Active devices can enable a wealth of advanced functions in photon manipulation, including frequency conversion [1–4], amplification [5–7], and isolation [8–15], which can hardly be achieved by conventional passive devices. Instead of realizing new functions based on established mechanisms, time has merged as a new degree of freedom to control waves [16,17]. The periodic modulation of optical properties in time and space enables photonic interband transition [18], quasi-phase matching for momentum and energy [19], unidirectional amplification [20], dispersion engineering [7,21], and parity-time (PT)-phase transition [6,22]. Several temporal counterparts of the physical effects are unveiled, e.g., Fresnel drag [23], Anderson localization [24], and Wood anomalies [25]. In addition, from the perspective of topology, time modulation enables a new synthetic dimension to excite high-dimensional topological states [26–30].

Despite the thriving development in theory, the realization of fast modulation becomes a formidable challenge that prevents active metamaterials and metasurfaces from being used in practical applications. Many efforts have been made to realize dynamic modulation by new materials and advanced nanofabrication [1,2,8–10,31–34]. The sudden change in the refractive index on the order of the subpicosecond has been realized by photonic excitation [31]. It enables frequency conversion at near-infrared frequencies [1]. However, some applications, e.g., isolation, require the periodic modulation of material properties in space and time simultaneously. The periodic modulation with a frequency ranging from several GHz to THz has been demonstrated based on photoelectric [8], photoacoustic [9,13], and nonlinear Kerr effects [10]. After all, the applications based on dynamic modulation are still hampered by the implementation of fast modulation at infrared and optical frequencies.

Parametric amplification (PA) is an irreplaceable mechanism that is used to generate coherent light at frequencies that are not sustained by the gain media in nature. In the past decades, PA has also revolutionized the generation of isolated attosecond pulses and is promoting the development of strong field physics [35]. Compared with amplifications based on semiconductor and rare-earth-doped media, PA is phase-sensitive and can reduce noise approaching quantum limits by locking the phase between pump and idler waves [36–39]. At present, PA at optical and infrared frequencies relies on second-order or third-order susceptibility of nonlinear materials. Due to the weak nature of nonlinearity, high-intensity and bulky pump sources are usually required, which hinders the integration and miniaturization of devices.

On the other hand, PA can be realized by time modulation. This mechanism is widely used at microwave frequencies. However, it is rarely employed at high frequencies since it requires a modulation frequency comparable to the operating frequency [40,41]. Such a frequency limitation is enforced by the Manley–Rowe relations [42]. To relax the restriction on the modulation frequency, we introduce the spatiotemporal modulation to realize cascaded phase matching between the signal and the amplified fundamental modes at half of the modulation frequency. As shown in Figs. 1(a) and 1(b), the permittivity of the medium is periodically modulated in both space and time. It enables the phase matching of energy and momentum at the same time. Hence, by choosing proper modulation frequencies, efficient cascaded frequency conversion is possible. As a result, a signal with the frequency ωinc much higher than the modulation frequency Ω can be amplified in a cascaded manner [Figs. 1(c) and 1(d)]. This remarkable advantage makes the realization of PA in a time-modulated system possible by using a low-frequency modulation.

In this work, we first show the eigenstate and eigenvalue of space-time modulated media. Second, the realization of cascaded PA is demonstrated by calculating the scattering coefficients of a space-time modulated slab. Next, the lasing nature of cascaded PA and the characteristics of gain mechanisms are analyzed. Finally, the cascaded parametric oscillator and the Si-waveguide-based cascaded PA are presented.

First, we show PA supported by time-modulated media with permittivity ε(t)=εs(1+2αcosΩt), where α is the modulation strength, Ω is the modulation frequency, εs is the background permittivity, and vs=(εsμ0)−1/2 is the phase velocity of unmodulated signal waves. The dispersion relation of the time-modulated media is shown in Fig. 2(a). Unlike the traditional photonic crystals, which open bandgaps in the frequency domain, the periodic time modulation forms the inverted bandgaps in the k space. In the inverted bandgaps, eigenfrequencies are complex numbers, and the corresponding eigenstate will be amplified with time, which is the well-known PA. The imaginary part of the eigenfrequency corresponds to the exponential growth rate of the PA and is used for characterizing the parametric gain of the time-modulated system.

From a different perspective, the temporal modulation with pump frequency Ω couples the forward propagating signal and the backward propagating idler at the frequencies of Ω/2 and −Ω/2, respectively [Fig. 2(a)]. The presence of signal waves stimulates the downward frequency conversion leading to the excitation of the idler waves, which in turn amplifies the signal waves through the up-conversion process. Hence, the generation of the signal and idler waves is reinforced by each other, and PA is supported.

As shown in Fig. 2(a), the inverted bandgaps also exist at high harmonic frequencies, e.g., 5.5 Ω, due to the repetition of dispersion relation beyond the first Brillouin zone. However, the magnitude of the electric field at high harmonic frequencies remains small for the modulation strength reaching up to 10% of static permittivity [Fig. 2(b)]. This is because the high harmonic modes are not the allowed modes in the unmodulated media, and their existence relies on the small perturbation. To amplify the signal at a frequency higher than the modulation frequency, the phase mismatching between the amplifying state and the light line needs to be compensated [shown as the blue arrow in Fig. 2(a)].

To compensate for the phase mismatching, we apply spatiotemporal modulation to amplify the signal at the frequency of 5.5Ω. We consider the media with permittivity ε(x,t)=εs(1+2αcosΩt+2βcosKx), where K=5Ks. As shown in Fig. 2(c), the inverted bandgaps create copies of themselves by shifting K due to the introduction of spatial modulation. As a result, the inverted bandgap appears near the light line. However, the amplitude of the high harmonic components is still small [Fig. 2(d)]. The reason for the inefficient amplification at the high frequencies can be attributed to the indirect coupling between the fundamental mode and the eigenstates of the unperturbed media. Hence, to increase the amplitude of the harmonic mode at the high frequency, proper spatial and temporal modulation frequencies should be introduced to directly couple the amplifying state and eigenstates near the light line.

In addition to PA, a distinct amplifying process is found to be available by introducing the traveling-wave-like modulation [43]. It has been demonstrated that the number of harmonic components is exponentially increased along the propagation if the modulation velocity equals the phase velocity of the unmodulated signal waves. Such an amplification process, named as luminal amplification (LA), leads to the exponential growth of energy along the propagation [43]. Next, we utilize LA to realize efficient cascaded PA.

Here, we consider a dispersionless space-time modulated medium with permittivity ε(x,t)=εs[1+2αcos(Kx−Ωt)+2βcosΩt],(1)where α and β are the modulation strengths. The modulation velocity is defined as vm=Ω/K. The second term in Eq. (1) represents the luminal modulation if the modulation velocity is within the range (1+2α+2β)−1/2≤vm/vs≤(1−2α−2β)−1/2. In this regime, the phase synchronism of all the harmonic components enables efficient frequency conversion in a cascaded manner for sinusoidal modulation. The capability of the efficient frequency conversion is described by the luminal gain in Appendix C. If the luminal gain is comparable with the parametric gain, the energy of the amplified fundamental mode should be efficiently transferred to the harmonic modes at high frequencies.

In Fig. 3, we show the eigenfrequencies and eigenstates of media with permittivity described by Eq. (1).

As shown in Fig. 3(a), multiple inverted and normal photonic bandgaps arise. They originate from the coupling between forward and backward propagating waves. As a reference, the dispersion relation of media with α, β→0 is shown as gray solid lines in Fig. 3(a). It satisfies ωm,n=±vskm, where (m, n) indicates the order of the harmonic modes, ωm,n=ω+(m+n)Ω is the corresponding harmonic frequency, and km=k+mK is the spatial frequency of the harmonic mode. As an example, Table 1 shows the interacting harmonic modes that form the photonic bandgaps. The coupling frequency and wavenumber are denoted as Δω and Δk, respectively. In general, the inverted (normal) bandgap will be generated if the modulation velocity is Δω/Δk>vs (Δω/Δk<vs), which corresponds to the superluminal (subluminal) modulation [42,44].Table 1.

Type of Photonic Bandgap^a

By increasing the modulation velocity from the subluminal regime to the luminal regime, the normal and inverted bandgaps are merged, as shown in Fig. 3(b). To further increase vm, the bandgaps are separated again [Fig. 3(c)].

The eigenstate of media with vm=vs is shown in Fig. 3(d). The amplitude of the high harmonic components is significantly improved. The corresponding wavenumbers are shown in Fig. 3(e). It is found that the amplitude of the harmonic components near the light lines is significantly larger than that of other modes.

To further check the performance of cascaded PA based on luminal modulation, we calculate the transmission from a spatiotemporal modulated slab.

In Fig. 4(a), the magnitudes of the reflection and the transmission coefficients increase with the modulation length. Due to the scattering loss, the system is static when the modulation length Lm is smaller than the threshold length lc. A similar relationship exists in time-varying media [see Fig. 9(a) in Appendix B].

When Lm>lc, the system is unstable. As shown in Fig. 4(b), the signal at the frequency of 10.5Ω is exponentially amplified after a period of time. To estimate the starting point of lasing, we fit the exponential growth rate of transmission waves. The cross point between the fitted line and static amplitude at the beginning is defined as the critical time for lasing, denoted as tc. After the critical time point, the amplifying state becomes the dominant one while the static states can be ignored.

The corresponding transmission spectrum is shown in Fig. 4(c). Since the high harmonic components are amplified by absorbing energy from the fundamental mode, the transmission spectrum is normalized by the amplitude of the electric field at the fundamental frequency, i.e., Etra(Ω/2), to characterize the luminal gain. A series of harmonic components are amplified at the same time. It demonstrates that the proposed modulation can amplify signals by a low-frequency pump. The proposed modulation for PA can be employed to generate a frequency comb.

We also solved the scattering coefficients of the amplifying state based on the Bloch–Floquet theory. In fact, the lasing state corresponds to a singular state of a space-time modulated slab. Hence, the scattering problem can be solved by an eigenvalue problem without incidence (for more details refer to Appendix A). The numerical and analytical results are in good agreement, as shown in Fig. 4(c).

The temporal and the spectral responses of cascaded PA show both characteristics of LA and PA [Figs. 4(b) and 4(c)]. In this section, the gain mechanisms of the cascaded PA through the space-time modulated media are analyzed.

The singular nature of the lasing state implies that the unstable state can be excited by the incidence at any harmonic frequency with non-negligible amplitude in Fig. 4(c). To demonstrate this, we calculate the response of the same modulation excited by an incidence at different harmonic frequencies based on finite-difference time-domain (FDTD) simulation and compare it with the analytical results. As shown in Fig. 5(a), the fitted exponential growth rate based on numerical simulations fluctuates around the theoretical results [see the gray dashed line in Fig. 5(a)]. The corresponding eigenstates are compared in Fig. 5(b). The consistency between numerical and analytical results further proves that the same state is excited. The only difference that is shown in the numerical results [Fig. 5(a)] is that the critical time point of lasing is delayed for input at high harmonic frequencies.

In addition to the modulation strength β, the parametric gain also depends on the modulation length. As shown in Fig. 5(c), the exponential growth rate increases with the modulation length, and the parametric gain tends to be saturated for a lengthened modulation range. The same behavior appears in PA [Fig. 9(c)]. The eigenstate with a larger exponential growth rate can be excited when the scattering loss in the slab is suppressed.

On the other hand, LA behaves as an efficient frequency converter in cascaded PA. As shown in Fig. 5(d), the highest order of frequency conversion in cascaded PA increases with the modulation strength α. The corresponding exponential growth rates ωi are shown in Fig. 11(a) in Appendix D. We assume that the harmonic components are efficiently amplified if the amplitude is larger than one percent of the magnitude at the fundamental frequency. For β=0, the luminal gain can be simply evaluated by the exponential relationship between the number of excited harmonic modes and the propagation length (for more details refer to Appendix C). However, for β≠0, the highest order of frequency conversion in cascaded PA does not exponentially increase with propagation [see Fig. 11(b) in Appendix D]. This can be attributed to the coupling with the time-growing state. The highest order of frequency conversion presents the capability to transfer energy from the amplified fundamental mode to high harmonic components, which are different from traditional LA [43].

To reduce the lasing threshold and improve the performance of the amplifier, we design a cascaded parametric oscillator (CPO) composed of a space-time modulated slab and a cavity formed by photonic crystals, as shown in Fig. 6(a). Here, the photonic crystals play a role in high reflection mirrors for all harmonic components. The resonant conditions are satisfied when the phase change of a round trip in the cavity equals 2mπ for m∈Z [see Fig. 12(b) in Appendix E]. The responses of the CPO with the varying length of cavities are presented in Fig. 6(b). It is found that the signal can be amplified when the phase conditions of resonance are fulfilled. The threshold length of lasing is decreased to 0.07lPA and is much smaller than the threshold length of lasing without a cavity.

In addition, we design a CPO operating at a single frequency. The photonic crystals are designed to only allow the transmission at the frequency of 10.5Ω and serve as high reflection mirrors for other harmonic components. The results show that the signal is amplified with time [Fig. 6(c)], and no parasite harmonic modes transmit through the CPO [Fig. 6(d)].

Intense amplifiers and light sources at mid-infrared frequencies are extremely desired for the applications of sensing and communications due to the strong characteristic resonances of molecules and the existence of atmospheric transmission windows [45]. Benefiting from the non-dispersive nature of silicon beyond ∼1100nm, silicon waveguides become the superior platform to achieve cascaded PA. As shown in Fig. 7(a), the dispersion curve of the TE0 mode is almost linear. The modulation velocity vm is set to equal the fitted group velocity of the guided mode to realize cascaded PA.

The signal at the frequency of 5.5Ω can be amplified by imposing spatiotemporal modulation given by Eq. (1), as shown in the inset of Fig. 7(b). The results show that cascaded PA is immune to the moderate dispersion of the guided mode. The spectrum of the transmission field is presented in Fig. 7(b).

In summary, we reported that PA can be achieved in a spatiotemporal modulated medium with a modulation frequency much lower than the oscillation frequency of electromagnetic waves. Benefitting from the strong interaction between harmonic modes based on luminal modulation, it has been demonstrated that the energy can be efficiently transferred from the amplified fundamental mode to the signal in a cascading manner. It is found that the lasing state cannot be excited if the parametric gain is unable to compensate for the scattering loss. Based on the characteristics of PA and LA, we presented that the parametric gain can be evaluated by the exponential growth rate of the amplifying state, and the luminal gain can be assessed through the efficiency of harmonic conversion from the amplified fundamental mode. Furthermore, we have shown that the threshold of cascaded PA can be significantly reduced by introducing a cavity resonating at all harmonic frequencies. Finally, realistic implementation of cascaded PA based on a Si waveguide is proposed. We show that the cascaded PA is immune to the moderate dispersion of the guided mode. Our design not only provides a practical solution for amplification and radiation based on spatiotemporal modulation at infrared and optical frequencies but also opens up a new opportunity to control, for example, spontaneous emission [46–49].

Acknowledgment. Q. Y. and Y. L. conceived the original idea. Q. Y. performed the numerical and analytical calculations, and all authors discussed the results and commented on the analyses. Q. Y. wrote the manuscript, and all authors provided feedback.