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Abstract
Topological edge states (ES) arise at the boundary between spatial domains with diverse topological properties in photonic crystals, which can transmit unidirectionally to suppress the backscattering and robustly to be immune to defects and disorders. In addition, optical devices with arbitrary geometries of cavities, such as lasers, are expected to be designed on the basis of ES. Herein, we first propose a topological cavity laser based on a honeycomb lattice of ring holes with the bearded interface in two-dimensional (2D) all-dielectric valley photonic crystals (VPhCs) at telecommunication wavelengths. Specifically, we construct a topological cavity using topological valley edge states (VES) and further study the lasing action of the optically pumped cavity with high-quality factors. Our findings could provide opportunities for practical applications of VES-based lasers as ultra-small light sources with the topological protection.
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1. Introduction
Since electronic topological phases were transferred to classical photonics by Haldane and Raghu in 2008 [1], topological photonics has been one of the promising research frontiers because they can provide a novel strategy for designing and implementing optical devices with a small footprint and the built-in topological protection [2–4]. The symbol of topological photonic systems is the emergence of topologically protected edge states (ES), which appear at the boundary between spatial domains with diverse topological properties in photonic crystals [5]. These ES can propagate unidirectionally and robustly which suppress the backscattering effectively in waveguides and is immune to defects and disorders. Various schemes for realizing photonic topological insulators based on quantum Hall (QH) [6–12], quantum spin Hall (QSH) [13–20], and quantum valley Hall (QVH) effect [21–36] have been proposed theoretically and demonstrated experimentally. To the best of our knowledge, the photonic topological insulator employing QH effect can be achieved based on gyromagnetic materials through strong magnetic fields in low-temperature environment [1]. The addition of magnetic fields leads to the inconvenience for photon integration, and this kind of photonic topological insulators can only work in the microwave regime owing to the weak magneto-optical effect. In order to overcome this shortcoming, the Floquet topological insulator (FTI) [8,9,37] by modulating the phase and space dimension has been reported, but the structure is complex. In addition, QSH effect are realized by implementing photonic spin-orbit interaction which need design complex metamaterials [15] and add the complexity of fabrication.
Recent developments extend to all-dielectric photonic crystals protected by the time-reversal symmetry, a band gap appears in the Dirac cone dispersion by deforming photonic crystal structures [18–22,29–32,35,36]. At the momentum space, the energy band structure of valley photonic crystals (VPhCs) has an energy extremum at the K’ or K point and a “Valley” shape near the K’ or K point. This valley is a degree of freedom whose function is similar to the spin degree of freedom in spintronic devices, but the strong spin-orbit coupling is not required. That is to say, VPhCs can provide the robust light transport without strong magnetic fields or the intricate photonic pseudospins’ construction in highly compact structures. Inversion-symmetry-broken VPhC structures can be manufactured more easily and precisely by nanofabrication techniques. Consequently, VPhCs are attractive platforms for low-loss photonic integrated circuits among various topological photonic systems.
On the other hand, topological photonics has been applied in laser physics to achieve more efficient and stable lasers using topologically protected states. Topological photonic lasers can be implemented in a one-dimensional (1D) Su–Schrieffer–Heeger (SSH) model [38–41], but there is no topology-protected transmission in these 1D systems due to zero-dimensional (0D) edges. As the fact that the magneto-optical effect in the optical system is so weak and the topological bandgap is narrower than the bandwidth of the laser, 2D topological lasers should avoid using magnetic materials [42]. Shortly thereafter, the method of stimulating the single-mode laser in ring-resonator arrays using non-magnetic topology-protected edge modes has been put forward, nevertheless, the intricate coupling is susceptible to the resonant motion [43,44]. Topological lasers relied on the photonic FTI concept are further investigated theoretically by twisting the spiral waveguide array along the propagation direction to generate the pseudomagnetic field [45]. In addition, a topological insulator laser with the Haldane lattice [46], a topological insulator vertical-cavity laser array [47], and a topological insulator laser pumped electrically at room temperature [48] are demonstrated in experiments. Very recently, the scheme is extended from topological ES to second-order 0D corner states based on photonic crystals nanocavities [49,50] or waveguide arrays [51] and topological bulk states based on the band-inversion mechanism [52]. VPhCs have many advantages as mentioned above, so that topological lasers based on valley edge states (VES) have attracted much attention [53–56]. For example, topological lasers based on Kagome photonic crystals cavities [53] and photonic honeycomb straight waveguide arrays [56] are theoretically proposed. A room temperature laser using the 0D corner state based on the 2D topological cavity [54] and an electrically pumped terahertz quantum cascade laser using the quasi-hexagonal photonic crystal [55] are demonstrated experimentally. The miniaturized laser at communication wavelengths plays a key role in promoting the development of integrated optical communications. Accordingly, communication-band lasers based on the Kagome lattice and the honeycomb lattice of triangular holes have been researched [53,54].
In this work, we propose a topological cavity laser at telecommunication wavelengths based on the 2D all-dielectric VPhC whose structure is a honeycomb lattice of ring holes with the bearded interface. The efficient robust and unidirectional transmission with suppression of backscattering is achieved using topological VES in a waveguide consisting of two VPhCs. We construct a topological cavity with high-quality factors based on VES and further study the lasing action of the optically pumped cavity using the four-level two-electron (FLTE) method. The simulation results could provide opportunities for practical applications of VES-based lasers as ultra-small light sources with the topological protection.
2. Valley photonic crystals
The designed valley photonic structure is a honeycomb lattice and details of the VPhC are given in Fig. 1(a). Two valley photonic crystals (VPhC1 and VPhC2) have the same lattice constant $(a = \textrm{385 nm)}$. Unperturbed air ring holes (green pattern in Fig. 1(a)) with the identical thickness (the difference between the outer and inner radius) ${d_0} = \textrm{50 nm}$ is embedded in the high refractive index Indium GaAs Phosphide (InGaAsP) membrane with 220 nm-thick layers. The VPhCs layer is placed between the SiO2 substrate with 2000-nm thickness and the air region. The refractive index of InGaAsP material is estimated to be 3.3. The VPhC1 (blue pattern in Fig. 1(a)) and the VPhC2 (red pattern in Fig. 1(a), as the inversion-symmetry partner of the VPhC1) can be obtained by altering the thicknesses of air ring holes in a primitive unit cell. The unit cell of the VPhC1 (VPhC2) comprises two inequivalent air ring holes, that is, thicker (finer) one ${d_1} = \textrm{80}\textrm{.5 nm}$ (${d_1} = \textrm{10 nm}$) and the finer (thicker) one ${d_2} = \textrm{10 nm}$ (${d_2} = \textrm{80}\textrm{.5 nm}$). The band structure for $H\textrm{z}$ out-of-plane transverse electric (TE) polarization of the valley photonic structure is captured by employing the finite-difference time-domain (FDTD) method. The red dashed line in Fig. 1(b) illustrates the band diagram of the primitive unit cell, and the grey shading indicates the light cone. There is a Dirac cone protected by temporal and spatial inversion symmetries shown near the K (K’) point of energy band, and it can be opened when one of symmetries is broken. For example, the spatial-inversion symmetry of our proposed photonic system (VPhC1 and VPhC2) is disrupted. The VPhC1 and the VPhC2 have the same band structure as the solid blue line shown in Fig. 1(b), that is because they are the inversion-symmetry partner of each other. In addition, the energy band structure has an energy extremum at the K’ or K point and a “Valley” shape near the K’ or K point at the momentum space as described in Fig. 1(b). When rings in two sublattices in the photonic system have different sizes, the spatial-inversion symmetry of this photonic system is broken and then the symmetry of K (K’) point in the Brillouin zone (BZ) changes from C6 to C3. Two degenerate modes split, and the band gap between 1258.45 nm and 1407.31 nm appears (blue region) owing to our particular design as given in Fig. 1(b).
Fig. 1. Construction characteristics and band structures of the all-dielectric VPhC. (a) Schematic diagram of the unperturbed (left) and perturbed (right) honeycomb lattice formed by air ring holes in the InGaAsP membrane. Ring holes’ thicknesses in green, blue and red pattern are ${d_0} = \textrm{50}\textrm{ }\textrm{nm,}$ ${d_1}({d_2}) = \textrm{80}\textrm{.5 (10)}\textrm{ }\textrm{nm,}$ and ${d_1}({d_2}) = \textrm{10 (80}\textrm{.5)}\textrm{ }\textrm{nm,}$ respectively. (b) Energy band structures of the unperturbed (red dashed lines) or perturbed (solid blue lines) unit cell in Fig. 1(a). Blue and gray regions indicate the valley-dependent bandgap and the air-light cone, respectively. The upper left insert displays the first BZ of the VPhCs. (c) and (d) Three-dimensional schematic diagram of the bearded-shaped and zigzag-shaped VPhCs. (e) and (f) Dispersion relation for VES of bearded and zigzag edge (purple lines). The dark grey region represents the bulk state. The height of the air ring holes $h = \textrm{220 nm}$ and the lattice constant $a = \textrm{385 nm}\textrm{.}$
As a result of the QVH effect owing to the effective magnetic field, the sign of the Berry curvature at the K valley is opposite to that at the K’ valley. The Chern number as the global integral of the Berry curvature for the whole BZ is zero, while the valley-dependent index CK (CK’) as the valley-dependent integration of the Berry curvature for the half BZ is nonzero. Therefore, the valley Chern index ${C_\textrm{V}} = {C_\textrm{K}} - {C_{\mathrm{K^{\prime}}}}$ can describe the topological property of the VPhC system. The corresponding valley Chern index Cv of two VPhCs have the opposite sign, i.e., ${C_\textrm{V}} < 0$ in the VPhC1 and ${C_\textrm{V}} > 0$ in the VPhC2. Topological edge states appear in the interface between spatial domains of different topological properties in the photonic crystal due to the bulk-edge correspondence. As a consequence, topologically valley-dependent states spanning the entire band gap can be acquired by constructing the boundary with photonic crystals having different valley Chern indices. Figure 1(c) and Fig. 1(d) show the three-dimensional schematic diagram of the bearded-shaped and zigzag-shaped VPhCs composed of the VPhC1 and the VPhC2 based on the QVH effect, respectively. The dark gray represents the high refractive index InGaAsP membrane and the gray blue indicates SiO2 substrate. We display the dispersion relation of the bearded (zigzag-shaped) interface in Fig. 1(e) and 1(f). Topologically valley-dependent states exist below the linear light cone and are also located in the band gap of the photonic crystal slab, so it can neither leak into the air nor scatter to the inside of the photonic crystal slab.
3. Unidirectionality and robustness of transmission for VES
In the next content, we investigate the unidirectional transmission of valley-dependent states in the bearded-shaped and zigzag-shaped VPhC waveguide. We use the FDTD method to calculate the light propagation and field distribution. The wavelengths of chiral sources cover from 1100 nm to 1600 nm. Figure 2(a) and (Fig. 3(a)) depict the structure of bearded-shaped (zigzag-shaped) VPhC waveguide with the straight boundary and the location of chiral sources labeled with a yellow star. In the bearded-shaped VPhC waveguide, the positive (negative) group velocity of VES at the K’ (K) valley as shown in Fig. 1(e) leads to the K’ valley-locked (K valley-locked) mode can spread in the forward (backward) direction, which is the momentum-valley locked effect. When the RCP (LCP) source excites photons, the positive (negative) chirality prevails so that the light propagates to the right (left) unidirectionally. According to the dispersion relation of the zigzag-shaped VPhC waveguide as shown in Fig. 1(f), the sign of group velocity at the K’ (K) valley is negative (positive). Therefore, we should use the LCP source as the excitation source if the light propagates to the right unidirectionally in the zigzag-shaped VPhC waveguide. Light beam's electric field intensity profiles for the bearded-shaped (zigzag-shaped) VPhC waveguide inspired by right/left-handed circularly polarized (RCP/LCP) sources at 1304 nm (1310 nm) can be seen in Fig. 2(c) and 2(d) and Fig. 3(d) and 3(c). Furthermore, results reveal VES in the bearded-shaped interface has stronger localization in transmission than that in the zigzag-shaped interface. Why are these results observed?
Fig. 2. Characteristics of the light transmission and field profiles for VES of bearded edge. (a) Schematic diagram of the valley-dependent waveguide composed of the VPhC1 (blue region) and the VPhC2 (red region). The location of chiral sources is labeled with a yellow star. (b) Forward (solid blue line) and backward (green chain line) transmittance curves for the one-way transport of light beam inspired by the RCP source, and the transmission contrast (red dashed line). (c) and (d) Light beam electric field intensity profiles inspired by RCP and LCP sources, respectively. (e) and (f) Structure of Ω-shaped topological waveguides constructed by the VPhC1 (blue region) and the VPhC2 (red region) and electric field intensity distribution excited by the RCP source represented by a yellow star.
Fig. 3. Characteristics of the light transmission and field profiles of VES of zigzag edge. (a) Schematic diagram of the valley-dependent waveguide composed of the VPhC1 (blue region) and the VPhC2 (red region). The location of chiral sources is labeled with a yellow star. (b) Forward (solid blue line) and backward (green chain line) transmittance curves for the one-way transport of light beam inspired by the LCP source, and the transmission contrast (red dashed line). (c) and (d) Light beam electric field intensity profiles inspired by LCP and RCP sources, respectively. (e) and (f) Structure of Ω-shaped topological waveguides constructed by the VPhC1 (blue region) and the VPhC2 (red region) and electric field intensity distribution excited by the LCP source.
The transmittance of light beam at different wavelengths in the straight VPhC waveguide has been acquired based on transmission characteristics. The forward transmittance ${T_F}$ curve (solid blue curve) and the backward transmittance ${T_B}$ curve (green chain curve) of the light beam inspired by the RCP source for straight boundary composed by the bearded-shaped VPhCs are respectively described in Fig. 2(b), which indicates the transmission rate of the forward propagation is much higher than that of the backward propagation. Wavelengths of the forward transmittance above 0.6 range from 1269.76 nm to 1405.69 nm and then the bandwidth is 135.93 nm. The highest value of the forward transmittance is 0.93 at 1304 nm.
The range of operating wavelength is in good agreement with the band gap, which confirms our forecast. According to ${C_T} = ({T_F} - {T_B})/({T_F} + {T_B})$ [30], the transmittance comparison is delineated by the red dashed curve in Fig. 2(b) and further indicates that the system have good unidirectional transmission characteristics and the transmittance curve for the structure of Fig. 3(a) are also displayed in Fig. 3(b). For the zigzag-shaped VPhC waveguide with the straight interface, wavelengths of the forward transmittance above 0.6 range from 1288.31 nm to 1350.33 nm and then the bandwidth is 62.02 nm. Although the transmittance comparison of the zigzag-shaped VPhC waveguide remains relatively close to 1 in the operating bandwidth range, the highest value of the forward transmittance is only 0.71 at 1310 nm. The beam at the working bandwidth scatters less into the VPhC structure than the beam whose central wavelength exceeds the operating bandwidth. Hence, most of the beam can be localized at the edge of the bearded-shaped VPhC waveguide when the wavelength of chiral sources at working bandwidth. Moreover, the portion of beam enters into the zigzag-shaped VPhC waveguide and the localization is relatively weak. Therefore, the bearded-shaped VPhC waveguide has good locality and is chosen as the structure of the resonant cavity or the laser cavity. There is no valley-to-valley scattering because of the vanishing field overlapping between two VES, which can be used to create the waveguide with the broadband robust transmission to resist sharp corners. To prove the robust topological propagation for VES of the bearded and zigzag edge, we further combine the VPhC1 and the VPhC2 to form a waveguide with the Ω-shape (interface with four sharp turns) VPhC interface (see Fig. 2(e) and Fig. 3(e)). As seen in Fig. 2(f) and Fig. 3(f), the topologically protected light beam is not affected by the backscattering despite sharp bends are encountered, resulting in the robust transmission.
4. Resonant cavity based on VPhCs
Based on VES as mentioned above, a topologically protected photonic triangle cavity with 120-degree bends can be realized. As can be viewed in Fig. 4(a), the cavity is constructed by forming a closed ring around the domain wall in which the triangular VPhC2 is embedded in the rectangular VPhC1. The spectra and Q factors are obtained by FDTD modeling. Figure 4(b) shows the optical spectrum as a function of the edge length of the resonant cavity that is equal to the product of the lattice constant and the amount of unit cells. The edge length of the cavity (L) is marked at the right of Fig. 4(b). The behavior of the resonant cavity is investigated: taking $L = \textrm{29}a$ as an example. Optical spectra of the resonator cavity have different characteristics when the resonant frequency is below or above 214 THz. To facilitate the description, two regions are labeled NT and T, respectively. The T region is on behalf of resonant modes with equal intervals and the NT region represents irregularly tight spaced resonance modes. Owing to the backscattering at the corner of the resonant cavity, optical spectra in the region NT reveal that several irregularly tight spaced resonance modes are stimulated. However, the spectra in the region T are in sharp contrast to those in the region NT. Multiple ring-resonant modes with a free-spectral range (FSR) about 10 nm appear by the excitation of edge modes with the low propagation loss. Additionally, resonant frequency depends on the edge length of the cavity, as shown in Fig. 4(c). The mode separation of the region T decreases as increasing the edge length of the cavity, which is a typical feature of the ring-type resonant. Figure 4(d) displays the relationship between Q factors and resonant modes’ frequencies of the cavity at different cavities. The research results manifest the Q factor tends to increase at the beginning and then decreases with increasing the frequency for the same cavity. Due to the different transverse loss at different resonant frequencies, the Q factor has been distinguished. At resonant frequencies located in the middle of the range, the higher Q factor is guaranteed by ring-resonator modes with the less transverse loss. The maximum Q factor of the cavity is 30984.8 at the wavelength (frequency) of 1418.73 nm (211.311 THz) when the edge length of the cavity is 21a. The maximum Q factor falls when L is unequal to 21a ($L < 21a$ or $L > 21a$), but reasons for the decrease are different in these two cases. The distance of two edges gradually gets shorter when the edge length of the cavity reduces, so light waves from edges of the cavity couple with each other and that result in increasing the loss. However, the edge mode propagation loss is so large that the maximum Q factor falls as the edge length of the cavity increases. Importantly, electric field distributions at the frequencies of 221.048 THz $(\lambda = 1356.23\textrm{ nm)}$ and 209.375 THz $(\lambda = 1431.84\textrm{ nm)}$ are respectively shown in Figs. 4(e) and 4(f), which display the light can be limited to the interface of VPhCs. The smooth light propagation along the entire cavity in Fig. 4(e) is the evidence of stimulating low propagation loss edge modes. While the irregular path of light propagation is shown in Fig. 4(f), it can be explained as the weak resistance to the backscattering at the resonator corner.
Fig. 4. Optical characteristics of the triangular VPhC cavity. (a) Sketch for the triangular VPhC cavity composed of the VPC1 and the VPC2. The black dashed line and the yellow star represent the boundary of the cavity and dipole sources, respectively. (b) Optical spectra as a function of edge lengths of resonant cavities. Edge lengths of cavities are marked on the right. (c) Dependence of resonant frequency on the edge length of the cavity. (d) Relationship between Q factors and resonant frequencies of cavities at different edge lengths. (e) and (f) Electric field profiles correspond to 29a at frequencies of 221.048 THz $(\lambda = 1356.21\textrm{ nm)}$ and 209.375 THz $(\lambda = 1431.84\textrm{ nm)}\textrm{.}$
5. Laser based on the VPhC cavity
To further study the lasing action in the VPhC cavity, III-V semiconductors InGaAsP is chosen as gain materials, and the cavity is optically pumped from above by the 1064 nm Gaussian-shaped continuous-wave laser, as shown in Fig. 5(a). We model the time evolution of the population density for gain materials by the FLTE method [57], and Fig. 5(b) shows the energy level distribution of the FLTE system. In this paper, the material parameters used in our simulations are obtained from the Ref. [53] and are shown in Table 1. ${\omega _a}\textrm{ (}{\omega _b})$ and ${\gamma _a}\textrm{ (}{\gamma _b})$ represent angular frequencies of transitions and dephasing rates from energy level 1 to energy level 2 (from energy level 0 to energy level 3), respectively. ${\tau _{mn}}\textrm{ }(m,\textrm{ }n = 0,\textrm{ }1,\textrm{ }2,\textrm{ }3)$ represents the decay time between energy level m and n.
Fig. 5. Laser actions in the VPhC cavity. (a) Three-dimensional schematic diagram of the VPhC cavity pumped optically from above by the 1064 nm Gaussian-shaped laser. (b) Schematic of the FLTE model for VPhC cavity lasers. (c) Evolution of population distributions N1 and N2 of the gain materials over time. (d) Evolution of the normalized emission spectral intensity versus the incident pump amplitude and frequency. (e) Emission spectral maximum intensity and linewidth to the pump amplitude. (f) Electric field profile for the edge mode at 1389.47 nm when the pump amplitude is $4 \times {10^6}\textrm{ }\textrm{V/m}.$ The edge length of the cavity is $L = 13a.$
Table 1. Material parameters for simulating laser based on the InGaAsP membrane.
In the next discussion, the character of laser is researched in the VPhC cavity with $L = 13a.$ The time-dependent populations distributions N1 and N2 for two lasing energy levels of gain materials are plotted in Fig. 5(c), respectively. When the pump amplitude is $|E |= 1.55 \times {10^6}\textrm{ V/m},$ the population inversion of levels 1 and 2 occurs at $t \approx 14\textrm{ }\textrm{ps}$ and keeps it steady until the simulation time is over. Figure 5(d) displays the normalized emission spectral intensity of the laser cavity versus the incident pump amplitude and frequency. A steep and narrow emission spectrum at 214.5 THz $(\lambda = 1397.63\textrm{ nm)}$ can be achieved when the pump amplitude is increased to $1.55 \times {10^6}\textrm{ V/m}.$ How the peak intensity and linewidth of the emission spectrum respectively vary with the incident pump amplitude is described in Fig. 5(e). With increasing the pump amplitude, the peak intensity increases while the linewidth decreases gradually. Simulation results about the peak intensity show a threshold behavior, which is the typical characteristic of the laser due to a transition from the spontaneous emission to the stimulated emission. We can also find that the spectrum linewidth is shrunken from 3.41 nm to 1.13 nm. Therefore, the generated laser based on the VPhC cavity has strong coherence properties. When the pumping amplitude is $|E |= 4\textrm{ } \times \textrm{ }{10^6}\textrm{ V/m,}$ the electric field profile for the trianglar cavity laser is depicted in Fig. 5(f), and the perfect laser along the edge of VPhC cavity can be obtained. Accordingly, the laser based on the VPhC cavity paves a new way for the realization of microminiaturization of laser sources.
6. Conclusions
We have designed the VPhC structure based on the all-dielectric honeycomb lattice of ring holes at telecommunication wavelengths. On account of the valley-spin locking effect, the waveguide consisting of two VPhCs can act as a one-way propagation optical device to manipulate the flow of light. Robust transport with the suppression of the backscattering has been observed in the topological Ω-shape VPhC interface. Furthermore, the ring-resonator mode appears in the topologically protected triangle cavity with high Q factors based on VES. Combined with the gain medium, we use the FLTE method to research the lasing behavior by population distributions transitions, emission spectra, and electric field profiles. As a result, our study provides a new type of nontrivial photonic topological platform to observe topological properties of states in non-Hermitian photonic systems. Finally, this topological laser can pave a new way for the realization of the stable, miniaturized laser source.
Funding
National Natural Science Foundation of China(61705159, 12104335); Special/Youth Foundation ofTaiyuan University of Technology;(tyutrc201387a); Fund for Shanxi Key Subjects Construction"1331 Project" Key Innovative Research Team.
Acknowledgments
The authors would like to acknowledge Yongkang Gong, Cardiff University, and Zhiyuan Gu, Taiyuan University of Technology, for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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