On space-time media that compute their own inverse

We show that it is possible to design a space-time medium such that the time scattering anticipates the space scattering in the medium and produces the exact inverse for the space scattering. Probing such a medium with a single broadband pulse thus also results in a single broadband pulse on the other side of the space-time heterogeneity. The transmission coefficient of such systems is unity. The reflection coefficient is non-zero, however, as both the time and spatial boundaries scatter waves in the backward direction. The time boundaries thus add energy. To construct a single pulse on the other side of a stack bounded by n interfaces, 2(n−1)−1 additional well-scaled and well-timed pulses need to be emitted following the main pulse, to cancel multiple reflections in the forward direction. Scattering at a time boundary doubles the number of wavepackets propagating in a medium. Furthermore, time refraction and reflection coefficients have the form required to produce well-scaled pulses. The required 2(n−1)−1 pulses can thus be efficiently generated by a single pulse interacting with n time boundaries. The contrasts across the time boundaries are the spatial contrasts in reverse. The time “thicknesses” of the boundaries are a simple function of the layer velocities.