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Abstract
A topological photonic crystal InGaAsP/InP core-shell nanowire array laser operating in the 1550 nm wavelength band is proposed and simulated. The structure is composed of an inner topological nontrivial photonic crystal and outer topological trivial photonic crystal. For a nanowire with height of 8 µm, high quality factor of 4.7 × 104 and side-mode suppression ratio of 11 dB are obtained, approximately 32.9 and 5.5 times that of the uniform photonic crystal nanowire array, respectively. Under optical pumping, the topological nanowire array laser exhibits a threshold 27.3% lower than that of the uniform nanowire array laser, due to the smaller nanowire slit width and stronger optical confinement. Moreover, the topological NW laser exhibits high tolerence to manufacturing errors. This work may pave the way for the development of low-threshold single-mode high-robustness nanolasers.
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1. Introduction
Nanowires (NWs) are considered as ideal structural units for miniaturized lasers due to the ultrasmall dimension, strong optical feedback, and excellent waveguiding properties [1]. So far, single NW lasers lying on a substrate have been widely demonstrated in a variety of materials such as ZnO, GaN, CdS, InP and GaAs, showing great potential in low-consumption on-chip microsystems [2–6]. In recent years, another promising structure, that is, the free-standing NWs, have attracted increasing attention in Si-based vertical direction surface-emitting lasers due to the ultrasmall footprint that tolerates large lattice mismatches. For example, Si-based vertical-emitting InGaAs, GaAs, and InGaAs/InP NW lasers have been demonstrated yet [7–10]. Moreover, ordered NW arrays are naturally photonic crystal (PhC) structures, enabling the realization of high-Q low-threshold PhC lasers. Up to date, PhC lasers have been reported in GaAs/InGaAs/GaAs, InP, and InGaN/AlGaN NW arrays [11–14]. One problem of the vertical NW lasers is the relatively poor cavity performance originating from the low refactive index difference between the NW bottom and substrate surface, which may increase the threshold and decrease the Q factor. In addition, multimode lasing have also been observed in vertical NW lasers [7,8]. Although the NW PhC structure can improve the lasing performance such as the Q factor and mode confinement, the relatively broad photonic band gap and sensitivity to defects limit the development of high-robustness single-mode vertical NW lasers.
Inspired by the discovery of the quantum Hall effect and topological insulators in condensed matter physics, topological photonics based on topological energy band theory is emerging, providing a promising way for light field manipulation and development of advanced photonic devices, e.g., topological lasers [15–19]. So far, researchers have constructed a variety of topological laser cavities based on different structures. In the one-dimensional (1D) systems, in addition to the 1D topological chains [20–23], the interface composed of nano-beams with different topological properties is also used to support a strongly local interface state [24]. In the two-dimensional (2D) systems, topological lasers can be classified into corner, edge and bulk states due to the hierarchical eigenstates that can exist in a topologically designed single 2D PhC platform [25–31]. In such laser systems, lasing occurs based on topologically protected edge states [32], forming running-wave modes that are robust to perturbations [25,33–35], which has great potential for achieving high quality factors, high power, low mode volume and and low thresholds [36].
Current topological lasers are typically built by placing foreign PhC arrays formed by periodic nanorods or nanoholes on traditional planar laser structures such as heterostructures and quantum wells. As III-V NWs are excellent gain medium and optical cavity, while ordered NW arrays are naturally PhC structures, it is possible to achieve attractive topological lasers based on ordered III-V NW array itself, without the need to introduce foreign dielectric PhC arrays. Compared with the traditional “top-down” etch method, the “bottom-up” grown NWs have advantages of simple fabrication process, smooth surfaces minimizing the photon and electron scattering, as well as the ability to form advanced heterostructures such as core-shell structures to enhance the lasing performance. Considering the physical properties of NWs, the topological NW array lasers have extra advantages of strong optical feedback and compatibility with Si substrate. However, up to date, the design and fabrication of topological NW array lasers are still very limited.
In this paper, topological PhC lasers based on InGaAsP/InP core-shell NW arrays are proposed and studied. The structure is formed by an inner topological nontrivial PhC and outer topological trivial PhC. The topological transition of 2D Zak phases are demonstrated by expanding or shrinking four NWs in the unit cell. By adjusting the NW and array parameters, a maximum Q factor of 4.7 × 104 and high side-mode suppression ratio (SMSR) of 13 dB are obtained. The threshold of the topological NW laser is 27.3% lower than that of the uniform PhC NW array laser due to a smaller NW slit width and stronger optical confinement. The topological PhC NW array laser also shows high manufacturing tolerance, enabling the realization of high-robustness nanolasers.
2. Structures and theory
2.1 Structures and methods
The unit structure of the topological PhC NW arrays is a square lattice containing four NWs, as shown in Fig. 1(a). A SiO2 mask with a thickness of 200 nm is placed on the Si substrate, and the height, core radius, and shell thickness of the InGaAsP/InP core-shell NW are 8 µm, 53 nm, and 20 nm, respectively. Inspired by the generalized 2D SSH (Su–Schrieffer–Heeger) model, two kinds of PhC slabs have been designed, which possess the identical band structure but different 2D Zak phases, as shown in Fig. 1(b). Four NWs are shrunk or expanded in a unit cell, which are spaced equally apart by distance d1 and d2 from the center of the unit cell in the x- and y-directions, respectively. 2d1 and 2d2 modulate the coupling strength between the nearest lattice sites. It is noticed that d1 + d2 = a/2, where a is the lattice constant, which is set to 418 nm. We define Δ = (d1 – d2)/2, which determines the lattice structure of a PhC if other parameters are fixed. The overall structure is surrounded by air. Figure 1(c) shows the electric field intensity distribution in the cross-section and longitudinal section of four NWs shrunk (expanded) in a unit cell. Photons oscillation is observed in the NWs slit waveguides, which is also reported in other dielectric nanostructures such as bowtie structures [37–39].
Fig. 1. Structure and properties of the topological PhC NW array. (a) Three-dimensional (3D) structure of the unit cell without tetramerization. (b) The 2D photonic SSH model and two representative PhC slabs with d1 = 73 nm, and d2 = 136 nm. (c) Cross-sectional and longitudinal electric field intensity distributions of four NWs shrunk (expanded) in a unit cell. (d) Band structures for Δ=−0.075 a (left), Δ=0 (middle), and Δ=0.075 a (right), which represent a shrunken, normal, and expanded lattice, respectively.
We assume that the modes are transverse-magnetic (TM) as the structure supports the fundamental TM-guided mode only. The photonic band structures of the unit cell are calculated by using the FDTD (Finite Difference Time Domain) method and Bloch boundary condition for three different Δ values: Δ=−0.075 a (left of Fig. 1(d)), Δ=0 (middle of Fig. 1(d)), and Δ=0.075 a (right of Fig. 1(d)), representing the shrunken, normal, and expanded lattices, respectively. When Δ=0, Dirac points and quadratic band touching appear at the X and M points, respectively, and a gapless double-degenerate band structure is obtained, which is due to band folding as the lattice constant is twice the actual periodicity. When Δ=-0.075 a or 0.075 a, the band gap opens and appears in the range of [191 THz, 200 THz]. In fact, the two energy band structures are identical as d1 and d2 are complementary. Nevertheless, they have different bulk polarization and are in different topological phases, being trivial for Δ=-0.075a and topological for Δ=0.075a.
For comparison, a uniform PhC NW array with the same NW number and material is designed and studied, as shown in Fig. 2. In order to obtain a lasing wavelength near 1550 nm, the lattice constant is set to 480 nm, as shown in Fig. 2(a). Figure 2(b) shows that calculated band structure of the designed PhC NW array, which shows a band gap range of [226 Hz, 300 Hz]. Photons oscillation is also observed in the NW slit waveguides, as shown in Fig. 2(c).
Fig. 2. (a) 3D structure of a uniform PhC NW array. (b) Band structure of the uniform PhC NW array. (c) Cross-sectional and longitudinal electric field intensity distribution between two adjacent NWs in the uniform PhC NW array.
2.2 Theory model and calculation
The SSH model represents a chain comprising two sublattices with alternating coupling strengths, distinguished as inter-cell and intra-cell coupling strengths, as depicted in Fig. 1(b). tx(ty) and Δtx(Δty) determine the dimerized patterns of coupling strengths in x(y) direction [40]. For simplicity, we consider a case where tx = ty = t and Δtx=Δty=Δt, namely the lattice is symmetric between x and y directions, and define t1 = t-Δt and t2 = t-Δt which describe the inter-cell coupling and intra-cell coupling in a similar way as shown in Fig. 1(b).
The photonic band structure and topological properties of the isotropic PhC can be well approximated by the tight-binding model, where the Hamiltonian quantity is
To demonstrate the existence of edge states, we calculate the projected band structure for the combined structure as shown in Fig. 3. The result shows that there are 1D edge states in the first band gap, and the edge modes are represented by read points.
Fig. 3. Corner (bulk) modes are denoted by blue (black) points, whereas the edge modes are represented by read points.
3. Results and discussions
3.1 Mode characteristics
The whole topological NW array structure is realized by placing a topological non-trivial PhC with 9 × 9 periods, surrounding by a five-layer trivial PhC. The Q-factor of the edge states has been optimized by tailoring the envelope function of the electric field profile in the cavity, as shown in Fig. 4(a). It can be seen that the Q factor first increases and then decreases as g gradually increases. The initial increase is because that the increase of g can reduce the reflection at the edge of the cavity and weaken the Bragg reflection intensity, resulting in low radiation loss and light leakage [42,43]. When the size of g further increases, the transverse loss caused by the finite size effect becomes larger, which leads to a decrease in Q [26]. At g = 20 nm, the edge-state mode supports a maximum Q factor of 1.58 × 104. The dependence of Q factor on cavity length (L) is also plotted in Fig. 4(b). Here, g is fixed to 20 nm. The Q factor reaches a maximum of 4.7 × 104 at L = 11 a, approximately 32.9 times that of the uniform PhC NW array laser (1.43 × 103). When the cavity length is smaller than 11 a, the Q factor drops as the cavity length decreases, as light in the cavity edges starts to couple from each other when the cavity edges become closer. For a long cavity length larger than 11a, the Q factor degrades due to the enhanced propagation loss of the edge mode.
Fig. 4. Dependence of Q factor on the distance between trivial PhC and topological nontrivial PhC (a) and cavity length (b).
The longitudinal modes of uniform and topological NW arrays are obtained based on the Padé approximation algorithm [44], as shown in Fig. 5(a). For a NW height of 8 µm, the SMSR of the topological NW array is about 11 dB, about 5.5 times that of the uniform NW array. The relatively poor single-mode performance of the uniform NW array is attributed to the PhC surface-emitting laser structure, which typically has at least two competing band-edge modes with high quality factors [27]. While the topological PhC employs a 2D broad-area mid-gap defect mode with a stabler mechanism for single-mode operation [28]. The dependence of the longitudinal modes of the topological PhC NW array on the NW height is shown in Fig. 5 (b). Interestingly, as the NW height increases, the edge-state wavelength of exhibits a redshift. As the NW height has no influence on the topological PhC cavity, the wavelength shift is probably attributed to the variation of F-P cavity formed by the two end facets of the NW. According to the wavelength formula λ=2nl/m, where n is the refractive index of cavity, l is the cavity length, and m is the number of longitudinal modes, the mode wavelength increases as the cavity length increases [45]. This means that the topological PhC NW array supports multi-cavity coupling, including a ring resonant cavity and F-P cavity. Hence, by simply adjusting the NW height (in experiment, the height can be easily controlled by growth time), different lasing wavelength can be obtained, without changing the material composition or the lattice constant.
Fig. 5. (a) Longitudinal modes of uniform and topological PhC NW arrays. (b) The dependence of the longitudinal modes on the NW height for the topological PhC NW arrays.
3.2 Threshold characteristics
To study the lasing characteristics of the two PhC lasers, a Gaussian-shaped continuous-wave laser with wavelength of 1064 nm is used to optically pump the NW arrays from above in the FDTD simulations, as shown in Fig. 6(a). Figure 6(b) is a four-level two-electron model, where Ni represents the population density of each energy level, τ30, τ32, τ21 and τ10 are the lifetimes of different decay channels, γa and γb represent the damping coefficients of Pa and Pb respectively, simulating the non-radiation loss [46,47]. Main material parameters are listed in Table 1 [48,49]. For the topological PhC NW array laser, g and L are set to 20 nm and 11 a, respectively.
Fig. 6. (a) 3D structure of an optically pumped topological PhC NW array laser (b) Schematic diagram of the Four-level two-electron model. (c) Evolution of population distributions Ni of the gain materials over time. (d) and (e) Emission spectrum and field distribution of the toplogical and uniform PhC NW arrays, respectively. (f) Threshold comparison between the topological and uniform PhC NW arrays.
Table 1. The material parameters in threshold characteristics simulation
The time-dependent populations distribution of the topological PhC NW array laser is shown in Fig. 6(c). When the pump amplitude is E = 3 × 107 V/m, the population inversion of Level 1 and Level 2 occurs at t = 0.4 ps, and then quickly remains stable until the end of the simulation time, while the population of Level 0 and Level 3 continue to oscillate over time. Consequently, the laser spectrum and electric field intensity distribution are obtained, as shown in Fig. 6(d). It is found that multiple lasing modes are excited, which is due to a relatively large value of dephasing rate γa used in the design so that the gain of the material covers a wide wavelength range of edge-states. And among the multiple excitation peaks, the lasing with edge-state wavelength at 1544 nm dominates, which has higher gain and lower loss than other resonance modes [50]. For the uniform PhC NW array laser, at a pump amplitude of E = 3 × 107 V/m, the laser spectrum and electric field intensity distribution are shown in Fig. 6(e). Comparing Fig. 6(d) and (e), it can be seen that the lasing intensity of the topological PhC NW array is much higher, which is attributed to the more concentrated energy of the single mode.
The threshold characteristics of the two PhC NW array lasers are shown in Fig. 6(f). With the increase of the pump amplitude, the output intensity curves of both structures show an inflection point, indicating a transition from the spontaneous emission to the stimulated emission. The threshold of the topological PhC NW laser is 27.3% lower than that of the uniform PhC NW array laser, which could be attributed to a smaller NW slit width and stronger optical confinement.
3.3 Robustness
The robustness of the topological PhC NW array laser is investigated by calculating the distributions of the edge modes when propagating through the waveguides with and without defects. The topological waveguide with straight line-shaped interface is first investigated. The structure is designed as trivial PhC and topological PhC are adjacent, Fig. 7(a) and (b) show the lattice order without and with defects. The results show that even if defects are introduced at the junction of the two structures, the light is still protected by topology and propagates along the edge, as shown in Fig. 7(d), which is similar to Fig. 7(c). Since the presence of the edge polarization is guaranteed by the topological property of the bulk, it cannot be eliminated by weak perturbations, which indicates the robustness of topological edge states [51].
Fig. 7. Line-shaped interface topology waveguides without (a) and with (b) defects. The location of the introduced defects are marked with a red dashed box. (c) and (d) Electric field intensity distributions in the waveguide without and with defects, respectively.
Next, the performance sensitivity of the topological PhC NW array laser and the uniform NW array laser to perturbations is compared. Here, the perturbation is defined as the percentage of the number of missing NWs in the overall number. Figure 8(a) shows the normalized electric field intensity distribution when the perturbation is 0%,1.19%, 2.38% and 3.57% at the edge. It is found that when the perturbation is smaller than 3.57%, the light propagates smoothly along the cavity. When the disturbance is increased to 3.57%, the edge state becomes rough. In contrast, the mode field shape of the uniform NW array laser changes significantly after introducing perturbations, as shown in Fig. 8(b). Finally, a disturbance of 3.57% is introduced to both lasers to investigate their threshold sensitivity to perturbations, as shown in Fig. 8(c) and (d), respectively. It is found that the thresholds of the topological PhC NW array laser with and without perturbations are close, suggesting high tolerance to manufacturing errors. By contrast, the threshold inflection point nearly disappears after introducing perturbations for the uniform NW array laser.
Fig. 8. (a) and (b) The normalized electric field intensity distribution of topological and uniform PhC NW array lasers with perturbation of 0%, 1.19%, 2.38%, and 3.57%, respectively. (c) and (d) The threshold of the topological and uniform PhC NW array lasers without and with 3.57% perturbation, respectively.
4. Conclusion
In summary, a topological PhC laser based on InGaAsP/InP core-shell NW array is proposed and studied by FDTD method. The structure is composed of an inner topological nontrivial PhC and outer topological trivial PhC. By adjusting the array parameters, high quality factor and large side-mode suppression ratio are obained. Compared with the traditional uniform PhC NW array laser, the topological NW laser exhibits a much lower threshold and higher manufacturing tolerence. This work demonstrates that the combination of NW arrays and topological photonics is a promising way for the development of high-performance and high-robustness photonic devices.
Funding
National Natural Science Foundation of China (61935003); State Key Laboratory of Information Photonicsand Optical Communications (IPOC2022ZT02, IPOC2022ZZ01).
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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