Space–time periodically driven systems have attracted widespread interest across the research areas of photonics [1–3], phononics [4,5], and condensed matter physics [6,7]. This is attributed to the rich physics and versatile functionalities that originate from breaking time-reversal and/or continuous time-translational symmetry. Particularly in photonics, such space–time periodic driving has provided a new perspective on the manipulation of light, including the observation of nonreciprocal propagation of light [8–10], amplification [11–13], and recently advancing topological photonics [14–17].

The driving-induced exotic behaviors of light in space–time periodic media can be understood by scrutinizing the photonic band structure in the Floquet picture. For example, nonreciprocal light propagation can be explained as a manifestation of asymmetric opening of the frequency band gap due to the directionality of traveling wave-like modulation [8,9,12]. For the case of superluminal modulation, the band can be gapped in momentum, and the parametric processes originate from the exponentially growing unstable solutions inside the momentum gap [11–13,18]. The emergence of topological photonic states is also deeply related to the opening and closing of the frequency or momentum gaps in response to changes in the driving parameters [16,17,19].

Time-varying photonic media have been primarily investigated with a lattice of driven optical resonators, analogous to tight-binding condensed matter systems [14,15]. However, in realistic experimental configurations, it is quite challenging and requires a sophisticated approach to properly modulate each of the optical resonators in the lattice. One of the alternative approaches is to modulate a continuous dielectric structure, but the method has been limited to the simplest case where the spatiotemporal modulation profile is given as a form of a single plane wave [8–13], or the temporal variation in the optical property is regarded as a weak perturbation to the nondriven system [16,17]. In addition, such simplifications hinder us from taking full advantage of the dispersion engineering capability that can be provided by a sophisticated structure.

In this Letter, we propose a systematic computational method for calculating the effective photonic band structure of dielectric media with space–time periodicity by adopting the generalized plane wave expansion method [20–22]. This method involves the multidimensional spatiotemporal Fourier transform of a unit cell, where an arbitrary profile of spatiotemporal modulation is decomposed into a series of plane waves. As a representative example, we consider one-dimensional photonic crystals consisting of two alternating dielectric regions: one with static permittivities and the other with time-variant permittivities. The method has been successfully utilized to show the simultaneous existence of frequency and momentum gaps as well as the mode field profiles near the gaps. Moreover, we discuss asymmetric formation of the photonic band structure by imposing directionality on the modulation, which is responsible for the nonreciprocal propagation and directional parametric amplification of light.

Consider a one-dimensional Floquet photonic crystal whose permittivity and/or permeability are given as arbitrary periodic functions in the space–time domain. For isotropic and nondispersive materials, Maxwell’s curl equations can be reduced as

Given that the relative permittivity $\varepsilon (x,t)$ and permeability $\mu(x,t)$ have the same spatial periodicity of $\Lambda$ and temporal periodicity of $T$, the material parameter matrix can be expanded into a two-dimensional Fourier series as

By rearranging Eq. (5), the eigenvalue problem can be established in two ways; for quasifrequency $\omega$ as the eigenvalue

Note that Eq. (6) is more adequate for obtaining complex-valued eigenfrequencies in the momentum gaps, while Eq. (7) is better suited for obtaining complex-valued wavenumbers in the frequency gaps. These eigenvalue equations are generally applicable to diverse space–time periodic photonic crystal structures regardless of the modulation strength. A similar approach has been taken in phononics [22], but here, we have improved that by linearizing the eigenvalue problems for faster calculations. In addition, later in this Letter, we will consider a super cell of Floquet photonic crystals and series-expand it for more efficient calculation of asymmetric band structures. We assume nonmagnetic materials hereafter, $\mu= 1$, for the sake of simplicity without loss of generality.

Three representative types of space–time periodic media are exemplified in Figs. 1 and 2. Figure 1(a) shows the permittivity profile and corresponding photonic band structure when the medium has spatial periodicity only; this structure is a conventional photonic crystal comprising two dielectric materials of ${\varepsilon _1} = 2$ with thickness $3\Lambda /4$ and ${\varepsilon _2} = 6$ with thickness $\Lambda /4$. The frequency gaps are opened as expected near the crossing points of the light lines horizontally shifted by integer multiples of $g = 2\pi /\Lambda$. In contrast, Fig. 1(b) depicts the case where the medium has temporal periodicity only; the spatially homogeneous yet time-varying permittivity is given as $\varepsilon (t) = {\varepsilon _c} + \Delta \varepsilon (t) = 6(1 + 0.6\sin \Omega t)$. The momentum band gap appears near the crossing points of the light lines vertically shifted by integer multiples of $\tilde \Omega = \Omega \Lambda /2\pi c$. For both cases, dashed gray lines in the lower panels of Figs. 1(a) and 1(b) indicate the light lines for the averaged refractive index over a space–time unit cell.

 

Fig. 1. Permittivity profiles (upper) and corresponding photonic band structures (lower) of (a) time-invariant photonic crystal and (b) homogeneous time-varying medium. The red (blue) arrow and the pink (cyan) ribbons indicate the reciprocal lattice constants and energy (momentum) band gaps, respectively. Dashed gray lines are the light lines of the homogenized media with averaged refractive indices.

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Fig. 2. Band structure and mode field analyses of Floquet photonic crystals. (a) Permittivity profile and (b) band structure of the Floquet photonic crystal. The color coding indicates the weight of static bands. (c)–(h) Spatiotemporal electric field profiles near the (c), (d) driving-induced momentum gap and (e)–(h) frequency gap.

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The Floquet photonic crystal, whose permittivity is periodic in both space and time, is shown in Fig. 2(a); it consists of a time-invariant region of ${\varepsilon _1} = 2$ with thickness $3\Lambda /4$ and a time-variant region of ${\varepsilon _2}(t) = {\varepsilon _c} + \Delta \varepsilon (t) = 6(1 + 0.6\sin \Omega t)$ with thickness $\Lambda /4$. We have included space harmonics up to $P = 20$ and only one time-harmonic $Q = 1$ in the calculation to trace a square wave-like spatial variation and a single sinusoidal temporal variation in the permittivity. The solid gray lines in Fig. 2(b) indicate the photonic band structure when $\Delta \varepsilon = 0$, which is equivalent to Fig. 1(a), and the dashed gray lines are the shifted replicas. At the crossings, gaps are opened vertically (frequency gap) or horizontally (momentum gap) according to from which bands the intersecting lines originate; the frequency gap opens if two lines originate from the bands with like-signed frequencies (${\omega _1}{\omega _2} \gt 0$), and the momentum gap opens otherwise (${\omega _1}{\omega _2} \lt 0$) [12,13]. In particular, the temporal variation additionally opens the frequency gap inside the continuum region in the static band structure, as highlighted with a pink ribbon. The momentum gap highlighted with cyan ribbons is at half the modulation frequency, as shown in Fig. 1(b). Except for the case where constituting materials are intrinsically dispersive, where such a co-existence of the frequency and momentum gaps is allowed only if the medium has both spatial and temporal periodicity.

The spatiotemporal plane wave expansion method described above can be employed for eigenmode analysis, as the eigenvector $\Phi$ contains the Fourier coefficients of the space–time harmonic components of the fields. As shown in Fig. 2(a), the weight of the static bands can be encoded in the effective Floquet bands by projecting the calculated Floquet eigenmodes onto the static ones. The static weight implies the relative energy density of each photonic band in the Floquet photonic crystal. One may exclude merely contributing Floquet modes by setting a threshold value of the static weight [6,20].

Figures 2(c) and 2(d) provide the electric field profiles at the band edges of the driving-induced momentum gap. Analogous to conventional photonic crystals, the field forms a standing wave at both edges. At the left edge [Fig. 2(c)], temporal field antinodes are observed at the moment when the permittivity reaches its minima, while at the right edge, the nodes are observed at the same time [Fig. 2(d)]. The driving-induced avoided crossing of the modes can also be seen by plotting mode profiles as in Figs. 2(e)–(h). The mode at (e) in the upper band continuously changes to (f) as it passes the frequency minimum.

Next, we take a closer look into the momentum gap induced by the temporal modulation. Figure 3(a) shows the photonic band structure near the momentum gap obtained by solving Eq. (7) for a range of modulation strengths. The opening of a gap can be quantified by the gap–midgap ratio, and it was found that the ratio is proportional to the temporal modulation strength $\Delta \varepsilon /{\varepsilon _c}$ as depicted in Fig. 3(b). Figure 3(c) shows the imaginary parts of the complex eigenfrequencies inside the momentum gap obtained by solving Eq. (6) for the same range of modulation strength as in Fig. 3(a). There always exists a pair of modes inside the momentum gap whose eigenfrequencies are complex conjugates of each other. The mode with the positive imaginary part of the eigenfrequency grows exponentially over time, while the mode with a conjugated frequency decays parametrically, as depicted in the insets. The peak value of the imaginary frequency is also proportional to the modulation strength, similar to the gap–midgap ratio [Fig. 3(d)]. At this point, it is worthwhile to note that the eigenmode profiles are necessary for evaluating the topological invariant (e.g., the Zak phase). The proposed method would be useful for topology analyses of arbitrary space–time periodic dielectric continua where the tight-binding approximation does not hold.

 

Fig. 3. (a) Widening of the momentum gap with increasing modulation strengths $\Delta \varepsilon /{\varepsilon _c}$. (b) Gap–midgap ratio of the momentum gap as a function of modulation strength. (c) Imaginary part of the eigenfrequencies for different modulation strengths. The insets show the spatiotemporal electric field profiles for the parametrically growing and decaying modes inside the momentum gap. (d) Peak values of the imaginary parts of the eigenfrequencies as a function of modulation strength.

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One of the key features of space–time periodic media is the nonreciprocal propagation of light. Nonreciprocity emerges when the spatiotemporal variation in the material parameters has a certain directionality, by which the time-reversal symmetry of the system can be broken. To impose such directionality in the Floquet photonic crystal configurations, we added a consecutive phase difference among the neighboring time-variant regions. Specifically, $N$ unit cells are gathered to constitute a super cell, and the $n$th time-variant region in the super cell is driven as ${\varepsilon _n}(t) = {\varepsilon _c} + \Delta \varepsilon \sin (\Omega t + 2\pi n/N)$. The temporal permittivity variations in the super cells of $N = 4$ and $N = 8$ are shown in the upper panels of Figs. 4(a) and 4(b), respectively. Due to the spatial phase control $2\pi n/N$, the spatial period of the media is larger by $N$ times, leading to $N$ times smaller reciprocal lattice constant $g^\prime = 2\pi /N\Lambda = g/N$. However, it should be noted that the $N$ unit cells in a super cell are identical except for their modulation phase $2\pi n/N$. In this case, the Fourier expansion of the permittivity can be reduced to

 

Fig. 4. Asymmetric formation of gaps due to the phased modulation of permittivity. (a) Asymmetrically opened energy gaps due to subluminal modulation with $g^\prime = 2\pi /4\Lambda$ and $\Omega = 0.05 \times 2\pi c/\Lambda$ (red arrow). (b) Asymmetrically opened momentum gaps due to superluminal modulation with $g^\prime = 2\pi /8\Lambda$ and $\Omega = 0.2 \times 2\pi c/\Lambda$ (blue arrow). The solid and dashed gray lines are the static band structure and its shifted replicas, respectively. The gray regions indicate the Brillouin zone of the Floquet photonic crystals.

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This particular expansion of the super cell reduces the sizes of the matrices and vectors in the eigenvalue problems Eqs. (6) and (7) and properly excludes the extraneous and multiple roots of the problems as well. The calculated photonic band structures with broken mirror symmetry are shown in Fig. 4. The solid gray lines are the band structures without temporal modulation, and the dashed gray lines are their replicas shifted by $(g^\prime ,\tilde \Omega)$. The modulation is called subluminal if the phase velocity of the spatiotemporal modulation, dictated by the slope $\Omega /g^\prime $, is lower than the phase velocity of the lowest band ${v_p} = \partial \omega /\partial k$ [Fig. 4(a)]. In this case, the positive frequency branches are crossing, and frequency gaps are asymmetrically opened, as highlighted by pink ribbons. Nonreciprocal propagation of light is observable in this type of Floquet photonic crystal, as the stop bands are asymmetrically positioned in the skewed Brillouin zone with respect to the propagation direction of light. On the other hand, superluminal modulation creates momentum gaps asymmetrically, as highlighted by cyan ribbons [Fig. 4(b)]. In momentum gaps, unstable solutions with complex eigenfrequencies exist that are responsible for the parametric amplification of light. Similar to the case of subluminal modulation, the amplification bands are asymmetric with respect to the propagation direction of light. It is worthwhile to note that $N$ must be set to at least three for the formation of asymmetric band structures and the observation of associated wave phenomena.

In conclusion, we have reorganized the generalized plane wave expansion method to systematically calculate the band structures of arbitrary space–time periodic photonic media. This method defines the unit cell of a photonic medium in the space–time domain and involves a two-dimensional Fourier transform of the material properties. One-dimensional Floquet photonic crystals have been analyzed as a representative example. In the calculated band structure, the temporal driving-induced opening of frequency and momentum gaps is clearly identified. Mode field analyses have been performed using this method and are a potentially powerful tool in analyzing topological band structures. In another example, by constructing a super cell with broken time-reversal symmetry, nonreciprocal light propagation and direction-dependent parametric amplification have been predicted. Therefore, this spatiotemporal plane wave expansion method will be helpful in developing novel photonic devices designed in the space–time domain. Moreover, this method is readily applicable to higher-dimensional Floquet photonic crystals [21,23] and will be of importance to Floquet engineering and topological analysis of space–time periodic photonic media.