Topological photonics has emerged as a novel approach to engineering 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-deg-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 observe a corner mode that is tightly localized in space while supporting a high 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.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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 . A promising approach of the development is to utilize innovative technological platforms, such as those based on photonic crystals (PhCs) . They may enable dense signal wiring as well as excellent control over waveguide group velocity and dispersion , 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 . In addition, helical waveguides have been realized in quantum spin Hall systems based on microring arrays  and metallic rods  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 [11–14]. 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 [15–18], 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 .
For implementing topological photonic nanocavities, the idea of recently introduced higher-order topological insulators (HOTIs) provides a promising starting point [20,21]. A 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 , phononic crystals , crystalline materials , electrical circuits , acoustic systems [26–29], and optical waveguide arrays , 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 nontrivial in the sense of its finite 2D Zak phase [31–33], . 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, making a stark contrast to conventional defect-based PhC nanocavities. The corner state is found to be tightly localized in both space and time and to indeed function as a high- nanocavity with a small mode volume. We experimentally verify the existence of such a nanocavity mode at a corner of the nontrivial 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  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, such as the Su–Schrieffer–Heeger model for polyacetylene . The band topology is characterized with a Zak phase , defined as integration of Berry connection within the band over the first Brillouin zone. Given that the unit cell under concern preserves inversion symmetry, a nontrivial Zak phase for the lowest band is associated with the presence of bulk dipole polarization within each unit cell . This in turn results in the emergence of edge polarization, which equals topologically protected in-gap states localized at the edges of the 1D material. Since the presence of the edge polarization is guaranteed by the topological property of the bulk, it cannot be eliminated by weak perturbations. Figure 1(a) schematically describes the situation that we consider here. A 2D PhC with nontrivial Zak phases in both and directions, or with a 2D Zak phase of , is terminated to form a 90-deg corner. Each component of the 2D Zak phase corresponds to the existence of edge polarization ( or ), or an optical edge mode at one of the 1D edges [32,38,39]. In this case, a corner charge (), 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 [21,33]. The presence of is topologically protected in a hierarchy of the bulk-edge and edge-corner correspondences as far as . A detailed theoretical description for the emergence of the corner modes is provided in Supplement 1. In what follows, we shall show actual designs of PhCs that translate the situation discussed above.
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-deg corner, together with 1D interfaces, is formed. Both PhCs are based on a common square PhC lattice with a period of , but differ in how to define the unit cell, as shown in Figs. 1(c)–1(e). Each unit cell consists of two air holes with different lengths of sides ( and ) in a GaAs slab. The two unit cells coincide when shifting the center of one of the unit cells by half a period for both and directions. As such, the two PhCs support the same band structure in the reciprocal space. Figure 1(d) shows a calculated band structure for and , using the 2D plane wave expansion method, assuming an effective refractive index of the slab to be 2.7. The band structure exhibits a complete 2D photonic bandgap for light with transverse electric polarization. The insets in Figs. 1(d) and 1(e) show field distributions of the modes of the lowest two energy bands at the X point. For the lowest band of the trivial PhC, an -wave-like, featureless field profile is found, while a -wave-like distribution is supported for its second-lowest band. On the other hand, the topological PhC exhibits an apparent band inversion, i.e., a -wave-like mode resides in its lowest band. Since the zero-energy mode at the point always possesses a featureless flat wave function, the -wave-like distribution of the topological PhC suggests the existence of a “twist” of band nature somewhere within the momentum space. A band inversion is also observed in the band connecting between the and M points (not shown). These results are reflected in the nontrivial 2D Zak phases of (, ) for the lowest energy band of the topological PhC. We note that the topological transition observed between the two PhCs occurs when , at which Dirac points and quadratic band touching appear, respectively, at the X and M points, as depicted in the corresponding band structure shown in Fig. 1(e). As far as , 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 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 modes other than those 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, which exhibits bidirectional transport, as it does not break time-reversal symmetry. Figure 2(c) shows a field profile of the waveguide mode calculated at the 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 generation of the higher-order topological state .
Now, we numerically investigate light confinement around the corner. Figure 3(a) shows a calculated excitation spectrum around the corner formed with the PhCs of and . In the simulator, we put a few sources of impulse excitation at various non-symmetric points around the corner and recorded frequency responses of electric and magnetic field components at various locations. The plotted curve is of a typical response. For comparison, we also computed excitation spectra for the bulk PhC and the straight edge and plotted them in the same figure. For the bulk spectrum, we observed a suppression of excitation within the bandgap with a frequency range from to , where expresses a wavelength. A shrunken bandgap is observed for the case of the straight edge, in which the interface waveguide mode exists and partially fulfills the bandgap. For the corner, a sharp resonance peak at a frequency of appears within the remaining mode gap. 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 factors and mode volumes when changing and in accordance with , computed by the 3D finite-difference time-domain method. We considered GaAs (refractive index, ) PhCs with a slab thickness of . For , the corner mode supports a high factor of 13,000 (limited by vertical light leakage) and a small mode volume of , demonstrating that it can indeed function as a nanocavity. As far as the photonic bandgap is opened (), we observed the corner mode with similar mode volumes. Meanwhile, an exponential decrease of factor is found as increases. This could probably arise from the weakened confinement in the slab by total internal reflection: the slab becomes more air-like with larger values. The complex behavior of the mode volumes might stem from complicated interplay of the light confinement effects by the photonic bandgap and total internal reflection. The current design does not provide any direct protection for and values. Nevertheless, its hallmark property, i.e., the topological protection on the existence of the mode, could be advantageous for applications under environments with harsh refractive index perturbations.
Finally, we experimentally characterize the nanocavity based on the corner mode. We fabricated the designed topological PhC in a 180-nm-thick GaAs slab containing five layers of InAs QDs as an internal light probe in the near infrared. The patterning of the structure was done with a combination of electron beam lithography and reactive ion etching. Airbridge PhCs were formed by dissolving a sacrificial AlGaAs layer underneath the GaAs slab. In the current fabrication condition, we found that it is difficult to clearly open small airholes with . Therefore, in the following, we focus ourselves on the study under the vanishing hole size limit of , allowing for avoiding digging excessively small holes while preserving the topological property of the PhC. Figure 4(a) shows a scanning electron microscope image of a fabricated sample with . A close-up image shows the formation of a corner at the interface between the topological and trivial PhC. We then optically characterized the frequency responses of the system by micro-photoluminescence (μPL) spectroscopy at 20 K. Using an objective lens, we irradiated tightly focused 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 . A bunch of peaks for is of the Fabry–Perot 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 () is well comparable to 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 factor of (calculated , see Supplement 1). For further confirmation of the origin of the resonance, we investigate position dependence PL intensities as summarized in Fig. 4(d). The resonant mode is tightly localized around the corner in both and 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 , we again observed a nice agreement between the theory and experiments. For 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 nontrivial 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 factor over 2000, which is tightly localized around the corner of the PhC. These results will be of importance for developing topological nanophotonic PICs with diverse functionalities.
Core Research for Evolutional Science and Technology (CREST); New Energy and Industrial Technology Development Organization (NEDO); Japan Society for the Promotion of Science (JSPS) (15H05700, 15H05868, 17F17326, 17H06138, 18H01154).
We thank Y. Hatsugai for fruitful discussion.
See Supplement 1 for supporting content.
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