Spectral causality and the scattering of waves

Abstract

Causality -the principle stating that the output of a system cannot precede the input -is a universal property of nature. Here, we extend the concept of causality, and its implications, from the temporal to the spectral domain, leveraging the peculiar properties of time-modulated non-Hermitian wave-physics systems, with particular emphasis on photonic systems. Specifically, we uncover the existence of a broad class of complex time-modulated metamaterials which obey the time-domain equivalent of the well-established frequency-domain Kramers-Kronig relations. We find that, in the scattering response of such time-modulated systems, the output frequencies are inherently prohibited from spectrally preceding the input frequencies, hence we refer to these systems as 'spectrally causal'. We explore the consequences of this newly introduced concept for several relevant applications, including broadband perfect absorption, temporal cloaking of an 'event', and truly unidirectional propagation along a synthetic dimension. By extending the concept of causality into the spectral domain and providing new tools to extend the field of temporally modulated metamaterials ("chrono-metamaterials") into the complex realm, our findings not only deepen our understanding of spectral scattering dynamics, but may also open unexplored opportunities and enable relevant technological advances in various areas of photonics and, more broadly, of wave physics and engineering.Significance Statement -The causality principle -an effect cannot temporally precede its cause -is a fundamental property of nature and underlies several constraints on the properties of physical materials. Here, we extend the notion of causality from the temporal to the spectral domain in temporally modulated wave-physics systems. Specifically, we uncover the existence of a broad class of time-modulated photonic systems that are "spectrally causal" in the sense that the output wave cannot contain oscillations with frequencies (colors) lower than the frequencies of the input wave oscillations. As an important consequence, wave reflections are then automatically minimized. This new class of timemodulated metamaterials has relevant implications for broadband perfect absorption, invisibility, unidirectional propagation, and may open unexplored opportunities in wave physics and engineering.

An Archimedes' screw for light

“…Temporal structuring of matter opens several new avenues for wave control: periodic modulations of material parameters can enable the design of topologically non-trivial phases 6 as well as Floquet topological insulators 7 and topological insulators with synthetic frequency dimensions 8 . In addition, appropriate tailoring of the temporal dependence of reactive elements can enable arbitrary energy accumulation 9 , whereas the introduction of time-modulated, non-Hermitian elements can lead to nonreciprocal mode-steering and gain 10 , as well as event cloaking and perfect absorption 11 , and surface-wave coupling on spatially flat interfaces 12 . In non-periodic systems, abrupt switching holds the key to new directions such as time-reversal 13 , time-refraction 14 and anisotropy-induced wave routing 15 , as well as frequency conversion 16 – 18 , bandwidth enhancement 19 and Anderson localization 20 .…”

Photonics of time-varying media

“…, where ε ∞ is the background relative permittivity and χðt; t 0 Þ is the time-dependent electric susceptibility, which must satisfy causality. Recently, the generalization of the Kramers-Kronig relations has been introduced for both adiabatic 46 and nonadiabatic 47 time-varying susceptibilities χ. Research on temporal switching with material dispersion dates back to Fante's and Felson's work in the 1970s.…”

“…Another interesting direction for non-Hermitian time-modulated systems relates to the temporal analog of the causality relations of conventional passive media. 46 As a well-known consequence of causality, imposed by demanding that the response function of a system is only non-zero at times following an input signal, the real and imaginary parts of the response function of a system must relate to each other through the Kramers-Kronig relations. However, the spectral analog of these relations can also be constructed: if one would like the spectral response of a system to be uniquely non-zero for frequencies higher or lower than the input frequency, such that only upconversion or downconversion occurs, then there exist temporal Kramers-Kronig relations, which can provide a recipe for how the Hermitian and the non-Hermitian components of a time-modulation must be related in order for such asymmetric frequency conversion to occur.…”

“…Related concepts have also been exploited for event-cloaking, broadband absorption, and synthetic frequency dimensions. 46,95 The introduction of temporal disorder constitutes yet another avenue where the inequivalence between space and time may bear new perspectives in long-standing problems such as Anderson localization. 87,[96][97][98] Interestingly, however, the conventional localization observed in spatially disordered media does not occur in a time-modulated system.…”

Using Time-Varying Systems to Challenge Fundamental Limitations in Electromagnetics

“…In some cases, such improvements may even exceed what is normally allowed by well-established physical bounds, as conventional physical bounds are typically derived under the assumption of linearity and time-invariance (LTI systems). Indeed, as mentioned above, time-variance in the system properties may enable various functionalities not present in a conventional setting, such as parametric gain [35] or spectral changes [19], [23], [24], [25], [26], [43], which are generally not accounted for when deriving conventional bounds for LTI systems, and can therefore be exploited to bypass such physical limits. Consequently, timevarying materials and components provide an exceptional opportunity to realize superior devices not restricted by various conventional limits.…”