Strongly unidirectional and weakly directional pulses

Abstract

The paper introduces new classes of spatiotemporal solutions to the wave equation – strongly unidirectional and weakly directional pulses, generalizing the definition of unidirectionality. The classes consist of families of solutions that arise under the Lorentz transformation of the common unidirectional solutions. We show that the parameter $-1<\beta <1$ of the Lorentz transformation is related to the opening angle ${\chi }_0=\arccos \left(\beta \right)$ of a cone within which the wave vectors of plane-wave constituents of a solution are localized. Strongly unidirectional and weakly directional pulses correspond to $\beta >0$ and $\beta <0$, respectively, and the common unidirectionality is described by $\beta =0$. Some general properties of the introduced classes of solutions are discussed with a representative example given.