  The advent of novel nonlinear materials has stirred unprecedented interest in
exploring the use of temporal inhomogeneities to achieve novel forms of wave
control, amidst the greater vision of engineering metamaterials across both
space and time. When the properties of an unbounded medium are abruptly
switched in time, propagating waves are efficiently converted to different
frequencies, and partially coupled to their back-propagating phase-conjugate
partners, through a process called time-reversal. However, in realistic
materials the switching time is necessarily finite, playing a central role in
the resulting temporal scattering features. By identifying and leveraging the
crucial role of electromagnetic momentum conservation in time-reversal
processes, here we develop a very general analytical formalism to quantify
time-reversal due to temporal inhomogeneities of arbitrary profile. Finally, we
deploy our analytic theory to develop a formalism, analogous to spatial
tapering theory, that enables the tailoring of a desired time-reversal spectral
response, demonstrating its use for the realization of broadband frequency
converters and filters.
