The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams

We present a rigorous, but mathematically relatively simple and elegant, theory of first-order spatio-temporal distortions, that is, couplings between spatial (or spatial-frequency) and temporal (or frequency) coordinates, of Gaussian pulses and beams. These distortions include pulse-front tilt, spatial dispersion, angular dispersion, and a less well-known distortion that has been called “time vs. angle.” We write pulses in four possible domains, xt, xω, kω, and kt; and we identify the first-order couplings (distortions) in each domain. In addition to the above four “amplitude” couplings, we identify four new spatio-temporal “phase” couplings: “wave-front rotation,” “wave-front-tilt dispersion,” “angular temporal chirp,” and “angular frequency chirp.” While there are eight such couplings in all, only two independent couplings exist and are fundamental in each domain, and we derive simple expressions for each distortion in terms of the others. In addition, because the dimensions and magnitudes of these distortions are unintuitive, we provide normalized, dimensionless definitions for them, which range from -1 to 1. Finally, we discuss the definitions of such quantities as pulse length, bandwidth, angular divergence, and spot size in the presence of spatio-temporal distortions. We show that two separate definitions are required in each case, specifically, “local” and “global” quantities, which can differ significantly in the presence of spatio-temporal distortions.