1. Introduction

Time crystals (TCs), a new form of matter that exhibits periodic changes in the time dimension, have recently attracted worldwide interest [1,2]. There have been a series of significant discoveries in the field of TCs over the last few years in both theory or experiment [3–7]. Theoretically, the original concept of quantum TCs has been gradually refined, and some new concepts such as discrete TCs have been proposed and studied [3]. Experimental studies have also flourished in many aspects. Research groups from University of Maryland, Harvard University, Yale University, and Google Research Team, have reported their work on realizing discrete TCs and observing their long-time behaviors [4–7].

With the rapid advancement of TCs, photonic time crystals (PTCs) have also gained increasing attention [8–14]. Traditional photonic crystals [15,16], or spatial photonic crystals (SPCs), are periodic structures with a refractive index that varies periodically in space. In SPCs, the spatial boundary conditions require continuity of the tangential electric and magnetic fields ($E$ and $H$) at an interface, and the frequency and energy are conserved. In contrast, for PTCs, in which the refractive index $n(t )$ changes periodically in time, the magnetic flux density B and the electric displacement D vary continuously over the time interface. Energy is not conserved while momentum k is conserved. Causality determines the PTC system’s dynamics [11]. The most important PTC feature is that there are gaps in the momentum k, rather than gaps in frequency for a SPC, in which the signals can be amplified [13]. A lot of applications have been developed that are based on PTC systems, including non-resonant amplifiers, non-reciprocal devices, and many others [11–14]. However, setting up PTC systems may not be easy because they require very fast modulation of the refractive index, i.e., on the time scale of a few temporal periods of the input signal or shorter. However, thanks to the advanced fabrication capabilities of modern technology, systems or materials with refractive indexes that vary on short time scales have been realized in the microwave and even optical regimes, which makes PTC systems practical and feasible [17–22].

Topology is a new degree of freedom for characterizing objects, and it was initially used to study the properties of spaces that are invariant under any continuous deformation [23,24]. With years of study, the concept of topology has been extended to various fields including photonic systems and developed for many important applications [25–29]. Topological invariance plays an important role in describing the topology of a system. In a one-dimensional (1D) photonic system, the topological invariant is characterized by the Zak phase, which is a kind of Berry phase defined along a 1D bulk band and is closely related to the appearance of the topological defect mode (TDM) or topological interface state (TIS) [28,29]. Since the TDM is topological and therefore should be robust to small perturbations such as disorder and defects, which is very useful in applications, it is worth to put some effort in investigating the properties of temporally-topological defect modes (TTDMs) in a PTC system.

Recently, some works on TTDMs or temporally-topological interface states (TTISs) in PTCs have been reported. For example, Lustig et al. studied the properties of topological band structures in PTC, and they found that the TTISs yield when an interface is generated between two TCs of different topologies [8]. Then, Ma et al. investigated the relationship between the topological phase transition and the $k$-gap size of a PTC system, and they predicted the existence of the TTIS in compound PTC system [9]. After that, Dong et al. proposed a kind of structure based on continuous PTCs, in which tunable and controllable multi-channel time-comb absorber can be achieved. The positions of the multiple TTISs, i.e., the multi-channel absorption peaks, can be adjusted by changing the refractive index, incident angle, and time period of the PTCs [30]. In this paper, we focus on the properties of TTDMs in topological PTC systems. TTDMs are governed by the topology of the system, making them different from normal defect modes. The cases of two-, three-, and multiple-sub-PTC for the topological PTC system are studied. Compared with the above-mentioned papers, topological systems of multiple sub-PTCs (more than two sub-PTCs) are investigated in our study. The field distributions indicate that the multiple TTDMs result from the coupling of the permitted states, and the number of TTDMs is one less than the number of sub-PTCs. The robustness of the topological PTC systems is also studied. The transfer matrix method (TMM), which is suitable for PTCs, is derived and used to obtain our results. The effects of structure and material parameters (including the modulation strength of the refractive index, the time duration, the period, and the number of the sub-PTCs) on the TTDMs are investigated. We note that the results in this paper have been verified by the finite-difference time-domain (FDTD) method. And the results for the two methods fit quite well. To the best of our knowledge, there have been no reports on these aspects yet for PTC systems. Such research may be useful in designing tunable band-stop or band-suppression filters for PTC applications.

2. Physical model and numerical method

We take a topological PTC system with two sub-PTCs as the basic model, for which the schematic is shown in Fig. 1(a). It consists of a cascade of two sub-PTCs, PTC₁ $\left[ {{{\left( {\frac{A}{2}B\frac{A}{2}} \right)}^N}} \right]$ and PTC₂ $\left[ {{{\left( {\frac{B}{2}A\frac{B}{2}} \right)}^N}} \right]$. Here, A and B represent the time segments (or temporal slabs) with corresponding refractive indexes ${n_A}$ and ${n_B}$ and time durations ${t_A}$ and ${t_B}$, while $\frac{A}{2}$ and $\frac{B}{2}$ are the time segments corresponding to ${n_A}$ and ${n_B}$ with durations ${t_A}/2$ and ${t_B}/2$, respectively. N is the period of the sub-PTC. In such a temporal arrangement, both PTC₁ and PTC₂ maintain inversion symmetry in time, which is important for studying the topology in a PTC system. $\left( {\frac{A}{2}B\frac{A}{2}} \right)$ is the unit cell of PTC₁, while $\left( {\frac{B}{2}A\frac{B}{2}} \right)$ is that of PTC₂. The unit cells for PTC₁ and PTC₂ have the same time period $T = {t_A} + {t_B}$, but they are reversed in arrangement. Thus, they have exactly the same bandgaps but different topologies or topological phases, resulting in the appearance of TTDMs or TTISs. The topological PTC system proposed in this paper is an analogue of the Su-Schrieffer-Heeger (SSH) model [8,9,29]. In addition, the $\frac{A}{2}$ and $\frac{B}{2}$ time segments in the middle [shown in Fig. 1(a)] can be seen as a joint defect segment, which is why the modes can be called TTDMs.

 

Fig. 1. (a) Schematic of the basic topological PTC system consisting of a cascade of PTC₁${\left( {\frac{A}{2}B\frac{A}{2}} \right)^N}$ and PTC₂ ${\left( {\frac{B}{2}A\frac{B}{2}} \right)^N}$. (b) The field profiles of electric displacements and magnetic flux densities D and B in m-th time segment.

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The dynamics of the PTC system can be studied via the TMM. Let us consider a linearly polarized wave incident on the PTC, as shown in Fig. 1(b). In each time segment, the frequency of a propagating wave is modified in accordance with $n(t )$, while the momentum k is preserved owing to the homogeneity of space, i.e., ${\omega _i} = kc/{n_i}\; ({i = m - 1,m,m + 1} )$, where k is the vacuum wavenumber of the wave and c is the speed of light in vacuum. The fields with and without the tilde (${\sim} $) identify the quantities just before and after the time interface, respectively. The fields with positive (+) and negative ($- $) symbols are called the forward (transmitted) and backward (time-reflected) D or B fields, with the time dependences $exp ({i\omega t} )$ and $exp ({ - i\omega t} )$, respectively. The fields inside the PTC region are considered as a combination of the forward and backward propagating waves in the time dimension [10,11]. The temporal boundary conditions require the total fields D and B to be continuous across the interface. Thus, we get

The relationship between D and B can be obtained via the Maxwell equation $\nabla \times B = \mu \partial D/\partial t$. This yields

Similar to the spatial phase delay due to propagation within each dielectric slab in an SPC, a temporal slab introduces a temporal phase delay as well. The phase delays for the fields propagating in the m-th time segment, i.e., the phase delays of the fields just after an m-1 temporal interface and the ones just before the m interface for a time duration of ${t_m}$, for the forward (transmitted) and backward (time-reflected) fields, can be written as

From Eq. (4), we note that the phase delay is $\delta = {\omega _m}{t_m}$ for a temporal slab with a time duration of ${t_m}$, and that $\delta = {k_m}{d_m}$ for a spatial one, where we assume ${k_m}$ and ${d_m}$ are the wavenumber and thickness of the spatial dielectric slab. Akin to the “optical thickness” in an SPC, we can define the “equivalent temporal optical thickness” in a PTC. In an SPC, $\delta = {k_m}{d_m} = \frac{\omega }{c}{n_m}{d_m} = 2\pi \frac{{{n_m}{d_m}}}{{{\lambda _0}}}$, where ${\lambda _0}$ is the wavelength in vacuum, and the optical thickness can be defined as ${L_s} = {n_m}{d_m}$. Correspondingly, in a PTC, we have $\delta = {\omega _m}{t_m} = \frac{{kc}}{{{n_m}}}{t_m} = 2\pi \left( {\frac{{c{t_m}/{n_m}}}{{{\lambda_0}}}} \right)$. Thus, we get ${L_t} = c{t_m}/{n_m}$ is the equivalent temporal optical thickness for a PTC system.

Combining Eqs. (1)-(4), we can build the relationship between the total fields D and B before and after an arbitrary m-th time segment. In matrix form, this is

For the finite PTC system, we can use the transfer matrix to relate the initial fields ${D_a}$ and ${B_a}$ (the fields in the initial temporal slab) to the final fields ${D_b}$ and ${B_b}$ (the fields in the final temporal slab) via

For the infinite PTC system, the Floquet (or Bloch) theorem holds because the problem is periodic. The fields for a unit cell of the infinite PTC can be written as

3. Numerical results and discussion

In the following, in Section 3.1, we first discuss the properties of the TTDM in the two-sub-PTC topological PTC system [PTC₁PTC₂]. Then in Section 3.2, we will study the properties of TTDMs in the three-sub-PTC topological PTC system [PTC₁PTC₂PTC₁], where the coupling of the TTDMs will be observed and studied. Finally, in Section 3.3, some discussions will be given for the multiple-sub-PTC topological PTC system. Four- and five-sub-PTC systems [PTC₁PTC₂PTC₁PTC₂] and [PTC₁PTC₂PTC₁PTC₂PTC₁] will be taken as examples. The robustness and application of such PTC systems will also be discussed in this section.

3.1 TTDM properties in the two-sub-PTC topological PTC system [PTC₁PTC₂]

Since the band structure is very important in studying a topological PTC system, we first study the relationship between the momentum k and the Floquet frequency $\mathrm{\Omega }$, as shown in Fig. 2. Figure 2(a) shows the band structure for the sub-PTC (complete PTC) [PTC₁] or [PTC₂], where the number and region of the $k$-gaps are indicated. Figure 2(b) and (c) show the imaginary part of the Floquet frequency $\mathrm{\Omega }$ versus the momentum k for the sub-PTC (complete PTC) [PTC₁] or [PTC₂] and the topological PTC (defective PTC) [PTC₁PTC₂], where ${\left( {\frac{A}{2}B\frac{A}{2}} \right)^N}$ [or ${\left( {\frac{B}{2}A\frac{B}{2}} \right)^N}$] and $[{\left( {\frac{A}{2}B\frac{A}{2}} \right)^N}{\left( {\frac{B}{2}A\frac{B}{2}} \right)^N}]$ are taken as the supercells for obtaining 2(b) and 2(c), respectively. In Fig. 2, the parameters are chosen as ${\varepsilon _A} = 6,\; \; {\varepsilon _B} = 1,\; \; {t_A} = {t_B} = 0.5{T_0},\; \; {T_0} = 2[{fs} ],\; \; N = 12$, all the time segments are considered non-magnetic (${\mu _A} = {\mu _B} = 1$), and the momentum k is normalized by ${k_0} = 2\pi /({{T_0}c} )$.

 

Fig. 2. (a) Real part (band structure) of the Floquet frequency $\mathrm{\Omega }$ for sub-PTC (complete PTC), where the $k$-gaps are numbered and indicated in grey. Imaginary part of the Floquet frequency $\mathrm{\Omega }$ for (b) sub-PTC (complete PTC) [PTC₁] or [PTC₂] and (c) topological PTC (defective PTC) [PTC₁PTC₂], respectively.

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From Fig. 2, we can see that the complete PTC [PTC₁] or [PTC₂] and the defective PTC [PTC₁PTC₂] share common $k$-gaps, as indicated in grey in the figure. For the complete PTC in Fig. 2(b), the Floquet frequency $\mathrm{\Omega }$ is real [Im($\mathrm{\Omega }$) = 0] and imaginary [Im($\mathrm{\Omega }$)$\ne $0] in the pass-band and band-gap regions, respectively. However, unlike the case of conventional SPCs, where growing states (or permitted states) are forbidden (by energy conservation) in the bandgap owing to the imaginary Bloch wavevector, the permitted states in PTCs exist not only in the pass-band region but also in the band-gap region [11]. In addition, there is no amplification for waves in the pass-band region, but there is exponential amplification for waves in the band-gap region, by noting that the real and imaginary values of the Floquet frequency in these two regions [8]. This point is fundamentally different between PTCs and SPCs. On the other hand, for the topological PTC [PTC₁PTC₂] shown in Fig. 2(c), $\frac{A}{2}$ and $\frac{B}{2}$ in the middle of the model can be seen as a joint defect segment, so the topological PTC can be regarded as a defective PTC. In the pass-band region, the behavior of the Floquet frequency $\mathrm{\Omega }$ is similar to that in the complete PTC, i.e., Im($\mathrm{\Omega }$) is near 0 except for a few small side lobes around the gap areas resulting from the asymmetry due to the presence of the defect. In the band-gap region, it is evident that there are new states in some of the k-gap, e.g., $k = 0.644{k_0}$ and $2.101{k_0}$ in the first and third $k$-gaps, respectively. These additional states result from the additional interaction between the transmitted and time-reflected waves from the two PTCs on the two sides of the defect segment, so they can be called TTDMs. Moreover, the imaginary part of the Floquet frequency for the TTDMs abruptly becomes zero [Im($\mathrm{\Omega }$) = 0], implying that there is no amplification for the TTDMs.

However, if we carefully examine Fig. 2(c), we find that, in the topological PTC system [PTC₁PTC₂], the TTDMs only exist in the first and third gaps, and are absent from the second and fourth gaps. Why are there no TTDMs in the second and fourth gaps? To answer this question, we study the topological properties (Zak phase and sign of gap) for the PTC system, as shown in Fig. 3.

 

Fig. 3. (a)-(d) The eigenfields $|D |$ for the band-edge states K, L, P, and Q, respectively. The band-edge states K, L, P, and Q are indicated by red points in (e) and (f). The red dashed lines in (a)-(d) mark the boundary between the unit-cells in PTC₁ and PTC₂. The band structure for (e) PTC₁ and (f) PTC₂. In both (e) and (f), the Zak phase is indicated below each band, and the sign of gap is indicated by $\tau $ inside each gap, with $\tau < 0$ in brown and $\tau > 0$ in purple. The $|{{t_M}} |$ and $|{{r_M}} |$ spectrums for (g) topological PTC $[{\left( {\frac{A}{2}B\frac{A}{2}} \right)^N}{\left( {\frac{B}{2}A\frac{B}{2}} \right)^N}]$, compared with that (h) normal defect PTC $[{{{({AB} )}^N}C{{({BA} )}^N}} ]$, where the blue solid and red dashed lines represent $|{{t_M}} |$ and $|{{r_M}} |$, respectively.

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The topological properties of the topological PTC system can be defined in a way akin to that in SPC [29]. As shown in Fig. 3, the Zak phase for a certain band can be obtained from the eigenfields of the band-edge states at each band. Taking band-3 in PTC₁ and PTC₂ as an example, Points K and L are the band-edge states for PTC₁, while Points P and Q are the band-edge states for PTC₂. Their eigenfields are shown in Fig. 3(a)-(d). We can see that the eigenfields for Points K, L, and Q are symmetric, and that for Point P is antisymmetric. If the band-edge points of a band have the same state (both symmetric or antisymmetric), the Zak phase of this band is 0 because there is no transition point on this band. If the band-edge points have different states, the Zak phase will be $\pi $ because at least one transition point must exist inside the band [29]. Thus, the Zak phase of band-3 is 0 for PTC₁ (K and L have the same state) and $\pi $ for PTC₂ (P and Q have different states). Beside this method, the Zak phase of each band can also be calculated by

According to Eq. (12), the sign of each gap can be obtained, indicated by $\tau $ inside each gap, with $\tau < 0$ in brown and $\tau > 0$ in purple, as shown in Fig. 3(e) and (f), respectively. To better understand the relationship between the topological and output properties of the topological PTC system, we plot the absolute values of the transmission and reflection coefficients, $|{{t_M}} |$ and $|{{r_M}} |$, for the topological PTC $[{\left( {\frac{A}{2}B\frac{A}{2}} \right)^N}{\left( {\frac{B}{2}A\frac{B}{2}} \right)^N}]$ in Fig. 3(g). For comparison, $|{{t_M}} |$ and $|{{r_M}} |$ for the normal defect PTC $[{{{({AB} )}^N}C{{({BA} )}^N}} ]$ are also studied in Fig. 3(h), where the equivalent optical thickness of the defect time segment C is set to be the same as that in the topological PTC, i.e., $c{t_c}/{n_c} = c({t_A}/2)/{n_A} + c({t_B}/2)/{n_B}$ with ${\varepsilon _C} = 3\; $ and ${\mu _C} = 1$. In Fig. 3, $N = 5$ for both panels (g) and (h), and other parameters are the same as those in Fig. 2. We find that, for the topological PTC, the TTDM emerges only for gaps with different topological properties (i.e., different signs for $\tau $), such as the first and third gaps, with $|{{t_M}} |\approx 1$ and $|{{r_M}} |\approx 0$. However, for the normal defect PTC, the TTDM appears regardless of whether the gaps have different or the same signs for $\tau $, with $|{{t_M}} |\approx 1$ and $|{{r_M}} |\approx 0$ as well. The performance of the topological PTC fits quite well with the behavior of the Floquet frequency $\mathrm{\Omega }$ in Fig. 2. As a result, we can explain why there are TTDMs in only the first and third gaps, and no TTDMs in the second and fourth gaps.

Next, we investigate the influence of the structure and material parameters on the TTDM in the first gap. For convenience, we define the factor of permittivity $\rho = {\varepsilon _A}/{\varepsilon _B}$, where ${\varepsilon _A}$ and ${\varepsilon _B}$ are the permittivity of time segments A and B; the factor of time duration $g = {t_A}/T$, i.e., ${t_A} = gT$, ${t_B} = ({1 - g} )T$, where $T = {t_A} + {t_B}$ is the unit cell time period. It is clear that these two factors relate to the modulation strength of the refractive index and the time duration in the topological PTC. Figure 4 shows the influence of factor $\rho $ on the TTDM, where $\rho = {\varepsilon _A}/{\varepsilon _B}$, ${\varepsilon _B} = 1$, $g = 0.5$, and other parameters are the same as those in Fig. 3. From Fig. 4(a), we find that the k of TTDM increases with the increase of $\rho $, while keeping $|{{t_M}} |\approx 1$. For detail, the value of k for the TTDM changes from $0.572{k_0}\sim 0.669{k_0}$ with the increase in $\rho $ from $2\sim 16$ as shown in Fig. 4(b). In Sec. 2, the equivalent optical thickness for a PTC system was obtained as ${L_t} = c{t_m}/{n_m}$. We have previously noted that $\frac{A}{2}$ and $\frac{B}{2}$ in the middle of the topological PTC can be regarded as a joint defect segment. By simple derivation for our case, we get the equivalent optical thickness for the defect segment as $\frac{1}{4}cT\left( {\frac{1}{{\sqrt \rho }} + 1} \right)$. It is obvious that the equivalent optical thickness for the defect segment decreases with the increase of $\rho $, leading to the decrease of the response wavelength $\lambda $, and thus causes the increase of k because $k = 2\pi /\lambda $. As a result, we can understand the performance of the TTDM in Fig. 4.

 

Fig. 4. (a) The $|{{t_M}} |$ spectrum of the two-sub-PTC topological PTC [PTC₁PTC₂] for different factor $\rho $, where $\rho = {\varepsilon _A}/{\varepsilon _B}$, and (b) the relationship between the k of TTDM and the factor $\rho $.

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Figure 5 shows the influence of factor g on the TTDM, where ${t_A} = gT$, ${t_B} = ({1 - g} )T$, $\rho = 6$, and other parameters are the same as those in Fig. 3. From Fig. 5, we can see that, the k of TTDM also increases with the increase of g, while $|{{t_M}} |\approx 1$ is maintained. When g increases from $0.1\sim 0.9$, the k of TTDM changes from $0.522{k_0}\sim 0.988{k_0}$. In addition, we study the influence of the period N of the sub-PTC on the TTDM, as shown in Fig. 6, where $\rho = 6$, $g = 0.5$, and other parameters are the same as those in Fig. 3. It can be easily seen from Fig. 6 that, the k of TTDM keeps unchanged with the increase of N, except for some increase of $|{{t_M}} |$ for the other modes in the gap. We can understand Figs. 5 and 6 similarly to Fig. 4, i.e., the increase of g results the decrease of the equivalent optical thickness for the defect segment, so the k of TTDM increases accordingly. However, the increase of N will not affect the equivalent optical thickness of the defect segment, so the k of TTDM keeps unchanged.

 

Fig. 5. (a) The $|{{t_M}} |$ spectrum of the two-sub-PTC topological PTC [PTC₁PTC₂] for different factor g, where $g = {t_A}/T$, and (b) the relationship between the k of TTDM and the factor g.

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Fig. 6. The $|{{t_M}} |$ spectrum of the two-sub-PTC topological PTC [PTC₁PTC₂] for different N, where N is the period of the sub-PTC, $\rho = 6$, and $g = 0.5$.

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From the results in this Section, we can conclude that the two-sub-PTC topological PTC may have potential applications in tunable narrow band-stop or band-suppression filters, in which the channel position can be adjusted by changing the modulation strength of the refractive index or the time duration in the topological PTC.

3.2 TTDM properties in the three-sub-PTC topological system [PTC₁PTC₂PTC₁]

Next, we focus on the properties of TTDMs in the three-sub-PTC topological PTC system [PTC₁PTC₂PTC₁]. The schematic of this type of PTC is shown in Fig. 7(a), with one sub-PTC [PTC₂] sandwiched between two sub-PTCs [PTC₁]. In this way, there are two interfaces with different topological properties on both sides, which may support two permitted states, and the coupling of the permitted states results in the appearance of two TTDMs in Gap-1 and Gap-3, as shown in Fig. 7(b). For more detail, the magnified views of TTDMs inside Gap-1 and Gap-3 are plotted in Fig. 7(c) and (d). In Fig. 7, all the parameters are the same as those in Fig. 3. To better understand this, we plot the field distributions of the TTDMs in the first gap for the two- and three-sub-PTC topological PTC systems [PTC₁PTC₂] and [PTC₁PTC₂PTC₁] in Fig. 8. In Fig. 8, each displacement field D is normalized by the initial field ${D_0}$, $k = 0.644{k_0}$ for the TTDM in the two-sub-PTC system, $k = 0.639{k_0}$ and $0.648{k_0}$ for the first and second TTDMs in the three-sub-PTC system, respectively, and other parameters are the same as those in Fig. 3. We can see clearly that the field peaks occur around the interfaces of the models for both the two- and three-sub-PTC systems. In addition, for the three-sub-PTC system in Fig. 8(b) and (c), the field distributions indicate the two TTDMs result from the different types of coupling. The first TTDM results from the even-mode coupling, while the second one is from the odd-mode coupling.

 

Fig. 7. (a) Schematic of the three-sub-PTC topological system [PTC₁PTC₂PTC₁], with two interfaces that support two TTDMs. (b) the $|{{t_M}} |$ spectrum of the topological PTC [PTC₁PTC₂PTC₁], and (c) and (d) are the magnified views of TTDMs inside Gap-1 and Gap-3 indicated in (b).

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Fig. 8. Field distributions of (a) the TTDM in the two-sub-PTC topological PTC [PTC₁PTC₂], and (b) the first TTDM and (c) the second TTDM in the three-sub-PTC topological PTC [PTC₁PTC₂PTC₁], where the red dash and grey dash-dotted lines indicate the interface and center of the PTC system, respectively.

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We now focus on the TTDMs in the first gap. The influences of the modulation strength of the refractive index $\rho $ and the time duration g on TTDMs are shown in Figs. 9 and 10, respectively. In Fig. 9, $\rho = {\varepsilon _A}/{\varepsilon _B}$, ${\varepsilon _B} = 1$, and $g = 0.5$; in Fig. 10, ${t_A} = gT$, ${t_B} = ({1 - g} )T$, and $\rho = 6$; other parameters for these two figures are the same as those in Fig. 3. From Figs. 9 and 10, it is obvious that the effects of $\rho $ and g are similar to those in the two sub-PTC case, except that there are two TTDMs in the three-sub-PTC system and only one TTDM in the two sub-PTC case.

 

Fig. 9. (a) The $|{{t_M}} |$ spectrum of the three-sub-PTC topological system [PTC₁PTC₂PTC₁] for different factor $\rho $, where $\rho = {\varepsilon _A}/{\varepsilon _B}$, and (b) the relationship between the k of TTDMs and the factor $\rho $.

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Fig. 10. (a) The $|{{t_M}} |$ spectrum of the three-sub-PTC topological system [PTC₁PTC₂PTC₁] for different factor g, where $g = {t_A}/T$, and (b) the relationship between the k of TTDMs and the factor g.

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Figure 11 shows the effect of the period N of the sub-PTC on the TTDMs, where $\Delta k$ is the TTDM separation, $\rho = 6$, $g = 0.5$, and other parameters are the same as those in Fig. 3. It is interesting that the separation $\Delta k$ drops dramatically when N changes from $2\sim 6$, and then continues to decrease with the increase of N, i.e., the change of N results in a change of the coupling strength of TTDMs, leading to the variation of the distance of the two TTDMs. This result indicates that the three-sub-PTC topological system may be useful for designing double-channel band-stop or band-suppression filters, in which the channel separation can be easily adjusted by changing the period N.

 

Fig. 11. (a) The $|{{t_M}} |$ spectrum of the three-sub-PTC topological system [PTC₁PTC₂PTC₁] for different N, where N is the period of the sub-PTC, $\Delta k$ is the TTDM separation, $\rho = 6$, and $g = 0.5$, and (b) the relationship between the separation $\Delta k$ and the period N.

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3.3 Discussion

We now discuss the cases of the multiple-sub-PTC topological PTC system. Here, we take four- and five-sub-PTC topological systems [PTC₁PTC₂PTC₁PTC₂] and [PTC₁PTC₂PTC₁PTC₂PTC₁], as examples. The $|{{t_M}} |$ spectrums of three-, four-, and five-sub-PTC topological systems are plotted in Fig. 12, where the parameters are the same as those in Fig. 3. We can see that the number of TTDMs is one less than the number of sub-PTCs, i.e., two, three, and four TTDMs for the three-, four-, and five-sub-PTC topological systems, respectively, identical to the number of interfaces in the system. In addition, the effects of the modulation strength of the refractive index, the time duration, and the period of the sub-PTCs for multiple-sub-PTC topological systems are found to be similar to those in two- or three-sub-PTC systems. Here we omit the detailed figures. These features make the topological PTC more practical for applying multiple-channel band-stop or band-suppression filters because the number of channels can be controlled by adjusting the number of sub-PTCs.

 

Fig. 12. The $|{{t_M}} |$ spectrum of the multiple-sub-PTC topological system for (a) three-sub-PTC [PTC₁PTC₂PTC₁], (b) four-sub-PTC [PTC₁PTC₂PTC₁PTC₂], and (c) five-sub-PTC [PTC₁PTC₂PTC₁PTC₂PTC₁] system.

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Then, we examine the robustness of the topological PTC system. Here, we take the case of two-sub-PTC topological system as an example, and the other cases can be studied in a similar way. Firstly, we study the effect of the refractive index shape in the two-sub-PTC topological system, as shown in Fig. 13(a). The deviation $\Delta $ is defined as $\Delta = ({k_{changed}} - {k_{origin}})/{k_{origin}}$, where ${k_{changed}}$ and ${k_{origin}}$ represent the wavenumber k for the TTDM with and without modulation (or change), respectively. As shown in the upper inset of Fig. 13(a), the shape of refractive index changes from a step-like pattern to a smooth one (a more realistic case) by introducing modulation intensity $\alpha $, where $\alpha = 0$ means no modulation. Here, we use the similar way with that in Ref. [8] to study the shape of refractive index. In Fig. 13(a), the inside inset shows the $|{{t_M}} |$ spectrum for the TTDMs with $\alpha = 0,\; 0.02,\; 0.04$, respectively. We find that the deviation $\Delta $ has only a small change under 2% when the modulation intensity $\alpha $ changes from $0$ to $0.05$. Secondly, we study effects of the randomness on the TTDM. Figure 13(b) and (c) show the deviation $\Delta $ of k for the TTDM through 100 times of calculations for temporal slabs with permittivity randomness of (${\pm} 5\%$) and (${\pm} 10\%$), and with time duration randomness of (${\pm} 5\%$) and (${\pm} 10\%$), respectively. The red dashed lines represent the average deviation $\bar{\Delta }$, and the upper insets in Fig. 13(b) and (c) show examples of the random configurations for the refractive index and time duration, respectively. We find that the average deviations $\bar{\Delta }$ are all under ${\pm} $0.5%. The standard deviations are also calculated, which are 0.096% and 0.18% for permittivity randomness of (${\pm} 5\%$) and (${\pm} 10\%$), and 0.48% and 1.08% for time duration randomness of (${\pm} 5\%$) and (${\pm} 10\%$), respectively. The results show that the topological system has good robustness on the random configuration of the refractive index and time duration for the temporal slabs in the system. Thirdly, we study the effects of small variations and defects on the TTDM. Figures 14(a)-(d) show the deviation $\Delta $ of wavenumber k in the TTDM for small variations or defects, by introducing small variations in all the temporal slabs A with $t_A^{\prime} = ({1 + {\eta_A}} ){t_A}$, small variations in all the temporal slabs B with $t_B^{\prime} = ({1 + {\eta_B}} ){t_B}$, temporal defect with ${t_{D1}} = {\eta _{D1}} \times {T_0}$ in position 1, and temporal defect with ${t_{D2}} = {\eta _{D2}} \times {T_0}$ in position 2, respectively. Here, ${\eta _A}$, ${\eta _B}$, ${\eta _{D1}}$, and ${\eta _{D2}}$ denote the variation strengths of the small variations or defects, respectively, the permittivity and permeability of the defects are ${\varepsilon _D} = 6$ and ${\mu _D} = 1$, and the upper insets in Fig. 14(c) and (d) indicate the positions of the defects. We note that, in Figs. 13 and 14, all the other parameters are the same as those in Fig. 3. From Fig. 14, we find that the deviations $\Delta $ have only small changes, within about ${\pm} 1\%$, ${\pm} 2\%$, $- 1\%$, and $- 2\%$ for ${\eta _A}$, ${\eta _B}$, ${\eta _{D1}}$, and ${\eta _{D2}}$ changing in the ranges of $- 5\%\sim 5\%$, $- 5\%\sim 5\%$, $0\sim 5\%$, and $0\sim 5\%$, respectively. Therefore, the results show that the topological system has good tolerances for the small variations and defects.

 

Fig. 13. (a) The deviation $\Delta $ of wavenumber k for the TTDM in the two-sub-PTC topological PTC system for different modulation intensity $\alpha $, where the upper inset shows the shape of refractive index changing from a step-like pattern to a smooth one by introducing $\alpha $, and the inside inset shows the $|{{t_M}} |$ spectrum for the TTDMs with $\alpha = 0,\; 0.02,\; 0.04$, respectively. The deviation $\Delta $ of k for the TTDM obtained through 100 times of calculations for (b) samples with permittivity randomness of (${\pm} 5\%$) and (${\pm} 10\%$), and (c) samples with time duration randomness of (${\pm} 5\%$) and (${\pm} 10\%$), respectively, where the red dashed lines represent the average deviations $\bar{\Delta }$. The upper insets in (b) and (c) show examples of the random configurations for the refractive index and time duration, respectively.

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Fig. 14. (a) The deviation $\Delta $ of wavenumber k for the TTDM in the two-sub-PTC topological PTC system for small variations or defects, by introducing (a) small variations in all the temporal slabs A with $t_A^{\prime} = ({1 + {\eta_A}} ){t_A}$, (b) small variations in all the temporal slabs B with $t_B^{\prime} = ({1 + {\eta_B}} ){t_B}$, (c) temporal defect with ${t_{D1}} = {\eta _{D1}} \times {T_0}$ in position 1, and (d) temporal defect with ${t_{D2}} = {\eta _{D2}} \times {T_0}$ in position 2, where ${\eta _A}$, ${\eta _B}$, ${\eta _{D1}}$, and ${\eta _{D2}}$ denote the variation strengths of the small variations or defects, respectively. The upper insets in (c) and (d) indicate the positions of the defects.

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We further note that, although all the results in this paper are in the momentum domain, devices based on the PTC system can be designed at a desired frequency [10,11]. Because the momentum is conserved in a PTC system, the input and output frequencies can be related via $k = 2\pi {f_{in}}{n_{in}}/c = 2\pi {f_{out}}{n_{out}}/c$, where ${f_{in}}$, and ${f_{out}}$ are the frequencies for the input and output regions, respectively, and ${n_{in}}$ and ${n_{out}}$ are their corresponding refractive indexes. We get ${f_{out}} = ({{n_{in}}/{n_{out}}} ){f_{in}}$, so the desired frequency can be obtained by selecting the proper contrast of ${n_{in}}/{n_{out}}$.

Last but not least, we discuss the feasibility of the PTC system. In fact, time evolution effects have already been observed in many fields such as water waves, acoustic waves, elastic waves, microwaves, and even optical waves [33–36], which is a necessary step in realizing a PTC system. In microwave domain, the PTC system has already been realized by two-dimensional artificial structures [21]. A periodic arrangement of metallic strips was utilized to build the time-varying metasurface, in which $k$-gaps and exponential wave growth were demonstrated. In the optical regime, it is challenging to realize a PTC system because the refractive index of a PTC material should change on a very small order (0.5 or smaller) and in an ultrafast time (on a femtosecond time scale). However, recent work on epsilon-near-zero (ENZ) materials [17–20] has demonstrated that the refractive index of such materials can be set to change on the order of $0.5$ in $10\sim 20$ fs, making ENZ materials excellent candidates for building the PTC system [22]. We believe that an optical PTC system can be realized in the very near future.

4. Conclusion

In summary, we have investigated the properties of TTDMs (or TTISs) in the topological PTC system. The PTC system is constructed by the cascade of multiple sub-PTCs that possess temporal inversion symmetries and different topologies. The cases of two-, three-, and multiple-sub-PTC are studied. It is demonstrated by TMM that, the TTDMs appear when the topological signs of the corresponding gaps are different. The positions of TTDMs can be adjusted by changing the modulation strength of the refractive index, the time duration, and the period of the sub-PTCs. Moreover, the number of TTDMs can be controlled by adjusting the number of sub-PTCs. In addition, the robustness of the systems is also studied, and the results show that the systems have good robust performances. Particularly, the topological PTC systems have pretty good robustness on the random configuration of the refractive index and time duration for the temporal slabs in the systems. Such research may provide a new degree of freedom for PTC applications, such as novel PTC lasers, tunable band-stop or band-suppression PTC filters, and many others, in the field of integrated photonic circuits for optical communications.