A moving medium drags light along with it as measured by Fizeau and explained by Einstein's theory of special relativity. Here we show that the same effect can be obtained in a situation where there is no physical motion of the medium. Modulations of both the permittivity and permeability, phased in space and time in the form of travelling waves, are the basis of our model. Space-time metamaterials are represented by effective bianisotropic parameters, which can in turn be mapped to a moving homogeneous medium. Hence these metamaterials mimic a relativistic effect without the need for any actual material motion. We discuss how both the permittivity and permeability need to be modulated in order to achieve these effects, and we present an equivalent transmission line model.

Time-varying media have recently emerged as a new paradigm for wave manipulation, due to the synergy between the discovery of highly nonlinear materials, such as epsilon-near-zero materials, and the quest for wave applications, such as magnet-free nonreciprocity, multimode light shaping, and ultrafast switching. In this review, we provide a comprehensive discussion of the recent progress achieved with photonic metamaterials whose properties stem from their modulation in time. We review the basic concepts underpinning temporal switching and its relation with spatial scattering and deploy the resulting insight to review photonic time-crystals and their emergent research avenues, such as topological and non-Hermitian physics. We then extend our discussion to account for spatiotemporal modulation and its applications to nonreciprocity, synthetic motion, giant anisotropy, amplification, and many other effects. Finally, we conclude with a review of the most attractive experimental avenues recently demonstrated and provide a few perspectives on emerging trends for future implementations of time-modulation in photonics.

Abstract:In advanced field theories there can be more than four dimensions to space, the excess dimensions described as compacted and unobservable on everyday length scales. We report a simple model, unconnected to field theory, for a compacted dimension realised in a metallic metasurface periodically structured in the form of a grating comprising a series of singularities. An extra dimension of the grating is hidden, and the surface plasmon excitations, though localised at the surface, are characterised by three wave vectors rather than the two of typical two-dimensional metal grating. We propose an experimental realisation in a doped graphene layer. One Sentence Summary: Plasmonic excitations of a singular metallic grating serve as a model for compacted dimensions.Main Text: A conventional two dimensional object is characterised by two quantum numbers. For example the frequencies of surface plasmons on a periodic surface are labelled by the components of their momentum projected onto the surface axes. We describe theoretically systems that instead require three quantum numbers to label them: the two conventional in-plane momenta plus a third momentum corresponding to a compacted dimension hidden from view inside a singularity. Compacted dimensions are ingredients of advanced string theories (1,2) where the extra dimensions in a 4+N dimensional theory are said to be compacted and so not directly observed on everyday length scales. As far as we know our singular surfaces are the only physically realisable model of this curious effect. We give two instances of how this might be done.We make use of the technique of transformation optics (3-5) which exploits the invariance of Maxwell's equations under a coordinate transformation: only the values of ε, µ are affected by the transformation. We use this theory to compact a dimension through a singular transformation that compresses one of the dimensions of a 3D system into one or more singular points. An example of the process is given (Fig. 1) for a 3D system (Fig. 1A), periodic in one of the dimensions and translationally invariant in the two other directions. The blue shaded areas are metallic and support surface plasmons (6) whose spectrum is characterised by three wave vectors: k x , k y , k u where k u is the wave vector heading out of the plane of the paper.Our intent is to show that the x dimension can be hidden using 2D conformal transformations where the x, y coordinates are represented by a complex number z = x + iy . Conformal transformations in 2D have the property of conserving the permittivity and permeability, ε, µ , in the plane of the transformation so that in this plane we are working with the same materials in all coordinate frames. Under each successive transformation the /page 2 spectral properties are preserved, and the modes once calculated in the initial frame can be found in the other frames through the properties of the transformation.In the first step we compress x = −∞ to a point at the origin,(1) which gives rise to Fig. 1B. This transfor...

Time has emerged as a new degree of freedom for metamaterials, promising new pathways in wave control. However, electromagnetism suffers from limitations in the modulation speed of material parameters. Here we argue that these limitations can be circumvented by introducing a travelingwave refractive index modulation, with the same phase velocity of the waves. We show how the concept of "luminal grating" can yield giant nonreciprocity, achieve efficient one-way amplification, pulse compression and frequency up-conversion, proposing a realistic implementation in double-layer graphene.Temporal control of light is a long-standing dream, which has recently demonstrated its potential to revolutionize optical and microwave technology, as well as our understanding of electromagnetic theory, overcoming the stringent constraint of energy conservation [1]. Along with the ability of time-dependent systems to violate electromagnetic reciprocity [2][3][4], realising photonic isolators and circulators [5][6][7][8], amplify signals [9], perform harmonic generation [10,11] and phase modulation [12], new concepts from topological [13][14][15] and non-Hermitian physics [16,17] are steadily permeating this field.However, current limitations to the possibility of significantly fast modulation in optics has constrained the concept of time-dependent electromagnetics to the radio frequency domain, where varactors can be used to modulate capacitance [18], and traveling-wave tubes are commonly used as (bulky) microwave amplifiers [19]. In the visible and near IR, optical nonlinearities have often been exploited to generate harmonics, and realize certain nonreciprocal effects [20]. However, nonlinearity is an inherently weak effect, and high field intensities are typically required.In this Letter, we challenge the need for high modulation frequencies, demonstrating that strong and broadband nonreciprocal response can be obtained by complementing the temporal periodic modulation of an electromagnetic medium with a spatial one, in such a way that the resulting traveling-wave modulation profile appears to drift uniformly at the speed of the wave. Exploiting acoustic plasmons in double-layer graphene (DLG), we show that unidirectional amplification and compression can be realistically accomplished in such luminal gratings, despite the intrinsic limitations in the modulation speed of graphene. Our results hold potential for efficient THz generation, loss-compensation and amplification of plasmons, overcoming the typical trade-off between plasmon confinement and loss.Bloch (Floquet) theory dictates that the wavevector (frequency) of a monochromatic wave propagating in a spatially (temporally) periodic medium can only Braggscatter onto a discrete set of harmonics, determined by the reciprocal lattice vectors. This still holds true when the modulation is of a travelling-wave type, whereby Bragg scattering couples Fourier modes which differ by a discrete value of both energy and momentum combined [2,7,[21][22][23][24]. As shown in Fig. 1 for a 1D s...

By exploiting singular spatial modulations of the graphene conductivity, we design a broadband, tunable THz absorber whose efficiency approaches the theoretical upper bound for a wide absorption band with a fractional bandwidth of 185%. Strong field enhancement is exhibited by the modes of this extended structure, which is able to excite a wealth of high-order surface plasmons, enabling deeply subwavelength focusing of incident THz radiation. Previous studies have shown that the conductivity can be modulated at GHz frequencies, which might lead to the development of efficient high-speed broadband switching by an atomically thin layer.

We present a general framework for the homogenization theory of space-time metamaterials. By mapping to a frame comoving with the space-time modulation, we derive analytical formulas for the effective material parameters for traveling-wave modulations in the low-frequency limit: electric permittivity, magnetic permeability, and magnetoelectric coupling. In doing so, we provide a recipe for the calculation of effective parameters of space-time-modulated media where the parameters follow a traveling-wave form of any shape and we show how synthetic motion can result in giant bianisotropy. This allows us to deepen the understanding of how nonreciprocity can be achieved in the long-wavelength limit and to completely characterize the different nonreciprocal behaviors available in space-time-modulated media. In particular, we show how the modulation speed, which can be subluminal or superluminal, together with the relative phase between electric and magnetic modulations, provide tuning knobs for the nonreciprocal response of these systems. Furthermore, we apply the theory to derive exact formulas for the Fresnel drag experienced by light traveling through traveling-wave modulations of electromagnetic media, providing insight into the differences and similarities between synthetic motion and moving matter. Since we exploit a series of Galilean coordinate transformations, the theory may be generalized to other kinds of waves.

Recent progress in nanophotonics and material science has inspired a strong interest in optically-induced material dynamics, opening new research directions in the distinct fields of Floquet matter and time metamaterials. Floquet phenomena are historically rooted in the condensed matter community, as they exploit periodic temporal drives to unveil novel phases of matter, unavailable in systems at equilibrium. In parallel, the field of metamaterials has been offering a platform for exotic wave phenomena based on tailored materials at the nanoscale, recently enhanced by incorporating time variations and switching as new degrees of freedom. In this Perspective, we connect these research areas and describe the exciting opportunities emerging from their synergy, hinging on giant wave-matter interactions enabled by metamaterials and on the exotic wave dynamics enabled by Floquet and parametric phenomena. We envision Floquet metamaterials in which nontrivial modulation dynamics, and their interplay with tailored material dispersion and nontrivial material properties such as anisotropy, non-Hermiticity and nonreciprocity, introduce a plethora of novel opportunities for wave manipulation and control.

Our ability to generate new distributions of light has been remarkably enhanced in recent years. At the most fundamental level, these light patterns are obtained by ingeniously combining different electromagnetic modes. Interestingly, the modal superposition occurs in the spatial, temporal as well as spatio-temporal domain. This generalized concept of structured light is being applied across the entire spectrum of optics: generating classical and quantum states of light, harnessing linear and nonlinear light-matter interactions, and advancing applications in microscopy, spectroscopy, holography, communication, and synchronization. This Roadmap highlights the common roots of these different techniques and thus establishes links between research areas that complement each other seamlessly. We provide an overview of all these areas, their backgrounds, current research, and future developments. We highlight the power of multimodal light manipulation and want to inspire new eclectic approaches in this vibrant research community.