Band structure and topological phase transition of photonic time crystals

Ji Ma; Zhi-Guo Wang

doi:10.1364/OE.27.012914

1. Introduction

Photonic time crystals (PTC), the physical parameters of which are periodically modulated in time, are temporal analogues of conventional photonic crystals [1–3]. A PTC system is seen as a promising platform for producing amplification and non-reciprocal behavior [4,5]. The breaking of time (space) transition symmetry corresponds to energy (momentum) conservation. That is, temporal (spatial) forward and backward propagating waves will strongly interfere with each other, and inducing a gap in wave vector (frequency) space when the system parameters have temporal (spatial) periodicity [6,7]. Thus, the PTC system can provide a new path for controlling electromagnetic (EM) waves.

In theory, PTC systems can realize all potential practical applications correspond to conventional photonic crystals. We know that the PTC system is required to occur on the time scale of the oscillation of the probe signal in order to observe its band structure features. Thus, setting up such a PTC platform is not easy, especially for high-frequency waves. In the past few years, many strategies are adopted to realize the PTC platforms in experiments [8–10]. It is exciting that a thin-film material with a permittivity that pulsates (uniformly in space) at optical frequencies is realized based on the high nonlinearity effect in epsilon-near-zero materials [10].

Topology in a new degree of freedom for describing an object and has drawn much research attention recently [11,12]. It is actually a mathematical concept with respect to quantities that are preserved under continuous deformations. Topology was extended to various physical systems, such as condensed matter, optics and acoustics, and many others [13–16]. Researchers utilize the topological invariant to characterize the topology of a system. Systems with the same topological invariants are topological equivalent. A topological phase transition occurs when the topological invariant of the system changes, which plays a key role in understanding topological edge states [17].

The topology of an object is robust against the presence of disorder or defects, thus inducing a sea of potential applications in optics, such as unidirectional lasing and photonic circuitry [18,19]. It is well-known that the topological invariant is described by the bulk band geometric phase in a photonic system, i.e., the first Chern number in a two-dimensional periodic system [20–23] and the Zak phase in a one-dimensional periodic system [24–26], which reveals the quantized collective behavior of the EM field on the bulk band. A rigorous relation between the surface states of one-dimensional photonic crystals and the Zak phases of the bulk bands was reported [27]. Inspired by that work, we introduce the topological phase into a PTC system.

The topological phases of a PTC system were proposed very recently [3], the authors calculated the topological invariant of bulk bands and obtained the localized interface state with two PTC of different topological invariants. In this work, we focus on the topological invariants of bulk bands, the $k -gap$ size and their relations. We then reveal the conditions of the topological phase transition of the PTC system. In addition, the dispersive medium for the first time, to the best of our knowledge, is included in the PTC system, which yields a temporal zero-averaged refractive index $k -gap$ that is insensitive to the modulation time scaling at a given modulation frequency.

2. Theoretical method

We consider EM wave propagation inside the PTC with time-dependent permittivity and constant permeability. The constitutive relations of the PTC can be written as $\vec{D} = ε_{0} ϵ (t) \vec{E},$ $\vec{B} = μ_{0} μ_{r} \vec{H} (μ_{r} = 1),$ where $ε_{0}$ and $μ_{0}$ are the permittivity and permeability in vacuum, respectively, and $ϵ (t)$ is the modulation function of the time-dependent permittivity. We restrict ourselves to the case where $ϵ (t)$ is a periodic step-like function:

ϵ (t) = {\begin{array}{l} ϵ_{1} & (n T \leq t \leq t_{1} + n T) \\ ϵ_{2} & (t_{1} + n T \leq t \leq (n + 1) T) \end{array},

where

T = t_{1} + t_{2}

is the modulation period.

t_{1} (t_{2})

is the duration correspond to the permittivity

ϵ_{1} (ϵ_{2}),

and

n

is an integer. In practice, experimenters typically use pump light to drive the materials for realizing the time-dependent permittivity (PTC system) and use a probe signal to study the properties of the PTC system [10]. The step-like permittivity is easier to realize in experiments than continuously varying one.

Using the fundamental solution to the wave equation and Eq. (1), the electric field of EM wave takes the form

E_{i} (t) = A_{0} e^{i (ω_{i} t + k r)} + B_{0} e^{- i (ω_{i} t - k r)},

where

A_{0}

and

B_{0}

are amplitudes, and

k

and

ω_{i} = k c / \sqrt{ϵ_{i}} (i = 1, 2)

denote the wave vector and angular frequency of the EM wave, respectively. One should note that the angular frequency of the EM wave varies with time while the wave vector remains constant. In other words, the EM wave inside the PTC system obeys conservation of momentum at the temporal boundary. Thus, the distinction between the PTC and conventional photonic crystals is the independent variable of the optical parameters, which is responsible for a wide range of exotic phenomena.

We assume the variation of permittivity occurs at the temporal boundary $t_{0}$ . Integrating the Maxwell equations from $t_{0}^{-}$ to $t_{0}^{+}$ yields

\int_{t_{0}^{-}}^{t_{0}^{+}} (\nabla \times \vec{E}) d t = - \int_{t_{0}^{-}}^{t_{0}^{+}} \frac{\partial \vec{B}}{\partial t} d t = - [\vec{B} (t_{0}^{+}) - \vec{B} (t_{0}^{-})]

\int_{t_{0}^{-}}^{t_{0}^{+}} (\nabla \times \vec{H}) d t = \int_{t_{0}^{-}}^{t_{0}^{+}} \frac{\partial \vec{D}}{\partial t} d t = \vec{D} (t_{0}^{+}) - \vec{D} (t_{0}^{-})

The left-hand side of Eq. (3) will vanish in the limit,

t_{0}^{+} - t_{0} = t_{0} - t_{0}^{-} = Δ t \to 0.

Thus, the continuity conditions of the EM fields at the temporal boundary should be

\vec{B} |_{t_{0}^{-}} = \vec{B} |_{t_{0}^{+}}

and

\vec{D} |_{t_{0}^{-}} = \vec{D} |_{t_{0}^{+}} .

Combining the Bloch theorem in time, one can obtain the following dispersion relation for the PTC:

\cos (Ω T) = \cos ω_{1} t_{1} \cos ω_{2} t_{2} - \frac{1}{2} (\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}}) \sin ω_{1} t_{1} \sin ω_{2} t_{2},

where

Ω

is the “Bloch frequency”, akin to the “Bloch wave vector” in the case of spatial periodicity, and

z_{i} = \sqrt{μ_{i} / ϵ_{i}} (i = 1, 2) .

When the absolute value of the right-hand side of Eq. (7) is smaller than or equal to unity, the corresponding Bloch frequencies are real numbers, which means the PTC supports these modes. In contrast, a value larger than unity, i.e., the Bloch frequency is nonzero and imaginary, corresponds to an evanescent mode in PTC.

Generally, the Zak phase is used to characterize the topological invariant of a one-dimensional photonic system. Consulting the definition of the Zak phase for conventional photonic crystals [27], the Zak phase of the bulk band in a PTC system can be written as

θ_{m}^{Z a k} = \int_{- π / T}^{π / T} [i \int_{u n i t c e l l} d t ε (t) u_{m, Ω}^{*} (t) \partial_{Ω} u_{m, Ω} (t)] d Ω,

where

i \int_{u n i t c e l l} d t ε (t) u_{m, Ω}^{*} (t) \partial_{Ω} u_{m, Ω} (t)

is the Berry connection and

u_{m, Ω} (t)

is the unit cell Bloch eigenstate on the

m th

bulk band with Bloch frequency

Ω

, which can be obtained using the transfer matrix method.

3. Results and discussion

The PTC system we are about to consider is shown in Fig. 1, where $T = t_{1} + t_{2}$ is the unit cell size. Here, $A$ and $B$ correspond to the two states of the PTC at time intervals $t_{1}$ and $t_{2}$ with refractive indices $n_{1} = \sqrt{ϵ_{1} μ_{1}}$ and $n_{2} = \sqrt{ϵ_{2} μ_{2}},$ respectively. The unit cell with inversion symmetry is marked by the dashed line in Fig. 1. We assume the probe signal is a plane wave, which will be refracted and reflected in the time domain. In the following calculations, the wave vector of the probe signal $k$ is normalized by $k_{0} = ω_{0} / c (ω_{0} = 1 G H z) .$ Moreover, we employ two ancillary parameters $g = n_{1} / n_{2}$ and $f = t_{1} / T,$ which correspond to the modulation strength and modulation frequency, respectively. According to the definition, zero (infinite) modulation strength means $n_{1} < < n_{2} (n_{1} > > n_{2}),$ thus the bandgap is infinite. In addition, when the modulation strength is unity, the system medium is uniform rather than time dependent. It is foreseeable that the properties of the band gap exhibit a certain symmetry with respect to $g = 1.$ Here, we focus on modulation strength ranging from 0.1 to 0.9. Similarly, a zero (unity) modulation frequency means $t_{1} < < t_{2} (t_{1} > > t_{2}),$ and the bandgap will disappear.

Fig. 1 Schematic diagram of the PTC system.

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3.1 Band structure of the PTC

In the limit of weak dispersion, we calculate the band structure of the PTC using Eq. (7), and the result is shown in Fig. 2(a). The basic physical parameters are $ϵ_{1} = 4,$ $ϵ_{2} = 9$ and $μ_{1} = μ_{2} = 1,$ and the modulation frequency is $f = 0.3.$ The calculated result indicates that the band structure exhibits bandgaps in wave vector space $(k -gap),$ which corresponds to a Bloch frequency with nonzero imaginary part. This results from scattering of the probe signal. Compared with previous research that treats permittivity as a continuous function of time [2,7], the size of the higher-order $k -gap$ here is no longer close to zero.

Fig. 2 Band structure of the PTC with different modulation frequencies: (a) $f = 0.3,$ (b) $f = 0.4,$ and (c) $f = 0.5.$ (d) The displacement field distribution of the compound PTC system constructed by ${(P T C 1)}_{10}$ and ${(P T C 2)}_{10} .$ The subsystems PTC1 and PTC2 correspond to the systems (a) and (c), respectively.

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For convenience, we marked the bulk band 2, $k -gap$ 2, and bulk band 3 with blue letters in Fig. 2. At the same time, we calculate the Zak phases of bulk bands 2 and 3 using Eq. (8), which is shown in red letters. The Zak phase reveals the topological properties of one bulk band and can be used to predict the existence of a localized state [28].

In Figs. 2(b) and 2(c), we examine the band structure when the modulation frequencies of the PTC system are $f = 0.4$ and $f = 0.5.$ The remaining physical parameters are the same as those used in Fig. 2(a). One can clearly see that the bulk bands 2 and 3 cross when the modulation frequency increases from 0.3 to 0.4, i.e., $k -gap$ 2 is closed. When the modulation frequency is increases further, $k -gap$ 2 opens again and is accompanied by a change of the Zak phase in bulk bands 2 and 3, which represents a topological phase transition in the PTC system. Therefore, the band structure and bulk band topological phase can be effectively tuned by choosing the appropriate modulation frequency. This makes PTC particularly interesting. Moreover, we also calculate the displacement field distribution of the compound PTC system by transfer matrix method, which is shown in Fig. 2(d). The compound PTC system constructed by two subsystems (PTC1 and PTC2) that undergo the topological phase transition. We find that a localized state forms at the interface between PTC1 (10 periods) and PTC2 (10 periods), and the localized interface state exponential decay on both sides of the interface. In fact, this result can be seen as an analogue of Su-Schrieffer-Heeger model [29] in PTC system.

3.2 The condition of topological phase transition in PTC system

One of the important features of a band structure is their gap size. This can be used to optimize the structure for specific applications, such as improving the properties of microwave antenna, vertical-cavity surface-emitting lasers, and others [30,31]. We find that it also reveals information on the topological phase transition of the PTC system. In the following, we focus our attention on the second $k -gap$ size and investigate the effects of changes in modulation frequency $f$ and modulation strength $g$ in detail.

Figure 3(a) shows a contour map of the second $k -gap$ size $Δ k$ in unit $k_{0} .$ The horizontal and vertical axes represent the modulation frequency $f$ and modulation strength $g,$ respectively. We take the basic parameter $ϵ_{2} = 4,$ followed by $ϵ_{1} = g ϵ_{2} .$ One can clearly find that the second $k -gap$ size varies obviously as the modulation parameters change. Thus, $k -gap$ size can be designed in a predictable manner. In addition, when the modulation strength and modulation frequency satisfy $g = f / (1 - f)$ (the dashed line in Fig. 3), the second $k -gap$ size is equal to zero, i.e., a topological phase transition occurs. Note that this constraint condition $g = f / (1 - f)$ corresponds to $n_{1} / t_{1} = n_{2} / t_{2},$ where $n_{i} / t_{i}$ has the physical meaning of optical thickness in PTC system. In other words, when the states $A$ and $B$ have the same optical thickness, the second $k -gap$ is closed. This is consistent with the conclusion of the conventional photonic crystal system. Therefore, bulk bands 2 and 3 have different topological phases on each side of the dashed line. According to this discovery, when the modulation frequency $f = 0.4,$ the occurrence of topological phase transition requires modulation strength $g = 0.67,$ which corresponds to the case shown in Fig. 2(b) exactly.

Fig. 3 Variation of the second $k -gap$ size $Δ k$ with modulation frequency $f$ and modulation strength $g :$ (a) $ϵ_{2} = 4$ and (b) $ϵ_{2} = 9.$

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Moreover, we also calculate the second $k -gap$ size for $ϵ_{2} = 9,$ and the result is shown in Fig. 3(b). In this case, the second $k -gap$ size increases given the same modulation parameters used in Fig. 3(a). Interestingly, the relations between the second $k -gap$ size and the modulation parameters $(f$ and $g)$ remain unchanged, which demonstrates the universality of the results obtained in Fig. 3(a).

We find the topological phase transition will be more complicated for higher order bandgaps. This begs the question: what is the relation between them? In what follows, we move our attention to explaining this phenomenon.

We know that the absolute value of the right-hand side of Eq. (7) must be smaller than or equal to unity when $\sin (k c t_{2} / \sqrt{ϵ_{2}}) = 0.$ That is to say the wave vector $k = m_{2} π \sqrt{ϵ_{2}} / c t_{2} (m_{2} \in N^{+})$ must be in the bulk band. Then, we have $\cos (k c t_{2} / \sqrt{ϵ_{2}}) = {(- 1)}^{m_{2}},$ $\cos (k c t_{1} / \sqrt{ϵ_{1}}) = {(- 1)}^{m_{1}} (m_{1} \in N^{+}) .$ Thus, $\cos (Ω T) = {(- 1)}^{m_{1} + m_{2}} .$ Namely, $Ω T = 0$ when $m_{1} + m_{2}$ is even, and $Ω T = \pm π$ when $m_{1} + m_{2}$ is odd. The topological phase transition is equivalent to the degeneracy of band edges. If $m = m_{1} + m_{2}$ is the order of the $k -gap,$ its degeneracy is just the times of topological phase transitions in the full parameter space. Since $m_{1}$ and $m_{2}$ are positive integer numbers, the topological phase transition time of the $m th$ $k -gap$ is $m - 1$ .

The results above reveal the condition of topological phase transition in PTC system. However, how to observe this topological phase transition in experiment? In conventional photonic crystals system, experimenters measure the topological phase transitions through the reflection phase changes [14]. By making analogies, we believe that the topological phase transition of the PTC system can be observed by measuring its reflection phase in the experiment too.

3.3 Band structure of the PTC with a dispersive medium

Finally, we examine the band structure of the PTC system when region $B$ is a dispersive medium. We assume that the dispersive permittivity and permeability are described using $ϵ_{2} = ϵ_{20} - α / k^{2}$ and $μ_{2} = μ_{20} - β / k^{2} .$ We subsequently take $ϵ_{20} = μ_{20} = 1,$ $α = β = 100$ and keeping the other physical parameters unchanged $(ϵ_{1} = 4, μ_{1} = 1) .$ This yields the band structure for different unit cell sizes with the same modulation frequency, which are shown in Figs. 4(a) and 4(b). We can find a special $k -gap$ whose size and edges are invariant to modulation time scaling for a given modulation frequency. One thing to note is that this special $k -gap$ is the only gap exhibiting this characteristic, which is different from the other $k -gaps$ that behave as Bragg gaps do. In effective medium theory, this special $k -gap$ is a zero-averaged refractive index gap; the wave vectors of the band edges are corresponding to the two solutions of the zero-averaged refractive index, which have no connection with the modulation time scaling. In contrast, other Bragg $k -gaps$ do not satisfy the zero-averaged refractive index condition and are therefore sensitive to the modulation time scaling.

Fig. 4 Band structure of the PTC with a dispersive medium for different unit cell sizes and the same modulation frequency: (a) $t_{1} = 4 n s, t_{2} = 8 n s,$ (b) $t_{1} = 3 n s, t_{2} = 6 n s .$

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4. Conclusions

In summary, we calculated the band structure of PTC, the topological invariants of bulk bands and the variation of the $k -gap$ size. We also obtained the correspondence between the topological phase transition and the modulation parameters of the PTC system. Moreover, a temporal zero-averaged refractive index $k -gap$ that is invariant to modulation time scaling for a given modulation frequency is obtained when the PTC system includes a dispersive medium. Our results provide a new path for controlling EM waves and provide topological insight into PTC systems.

Funding

National Key R&D Program of China (2017YFA0303400) and National Natural Science Foundation of China (NSFC) (91630313).

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Band structure and topological phase transition of photonic time crystals

Abstract

1. Introduction

2. Theoretical method

3. Results and discussion

3.1 Band structure of the PTC

3.2 The condition of topological phase transition in PTC system

3.3 Band structure of the PTC with a dispersive medium

4. Conclusions

Funding

References

Cited By

Figures (4)

Equations (8)

Optics Express