1. Introduction

Although its roots can be traced back to several decades ago [1–4], the study of time-varying photonic materials has recently emerged as one of the most active areas of optics and photonics, potentially opening intriguing directions for new physics and applications [5–11]. Just as spatially-varying artificial materials and structures, such as metamaterials, metasurfaces, and photonic crystals, have enriched the photonic design toolbox to gain better control over the light flow and have led to the unveiling of many unexplored physical phenomena [12], time-varying engineered materials are now being explored by many research groups worldwide as the next stepping stone to achieve a better level of control of wave phenomena [9]. For instance, space-time-modulated structures can realize nonreciprocal effects without the need for magneto-optical materials [5,13], which has been one of the main motivating factors for the recent surge of interest in this field. Moreover, a variety of other interesting effects and concepts based on time-varying systems have been reported in recent years, such as antireflection temporal coatings [14], spacetime cloaking [15], inverse prisms [16], nonreciprocal amplification [17], static-to-dynamic field conversion [18], temporal Wood anomalies [19], correlated-photon-pair generation [20,21], Fresnel drag [22], spectral causality [23], topological phase transitions [24], among many others [7]. This short list provides a glimpse into the rich physics that can be accessed by bringing the temporal dimension into play. Importantly, many of these works are based on the concept of space-time duality [25], namely, the duality of space and time in the wave equation, which provides a conceptual foundation for time-varying systems and facilitates the discovery of new effects and potential applications based on time-varying photonics. However, while analogies between space-varying and time-varying systems are certainly insightful, the two domains have certain distinct characteristics that make modulations in time significantly more challenging to physically realize, especially if large and fast temporal modulations are required, as further discussed in the following.

Some of the proposed applications of time-varying photonics do not necessarily require ultra-fast temporal variations and/or large modulation strengths, thus their experimental realization is arguably more feasible. This is true, for instance, for optical modulators and optical isolators based on traveling-wave modulations, where the frequency of the temporal perturbation can be significantly smaller than the operational frequency and the refractive index change can be low [13,26–28] (with the caveat that, for spacetime-modulated optical isolators, the length of the device trade-offs with the coupling strength between modes [13] and the modulation frequency [29]). Conversely, for other time-varying systems that have recently been the subject of intense theoretical interest, such as photonic time crystals [30–34] and broadband optical parametric amplifiers based on momentum bandgaps [35], the modulation frequency is comparable or larger than the operational frequency, and the modulation strength needs to be sufficiently large to observe the relevant phenomena. For instance, the relative width of the momentum gaps in a time crystal is proportional to the relative modulation strength [36,37] and, therefore, the typical relative permittivity change for these systems is around 10% [38] and can be as high as 200% [32]. This ultimately raises the question about whether such fast modulation speeds and large modulation strengths can be physically realized with current or future experimental platforms. Indeed, while the originally proposed time crystals in atomic systems [39] have attracted considerable experimental efforts based on various platforms [40–44], experimental demonstrations of photonic time crystals with momentum bandgaps have been scarce and limited to radio frequencies [36]. Although there have been recent promising advances in dynamic optical materials – especially epsilon-near-zero media [45] such as transparent conducting oxides – with experiments showing large femtosecond-scale variation in the refractive index [46,47], we will show that modulating such materials at the required frequencies implies large power consumption and, moreover, the desired effects are often obscured by less exotic nonlinear phenomena.

More broadly, in this article, we discuss two key physical aspects of time-varying materials, namely, temporal dispersion and external power requirements, both of which have important implications for the physical realization and observation of photonic time-varying systems. We argue that, despite the often-invoked duality between space and time, the characteristics of the temporal dimension – particularly its relation to causality and energy conservation – can bring up challenges that are unique to time-varying systems.

2. Dispersion constraints in time-varying materials

Most of the studies on time-varying photonics so far have assumed a rather simple dynamic material model, where the time-varying material is non-dispersive and can respond to the external modulation instantaneously. The time-domain constitutive relation for the induced polarization that results from such an approach can be written as $\textbf {P}(t)=\epsilon _0\chi (t)\textbf {E}(t)$ [31,48], where $\textbf {E}$ and $\textbf {P}$ are respectively the electric field and the electric polarization density, while $\chi (t)$ is the time-varying, isotropic, non-dispersive electric susceptibility and $\epsilon _0$ is the free-space permittivity. A time-varying non-dispersive loss (and/or gain) profile can then be modeled by a time-varying conductivity $\sigma (t)$ [49] that relates the electric field to the electric current density $\textbf {J}(t)$, i.e., $\textbf {J}(t)=\sigma (t)\textbf {E}(t)$. Such simplified relations can be physically interpreted as representing fields that exist in a quasi-steady-state at every time instant and that can acquire a new quasi-steady-state instantaneously as a result of the applied modulation. Clearly, this dynamic model is overly simplistic and, while it can provide approximately correct results in certain cases, it requires proper justification in its usage. Since all physical materials posses a causal, non-instantaneous polarization response with a finite temporal duration, the material constitutive parameters are inherently dispersive, i.e., frequency-dependent. In the linear time-invariant case, this translates into the constitutive relation, $\textbf {P}(t)=\epsilon _0 \int _{-\infty }^{\infty }\chi (t-t')\textbf {E}(t')dt'$, which means that $\textbf {P}$ at time $t$ depends on $\textbf {E}$ not only at $t$ but also at previous time instants (causality implies that $\chi (t-t') = 0$ for $t - t' \leq 0$, namely, no polarization is induced before the field is applied). If the material is time-variant, one can generalize this convolution relation using a time-varying susceptibility kernel, i.e.,

A major consequence of material dispersion is that the time-varying electric susceptibility depends explicitly on two independent time variables, $t$ and $t'$, as in Eq. (1), rather than their difference $t-t'$ as in the time-translation invariant case. Importantly, $\chi (t,t')$ is still causal at every $t$-time instant, so that $\chi (t,t') = 0$ for $t - t' \leq 0$, which results in the generalization of the well-known Kramers-Kronig relations (Hilbert transforms) for time-varying systems, as was studied in Ref. [53] and more recently in Ref. [55]). To gain more physical insight, one can resort to standard differential equation model that relate $\textbf {P}$ to $\textbf {E}$. For instance, for the case of a time-varying dispersive isotropic material of Drude-Lorentz type, the relevant differential equation reads

If one considers temporal modulation of the above material, both $\omega _0$ and $\omega _p$ can get modulated. The most obvious option is a modulation of $N_1 - N_2$, achieved through absorption of light via all-optical modulation or through carrier injection/depletion in an electro-optical modulator, such as Si [57], ITO [58] or graphene [27] based modulators. For a Drude-type dispersive system, $\omega _p$ can also be changed by simply re-arranging carriers inside a non-parabolic conduction band, leading to an increase of the effective mass as the carrier temperature increases [59]. In most of the faster modulators, such as the electro-optic ones based on Pockels [60] or Kerr [61] effects, acousto-optic ones [62], and many all-optical ones [63], both the effective resonant frequency $\omega _0$ and, to a lesser degree, $f \omega _p^2$ change. This happens when the electronic states shift and mix, which changes $\omega _0$ and (to a lesser degree) the oscillator strength $f$. On the microscopic level, this shifting and mixing can be directly caused by the electric or optical field via DC or AC Stark effects [64] or, as in the case of acousto-optic and (partially) Pockels effects, the shifting/mixing is mediated by the re-arrangement of the lattice ions. But overall, one can always model the change of dielectric permittivity via these two parameters – mean resonant frequency and plasma frequency (strength of the transition).

In general, the electric susceptibility in Eq. (1) can be found by determining the causal Green’s function $G(t,t')$ of the differential operator on the left side of Eq. (2) (assuming again an isotropic medium)

The situation becomes more involved if an external time modulation affects the dynamics of the carriers that are already in motion due to the probe wave. This may be the case, for instance, if $\omega _0$ is modulated in time (for example using any effect that shifts the relevant energy levels). In this case, $\chi (t,t')$ cannot simply be found, in general, by replacing the time-invariant $\omega _0$ with the time-varying one, and $G(t,t')$ in Eq. (2) will need to be evaluated (numerically in most cases). However, if the time scale of the modulation is sufficiently large compared to the period of the probe wave, one can still use the adiabatic assumption to approximate the time-varying susceptibility as $\chi (t,t') \approx f(\omega _0(t),\omega _p,\gamma,t-t')$ [6]. In this regard, to emphasize the distinction between the adiabatic and non-adiabatic regimes for time-varying systems, in Fig. 1 we provide full-wave simulation results based on the finite-difference time-domain (FDTD) method for slow and fast time-varying systems, using a commercially available software [65] and by incorporating the auxiliary differential equation method [66–68] to implement the time-varying dispersive material as a custom material plugin (Eq. (2) is solved at each time instant to produce the polarization density and displacement field, which then enter Maxwell’s equations to update the fields at the next time instant). Specifically, in Fig. 1, we show the polarization density calculated analytically using Eq. (1) and the adiabatic approximation for $\chi (t,t')$, compared with the exact full-wave simulation results. It is evident that while the adiabatic approximation leads to accurate results if the time scale of the modulation is large compared to the time scale of the signal wave (and the time scale of the non-instantaneous response of the dispersive material) (Figs. 1(a) and (b)), the approximation starts to break down once the modulation time scale becomes smaller, on the order of the signal wave period (Figs. 1(c) and (d)) since the field does not have enough time to interact fully with the dispersive material and reach a new steady state before the material properties are drastically changed again. As a result, for a fast modulation, an adiabatic approximation may significantly overestimate the strength of the resulting polarization density modulation, as seen in Fig. 1(d), which in turn would lead to an overestimation of the scattering intensity from temporal interfaces and other related effects, such as the size of momentum bandgaps.

 

Fig. 1. (a) For a slow, adiabatic time modulation, the time scale $\Delta t$ (period) of the modulation is much larger than the time period $\Delta t'$ of the excitation field. (b) As a result, the adiabatic approximation holds and the time-varying dispersive susceptibility is well approximated by the time-invariant dispersive susceptibility function but with time-variant parameters (in this case $\omega _0(t)$). The resulting electric polarization density, calculated with and without adiabatic approximation, is plotted in the figure. (c) If the time scale $\Delta t$ is comparable to $\Delta t'$, (d) the adiabatic approximation becomes inaccurate as the field does not have enough time to interact fully with the dispersive material, and reach a new steady state while the material properties are varied. As a result, the adiabatic approximation may overestimate the strength of the resulting polarization density modulation and of various effects that depend on it, e.g., scattering at temporal interfaces, harmonic generation, and momentum bandgaps in time crystals. (e),(f) Spectra of the polarization density for the slow and fast modulation cases, respectively. Material properties for the two cases are as follows: $\omega _p = 2\pi ~ 400 ~ \textrm {THz}$, $\gamma = 2\pi ~ 2 ~\textrm {THz}$, while the excitation signal is monochromatic with a frequency of 390 THz. The modulation frequency is 8 THz and 195 THz for the slow and fast modulation cases, respectively.

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To further elucidate these observations, Figs. 1(e) and (f)) show the spectrum of the polarization density for the slow and fast modulation cases, respectively, obtained through both full-wave simulations and the adiabatic approximation. The amplitudes of the generated harmonics can be used to assess the modulation strength experienced by the propagating wave in the time-varying medium. In the slow, adiabatic case (Fig. 1(e)), the effect of dispersion is weak and the adiabatic approximation is accurate. Conversely, for the fast, non-adiabatic modulation ((Fig. 1(f))), the resulting polarization modulation strength is significantly affected by dispersion, as evidenced by the fact that the adiabatic approximation largely overestimates (by more than 100%) the intensity of the generated harmonics.

Another important implication of material dispersion arises in relation to the field boundary conditions at a temporal interface (especially in the context of fast temporal modulations, such as time-switching), which has been the subject of debate in the literature [69,70], and was recently further clarified and reviewed in Refs. [10,71]. Across a spatial boundary the tangential component of E and the normal component of D are continuous, whereas the normal component of E and the tangential component of D are discontinuous (Fig. 2(a))). Interestingly, however, this needs to be revisited in the case of a spatial interface of a spatially-dispersive (nonlocal) material with a spatially extended material response, physically originating from a spread-out charge distribution. This implies that all the fields should be treated as continuous, and gradually changing, across the interface (Fig. 2(b)) [72,73].

 

Fig. 2. (a) At the interface of a spatially-nondispersive (local) material the normal component of the electric field and the tangential component of the displacement field are discontinuous, (b) whereas at the interface of a nonlocal material, with a spatially spread-out material response, these fields should be treated as continuous. (c) Similarly, at the temporal interface of a temporally-nondispersive material, the electric field is discontinuous, (d) whereas at the temporal interface of a temporally-dispersive material, with a temporally spread-out material response, the electric field is continuous due to the non-zero response time of the material. The subscripts "n" and "t" denote the normal and tangential components of the fields, respectively, while the subscripts "1" and "2" denote the fields before and after the spatial/temporal interface, respectively. Full-wave FDTD simulation results (envelope of the electric field) for a time-switched, spatially-homogeneous, (e) nondispersive and (f) dispersive (Lorentz-type) material. Dashed white lines indicate the time-instant when the time-switching of the susceptibility occurs (temporal boundary). For (e), the nondispersive electric susceptibility is switched from 6.42 to 10.05, while for (f) $\omega _p/2\pi$ is switched from 400 THz to 500 THz, with $\omega _0 = 2\pi ~ 430 ~ \textrm {THz}$ and $\gamma = 2\pi ~ 0.02 ~ \textrm {THz}$. In both cases, the incident signal has a bandwidth of 8.8 THz with a central frequency of 400 THz.

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On the other hand, across the temporal interface of a time-switched nondispersive material, D is continuous whereas E is discontinuous (Fig. 2(c)) [1,20,69]. The continuity of D can be directly proven by integrating Ampere’s Law over a vanishing time interval across the temporal boundary. However, similar to the spatially dispersive case, if the time-switched material is temporally dispersive, E will be continuous across the temporal interface as well (Fig. 2(d)) [70,71] due to the non-zero, finite response time of the medium. This can be understood directly from Eq. (1) [70]: due to the finiteness of $\chi$ and $\textbf {E}$, the resulting $\textbf {P}$ is continuous at the temporal boundary even if the integrand of (1) is discontinuous; then, since we already established that $\textbf {D}=\epsilon _0 \textbf {E}+\textbf {P}$ is continuous, $\textbf {E}$ is also continuous. As a result, a propagating wave will effectively experience a relatively gradual change even if time-switching occurs instantaneously. To clarify this point further, in Figs. 2(e) and 2(f), we provide FDTD simulation results (envelope of the electric field) for time-switched non-dispersive and dispersive materials, respectively. It is evident that the electric field experiences a continuous change in its amplitude in the dispersive case even though the material property is switched instantaneously. Hence, special care needs to be taken when determining the boundary conditions in a time-varying system in the presence of dispersion. This is necessary to develop a physically consistent model that is able to accurately predict the behavior of waves at time interfaces, as was already recognized in the literature on rapidly growing plasmas [70]. For instance, the results in Figs. 2(e) and 2(f) indicate that using a non-dispersive model for a dispersive material may lead to an overestimation of the reflection from a temporal interface and, therefore, of the width of momentum bandgaps if the temporal variation is periodic.

3. Power constraints in time-varying materials

As mentioned in the introduction, the space-time duality in optics has been a fertile ground in recent years and many of its aspects have been successfully explored. At the same time, while the wave equation looks the same for temporal and spatial variables and there are exact correspondences between the physical phenomena in the two domains, e.g. diffraction and dispersion, there exists one critical difference between perturbations in time and space. The evolution in space domain is characterized by the wave-vector $\boldsymbol {k}$, whereas temporal evolution involves frequency $\omega$. Since $\hbar \boldsymbol {k}$ is a momentum, to accomplish change in spatial domain one must supply momentum, which is rather simple – any stationary perturbation $\Delta n(r)$ of the refractive index (periodic or not) amounts to a momentum change on the scale of $\Delta n ~ \omega /c$. The relative change of index $\Delta n/n$ can be 100% or more, thus enabling a wide range of modern photonic structures and devices, such as photonic crystals, metasurfaces, etc. What is most important, the momentum is "supplied" without expending energy. In temporal domain, however, $\hbar \omega$ happens to be the energy, hence "shaping" light in temporal domain invariably involves a transfer of energy. Specifically, changing the refractive index in time $\Delta n(t)$ always involves prodigious amounts of energy.

One can recognize the scale of the effort required to change the refractive index by a large amount by first noting that the range of refractive indices for all materials in the visible-near IR range is rather small, somewhere between 1.4 and 3.4 [74]. Low-index materials, like $\mathrm {SiO_2}$, have bandgaps of about 10 eV, while high-index materials, such as GaAs or Si, have bandgaps of about 1 eV. Hence, in order to achieve a large (e.g., 100%) change of index one must change the bandgap by a few eV. This can be confirmed by using the empirical Moss rule [75], which states that $n^4 E_{\mathrm {gap}} \sim 95$ eV for many materials. Now, changing the bandgap in real time by 1 eV implies changing the bonding energy of each bond by a commensurate amount, and with about few times $10^{23}$ $\mathrm {cm^{-3}}$ valence electrons, the energy required to change the refractive index by 10-20% is about $10^4 \mathrm {J/cm^3}$ (electric field in excess of $10^8 \mathrm {V/cm}$), i.e., more than enough to cause a breakdown. Furthermore, even if only as little as 0.1% of the required energy ends up being dissipated inside the material, it would take less than 100 modulation cycles to raise the temperature by 1000 K and melt the material. This suggests that, for relatively large values of $\Delta n/n$, only individual or infrequent changes to the refractive index, as for example in time-switched metamaterials [16,76,77], may be feasible.

Getting into the specifics, different mechanisms can be conceivably used to change the refractive index, but most of them, such as electro-optic, acousto-optic, let alone thermo-optic, are limited in speed, so no photonic time crystals at optical frequencies can arguably be realized using them, as they require modulation frequencies on the order of the propagating wave frequency (instead, the first two of these mechanisms may be used in applications such as time-modulated optical isolators, which require modulation frequencies significantly smaller than the operational frequency, just enough to couple different modes [13]). Neither electro-optic nor acousto-optic modulation can change the refractive index by more than a fraction of a percent, and the speed of electro-optic modulators can reach 100’s of GHz at best [78].

All-optical modulation of the refractive index is the method most actively pursued in this context [79]. Confirming our back-of-the-envelope estimates made above, it is well-established that all attempts to change the refractive index via the instantaneous third-order optical Kerr effect result in $\Delta n/n$ saturating at less than 1% at power densities well in excess of 1 TW/$\mathrm {cm^2}$ [80]. Such power densities are available only in femtosecond laser pulses with low duty cycle, hence one cannot expect to realize photonic time crystals with non-negligible momentum bandgaps (or other time-varying systems with deep modulations) using conventional nonlinear optical materials.

Recently, however, different types of nonlinearities have been the subject of great interest – those in transparent conductive oxides (TCO’s), such as ITO [46,81,82] and AZO [47,83], where much larger relative changes of the index have been achieved close to the epsilon-near-zero frequency. The dominant nonlinearities in TCOs have a different origin from the familiar Kerr materials, and they are associated with the non-parabolic dispersion of the conduction band and the resulting change of the average effective mass of the electron sea due to intraband absorption [59,84,85]. Large (on the order of 100%) changes of refractive index have been indeed observed in these materials, and these changes are fast (few 100’s of fs), albeit still two-to-three orders of magnitude too slow to achieve effects such as time reversal through modulation at twice the signal frequency. Faster nonlinearities are available, but are usually much weaker as discussed below. One can estimate the power requirements for TCO’s from simple considerations: the change in refractive index or permittivity originates from the change of avarage effective mass, which itself depends on the energy inside the band [59]. Therefore, in the most optimistic case, the relative change of the permittivity can be found to be proportional to the relative change $\delta U$ of the energy of carriers:

 

Fig. 3. Relative momentum-bandgap size (gap/mid-gap ratio) for a frequency-dispersive photonic time crystal, as a function of relative permittivity change (lower horizontal axis) and the corresponding required power density (upper horizontal axis) calculated from Eq. (5) for a modulation frequency $\omega _{\mathrm {mod}} = 10^{13} \mathrm {rad/s}$. A lossless Drude-type dispersive material is assumed and its plasma frequency $\omega _p$ is periodically varied in time as $\omega _p^2 = \omega _{p0}^2 + \Delta \omega _p^2(\cos {(\omega _{\mathrm {mod}}t)}-1)$, where $\omega _{p0} = 1.225\omega _p$, $\Delta \omega _p = 0.707\omega _p$, and $\omega _{\mathrm {mod}} = 4\omega _p$. The insets show the $\omega -k$ dispersion diagram of the photonic time crystal for two values of periodic permittivity modulation, $\Delta \epsilon$, equal to $0.667\epsilon _0$ and $0.333\epsilon _0$ (at frequency $\omega _{p0}$), which both require a supplied power density of more than 50 $\mathrm {TW/cm^3}$ (by assuming a conducting material with $\epsilon _\infty = 1$, $\omega _p = 2.5 \times 10^{12} \mathrm {rad/s}$, $N = 10^{20} \mathrm {cm^{-3}}$, and $E_F = \hbar v_F k_F = 1$ $\mathrm {eV}$). The light-red shaded regions denote the partial momentum bandgaps.

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Moreover, when the time crystal is formed all-optically via the third-order susceptibility $\chi ^{(3)}$, the time-crystal related effects are almost always obscured by the less exotic phase-conjugation phenomena. Consider the general four-wave mixing process in which two counter-propagating pump fields $E_{\mathrm {p\pm }} = A_{\mathrm {p}} e^{j(\pm \boldsymbol {k}_\mathrm {p}\cdot \boldsymbol {r}-\omega t)}+c.c.$ (where $c.c.$ stands for complex conjugate) interact with the signal wave $E_{\mathrm {s}} = A_\mathrm {s} e^{j(\boldsymbol {k}_\mathrm {s}\cdot \boldsymbol {r}-\omega t)}+c.c.$, and generate a third-order nonlinear polarization $P_\mathrm {NL}$, which includes three terms proportional to $\chi ^{(3)}$ due to the mixing of these three waves [86]. While at first glance it may appear that all three terms are equal, this is certainly not the case. The order in which the waves mix is critical.

The first term is $P_{\mathrm {TR}} \sim \chi ^{(3)}(\omega ; -\omega, \omega, \omega )A_{\mathrm {p+}}A_{\mathrm {p-}}A_\mathrm {s}^*$ where the subscript "TR" stands for "time reversal". What this physically means is that the two pump waves mix and produce a uniform intensity pattern oscillating at frequency $2\omega$, which modulates the index at this frequency thus forming a time crystal in which Floquet states become possible. The signal wave then scatters off the time crystal into a time-reversed wave. Note that the origin of this "fast" nonlinearity in TCOs is ballistic motion of electrons inside the non-parabolic conduction band and this ultrafast nonlinearity is relatively weak, as discussed in Ref. [85].

The other two terms are $P_{\mathrm {DG}} \sim \chi ^{(3)}(\omega ; \omega, -\omega, \omega )[A_{\mathrm {p+}}A_{\mathrm {s}}^*A_{\mathrm {p-}} + A_{\mathrm {p-}}A_{\mathrm {s}}^*A_{\mathrm {p+}}]$, where the subscript "DR" stands for "dynamic grating" [87]. Here, it is the interference of pump and signal beams that produces the grating. For a CW pump and probe, the grating is stationary, but, since in most experiments short pulses are used, the grating is actually dynamic on the scale of the pulse length, which, compared to optical frequencies, is quite slow. As already mentioned, the origin of this "slow" nonlinearity is an increase in the energy inside the nonparabolic band (increase of electron temperature) due to absorption [85], and the strength of this grating is determined by the characteristic response time of the medium. In TCO’s, this response time is the time it takes to cool photoexcited electrons to the lattice temperature $\tau _{\mathrm {el}}$ (typically on the scale of few hundreds of fs) [85]. The "slow" nonlinearity $\chi ^{(3)}(\omega ; \omega, -\omega, \omega )$ is much larger than the fast nonlinearity $\chi ^{(3)}(\omega ; -\omega, \omega, \omega )$, because the index change is integrated over the shorter of pulse length and characteristic time. In particular, in TCO’s the ratio of "slow" to "fast" nonlinearities is on the scale of $2\tau _{\mathrm {el}}/\tau _\mathrm {s}$, where $\tau _\mathrm {s}$ is a momentum scattering time on the scale of a few fs. Hence, the "slow" nonlinearity is about 100 times stronger than fast one. In the more general case, the ratio of the two nonlinearities is $2T_1/T_2$, where $T_1$ is the characteristic lifetime and $T_2$ is the (usually much shorter) coherence time in the nonlinear medium. Therefore, in a medium like TCO and many others the response is always dominated by the relatively slow dynamic grating and not by the time crystal oscillating at $2\omega$. We want to stress that, as far as measurements go, the two responses are externally indistinguishable. One can modify the geometry and use a very thin sub-wavelength layer of nonlinear medium to obtain a forward-propagating "negative refraction" wave in addition to the backward-propagating "phase conjugated" wave [88,89], but the response due to the slow dynamic grating will always dominate. The only way to obtain a "pure" time crystal response is to operate near the two-photon absorption edge in transparent materials, such as Si or GaAs, to maximize $\chi ^{(3)}(\omega ; -\omega, \omega, \omega )$ while avoiding absorption associated with the slow nonlinearity $\chi ^{(3)}(\omega ; \omega, -\omega, \omega )$, but even then the effect remains quite weak.

It is our belief that one of the most promising paths to observe a time crystal in the optical range is to use not the third but the second order ($\chi ^{(2)}$) nonlinear process, in which a standing pump wave at the second harmonic frequency $2\omega$ interacts with counter-propagating waves at the fundamental frequency, which are of course time reversed. This process is instantaneous and is not obscured by spurious dynamic-grating effects. Such an arrangement was first proposed in 1996 [90,91] and named “Transversely Pumped Counterpropagating Optical Parametric Oscillation and Amplification” (TCOPA). It was demonstrated a decade later [92] when twin counter-propagating photons – time-reversed replicas of each other – were produced. The advantage of the TCOPA scheme is that the pump wave is contained inside a resonant cavity which increases the amplitude, $E_{2\omega }$, and thus the effective index change $\Delta n \sim 1/2 ~ \chi ^{(2)}E_{2\omega }$. The transverse geometry facilitates effective phase matching in cubic III-V crystals with $\chi ^{(2)} > 100$ pm/V, hence one can effectively achieve $\Delta n/n$ of a few percent (and momentum bandgaps of commensurate relative size) with a power density of less than 100 GW/$\mathrm {cm^2}$ inside the cavity, i.e., with only a few GW/$\mathrm {cm^2}$ of incident power.

4. Conclusions

In summary, in this article we have discussed various aspects of time-varying photonics that have important implications for the modeling and practical realization of photonic time-varying materials and time crystals. While the space-time duality in wave physics has provided valuable insight and facilitated the discovery of various new effects, the temporal domain is fundamentally different from the space domain due to its direct relation to the principles of causality and energy conservation: (i) causality implies that any material (except vacuum) has a non-instantaneous, dispersive, electromagnetic response, and (ii) temporal evolution is characterized by frequency $\omega$ and, therefore, shaping light in the temporal domain requires a transfer of energy $\hbar \omega$.

In the first half of this article, we have reviewed how temporal dispersion is modeled in time-varying systems and how it affects the physical behavior of fields in temporally modulated materials and at temporal interfaces. We showed that, in a dispersive time-varying scenario, a trade-off typically exists between the modulation speed and the strength of the resulting polarization density modulation and, hence, of various effects that depend on it. An adiabatic approximation may lead to largely overestimating these effects. Moreover, in dispersive time-varying materials, all fields are continuous across a temporal interface even when the material properties are modified instantaneously. Then, in the second part of the article, we have discussed the energy requirements to change the refractive index in time based on general considerations and for specific material platforms, focusing on epsilon-near-zero media such as transparent conducting oxides. We found that a modulation sufficiently large and fast to observe a time-crystal response in the optical range requires very high levels of input power, available only in femtosecond lasers, which makes the observation of these effects possible but challenging, and limited to short times. Moreover, we argued that time-crystal effects based on third-order nonlinearities are almost always obscured by less exotic phase-conjugation phenomena, whereas second-order nonlinearities may offer an interesting path to observe these effects at optical frequencies.

We believe that the rich physics of time-varying systems may open intriguing opportunities for new and advanced functionalities in photonics, which the research efforts so far have provided a valuable glimpse into. However, physical considerations, constraints, and experimental challenges must be carefully assessed before useful applications can emerge from these research efforts.