One-dimensional time-Floquet photonic crystal

Using the Floquet Hamiltonian derived based on the time-dependent perturbation theory, we investigated the quasienergy bands of a one-dimensional time-Floquet photonic crystal. The time-Floquet photonic crystal contains two alternating layers labeled as A and B, and the permittivity of A layer is modulated periodically in time. We showed that the quasienergy bands are reciprocal when the modulation function is a function of time only, while the quasienergy bands could be nonreciprocal when the permittivity is modulated in both time and space through an unique combination. In the former case, the coupling between the positive (negative) and positive (negative) bands results in quasienergy gaps, while the coupling between the positive and negative bands leads to pairs of exception points, when the modulation is on the real part of the permittivity. In the latter case, the coupling between the positive (negative) and positive (negative) bands still results in quasienergy gaps. However, the coupling between the positive and negative bands leads to quasienergy gaps at a small modulation speed and pairs of exceptional points at a high modulation speed.

Periodically modulating the parameters in time breaks the time-reversal symmetry, enabling the time-Floquet system to support topologically nontrivial states that are topologically trivial in the static case, i.e., the Floquet topological states [17,18], and to possess nonreciprocal band structures [19,20]. Recently, the Floquet topological states in various classical wave systems were studied [21][22][23][24][25] and experimentally observed [26][27][28][29][30], and several types of nonreciprocal devices based on the time-modulated elements were designed [31][32][33]. The periodic time-modulation also brings about the discrete translational symmetry in time, which makes the frequency can be converted through adding or subtracting an integer times of the modulation frequency [34], in analogy to the wave vector conversion of the classical waves by a lattice structure. On the other hand, the time modulation requires an external driving in principle, which could make the total energy of the system not necessarily conservative thus introduce non-Hermiticity to the system [35][36][37][38], even though there is no gain or loss material used. The coexistence of the non-reciprocity, frequency conversion and non-Hermiticity in the time-Floquet system enables the realization of various attractive applications, such as the broadband nonreciprocal wave amplification [39][40][41][42] and frequency-selective wave filtering [43].
In our previous work [35], we have studied the quasienergy bands of a homogeneous medium with a time-periodic complex permitttivity. Here, we extended the study to the case that the periodic time-modulation is applied to the one-dimensional (1D) photonic crystal (PC) with two alternating layers A and B. While the permittivity of B layer is static, the permittivity of A layer is time-modulated as ) , ( t z a    , where a  is the static part and is the time-periodic modulation function of variables time t and spatial coordinate z. We calculated and analyzed the quasienergy bands using the Floquet Hamiltonian which is derived from the Maxwell's equations based on the time-dependent perturbation theory. In the vanishing  limit, the bands of the 1D PC in the static case copy themselves and shift up and down in the quasienergy space by nΩ to generate the bands of order n, where n is an integer and Ω is the modulation frequency. Bands of different orders cross and form diabolic points. Here, we call the bands with positive frequencies in the static case (or bands of zeroth order) and their non-zeroth order counterparts positive bands, and the bands with negative frequencies in the static case and their non-zeroth order counterparts negative bands. order are located at Ω/2 (-Ω/2) in the quasienergy space, similar to the case of a homogeneous medium with time-periodic real permittivity [34].
When t and z are not separable variables in the modulation function, the quasienergy bands could be nonreciprocal. We investigated the simplest case that is small, but

II. Formulation of the Floquet Hamiltonian
The unit cell of the 1D time-Floquet PC we consider is schematically shown in Fig. 1, which contains two dielectric layers. The static relative permittivities of the A and B layers are a  and b  , respectively. Consider the perimittivity of the A layer is modulated periodically in time, the time-dependent relative permittivity of the PC is expressed as with T being the modulation period.
We first consider the modulation function ) For the sake of mathematical simplicity, we set the permittivity 0  , permeability 0  and light speed c in vacuum as unity. Also we consider all the materials are nonmagnetic, namely the relative permeability is 1. Without loss of generality, we let the electric field to be polarized along the x direction. According to the Maxwell's equations, we can obtain the Schrodinger-like equation as For the Bloch wavenumber q, the time-Floquet solution can be written as where Q is the quasi-energy and ) , ( ) , is the time-periodic function. We use the eigenmodes of the static bands ( where jm c is the expansion coefficient, Substituting Eq. (4) into Eq. (2), we obtain Eliminating iQt iqz e  on both sides and using Eq. (4), Eq. (5) becomes Inner product from left on both sides and using the orthonormality of the static eigenmodes ( where j  is the eigenfrequency of the jth band in the static case, ij  is the Kronecker . In the matrix form, Eq. (8) is rewritten as In the static case, the bands are reciprocal, namely . If we choose the center of the A or B layer as the origin, then the static PC possesses the inversion symmetry, which leads ) , and note that the electric field is a vector while the magnetic field is a pseudovector, the normalized electromagnetic In Eq. (11), we have used the Bloch theorem

III. Coupling between the positive and positive bands
Consider that .
The static band dispersion and the corresponding eigenmodes of the photonic crystal can be calculated using the transfer matrix method [44], see details in the appendix. The band diagram in the static case is shown in Fig. 2 For a finite  , the Floquet bands can be numerically calculated using Eq. (9) by truncating the band order and the band index to finite numbers. Because of the nonzero matrix  , there are couplings between the bands of different orders, which will lift the diabolic points to generate quasienergy gaps, EPs or a mixture of the two.
From Eqs. (9) and (10), we can find that the coupling are vanishing small when the order difference of the two bands is greater than 1.
According to Eqs. (9) and (10), when  is small, the quasienergy band in the vicinity of the diabolic point formed by bands 1 , j and 0 , l (point A for example) can be well described by the following When l  and j  have the same sign, Therefore, there is a quasienergy gap with gap size given by In Fig. 3(a), we plotted the quasienergy bands evolved from the bands 0  Fig. 3(a). There is also a band gap opening at the point C, as shown in Fig. 3(b). Because the coupling between the bands differing by an order 2 is much weaker, the gap size is several orders smaller.
. Therefore, the quasienergies in the vicinity of the diabolic point become a complex conjugate pair in between a pair of EPs. According to Eq. (13), the EPs are located at  Fig. 4(b). of the static band, which are marked by the red arrows in Fig. 5(a) and very close to the wavenumbers of the two EPs. The positive (negative) value of the imaginary part of the quasienergy denotes the decay (amplification) rate of the fields in time [34]. From Eq. (17), we see that the maxim imaginary part of the quasienergy is about

IV. Coupling between the positive and negative bands
with the results as shown in Fig. 5(b).
When the modulation frequency is When  is pure imaginary, it is easy to know that the diabolic points formed by the bands differing by an order 1 will be lifted to open quasienergy gaps.

V. Nonreciprocal band structures
As discussed previously, the quasienergy bands are reciprocal when the time-modulation function is in the form of . Substituting Eqs. (19) and (20) into Eq. (2), we arrive at are defined in Eq. (3), and . 0 Then Eq. (21) is rewritten as is the Pauli matrix, then we obtain In a matrix form, Eq. (25) is written as where 2 H  is defined in Eq. (10), and The quasienergies are obtained as the eigenvalues of the Floquet Hamiltonian Similarly, the normalized electromagnetic fields of the static modes ) , ( q  and can be given by

VI. Conclusions
In summary, we studied the quasienergy bands of the one-dimensional time-Floquet To the contrary, when the time-modulation is on the imaginary part of the permittivity, the former forms a pair of EPs, while the latter generates a quasienergy gap.
We also investigated the case that t and z are not separable variables in the modulation function. We showed that the quasienergy bands could be nonreciprocal in this case.
When the modulated permittivity is pure real, the coupling between the positive and positive bands still results in quasienergy bands. However, the coupling between the positive and negative bands can lead to quasienergy gaps for a small modulation speed and EP pairs for a high modulation speed, which is also similar to the case of a time-modulated homogeneous medium.
The photonic band of AB layered lattice in the static case can be calculated using the transfer matrix method as [44] ), sin( ) sin( 2 According to the definition,