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Here, we show that isorefractive spacetime crystals with a travelling-wave modulation may mimic rigorously the response of moving material systems. Unlike generic spacetime crystals, which are characterized by a bi-anisotropic coupling in the co-moving frame, isorefractive crystals exhibit an observer-independent response, resulting in isotropic constitutive relations devoid of any bianisotropy. We show how to take advantage of this property in the calculation of the band diagrams of isorefractive spacetime crystals in the laboratory frame and in the study of the synthetic Fresnel drag. Furthermore, we discuss the impact of considering either a Galilean or a Lorentz transformation in the homogenization of spacetime crystals, showing that the effective response is independent of the considered transformation.

We apply a velocity-field approach to investigate the interaction between spiral waves and the travelling wave modulation of system excitability which leads to a prediction: the direction of the straight-line drift of spiral waves is linearly adjusted by the propagation direction of the travelling waves. Direct numerical computations of the Oregonator model and the formulas of drift-velocity field confirm the validity and robustness of our theoretical prediction.

We introduce a homogenization approach to characterize the dynamical response of a generic dispersive spacetime crystal in the long-wavelength limit. The theory is applied to dispersive spacetime platforms with a travelling-wave modulation. It is shown that for long wavelengths the effective response may be described by a frequency dependent permittivity. Due to the active nature of spacetime systems, the permittivity is not bound by the same constraints as in standard time-invariant metamaterials. In particular, we find that dispersive spacetime crystals can exhibit rather peculiar physics, such as an anomalous (non-Foster) permittivity dispersion with a negative stored energy density, alternate between gain and loss regimes, and present multiple resonances in the quasistatic regime. Furthermore, it is verified with numerical simulations that the effective theory captures faithfully the exact dispersion of the first few photonic bands.

The control of reflection and refraction at interfaces using engineered media is central to numerous optical technologies, with negative refraction and the suppression of backscattering representing two prominent research frontiers. In this work, we demonstrate that an effective negative refraction accompanied by an absence of backscattering can be realized at a moving spatiotemporal interface when temporal and spatial reflections occur concurrently. While such spatiotemporal co-reflection is prohibited in one-dimensional linear dispersive media, we show that it becomes permissible under oblique incidence within a specific range of traveling-wave modulation velocities. Leveraging this mechanism, we propose a spatiotemporal flat lens capable of nonreciprocal electromagnetic wave focusing. These findings provide a framework for developing advanced spatiotemporal metamaterials and time-varying metasurfaces.

Studying the topology of spatiotemporal media poses a fundamental challenge: their remarkable properties stem from breaking spatial and temporal symmetries, yet this same breaking obscures their topological characterization. Here, we show that space-time symmetries persist in crystals with travelling-wave modulation, enabling the study of their topological properties and the prediction of spatiotemporal interface states. Using a Lorentz transformation to the frame comoving with the modulation, we identify a conserved joint parity-time-reversal symmetry in the new variables that enforces the quantization of the Zak phase, elevating it to a \\(Z_2\\) topological invariant. We then calculate the associated interface states and uncover unique features arising from time-varying effects, including selective directional excitation, propagation along moving boundaries, frequency-converted replicas, and broadband amplification even in the absence of momentum gaps.

Here, we show that isorefractive spacetime crystals with a travelling-wave modulation may mimic rigorously the response of moving material systems. While generic spacetime crystals are characterized by a bi-anisotropic coupling in the co-moving frame, isorefractive crystals have a response that is observer independent, which leads to isotropic constitutive relations free of bianisotropy. We show how to take advantage of this property in the calculation of the band diagrams of isorefractive spacetime crystals in the laboratory frame and in the study of the synthetic Fresnel drag. Furthermore, we discuss the impact of considering either a Galilean or a Lorentz transformation in the homogenization of spacetime crystals, showing that the effective response is independent of the considered transformation.

Here, we investigate the effective response of three-dimensional spacetime crystals formed by spherical scatterers under a travelling-wave modulation. We develop an analytical formalism to homogenize the spacetime crystals that extends the renowned Clausius-Mossotti formula to time-varying platforms. Our formalism shows that travelling-wave spacetime crystals can be used to engineer a wide range of classes of nonreciprocal bianisotropic couplings in the long wavelength limit. In particular, our theory reveals the possibility of realizing a purely isotropic Tellegen response in crystals formed by interlaced sub-lattices of scatterers subjected to different modulation velocities. Furthermore, we introduce a class of generalized Minkowskian crystals that displays invariance under arbitrary Lorentz boosts aligned with a fixed spatial direction. We prove that such systems are formed by pseudo-uniaxial materials with the principal axis aligned parallel to the modulation velocity. The electromagnetic response of such generalized Minkowskian crystals is indistinguishable from that of moving photonic crystals.

Time-variant systems have recently garnered considerable attention due to their unique potentials in manipulating electromagnetic waves. Here, a novel class of topological spacetime crystals is introduced, with a traveling-wave modulation that mimics certain aspects of physical motion. Challenging intuition, our findings reveal that, even though such systems rely on a linear momentum bias, it is feasible to engineer an internal angular momentum and non-trivial topological phases by leveraging the symmetry of its structural elements. Furthermore, these platforms exhibit a gauge degree of freedom associated with the arbitrariness in the choice of the coordinate transformation that eliminates the time dependence of the system Hamiltonian. The topology of the system is intricately governed by a synthetic magnetic potential whose field lines can be controlled by manipulating material anisotropy. Remarkably, the proposed spacetime crystals host an unconventional class of scattering-immune edge states, whose oscillation frequency adapts continuously along the propagation path, shaped by the geometric attributes of the path itself.

This work theoretically and experimentally studies metasurfaces with spatially-discrete, traveling-wave modulation (SD-TWM). A representative metasurface is considered consisting of columns of time-modulated subwavelength unit cells, referred to as stixels. SD-TWM is achieved by enforcing a time delay between temporal waveforms applied to adjacent columns. In contrast to the continuous traveling-wave modulation commonly assumed in studies of space-time metasurfaces, here the modulation is spatially discretized. In order to account for the discretized spatial modulation, a modified Floquet analysis is introduced based on a new boundary condition that has been derived for SD-TWM structures. The modified Floquet analysis separates the scattered field into its macroscopic and microscopic variations. The reported theoretical and experimental results reveal that the electromagnetic behavior of an SD-TWM metasurface can be categorized into three regimes. For electrically-large spatial modulation periods, the microscopic field variation across each stixel can be neglected. In this regime, the space-time metasurface allows simultaneous frequency translation and angular deflection. When the spatial modulation period on the metasurface is electrically small, the microscopic variation results in new metasurface capabilities such as subharmonic mixing. When the spatial modulation period of the metasurface is wavelength-scale, the metasurface allows both subharmonic mixing and angular deflection to be achieved simultaneously. To verify our analysis, a dual-polarized, spatio-temporally modulated metasurface, is developed and measured at X-band frequencies.

Creating materials with time-variant properties is critical for breaking reciprocity that imposes fundamental limitations to wave propagation. However, it is challenging to realize efficient and ultrafast temporal modulation in a photonic system. Here, leveraging both spatial and temporal phase manipulation offered by an ultrathin nonlinear metasurface, we experimentally demonstrated nonreciprocal light reflection at wavelengths around 860 nm. The metasurface, with traveling-wave modulation upon nonlinear Kerr building blocks, creates spatial phase gradient and multi-terahertz temporal phase wobbling, which leads to unidirectional photonic transitions in both momentum and energy spaces. We observed completely asymmetric reflections in forward and backward light propagations within a sub-wavelength interaction length of 150 nm. Our approach pointed out a potential means for creating miniaturized and integratable nonreciprocal optical components.