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Abstract
A topological photonic crystal InGaAsP/InP core-shell nanowire array laser with bulk states operating in the 1550 nm band is proposed and simulated. By optimizing the structure parameters, high Q factor of 1.2 × 105 and side-mode suppression ratio of 13.2 dB are obtained, which are 28.6 and 4.6 times that of a uniform nanowire array, respectively. The threshold and maximum output are 17% lower and 613% higher than that of the uniform nanowire array laser, respectively, due to the narrower nanowire slits and stronger optical confinement. In addition, a low beam divergence angle of 2° is obtained due to the topological protection. This work may pave the way for the development of high-output, low-threshold, low-beam-divergence nanolasers.
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1. Introduction
Semiconductor nanowires (NWs) naturally serve as gain media, resonant cavities, and waveguides, concurrently enabling functions for photon emission, transmission, and coupling. NW lasers have attracted widespread attention with their extremely small size and low power consumption [1]. Single NW lasers lying on a substrate have been widely demonstrated in a variety of materials such as ZnO, CdS, GaN, InP and (In)GaAs, showing great potential in low-consumption on-chip microsystems [2–7]. In recent years, vertical NW lasers have attracted increasing attention due to their tiny footprints and scalability [6,8–10]. Particularly, ordered NW array naturally forms a photonic crystal (PhC) structure, which can achieve low threshold and high quality factor (Q factor) lasers. To date, PhC lasers based on GaAs/InGaAs/GaAs, InP, and InGaN/AlGaN NW arrays have been reported [11–14]. However, existing NW array lasers still have certain limitations. Due to the in-plane weak optical confinement, the threshold gain is generally high. Moreover, it is difficult to obtain large output power and small beam divergence angle at small sizes due to the strong scattering loss and laser mode competition [15]. So far, the development of high-output, low-threshold, low-beam-divergence NW array lasers still faces great challenges.
Topological insulators have become the focus of substantial research efforts in recent years owing to their novel edge/ interface states protected by the topological property of the bulk band dispersion [16–18]. Most recently, photonic topology has been employed in laser physics and devices. Current topological lasers are usually constructed by placing foreign PhC arrays formed by periodic nanorods or nanopores on traditional planar laser structures such as heterostructures and quantum wells [19–22]. Nanostructures made from high refractive index dielectric materials possess a distinct potential for achieving topological order in light propagation due to their well-designed resonant elements and lattice arrangements [23–25]. For instance, III-V semiconductor NWs, known for their excellent gain media and optical cavities, offer an attractive platform for realizing topological lasers. Furthermore, “bottom-up” grown NWs, which are compatible with Si substrates and exhibit smooth surfaces, minimize photon and electron scattering and can form advanced heterostructures, including core-shell configurations, to enhance laser performance. 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 [21,26–31]. The physical mechanism of topological cavity mode generation mainly involves topological phase transition and symmetry protection in quantum mechanics. In recent years, topological PhC NW array lasers based on corner and edge states have been reported, enabling the development of single-mode, low-threshold and high-robustness semiconductor lasers [32,33]. However, topological PhC NW array lasers with bulk states, which have great potential in lidars due to the excellent output power and beam divergence angle, have not been studied yet.
In this paper, a topological PhC laser based on InGaAsP/InP core-shell NW array with bulk states is proposed and studied. Two PhC slabs with the same band structure but different two-dimensional (2D) Zak phases are obtained by expanding or shrinking four NWs. The structure of a topological PhC NW array laser consists of a topological nontrivial region surrounded by a topological trivial region. By adjusting the area of the topological region and the height of the NWs, a maximum Q factor of 1.2 × 105 and a high side-mode suppression ratio (SMSR) of 13.2 dB are obtained. Due to its smaller NW slit width and stronger optical confinement, the light-matter interaction is enhanced and a higher gain for lasing is obtained. The threshold of the topological PhC NW array laser is 17% lower than that of the uniform PhC NW array laser, and the maximum output intensity is 613% higher. In addition, a beam divergence angle of as low as 2° is obtained.
2. Structures
The unit structure of the topological PhC NW array is a square lattice containing four NWs with a lattice constant (a) of 480 nm, as shown in Fig. 1(a). A layer of SiO2 mask with a thickness of 200 nm is placed on the Si substrate, and InGaAsP/InP core-shell NWs are grown out of the hole [14,34]. The radius of the InGaAsP core is 55 nm, which is the outer radius of the hexagon, the refractive index is 3.5, and the thickness of the InP shell is 20 nm, which can be acted as a passivation layer and potential barrier, providing strong carrier confinement. Inspired by the generalized 2D SSH model, two PhC slabs with identical band structures but differing 2D Zak phases are designed, as depicted in Fig. 1(b). Four NWs within a unit cell exhibit shrunken (blue), normal positions (yellow), and expanded (green), with respective distances from the unit cell center of 106 nm, 120 nm, and 233 nm. Figure 1(c) is a 3D structure diagram of the topological PhC NW array proposed in this paper. The topological domain is surrounded by the trivial domain. For comparison, a uniform PhC NW array with the same material is designed, as shown in Fig. 1(d). In order to achieve a lasing wavelength near 1550 nm, the lattice constant of the uniform PhC NW array is set to 500 nm.
Fig. 1. (a) Three-dimensional (3D) structure of the unit cell without tetramerization. (b) Four NWs shrink (blue), normal (yellow) and expand (green) in a unit cell. (c) 3D structure of topological PhC NW array. (d) 3D structure of a uniform PhC NW array.
3. Results and discussions
3.1 Band gap characteristics
In the simulations, electric dipole source is used, as the structure supports the fundamental TM-guided mode only. Using the Finite-Difference Time-Domain (FDTD) method and employing Bloch boundary conditions, we compute the band structures for the contracted, normal, and expanded lattices, as shown in Fig. 2. The red dots represent the band structure for the normal lattice, Dirac points and quadratic band touching appear at the X and M points, respectively, and a gapless double-degenerate band structure is obtained [21,35,36]. The black dots represent the band structures for the contracted and expanded lattices, which exhibit a same band gap in the range of [166 THz, 189 THz]. Figure 2(b) presents the band structure of the designed uniform PhC NW array, exhibiting a band gap range of [220 THz, 287 THz].
3.2 Mode characteristics
Figure 3(a) is a 2D schematic diagram of the topological PhC NW array that we consider. A topologically nontrivial 2D PhC with 26 × 26 periods in a square shape is surrounded by a three-layer trivial PhC. The overall structure is surrounded by air. By adjusting the length (L) of the topological region, the Q factor of the topological PhC NW array with bulk state is optimized, as shown in Fig. 3 (b). The black and red lines in Fig. 3 (b) represent the calculated results of the Q factor and resonance wavelength of the bulk state for different values of L. It is worth noting that the overall periodicity of the PhC remains unchanged, and the NWs itself in the PhC arrays does not form a cavity since the diameter of the InGaAsP NWs are below the diffraction limited critical diameter. The bulk state shows a clear redshift as L gradually increases, which may be due to the increase in the topological area equivalent to the increase in the topological cavity. According to the wavelength formula
where n is the refractive index of cavity, g is the cavity length, and m is the number of longitudinal modes, the mode wavelength increases as the cavity length increases [37]. The maximum Q-value is observed at L = 26a, approximately equals to 1.2 × 105, which is 28.6 times that of the uniform PhC NW array of the same dimensions. When L < 26a, the Q factor increases with L, as the bulk state is confined within the topological domain, leading to an increased overlap between the optical field and the gain medium and increase in the Q value. However, when L > 26a, the Q factor decreases with L due to the corresponding reduction in the trivial regions as the topological domain expands, leading to an increase in light leakage. When L = 32a, the topological PhC no longer has topological properties, transforming into a uniform PhC NW array laser with a lattice period of 290 nm.Fig. 3. (a) 2D structure of a topological PhC NW array. (b) Dependence of Q factor and bulk state wavelength on the length of the topological region.
Next, the transverse modes between the NWs in the topological and uniform PhC NW array structures are studied. The left and right plots of Fig. 4(a) represent the electric field intensity distribution in the longitudinal section when the four NWs shrunk or expanded in a topological photonic crystal nanowire array structure, respectively. It is found that photons oscillate in the NWs slits and the electric field intensity is stronger when the NWs are closer to each other. Figure 4(b) shows the longitudinal electric field intensity distribution between NWs in a uniform PhC NW array structure, and it is found that photons can also oscillate in NWs slits. The above shows that the Fabry-Perot (F-P) cavity of our designed topological and uniform PhC NW array structure is located in the NWs slit, which is also reported in other dielectric nanostructures [38,39].
Fig. 4. (a) Cross-sectional and longitudinal electric field intensity distributions of four NWs shrunk (expanded) in a unit cell. (b) Cross-sectional and longitudinal electric field intensity distribution between two adjacent NWs in the uniform PhC NW array. The respective electric field intensity distribution map corresponds to the red dashed line at the top.
Here, the Padé approximation algorithm [40] is employed to investigate the longitudinal modes of two PhC NW arrays. The longitudinal mode of the topological PhC NW array cavity based on different NW heights is shown in Fig. 5(a). It is observed that as the NW height increases, the bulk state wavelength exhibits a redshift. This phenomenon is similar to the edge-state topological PhC NW array laser reported in the previous study [33]. This is because the change in NW height has no effect on the length of the topological cavity, but it changes the length of the F-P cavity at the two ends of the NW. Consequently, it can be deduced from Eq. (1) that the NW height has an impact on wavelength tuning. This means that the topological PhC NW array supports multi-cavity coupling, including topological cavities and F-P cavities. Therefore, in experiments, different lasing wavelengths can be obtained by simply adjusting the NW height.
Fig. 5. (a) The longitudinal mode of the topological PhC NW array cavity based on different NW heights. (b) An enlarged view of the red dashed box in (a). (c) The dependence of the SMSR on the NW height. (d) Longitudinal modes of uniform and topological PhC NW arrays.
Figure 5(b) is an enlarged view of the red dashed box in Fig. 5(a). It can be seen that the side mode intensity of the topological PhC NW array cavity decreases first and then rises as the height of the NW increases. This directly affects the SMSR of the cavity, as shown in Fig. 5(c). As the NW height increases, the rising rate of the SMSR decreases from fast to slow. When the height of the NW is 12 µm, it reaches a maximum value of 13.2 dB. When the height of the NW is infinite, the SMSR decreases. This behavior is attributed to the increase in NW height, which leads to an increase in the effective mode volume of the laser [41,42], and the local effect of photons is enhanced, causing more photons to be concentrated in the main mode. However, when the height of the NWs increases to a certain extent, other factors (such as material loss, etc.) will have a dominant effect on the performance of the laser and have a negative impact on the SMSR.
Due to the multi-mode oscillations in the lasers, we also compare the longitudinal modes of the topological and uniform PhC NW array, as shown in Fig. 5(d). When the NW height is 12 um, the SMSR of the topological PhC NW array is about 4.6 times that of the uniform PhC NW array. The relatively good single-mode performance of the topological PhC NW array is due to the fact that the topological PhC adopts a 2D broad-area mid-gap defect mode with a stabler mechanism for single-mode operation [30]. Topological-cavity surface-emitting laser employs a 2D broad-area mid-gap defect mode, whose design is consistent with the historical success and development of semiconductor lasers (distributed feedback laser, vertical-cavity surface-emitting laser and photonic-crystal surface-emitting laser): from one to two dimensions, from the band-edge to mid-gap mode and from edge emitter to surface emitter. While the uniform NW array belongs to the PhC surface-emitting laser structure, which usually has at least two competing band-edge modes with high Q factors [29].
3.3 Threshold characteristics
The FDTD model of laser dynamics of four-level two-electron atomic system is used to numerically simulate the two lasers. The model incorporates simplified quantized electron energies, providing four energy levels for each of the two interacting electrons. The four-level diagram is shown in Fig. 6.
Energy levels 1 and 2 correspond to the dipole Pa, and energy levels 0 and 3 correspond to the dipole Pb. 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 [43]. Transitions between the energy levels are controlled by the coupled rate equation and the Pauli Exclusion Principle (PEP) [44]. Coupling the rate equation together with Maxwell's equations, an active regional dispersion model is established. The rate equation [45] is
In topological PhC NW array laser and uniform NW array laser, InGaAsP is chosen as gain materials, and the cavity is optically pumped from above by the 1064 nm Gaussian-shaped continuous-wave laser. Main material parameters are listed in Table 1 [47–50].
Table 1. Main parameters used in simulation.
The carrier populations distribution of energy level 1 and 2 in the active region of the topological PhC NW array laser is plotted in Fig. 7(a), where the pump amplitude is |E|=3 × 107 V/m. It can be seen that population inversion is realized in a very short time (less than 0.75 ps), and remains stable until the end of the simulation time. Indeed, this is achieved mainly because of the effective way of pumping in which the pump resonantly excites the active region. Figure 7(b) and (c) show the laser spectrum and electric field intensity distribution of topological and uniform PhC NW array lasers, respectively, with a pump amplitude of 3 × 107 V/m. Multiple lasing modes are excited, which is due to a relatively large value of dephasing rate γa used in the design [51]. For the topological PhC NW array laser, the bulk state lasing wavelength of 1546 nm dominates, which has lower losses and higher gains compared with other resonance modes. Moreover, by comparing Fig. 7(b) and (c), it can be further verified that the SMSR of topological PhC NW array laser is higher than that of uniform PhC NW array laser.
Fig. 7. (a) Evolution of population distributions N1 and N2 of the gain materials over time. (b) and (c) Emission spectrum and field distribution of the toplogical and uniform PhC NW arrays, respectively. (d) Threshold comparison between the topological and uniform PhC NW arrays.
The threshold characteristics of the topological and uniform NW array laser are shown in Fig. 7(d). It can be seen that the threshold value of the topological laser is 17% lower than that of the uniform laser, and the stable output intensity of the topological laser is 613% higher. This is due to the topological properties that enable effective light confinement and localization, reduce the energy required to achieve lasing, and enhance light-matter interaction, resulting in lower gain for lasing. In addition, the smaller NWs gap in the topological PhC NW laser leads to a larger mode restraint factor, which further reduces the threshold [52].
3.4 Far-field characteristics
Since both the topological and uniform PhC NW array lasers are surface-emitting lasers, the light confined in the PhC plane undergoes coherent oscillation and is coupled to the vertical direction through first-order Bragg diffraction [53]. Therefore, the relationship between the beam divergence angle and the PhC area is neccessary to be investigated. The results are shown in Fig. 8(a). Note that when studying topological PhC NW array laser, the area of the topological region is changed, and the width of the trivial region is set to 3a, which ensures that the confined light in the bulk state does not leak out and maintains that the optical field confinement is essentially the same. It can be observed that the beam divergence angles of both the two lasers decrease with the increase of the PhC area. This phenomenon can be attributed to that the increase of the PhC area improves the dispersion relationship of the photon modes, resulting in lower group velocity and better light confinement [30,34,54]. Additionally, since the confined optical field is localized in the topological region, the increase in the length of the topological region increases the number and size of the resonators. Hence, the emission of light in a specific direction is more controlled and concentrated, resulting in a reduction of beam divergence angle. With the same PhC area, the beam divergence angle of the topological PhC NW array laser is much smaller than that of the uniform NW array laser, which can reach a rather low value of ∼2° at a relatively large PhC area. Figure 8(b), (i) and (ii) represent the far-field projections of the topological and uniform PhC NW array lasers, respectively, with a same PhC area of 1024a2. It is evident that the beam divergence angle of the topological laser is much smaller. This could be attributed to the topological protection mechanism [51], leading to lower losses and more concentrated confinements on the optical modes, as can be observed from (i) and (ii) in Fig. 8(c).
Fig. 8. (a) The relationship between the beam divergence angle and the PhC area for topological and uniform PhC NW array lasers. (b) (i) and (ii) represent the far-field projections of the topological and uniform PhC NW array lasers, respectively. The PhC area as 1024a2. (c) (i) and (ii) represent the electric field intensity distribution of the topological and the uniform PhC NW array lasers, respectively.
4. Conclusion
In summary, a topological PhC InGaAsP/InP core-shell NW array laser with bulk states operating in the 1550 nm band is proposed and studied by the FDTD method. The structure consists of a topological nontrivial region surrounded by a topological trivial region. By adjusting the area of the topological region and the height of the NWs, high Q factor and large SMSR can be obtained. Compared with traditional uniform PhC NW array laser, the topological NW array laser exhibits lower threshold, higher output, and lower beam divergence angle. This work demonstrates that the combination of NW array and topological photonics is a promising way for the development of high performance nanophotonic devices.
Funding
National Natural Science Foundation of China (61935003); State Key Laboratory of Information Photonics and 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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