| We investigate the spatiotemporal evolution of single-cycle and few-cycle ultrashort electromagnetic pulses. Exact solutions of the free space full vector Maxwell's equations are provided. These solutions are nonsinusoidal, nonstationary and nonseparable in space and time. They describe space-time localized wavepackets representing transmission of electromagnetic energy in free spare. In a manner similar to dispersion effects on broadband signals, diffraction effects on transversely confined ultrashort pulses in the single-cycle region cause significant pulse reshaping and waveform distortion as a consequence of spacetime coupling. Particularly in the paraxial region, the diffraction-induced Gouy phase shift plays a significant role in free space propagation to change the temporal waveform of the single-cycle pulses. As a result of the Gouy phase shift, the single-cycle pulse experiences polarity and time reversals as it passes through the focus. When such a pulse propagates in quadratic phase media, the accumulated Gouy phase shift results in nonstationary variation of absolute phase and temporal waveform distortion. We show that the physical origin of the Gouy phase shift is the transverse confinement of the extended wave and hence is related to the uncertainty principle of the fundamental physics. In the nonparaxial region, polarization of the electric field of single-cycle pulses cannot be represented by any of a simple ellipse, circle or straight line. In general the tip of the electric field vector moves along a cardioid and the polarization state changes during propagation.; We also investigate the higher-order transverse modes of single-cycle pulses, particularly Hermite-Gaussian and Laguerre-Gaussian modes. The effects of space-time coupling is more significant in the higher-order transverse modes. These effects includes dark pulses, dark ring pulses, spiral pulses and vortex strings. Since Hermite-Gaussian or Laguerre-Gaussian modes of single-cycle pulses form a complete set of pulsed-beam basis functions for two-dimensional space, the study of these higher-order modes has practical interest in the time-domain synthesis of ultrashort pulses radiated, scattered, and diffracted from arbitrary well-collimated aperture distributions. For well-collimated pulsed distributions, such a time-domain approach can yield great efficiency owing to the fast convergence and hence reduce computational tasks. Those applications can be implemented through time-domain continuous wavelet transforms. |
