Impact of the boundary's sharpness on temporal reflection in dispersive media

We investigate the impact of the finite rise time of a spatiotemporal boundary inside a dispersive medium used for reflection and refraction of optical pulses. We develop a matrix approach in the frequency domain for analyzing such spatiotemporal boundaries and use it to show that the frequency range over which reflection can occur is reduced as the rise time increases. We also show that total internal reflection can occur even for boundaries with long rise times. This feature suggests that spatiotemporal waveguides can be realized through cross-phase modulation even when pump pulses have relatively long rise and fall times.