The time-diffraction technique introduced by Porras recently is motivated in this article in terms of the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation and it is used to derive finite-energy spatiotemporally confined subluminal, luminal and superluminal Airy wave packets. In addition, a novel exact finite-energy luminal Airy splash mode-type solution to the scalar wave equation is derived using Bateman's conformal transformation.

Propagation of time-truncated Airy-type pulses in media with quadratic and cubic dispersion.

In this paper, we describe analytically the propagation of Airy-type pulses truncated by a finite-time aperture when second- and third-order dispersion effects are considered. The mathematical method presented here, which is based on the superposition of exponentially truncated Airy pulses, is very effective and allows us to avoid the use of time-consuming numerical simulations. We analyze the behavior of the time-truncated ideal Airy pulse and also the interesting case of a time-truncated Airy pulse with a "defect" in its initial profile, which reveals the self-healing property of this kind of pulse solution.

Discrete fractional Fourier transform as a fast algorithm for evaluating the diffraction pattern of pulsed radiation.

A technique is proposed for computing the field radiated from a rectangular aperture. This technique, based on the discrete fractional Fourier transform, avoids the complexities of computing the diffraction pattern by the direct evaluation of the Fresnel integral. The advocated approach provides a fast and accurate computational tool, especially in the case of evaluating pulsed fields radiated through two-dimensional screens of complex amplitude. A detailed numerical study that demonstrates the efficacy of this approach is carried out.

Hopf-Ranãda linked and knotted light beam solution viewed as a null electromagnetic field.

The Hopf-Ranãda linked and knotted light beam solution, which has been interpreted physically and extended analytically by Irvine and Bouwmeester recently, is viewed in this Letter as a null electromagnetic field. It is shown, in particular, that the Hopf-Ranãda solution is a variant of a luminal null electromagnetic wave due originally to Robinson and Troutman and reported by Bialynicki-Birula recently. This analogy is motivated by means of a method due to Whittaker and Bateman, and a relationship to well-known scalar luminal localized waves is examined.

Accelerating Airy wave packets in the presence of quadratic and cubic dispersion.

The equation governing quadratic and cubic transparent dispersion within the framework of the slowly varying envelope approximation is shown to admit an infinite-energy uniformly moving Airy wave packet solution, as well as a square-integrable accelerating Airy solution. Some insight is provided regarding the local acceleration dynamics in the latter case and comparisons are made with the "accelerating" beam solution introduced by Siviloglou and Christodoulides and experimentally demonstrated by Siviloglou, Broky, Dogariu, and Christodoulides recently. It is shown, in particular, that under certain parametrizations, the presence of cubic dispersion can increase the "depth of penetration" of a wave packet. In other words, a pulse can propagate for a larger range without sustaining significant dispersive distortion than in the presence of quadratic dispersion alone. Finally, imaging properties of accelerating airy wave packets are discussed.

(2+1)-dimensional X-shaped localized waves.

A hybrid spectral superposition method is presented that allows a smooth transition between two seemingly distinct classes of localized wave solutions to the homogeneous scalar wave equation in free space; specifically, luminal or focus wave modes, and superluminal or X waves. This representation, which is based on superpositions of products of forward plane waves moving at a fixed speed v>c and backward plane waves moving at the speed c, is used to construct a large class of finite-energy superluminal-type X-shaped localized waves. The latter are characterized by arbitrarily high-frequency bands and are suitable for applications in the microwave and optical regime. In the limiting case v-->c, one recaptures the well-known focus wave mode-type localized wave solutions. A modified hybrid spectral representation, based on superpositions of products of forward plane waves moving at a fixed speed c and backward plane waves moving at the speed v>c, allows in the limit v-->c a smooth transition from superluminal localized waves to paraxial luminal pulsed beams. Although the proposed methods are applicable to a (n+1)-dimensional, n> or =2, scalar wave equation, the discussion will be limited to the case n=2 for simplicity; also, so that comparisons can be made to related recent results in the literature.

Paraxial localized waves in free space.

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.

Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos-Hänchen effect.

A study of X waves undergoing frustrated total internal reflection at a planar slab is provided. This is achieved by choosing the spectral plane wave components of the incident X wave to fall on the upper interface at angles greater than the critical angle. Thus, evanescent fields are generated in the slab and the peak of the field tunneling through the slab appears to be transmitted at a superluminal speed. Furthermore, it is shown that for deep barrier penetration, the peak of the transmitted field emerges from the rear interface of the slab before the incident peak reaches the front interface. To understand this advanced transmission of the peak of the pulse, a detailed study of the behavior of the evanescent fields in the barrier region is undertaken. The difference in tunneling behavior between deep and shallow barrier penetrations is shown to be influenced by the sense of the Goos-Hänchen shift.

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.

Localized pulses exhibiting a missilelike slow decay.

We investigate the quasi-missile behavior of known localized wave solutions, such as the modified power spectrum and splash pulses. We demonstrate that source-free localized waves can exhibit slow decay rates analogous to Wu's missile solutions, which are characterized by an amplitude decay rate slower than 1/R over an unlimited range. When excited from a finite aperture, the missilelike decay is not exhibited by all localized waves showing such behavior in the source-free situation. On the other hand, localized wave missiles generated from a finite aperture have peaks that exhibit quasi-missile decay. In an extended intermediate range between the near- and the far-field regions, these pulses decay at a rate slower than 1/R before switching to the usual 1/R decay.

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.

Temporal focusing by use of composite X waves.

It is shown that highly focused pulses can be shaped by exciting a finite aperture with a spread-out pulse train of X waves. The basis of the proposed scheme is that the peaks of X waves, characterized by different apex angles, travel at different velocities. This property allows one to vary the temporal starting points of the initial excitations of a sequence of X waves so that all their peaks meet at a chosen focusing point. It is demonstrated that this simple criterion can be effective in producing a highly focused, composite X-wave pulse that exhibits a slower decay behavior than the individual X-wave components used in synthesizing it.