The demand for larger capacity in semiconductor devices is driving the development toward multichannel functionalization. In order to increase the multiplexing communication capability of devices, various methods have been proposed to improve the efficiency of information transmission. These include wavelength division multiplexing (WDM) [1–3], polarization division multiplexing (PDM) [4–6], spatial division multiplexing (SDM) [7–12], modular division multiplexing (MDM) [13–16], orbital angular momentum multiplexing (OAMM) [17–20], and hybrid multiplexing, which are based on the principle of combining multiple signals into a shared channel for propagation [21–23]. How to achieve stable output resources of multichannels in the same semiconductor micro/nano device has become one of the hotspots of researchers’ attention [24–27].

Topological photonics is widely introduced into many different photonic systems because of its strong stability and tunability, including the unidirectional transmission optical waveguide [28], the topological photonic crystal fiber [29], and the topological photonic crystal cavity [30]. The topological photonic crystal cavity is one of the most important areas for photonic crystal research. The modes can be designed in many different ways, including the one-dimensional (1D) optical waveguide, which was designed by the Su–Schrieffer–Heeger (SSH) model [31,32] and the Jackiw–Rebbi model [33], the two-dimensional (2D) Dirac-vortex topological optical cavity, which was designed based on the Jackiw–Rossi model [34], and high order topological modes designed based on the Benalcazar–Bernevig–Hughes (BBH) model [35]. The Dirac-vortex topological cavity is realized by applying generalized Kekulé modulation in a hexagon lattice at telecommunication wavelength. The Dirac-vortex cavity has attracted wide attention from researchers of surface emitting lasers because of its excellent properties, such as arbitrarily extended mode region, arbitrary mode degeneracy, high compatibility, and its larger free spectral range (FSR), which means that the Dirac-vortex topological cavity can achieve stabler single-mode emission and a wider spectral tuning range in a large-area photonic crystal structure [34].

In this paper, the dual-channel topological Dirac-vortex single-mode and multimode emission capability and property are studied, based on generalized Kekulé modulation in a hexagon lattice. We can open two Dirac points of different wavelengths by adjusting the size of hexagon, so the structures can generate two bandgaps at different wavelengths within a 2π angle. In addition, we study the Dirac-vortex cavity with different winding numbers, summarize the emission characteristics and mode distribution of the cavity with winding numbers, and obtain the general law in midgap. The characteristics of dual-channel emission can make it useful as the signal emission source of WDM devices, while the characteristics of multimode emission make it useful as the signal emission source of MDM devices. The two characteristics of dual-channel and multimode emission can coexist, making the Dirac-vortex cavity useful for mixed multiplexing of WDM and MDM. These features greatly improve the information transmission ability of the Dirac-vortex cavity and unlock new options for the application of the Dirac-vortex cavity.

To start, we calculated and found 2D lattice structures with two Dirac points from various band structures by the finite element method (FEM). In this paper, the 2D hexagonal photonic crystal lattice model is adopted. The refractive index of silicon is 3.4, and the holes are filled with air, as shown in Fig. 1(a); the chrysanthemum color is silicon, and the white color is air. The side length of the supercell lattice is a, the side length of the unit cell is d=a/3, and the initial side length of the hexagon is r=0.8d. There are two Dirac points (P1 and P2, the points are indicated by the blue arrows) with different frequencies at point K in the transverse magnetic (TM) band diagram of the unit structure, as shown in Fig. 1(c). In this paper, the energy bands of other hexagon lattices with different shapes are studied, and it is found that there are two Dirac points in Kagomé and circular crystals. See Appendix A for details of band distribution.

Next, we expanded the size of the unit cell (dashed hexagon) into a supercell (solid hexagon) structure and used the generalized Kekulé modulation method to open the bandgap at two Dirac points at the same time. Figure 1(b) shows the morphology of the supercell after the size change. The size change displacement dr=dr0⋅cos[G·(x+y)] is applied to the supercell, thereby controlling the expansion or contraction of the air hexagon. The unit cell at coordinates (x,y) indicates air hole position. Here, dr0=0.05a is selected, the wave vector G=K+−K−=8π/3a, and Dirac points at K±=±4π/3a. K± are wave vectors in the Brillouin zone. G indicates the coupling degree of two Dirac points in the supercell. The bandgap at two Dirac points P1 and P2 is successfully opened within the 2π phase. Figures 1(d) and 1(e) show the band structure of supercells near the two Dirac points P1 and P2, respectively; a 6.8% bandgap is opened at low frequency P1, and a 1.3% bandgap is opened at high frequency P2, where gray lines show the band with no size change, the gray quadruple degenerate Γ point is caused by the folding of Dirac points (K± points) after the lattice structure is expanded, and the black lines in the figures are the band after modulation in supercell. That is, the two Dirac points P1 and P2 of the unit cell located at the K± points are folded to the Γ point, forming two double Dirac cones. The double bandgap structure is the basis for the design of dual-channel topological Dirac-vortex cavities.

In order to make a dual-channel topological Dirac-vortex cavity, we arranged a library of hexagonal supercell lattices by following the spatial function formula for the vortex modulation, rk(x,y)=r+dr(x,y)cos[G·(x+y)+w·θ(x,y)].(1)

rk(x,y) is the length of the side of the hexagonal unit cell at coordinates (x,y). dr(x,y)=dr0·tanh(|x2+y2R|α), where dr0 is the maximum amplitude of air hole modulation. x2+y2 is the distance from the center of the unit cell to the center of the cavity position coordinate in space, R is the potential well radius of the local mode of the cavity, α is shape factor, and w is the winding number of the whole photonic crystal. When function modulation is satisfied, the topological Dirac-vortex cavities can be made. We modulated the lattice composed of the Kagomé structure in the same way; see Appendix B for details. It was found that the Kagomé structure also opened two Dirac point bandgaps, indicating that this modulation method has the same modulation capability for other structures with double Dirac points.

The cavity formation is a matter of arranging supercells angularly around the cavity center (0,0), and the phase distribution of the supercells is continuous in practice when the size of the overall cavity is large enough, as illustrated in Fig. 2(a). The study on properties of the emitted optical field for the cavity with the overall size of the cavity and the discretion of the phase is shown in Appendix E. The coordinate origin is defined as the center of the cavity, the phase of any hexagonal structure is relative to the center of the cavity, that is θ, the size of any hexagonal structure satisfies r+dr, and dr satisfies Eq. (1), when a series of hexagons are arranged in this way to fill the entire space. The cavity shown in Fig. 2(a) is obtained. Here a=0.404μm, r=0.8d, dr0=0.05a, and R=2a. Six supercells with several representative phases θ(x,y) are given here as examples. The eigenvalue mode index and the electromagnetic field distribution of the Dirac-vortex topological cavity are obtained by the 2D FEM simulating. Figures 2(b) and 2(d) show the normalized spectrum of cavity modes at the frequencies of P1 and P2, where the red solid circle is the topological single mode index, and the gray hollow circle is the body mode index. Because both topological modes are TM modes, we can show the z component distribution of the simulated topological mode electric field in Figs. 2(c) and 2(e). The entire electric field does not have symmetry about the center of the cavity. The electromagnetic field at the frequency of point P1 is basically distributed in dielectric, and the optical spot is larger, while the frequency of point P2 is relatively higher, so the field mode distribution is more concentrated. Both the topological modes’ polarization distributions in the magnetic field are vortices, as shown by the black arrows in the enlarged illustration. Simulation studies have also been conducted on the Dirac vortex cavity composed of the Kagomé structure, as shown in Appendix C. The results show that the Dirac-vortex cavity can also generate two topological modes of low frequency and high frequency, which indicates that the dual-band Dirac vortex cavity construction method described in this paper is universal.

We study the relationship of the topological mode diameter, FSR, and far-field divergence half-angle with the Dirac-vortex cavity diameter 2R. The scaling laws and mode distribution with different cavity diameters at high and low frequencies are shown in Fig. 3. When computing mode diameter L, the boundary of the mode is defined at the outer edge, where the field intensity drops to 1/e of the central maximum intensity. Figures 3(a) and 3(c) show the scaling laws between effective mode diameter and cavity diameter; the FSR simulation results do not particularly agree with the theoretical values at small-mode diameter due to the breakdown of chiral symmetry. When the cavity mode increases, the FSR results gradually approach a theoretical straight line. The FSR and far-field divergence half-angle at low frequency and high frequency are calculated. Here, the effective mode diameter L is derived from the mode area. It can be seen from the results that the scaling law of the high-frequency mode is consistent with that of the low-frequency mode, and the effective mode diameter is linear to the cavity diameter. Theoretically, the relationship between the mode diameter and the cavity diameter satisfies 2R∝Lα/α+1. In this model, the shape factor α is 4, so theoretically the scale value between the mode diameter and the cavity diameter is 0.8. The far-field divergence half-angle at lower frequency is larger than that at the higher frequency; that is because the bandgap is larger at lower frequency, and the mode area in the cavity is smaller.

We show the mode spectrum and near- and far-field distribution at the two frequency bands in Figs. 3(b) and 3(d). The topological mode gradually approaches the center of the bandgap with the increase of cavity diameter, and the bulk modes gradually enter the inside of the bandgap. Comparing the spectrum distribution within the two types of bandgaps, we find that when the cavity diameter of the high-frequency mode increases, the number of bulk modes entering the bandgap is less than that of the low-frequency mode, indicating that the high-frequency mode has better unimodularity and less interference from other modes under the condition of larger cavity diameter. The far-field polarization images can be further distinguished by the topological number of far fields in donuts. Appendix D shows the near and far fields of all modes near the topological mode.

To further study the relationship between the Dirac-vortex topological cavity mode and winding number in the bandgaps, the near and far fields of cavities with radius R=5a and winding number w=1, 2, 3 are calculated in Fig. 4. Figures 4(a)–4(c) show the low-frequency result, and Figs. 4(d)–4(f) show the high-frequency result. The upper part shows the mode electric-field distribution, and the lower part shows the far field and x linear polarization (XLP) component of the far field. Red lines indicate the mode frequency. It can be seen that the number of topological modes of the two frequencies is consistent with the magnitude |w| in both bandgaps. This is mainly due to the solution of the Dirac equation. Both near-field distributions with w=1 are relatively uniform. As w increases, the near-field distribution shows a divergent trend, while the near field and the far field of x polarization at w=2 exhibit complementary trends. The first two near fields of w=3 exhibit complementary trends with the far field of x polarization, while the third mode is the formation of near-field mode coupling. The difference is that the high-frequency mode field is locally denser than the low-frequency mode field.

The properties of far-field spots are basically consistent at both frequencies, and the far field of the multimode also presents the characteristics of the vector. The difference is that the far-field divergence half-angle at high frequencies is much smaller than that at low frequencies with the same winding number. That is because the effective mode area of the high-frequency mode is larger than that of the low-frequency mode. Interestingly, we note that when w=3, the far-field XLP components of the three modes are divided into six, six, and eight parts, respectively, which indicates that the topological numbers of the far field of the three modes are 3, 3, 4, respectively.

In order to further explore the underlying mechanisms of topological modes with high winding numbers, we further calculated Dirac-vortex cavities with winding numbers of 4 and 8. Figures 5(a) and 5(b) show the low-frequency modes, and Figs. 5(c) and 5(d) show the high-frequency modes. It can be seen that when the cavity diameter is the same, the mode location area of the mode with w=8 is generally larger than that of the mode with w=4, while the far-field divergence half-angle is larger for w=8. The reason for the increase of the far-field divergence angle is that the increase of the winding number increases the orbital angular momentum of the far-field mode, which is not in conflict with the conclusion above. By observing the linear polarization components of the far-field mode, it is found that the number of far-field topological nuclei of the four topological states with w=4 is 4, 4, 5, 5, and the number of far-field topological charges of w=8 is 8, 8, 9, 9, 10, 10, 11. The number of topological charges increases gradually.

A series of extensive calculations and principle summaries of the far-fields’ topological charge with w is shown in Fig. 5(e). By summarizing the far field of each mode with a high winding number, we found that the topological number of the far-field mode was not completely consistent with the topological mode number of the cavity structure. The minimum topological number of the far field was consistent with the cavity winding number, and the number began to increase in pairs. When w is an odd number, the maximum number of topological charges is |w|+(|w|−1)/2; when w is an even number, the maximum number of topological charges is |w|+(|w|−2)/2. The sign of the topological charge is consistent with w. This is mainly due to the structure not strictly satisfying chiral symmetry, resulting in near-field coupling.

Next, the properties of the multimode cavity mode are further analyzed. The cavity selected to be calculated has a radius R of 0, and the wavelength is 1550 nm. Figure 6 shows the cavity mode spectrum of different winding numbers. It can be seen that the mode number is still consistent with the winding number. With the increase in the winding number, the states outside the bandgap gradually approach topological modes. This is because the increase in the winding number will be accompanied by greater orbital angular momentum, which makes the cavity binding ability relatively weak and increases the effective mode area of the cavity mode. When w=32, it is difficult to distinguish between the topological mode and the body mode.

In order to further verify the experimental feasibility of the two-channel Dirac-vortex cavity, we use the plane wave expansion method (MPB) and finite-difference time-domain method to simulate the 3D dual-wavelength emission of the topological cavity model. GaAs was selected as the photonic crystal layer material with a refractive index of 3.52, and SiO2 was used as the substrate with a refractive index of 1.45. First, MPB was used to calculate the band structure of the 3D model of the hexagon lattice. The hexagonal holes in the lattice structure are with r=0.8d, dr=0.05a; a is the unit cell size, and the cavity thickness is 630 nm. Figure 7(a) shows the band structure calculation results; it can be seen that there are two Dirac points in the 3D model, which makes it possible for the 3D model to achieve dual-wavelength emission. Then, we build a 3D model of a Dirac cavity with dual-wavelength emission by using the finite-difference time-domain method. The electric dipole is placed inside the cavity as a light source to simulate the TM mode optical field, which covers the low-frequency (1550 nm) and high-frequency (980 nm) bands, respectively. The Q factor results of the cavity mode are shown in Fig. 7(b). The Q factor is 233 at low frequency and 275 at high frequency. Figures 7(c) and 7(d) show the far-field morphologies of high-frequency and low-frequency modes. Because the bandgap near the Dirac point at high frequency is very small, the mode field needs a relatively large photonic crystal structure to wrap it, so the far field at high frequency is not as obvious as the vector characteristics at low frequency.

In this paper, we present a dual-channel emission model for a Dirac-vortex cavity in the hexagon lattice. First, the bandgap at two Dirac points of the photonic crystal supercell is opened simultaneously by generalized Kekulé modulation in the hexagon lattice, and the Dirac-vortex cavity is constructed at both low and high frequencies by the potential well function. Second, the scaling laws for the cavity diameter, FSR, and the far-field divergence half-angle with mode diameter are studied at both low and high frequencies. Then, by changing the winding number of the Dirac-vortex cavities, multiple topological modes are generated in the bandgap. The number of modes is consistent with the winding number. The properties of multiple topological modes for the Dirac-vortex cavities are summarized, and the general laws of the far-field orbital angular momentum and winding number are found. Moreover, the topological mode integrated emission of any number of dual-band Dirac-vortex cavities and any angular momentum can be realized. Therefore, this work increases the number of channels in the Dirac-vortex cavity and explores the abilities of the Dirac-vortex cavity in communication and multiplexing.

Acknowledgment. The authors are also grateful to Henbin Cheng for inspirational discussions.