Goos-Hänchen shift at a temporal boundary

We discover a pronounced temporal shift of the peak of an optical pulse upon total internal reﬂection of the pulse from a sharp temporal boundary propagating in a homogeneous, isotropic, weakly dispersive linear medium. We derive an analytical expression for this shift and juxtapose the discovered effect to the spatial Goos-Hänchen shift occurring on reﬂection of a beam from an interface separating two homogeneous, isotropic, conservative linear media. In particular, we show that, in contrast to the spatial shift, the sign of the temporal shift is dictated by that of the group-velocity mismatch between the pulse and the temporal boundary, implying the possibility of a delay or advancement of the pulse upon reﬂection. Our analytical results, which are in excellent agreement with our numerical simulations, shed light on the fundamental aspects of the interaction of wave packets with temporal boundaries in material media