- Photonics Research
- Vol. 10, Issue 5, 1244 (2022)
Abstract
1. INTRODUCTION
With further development of photonic crystals (PhCs) armored by topological understanding for condensed matter, people have found miscellaneous photonic counterparts of topological phases [1–5]. Among them, the topological edge states generated on the interface between different topological phases promise superior features, such as robustly smooth transmission, backscattering suppression, and defect immunity despite rather strong perturbation of the local boundary. Following quantum Hall phases [2], more intricate topological phases, such as quantum spin Hall phases [3,5] and quantum valley Hall phases [4], are also invented in the context of analog PhC systems, the two of which, respectively, exploit the dichroism freedom by redefining pseudospin/valley concepts in classical wave setups. Such configurable symmetrical lattices, furthermore, provide easy access to topological crystalline insulators (TCIs), for example, those with synchronous rotation giving rise to high tunability in practical realization [6]. Specifically for a kagome lattice of broken inversion symmetry, distinct valley states will emerge in the first Brillouin zone (FBZ) and produce their Berry curvature of opposite values [4,7,8]. Such a valleytronics concept calls for bulk valley states locked to their chiralities, which are possible to couple into and out of communication devices, such as valley filters and valley sources, respectively [4,7–10].
Nevertheless, a concept of corner states from higher-order topology that is one further dimension lower than the edges in a two-dimensional (2D) setup [11–15] has added new bricks to the premise for topological information devices. Among the class of higher-order phases, one type of topology is measured by the fractional bulk polarization (or the position of Wannier centers) [15]. For instance, 0D corner states, other than the 1D edge ones, emerge in the second-order TCIs, whose spatial positions are associated with Wannier centers determined from the polarization value [16–19].
Peculiar to the classical analog for topological quantum physics, the spatial vortex, i.e., the wave function with an undefined phase in certain spatial locations, remains less explored in the context of topological photonics despite its mechanical power to manipulate macroparticles. The vortex flow of electromagnetic waves, also defined as the orbital angular momentum (OAM) of light, may open up new avenues to exert optical torques to matters in a noninvasive manner. In this paper, we will reveal such a possibility by designing a valley higher-order topological insulator (HOTI) in a triangular lattice with symmetry, which is fueled by synchronous rotation of each unit cell. By observing the phase of electric fields near and points, we recognize a valley selection feature discussed previously [4]. We also find that the synchronous rotation mechanism of unit cells induces a band inversion at valleys, which leads to a topological phase transition in our photonic system. This topological transition can be characterized by the extended 2D bulk polarization related to the Zak phase [15,20,21]. In the electric field of the valley HOTI, pointwise corner states are predicted by the 2D bulk polarization. Furthermore, not only does our proposed HOTI have a vortex edge state locked to one of the dichroic valleys [22,23], but also it supports a topologically corner state. Using chiral point sources of different frequencies, our simulations verify that the electromagnetic waves shape into high-quality corner states and robust edge states. Our idea can be extended to higher or synthetic dimensions, which contributes to an experimentally feasible platform for HOTI in the photonic vortex system [4,8,9,24–26].
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2. THEORY AND MODEL
We propose a 2D PhC in a triangular lattice with symmetry, the unit cell of which is composed of six identical pure dielectric cylinders embedded in air as shown in the left panel of Fig. 1(a). Additionally, the maximal Wyckoff points in the unit cell are represented by labels , , and in real space. The dielectric permittivity is , is the lattice constant, and and are the lattice vectors with cylinder diameter , the lattice constants , and . The synchronous rotation angle of the dielectric cylinders in the unit cell is represented by , shown in the right panel of Fig. 1(a) with counterclockwise rotation as the positive direction of rotation whose maximum rotation angle is 60°.
Figure 1.(a) Left: Schematic of unrotated sampled PhC with lattice constant
In this paper, a finite element method is used to calculate the PhC dispersion and to solve for the related electric fields. In a -symmetric lattice, the photonic FBZ contains a pair of and points in its vertices, which are named valley points [27,28] as shown in the Fig. 1(c) inset. Here, the valley states at and , connected by time-reversal (TR) symmetry [29,30], are both linearly dispersed, which are, hence, named Dirac points [31]. We only focus on the eigenstates near these two valley points and refer valleys and to Dirac points throughout our whole paper to be succinct.
Let us focus on the properties of the -valley state, i.e., and for its lower and higher band in frequency, respectively, whereas the counterparts for the valley can be deduced by TR symmetry (cf. Appendix B) [17,29,33]. We find that the photonic valley states are chiral in the sense of phase singularity, which can be readily seen from the electric fields in Fig. 2 where the top and bottom panels display the phase and amplitude distributions, respectively. In the positions of maximal Wyckoff [13] and , the electric amplitudes vanish, and, thus, the phases become singular for the chiral valley states [34]. Note that in our PhC unit of symmetry, it has three maximal Wyckoff positions: at the center of the unit cell, and and at the vertices of it [cf. Fig. 1(a) and Appendix D]. The electric fields above reveal a typical feature of the vortex field, aligning in flow directions defined by time-averaged Poynting vectors [35], which are represented by the arrows of the lower panels in Fig. 2. Therefore, we can control the chirality of the valley vortex by choosing the source chirality. Other than such valley-chirality locking, we also note that the state in field distribution in Fig. 2(a) actually supports a whole circle of zero amplitude and singularity, and that in Fig. 2(b) a Y-type singularity curve, other than discrete singularity points. Recently, it has been suggested that the HOTI state can be evaluated by integrating the Berry connection in the FBZ, which is actually the Zak phase along the wave-vector direction [17,20,21,33]. The 2D Zak phase is connected to the fractional polarization through for where its Zak phase or polarization is completely determined by the bulk property. In the 2D system, the bulk polarization is defined in terms of the Berry connection as [36]
Figure 2.Electric-field distribution
Figure 3.(a) Bulk polarization changes when the unit cells rotate synchronously. Red circles for
Therefore, our HOTI supports, thus, defined corner states in the bandgap, which appear at the maximal Wyckoff positions of the unit cell. Generally, in -symmetric lattices, given a choice of unit cell, there exist special high-symmetry points (HSP) in the unit cell, which are called the maximal Wyckoff position (cf. Appendix D). As in Eq. (1), the nontrivial second-order topology and emergence of the valley-selective corner states are theoretically characterized by the nontrivial bulk polarizations and the associated Wannier centers. Here, the Wannier center refers to the center of the maximally localized Wannier function and for nontrivial polarization insulators, the Wannier center is located at the same position with the maximal Wyckoff position in the unit cell [12,13].
3. NUMERICAL RESULTS AND DISCUSSION
To investigate the concept of valley-selective HOTI, we construct nanodisks made of two types of triangular lattices with distinct polarizations. When , the eigenspectra of our nanodisk are shown in Fig. 4(a). The two colored curves indicate the eigenfrequency functions with for the two types of vertices [up-corner I (U-I) and up-corner II (U-II) for shorthand, respectively]. Here, we refer to the PhC with as the up-triangular PhC (UPC) and as the down-triangular PhC (DPC). A schematic for our simulation is shown in the left panel of Fig. 4(b) where the UPC is surrounded by the DPC to interface a zigzag edge mode. The eigenfrequencies of a bulk-edge corner in the UPC structure are shown in the right panel of Fig. 4(b) where the U-I and U-II corner states both are triply degenerate. In the insets of Figs. 4(b) and 4(e), Wannier centers are colored at the corners of the UPC structure. As the electric field shows in Fig. 4(c), Wannier center representation, illustrated by the red dots in the UPC structure, reveals the valley selectivity of U-I corner states. Additionally, the blue dots in the UPC structure reveal the valley selectivity of the U-II corner state. When the D-I and D-II corner states, respectively, appear below and above the edge state as shown in Fig. 4(d). From the eigenfrequency distribution of the DPC structure, we find that D-I and D-II corner states each also have three degenerate corner states, and the Wannier center configurations (in red dots) of the corner UPC structure are shown in the right panel of Fig. 4(e); whereas the electric field in Fig. 4(f) shows, Wannier centers (cf. , points in the picture) of the UPC and DPC are both fired by corner states. We speculate that in the DPC case [cf. the left panel of Fig. 4(e)] the zigzag boundary appears to disrupt and, hence, the corner states of the two models are excited mixedly at the same time. Moreover, the amplitude of the D-I corner electric field () is higher than that of the D-II corner electric field (). For a further view of valley-selective corner states, we construct two kinds of hexagonal nanodisks forming armchair edges. Along some position of the armchair edge, the corner state of a polarization model can be excited separately, whose position is determined by its polarization value of the unit cell (cf. Appendix E). We remark that the valley selectivity behaves globally, which should apply beyond the UPC and the DPC cases here.
Figure 4.Up-corner states and down-corner states in a triangular nanodisk with opposite polarization. (a) Eigenfrequency evolution spectrum when
Now, we set up full-wave simulation to verify the corner and edge states above in one where the valley dependence of OAM chirality can be exploited to achieve unidirectional excitation of valley chiral states. In Fig. 5, we consider chiral line sources (in blue pentagrams where we choose an LCP OAM source) with a chiral phase, which are fired near the bottom of our PhC with three zigzag boundaries. By switching the source frequency, we can directly control the appearance of edge and corner states as shown in Figs. 5(a) and 5(b). In the superunit where the UPC is surrounded by the DPC, U-I and U-II corner states are, respectively, excited at frequencies and at the same frequencies as in Fig. 4(c). We note that corner states rely more sensitively on frequency parameters than edge ones do. Since the corner state transmits with loss, the electric amplitude of the corner state near the source remains higher than the further one. We choose frequency to fire the edge states, and our simulation shows that electromagnetic waves propagate smoothly along the interface even around sharp corners. It will promise new methods for streering electromagnetic waves along arbitrarily cornered pathways (cf. Appendix G). In the nanodisk where the DPC is surrounded by the UPC, D-I and D-II corner states are excited at and , respectively. In addition, the edge states are excited at . Our results then show that corner states can be selectively excited by tuning the source frequency in addition to valley selection.
Figure 5.Simulated electric-field
4. CONCLUSION
To summarize, we numerically realized a valley-type second-order topology due to unit-cell rotation characterized by the nontrivial bulk polarization. Specifically, the corner states are found to be valley dependent and, therefore, enable flexible manipulation on the wave localization. Thus, topological switches by valley selection of the corner states were numerically demonstrated in our paper. Our valley HOTI and the valley-selective corner states provide preliminary understanding on the interplay between the higher-order topology and the valley degree of freedom, which may find potential applications in valleytronics for future information carriers, such as waveguides, couplers, and topological circuit switches in the terahertz regime [24,30,37–42].
Acknowledgment
Acknowledgment. R.Z., H.L., Y.W., Z.L. and Z.Y. thank the Central China Normal University. Y.L. and D.-H.X. thank the Chutian Scholars Program in Hubei Province, the Hubei Key Laboratory of Ferro- and Piezoelectric Materials and Devices (Hubei University), and the Institute of Physics Carers’ Funds (IOP, U.K.). R.Z. and D.-H.X. proposed the idea. R.Z. performed the calculation, produced all the figures, and wrote the paper draft. H.L. and Y.L. led the project and revised the whole paper thoroughly. R.Z., Y.L. and D.-H.X. put inputs together from all other coauthors in the paper revision.
APPENDIX A: COMPLETE BAND DIAGRAM
The nontrivial bandgaps were distributed in the range of 5.5–6.5 THz, i.e., seventh band, which is what we focused on in the main text. We then showed the complete band diagram of the PhCs for three rotation angles. When and , four complete bandgaps were produced. The green regions indicated the nontrival bandgap, and the brown regions indicated the trivial bandgap as shown in Figs.
Figure 6.TM mode dispersion diagrams when the PhC unit cell rotates for different angles. (a)–(c) When
APPENDIX B: TR SYMMETRY AND ROTATION SYMMETRY
We first reviewed in concept TR symmetry (TRS), then rotation symmetry, and finally the interplay of the two of them. Using these constraints, we then constructed the complete set of invariants for -symmetry insulators. Insulators in this class have TRS with a Bloch Hamiltonian satisfying [
Invariant points under rotation with symmetry, in -symmetric TCIs, there are only three threefold HSPs: , and . These points are shown in Fig.
Figure 7.(a) Schematic showing our choice of lattice vectors
Finally, we inspect the interplay between TRS and rotation symmetry. The two operators commute
APPENDIX C: QUANTIZATION OF POLARIZATION
In this appendix, we review the quantization of polarization due to symmetry [
Without loss of generality, we choose our lattice vectors and reciprocal lattice vectors for each symmetry to be those shown in Fig.
Since are defined as mod , the constraints from the above equations imply that with symmetry, are quantized to be , and the difference of the two polarization components is a multiple of the integer charge . Therefore, the two polarization components are the same,
APPENDIX D: UNIT CELLS AND MAXIMAL WYCKOFF POSITIONS
In -symmetric lattices, given a choice of unit cell, there were special high-symmetry points within the unit cell, called maximal Wyckoff positions, that were invariant under rotations (about the center of the unit cell) up to lattice translations. Let us take the symmetric lattice as an example (Fig.
APPENDIX E: SELECTION OF VALLEY HOTI CORNER STATES
Cutting the lattice structure in different directions, at least, two types of boundaries can be formed: the zigzag and the armchair types. In the eigenmodes, the UPC and DPC are bounded by each other to create edges [
The valley selectivity was further manifested in the armchair edges where the corner states surprisingly only emerged at three (out of six) corners as shown in Figs.
Figure 8.Dual-polarization models (UPC and DPC) select the corner state in the armchair boundary. (a) Left panel: Schematic structure for armchair edges with the DPC surrounding the UPC. Right panel: UPC eigenfrequency distribution of the bulk-edge-corner states. The red and blue dots indicate the positions of U-I and U-II corner states, and the brown dots indicate those of corner states. (b)–(d) Electric-field distribution of U-I and U-II corner states. (e) Left panel: Schematic for armchair edges with the UPC surrounding the DPC. Right panel: DPC eigenfrequency distribution of the bulk-edge-corner states. The red and blue dots indicate the positions of D-I and D-II corner states, and the brown dots indicate those of the corner states. (f)–(h) Electric-field distribution of D-I and D-II corner states. Note that the U-I corner states both appear at the same
APPENDIX F: THE BAND STRUCTURES OF NANORIBBON SUPERCELLS
In this appendix, we calculate the projected band of the ribbon surpercells for our valley photonic crystal (VPC). Consider four types of supercells, comprising the zigzag and armchair interfaces between the UPC and the DPC. From calculation, we know that in the trivial bandgap between bands 1 and 2, rotation of unit cells cannot induce topological edge states. Here, we focus on the edge states of the nontrivial bandgap between band 7 and band 8 [cf. Figs.
Figure 9.Projection bands for the zigzag and armchair interfaces between the UPC and the DPC and its unit-cell layout. (a) The corner state of the zigzag interface appears below the edge state. (b) The corner state of the zigzag interface appears above and below the edge state. (c) and (d) The corner state of the armchair interface all appears below the edge state and the gray area is marked according to the frequency interval of corner states. Note that we omit two edge dispersion curves in (c) and (d) resulting from irrelevant interaction due to perfect electrical conductor (PEC) boundaries along the
APPENDIX G: TRANSPORT OF VALLEY EDGE STATES
In order to further verify the edge-transmission features of our design, we test five boundary types for waveguides and use the chiral OAM source in Figs.
Figure 10.Defected waveguides of various shapes excited by an OAM source, which is an LCP OAM one. (a) In-line waveguide along the zigzag interface; (b)–(e) curved waveguides along the zigzag interface; (f) transmission spectrum for three zigzag edge states (a), (b), and (d) [cf. also the edge mode of Fig.
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