Photonic crystal nanocavity based on a topological corner state

Topological phonics has emerged as a novel approach to engineer the flow of light and provides unprecedented means for developing diverse photonic elements, including robust optical waveguides immune to structural imperfections. However, the development of nanoscale standing-wave cavities in topological photonics is rather slow, despite its importance when building densely-integrated photonic integrated circuits. In this Letter, we report a photonic crystal nanocavity based on a topological corner state, supported at a 90-degrees-angled rim of a two dimensional photonic crystal. A combination of the bulk-edge and edge-corner correspondences guarantees the presence of the higher-order topological state in a hierarchical manner. We experimentally observed a corner mode that is tightly localized in space while supporting a high Q factor over 2,000, verifying its promise as a nanocavity. These results cast new light on the way to introduce nanocavities in topological photonics platforms.

Faster, denser and more energy-efficient photonic integrated circuits (PICs) have been under intensive development, as witnessed in recent rapid progress in e.g. silicon photonics [1]. A promising approach of the development is to utilize innovative technological platforms, such as those based on photonic crystals (PhCs) [2]. They may enable dense signal wiring as well as excellent control over waveguide group velocity and dispersion [3], although often suffer from non-negligible loss and back reflection, the origins of which are closely linked to unavoidable fabrication imperfections.
In this context, novel PIC platforms based on topological photonics gather great attention due to their potential for robust waveguiding immune to perturbations [4,5]. Indeed, unidirectional, back-reflection-free light propagation has been demonstrated in a microwave quantum Hall system with broken time-reversal symmetry using a gyromagnetic PhC [6]. In addition, helical waveguides have been realized in quantum spin Hall systems based on microring arrays [7], metallic rods [8] and on PhC systems emulating crystalline topological insulators [9,10]. Photonic quantum valley Hall systems have also been investigated and employed for reflection-less waveguiding even under the presence of sharp waveguide bends [11][12][13][14]. In contrast to such flourish in topologically-protected waveguides, the development of compact optical resonators in topological photonics is much slower, despite their importance for diverse PIC functions including routing and sorting of signal bits. There are demonstrations of relatively-large topological microresonators, some of which have been employed for topological semiconductor lasers [15][16][17][18], but nanoscale stationary photonic cavities still lack in the device list of topological photonics, except for our previous demonstration based on a topological defect in a one dimensional (1D) PhC [19].
For implementing topological photonic nanocavities, the idea of recently-introduced higher order topological insulators (HOTIs) provides a promising starting point [20,21]. The perimeter of a HOTI supports lower dimensional interface states: for the 2D case, topological 0D states are localized at the corners of the HOTI. In this way, HOTIs can naturally introduce stationary cavity modes into the system. HOTIs and topological 0D corner states have been experimentally investigated using microwave circuits [22], phononic crystals [23], crystalline materials [24], electrical circuits [25], acoustic systems [26][27][28], and optical waveguide arrays [29], but so far not using PhCs.
In this Letter, we report the design and fabrication of a PhC nanocavity based on a topological corner state. We consider a corner of a slab-type 2D PhC that is topologically non-trivial in the sense of its finite two-dimensional Zak phase [30][31][32], = ( , ). The presence of the corner mode as a higher-order topological state is guaranteed by a combination of the bulk-edge and edge-corner correspondences in a hierarchical manner. The corner state is found to be tightly localized both in space and time and to indeed function as a high Q nanocavity with a small mode volume. We experimentally verify the existence of such nanocavity mode at a corner of the non-trivial PhC. Our results provide an important step for developing topological nanophotonic circuitry that can robustly manipulate light at will in the micro/nanoscale. The celebrated bulk-edge correspondence [33] refers to the connection between topological properties of bulk bands of gapped materials and the existence of edge modes. The simplest case considers the lowest and second lowest energy bands of a gapped 1D material, like the Su-Schrieffer-Heeger model for polyacetylene [34]. The band topology is characterized with a Zak phase [35], defined as the integration of Berry connection within the band over the first Brillouin zone. Given that the unit cell under concern preserves inversion symmetry, a non-trivial Zak phase for the lowest band is associated with the presence of bulk dipole polarization within each unit cell [36]. This in turn results in the emergence of edge polarization, which equal topologically-protected in-gap states localized at the edges of the 1D material. Figure 1(a) schematically describes the situation that we consider here.
A 2D PhC with nontrivial Zak phases both in the x and y direction, or with a 2D Zak phase of = ( , ), is terminated to form a 90-degrees corner. Each component of the 2D Zak phase corresponds to the existence of edge polarization (px or py), or an edge mode at one of the 1D edges [31,37]. In this case, a corner charge (Qxy), or a corner mode, is deterministically generated as a convergence of the two sets of the 1D interface polarization, according to the edge-corner correspondence [21,32]. The presence of Qxy is topologically protected in a hierarchy of the bulk-edge and edge-corner A schematic of the slab-type topological PhC that we consider here is shown in Fig. 1(b). A topologically-nontrivial 2D PhC (red colored) in a square shape is surrounded by a trivial PhC (blue).
Along with the interface between the two PhCs, a 90-degree corner, together with 1D interfaces, is formed. Both the PhCs are based on a common square PhC lattice with a period of a, but differ in how to define the unit cell, as shown in Fig. 1 We note that the topological transition observed between the two PhCs occurs when d1 = d2, at which Dirac points and quadratic band touchings appear respectively at the X and M points, as depicted in the corresponding band structure shown in Fig. 1(e). As far as d1 > d2, the PhC colored in red remains topological. Before studying properties of the corner, we verify the bulk-edge correspondence in our case by investigating a straight edge of the topological PhC. The situation under the discussion is schematically drawn in Fig. 2(a). Here, the trivial PhC functions as a trivial bandgap material suppressing the leakage of photons from the topological PhC. The abrupt interface between the two PhC forbids the emergence of other modes than that predicted from the bulk-edge correspondence.
Along with the interface, we computed a projected band structure as shown in Fig. 2(b). As expected, an in-gap waveguide mode (red line) appears within the mode gap. A part of the dispersion curve lies well below the light line (gray), thus supporting a waveguide mode confined within the slab. Figure   2(c) shows a field profile of the waveguide mode calculated at the Γ point. The localization of the wave at the edge elucidates that the waveguide mode is indeed the edge mode of the topological PhC.
Importantly, the 1D topological edge state shows the gapped dispersion curve, which is a prerequisite for the generation of the higher-order topological state [21]. We computed the field profile of the mode causing the peak as shown in Fig. 3(b) and found that it is actually a corner mode as it tightly localizes at the corner. Figure 3(c) shows the evolutions of Q factors and mode volumes when changing d1 and d2 in accordance with d1 + d2 = 0.8a, computed by the 3D finite difference time domain method. We considered GaAs (refractive index, n = 3.4) PhCs with a slab thickness of 0.5a. For d1 = 0.6a, the corner mode supports a high Q factor of 13,000 and a small mode volume of 0.37(λ/n) 3 , demonstrating that it can indeed function as a nanocavity. As far as the photonic bandgap is opened (0.5a < d1), we observed the corner mode with similar mode volumes.
Meanwhile, an exponential decrease of Q factor is found as d1 increases. This could probably arise from the weakened confinement in the slab by total internal reflection: the slab becomes more air-like shows the formation of a corner at the interface between the topological and trivial PhC. We then optically characterize the frequency responses of the system by micro-photoluminescence (μPL) spectroscopy at 20 K. Using an objective lens, we focus pump laser light oscillating at 808 nm on the sample. PL signals were collected by the same lens and analyzed by a spectrometer. Figure 4(b) shows observed spectra for the samples designed with d1 = 0.6a. A bunch of peaks for > 1,100 nm is of the Fabry-Pérot resonances originated from the waveguide mode based on the 1D edge states. Indeed, the fringe interval is confirmed to be dependent on the waveguide length (not shown). Separated from the peaks originated from the waveguide modes, we observe a sharp isolated resonance at 1,079 nm, which is likely to stem from the corner mode. The measured frequency separation of the peak from the waveguide mode edge (~ 21 meV) is well comparable with that simulated numerically (24 meV). The linewidth of the peak is deduced to be 0.43 nm by fitting with a Lorentzian function, as shown in Fig.   4(c). The isolated peak supports a relatively-high experimental Q factor of ~2,500. For further confirmation of the origin of the resonance, we investigate position dependence PL intensities as summarized in Fig. 4(c). The resonant mode is tightly localized around the corner both in the x and y directions. The distributions of the mode match well with those calculated, after taking into account the spatial resolution of the μPL setup. When measured another sample of d1 = 0.7a, we again observed a nice agreement between the theory and experiments. From the reasons above, we conclude that we observed the emission from the designed corner modes in our PhCs. In summary, we have demonstrated a nanocavity based on a topological corner state. We utilized a slab-type PhC with a non-trivial 2D Zak phase, which is suitable for PIC applications based on transverse-electric polarization. A cascade of the bulk-edge and edge-corner correspondences guarantees the existence of the corner modes, opening a deterministic route to introduce nanocavities into topological nanophotonics platforms. Experimentally, we observed a corner mode with a high Q factor over 2,000, which is tightly localized around the corner of the PhC. These results will be of importance for developing topological nanophotonic PICs with diverse functionalities.
During the completion of the manuscript, we become aware of related works in arXiv [38][39][40][41], though none of which measures topological corner states in 2D PhCs in the optical domain.