Photonic time crystals

When space (time) translation symmetry is spontaneously broken, the space crystal (time crystal) forms; when permittivity and permeability periodically vary with space (time), the photonic crystal (photonic time crystal) forms. We proposed the concept of photonic time crystal and rewritten the Maxwell’s equations. Utilizing Finite Difference Time Domain (FDTD) method, we simulated electromagnetic wave propagation in photonic time crystal and photonic space-time crystal, the simulation results show that more intensive scatter fields can obtained in photonic time crystal and photonic space-time crystal.


Results
Electromagnetic wave propagates in 1D photonic time crystal. Electromagnetic wave propagation is described by the Maxwell's equation, in photonic time crystal, permittivity and permeability are the function of time, which vary with time periodically, for free source case, the Maxwell's equations ∇ × → → = E r t ( , ) μ μ −∂ → → → ∂ r t H r t t ( ( , ) ( , ))/ r 0 and ε ε ∇ × → → = ∂ → → → ∂ H r t r t E r t t ( , ) ( ( , ) ( , ))/ r 0 can be rewritten as r r 0 0 where → → E r t ( , ) and → → H r t ( , ) are the time harmonic electric field and time harmonic magnetic field, respectively, ε 0 and μ 0 are the permittivity and permeability in free space, respectively. ε → r t ( , ) r and μ → r t ( , ) r are time-and space-dependent relative permittivity and permeability, respectively. Utilizing FDTD method 20 , Eqs (1) and (2) can be discretized. For 1D photonic time crystal, ∂/∂x = 0, ∂/∂y = 0, the discrete electric field and magnetic field can be written as (see Supplementary  where t = (n + 1/2)Δt, Δt is time step, n is total number of time step, k is the position of grid cell. The permittivity in any grid cell is equal at same time, so ε n (k + 1/2) = ε n (k) = …, so is the permeability in any grid cell, it can be written as μ μ In Fig. 1, we simulated electromagnetic wave propagation in 1D photonic time crystal and photonic non-time crystal (conventional dielectric), the total number of grid cell is 200, the space increment is 0.015 m, the total number of time step is 360, the time step is 5 × 10 −11 s. To simplify, suppose that the relative permittivity varies with time periodically and relative permeability is a constant, namely, the relative permittivity and permeability of the photonic time crystal are ε(t) = 5 + sin ωt and μ(t) = 1, respectively, Fig. 1a and b show the electromagnetic wave propagation in 1D photonic time crystal, the source is a sine signal, E z0 = sin ωt, ω = 2πf, the frequency f = 1.0 × 10 9 Hz, the amplitudes vary with space and time. For comparison, we simulated electromagnetic wave propagation in 1D photonic non-time crystal in Fig. 1c and d, the relative permittivity and permeability of the photonic non-time crystal are ε(t) = 6 ≥ 5 + sin ωt and μ(t) = 1, respectively, the amplitudes are invariant. Figure 1e and f show the electromagnetic wave propagation in 1D photonic non-time crystal, the relative permittivity and permeability are ε(t) = 4 ≤ 5 + sin ωt and μ(t) = 1, respectively, the amplitudes are invariant too. We concluded that in the photonic time crystal, the impedances don't match with each other, electromagnetic wave is scattered everywhere, so the amplitudes vary with space and time, the amplitudes in photonic time crystal ( Fig. 1a and b) are smaller than those in photonic non-time crystal (Fig. 1c-f).
Electromagnetic wave scattered by a 2D photonic time crystal. For 2D photonic time crystal (TM wave and TE wave), ∂/∂z = 0, the discrete electric field and magnetic field can be obtained (see Supplementary  Information). In Fig. 2, we simulated the electromagnetic wave scattered by a 2D photonic time crystal and 2D photonic non-time crystal cylinder, the total number of grid cell is x = 100, y = 50, the space increment is Δx = Δy = 0.003 m, the total number of time step is 300, the time step is 5.0 × 10 −12 s, the radius of the cylinder is 0.01 m, the center of the cylinder is located at x = 80, y = 25. Figure 2a-c show the electromagnetic wave scattered by a 2D photonic time crystal cylinder, the relative permittivity and permeability of the cylinder are ε(t) = 5 + sin ωt and μ(t) = 1, respectively, the source is also a sine signal, H z0 = Asin ωt, the frequency f = 5.0 × 10 9 Hz. For comparison, Fig. 2d-f show the electromagnetic wave scattered by a 2D photonic non-time crystal cylinder, the relative permittivity and permeability of the cylinder are ε(t) = 6 and μ(t) = 1, respectively. Electromagnetic wave propagates in 3D photonic time crystal. For 3D photonic time crystal, the discrete electric field and magnetic field can be obtained (see Supplementary Information). In Fig. 3, we simulated electromagnetic wave propagation in 3D photonic time crystal and photonic non-time crystal, the total number of grid cell in three direction is x = 50, y = 24, z = 10, correspondingly, the space increment Δx = Δy = Δz = 200 m, the total number of time step is 500, the time step is 3.3 × 10 −12 s. Figure 3a and b show electromagnetic wave propagation in 3D photonic time crystal, the relative permittivity and permeability of the photonic time crystal are ε(t) = 5 + sin ωt and μ(t) = 1, respectively, the frequency of the source is f = 1.0 × 10 9 Hz. For comparison, Fig. 3c and d show the electromagnetic wave propagation in 3D photonic non-time crystal, the relative permittivity and permeability of the photonic non-time crystal are ε(t) = 6 and μ(t) = 1, respectively. Figure 4e and f show the electromagnetic wave propagation in 3D photonic non-time crystal, the relative permittivity and permeability of the photonic non-time crystal are ε(t) = 4 and μ(t) = 1, respectively. Fig. 4, we simulated electromagnetic wave propagation in 2D photonic space-time crystal (the permittivity of the periodical array dielectrics varies with time periodically in photonic space crystal) and photonic space crystal. The photonic crystals consist of 7 × 7 periodical array dielectrics surrounded with air, the side length is 1.5 × 10 −7 m. The relative permittivity of the air is ε = 1. A central horizontal line of seven periodical structures are removed to form a central wave guide. The space increment Δx = Δy = 1.5 × 10 −8 m, the total number of time step is 1000, the time step is 3.5 × 10 −17 s. Figure 4a shows electromagnetic wave propagation in 2D photonic space-time crystal, the relative permittivity and permeability of the period array dielectrics are ε(t) = 12.25 + sin ωt and μ(t) = 1, respectively, the frequency of the source is f = 1.9 × 10 14 Hz. For comparison, Fig. 4b shows the electromagnetic wave propagation in 2D photonic space crystal, the relative permittivity and permeability of the periodical array dielectrics are ε(t) = 13.25 and μ(t) = 1, respectively. Figure 4c shows the electromagnetic wave propagation in 2D photonic space crystal, the relative permittivity and permeability of the periodical array dielectrics are ε(t) = 11.25 and μ(t) = 1, respectively. The periodical array dielectrics in Fig. 4a are darker than those in Fig. 4b and c, this is because the permittivity of the periodical array dielectrics is not uniform in photonic space-time crystal, and the scatter fields in photonic space-time crystal are more intensive than those in photonic space crystal.

Electromagnetic wave propagates in photonic space-time crystal. In
In Fig. 5, we simulated electromagnetic wave propagation in 2D photonic space-time crystal and photonic space crystal. Figure 5a shows the relation between transmission coefficient and the frequency in 2D photonic space-time crystal, the permittivity and permeability of the periodical array dielectrics are ε(t) = 12.95 + sin ωt and μ(t) = 1, respectively. Figure 5b shows the relation between transmission coefficient and the frequency in 2D photonic space crystal, the permittivity and permeability of the periodical array dielectrics are ε(t) = 13.95 and μ(t) = 1, respectively. Figure 5c also shows the relation between transmission coefficient and the frequency in 2D photonic space crystal, the permittivity and permeability of the periodical array dielectrics are ε(t) = 11.95 and μ(t) = 1, respectively. The band gaps in Fig. 5a are larger than those in Fig. 5b and c, this is because the permittivity of the periodical array dielectrics in photonic space-time crystal is not uniform, and the scatter fields in photonic space-time crystal are more intensive than those in photonic space crystal, namely, the larger band gaps can be obtained in photonic space-time crystal.
Theoretically, the field-dependent dielectric [21][22][23] can be designed as a photonic time crystal, yet, in high frequency, the permittivity varying with time is not obvious 23 , it is very difficult to make the period of electromagnetic field equal to that of permittivity. As for some heat diffusion materials, such as silicon and germanium, the heat conductivity, mass density and specific heat vary with temperature, by adjusting temperature periodically, one might make the period of the temperature field equal to that of material parameters. Whereas, for some acoustic wave materials, one can also adjust mass density and bulk module periodically to design acoustic time crystal. By the same method, other time crystals, like mass diffusion time crystal, could be designed too. It should be pointed out that reference 8,12,13 described the discrete time crystal whose period is the integer multiple of the drive period and robustness against external

Conclusion
In this work, we proposed the concept of the photonic time crystal and photonic space-time crystal, and simulated electromagnetic wave propagation in 1D, 2D, 3D photonic time crystal and photonic space-time crystal, the simulated results indicate that the scatter fields in photonic time crystal are more intensive than those in photonic non-crystal, and the band gaps in photonic time crystal are larger than those in photonic space crystal. The method we adopted provides the possibility for further investigation in other time crystal and space-time crystal.