Researching | Uniform-velocity spacetime crystals

Advanced Photonics, Volume. 1, Issue 5, 056002(2019)

Uniform-velocity spacetime crystals

Zoé-Lise Deck-Léger^1、*, Nima Chamanara¹, Maksim Skorobogatiy², Mário G. Silveirinha³, and Christophe Caloz¹

Author Affiliations

¹Polytechnique Montréal, Department of Electrical Engineering, Montréal, Quebec, Canada

²Polytechnique Montréal, Department of Engineering Physics, Montréal, Quebec, Canada

³Universidade de Lisboa - Instituto Superior Técnico and Instituto de Telecomunicações, Department of Electrical Engineering, Lisbon, Portugal

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$Representation of two canonical spacetime crystals. Here, the variable z is intended to represent the hyperspace, i.e., the three dimensions of space. The white and gray regions correspond to media with refractive indices ni and nj, respectively. (a) Double-period (pA and pB) structure, characterized by two velocities, υm1 and υm2, which may be interpreted as acceleration at the corner. (b) Single-period (p) structure, characterized by a unique and uniform velocity, υm.$

Fig. 1. Representation of two canonical spacetime crystals. Here, the variable $z$ is intended to represent the hyperspace, i.e., the three dimensions of space. The white and gray regions correspond to media with refractive indices $n_{i}$ and $n_{j}$ , respectively. (a) Double-period ( $p_{A}$ and $p_{B}$ ) structure, characterized by two velocities, $υ_{m 1}$ and $υ_{m 2}$ , which may be interpreted as acceleration at the corner. (b) Single-period ( $p$ ) structure, characterized by a unique and uniform velocity, $υ_{m}$ .

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$Scattering at a spacetime interface. The white and gray regions correspond to media i and j, with refractive indices ni and nj (ni<nj). (a) Sketch of a wave incident on a subluminal interface. (b) Same as (a) but for a superluminal interface. (c) Scattering from a subluminal (υm<c) interface in a spacetime diagram. The blue arrows represent wave trajectories (referring to a specific phase point of the waveform), and the black arrows represent scattering coefficients. The dashed lines correspond to incidence from the right. The laboratory and moving frames are superimposed with common origin, and the moving frame has the same velocity as the interface (υf/c=υm/c). (d) Same as (c) but for a superluminal interface (υm>c), with moving frame having the inverse velocity of the interface (υf/c=c/υm).$

Fig. 2. Scattering at a spacetime interface. The white and gray regions correspond to media $i$ and $j$ , with refractive indices $n_{i}$ and $n_{j}$ ( $n_{i} < n_{j}$ ). (a) Sketch of a wave incident on a subluminal interface. (b) Same as (a) but for a superluminal interface. (c) Scattering from a subluminal ( $υ_{m} < c$ ) 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 ( $υ_{f} / c = υ_{m} / c$ ). (d) Same as (c) but for a superluminal interface ( $υ_{m} > c$ ), with moving frame having the inverse velocity of the interface ( $υ_{f} / c = c / υ_{m}$ ).

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Fig. 3. Spacetime-inversion symmetry of subluminal (SUB) and superluminal (SUP) structures. (a) Interfaces. (b) Slabs.

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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.

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Fig. 5. Generalization of the Stokes principle. (a) Subluminal regime. (b) Superluminal regime.

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Fig. 6. Frequency transitions at a spacetime interface corresponding to Fig. 2. (a) Subluminal case. (b) Superluminal case.

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Fig. 7. Multiple-reflection description of the scattering phenomenology in spacetime slabs. Changes in line type (solid $\leftrightarrow$ dashed) denote phase reversals. (a) Subluminal slab, with phase change occurring upon reflection to a lower impedance medium ( $η_{j} < η_{i}$ ), 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 ( $η_{k} > η_{j}$ ), according to Eqs. (22).

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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 $\leftrightarrow$ 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.

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Fig. 9. Bilayer spacetime crystal with spacetime unit cell and out-of-gap wave trajectories. (a) Subluminal equal-length crystal, with $ℓ_{1} = ℓ_{2}$ . The slopes of the triangles ① and ② are $n_{1} = c t_{1}^{+} / z_{1}^{+}$ and $n_{2} = c t_{2}^{+} / z_{2}^{+}$ , so $z_{1}^{+} = c t_{1}^{+} / n_{1}$ and $z_{2} = c t_{2}^{+} / n_{2}$ . Substituting these lengths into the expression for the slope $n_{av} = (c t_{1}^{+} + c t_{2}^{+}) / (z_{1}^{+} + z_{2}^{+})$ , yields Eq. (51). (b) Superluminal equal-duration crystal, with $d_{1} = d_{2}$ . From the slopes of the triangles ① and ②, given in (a), we have $c t_{1}^{+} = z_{1}^{+} n_{1}$ and $c t_{2} = z_{2}^{+} n_{2}$ . Substituting these durations into the expression for the average slope, also given in (a), yields Eq. (55).

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Fig. 10. Linear approximation of the dispersion diagram of bilayer crystals with $n_{2} / n_{1} = 1.5$ and equal electrical lengths. (a) Subluminal case, with $υ = (1 / 3) c$ . (b) Superluminal case, with $υ = 2 c$ .

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Fig. 11. Bilayer spacetime crystal with spacetime unit cell. (a) Subluminal regime. (b) Superluminal regime.

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Fig. 12. Dispersion diagram of bilayer crystals with $n_{2} / n_{1} = 1.5$ and equal phases ( ${\bar{φ}}_{j} = {\bar{φ}}_{k}$ ). 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 $υ = (1 / 3) c$ . (b) Superluminal case, with $υ = 3 c$ .

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Fig. 13. Examples of spacetime crystal truncation by a pair of spactime interfaces of velocities $υ_{L}$ and $υ_{R}$ for the left and right interfaces, respectively. Top row: the two interfaces have the same velocity, $υ_{L} = υ_{R}$ . Bottom row: the two interfaces have different velocities, $υ_{L} \neq υ_{R}$ . (a) Purely spatial truncation, $υ_{L} = υ_{R} = 0$ . (b) Truncation with velocity different from the modulation velocity, $υ_{L} = υ_{R} \neq υ_{m}$ . (c) Comoving truncation, $υ_{L} = υ_{R} = υ_{m}$ . (d) Antiparallel truncation, $υ_{L} \neq υ_{R} \neq υ_{m}$ . (e) Spacetime cavity with piecewise constant velocities. (f) Spacetime cavity with continuously varying velocity.

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$Scattering from two canonical truncated spacetime crystals. In both cases, the crystal is subluminal, and the medium surrounding it is a simple nondispersive dielectric medium of refractive index n0. Top row: dispersion diagrams with transition frequencies. Bottom row: spacetime diagrams with scattered waves, with labels corresponding the solutions of the top panels. Note that the drawn scattered waves correspond to the waves seen in the laboratory frame and would completely different in the moving frame. (a) Comoving truncation. A moving-frame observer would see a stationary crystal bounded by stationary interfaces, and hence measure a unique frequency everywhere, inside and outside the crystal. The arrows indicate the up and down frequency and wavenumber transitions at the two interfaces. (b) Purely spatial truncation. A moving-frame observer would see a stationary crystal bounded by moving interfaces, and hence measure an infinity of frequencies.$

Fig. 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 $n_{0}$ . 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.

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Fig. 15. Transmission and reflection coefficients for an $N = 15$ -layer crystal [corresponding to Fig. 13(c) and Fig. 14(a)] with ${\bar{ϕ}}_{1,2} = π / 2$ . The gap centers $ω_{i}$ , $k_{i}$ are provided in Appendix E (Sec. 12). (a) Subluminal case, with $υ = 1 / 3 c$ and $n_{2} / n_{1} = 2$ , with attenuation in the bandgaps. (b) Superluminal case, with $υ = 3 c$ and $n_{2} / n_{1} = 1.2$ with amplification in the bandgaps.

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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 $c t^{'}$ axis and the media trajectories are parallel to the $c t$ axis. (a) Laboratory frame viewpoint. Media appear at rest. Wave velocity is independent of direction: $υ_{j}^{+} = υ_{j}^{-}$ (b) Moving frame viewpoint. Media appear to be moving in $- z$ direction. Wave velocities are direction dependent: $υ_{j}^{+} \neq υ_{j}^{-}$ , with $| υ_{j}^{-} | > | υ_{j}^{+} |$ .

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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).

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Fig. 18. Successive application of time reversal ( $T$ ) and space reversal ( $P$ ).

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Fig. 19. Construction to find the frequencies aligned with the bandgaps. (a) Subluminal regime. (b) Superluminal regime.

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Table 1. Duality transformations between the subluminal and superluminal regimes.

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Table 1. Duality transformations between the subluminal and superluminal regimes.


Subluminal regime	Superluminal regime
Space	$z$	Time	$c t$
Time	$c t$	Space	$z$
Modulation velocity	$v_{m} / c$	Inverse velocity	$c / v_{m}$
Refractive index	$n$	Inverse index	$1 / n$
Phase velocity	$v / c = 1 / n$	Inverse phase velocity	$c / v = n$
Length	$ℓ$	Duration	$c d$
Wavelength	$λ$	Period	$c T$
Frequency	$ω / c$	Wavenumber	$k$

Table 2. Summary of the scattering formulas, derived in Sec. 3, for a spacetime interface.

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Table 2. Summary of the scattering formulas, derived in Sec. 3, for a spacetime interface.


Subluminal	Superluminal
Coefficients (Fig. 2)
$τ_{j i} = \frac{2 η_{j}}{η_{i} + η_{j}} (\frac{1 - α_{i}}{1 - α_{j}})$ [Eq. (18b)]	$ξ_{j i} = \frac{η_{i} + η_{j}}{2 η_{i}} (\frac{1 - α_{i}}{1 - α_{j}})$ [Eq. (22b)]
$γ_{i j i} = \frac{η_{j} - η_{i}}{η_{i} + η_{j}} (\frac{1 - α_{i}}{1 + α_{i}})$ [Eq. (18a)]	$ζ_{j i} = \frac{η_{i} - η_{j}}{2 η_{i}} (\frac{1 - α_{i}}{1 + α_{j}})$ [Eq. (22a)]
Frequencies (Fig. 6)
$ω_{i}^{-} = ω_{i}^{+} \frac{1 - α_{i}}{1 + α_{i}}$ [Eq. (26b)]	$k_{j}^{-} = k_{i}^{+} \frac{1 - 1 / α_{j}}{1 + 1 / α_{i}}$ [Eq. (28b)]
$ω_{j}^{+} = ω_{i}^{+} \frac{1 - α_{i}}{1 - α_{j}}$ [Eq. (26b)]	$k_{j}^{+} = k_{i}^{+} \frac{1 - 1 / α_{i}}{1 - 1 / α_{i}}$ [Eq. (28b)]
$α_{i, j} = v_{m} / v_{i, j}$

Table 3. Summary of spacetime slab results.

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Table 3. Summary of spacetime slab results.


Subluminal regime	Superluminal regime
Phase (Fig. 7)
$φ_{j}^{\pm'} = {\bar{φ}}_{j}^{'} \pm Δ φ_{j}^{'}$ [Eq. (35a)]
${\bar{φ}}_{j} = ω_{e} ℓ_{j} \frac{v_{j}}{v_{j}^{2} - v_{m}^{2}}$ [Eq. (38b)]	${\bar{φ}}_{j} = k_{e} d_{j} \frac{v_{j} v_{m}^{2}}{v_{j}^{2} - v_{m}^{2}}$ [Eq. (41b)]
$Δ φ_{j} = \frac{{\bar{φ}}_{j} γ^{2} (1 - v_{j}^{2} / c^{2}) v_{m}}{v_{j}}$ [Eq. (38c)]	$Δ φ_{j} = \frac{{\bar{φ}}_{j} γ^{2} (1 - v_{j}^{2} / c^{2}) c^{2}}{v_{j} v_{m}}$ [Eq. (41c)]
Coefficients (Fig. 7)
$Γ_{i k i} = γ_{i j i} \frac{1 - e^{2 i {\bar{φ}}_{j}}}{1 - {\bar{γ}}_{j i j} γ_{j k j} e^{2 i {\bar{φ}}_{j}}}$ [Eq. (42)]	$Z_{k i} = ζ_{k j} ξ_{j i} e^{- i φ_{j}^{-}} (e^{i {\bar{φ}}_{j}} - 1)$ [Eq. (47)]
$T_{k i} = \frac{τ_{k j} τ_{j i} e^{i φ_{j}^{+}}}{1 - {\bar{γ}}_{j i j} γ_{j k j} e^{2 i {\bar{φ}}_{j}}}$ [Eq. (44)]	$Ξ_{k i} = [1 + ξ_{k j} ξ_{j i} (e^{i {\bar{φ}}_{j}} - 1)] e^{- i φ_{j}^{-}}$ [Eq. (47)]
Bragg interference condition (Fig. 8)
$ℓ_{j} = \frac{λ_{j}^{+}}{4} (1 + \frac{v_{m}}{v_{j}})$ [Eq. (49)]	$d_{j} = \frac{T_{j}^{+}}{4} (1 + \frac{v_{j}}{v_{m}})$ [Eq. (50)]

Table 4. Average refractive index for different modulation velocities, from (53), with $ℓ_{1} = ℓ_{2}$ (upper half) and Eq. (57), with $d_{1} = d_{2}$ (lower half).

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Table 4. Average refractive index for different modulation velocities, from (53), with $ℓ_{1} = ℓ_{2}$ (upper half) and Eq. (57), with $d_{1} = d_{2}$ (lower half).


Modulation regime	Velocity ( $v_{m}$ )	Average index ( $n_{av}^{+}$ )	Average index ( $n_{av}^{-}$ )
Spatial	0	$\frac{1}{2} (n_{1} + n_{2})$	$\frac{1}{2} (n_{1} + n_{2})$
Subluminal upper limit	$v_{2} = \frac{c}{n_{2}}$	$n_{2}$	$n_{2} \frac{3 n_{1} + n_{2}}{n_{1} + 3 n_{2}}$
Superluminal lower limit	$v_{1} = \frac{c}{n_{1}}$	$n_{1}$	$n_{1} \frac{3 n_{2} + n_{1}}{n_{2} + 3 n_{1}}$
Temporal	$\infty$	$\frac{1}{2} {(\frac{1}{n_{1}} + \frac{1}{n_{2}})}^{- 1}$	$\frac{1}{2} {(\frac{1}{n_{1}} + \frac{1}{n_{2}})}^{- 1}$

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Zoé-Lise Deck-Léger, Nima Chamanara, Maksim Skorobogatiy, Mário G. Silveirinha, Christophe Caloz, "Uniform-velocity spacetime crystals," Adv. Photon. 1, 056002 (2019)

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Paper Information

Category: Research Articles

Received: Aug. 12, 2019

Accepted: Oct. 8, 2019

Published Online: Nov. 1, 2019

The Author Email: Deck-Léger Zoé-Lise (zoe-lise.deck-leger@polymtl.ca)

DOI:10.1117/1.AP.1.5.056002

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