1. Introduction

Wave propagation in time-varying media has recently reemerged as one of the most active areas of research in applied electromagnetics and photonics, and especially within the metamaterials scientific community [1,2]. Although the first studies on this broad topic date back as early as the late 1950s [3,4], recent investigations have leveraged conceptual and technological advances in the field of metamaterials to control and manipulate light in new ways in both the spatial and temporal dimensions.

Waves in dynamic materials exhibit rich physics, with interesting dualities between space and time (albeit with some important differences [5]). For instance, wave propagation in homogeneous time-varying media conserves the momentum of light, while frequency changes [3,6], the dual with respect to wave propagation in space-varying time-invariant linear media. Moreover, an abrupt change of the medium’s properties induces the generation of reflected and refracted waves [7,8], whereas a periodic modulation of the refractive index can lead to parametric amplification effects [4,9–11]. (Interestingly, these amplification phenomena have recently been connected with parity-time symmetry and related concepts [7,12]). Scattering by finite objects with time-modulated material properties has also been recently studied [13,14], in parallel with the analysis of time-varying electric and magnetic dipoles and dipolar meta-atoms [15,16]; in these scenarios, the scattering/radiation process leads to harmonic generation via the coupling of the incident field with the Floquet dynamics of the scatterer’s time-periodic refractive index and the corrective radiation terms of the modulated dipole moments, respectively. The theoretical analysis of waves in time-varying systems has also been recently extended to materials with non-negligible temporal dispersion [17–23], which exhibit new intriguing physical effects. For example, wave propagation in time-varying dispersive media with a discontinuity of the electron density has been recently studied showing an interesting second-order effect, namely, the emergence of two shifted frequencies, which implies the need for two extra temporal boundary conditions [19]. In addition, electromagnetic pulses in time-varying dispersive media have also been analyzed in [24], while the temporal equivalent of a reflectionless medium based on the spatial Kramers-Kronig relations [25] has been recently proposed in [26] using time-varying non-Hermitian dispersive media. Moreover, an extension of Mie theory has been recently established for the scattering by time-modulated dispersive spheres in [27], and different homogenization methods have been extended to the time-varying case to derive effective constitutive parameters in the long-wavelength regime [7,28–32].

In addition to significant theoretical advances, this field has also seen increased experimental efforts across the electromagnetic spectrum. For example, microwave systems using varactors have been used for the first experimental demonstration of an electromagnetic time-crystal exhibiting momentum band-gaps [33,34], while abrupt changes of the refractive index to realize temporal interfaces have been achieved via the use of switches in transmission-line circuits [35], as well as using epsilon-near-zero materials at optical frequencies (e.g., indium tin oxide) modulated by ultrafast laser pulses [36–39].

The rich physics of wave scattering and propagation in temporal metamaterials may open new opportunities for various applications, for example to enable efficient frequency mixing [40], enhanced matching networks [41,42], magnetic-free isolators [43–46], spread-spectrum selective camouflaging [47] and antireflection coatings [48]. Particularly promising is the fact that some of the features enabled by temporal modulations may even allow surpassing well-established fundamental performance bounds of linear time-invariant electromagnetic systems, such as the Bode-Fano limit on broadband impedance matching, the Rozanov bound on electromagnetic absorption, and the Chu limit on the bandwidth of small antennas [49–56]. Many of these limits and bounds are related to bandwidth and dispersion constraints, which can be altered and possibly relaxed by breaking the assumptions of linearity and/or time-invariance. The principle of causality and its implications, in particular the well-known Kramers-Kronig relations, play an important role in this context, and their generalization to time-varying dispersive constitutive relations is crucially important to understand the bandwidth/dispersion properties of time-varying media. In this review article, we focus on these important aspects of the field of time-varying metamaterials. We provide a brief overview of how linear, dispersive, time-varying media can be modeled, and we discuss the generalization of the Kramers-Kronig relations for the electric susceptibility of these materials. In addition, we review the Kramers-Kronig-like relations that are satisfied by nonlinear susceptibilities, as this is particularly relevant to the implementation of time-varying systems at optical frequencies, which requires, in most cases, nonlinear optical processes.

2. Linear time-varying media

2.1 General framework for linear processes

For an isotropic medium that is both temporally dispersive and time-varying, we can extend the standard constitutive relation between the polarization density $P$ and the electric field $E$, inspired by time-varying linear circuits, in the form of a linear functional as [20]

The equivalent kernels $R(\xi,t)$ and $h(t,\tau )$, representing a time-varying response function, are characterized by two time variables, which correspond to two frequencies variables, somewhat analogous to the multi-frequency nonlinear susceptibilities, $\chi ^{(n)}$, used in nonlinear optics (for which the $n$-th order nonlinear polarization is defined as a multiple convolution, with multiple fields and multiple time variables, as discussed in Section 3.1) [57]. Through Fourier transforms with respect to the time variables, we can then define, following [17], a temporal complex susceptibility as

By substituting the inverse Fourier transform corresponding to Eq. (3) into Eq. (1), and after a few steps, we obtain

Finally, by Fourier transforming Eq. (5) with respect to the $t$ time variable, we can obtain an expression for the frequency-domain polarization density as

Clearly, $P(t)$, $E(t)$, $R (\xi,t)$ are real functions, therefore the symmetry relation known as “reality condition” is satisfied with respect to frequency $\Omega$

The reality condition is instead not generally satisfied for the two-frequency susceptibility, i.e., ${\bar \chi _T}^ *(\Omega,\omega ) \neq {\bar \chi _T}(\Omega,-\omega )$, as this is the Fourier transform of the function $\chi _T (\Omega,t)$, which is not necessarily real-valued.

2.2 Dispersion models

After discussing some of the general properties of the susceptibility of linear, dispersive, time-varying systems, in this section we review some of the main models that can be used to derive this susceptibility. We start by considering a polarizable medium with bound electrons, which act as damped harmonic oscillators driven by the Lorentz force exerted by an applied electromagnetic field. This leads to the standard Lorentz model for non-conducting media or the Lorentz-Drude model if free electrons are also included. A corresponding model for the susceptibility of dispersive time-varying material can then be derived from the same microscopic equation of motion for the electrons but with some properties of the oscillator/material changed in time by an external agent. Specifically, for a time-varying Lorentz medium, the equation of motion for a bound electron of mass $m$ and charge $-e$ can be written as

At this point, to determine the susceptibility of the material, one needs an equation to relate the induced polarization density to the microscopic dipole moments, which, in the time-invariant case, is simply $P = N p$. If instead $N(t)$ is time-dependent, two different models have been proposed in the literature. In Ref. [17], an instantaneous response was assumed, $P(t)=N(t)p(t)$, which was argued to be valid under a low-density approximation, such that the movement/velocity of electrons is not affected by their density. With this assumption, the polarization density is simply given by

In [17], the case of a conducting material with both a time-varying electron density and a time-varying damping rate was also considered, assuming again an instantaneous response $P(t)=N(t)p(t)$. Going back to the differential equation for the induced dipole moment, it was shown that the resulting susceptibility may be quite different from the one obtained by simply using a time-varying damping rate in the conventional Drude model. (This also has intriguing implications related to causality, as discussed in the next section.) If we consider, as in [17], $\omega _{0}=0$, $ \Gamma(t)=2 \Gamma_0 /\left(1+\Gamma_0 t\right)$, the temporal complex susceptibility is given by

In Fig. 1, this model is compared with the conventional Drude model with time-varying damping rate $\Gamma (t)$, for different values of $\Gamma _0$ and fixed plasma frequency. Clearly, the new model differs from the conventional one, while the results tend to agree asymptotically for large values of $t$.

 

Fig. 1. Imaginary and real parts of the permittivity $\varepsilon (\omega,t)=1+\chi _{T}(\omega,t)$, plotted with respect to normalized time, for a conducting medium with time-varying damping rate. The plasma frequency and incident wave frequency are fixed, $\omega_p=10^{16} \mathrm{rad} / \mathrm{s}$ and $\omega=\omega_p / 2$. The plots compare the conventional Drude model with time-varying damping term with the corrected model derived from the microscopic equation of motion with time-varying parameters. As discussed in the text, the imaginary and real parts of the modified model are not related by Kramers-Kronig relations. The figure is reprinted from [17] which is under the terms of the Creative Commons Attribution 4.0 licence and is used with permission.

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All the results above were derived assuming an instantaneous relation between induced dipole moments and polarization density, leading to Eq. (9). An alternative model for a Lorentz material with time-varying oscillator density has been recently suggested in various papers in the literature [18,19,21,27]. Following [21], the polarization density is assumed to be connected with the dipole moments via the relation: $P(t) = \int _{ - \infty }^t {{\textstyle {{dN(t')} \over {dt'}}}} p(t,t')dt'$, where $p(t,t')$ is the dipole moment at instant $t$ of an oscillator appearing at instant $t'$. Therefore, the induced polarization is a superposition of contributions from oscillators appearing at different instants of time and, since the principle of causality has to be satisfied, $0<t'<t$.

As shown explicitly in [21], by substituting the relation between dipole moment $p(t)$ and electric field into this expression for $P(t)$ and integrating by parts, one obtains a relation between polarization density and electric field that is different from Eq. (9) with respect to how the electric field at each moment interacts with the available electron density:

This relation is assumed directly in this form (albeit with a change of variables) in [27]. By then comparing Eqs. (12) and (1) and through a Fourier transform with respect to the $\xi$ time-variable, the corresponding temporal complex susceptibility can be found. However, here we give directly the two-frequency susceptibility ${\bar \chi _T}(\Omega, \omega - \Omega )$, in the form relevant for Eq. (6), which has a more straightforward expression

Note that this is different from the two-frequency susceptibility that can be obtained from Eq. (10). By substituting this into Eq. (6), we can also find an interesting expression for the induced polarization density in terms of a frequency-domain convolution, consistent with the results of [18] :

Reference [18] also proposes a $RLC$ circuit analogy for this model of a Lorentz medium with time-varying oscillator density, which results in the following differential equation for the charge (polarization) response of the circuit (material):

We also note that Ref. [18] defined three equivalent kernels (impulse responses) for dispersive time-varying linear media, two of them being the same as the ones we defined in Section 2.1 and an extra one, $hc(t',\tau )=h(t'+\tau,\tau )$, and it also defined impedances and transfer functions for the corresponding circuit models. The kernel used in the core of this review article corresponds to the one in Ref. [18] denoted as $c(t,\xi )=R(\xi,t)$ (central column of Fig. 2). We stress, however, that these different kernels are simply different representations of the same response with varied retardations, and they yield identical polarization outputs for the same linear time-variant system. Representative calculations of these kernels for a specific example are shown in Fig. 2, together with the corresponding one- and two-frequency susceptibilities (details are provided in the caption).

 

Fig. 2. Time-varying impulse responses and frequency-domain susceptibilities under different representations, $h(t,\tau )$, $c(t,\xi )=R(\xi,t)$ and $hc(\xi,\tau )$, as defined in [18], for a medium with time-varying dispersive Lorentzian response. The particular numerical example considered here assumes that the polarization response follows the model in Eqs. (12)-(14), with a time-varying plasma frequency $\omega _p^2(t) = \omega _{p0}^2(1 + \Delta \cos (\Omega t))$, where $\omega _{p0}$ and $\Gamma$ are chosen such that if $\Delta =0$ (i.e., the system is linear and time-invariant) then: $\chi _{T} (\Omega ) = (3-0.1i)$ with a resonant frequency $\omega _0 =5 \Omega$. The numerical results presented in this figure are calculated with a modulation depth, $\Delta = 0.9$. The kernel used in the core of this review article is $c(t,\xi )=R(\xi,t)$ (central column), while the other kernels are different representations of the same response with varied retardations, which result in the same polarization output. (a) Impulse responses in the time domain, defined by two time variables. (b) Time-varying complex susceptibilities with the first temporal axis transformed to frequency. (c) Time-varying complex susceptibilities with the second temporal axis transformed to frequency. (d) Two-frequency susceptibilities with both time axis transformed to the corresponding frequency variables. Only the real parts of the Fourier-transformed kernels are shown. The figure is reprinted with permission from [18] © 2021 American Physical Society.

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Based on the time-varying Lorentz model discussed in the previous paragraphs, wave propagation under a temporal discontinuity of electron/oscillator density was also investigated in [19,22]. These studies indicated that, in contrast to nondispersive time-varying media, two forward and two backward waves are generated at the temporal discontinuity in response to a forward-only initial wave [19]. This highlights the importance of including the dispersive properties of a time-varying medium to unveil and elucidate new physical effects. We also note that determining which of the two models discussed above should be used in specific situations remains an open question, which we believe will ultimately be addressed by comparison with experimental results. The general theoretical framework discussed in this review article, and in the relevant references, can however be used for both of these, and other, phenomenological models of the response function.

Finally, in addition to the models discussed above, Refs. [23,59] discussed the case of a Lorentz medium with a time-varying oscillator frequency, $\omega _0^2(1+\delta \omega f(t))$. By assuming a weak perturbation $|\delta \omega | \ll 1$ for all times, it was found that, in this perturbative regime, the two-frequency susceptibility with a first-order correction takes the following form

2.3 Kramers-Kronig relations for linear time-varying susceptibilities

The electric susceptibility must be causal, as it represents an input-output relation between electric field and induced polarization. According to Titchmarsh’s theorem, if a linear impulse response is causal and sufficiently well-behaved (i.e., square integrable, which is the case for physical materials with a response that is bounded and decays sufficiently fast in the high-frequency limit), then its Fourier transform (transfer function) is analytic in the upper half of the complex frequency plane and its real and imaginary parts are related by Kramers-Kronig relations [60,61]. These considerations also apply to the response of linear time-varying dispersive media, as discussed in [18] (albeit with some notable exceptions, as noted below). Taking also into account the reality condition in Eq. (7) for the temporal complex susceptibility, we get the following Kramers-Kronig relations

Since the reality condition doesn’t necessarily hold, as ${\bar \chi }_{T} = (\Omega,\omega )$ is the Fourier transform of a complex function, the integral cannot, in general, be reduced to positive frequencies only. Additionally, we note that Kramers-Kronig relations with respect to both frequency variables (hence, with a double integral) cannot be derived since causality only requires the response function $R(\xi,t)$ to vanish for $\xi <0$, and not for $t <0$ (see Section 2.1). Indeed, an important case of time modulation is the Floquet case in which the modulation is periodic, and hence $R(\xi,t)$ is periodic with respect to $t$.

In the Kramers-Kronig relations above, it is assumed that the high-frequency asymptotic behavior is ${\chi _T}(\Omega,t) = \mathcal {O}({\Omega ^{ - N}})$, and ${\bar \chi _T}(\Omega,\omega ) = \mathcal {O}({\Omega ^{ - N}})$, with $N \in {\mathbb {Z}^ + }\backslash \{ 0,1\}$. Consistent with the models discussed in the previous sections, it is often found that the asymptotic behavior is essentially the same as in the stationary case, where $N=2$. If $N>2$, additional Kramers-Kronig relations could be found in the same way as in nonlinear optics [57]. As in the time-invariant case, Eqs. (17)–(20) imply that the time-varying absorption losses of a temporal metamaterial are deeply related to the real part of the susceptibility and therefore to propagation/refraction properties (and their frequency dispersion), and vice versa. If the imaginary part of the temporal susceptibility is a separable function, such that ${\mathop {\rm Im}\nolimits } {{\chi _T}(\Omega,t)} = X(\Omega )Y(t)$, then the real part is also a separable function and has the same time dependence as the imaginary part. This means that if the time-varying absorption properties of the material (e.g., a transparent conducting oxide under an optical pump) can be represented as a separable function, then the refractive index of the material would change as quickly, and with the same temporal profile, as the dynamical change of its absorption. Clearly, however, the separability condition is not necessarily true, since the $\Omega -$context and $t-$context of the complex temporal susceptibility can be intermingled and the time profile of the real part of the susceptibility might be different from that of the imaginary part according to Eq. (17).

We also note that, while Eq. (10) (Lorentz model with time-varying oscillator density) directly satisfies the Kramers-Kronig relations (17), (18), this is not the case for Eq. (11) (Drude model with time-varying electron density and damping rate). In fact, the case represented by Eq. (11) illustrates the consequences of not satisfying the conditions of Titchmarsh’s theorem. In particular, one can show that, for this case, the time-varying response function is $R(\xi,t) = {\textstyle {{N(t){e^2}} \over {{\varepsilon _0}m}}}\xi \left ( {1 - {\textstyle {{{\Gamma _0}\xi } \over {1 + {\Gamma _0}t}}}} \right )$, which is not square-integrable, i.e., $\int _0^\infty {{{\left | {R(\xi,t)} \right |}^2}d\xi } = \infty$ (in addition, ${\chi _T}(\Omega,t)$ is not square integrable with respect to $\Omega$, consistent with Parseval/Plancherel’s Theorem). Hence, although the response function is still causal, Kramers-Kronig relations cannot be derived, and therefore the real and imaginary parts of this susceptibility can be unrelated [61] (note how this is different from the time-invariant lossy Drude model, which is also not square integrable due to a pole at zero frequency, but a simple multiplication by $\Omega$ is sufficient to ensure square integrability, leading to the standard modified Kramers-Kronig relations for time-invariant conducting media [57]).

3. Nonlinear time-varying optical media

3.1 General framework for nonlinear processes

The implementation of various time-varying systems at optical frequencies requires large modulation depths and ultra-fast refractive-index changes. For example, in a photonic time–crystal, a fast periodic modulation of the refractive index results in the opening of momentum band-gaps at $\omega / \Omega =N/2$, where $\omega$ is the propagating wave frequency, $\Omega$ is the modulation frequency, and $N \in {\mathbb {Z}}$, with the width of the band-gaps proportional to the strength of the modulation [9–11,28]. In Fig. 3, an example of momentum gap (for $N=1$) is illustrated, calculated using the coupled-wave theory introduced in [10].

 

Fig. 3. Dispersion diagram (Brillouin diagram) near the first momentum band-gap of a photonic time-crystal. Solid and dotted lines represent, respectively, the real and imaginary parts of the eigenfrequency, normalized by the modulation frequency $\Omega$. Dashed lines correspond to the light cones for wave propagation in a time-invariant material with effective permittivity equal to $\varepsilon_{\text {eff }}=T / \int_0^T d \tau / \varepsilon(\tau)$. The parameter $\theta _1$ is the coefficient of the first harmonic of the Fourier series of $\varepsilon ^{-1}$. The figure is reprinted with permission from [10] © 2022 Optica Publishing Group.

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Arguably, such strongly time-varying systems can only be implemented at optical frequencies by using all-optical modulations based on second- or third-order nonlinear optical processes. For instance, all–optically modulated transparent conducting oxides have been suggested as a promising experimental platform to demonstrate various effects induced by strong temporal modulations, including the emergence of momentum gaps if sufficiently fast and deep modulations, with high repetition rates, can be realized [5,62]. Notably, Ref. [39] identified specific nonlinear processes, and the corresponding susceptibilities, that may induce photonic time-crystal effects, and discussed their relation to standard optical parametric generation/amplification, four wave mixing, and phase conjugation. In particular, Ref. [39] considered: (1) a second-order nonlinear parametric process, ${{\chi ^{(2)}}(\omega, - 2{\omega _0})}$, in which the permittivity is modulated by the pump at a frequency $2\omega _0$ to obtain a first-order momentum gap at $\omega = \omega _0$; and (2) the following third-order processes: (a) ${{\chi ^{(3)}}(\omega, - {\omega _0}, - {\omega _0})}$, a fast weak process in which two pump photons modulate the permittivity at frequency $2\omega _0$ and the signal photon scatters into the idler, and (b) ${{\chi ^{(3)}}( - {\omega _0},\omega, - {\omega _0})}$, a slow stronger process in which a moving grating is produced by one pump and one signal photon, and another pump photon scatters off the grating into the idler (a process related to standard four-wave mixing or phase conjugation). We refer the interested reader to [39] for additional details. We just note that, from a practical viewpoint, while the second-order susceptibility results in high modulation strength, the efficiency of the resulting effects (e.g., amplification in a photonic time-crystal) strongly depends on phase matching. On the other hand, phase matching is less critical for the third-order susceptibilities, but the modulation is weaker as the order of nonlinearity increases [39].

Given the relevance of nonlinear optical processes in the field of time-varying metamaterials, it is useful to consider their dispersion and causality properties within this context. The general time-domain relation between the nonlinear polarization density and the applied electric fields for a nonlinear optical medium (in the perturbative regime and assuming a dipole approximation) can be written as a multiple convolution [57]

Similarly with Section (2), the principle of causality dictates that $R({\xi _1}, \ldots,{\xi _n})=0$ for ${\xi _1}, \ldots,{\xi _n} < 0$, and therefore the limits of integration can be reduced to the range from $0$ to $+\infty$. The $n$th-order nonlinear frequency-domain susceptibility is thus defined as

$P^{(n)}(t)$ and $E(t)$ are real fields; therefore, from Eqs. (21) and (22) the reality condition for the nonlinear susceptibilities can be determined as

3.2 Kramers-Kronig relations for nonlinear susceptibilities relevant to photonic time crystals

Integral relations similar to the Kramers-Kronig relations for the linear response can also be derived for some, but not all, nonlinear optical interactions [63]. In particular, in nonlinear cases where all input frequencies are kept fixed except one, Kramers-Kronig relations can be directly derived as in the linear case (indeed, as an example, the relation $P^{(3)} = {{\chi ^{(3)}}(\omega _1 , \omega _2, \omega _3)} E(\omega _1)E(\omega _2)E(\omega _3)$ is linear in the field $E(\omega _1)$ if $\omega _1$ is independent of $\omega _2,\omega _3$). This is indeed the case for the nonlinear susceptibilities that we mentioned in the previous section to realize photonic time-crystal effects: a second-order nonlinear process with fixed pump frequency equal to twice the frequency of the desired first-order momentum band-gap, $2\omega _{0}$, or a third-order nonlinear effect with pump frequency the band-gap frequency, $\omega _0$. For these cases, where the pump has a fixed frequency, one can therefore derive Kramers-Kronig relations that apply to the material response as the incident (signal) frequency is swept while the pump frequency is kept fixed (assuming suitable integrability conditions as in the linear case). The Kramers-Kronig relations for the second-order nonlinear susceptibility of interest, ${{\chi ^{(2)}}(\omega, - 2{\omega _0})}$, are found as

The third-order nonlinear susceptibilities of interest are, instead, ${{\chi ^{(3)}}(\omega, - {\omega _0}, - {\omega _0})}$ and ${{\chi ^{(3)}}( - {\omega _0},\omega, - {\omega _0})}$, as mentioned above. For these susceptibilities, the following Kramers-Kronig relations hold:

These relations indicate that the real and imaginary parts of these specific nonlinear susceptibilities, for a fixed pump at $\omega _0$ or $2\omega _0$, are connected as a result of the principle of causality (and assuming again square integrability [61]). Similarly to the linear case [64], we speculate that these relations, together with relevant sum rules, could potentially be used to derive limits and trade-offs on nonlinear processes relevant for optical time-varying systems.

We also briefly note that for nonlinear responses where a pair or more of the input frequencies are mutually dependent and are varied at the same time (as in the case of harmonic generation, e.g., $P^{(3)}(3\omega ) = {{\chi ^{(3)}}(\omega, \omega, \omega )}E(\omega )E(\omega )E(\omega )$), Kramers-Kronig relations may still be derived if the nonlinear susceptibility satisfies the so-called “Scandolo theorem” [57,65]. This theorem allows the choice of a one-dimensional space to be embedded in the $n-$dimensional space defined by all the frequency variables for a $n$-th order nonlinear susceptibility, such that a single Kramers-Kronig integration can be applied for all (or some of) the involved frequencies. These nonlinear effects are then essentially described as an effective first-order susceptibility for the purpose of deriving and applying Kramers-Kronig relations. We refer the interested reader to Ref. [57] for a detailed discussion of this topic. Finally, we note that, unlike the linear case, the imaginary part of nonlinear susceptibilities is not always directly related to absorption (or gain); in some cases (e.g., the imaginary part of harmonic generation susceptibilities), it simply represents a phase relationship between nonlinear polarization and applied fields, without implying any time-averaged absorbed/gained power [66].

4. Discussion

Kramers-Kronig relations are a fundamentally important result in the study of the optical properties of natural and artificial materials. Since passivity implies causality but causality does not necessarily imply passivity [67], these relations are automatically valid for passive materials, whereas the situation is more subtle for active media. In fact, Kramers-Kronig relations cannot always be established for causal, active media, due to the possibility of absolute instabilities [68]. Additionally, even if Kramers-Kronig relations can be established for stable active media, they may not impose fundamental limitations to the dispersion/bandwidth properties of the material response (e.g., anomalous dispersion may be possible even in low-loss regions) [68–70].

Time-varying metamaterials may be considered another class of active media that dynamically change their properties due to an external agent (which may also transfer energy in and out of the electromagnetic system). In this article, we have first reviewed general properties of linear, dispersive, time-varying materials and their relation to the principle of causality. We discussed several models for the response function of these media, with a focus on two dispersion models for Lorentz materials with a time-varying electron/oscillator density $N(t)$. The first one [17] assumes an instantaneous response for the polarization density $P(t)=N(t)p(t)$, whereas the second one [18,19,21] considers that the induced polarization is a superposition of contributions from oscillators appearing at different instants of time. We reiterate that the development of phenomenological models for the response function of time-varying media is a relatively new research area, and we anticipate the emergence of numerous models tailored to address specific instances of time modulation. We believe such an undertaking will need to involve a combination of theoretical and experimental approaches to determine which models and approximations should be used in specific scenarios.

As in the stationary case, also for time-varying linear materials the implications of the principle of causality establish the appropriate relations between the real and the imaginary part of the electric susceptibility, but not in all possible cases, as some models of time-varying dispersive media may lead to causal but non-square-integrable response functions. Additionally, given the importance of nonlinear optical effects to realize temporal metamaterials at optical frequencies, we have reviewed the Kramers-Kronig relations of the relevant second- and third-order (perturbative) susceptibilities that may be used to induce parametric amplification and related photonic-crystal effects [10,39]. Indeed, although temporal modulation can be created through many different methods and materials, such as injection or depletion of charge carriers or electro-optical modulation, the most prominent platforms for fast and strong modulations of the refractive index at optical frequencies are based on all-optical nonlinear effects. In particular, the strong nonlinear response of transparent conducting oxides in their epsilon-near-zero regime have attracted significant interest in this field [36–38,62]. We also emphasize that very strong nonlinear light–matter interactions may not necessarily follow a perturbative model associated with a power series expansion of the polarization density. In such cases, the relevant phenomena may not necessarily be representable by square-integrable susceptibilities and, therefore, may not satisfy any Kramers-Kronig-like relation. Incidentally, we also note that interesting considerations about the energy requirements for the temporal modulation of materials at optical frequencies can be found in a recent paper by Khurgin [71]. Additionally, it is worth noting that the specific form of the Kramers-Kronig relations depends on the asymptotic behavior of the susceptibilities for very high frequencies (much larger than any resonant frequencies of the material). The material response, and hence the susceptibility, has to go to zero for infinite frequency in all cases [61], but the specific asymptotic behavior might differ for different time-varying susceptibility models, which implies that higher–order Kramers-Kronig relations may be constructed as for certain nonlinear optical phenomena [57].

The extension and translation of the dispersion models and Kramers-Kronig relations discussed in this paper to other domains of time-varying wave physics, for example, for time-varying acoustic and elastic materials, is an interesting research direction. While Kramers-Kronig-like relations exist in these domains [72], a universal model for the high-frequency asymptotic behavior of acoustic/elastic properties is not available, as the high-frequency behavior of acoustic/elastic media cannot be related to a universal reference case, which in electromagnetics is simply the case of waves in vacuum [67]. Instead, the asymptotic relations in acoustics/elastics are often based on empirical assumptions and may vary depending on the particular model used [73].

Finally, we believe that some of the most interesting research questions in this field are about the possible generalization of fundamental limitations based on causality, time-invarience and passivity to the case of time-varying metamaterials, including the Bode–Fano limit on broadband impedance matching [74,75], the Rozanov bound on electromagnetic absorption [50,56,76], and causality limits on material properties and general scattering problems [64,77].