{'fraction_non_alphanumeric': 0.03649251772118666, 'fraction_numerical': 0.014045681281176161, 'mean_word_length': 5.1493139628732845, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Higher-order photonic topological states in surface-wave photonic crystals

Photonic topological states have revolutionized our understanding on the propagation and scattering of light. Recent discovery of higher-order photonic topological insulators opens an emergent horizon for zero-dimensional topological corner states. However, the previous realizations of higher-order photonic topological insulators suffer from either a limited operational frequency range due to the lumped components involved or a bulky structure with a large footprint, which are unfavorable for future integrated photonics. To overcome these limitations, we hereby experimentally demonstrate a planar surface-wave photonic crystal realization of two-dimensional higher-order topological insulators. The surface-wave photonic crystals exhibit a very large bulk bandgap (a bandwidth of 28%) due to multiple Bragg scatterings and host one-dimensional gapped edge states described by massive Dirac equations. The topology of those higher-dimensional photonic bands leads to the emergence of zero-dimensional corner states, which provide a route toward robust cavity modes for scalable, integrated photonic chips and an interface for the control of light-matter interaction.

The quest for topological zero-dimensional (0D) cavity modes in 2D photonic systems [24][25][26], which may provide a route toward the build-up of robust and scalable integrated photonic circuits, was unsuccessful until very recently [27,28]. Such an achievement is realized using the higher-order topological insulators [27][28][29][30][31][32][33][34][35][36][37]. Unlike the conventional D-dimensional topological insulators which lead to (D-1)-dimensional topological edge states, a D-dimensional higher-order topological insulator gives rise to (D-2)-dimensional (or even lower-dimensional) topological boundary states, in addition to the (D-1)-dimensional topological edge states, beyond the conventional bulkboundary correspondence. Through the concept of higher-order topological insulators, it has been demonstrated that topological 0D corner states can emerge in microwave circuits with lumped components [28] and coupled optical waveguides [27]. However, the microwave circuits with lumped components can work at low frequencies (normally up to several GHz), and the coupled optical waveguides have lattice constants much larger than the wavelength ( ≈ 40 ) and a small bandgap (2%). These photonic systems are hence unfavorable for future integrated photonics.
Here, we demonstrate a second-order PTI which realizes photonic quantum spin Hall effect through topological crystalline order and gives rise to gapped edge states and topological corner states in surface-wave photonic crystals (PCs) with ≈ 0.5 . The surface-wave PC design provides a very large photonic bandgap (a bandwidth of 28% at the M point below the light-line) as induced by multiple Bragg scatterings. The symmetry-guided approach enables us to tune the photonic bandgap and band topology in both the bulk and the edges, simultaneously. In particular, the orthogonal edges in a finite square structure are described by 1D massive Dirac equations with tunable Dirac masses, which leads to topological localization of light at the corners connecting the orthogonal edges through the Jackiw-Rebbi mechanism [38,39] (see Fig. 1a). The gapped topological edge states, as well as the topological corner states within the edge bandgap, are directly observed and characterized in our experiments. Compared with the coupled optical waveguide arrays which have a large lattice constant ( ≈ 40 ) and small photonic bandgap (2%) and the microwave circuits with lumped components working at low frequencies, the present surface-wave PCs with a small lattice constant can lead to pronounced advantages [24][25][26] such as large photonic bandgap, miniature structures, and the compatibility for the integration of optoelectronic devices.
The 0D topological corner states in PCs provide an efficient way for scalable integration of cavity modes with identical and robust frequency for integrated photonics and optoelectronics.

RESULTS
As depicted in Fig Fig. 1b), which can also be explained by the Brillouin zone folding mechanism (See Supplementary Information). When θ deviates from 0° or 90°, the four-fold Dirac degeneracy splits into two pairs of doubly-degenerate bands, and a photonic bandgap appears. For instance, the photonic band structure for θ =45° is shown in Fig. 1b (the red and green curves for the "conduction" and "valence" bands, respectively).
The evolution of the band edges at the M point with the rotation angle θ is shown in Fig. 1c.
These band edges consist of two pairs of photonic states of opposite parities (see Fig. 1d): the evenparity, quadrupole-like modes (dx 2 -y 2 and dxy) and the odd-parity, dipole-like modes (px and py). When θ goes through 0° or 90°, the photonic bandgap experiences a parity switch, indicating a topological phase transition [38]. There are two topologically distinct phases: the normal insulator phase (NIP) and the topological insulator phase (TIP). The latter is a topological crystalline insulator which mimics the quantum spin Hall effect in photonic systems, as shown in the following.
We now investigate the topological edge states emerging at the interface between the NIP and TIP PCs. Since the PCs may not have C4 rotation symmetry, the edge states for the interfaces along   Fig. 2a. The electromagnetic field profile measured directly by a near-field scanning system (see Supplementary Information) is shown in Fig. 3c. The measured field profile and the corresponding simulation (Fig. 3d) indicate light flow along the edges since the exciting frequency 13.24 GHz is in the bulk bandgap.
We further measure a sample consisting of a TIP PC with θ2 =50° surrounded by a NIP PC with θ1 =-25°, as illustrated in Fig. 4a. We measure the responses of the edge and bulk states in similar means as in Fig. 3. The results in Fig. 4b

CONCLUSION AND DISCUSSIONS
Exploiting surface-wave PCs, we realized experimentally a higher-order photonic topological state with topological edge states and corner states. The concurrent emergence of topological edge states within the bulk band gap and corner states within the edge band gap demonstrates the higherorder topology in our PCs. In particular, the strongly confined topological corner states provide an efficient approach toward scalable integration of degenerate cavity modes in photonic chips. Besides, our surface-wave PCs offer a versatile platform for the realization of higher-order topology through multiple Bragg scatterings which can yield photonic bandgaps much larger than those in the coupled optical waveguide arrays or other analogs with perturbative tight-binding couplings. The surfacewave PC architecture can also be extended to higher frequency regimes. Our study thus opens a pathway toward higher-order photonic topology with large bandwidths for future photonic science and applications.    The red star represents the location of the point source. The green and blue dots denote the locations of the probes.
(b) Normalized magnetic field intensity |Hz| 2 at the two edge probes (the blue and green regions for the xand yedges, respectively) and at the bulk probe (located at the center of the sample). (c) and (d) Measured and simulated magnetic field intensity distribution |Hz| 2 over the sample at 12.71 GHz (marked by the black dashed line in Fig. 3b) excitation, respectively. The interface between the TIP and the NIP PCs is labeled by the white dashed lines.