Directional Excitation of Multi-Dimensional Coupled Topological Photonic States Based on Higher-Order Chiral Source

Abstract

The topological phase of matter brings extra inspiration for efficient light manipulation. Here, we propose two-parameter tunable topological transitions based on distorted Kagome photonic crystals. By selecting specific splicing boundaries, we successfully visualize several diverse types of robust edge states and corner states. Through introducing optical vortices with tunable orbital angular momentum, we demonstrate the directional excitation of multi-dimensional topological states as needed. Furthermore, we have studied the coupling effects of multi-dimensional photonic states and the modulation of source in three typical areas. This work provides an instructive avenue for manipulating light in integrated topological photonic devices.

1. Introduction

The concept of topological phase has created another upsurge in the study of photonic crystals (PCs), as it has brought about a novel approach to robust light transmission and localization [1,2,3,4,5,6,7,8,9,10,11,12]. Corresponding topological photonic structures and devices have already been demonstrated to have great potential in laser [5,13], quantum teleportation [14], optical nonlinearity [15,16], etc., applications. As one of the most fundamental physical quantities in optics and photonics, the angular momentum of light, including spin angular momentum and orbital angular momentum (OAM), is of great research interest in information communication, storage, and processing [17,18,19]. By invoking the spin or pseudospin degree of freedom in topological photonic boundary states, we can enable chiral light to propagate unidirectionally without backscattering. When splicing distorted PCs with different quantized bulk polarizations, there exists spin-momentum-locked edge states, known as the quantum spin Hall effect (QSHE) [20,21,22,23]. If we splice different valley PCs, breaking inversion symmetry, valley-locked edge states can be found, known as the quantum valley Hall effect (QVHE) of light [24,25,26,27,28,29]. More recently, the traditional bulk-boundary correspondence has been extended to a higher version. For higher-order topological insulators (HOTIs), the photonic structure can host both gapped-edge states and lower-dimensional corner states [30,31,32,33,34,35]. When combing the photonic HOTIs and OAM degree of freedom, the manipulation of multi-dimensional topological photonic states becomes flexible and efficient [36,37,38,39,40]. By changing the chirality or spatial position of the source, the selected topological edge states and corner states can be directionally excited as needed. In addition, studies have reported that those photonic topological states overlapping in spectral and spatial space will interact with each other, leading to higher-quality coupled optical modes and richer means of regulation [41,42,43].

In this work, we proposed a two-parameter tunable topological PCs, in which diverse types of topological edge states and corner states can be activated. By introducing chiral sources with different topological charge, we demonstrate the directional excitation of multi-dimensional topological photonic states. And the OAM conversion in edge states and corner states show excellent mouldability. Furthermore, we have also visualized the coupling effects of multi-dimensional photonic modes and the influence of different spatial location sources on the well-designed HOTIs. Our work provides inspiration for the development and design of low-loss on-chip photonic devices.

2. Results and Disscussion

In this work, we constructed Kagome lattice PCs in an air background. As shown in Figure 1a, the orange hexagon is the Kagome lattice’s unit cell, and the green solid lines and the black dashed lines represent the intra-unit cell coupling and the inter-unit cell coupling, respectively. For the triangular clusters within the structure, each unit cell contains three identical cylindrical rods. In these two-dimensional hexagonal C₃-symmetric PCs, the lattice constant $a=18 \mathrm{m}\mathrm{m}$, and the hexagonal unit cell side length $l=a/\sqrt{3}$. For the dielectric rods with relative permittivity, $ϵ=15$, and radius $r=1.8 \mathrm{m}\mathrm{m}$. The distance between the center of the cell and the center of each dielectric column is denoted as $d$. Based on numerically simulated COMSOL Multiphysics 6.1 software, we calculated the band structure of transverse magnetic (TM) modes and corresponding electric field distributions. In the design, we proposed two-parameter tunable topological transitions based on distorted Kagome PCs. The crucial structural parameters here are the rotation angles $\theta$ and the distance $d$ of dielectric rods from the center of the unit cell. As shown in Figure 1b,c, the unperturbed Kagome lattices with $d=0.5l$ and $\theta =0$ show an obvious two-fold degenerate Dirac cone at the K point of the Brillouin zone, which is protected by structural symmetry. When we change the parameters $d$ or $\theta$, the degeneracy is lifted with topological phase transition [44]. The higher-order topological phases of the first bandgap can be identified by the symmetry representations of the high-symmetry points (HSP) in the Brillouin zone [45], which provides an important theoretical basis and research direction for us to deeply understand the topological properties of PCs.

The topology of PCs with distinct rotational symmetries can be described by corresponding topological indexes. The unit cells are in the topological trivial phase with a higher-order topological index ${\chi }^{\left(n\right)}=[0, 0]$, while others are in a higher-order topological phase with at least one non-zero topological metric element. Our proposed PCs are protected by C₃ rotational symmetry, and their topological invariants are denoted as follows [46]:

$${\chi }^{\left(3\right)}=\left(\right[{K_1}^{\left(3\right)}], [{K_2}^{\left(3\right)}\left]\right)$$

(1)

A higher-order topological index ${\chi }^{\left(3\right)}=\left[-\mathrm{1,1}\right]$ or $[-1, 0]$ represents higher-order topological phases that host gapped edge states and in-gap corner states under C₃ rotational symmetry. Figure 1b visualizes the variation in the first bandgap at the K-valley with respect to the geometrical parameters $d$ and $\theta$. The x-axis of the phase diagram denotes $d\times cos \theta$, while the y-axis denotes $d\times sin \theta$, with the variation in color shade indicating the size of the bandgap. The white areas correspond to a closed bandgap, while areas of other different colors represent three distinct topological phases. When $0<d<0.5l$, the system is always in the trivial phase ${\chi }^{\left(3\right)}=\left[\mathrm{0,0}\right]$, as marked in yellow. In region $0.5l<d<l/\sqrt{3},$ a different rotation parameter $\theta$ will induce three different topological phases. When $l/\sqrt{3}<d<l$, the topological index of the PCs alternates periodically between ${\chi }^{\left(3\right)}=\left[-\mathrm{1,1}\right]$ (marked in red) and ${\chi }^{\left(3\right)}=\left[-\mathrm{1,0}\right]$ (marked in blue) as the rotation parameter $\theta$ varies at intervals of $30°$.

To demonstrate the higher-order topology phase, we selected three typical structural units: PC1 ($d1=0.7l$, $\theta 1=-30°$), PC2 ($d2=0.7l$, $\theta 2=30°$), and PC3 ($d3=0.3l$, $\theta 3=30°$). As shown in Figure 1c,d, although these three selected unit cells share the same band structure, their first bands have distinct topological properties. For the first band at the K point, the corresponding phase distributions of the electric field $E_z$ are shown in Figure 1d, which reveal two opposite topological phases and one topologically trivial phase.

In order to study the bulk-edge correspondence of various types of PCs, we splice these three units into a zigzag interface, and the projected band diagrams of the calculated edge states are shown in Figure 2a,b. Here, Type-I denotes the structure of PC3 above PC1, and the Type-II denotes the structure of PC1 above PC2. These two types of edge states show different spectral characteristics and field distributions. In the projected band diagram, the bulk states are represented by shades of gray, while the edge states of Type-I and Type-II are represented by green and blue lines. The simulated results show the existence of one-dimensional topological edge states (TESs) within the bandgap, which is consistent with prediction. It is worth noting that Type-II has two valley-dependent edge states crossing each other, with opposite group velocities and phase vortices near the K valley or K′ valley, which are typical characteristics of valley pseudospin-momentum locking. Meanwhile, based on the above three kinds of PCs, we have also calculated the band diagrams of the TES of other splicing structures, as shown in Figure S1.

OAM is one of the most fundamental physical quantities in photonics. It has enabled revolutionary advancements in optical communication, allowing for the multiplexing of multiple data channels through different OAM modes, thereby significantly enhancing the data-carrying capacity of optical systems. However, most of the research on bulk-edge correspondence is limited to first-order OAM chiral sources. The interaction between higher-order OAM chiral sources and the TESs of Kagome PCs has not been mentioned, and there is a lack of effective regulation. Therefore, we study the impact of the chiral sources with different OAM on the manipulation of edge states. Since the eigenmodes of the type-II TES at the eigenfrequency ($f_{TCS}$) of the type-I TCS have an OAM with different proportions, we perform a decomposition of the $E_z$ field near the excitation source of the splicing line on the basis of OAM [47]

$$E_z\left(\phi \right)={\sum }_m{a}_m{e}^{im\phi }$$

(2)

where $a_m$ is the weighting factor of the carrying mth-order OAM, $\phi$ is the polar angle of the circle near the excitation source of the splicing line, as shown in Figure 2b, and the OAM modes with different orders are orthogonal to each other. When the eigenfrequency of the Type-II TES is $f_{TCS}$, there exist two wave vectors, $k_1=0.466\left(2\pi ∕a\right)$ and $k_2=0.534\left(2\pi ∕a\right)$, which are symmetric about $k_x=0.5\left(2\pi ∕a\right)$ and have opposite energy flow directions, as shown in Figure 2b. We performed decomposition on these two eigenmodes and found that for the mode of order $m=1$ ($m=-1$), the wave vector $k_1=0.534\left(2\pi ∕a\right)$ $\left(k_2=0.466\left(2\pi ∕a\right)\right)$ is dominant. Thus, we predict that the light propagates to the right side (left side). For the mode of order $m=2$ ($m=-2$), the wave vector $k_1=0.466\left(2\pi ∕a\right)$ $\left(k_2=0.534\left(2\pi ∕a\right)\right)$ is dominant. Thus, we predict that the light propagates to the left side (right side). The simulation results are basically consistent with the theory, as shown in Figure 2c. We place the chiral source marked with a pentagram at the position circled in the inset of Figure 2b without changing the supercell geometry, and the OAM ($m=\pm 1, \pm 2)$ is considered. When $m=1$, the chiral light propagates unidirectionally to the right without backscattering; when $m=-1$, the one-way transmission direction is the opposite. The transmission spectrum in the lower half of Figure 2c further demonstrates the phenomenon of unidirectional excitation in TESs within a wide frequency range, which is similar to the QVHE of light. Furthermore, we have calculated the case of $\left|m\right|=2,$ where the direction of propagation of the edge states is reversed to the case of $\left|m\right|=1$. We have also conducted a detailed study on the selective regulation of the transmission direction of Type-II TES by higher-order OAM chiral sources, as shown in Figure S2. Consequently, OAM with different signs and orders can achieve the directional excitation of the Type-II TES. In addition, the difference in the position of the spatial chiral source will also lead to the selective excitation of the TES, as shown in Figure S3.

In addition, the photonic structure can also host both gapped TESs and lower-dimensional TCSs. We construct a structure consisting of PC3 and PC2, as shown in Figure 3a. It shows the calculated eigenmodes of the structure, with black dots denoting the bulk states, yellow dots denoting the edge states, and red or blue dots denoting the type-I or type-II topological corner states (TCSs), respectively. Type-I TCSs result from nearest-neighbor interactions and are protected by the generalized chiral symmetry. As for type-II TCSs, their formation is attributed to the localization of TESs at the corners. This localization is induced by long-range interactions, so its light field distribution is on both sides of the splicing corner. Here, we focus on type-I TCS with better localizability in trapezoidal splicing with a frequency of 6.481 GHz. The intensive cavity–waveguide interaction of the topological coupled cavity–waveguide system provides a strong optical localization, high-quality factor, and excellent robustness compared to conventional corner state cavities. Therefore, we expect the waveguide–cavity coupling to reamplify the dependence of the excitation of topological corner states on the order of the external source. Therefore, we propose a waveguide–cavity coupling system consisting of PC1 (blue region), PC2 (red region), and PC3 (yellow region), with the distance between the TES waveguide and the TCS cavity $L=7a∕\sqrt{3}$. The edge states of the structure excite the corner states of the corner structure at the same time, and the unidirectionality is weak. By modifying the size of the cell medium column in the corner cavity (the yellow area of the trapezoidal structure constituted by PC3 units in Figure 3a), the frequency of the corner state is reduced, and there is the possibility of unidirectional excitation by the edge state. The edge states in this structure support corner states ranging from 6.279 GHz to 7.266 GHz. The corner state frequency of the corner structure is exactly within its encompassing range, and as shown in Figure 3b, the designed structure can successfully achieve coupling effects between the TES waveguide and the TCS cavity. As shown in Figure 3c,d, a right-handed source with OAM $m=+1$ is placed under the splicing interface. For the direct excitation of corner states, the intensities at two splicing corners are nearly equal and both are weak. Here, we define the Q factors as $Q=f/\mathsf{\Delta }f$, where $f$ is the resonant frequency, and $\mathsf{\Delta }f$ is full-width at half maximum. For the direct excitation of corner states, the Q factor is about 800, while for edge-coupled corner states, the Q factor is about 5000. More importantly, the intensities at the splicing corner are significantly enhanced by three orders of magnitude. In my opinion, this may be relevant to the mode coupling. For two-dimensional structures, the zero-dimensional corner states are localized modes. When the excitation source is far away from the nanocavity, the coupling efficiency of the EM field is usually low. The one-dimensional edge states are transmission modes, which can enable light to efficiently propagate unidirectionally without backscattering. Furthermore, by introducing coupling effects between corner states and edge states, the coupled eigenstate can support higher-quality modes. Therefore, the corner state in edge-coupled cavities is significantly enhanced.

Furthermore, we adjust the OAM of the chiral source when maintaining the supercell geometry and the position of the excitation source in the structure. As shown in Figure 4, when we change the OAM of the chiral source, the relative intensity at different splicing corners can also be manipulated. When $m=0$, the intensity of corner states at both sides are nearly equal, owing to the bidirectional excitation of TES; the energy coupled in the TCSs cavities is nearly the same. When $m=1$, due to the unidirectional excitation coupling of the TESs, the intensity of the right splicing corner is much stronger than the left one, which presents an excellent switch ratio. While for $m=2$, the result is opposite, the left splicing corner is higher. Although the electric field intensity of the excited corner state decays as the order of source $\left|m\right|$ increases, the electric field intensity and selective excitation are strongly improved compared to the conventional corner state cavity structure (as shown in Figure S4 in the Supporting Information). Moreover, when changing the chirality of the source, the selective excitation of the corner states is also strictly inverted. The corner state at the right intersection is selectively excited in the case of $m=1$, whereas at the left intersection, the case is $m=-1$. The frequencies and electric field strengths excited by both remain highly consistent, with only a change in position (as shown in Figure S5). Therefore, the higher-order sources can effectively control the selective excitation of corner states in edge-coupled structures. The multi-dimensional topological states can be flexibly and efficiently manipulated by a chiral source with a different OAM.

In addition, we placed the chiral source in three typical areas near the splicing interface, and the transmission and localization of light can be manipulated accordingly. As shown in Figure 5a, the propagation direction of the edge state changes alters the position of the source marked by the triangle (above the interface), quadrilateral (on the interface), and star (below the interface), respectively. We quantitatively analyzed the cause of this phenomenon by calculating the distribution of the Stokes parameters near the Type-II splicing line [42], as shown in the Figure S6. For instance, the edge state propagates unidirectionally to the left, when a source with $m=1$ is placed at position 1, whereas it is positioned to the right for the source at position 3. In the case that the source is at position 2, the edge states propagate in both directions at position 2. These variations also occur for higher-order sources, but with distinct phenomena. However, the unidirectional transport of edge states reverses in both cases when the OAM of the excitation source changes to its opposite (as shown in Figure S7). This calculation demonstrates that the unidirectional selective excitation of the first-order source is the best in the case of changing the source position. Therefore, we try to selectively excite the corner state using the first-order source in the edge-corner coupling structure. Simulation results are shown in Figure 5b, where the unidirectional excitation of the corner state agrees with the selectivity of the boundary state. We find in Figure 5c that the electric field strength is enhanced as the excitation source moves down, despite there being no significant change in the quality factor.

3. Summary

In summary, we have proposed two-parameter tunable topological transitions based on distorted Kagome PCs. Based on several diverse types of robust edge states and corner states, we demonstrate the coupling effects between the multi-dimensional photonic topological states. More importantly, we introduce optical vortices with tunable OAM to the designed HOTIs. And the directional excitation of the corresponding multi-dimensional topological states shows excellent mouldability and high efficiency. Our results demonstrate controllable manipulation of on-chip photons, which may support potential applications in nanophotonics.