Simultaneous Appearance of Different Topological Phases in a Single Photonic System: Coexisting Phases Characterized by Bulk Polarization and Valley‐Chern Number Enabling Dual‐Band Second‐Order Topological States

Abstract

Topological phases appearing in photonic systems have recently attracted much attention in the community of photonics due to the fundamental and practical significances. Despite extensive investigations, the preceding results show that the photonic systems exhibit single topological phases with few exceptions. This work presents the simultaneous appearance of different topological phases in a single photonic crystal structure, which are characterized by bulk polarization and valley‐Chern number. Square‐lattice photonic crystals with unit cells consisting of four dielectric rods, one of which has the size different from the other three ones, exhibit the topological phase characterized by bulk polarization in the first bandgap, depending on the distance between the dielectric rods. Due to the broken inversion symmetry, in the meantime, the photonic systems show the valley‐Hall topological phase in the fourth bandgap. The proposed structures have the advantage of flexible design for generating the dual‐band topological edge and corner states compared with other photonic systems exhibiting single topological phases. From the fundamental and practical significances, the presented results will trigger the extensive investigations for exploring the photonic systems with multiple topological phases and find important practical applications.

Multiple Second‐Order Topological Photonic Systems for Multichannel Beam Splitting and Wavelength Demultiplexing

“…We calculate the band structures along a path of the first Brillouin zone (BZ), denoted as red lines in Figure 1d, by using the plane‐wave expansion method [ 58 ] and here we only consider the transverse magnetic (TM) mode. The band structures of the expanded and shrunken unit cells are exactly the same as each other (Figure 1e,f), while the first and second bands are inverted, as can be seen from the eigenmode parities [ 18,50 ] at high symmetric points of the first BZ marked with + and – symbols (for details, see Supporting Information S1). Consequently, the pair of topologically nontrivial (denoted as A2) and trivial unit cells (A1) with identical first bandgaps are obtained.…”

“…The square rods with side length b of the unit cell A1 (f) are equally separated from each other by d and A2 is the expanded counterpart of A1 (see, e.g., refs. [18,50]). In the unit cells B1 (g) and B2 (h), the side length of the rods and the distance between the center of each rod and the center of the unit cell are denoted as b 1 , d 1 , b 2 , and d 2 , respectively.…”

“…Photonic spin-Hall systems with broken inversion symmetry exhibit spin-valley-locked edge states, [46][47][48][49] which can be used for valley-dependent spin flow, slow light propagation, and spin-valley-coupled Klein tunneling. Recently, simultaneous appearance of two topological phases with different natures characterized by valley-Chern number and bulk polarizations in a single system has been reported, [50] which can be applied for nonlinear frequency conversion. Multiple topology of composite system with different topological phases has also been discussed in a few reports: in refs.…”

Tunable Second‐Order Topological States in Rhombic Photonic Crystals

“…The well‐known square lattice PCs composed of four rods exhibit second‐order topological states in the first bandgap, [ 28,35,47 ] whose topological invariants (bulk polarizations) are quantized due to inversion symmetry. [ 27 ] Naturally, we can anticipate that the deformation of the above unit cell keeping its inversion symmetry might still result in well‐quantized bulk polarizations, which is the generalized version of the square lattice second‐order topological systems.…”

“…Unlike initial steps [5][6][7][8][9] focused on seeking for photonic analogs of fundamental concepts of topological insulators, the realm of topological photonics nowadays has been extending to the other systems such as nonlinear, [10][11][12] non-Hermitian, [13,14] and asymmetric ones, [15][16][17] enabling to find new applications such as nonlinear frequency conversions, [18][19][20][21] topological lasing, [22,23] and others. [24,25] Recently discovered higher order topological phases [26,27] have ignited the extensive investigations on topological photonics [28][29][30][31][32][33][34][35][36][37] for their theoretical and practical significances. However, the generalization of higher-order topology of square lattice photonic crystals (PCs) to rhombic ones has still not been reported.…”