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Abstract
In this opinion article, we briefly summarize some of the background materials and recent developments in the field of temporal and spatiotemporal media and provide our opinion on some of potential challenges, opportunities, and open research questions for manipulation of fields and waves in four dimensions.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
At the atomic (quantum) level, light-matter interaction is ultimately the result of polarizable atoms individually responding to quantized particles (photons) of local electromagnetic (EM) radiation with a given probability (expectation value). From a macroscopic (classical) deterministic perspective, this multipolar response—primarily dipolar response, since quadrupoles and higher order terms can be often neglected—can be collectively parameterized by bringing in the notion of electric and magnetic polarization vectors, which relate to the driving electric and/or magnetic field vectors through several relevant transfer functions (susceptibilities). In the most general linear case contemplating anisotropy and spacetime nonlocality and inhomogeneity, these susceptibilities become second-rank tensorial Fredholm-integral operators, $\overline{\overline \chi } \; ({{\boldsymbol r},\; t;\; {\boldsymbol r^{\prime}},\; t^{\prime}} )$ that are a function of both the prime (source) and unprimed (observation) spacetime coordinates. Examples of these operators are, for instance, the linear susceptibilities, which reduce to the real scalar constants ${\chi ^{(1 )}},\; $ the simplest possible scenario of homogeneous local isotropy. Alternatively, similar operators perform the mapping onto the domain of the electric and magnetic flux densities, giving rise to the ubiquitous concepts of linear permittivity (ε), permeability (µ), conductivity (σ), chirality ($\xi )$, nonlinear susceptibilities (χ(a), a = 2,3…), etc. [1,2]. In view of the above, the possibilities to manipulate light—within the bounds of Maxwellian electrodynamics— by tailoring these material parameters (operators) in the spacetime manifold appear, at least theoretically, limitless. The ability to control and engineer EM wave-matter interactions has been of interest within the scientific (academic and industrial) communities in many different fields including microwave engineering and photonics, and other non-EM fields such as acoustics and mechanics. When dealing with EM waves, one can manipulate wave-matter interactions by properly designing spatial inhomogeneities along the pathway where the wave propagates. Such inhomogeneities are then carefully crafted taking into account their macroscopic parameters mentioned above. This has been the fundamental pillar for the development and progress of the field of light-matter interactions allowing the creation of numerous devices and components ranging from lenses and antennas [3], to sensors [4], and photonic circuits [5], among others.
Such spatial manipulation of wave-matter interactions has been mastered by the field of metamaterials and metasurfaces [6,7], demonstrating how they can enable unconventional EM responses of media with extreme parameters of ε and µ including near zero or negative values [8–14]. Since its introduction several decades ago, the field of metamaterials and its predecessors, viz. complex media [15–17] and artificial dielectrics [18,19], has caught the attention of the scientific community and has evolved at an unprecedent speed. This has been possible due to the multidisciplinary nature of this field, which nowadays involves researchers from physics, engineering, material sciences, chemistry, mathematics, and computational science, just to name a few, allowing the implementation of these metamaterial platforms in applications such as lens-antennas [20–22], signal processing [23–25], invisibility cloaking [26–29], neural-network based designs [30,31] and analog computing [32–37].
In addition to controlling wave-matter interactions using artificial media via spatial inhomogeneities in three dimensions (x,y,z), another interesting paradigm that is becoming a hot topic is the ability to bring time (t) as an additional degree of freedom in material parameters, enabling the introduction of metamaterials in four dimensions (4D metamaterials x,y,z,t) [38,39]. The curiosity and interest in exploring wave propagation in spatiotemporally modulated media, or in general time-dependent EM platforms, can indeed be traced back to decades ago. Within the plasma physics community, the exploration of photon acceleration using flash ionization of plasmas via time-dependent media has been a hot topic [40]. Additionally, non-stationary radiofrequency (RF) elements in electrical signal modulation have been around for decades: a well-known example is the voltage-controlled oscillator utilized in frequency modulations. In a broader sense, linear time-varying circuits have attracted the attention of engineers since the 1950s (from parametric amplifiers and nonreciprocal devices [41] in microwave engineering, to variable switching networks for control engineering and relevant mathematical description tools—state-space theory [42] and functional analysis [43]—or the time-variance equivalence in nonlinear-circuit linearization [44]). In telecommunication engineering, a fundamental field of research is focused on EM radiation in environments where media vary in time such as, for instance, a cell phone while in motion where the temporal and spatial change of obstacles nearby needs to be considered: linear time-variance is thus a key ingredient in mobile communications [45,46], so the signal processing community has devoted much effort into its study [47]. More pertinent to electromagnetic interactions and matter, travelling-wave parametric amplifiers [48,49], phase modulators based on the Pockels effect [50] and on liquid crystals [51], piezo-optic transducers [52] and the electro-optic modulation of silicon by charge carriers [53] have been present for quite a long time.
Some of the pioneering works in the field of time-varying EM media can be attributed to Morgenthaler back in the 1950’s and Fante in the 1970’s with the study of wave propagation within a medium with a time modulated ε(t) as a single step function [54,55], i.e., wave propagation in an induced temporal boundary, as the temporal analogue of a spatial boundary between two media with different impedances. Other works on time-varying platforms in the 1960’s and 1970’s include the work of Cassedy and Oliner [56,57], Cassedy [58], Holberg and Kunz [59], Felsen and Whitman [60], and Yablonovitch [61], just to name a few. In recent years, these pioneering works have gained the attention of the scientific communities and many recent studies have been proposed considering time-varying properties of media. Examples include wave propagation in spatiotemporal dielectric media [62] and double-periodic (space-time periodic) microstructures [63], holography and time reversal with water waves [64], effective medium concepts in the time domain [65,66], non-Foster inspired temporal media [67] space-time gradient metasurfaces [68], temporal aiming and inverse prism [69,70], generalization of the Kramers-Kroning relations [71], Fresnel drag [72], antireflection temporal coatings [73,74], filters [75,76], temporal Brewster angle [77], temporal photonic crystals [78,79], exceptional points in time-varying media [80], loss compensation in time-modulated elastic metamaterials [81], non-reciprocity [82], energy accumulation [83], ultra-fast light polarization switching [84], frequency conversion by rapid ionization [85], temporal meta-atoms [86,87] and the study of the effects of dispersion in time-modulated media [88], to name a few. As an aside, spatiotemporally modulated (synthetically-moving) media [56,58] can effectively reproduce some of the relativistic effects [72]. In this regard, a modulated cloak has been explored to compensate for the Doppler shift that results from the relativistic motion of an object [89].
Parity-time (PT)-symmetry has also been playing a prominent role in many novel time-varying platforms [90–93]. Coherent perfect absorption lasing, phase transitions, and anisotropic transmission resonances have been demonstrated without the presence of balanced interactions between time-invariant material gain and loss, on the understanding that non-Hermitian exceptional points of degeneracy can be prompted by parametrically-driven time-Floquet modulations [94]. Strikingly, PT-symmetry has also been proved in Hermitian systems with slow aperiodic time-modulation [91], whereas in [92], the dual of spatial PT-symmetry was explored on the basis of non-Hermitian temporal slabs. Topological photonics is yet another hot research topic that nourishes from dynamic driving, with examples including topological nontrivial phases in photonic time crystals [95], a photonic Chern insulator in periodically-pump nonlinear crystals [96], or the nontrivial topological winding of a non-Hermitian energy band by modulation of a ring resonator [97]. More examples of intriguing physics include the experimental observation of the dynamic Casimir effect in a superconducting circuit of time-varying electrical length [98], and the identification of a new gain mechanism in time-dependent media based on the compression of lines of force rather than on parametric amplification [99].
The window of opportunities offered by the extra degree of freedom, time, seems overwhelmingly far-reaching. For instance, in [100] a shocklike modulation of the refractive index in a photonic crystal displayed new physics such as the capture of light at the shock wave front and tunable pulsed reemission with bandwidth narrowing; while in [101], an adiabatic modulation was shown to stop a light pulse with bandwidth compression and later recover it without distortion. A sudden switch in the properties of the propagating medium can also prove useful for all-linear time reversal, up to recently reliant on nonlinear phenomena like four-wave mixing: spectral inversion of a spin-wave packet was accomplished in [102] after a square pulse of magnetic bias spatially modulates the magnetic field in a magnonic crystal. Though not related to electromagnetics, this approach of abruptly “shocking” the properties of the material in order to achieve time reversal was experimented for water waves in [64], where a space-homogeneous step-like perturbation of the effective gravity realized an instantaneous time mirror described in terms of Cauchy sources. An entirely different conception based on a parametrically-modulated dielectric sheet [103] showed negative refraction for the far field (i.e., without the infinite resolution of Veselago lens [13,15]) and its link with time reversal.
By observing the studies above, one can notice how the field is rapidly evolving and thereby enabling scientific progression from both the application and fundamental points of view. This is indeed key for the field to become a new paradigm for arbitrary control of fields and waves, as potential applications can inform the needs in theoretical studies and vice versa. For instance, most of the current studies have been focused on wave propagation within unbounded time-modulated media or structures with some time-modulated components. However, one can envision the field can potentially evolve into time-modulated systems involving multiple, yet somehow, interconnected individual time-modulated devices. While this might potentially become a challenge in terms of its design and modelling, it could also offer exciting possibilities due to higher degrees of freedom. It is here when the multidisciplinary nature of the field of metamaterials/metasurfaces can come into play, as some of the knowledge from signal processing and wireless communications (as explained before) could potentially be implemented in time-varying EM metamaterials. Things get even more exciting (and challenging) in coalescence with space gradients, for example, a space-time crystal, e.g. under non-uniform motion or a superluminal synthetic modulation. This is just an example of the many exciting possibilities worth pursuing, and some recent works have already explored the answers to such questions [66,92].
Arguably one of the main limitations concerning time metastructures is related to the modulation speed of the materials. On this subject, it has been recently proposed that these limitations can be overcome by introducing a traveling-wave modulation drifting at the phase velocity of the wave [104]. In another front, transparent conducting oxides near their epsilon near-zero (ENZ) frequency have emerged over the last few years as a realistic venue to provide sub-picosecond strongly nonlinear responses allowing for all-optical switching and ultrafast frequency conversion [105–108]. As in the case of stationary (time-invariant) metamaterials, where 2D structures—metasurfaces—have proved more achievable than their 3D (volumetric) variants, here also the compactness and richness in wave-matter interactions offered by time-varying metasurfaces seems to be already translating into the preponderance of 2D (2 space) +1D (time) structures—or even 2D (2 space) +1D (time) +1D (synthetic dimension) if one extra synthetic dimension is included—over their volumetric (4D) spatiotemporal counterparts, and many more examples can already be found in the literature on the field of active metasurfaces [68,109–115]. The experimental demonstration of time-varying EM metastructures should involve a careful selection of such metastructures and operational spectral ranges. In this realm, as mentioned above, in recent years there has been an enormous amount of efforts worldwide in the field of tunable metasurfaces [116,117]. Such progress has only been possible thanks to the research both in the fabrication techniques and in the theoretical aspects behind the materials/structures that can be used to achieve tunability. In this context, while fabrication of time-modulated media is under development, one can envision that the field could leverage all the knowledge and investment in tunable metamaterials and metasurfaces [115].
In terms of applications, time-varying (4D) metamaterials could open further opportunities in fields such as computing and signal processing, filters, nonreciprocal devices, frequency conversion, signal amplification, and surface-wave generation. These applications could potentially find new opportunities in telecommunications systems, making them interesting from applied points of view. However, it is important to note that in some sense the field is currently in its early stages, and scientific progress requires both application and curiosity-driven research. In this context, one needs to consider that adding the extra degree of freedom of time for the manipulation of EM wave propagation indeed opens a vast space of opportunities and possibilities to explore. For instance, what would happen if we are able to manipulate both ε and µ in 4D and then consider either a multilayered media or metasurfaces with different unit cells such that ε(x,y,z,t) and µ(x,y,z,t) are modulated using different spatiotemporal functions (multiple steps, sinusoidally, etc.) in the same structure? One can also add another possibility, which is adding temporally anisotropy to both EM parameters: what exciting physical responses will exist in such scenarios? In terms of their design and optimization, could we explore artificial intelligence techniques, machine learning and neural networks in the design of 4D metamaterial? Would this be a potential solution in terms of future scalability of the field as 4D systems? Also, as mentioned above, in recent years we have seen a tremendous progress in topological systems [23,95,118], hence one could ask: would it be possible to combine topological systems and explore them using spatiotemporal media? What exotic EM phenomena may occur in those systems?
In conclusion, in our opinion 4D metamaterials and metasurfaces can open new paradigms for a full manipulation of wave-matter interactions. There are many challenges ahead of us, of fundamental theory and modeling nature, technological and of materials science. But not only do we envisage, in due time, real-life usage of many of the dynamic-modulation-induced unique effects discussed above; we are also optimistic about the additional exotic applications that will follow in the near future derived from already established knowledge. We anticipate that new physical paradigms are still to come. And let us not forget that the field of reconfigurable metastructures [119–121] is in some contexts already active enough to transform into suitable paradigms for time-varying media with many opportunities to explore, both from the theoretical and from the applied points of view.
Funding
Simons Foundation (733684); Newcastle University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. Collin, Foundations for Microwave Engineering, Second (John Wiley & Sons: Hoboken, 2001).
2. M. Born and E. Wolf, Principles Of Optics, 7th ed. (Cambridge University Press, 1999).
3. R. E. Colling and F. J. Zucker, Antenna Theory (McGraw-Hill, 1969).
4. M. F. S. Ferreira, E. Castro-Camus, D. J. Ottaway, J. M. López-Higuera, X. Feng, W. Jin, Y. Jeong, N. Picqué, L. Tong, B. M. Reinhard, P. M. Pellegrino, A. Méndez, M. Diem, F. Vollmer, and Q. Quan, “Roadmap on optical sensors,” J. Opt. 19(8), 083001 (2017). [CrossRef]
5. A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 X 4-port universal linear circuit,” Optica 3(12), 1348 (2016). [CrossRef]
6. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (2006).
7. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University Press, 2009).
8. A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
9. V. Pacheco-Peña, V. Torres, B. Orazbayev, M. Navarro-cía, M. Beruete, and N. Engheta, “Advances in ε -Near-Zero Metamaterial Devices,” Opt. Photonics News 094009(December), 243503 (2015).
10. I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017). [CrossRef]
11. A. Ciattoni, C. Rizza, A. Marini, A. Di Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser & Photonics Reviews 10(3), 517–525 (2016). [CrossRef]
12. P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7(10), 791–795 (2013). [CrossRef]
13. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef]
14. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(October), 157403-1 (2006).
15. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
16. N. Engheta (Guest Editor), “Special Issue of “Journal of Electromagnetic Waves and Applications (JEWA)” on the topic of “Wave Interaction with Chiral and Complex Media,”“ 6(No. 5/6 May and June), (1992).
17. A. H. Sihvola, A. J. Viitanen, I. V. Lindell, and S. A. Tretyakov, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House Antenna Library, 1994).
18. W. Rotman, “Plasma simulation by artificial dielectrics and Parallel-plate media,” IRE Trans. Antennas Propag. 10(1), 82–95 (1962). [CrossRef]
19. W. E. Kock, “Metallic Delay Lenses,” Bell Syst. Tech. J. 27(1), 58–82 (1948). [CrossRef]
20. E. Lier, D. H. Werner, C. P. Scarborough, Q. Wu, and J. A. Bossard, “An octave-bandwidth negligible-loss radiofrequency metamaterial,” Nat. Mater. 10(3), 216–222 (2011). [CrossRef]
21. V. Pacheco-Peña, V. Torres, B. Orazbayev, M. Beruete, M. Navarro-Cía, M. Sorolla, and N. Engheta, “Mechanical 144 GHz beam steering with all-metallic epsilon-near-zero lens antenna,” Appl. Phys. Lett. 105(24), 243503 (2014). [CrossRef]
22. Z. Wei, Y. Cao, Z. Gong, X. Su, Y. Fan, C. Wu, J. Zhang, and H. Li, “Subwavelength imaging with a fishnet flat lens,” Phys. Rev. B 88(19), 195123 (2013). [CrossRef]
23. F. Zangeneh-Nejad and R. Fleury, “Topological analog signal processing,” Nat. Commun. 10(1), 2058 (2019). [CrossRef]
24. H. Kwon, A. Cordaro, D. Sounas, A. Polman, and A. Alù, “Dual-Polarization Analog 2D Image Processing with Nonlocal Metasurfaces,” ACS Photonics 7(7), 1799–1805 (2020). [CrossRef]
25. H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal Metasurfaces for Optical Signal Processing,” Phys. Rev. Lett. 121(17), 173004 (2018). [CrossRef]
26. H. Li, M. Rosendo-López, Y. Zhu, X. Fan, D. Torrent, B. Liang, J. Cheng, and J. Christensen, “Ultrathin Acoustic Parity-Time Symmetric Metasurface Cloak,” Research 2019, 1–7 (2019).
27. L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Three-dimensional broadband omnidirectional acoustic ground cloak,” Nat. Mater. 13(4), 352–355 (2014). [CrossRef]
28. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005). [CrossRef]
29. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]
30. X. Han, Z. Fan, Z. Liu, C. Li, and L. J. Guo, “Inverse design of metasurface optical filters using deep neural network with high degrees of freedom,” InfoMat 3, 432–442 (2021). [CrossRef]
31. C. C. Nadell, B. Huang, J. M. Malof, and W. J. Padilla, “Deep learning for accelerated all-dielectric metasurface design,” Opt. Express 27(20), 27523 (2019). [CrossRef]
32. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing Mathematical Operations with Metamaterials,” Science 343(6167), 160–163 (2014). [CrossRef]
33. F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20(7), 073001 (2018). [CrossRef]
34. N. Mohammadi Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363(6433), 1333–1338 (2019). [CrossRef]
35. A. Yakovlev and V. Pacheco-Peña, “Enabling high-speed computing with electromagnetic pulse switching,” Adv. Mater. Technol. 5(12), 2000796 (2020). [CrossRef]
36. R. G. MacDonald, A. Yakovlev, and V. Pacheco-Peña, “Amplitude controlled electromagnetic pulse switching using waveguide junctions for high-speed computing processes,” arXiv: 2205.04814 (2022).
37. A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-Index Dielectric Metasurfaces Performing Mathematical Operations,” Nano Lett. 19(12), 8418–8423 (2019). [CrossRef]
38. C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials — part i: general concepts,” IEEE Trans. Antennas Propag. 68(3), 1569–1582 (2020). [CrossRef]
39. C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials—part ii: theory and applications,” IEEE Trans. Antennas Propag. 68(3), 1583–1598 (2020). [CrossRef]
40. M. R. Shcherbakov, K. Werner, Z. Fan, N. Talisa, E. Chowdhury, and G. Shvets, “Photon acceleration and tunable broadband harmonics generation in nonlinear time-dependent metasurfaces,” Nat. Commun. 10(1), 1345 (2019). [CrossRef]
41. A. K. Kamal, “A parametric device as a nonreciprocal element,” Proc. IRE 48(8), 1424–1430 (1960). [CrossRef]
42. L. A. Zadeh and C. A. Desoer, “Linear System Theory: The State-Space Approach,” (1963).
43. L. A. Zadeh, “Frequency Analysis of Variable Networks,” Proc. IRE 38(3), 291–299 (1950). [CrossRef]
44. E. Verriest and T. Kailath, “On generalized balanced realizations,” IEEE Trans. Autom. Control 28(8), 833–844 (1983). [CrossRef]
45. G. Matz and F. Hlawatsch, “Fundamentals of Time-Varying Communication Channels,” in Wireless Communications Over Rapidly Time-Varying Channels (Elsevier, 2011), pp. 1–63.
46. G. Matz and F. Hlawatsch, “Time-varying communication channels: Fundamentals, recent developments, and open problems,” in 14th European Signal Processing Conference (IEEE, 2006), (Eusipco), pp. 1–5.
47. E. J. Baghdady, Lectures on Communication System Theory (McGraw-Hill, 1961).
48. A. L. Cullen, “A Travelling-Wave Parametric Amplifier,” Nature 181(4605), 332 (1958). [CrossRef]
49. M. R. Currie and R. W. Gould, “Coupled-Cavity Traveling-Wave Parametric Amplifiers: Part I-Analysis,” Proc. IRE 48(12), 1960–1973 (1960). [CrossRef]
50. H. X. Li Mo and Tang, “Strong Pockels materials,” Nat. Mater. 18(1), 9–11 (2019). [CrossRef]
51. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15(6), 326–328 (1990). [CrossRef]
52. P. Etchegoin, J. Kircher, M. Cardona, C. Grein, and E. Bustarret, “Piezo-optics of GaAs,” Phys. Rev. B 46(23), 15139–15149 (1992). [CrossRef]
53. R. Soref and B. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
54. F. Morgenthaler, “Velocity Modulation of Electromagnetic Waves,” IEEE Trans. Microwave Theory Tech. 6(2), 167–172 (1958). [CrossRef]
55. R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19(3), 417–424 (1971). [CrossRef]
56. E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I—Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963). [CrossRef]
57. E. S. Cassedy, “Dispersion relations in time-space periodic media part II—Unstable interactions,” Proc. IEEE 55(7), 1154–1168 (1967). [CrossRef]
58. E. S. Cassedy, “Waves guided by a boundary with time-space periodic modulation,” Proc. Inst. Electr. Eng. 112(2), 269–279(10) (1965). [CrossRef]
59. D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14(2), 183–194 (1966). [CrossRef]
60. L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18(2), 242–253 (1970). [CrossRef]
61. E. Yablonovitch, “Self-Phase Modulation of Light in a Laser-Breakdown Plasma,” Phys. Rev. Lett. 32(20), 1101–1104 (1974). [CrossRef]
62. F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, “Dynamics of light propagation in spatiotemporal dielectric structures,” Phys. Rev. E 75(4), 046607 (2007). [CrossRef]
63. K. A. Lurie and S. L. Weekes, “Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure,” Journal of Mathematical Analysis and Applications 314(1), 286–310 (2006). [CrossRef]
64. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12(10), 972–977 (2016). [CrossRef]
65. V. Pacheco-Peña and N. Engheta, “Effective medium concept in temporal metamaterials,” Nanophotonics 9(2), 379–391 (2020). [CrossRef]
66. P. A. Huidobro, M. G. Silveirinha, E. Galiffi, and J. B. Pendry, “Homogenization Theory of Space-Time Metamaterials,” Phys. Rev. Appl. 16(1), 014044 (2021). [CrossRef]
67. V. Pacheco-Peña, Y. Kiasat, B. Edwards, and N. Engheta, “Salient Features of Temporal and Spatio-Temporal Metamaterials,” in 2018 International Conference on Electromagnetics in Advanced Applications (ICEAA) (IEEE, 2018), pp. 524–526.
68. Y. Hadad, D. L. Sounas, and A. Alu, “Space-time gradient metasurfaces,” Phys. Rev. B 92(10), 100304 (2015). [CrossRef]
69. V. Pacheco-Peña and N. Engheta, “Temporal aiming,” Light: Sci. Appl. 9(1), 129 (2020). [CrossRef]
70. A. Akbarzadeh, N. Chamanara, and C. Caloz, “Inverse prism based on temporal discontinuity and spatial dispersion,” Opt. Lett. 43(14), 3297 (2018). [CrossRef]
71. D. M. Solís and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: A generalization of the Kramers-Kronig relations,” Phys. Rev. B 103(14), 144303 (2021). [CrossRef]
72. P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, and J. B. Pendry, “Fresnel drag in space–time-modulated metamaterials,” Proc. Natl. Acad. Sci. 116(50), 24943–24948 (2019). [CrossRef]
73. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7(4), 323 (2020). [CrossRef]
74. Y. Zhang, L. Peng, J. Wang, and D. Ye, “Ultra-wideband Antireflection Temporal Medium,” arXiv:2108.05178 (2021).
75. D. Ramaccia, A. Alù, A. Toscano, and F. Bilotti, “Temporal multilayer structures for designing higher-order transfer functions using time-varying metamaterials,” Appl. Phys. Lett. 118(10), 101901 (2021). [CrossRef]
76. G. Castaldi, V. Pacheco-Peña, M. Moccia, N. Engheta, and V. Galdi, “Exploiting space-time duality in the synthesis of impedance transformers via temporal metamaterials,” Nanophotonics 10(14), 3687–3699 (2021). [CrossRef]
77. V. Pacheco-Peña and N. Engheta, “Temporal equivalent of the Brewster angle,” Phys. Rev. B 104(21), 214308 (2021). [CrossRef]
78. J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93(6), 063813 (2016). [CrossRef]
79. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function (t),” Physical Review A - Atomic, Molecular, and Optical Physics 79(5), 1–13 (2009).
80. T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66(10), 5300–5307 (2018). [CrossRef]
81. D. Torrent, W. J. Parnell, and A. N. Norris, “Loss compensation in time-dependent elastic metamaterials,” Phys. Rev. B 97(1), 014105 (2018). [CrossRef]
82. X. Guo, Y. Ding, Y. Duan, and X. Ni, “Nonreciprocal metasurface with space–time phase modulation,” Light: Sci. Appl. 8(1), 123 (2019). [CrossRef]
83. M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, “Time-Varying Reactive Elements for Extreme Accumulation of Electromagnetic Energy,” Phys. Rev. Appl. 11(1), 014024 (2019). [CrossRef]
84. L. H. Nicholls, F. J. Rodríguez-Fortuño, M. E. Nasir, R. M. Córdova-Castro, N. Olivier, G. A. Wurtz, and A. V. Zayats, “Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials,” Nat. Photonics 11(10), 628–633 (2017). [CrossRef]
85. A. Nishida, N. Yugami, T. Higashiguchi, T. Otsuka, F. Suzuki, M. Nakata, Y. Sentoku, and R. Kodama, “Experimental observation of frequency up-conversion by flash ionization,” Appl. Phys. Lett. 101(16), 161118 (2012). [CrossRef]
86. G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1(2), 023014 (2019). [CrossRef]
87. V. Pacheco-Peña and N. Engheta, “Spatiotemporal Isotropic-to-Anisotropic Meta-Atoms,” New J. Phys. 23(9), 095006 (2021). [CrossRef]
88. D. M. Solís, R. Kastner, and N. Engheta, “Time-varying materials in the presence of dispersion: plane-wave propagation in a Lorentzian medium with temporal discontinuity,” Photonics Res. 9(9), 1842 (2021). [CrossRef]
89. D. Ramaccia, D. L. Sounas, A. Alù, A. Toscano, and F. Bilotti, “Doppler cloak restores invisibility to objects in relativistic motion,” Phys. Rev. B 95(7), 075113 (2017). [CrossRef]
90. V. Pacheco-Peña and N. Engheta, “Temporal metamaterials with gain and loss,” arXiv:2108.01007 (2021).
91. H. Li, H. Moussa, D. Sounas, and A. Alù, “Parity-time Symmetry Based on Time Modulation,” Phys. Rev. Appl. 14(3), 031002 (2020). [CrossRef]
92. Z. Hayran, A. Chen, and F. Monticone, “Spectral causality and the scattering of waves,” Optica 8(8), 1040 (2021). [CrossRef]
93. H. Li, S. Yin, E. Galiffi, and A. Alù, “Temporal Parity-Time Symmetry for Extreme Energy Transformations,” Phys. Rev. Lett. 127(15), 153903 (2021). [CrossRef]
94. T. T. Koutserimpas and R. Fleury, “Nonreciprocal Gain in Non-Hermitian Time-Floquet Systems,” Phys. Rev. Lett. 120(8), 087401 (2018). [CrossRef]
95. E. Lustig, Y. Sharabi, and M. Segev, “Topological aspects of photonic time crystals,” Optica 5(11), 1390 (2018). [CrossRef]
96. L. He, Z. Addison, J. Jin, E. J. Mele, S. G. Johnson, and B. Zhen, “Floquet Chern insulators of light,” Nat. Commun. 10(1), 4194 (2019). [CrossRef]
97. K. Wang, A. Dutt, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, “Generating arbitrary topological windings of a non-Hermitian band,” Science 371(6535), 1240–1245 (2021). [CrossRef]
98. G. and P, A. and S, M. and J, J. R. and D, T. and N, F. and D, P. Wilson C, and M. and Johansson, “Observation of the dynamical Casimir effect in a superconducting circuit,” Nature 479(7373), 376–379 (2011). [CrossRef]
99. J. B. Pendry, E. Galiffi, and P. A. Huidobro, “Gain mechanism in time-dependent media,” Optica 8(5), 636–637 (2021). [CrossRef]
100. E. J. Reed, M. Soljačić, and J. D. Joannopoulos, “Color of Shock Waves in Photonic Crystals,” Phys. Rev. Lett. 90(20), 203904 (2003). [CrossRef]
101. M. F. Yanik and S. Fan, “Stopping Light All Optically,” Phys. Rev. Lett. 92(8), 083901 (2004). [CrossRef]
102. V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, “All-linear time reversal by a dynamic artificial crystal,” Nat. Commun. 1(1), 141 (2010). [CrossRef]
103. J. B. Pendry, “Time Reversal and Negative Refraction,” Science 322(5898), 71–73 (2008). [CrossRef]
104. E. Galiffi, P. A. Huidobro, and J. B. Pendry, “Broadband Nonreciprocal Amplification in Luminal Metamaterials,” Phys. Rev. Lett. 123(20), 206101 (2019). [CrossRef]
105. J. B. Khurgin, M. Clerici, V. Bruno, L. Caspani, C. DeVault, J. Kim, A. Shaltout, A. Boltasseva, V. M. Shalaev, M. Ferrera, D. Faccio, and N. Kinsey, “Adiabatic frequency shifting in epsilon-near-zero materials: the role of group velocity,” Optica 7(3), 226–231 (2020). [CrossRef]
106. Y. Zhou, M. Z. Alam, M. Karimi, J. Upham, O. Reshef, C. Liu, A. E. Willner, and R. W. Boyd, “Broadband frequency translation through time refraction in an epsilon-near-zero material,” Nat. Commun. 11(1), 2180 (2020). [CrossRef]
107. M. Z. Alam, I. de Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016). [CrossRef]
108. L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced Nonlinear Refractive Index in ε-Near-Zero Materials,” Phys. Rev. Lett. 116(23), 233901 (2016). [CrossRef]
109. Z. Wu, C. Scarborough, and A. Grbic, “Space-Time-Modulated Metasurfaces with Spatial Discretization: Free-Space N-Path Systems,” Phys. Rev. Applied 14(6), 064060 (2020). [CrossRef]
110. Y. Mazor and A. Alù, “One-Way Hyperbolic Metasurfaces Based on Synthetic Motion,” IEEE Trans. Antennas Propag. 68(3), 1739–1747 (2020). [CrossRef]
111. E. Galiffi, Y.-T. Wang, Z. Lim, J. B. Pendry, A. Alù, and P. A. Huidobro, “Wood Anomalies and Surface-Wave Excitation with a Time Grating,” Phys. Rev. Lett. 125(12), 127403 (2020). [CrossRef]
112. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science (1979) 364(6441), 2 (2019).
113. S. Taravati and C. Caloz, “Mixer-Duplexer-Antenna Leaky-Wave System Based on Periodic Space-Time Modulation,” IEEE Trans. Antennas Propag. 65(2), 442–452 (2017). [CrossRef]
114. Z. Liu, Z. Li, and K. Aydin, “Time-Varying Metasurfaces Based on Graphene Microribbon Arrays,” ACS Photonics 3(11), 2035–2039 (2016). [CrossRef]
115. L. Zhang, X. Q. Chen, S. Liu, Q. Zhang, J. Zhao, J. Y. Dai, G. D. Bai, X. Wan, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Space-time-coding digital metasurfaces,” Nat. Commun. 9(1), 1–11 (2018). [CrossRef]
116. J. Shabanpour, “Programmable anisotropic digital metasurface for independent manipulation of dual-polarized THz waves based on a voltage-controlled phase transition of VO 2 microwires,” J. Mater. Chem. C 8(21), 7189–7199 (2020). [CrossRef]
117. X. G. Zhang, Q. Yu, W. X. Jiang, Y. L. Sun, L. Bai, Q. Wang, C. W. Qiu, and T. J. Cui, “Polarization-Controlled Dual-Programmable Metasurfaces,” Adv. Sci. 7(11), 1903382 (2020). [CrossRef]
118. O. Ilic, I. Kaminer, B. Zhen, O. D. Miller, H. Buljan, and M. Soljačić, “Topologically enabled optical nanomotors,” Sci. Adv. 3(6), e1602738 (2017). [CrossRef]
119. X. Wang, A. Díaz-Rubio, H. Li, S. A. Tretyakov, and A. Alù, “Theory and Design of Multifunctional Space-Time Metasurfaces,” Phys. Rev. Applied 13(4), 044040 (2020). [CrossRef]
120. I. Liberal, Y. Li, and N. Engheta, “Reconfigurable epsilon-near-zero metasurfaces via photonic doping,” Nanophotonics 7(6), 1117–1127 (2018). [CrossRef]
121. K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A Reconfigurable Active Huygens’ Metalens,” Adv. Mater. 29(17), 1606422 (2017). [CrossRef]