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 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.

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.

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.