(a) Path taken by an optical or quantum wave through two adjoining media with different refractive indices and . (b) Trajectory of a wave packet in -dimensional spacetime, with time measured in terms of propagation distance. The wave packet exhibits a refractionlike phenomenon with the refraction angles satisfying a modified law of refraction given by Eq. (12b). (c) Trajectory of an optical wave in -dimensional spacetime when crossing between two nondispersive media. The equivalent law describing its refraction in spacetime is different [see Eq. (12a)]. (d) Refractive index as a function of the spatial coordinate.
Refraction in spacetime for a Gaussian wave packet. The coordinate is measured in terms of an arbitrary unit of length , and . The parameters used in the simulation are , , , and . The wave packet is refracted away from the normal as a result of the positive potential step. The white lines are plots of the trajectories for the incident, reflected, and the refracted rays obeying the law of refraction in spacetime [Eq. (12b)].
Refraction in spacetime for a Gaussian wave packet that broadens significantly during the scattering event. The parameters are the same as in Fig. 2, except that . The interference fringes between the incident and reflected components are seen to be curved, in contrast to the phase fronts in Fig. 2 which are straight.
Birefringence in spacetime of a Gaussian wave packet prepared in a field-free region as a linear superposition of spin- and spin- components with equal weight. The parameters are , , , and . The incident wave packet crosses into a region with finite magnetic field at approximately (that is, ). The two distinct paths visible for the transmitted wave correspond to its separated spin- and spin- components.
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