Since the discovery of photonic crystals (PCs) that support energy band gaps (EBGs) [1,2], the design of materials with photonic band structures has led to the realization of test platforms for fundamental physics and broad engineering applications [3]. These advancements can be clearly seen in the observation of exotic light propagation characteristics, such as the realization of left-handed materials [4], extreme group and phase velocities [5,6], and topologically protected photonic edge states [7,8]. In addition, the EBGs in PCs have been utilized to create integrated and miniaturized photonic platforms for manipulating the flow of light [9]. Within the frequency range of photonic EBGs, photons are localized in the defective region of spatially periodic structures due to the absence of available photonic states in the surroundings. The ability to confine photons on a wavelength scale has been exploited in diverse areas of physics and engineering, especially for enhancing spontaneous emission of photons [1,2,10], implementing micro/nanolasers [11,12], boosting optical nonlinear effects [13,14], and developing photonic crystal fibers [15] and on-chip photonic integrated circuits [16–18].

In line with the diverse research activities on PCs, time-variant media have been investigated due to their distinct functionalities, such as nonreciprocal propagation, frequency conversion, Floquet engineering, and light amplification [19–28]. Among these, one of the most significant developments is the observation of nonreciprocal wave propagation in media with traveling-wave-like spatiotemporal modulation. In such media, an asymmetrical band structure can be formed with respect to reversal of a wavevector, and this asymmetry makes it possible to observe the nonreciprocal propagation of waves [19–23]. Furthermore, spatiotemporal modulation can modify not only the symmetry of band structures but also their topological properties. For example, warping of the band structure by spatiotemporal modulation can change the Chern number of bands, allowing the observation of Floquet topological insulators [24,25] and the modulation-induced anomalous Hall effect [26,27]. All these advancements indicate that spatiotemporal modulation of media has the potential to provide an unprecedented approach to designing band structures for photons, phonons, and potentially electrons.

Compared with spatially periodic PCs, a completely distinct dispersion relation has been predicted for media with a temporally periodic property [29–31]. In these media, momentum band gaps (MBGs) can be opened, leading to parametric amplification of electromagnetic waves. The concept of MBGs dates back to the 1960s; in the seminal work by Cassedy [32,33], traveling-wave-type permittivity modulation was considered for an otherwise spatially homogeneous medium. Particularly, it was shown that MBGs emerge when a medium is modulated in the form of a simple harmonic wave at a superluminal phase velocity, and unstable solutions of exponentially growing and decaying oscillatory waves with complex-valued frequencies exist [34,35]. Since then, discussions have been extended to take into account more complicated cases of a slab with finite thickness [36–39], intrinsically dispersive media [40], and even phononic lattices [23,41]. These early works showed that MBGs are opened when positive-frequency and negative-frequency branches of an unmodulated dispersion curve are coupled via the modulating harmonic wave.

Previously, the properties of time-variant media were investigated mostly for cases in which the spatiotemporal modulation was in the form of a simple harmonic wave. However, such an assumption is inadequate for the description of potential configurations with sophisticated spatial structures. Here we generalize the analysis to include spatiotemporal crystals (SCs), i.e., time-variant media with PC-like spatial structures, by considering multiple spatial frequencies of the spatiotemporal modulation. This generalization allows us to take full advantage of the dispersion engineering capability in PCs, as well as the additional degree of freedom from spatiotemporal modulation. Therefore, the design of SCs makes it possible to engineer the MBG within which parametric amplification occurs. As a platform for the construction of a parametric oscillator, a finite-sized SC is proposed and analyzed. The SC is composed of a one-dimensional array of thin slabs with temporally periodically modulated permittivity. Characterization of the finite-sized SC reveals that parametric oscillation is observed when a transition threshold is reached by increasing the normalized permittivity variation. Furthermore, spatial phase control of the temporal modulation leads to asymmetric formation of MBGs and directional radiation of oscillations at distinct frequencies from the finite-sized spatiotemporal crystal.

The periodic temporal modulation of the permittivity allows the existence of additional temporally scattered modes. The frequencies of these modes are shifted by integer multiples of the modulation frequency from the original frequencies. The collection of these additional modes is represented by dispersion curves that are shifted with respect to that calculated from a time-invariant PC. Consequently, the initial and all of the shifted dispersion curves may intersect each other [as schematically shown in the right panels of Fig. 1(d)]. Near the crossings where mode coupling exists, the modes can repel each other to create an MBG or an EBG. Note that in this paper, the term band gap is used to describe both band gaps and mode gaps. Specifically, MBGs are observed when strongly interacting modes in the dispersion curves have frequencies of opposite signs [for example, see A and B in Fig. 2(b)], while EBGs are observed when strongly interacting modes in the dispersion curves have frequencies of the same sign [for example, see B in Fig. 2(d)]. This is consistent with previous knowledge on the conditions for band gap formation [35]. Superluminal modulation mediates interactions between the modes with frequencies of opposite signs, whereas subluminal modulation mediates interactions between the modes with frequencies of the same sign. The opening of EBGs or MBGs is clearly illustrated in the middle panels of Figs. 2(b)–2(d). Depending on the Floquet–Bloch modes participating in the interaction, an MBG can be labeled with (m1, s1; m2, s2). For example, the MBGs denoted by A and B in Fig. 2(b) are formed as a result of the interaction between the positive- and negative-frequency lowest-order modes and are classified as (1, 0; −1, 1) and (1, 1; −1, 2), respectively.

When considering only the lowest-order photonic band that starts from the origin of the dispersion diagram, MBGs are generally located at odd integer multiples of half the modulation frequency, which can be simply inferred from the axial symmetry of the lowest-order band of a PC with respect to reversal of frequency. Because of the spatial periodicity, however, the lowest-order photonic band deviates from the linear relation with increasing wavenumber and exhibits zero group velocity at the band edge (vg=∂ω/∂k→0). Therefore, over a certain modulation frequency (i.e., fm>fc, where fc denotes the cutoff frequency of the lowest-order PC band), the dispersiveness of the lowest-order photonic band results in the absence of MBGs at the odd integer multiples of half the modulation frequency [for example, note the absence of an MBG at fm/2 in Fig. 2(d)]. Furthermore, the presence of higher-order photonic bands in a PC leads to the formation of MBGs whose frequencies cannot be expressed as odd integer multiples of half the modulation frequency [for example, see the MBG denoted by A in Figs. 2(c) and 2(d)]. The MBG frequencies are not expressed as odd integer multiples of half the modulation frequency when the following two conditions are satisfied: (1) m1m2<0 and (2) |m1|≠|m2|. The first condition states that the interacting modes should have frequencies of opposite signs, which is a necessary condition for the formation of an MBG. The second condition requires that the interacting modes have different mode orders. In this case, the two interacting modes generally do not form a standing wave and thus create an MBG with nonzero group velocity [for example, see the MBG denoted by A in Fig. 2(e)]. Exceptions may exist in the case of accidental group velocity matching (vg,m1=−vg,m2); for example, see the MBG formed at the edge of the Brillouin zone denoted by B in Fig. 2(e).

Within the MBG, there exist two eigenfrequencies that are complex conjugates of each other with the same real part (describing the MBG frequency); one of the two imaginary parts is related to the parametric gain, while the other is related to the loss. The imaginary eigenfrequency, which gives the parametric gain within the MBG, is plotted below the bottom panels of Fig. 2. The two-dimensional Fourier-transformed field shows agglomerated spots within the MBG, which is related to the exponentially growing field [for example, see the bottom panel in Fig. 2(b)]. From the simulations, the width of the MBG and the imaginary eigenfrequency are found to increase with the normalized permittivity variation. When considered with the radiative loss of a finite-sized SC, the parametric gain determines the transition threshold for oscillation.

Once the normalized permittivity variation becomes larger than the transition threshold [Δε/εc]th,N, the parametric oscillation is initiated from the seed field, as the radiative loss of the finite-sized SC is compensated by the parametric gain at MBG frequencies. The seed field is generated from the transient evolution of the input field in the SC. In realistic situations, the seed field for parametric oscillation can also be provided by any noise process. Because no intrinsic loss is assumed in these examples, the total loss is determined solely by the radiative loss of the finite-sized SC. Qualitatively, the conclusions derived in this paper are not affected by the addition of material loss and dispersion. Above the transition threshold, the field amplitudes at MBG frequencies grow exponentially with normalized permittivity variation [see Fig. 4(b) for the finite-sized SC with N=8]. Similar tendencies are observed, but with a lower transition threshold, for the finite-sized SC with N=10, as the parametric gain tends to increase with the number of unit cells in the finite-sized SC [Fig. 4(c)].

After the initial transient evolution of the input field, the transmitted field can be decomposed into frequency mixing and parametrically oscillating components. Specifically, the exponentially growing field can be expanded as ∑qaqe2πgqte−j2π[(2q−1)/2]fmt+c.c., where gq describes the temporal field growth rate. The transition from frequency mixing to parametric oscillations can be clearly seen by plotting two-dimensional maps of the temporal field growth rate gq at MBG frequencies. These temporal field growth rates are plotted in Fig. 4(d) as a function of the normalized permittivity variation and the number of unit cells, from which the transition threshold values can be determined. Below the transition threshold [Δε/εc]th,N, where the frequency mixing dominates, the temporal field growth rate is zero [blue shaded region in Fig. 4(d)]. Above the transition threshold, the transmitted field amplitude grows exponentially in time, with major frequency components associated with the MBGs. From the mapping, several observations can be made: (1) the transition threshold tends to decrease with the number of unit cells, (2) the rises and falls in the transition threshold with the number of unit cells are related to the change in the spatial mode overlap with the structure, and (3) the temporal field growth rate increases monotonically with the normalized permittivity variation above the transition threshold.

SCs are a special type of PC that can be constructed by implementing temporal permittivity variation in PCs. SCs are characterized by richer dispersion characteristics and distinctive functionalities compared to PCs. The most intriguing features of time-variant media such as SCs are the formation of MBGs and the observation of parametric oscillations. Compared with time-variant media modulated in the form of a simple harmonic wave, the design of SCs makes it possible to engineer the properties of MBGs within which parametric amplification occurs. For example, the position, width, and slope of an MBG can be predicted by band structure analysis, such as the ST-PWEM and numerical simulations. Based on the understanding of MBG formation in SCs, we have shown that a finite-sized SC can be configured as a parametric oscillator. The characterization of finite-sized SCs shows that the transition from frequency mixing to parametric oscillation occurs above the transition threshold determined by the normalized permittivity variation and the number of unit cells. When combined with the traveling-wave-like modulation, the MBGs are asymmetrically positioned with respect to reversal of a wavevector, and direction-dependent frequencies of parametric oscillations observed. With the added temporal controllability, the proposed structure would enable simultaneous engineering of energy and MBGs and provide a guideline for implementation of advanced dispersion-engineered parametric oscillators [47–50].

Acknowledgment. B.M., S.L., and J.P. conceived the original idea. S.L. and J.P. performed numerical calculations and theoretical analyses. S.L., J.P., H.C., B.K., Y.W., C.D., and B.M. discussed the results and commented on the analyses. J.P., B.M., S.L., and C.D. wrote the manuscript, and all authors provided feedback.