Fig. 1. (a) Schematic of scattering from a space-time modulated slab. (b) The profile of space-time modulated permittivity. (c) The signal can be amplified with time by compensating for the phase mismatching through spatiotemporal modulation. (d) The transmission spectrum. A series of harmonic components with spaced frequency Ω are excited, and the signal is amplified in a cascaded manner.

Fig. 2. PA in time-varying media and quasi-phase matching for amplification at the high harmonic frequencies. (a) The dispersion relation of the photonic Floquet media with sinusoidally modulated permittivity ε(t)=εs (1+2α  cos Ωt), where α=0.1, εs=1.462ε0, and Ks=Ω/vs. The shadowed area shows the first Brillouin zone. The red dashed lines indicate the light lines and the blue arrow shows the phase mismatching between the inverted bandgap and the light line. (b) The amplitude of the high harmonic modes as the modulation strength α increases. (c) By introducing spatial modulation with permittivity ε(x,t)=εs(1+2α  cos Ωt+2β  cos Kx), the inverted bandgaps generate copies of themselves by shifting K. (d) The amplitude of the high harmonic modes as the modulation strength β increases, where α=0.01. Note that the corresponding eigenfrequencies of the eigenstates shown in (b) and (d) are complex numbers.

Fig. 3. Eigenvalues and eigenstates of space-time modulated media. The dispersion relation of the space-time modulated media with the modulation velocity in (a) the subluminal regime [Ω/K=(1.023εsμ0)−1/2<vs], (b) the luminal regime (Ω/K=vs), and (c)the superluminal regime [Ω/K=(0.978εsμ0)−1/2>vs]. The permittivity is given by Eq. (1), where α=β=0.01. The real and imaginary parts of the eigenfrequencies are depicted in the top and bottom panels, respectively. The colored dots show the real part of the eigenfrequencies in the inverted bandgaps in the top panels. The gray solid lines show the dispersion curves with α, β→0. The red dashed lines in (a) and (b), as well as the blue dashed line in (c), depict the light lines. (d) A typical eigenstate in the inverted bandgap of (b). (e) The corresponding harmonic frequencies and wavenumbers of eigenstate in (d).

Fig. 4. Cascaded PA supported by a space-time modulated slab. (a) The amplitude of the reflection and the transmission coefficients as the modulation length Lm increases. lc represents the threshold length of modulation for lasing. Here, ωinc=Ω/2. (b) The temporal response of the space-time modulated slab with Lm=1.1lPA>lc based on FDTD simulations. The critical time point of lasing is estimated by the cross point between the fitted amplified amplitude and the static amplitude before lasing (black dashed lines), denoted as tc. The numerically fitted and analytically calculated exponential growth rates ωi are given. (c) The corresponding transmission spectrum. The analytical and numerical results are shown by circles and the solid line, respectively. The incident frequency is equal to 10.5Ω in (b) and (c). The permittivity of the modulated slab is given by Eq. (1) with K=Ks, α=0.05, β=0.1, and εs=1.462ε0.

Fig. 5. (a) Exponential growth rate ωi and critical time point tc of lasing are calculated by FDTD simulations with the incidence at different frequencies. The dashed line shows the analytically calculated ωi. (b) The amplitude of the transmission field calculated by the Bloch–Floquet theory (diamonds) and FDTD simulations (light gray dots). The error bars show the mean and the standard deviation of the field amplitude excited by an incidence at different frequencies. (c) The exponential growth rate as the modulation length increases. (d) The highest order of frequency conversion as the modulation strength α increases.

Fig. 6. (a) Schematic of the CPO composed of a space-time modulated slab and a cavity formed by photonic crystals. (b) The temporal response of the space-time modulated slab with varying modulation lengths. Here, α=β=0.005 and K=Ks. The number of periods is equal to 3 for both PCs on the right and left sides. (c) The temporal response and (d) the transmission spectrum of the CPO operating at a single frequency. The photonic crystal is designed to only allow transmission at the frequency of 10.5Ω. Here, α=0.049, β=0.05, K=0.95Ks, and εs=3.52ε0. The number of periods is equal to 40 for both PCs. For the parameters of the photonic crystals and the band structure see Figs. 12(a) and 12(c) in Appendix E.

Fig. 7. Cascaded PA based on the Si waveguide. (a) The dispersion relation of an unmodulated Si waveguide with the width of waveguide dWG=2πc0/(nSiΩ) and the refractive index of Si nSi=3.5. The fitted velocity of the TE0 mode is equal to 0.996c0/nSi and is set as the modulation velocity vm to achieve cascaded PA. (b) The transmission spectrum of Si waveguide with space-time modulation described by Eq. (1). Here, α=0.04, β=0.05, and Lm=1.5lPA.

Fig. 8. Dispersion curve of the time-Floquet media with permittivity ε(t)=εs(1+2α cos Ωt). (a) The real part and (b) the imaginary part of the eigenfrequency are shown.

Fig. 9. (a) Dependence of the amplitude of the reflection and the transmission coefficients on the modulation length Lm. (b) The reflection and transmission fields for Lm=lPA. (c) The imaginary part of the eigenfrequency as the modulation length increases for Lm>lPA. Here, α=0.01.

Fig. 10. (a) The frequency up-conversion effect of the LA and the characterization of the luminal gain. (b) The luminal gain with varying excitation frequencies and modulation strengths. (c) The relationship between the highest order of frequency conversion and phase mismatching.

Fig. 11. (a) The exponential growth rate of the cascaded PA as modulation strength α increases. (b) The number of harmonic components in the cascaded PA as the modulation length increases. Here, α=0.02 and β=0.1. (c) The dependence of the cascaded PA on phase mismatching dK=K/Ks−1.

Fig. 12. (a) The bandstructure of photonic crystals in Fig. 5(b). (b) The phase change of light propagating one round trip in the cavity. The cavity is resonant when this phase shift equals 2π. Here, Λ=0.6429(2πc0/Ω),da=nbΛ/(na+nb), and db=naΛ/(na+nb). (c) The bandstructure of photonic crystals in Figs. 5(c) and 5(d). Here, Λ=0.3197(2πc0/Ω), da=0.8502Λ, and db=0.1498Λ.