Topological Cavity Chains via Shifted Photonic Crystal Interfaces

Abstract

Recent advances in topological photonics provide unprecedented opportunities to realize a photonic cavity. A recent work shows that the electromagnetic wave can be effectively trapped via the shifted photonic crystal interfaces (SPCIs), which offers an alternative approach to realizing the photonic cavity. Here, we proposed one-dimensional topological insulators based on an SPCIs-induced cavity chain, which is analogous to the Su–Schrieffer–Hegger model and is compatible with the silicon-on-insulator platform. Owing to the asymmetry feature of SPCIs-induced cavities, the topological cavity chains can be either realized by alternating the cavity modes or by tuning the distance between two cavities. The nontrivial band topology of SPCIs-induced cavity chains is further confirmed by observing topological end states, which exhibit robustness against geometrical imperfections. Our work holds promises for designing robust photonic devices, which may find potential applications in future integrated photonics.

1. Introduction

As an essential building block of photonic devices, a photonic cavity that can strongly confine light has attracted much attention in past decades [1,2,3,4,5,6,7,8,9,10]. One of the most promising platforms is photonic crystals (PCs), which are periodic structures of electromagnetic materials with photonic bandgap and can control light propagation effectively [11]. Various mechanisms have been exploited for building photonic cavities, including distributed Bragg reflection [2,6], total internal reflection [8], Fano resonance or bound states in the continuum [12], and so on. Particularly, recent advances in topological photonics provide new insight for designing novel topological photonic cavities [13,14,15,16,17,18,19,20,21,22]. For example, geometry-independent topological photonic cavities based on gyromagnetic materials have been demonstrated, which find potential application in nonreciprocal single-mode lasing [17]. Another topological cavity with broken time-reversal symmetry is realized in topological photonic crystals with a dislocation, which is protected by the concurrent wavevector space and real-space topology [18]. However, due to the weak response of gyromagnetic materials at optical frequencies, it is highly desirable to realize topological photonic cavities based on all-dielectric materials. A representative scheme to realize all-dielectric topological photonic cavities is to utilize a coupled ring-resonator array, which produces topological edge-mode lasing in non-magnetic material. In 2017, the discovery of the high-order topology, in which the dimension of topological states can be more than one dimension lower than that of the bulk states, opened a new avenue to realize an all-dielectric topological cavity. Inspired by the higher-order band topology, the promise of a higher-order corner state as a nanocavity has been experimentally verified [19]. Very recently, Zhan et al. proposed a new scheme to trap light via shifted photonic crystal interfaces (SPCIs) in square photonic crystals, where the cavity mode is a physical manifestation of the Jackiw–Rebbi soliton [22].

On the other hand, as the prototype of one-dimensional topological insulators, the Su–Schrieffer–Heeger (SSH) model, originally introduced to describe polyacetylene chain, provides a concise but rich physics to understand band topology. Institutively, the SSH model is in a nontrivial (trivial) regime when the intercell hopping is larger (smaller) than the intracell hopping. Thanks to its structural simplicity, the photonic SSH model has been extensively studied in cavity chain systems by taking different photonic modes as a basic photonic resonator [23,24,25], including photonic cavity, plasmonic nanoparticles, and so on. A benefit from the topologically protected edge states in the SSH model that has been reported is that the topological photonic cavity chain can enable giant enhancement of nonlinear harmonic generation [26], high-Q nanocavities [27,28], and an on-chip wavelength sensor [29]. Nevertheless, in all existing works, the only way to realize a topological cavity chain is to enlarge the intercell hopping, which is required for tuning the distance between neighboring cavities. Moreover, for most existing works, photonic cavities designed by introducing the defect into PCs usually do not exhibit topological origins.

In this work, we take SPCIs-induced cavities as basic resonators and study the topological properties of the SPCIs-induced cavity chain with different configurations. We find that the topological phase transition of the SPCIs-induced cavity chain can be triggered either by tuning the coupling strength or by changing the cavity mode. Remarkably, a kind of topological cavity chain with identical spacing between two neighboring cavities, which originated from the asymmetry distribution of the SPCIs-induced cavity modes, has been identified. Our work holds promise for designing robust photonic devices, which may find potential applications in future integrated photonics.

2. Results and Discussion

We start by revisiting the SSH model. As shown in Figure 1a, the effective Hamiltonian of the SSH model is given as follows:

$$H\left(k\right)=\left(\begin{array}{cc}{\omega }_0& t_1+t_2{e}^{-ika}\\ t_1+t_2{e}^{ika}& {\omega }_0\end{array}\right),$$

where ${\omega }_0$ denotes the eigen frequency of the resonator and $k$ and $k$ refer to the wave number and lattice constant, respectively. To introduce the SPCIs-induced cavity as the basic resonator, we first consider two-dimensional PCs with circular air holes arrayed in a square lattice in a silicon background ($\epsilon =11.9$). As shown in Figure 1b, the radii of the air hole are denoted as $r=0.5a_0$, where $a_0$ is the lattice constant. Throughout this work, we only consider the transverse–magnetic (TM) mode and all simulations are carried out with the commercial software COMSOL Multiphysics (version 6.3). A complete band gap is observed in the photonic band structure of the square PCs, which is an essential condition to trap electromagnetic waves. To illustrate the cavity mode, we construct a finite supercell and divide the supercell into three regions [from region I to III; see Figure 1c]. It is evident that shrinking or expanding regions II and III with a distance of $g(0<g\le {\displaystyle \frac{a_0}{2}})$ gives two interfaces along the $y$-axis, which are termed as shifted photonic crystal interfaces (SPCIs). According to Ref. [22], each SPCI supports two pairs of interface states owing to the mismatch of the Wannier center between two PCs. It is reported that the dispersion of SPCIs becomes gapped when $g\ne {\displaystyle \frac{a_0}{2}}$, and the gap size is well described by a finite Dirac mass. In particular, placing two SPCIs with opposite Dirac mass leads to a photonic bound state according to the Jackiw–Rebbi theory, which can be utilized as photonic cavity modes. For convenience, the cavity mode formed by two expanded (shrunken) SPCIs is termed a type-I (type-II) SPCIs-induced cavity. As an example, we replot the eigen electric field patterns of type-I and type-II SPCIs-induced cavity modes with $g=0.3281a_0$ in Figure 1d. It is seen that the type-I (type-II) SPCIs-induced cavity is odd (even) symmetric with respect to the $y$-axis, and the electric field mainly localizes along $+x(-x)$ direction, indicating both cavity modes are asymmetric along $x$-axis.

To construct a cavity chain, a supercell formed by placing two identical SPCIs-induced cavities together is designed to satisfy the periodic boundary condition. As indicated by the dashed box in Figure 1e, the supercell host mirrors symmetry along the $x$-direction. The lattice constant for the supercell is $a=20a_0$, and the distance between two neighboring cavities within a primitive cell (neighboring primitive cells) denoted as $d_1$ ($d_2$) are varied. It is noteworthy that the SPCIs-induced cavity plays the role of the basic resonant unit cell, and the overlapping integral of field between two cavities mimics the coupling strength.

We first study the mode splitting effect of the coupled SPCIs-induced cavity versus shifted parameter $g$ by setting $d_1=d_2$. As shown in Figure 2a, the frequency of the type-I (type-II) SPCIs-induced cavity mode gradually increased (decreased) accompanied by the increase in $g$. Note that the presence of the mirror symmetry enables a classification of even and odd states. Hence, there are two nearly degenerated modes, of which the splitting frequency is hardly enough to distinguish owing to the weak coupling strength (see the zoom-in plot in the inset). By adopting the periodic boundary condition, we implement the eigen calculation of the coupled SPCIs-induced cavity system. The band structures with the eigen field patterns for type-I and type-II coupled SPCIs-induced cavity systems with $g=0.3281a_0$ are displayed in Figure 2b and Figure 2c, respectively. Although the mode splitting is very weak for both systems, it is observed that the frequency gap of the former is an order of magnitude larger than that of the latter. Note that, for a one-dimensional system, the topological properties of the system can be described by the Zak phase. The Zak phase ${\theta }_{zak}$ of each band is given as follows:

$${\displaystyle \frac{{\theta }_{zak}}{\pi }}={\displaystyle \frac{1}{2}}\left[\eta \left(k_x=0\right)-\eta \left(k_x=\pi /a\right)\right]:\mathrm{m}\mathrm{o}\mathrm{d}2,$$

where $\eta$ is the mirror parity of the eigenmodes at the high-symmetry point.

Moreover, for the type-I coupled SPCIs-induced cavity system, the mirror eigenvalue of the electric field of the lower band at $k_x=0$ and $k_x=\pi$ are the same. In contrast, for type-II coupled SPCIs-induced cavity systems, the mirror eigenvalues of the electric fields of the lower band at $k_x=0$ and $k_x=\pi$ are opposite, indicating a parity inversion. According to the band topology theory, we claim that the tiny gap exhibits with nontrivial band topology, which is described by a nonzero Zak phase. We remark that the asymmetric field distribution of the SPCIs-induced cavity results in the inequivalent strengthening of intercell and intracell hoppings, which plays a key role in the topological phase transition of the cavity chains.

Generally, large intercell hopping means large field integral overlapping, which is required for decreasing the distance between two resonators within neighboring primitive cells. Following this method, we study the mode-splitting effect of type-II coupled SPCIs-induced cavities by tuning the distance between two cavities. As shown in Figure 3a, the band gap first decreases accompanied by the increasing in $d_1$, closes at around $d_1=15a_0$, and increases slowly when $d_1\ge 16a_0$. Different from previous studies, the band gap does not close at $d_1=d_2=10a_0$. This is because the overlapping integral of SPCIs-induced cavities within a primitive cell is not equivalent to that between two primitive cells, which originates from the asymmetry field distributions of the SPCIs-induced cavity.

Guided by the phase diagram, we proceed to study the band topology of type-I coupled SPCIs-induced cavity system by selecting different geometric configurations. We first construct a type-I coupled SPCIs-induced cavity system by setting $d_1=4a_0$ and $d_2=16a_0$. The eigen calculation results in Figure 3b show that the two bands formed by two coupled SPCIs-induced cavities are separated by a large band gap owing to the strong coupling, which can be further verified by the eigen field pattern. It is seen that the electric field patterns of the first band at $k_x=0$ and $k_x=\pi$ are antisymmetric, suggesting that they are of the same mirror parity. We then consider another type-I coupled SPCIs-induced cavity by exchanging $d_1$ and $d_2$. The band structure depicted in Figure 3c shows that the two bands formed by two cavities are hardly distinguished. Nevertheless, the zoom-in shows a tiny gap separating two nearly flat bands around normalized frequency 0.2687, indicating that the intercell hopping is slightly larger than the intracell hopping. The band topology can be further confirmed by examining the eigen electric field patterns at high symmetry points at the right panel of Figure 3c. It is seen that the electric fields of the first band at $k_x=0$ and $k_x=\pi$ are odd- and even-symmetric with respect to the mirror line, respectively, indicating a parity inversion has occurred.

According to the bulk-edge correspondence, the nontrivial band topology manifests itself as the boundary states. To this end, we construct a finite cavity chain by placing seven type-II coupled SPCI-induced cavities with $d_1=d_2=10a_0$ and implement the eigen calculations. As shown by the eigen spectra in Figure 4a, a tiny gap (~${10}^{-6}c/a$) separates two bands, in which two edge states are verified by the eigen field distribution [see Figure 4b]. Inherited from the even symmetry feature of the cavity, it is seen that the topological edge state of the cavity chain is also of even symmetry. In order to display the field pattern more clearly, we extract the electric field value at $y=0.01a$ (see the dashed lines in Figure 4b) and find that the electric field is almost confined in the cavities at the leftmost or rightmost end of the cavity chain. In addition, a typical bulk state of the cavity chain featured with extended field distribution is observed in Figure 4c, which is evidence of the large intercell coupling between two primitive cells in the topological cavity chain.

In parallel, we construct another topological cavity chain by placing seven type-I coupled SPCI-induced cavities with $d_1=16a_0$ and $d_2=4a_0$ and implement the eigen calculations. The eigen spectra and the eigen field patterns of topological edge states are depicted in Figure 4c and Figure 4d, respectively. Compared with the former type-II topological cavity chain, a larger topological gap (~${10}^{-5}c/a$) with two topological edge states is observed in the eigen spectra owing to the stronger intercell coupling. Moreover, the strong intercell coupling makes bulk states more dispersive [see the inset of Figure 4d]. The topological edge states are further identified by the electric field patterns, which are almost localized at the leftmost or rightmost end of the cavity chain. In contrast to the former case, the topological edge states exhibit odd symmetry, which is inherited from the type-I SPCIs-induced cavity mode. Furthermore, by examining the electric field value $y=0$ (see the dashed lines in Figure 4d), it seems that topological edge states are not as localizable as that in Figure 4b despite the larger band gap, which is mainly attributed to the strong intercell coupling.

To verify the robustness of the topological edge states in SPCIs-induced cavity chain, we investigate the influence of disorders on the topological edge states via the following two ways: (1) variation of the resonant frequencies of all cavities except the one on the edge [see Figure 5(a1,c1)], and (2) variation of the resonant frequencies of all cavities [see Figure 5(a2,c2)]. Note that the variation of the resonant frequencies is introduced by randomly changing the radius of the air holes around the cavity, and the deviation of the hole radius is limited to 2%. To quantify the disorder strength, we define a perturbation term $\delta r_i=0.02\beta {\zeta }_i{a}_0$, where $\beta (0\le \beta \le 1)$ and ${\zeta }_i(-1\le \zeta \le 1)$ refer to the disorder factor and a random number of the $i$th perturbated hole, respectively. The calculated eigen spectra of the type-II SPCIs-induced cavity chain with $d_1=d_2=10a_0$ as a function of the disorder factor $\beta$ are shown in Figure 5b. It is seen that the frequency of the topological edge states is almost unchanged while the frequencies of the bulk states are shifted apparently when the perturbation is introduced to all cavities except the edge one. Furthermore, when the resonant frequencies of all cavities are perturbated, it is observed that the frequency of the topological edge states is no longer pined. However, it should be emphasized that the edge states still exist due to the protection of nontrivial band topology. The eigen spectrum of type-I SPCIs-induced cavity chain with $d_1=16a_0$ and $d_2=4a_0$ are plotted in Figure 5d. Apparently, the results are similar to those for the type-II SPCIs-induced topological cavity chain.

3. Conclusions

We designed two kinds of photonic cavity chains in which a cavity is induced by SPCIs and studied its topology properties by mapping it to the SSH model. We found that the topological phase transition of both SPCIs-induced cavity chains can be triggered either by tuning the coupling strength or by changing the cavity mode. Remarkably, the topological type-II SPCIs-induced cavity chain with identical spacing distances between two neighboring cavities, which originated from the asymmetry distribution of the SPCIs-induced cavity modes, has been identified. The robustness of the topological edge states is verified by implementing a disorder testament. Our work holds promise for designing robust photonic devices, which may find potential applications in future integrated photonics.