Fig. 1. Representation of two canonical spacetime crystals. Here, the variable is intended to represent the hyperspace, i.e., the three dimensions of space. The white and gray regions correspond to media with refractive indices and , respectively. (a) Double-period ( and ) structure, characterized by two velocities, and , which may be interpreted as acceleration at the corner. (b) Single-period () structure, characterized by a unique and uniform velocity, .
Fig. 2. Scattering at a spacetime interface. The white and gray regions correspond to media and , with refractive indices and (). (a) Sketch of a wave incident on a subluminal interface. (b) Same as (a) but for a superluminal interface. (c) Scattering from a subluminal () interface in a spacetime diagram. The blue arrows represent wave trajectories (referring to a specific phase point of the waveform), and the black arrows represent scattering coefficients. The dashed lines correspond to incidence from the right. The laboratory and moving frames are superimposed with common origin, and the moving frame has the same velocity as the interface (). (d) Same as (c) but for a superluminal interface (), with moving frame having the inverse velocity of the interface ().
Fig. 3. Spacetime-inversion symmetry of subluminal (SUB) and superluminal (SUP) structures. (a) Interfaces. (b) Slabs.
Fig. 4. Graphical description of the interluminal regime in spacetime diagram. (a) Codirectional case, with a single scattered wave. (b) Contradirectional case, with three scattered waves.
Fig. 5. Generalization of the Stokes principle. (a) Subluminal regime. (b) Superluminal regime.
Fig. 6. Frequency transitions at a spacetime interface corresponding to Fig. 2. (a) Subluminal case. (b) Superluminal case.
Fig. 7. Multiple-reflection description of the scattering phenomenology in spacetime slabs. Changes in line type (solid dashed) denote phase reversals. (a) Subluminal slab, with phase change occurring upon reflection to a lower impedance medium (), according to Eqs. (18). Note that the slope of the trajectories has been altered for representation convenience. (b) Superluminal slab, with phase change occurring upon reflection to a higher impedance medium (), according to Eqs. (22).
Fig. 8. Graphical Bragg-like interference argument. The light and dark blue trajectories correspond to the maxima and minima of the incident wave, and changes in line type (solid dashed) denote phase reversals. (a) Subluminal case, with constructive and destructive interference in reflection and transmission, respectively. Note that the slope of the trajectories has been altered for representation convenience. (b) Superluminal case, with constructive interference in both the later forward and later backward waves.
Fig. 9. Bilayer spacetime crystal with spacetime unit cell and out-of-gap wave trajectories. (a) Subluminal equal-length crystal, with . The slopes of the triangles ① and ② are and , so and . Substituting these lengths into the expression for the slope , yields Eq. (51). (b) Superluminal equal-duration crystal, with . From the slopes of the triangles ① and ②, given in (a), we have and . Substituting these durations into the expression for the average slope, also given in (a), yields Eq. (55).
Fig. 10. Linear approximation of the dispersion diagram of bilayer crystals with and equal electrical lengths. (a) Subluminal case, with . (b) Superluminal case, with .
Fig. 11. Bilayer spacetime crystal with spacetime unit cell. (a) Subluminal regime. (b) Superluminal regime.
Fig. 12. Dispersion diagram of bilayer crystals with and equal phases (). The solid curves correspond to the exact solution [Eqs. (86) and (89)], with the black and blue parts, respectively, corresponding to the real and imaginary parts, and the dashed curves to the linear approximation [Eq. (63)]. (a) Subluminal case, with . (b) Superluminal case, with .
Fig. 13. Examples of spacetime crystal truncation by a pair of spactime interfaces of velocities and for the left and right interfaces, respectively. Top row: the two interfaces have the same velocity, . Bottom row: the two interfaces have different velocities, . (a) Purely spatial truncation, . (b) Truncation with velocity different from the modulation velocity, . (c) Comoving truncation, . (d) Antiparallel truncation, . (e) Spacetime cavity with piecewise constant velocities. (f) Spacetime cavity with continuously varying velocity.
Fig. 14. Scattering from two canonical truncated spacetime crystals. In both cases, the crystal is subluminal, and the medium surrounding it is a simple nondispersive dielectric medium of refractive index . Top row: dispersion diagrams with transition frequencies. Bottom row: spacetime diagrams with scattered waves, with labels corresponding the solutions of the top panels. Note that the drawn scattered waves correspond to the waves seen in the laboratory frame and would completely different in the moving frame. (a) Comoving truncation. A moving-frame observer would see a stationary crystal bounded by stationary interfaces, and hence measure a unique frequency everywhere, inside and outside the crystal. The arrows indicate the up and down frequency and wavenumber transitions at the two interfaces. (b) Purely spatial truncation. A moving-frame observer would see a stationary crystal bounded by moving interfaces, and hence measure an infinity of frequencies.
Fig. 15. Transmission and reflection coefficients for an -layer crystal [corresponding to Fig. 13(c) and Fig. 14(a)] with . The gap centers , are provided in Appendix E (Sec. 12). (a) Subluminal case, with and , with attenuation in the bandgaps. (b) Superluminal case, with and with amplification in the bandgaps.
Fig. 16. Spacetime interface represented in two inertial frames. The arrows represent the trajectories of the media particles. In both (a) and (b), the interfaces of the spacetime variation are parallel to the axis and the media trajectories are parallel to the axis. (a) Laboratory frame viewpoint. Media appear at rest. Wave velocity is independent of direction: (b) Moving frame viewpoint. Media appear to be moving in direction. Wave velocities are direction dependent: , with .
Fig. 17. Graphical derivation of the travel length or duration across the unit cell of the crystal. (a) Subluminal regime (length). (b) Superluminal regime (duration).
Fig. 18. Successive application of time reversal () and space reversal ().
Fig. 19. Construction to find the frequencies aligned with the bandgaps. (a) Subluminal regime. (b) Superluminal regime.
Subluminal regime | Superluminal regime | Space | | Time | | Time | | Space | | Modulation velocity | | Inverse velocity | | Refractive index | | Inverse index | | Phase velocity | | Inverse phase velocity | | Length | | Duration | | Wavelength | | Period | | Frequency | | Wavenumber | |
|
Table 1. Duality transformations between the subluminal and superluminal regimes.
Subluminal | Superluminal | Coefficients (Fig. 2) | | [Eq. (18b)] | [Eq. (22b)] | [Eq. (18a)] | [Eq. (22a)] | Frequencies (Fig. 6) | | [Eq. (26b)] | [Eq. (28b)] | [Eq. (26b)] | [Eq. (28b)] | | |
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Table 2. Summary of the scattering formulas, derived in Sec. 3, for a spacetime interface.
Subluminal regime | Superluminal regime | Phase (Fig. 7) | | [Eq. (35a)] | [Eq. (38b)] | [Eq. (41b)] | [Eq. (38c)] | [Eq. (41c)] | Coefficients (Fig. 7) | | [Eq. (42)] | [Eq. (47)] | [Eq. (44)] | [Eq. (47)] | Bragg interference condition (Fig. 8) | | [Eq. (49)] | [Eq. (50)] |
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Table 3. Summary of spacetime slab results.
Modulation regime | Velocity () | Average index () | Average index () | Spatial | 0 | | | Subluminal upper limit | | | | Superluminal lower limit | | | | Temporal | | | |
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Table 4. Average refractive index for different modulation velocities, from (53), with (upper half) and Eq. (57), with (lower half).