Open Access
Add to BibSonomy
Abstract
As a novel branch of topology, non-Hermitian topological systems have been extensively studied in theory and experiments recently. Topological parity-time (PT)-symmetric semiconductor stripe laser arrays based on the Su-Schreiffer-Heeger model are proposed. The degree of non-Hermicity can be tuned by altering the length of the cavities, and PT symmetry can be realized by patterned electrode. Three laser arrays working in different non-Hermitian phases are analyzed and fabricated. With the increasing degree of non-Hermicity, the peaks of output intensities move from the edge to the bulk. The proposed semiconductor stripe laser array can function as an active, flexible, and feasible platform to investigate and explore non-Hermitian topology for further developments in this field.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Topological photonics originated from topological insulators in condensed matter physics [1–7] is a cutting-edge and active research nowadays. After the first experimental observation of photonic topological insulator [8], numerous theoretical and experimental studies have spurred. In analogy to integer quantum-Hall effect in quantum mechanics, resonant photonic cavities on yttrium iron garnet substrate under an external magnetic field that break time-reversal symmetry (TRS) have been demonstrated to exhibit topological lasing [9]. Topological laser based on Haldane model [10] has also been fabricated and demonstrated [11]. Stemming from quantum spin-Hall effect, crystalline model offers another approach to organize topological photonic systems [12]. Photonic crystals based on crystalline model break pseudo-TRS successfully, inspiring ground-breaking studies in dielectric and laser platforms [13–15]. It has also been demonstrated that coupled ring-resonator laser, establishing a synthetic magnetic field through spin-like counter-propagating modes, yields topological characteristics [8,16–18]. Likewise, topological modes in valley photonic crystal have been analyzed and demonstrated [19–21], inspired by topological valley-Hall transport in bilayer graphene [22,23]. Besides, one-dimensional photonic lattices can also possess topological feature including Aubry-Andre or Harper model [24,25], Kitaev model [26], quasiperiodic Harper chain [27–29] and Su-Schreiffer-Heeger (SSH) model [30–37].
Based on the research aforementioned in addition with experimental feasibility of photonic structures, we can interpret that the photonic systems have been extensively utilized to investigate novel topological properties. Therefore, topological photonics can serve as an effective platform to verify theoretical assumptions and motivate new discoveries in topological physics. In practice, photonic systems possess intrinsic loss, which deviates from ideal theoretical analysis. Consequently, the introduction of non-Hermitian operator involving gain and loss is pivotal for further study and technical application of topological photonics. After parity-time (PT) symmetry was firstly proposed [38], non-Hermitian physics has stepped into a new stage. For operators that are PT-symmetric, they have phase in which the eigenvalues are real [39]. This indicates that there are quantities observable and measurable in non-Hermitian systems and boosts research in the optical realm with PT symmetry [40–50]. Accordingly, studies on non-Hermitian topological photonics along with PT symmetry have been broadly conducted [51–62], since non-Hermicity could work as an extra dimension of freedom to enrich the research of topology.
Among these studies, one-dimensional complex SSH model with PT symmetry is under great investigation. Theoretical predictions [63–68] and experimental realizations based on passive waveguides [32,69–72], nanocavity arrays [73], active mechanical metamaterial [74] and active microring resonators [35] have been reported recently. Different opinions on the topological phases in this model that have no counterparts in Hermitian systems also arise. According to the definition of topological transition, where bandgap closes and reopens [68], some researchers believe that topological phase transition takes place through altering the degree of non-Hermicity [67]. While others hold that topological phase should be indicated by global topological invariant in complex model [51,64] and all the non-Hermitian phases are topologically identical [65,66]. In parallel, more definitions spring out to characterize non-Hermitian topology, such as non-Bloch band theory [57,58], non-Hermitian skin effect [75,76] and so on.
In this work, we adopt semiconductor stripe laser arrays as the platform to observe different non-Hermitian phases in topologically non-trivial setting, where the intrinsic loss of photonic material and optical gain offered by electrical injection are easily attainable. The spectra in momentum space with different degree of non-Hermicity are calculated by the method of coupled mode theory (CMT) with tight-binding approximation (TBA). Different quantum phases in PT-symmetric topological system are introduced based on these spectra, as presented in [64,70]. In the proposed experimental platform, complex mode spectra and mode profiles in different degree of non-Hermicity are analyzed. Mode characteristics are obviously varied with different on-site gain or loss in real space, corresponding to different phases in momentum spectra. With alternate injection on dimerized ridge waveguides in stripe laser array, we obtain a one-dimensional PT-symmetric SSH lattice in experiments. By tuning the loss of the device, through altering the length of the cavities [49], we manipulate the stripe laser arrays working in different quantum phases with different degree of non-Hermicity. The near-field patterns near thresholds of three stripe laser arrays with different loss are observed. The output profiles of three stripe laser arrays, which correspond to different quantum phases, differ from each other near thresholds and the peaks of the intensity profiles gradually move from the edge to the bulk of the device with increasing degree of non-Hermicity.
2. Different phases in non-Hermitian SSH model
A typical PT-symmetric SSH model is plotted in Fig. 1. To satisfy PT symmetry, sites in every dimer possess complex refractive index that is symmetric in the real part and antisymmetric in the imaginary part, respectively [66,71]. The coupling coefficient between intra-dimer is κa and the coupling coefficient between inter-dimer is κe.
Fig. 1. The schematic of PT-symmetric SSH model. One dimer is represented by dashed square, where the red and gray squares represent sites with gain and loss, respectively. κa and κe are the coupling coefficient intra and inter dimer, respectively.
CMT provides a method to calculate modes in the system with multiple cavities or waveguides [35,71]. With TBA, the corresponding Hamiltonian in momentum space can be obtained:
Ĥm(k) can be regarded as the Bloch Hamiltonian of the PT-symmetric SSH model under periodic boundary condition. +γ and -γ represent the corresponding gain and loss in the PT-symmetric setting, respectively. ζ is the real part of on-site eigenvalue. According to [64,70], band diagrams obtained by Ĥm(k) can be divided into three different quantum phases with respect to the degree of non-Hermicity. The three phases are distinguished by γ =κe-κa (κa <κe) and γ =κe + κa. The band diagrams within the first Brillouin zone of three quantum phases are plotted in Fig. 2(a)-(c). In Fig. 2(a), the value of γ is smaller than κe-κa, it corresponds to unbroken PT phase I where the whole spectrum in momentum space is real. In the phase with κe-κa < γ < κe + κa, the spectrum in momentum space starts to be complex as shown in Fig. 2(b). Therefore, the phase is defined as partially broken PT phase II. When γ is further increased and greater than κe + κa, the system works in phase III, corresponding to completely broken PT phase shown in Fig. 2(c). The whole spectrum becomes complex in this phase.
Fig. 2. Band diagrams and mode spectra of the infinite PT-symmetric SSH model and finite PT-symmetric topological semiconductor dimerized stripe laser array in different quantum phases under periodic boundary condition and open boundary condition, respectively. The value of κa and κe are 1.62 cm−1 and 6.15 cm−1 (κa <κe), respectively. (a) and (d): unbroken PT phase I with γ=3.13 cm-1 (<6.15-1.62 = 4.53 cm-1). (b) and (e): partially broken PT phase II with γ=5.99 cm-1 (>4.53 cm-1, < 6.15 + 1.62 = 7.77 cm-1). (c) and (f): completely broken PT phase III with γ=9.79 cm-1 (>7.77 cm-1). Orange circles in (d)-(f) denotes the amplified edge mode. Re: real parts of the eigenvalues or propagation constants. Im: imaginary parts of the eigenvalues or propagation constants.
The infinite SSH model discussed above cannot be realized experimentally and the topological protection may be weakened in a finite situation. However, finite-size effects on topological interface states have been analyzed theoretically [77] and topological edge states have been demonstrated in finite topologically nontrivial SSH chains experimentally [35,71]. Therefore, in this context, we adopt semiconductor stripe laser arrays, which consists of 6 dimers, as the experimental platform to study and observe different non-Hermitian phases in finite PT-symmetric SSH chain.
The stripe laser array is composed of 12 ridge waveguides and terminates with topologically nontrivial dimerization [35,71,72]. Following CMT, the real-space Hamiltonian of our design is a 12 × 12 matrix because of its finite size:
β is mode propagation constant in each ridge waveguide. Mode spectra and distributions in the designed structure can be obtained by solving:
U represents the field amplitudes in each ridge waveguide. βs is the corresponding eigenvalue of the characteristic equation. The mode spectra in different phases are presented in Fig. 2 (d)-(f), which correspond to the spectra in momentum space, respectively. Figure 3 plots the normalized intensity distributions of eigenmodes including two edge modes and two bulk modes.
Fig. 3. Normalized intensity distributions of eigenmodes. (a) Normalized intensity distribution of amplified edge mode that marked by orange circles in three phases, where the black dashed line satisfies y = (κa /κe) N. N is the number of waveguides. (b) Normalized intensity distribution of the other edge mode, which mainly locates in the lossy waveguides. (c) and (d) show the normalized intensity distributions of two bulk modes in phase II and III, which concentrates on the bulk of the lattice.
In Fig. 2(d), where γ < κe-κa, only one mode with positive imaginary propagation constant exists. The normalized intensity distribution of the amplified mode is plotted in the Fig. 3(a), which decays exponentially from edge to bulk of the lattice. It indicates that only the amplified edge mode will lase near threshold. At the same time, other modes remain neutral or lossy, including the other edge mode. Normalized intensity distribution of the other edge mode is plotted in Fig. 3(b), the field of this mode concentrates mainly on the lossy waveguides and cannot be observed with an increasing injection current. It corresponds to unbroken PT phase I in Fig. 2(a). Figure 2(e) presents the mode spectrum where κe-κa < γ < κe + κa. Two more mode propagation constants become complex in real space. Modes with positive imaginary propagation constants will all contribute to lasing, including the amplified edge mode and other bulk modes. The normalized intensity distributions of the lasing bulk modes are plotted in Fig. 3(c) and (d), which concentrate on the bulk rather than the edge of the array. This spectrum corresponds to partially broken PT phase II in Fig. 2(b). When γ is increased and greater than κe + κa, the system works in phase III corresponding to completely broken PT phase shown in Fig. 2(c). Half of the eigenmodes possess positive imaginary propagation constants in real space, as shown in Fig. 2(f). Compared to previous two phases, more bulk modes possess gain and may lase with an increasing injection current. All other bulk modes distribute in similar manners as presented in Fig. 3(c) and (d). It is not hard to predict that increasing degree of non-Hermicity may result in distinguishable lasing profiles in experiments due to these amplified bulk modes. The edge-location of the intensity may be altered, as the increasing number of amplified bulk modes from unbroken PT phase I to completely broken PT phase III.
Based on analysis above, the mode characteristics are quite different in each quantum phase. However, along with fixed values of κe and κa and increasing value of γ, topological edge mode always exists marked by orange circles in Fig. 2(d)-(f), indicating topological phase transition does not take place in the process. This is in correspondence with recent reports that the global topological invariant, which depends on the ratio of coupling coefficients κe and κa, determines the topological phase of the lattice [51,64–66,71], in spite of the fact that the edge mode is gradually obscured by more amplified bulk modes.
3. Non-Hermitian SSH semiconductor stripe laser arrays
The schematic of our proposed design is plotted in Fig. 4(a) and (b). The epitaxy is based on asymmetric super large optical cavity structure and the central lasing wavelength is 980 nm [78]. Every dimer consists of two identical ridge waveguides operating in single-mode regime, of which the height h and width w is 1.2 µm and 5.0 µm, respectively. The distance between ridge waveguides in a dimer d1 is set to be 5.0 µm, while the distance between different dimers d2 is 2.0 µm. In addition, to realize PT symmetry, gain and loss are introduced into the ridge waveguides alternately by patterned electrodes.
Fig. 4. Structure of PT-symmetric topological semiconductor stripe laser array. (a) The schematic of PT-symmetric topological semiconductor stripe laser array. Ti/Pt/Au as p-side metal stacks (top) and AuGeNi/Au as n-side metal stacks (bottom), respectively (yellow); SiO2 as insulator (blue); Heavily-doped AlGaAs as cladding layer (light gray); Lightly-doped AlGaAs as waveguide layer (dark gray); GaAs/GaInAs double quantum wells as active region (red). (b) The geometric parameters of one dimer at the edge. (c) SEM of the device and the cross section is shown in the inset. In the fabricated device, the height and width of the ridge waveguide is 1.25 µm and 5.08 µm, respectively. The distance of intra dimer is 5.00 µm, and the distance between different dimers is 1.96 µm.
In fabrication, the dimerized waveguide arrays are obtained by standard photolithography process and inductively coupled plasma etching. The insulator layer is grown on the top of wafer through plasma enhanced chemical vapor deposition. Electrode patterns are formed by etching the insulator layer on each ridge waveguides alternately based on reactive ion etching. Then, the metal stacks on p and n side are sputtered on the wafer after polishing the substrate. Therefore, with current injection, the optical gain is introduced into the pumped waveguides where the insulator layer is etched off. On the contrast, the light in the unpumped waveguides with insulator layer between epitaxy and p-side metal stacks experiences intrinsic loss. Figure 4(c) shows the scanning electron microscope (SEM) image of the device. The deviation of size parameters is introduced by fabrication error inevitably. However, the deviations are less than 0.1 µm, which do not disable the theoretical calculations in the context.
After the geometric parameters w, h, d1 and d2 are determined, the coupling coefficients intra and inter dimer can be obtained based on super mode theory [79],
Equation (4) can be obtained through CMT when two identical single-mode waveguides are considered. βc1 and βc2 are the propagation constants of supermodes in the coupled two-waveguide system, which can be obtained by finite element method (FEM). Figure 5 presents the relationship between coupling coefficient and distance between two waveguides in our design. Therefore, the intra-dimer coupling coefficient (κa) is smaller than inter-dimer coupling coefficient (κe), and the value of them are 1.62 cm−1 and 6.15 cm−1 as presented in Fig. 5, respectively.
Fig. 5. Coupling coefficient versus distance between adjacent waveguides with the ridge width of 5.0 µm and height of 1.2 µm.
After the determination of coupling coefficients, we obtain devices working in different quantum phases by tuning the loss of them. Intrinsic loss in an unpumped waveguide is composed of internal loss αi and radiation loss αm:
The components of radiation loss are presented in Eq. (6). L is the cavity length of the stripe laser array. Rf and Rr represent the reflectivity of front facet and rear facet of the stripe laser array, respectively. The value of αi is 0.58 cm-1 in our design [78]. The steady-state gain in the pumped waveguide is intrinsically nonlinear and not easy to manipulate [80]. Therefore, we can alter the length of the cavities and the coating on the facets to tune the intrinsic loss of the unpumped waveguides and maneuver the stripe laser arrays working in different quantum phases near their thresholds.
According to Eq. (5) and (6), the cavity lengths and coating on the facets are deliberately designed to verify theoretical analysis aforementioned. After metallization process, each stripe laser array is cleaved and the cavity length of the device is determined simultaneously. In this context, we take advantage of the flexibility of cleaving and avoid facet coating process to simplify the fabrications. Consequently, three devices working in different quantum phases are fabricated. For the stripe laser array possessing intrinsic loss of 3.13 cm-1, which corresponds to unbroken PT phase I in Fig. 2(d), the cavity length is set to be 2.0 mm. The second stripe laser array with intrinsic loss of 5.99 cm-1 is cleaved to be 1.0 mm in length. It works in partially broken PT phase II near its threshold, as shown in Fig. 2(e). Another stripe laser array is set to be 0.6 mm in length. Its intrinsic loss is 9.79 cm-1 and corresponds to completely broken PT phase III in Fig. 2(f).
In the experimental demonstration, the light-current-voltage (LIV) characteristics of the three devices are tested. Threshold current of each stripe laser array can be extracted from the results of LI tests. The far-field pattern of the laser array working in unbroken PT phase I is also measured. LI and far fields are measured by the COS Tester of Raybow Opto. The far-field pattern measurement is performed with a detector (Silicon PIN photodiode in COS Tester RB-CT1000 of Raybow Opto), which locates around 0.4 meters away from the device. Therefore, the far-field pattern can be approximated to the Fourier transform of the near-field distribution, serving as an identifier of the output mode. The near-field patterns of the devices are measured near thresholds to investigate the output profiles of them. We collect the near-field pattern of the sample at the facets from top views by InGaAs charge-coupled device (CCD) camera of HAMAMATSU C10633. A set of lenses with a neutral density filter is utilized to focus on the top of facets and collect the output profile clearly.
4. Results and discussions
The threshold current is obtained by linear fitting of LI curve where stimulated radiation dominates, as shown in Fig. 6 (short dashed lines). The value of threshold could be defined as the intercept of the fitting line and x axis marked by stars in Fig. 6(a)-(c) according to Eq. (7) [81]:
Fig. 6. LIV results of three stripe laser arrays in different quantum phases. (a) Laser array with cavity length of 2.0 mm. (b) Laser array with cavity length of 1.0 mm. (c) Laser array with cavity length of 0.6 mm. The short dashed lines are the linear fitting curves of each LI curve where stimulated emission power increases. The stars highlight the intercepts of the fitting lines and x axes and correspond to values of threshold current.
The threshold currents of three devices are around 250 mA, 170 mA and 230 mA, respectively.
Next, the near-field patterns near their thresholds are captured. The output profiles of three devices are presented in Fig. 7(a)-(c). Black solid lines in Fig. 7(d)-(f) are pixels distributions collected by CCD graphs along white dashed axes in Fig. 7(a)-(c), which can directly represent the output intensity distributions along the non-Hermitian SSH waveguide arrays. We also plot the theoretical result of intensity profile in bar diagrams in Fig. 7(d) to compare it with experimental result visually. The theoretical result can be obtained by solving Eq. (3), where the eigenvectors represent the field distributions of corresponding eigenmodes as shown in Fig. 3. Here, we only present the calculated result in unbroken PT phase I since there is only an edge mode possess modal gain in this phase. In phase II and III, two and more bulk modes experience amplification other than the edge mode, as shown in Fig. 2(e) and (f). Output profiles may be affected drastically by mode competition [82] in these phases, which is complicated and should be studied in the future work.
Fig. 7. The measured and calculated output profiles along the SSH waveguide arrays (white dashed axes) in different phases. (a)-(c): the near-field patterns of three laser arrays are captured by CCD camera at 260 mA, 180 mA and 240 mA, respectively. (d)-(f): black solid lines are corresponding pixels distributions which can reflect the output profiles along white axes in (a)-(c). Bars in (d) present the calculated output profile by means of CMT.
Figure 7(a) and (d) present the measured near-field pattern of the laser array with cavity length of 2.0 mm. The experimental measured intensity profile of laser array dominantly locates at the edge of the stripe laser array and decays drastically to the opposite end. Only the amplified edge mode possesses modal gain in the laser array with γ = 3.13 cm-1, as plotted in Fig. 2(d). Hence, the theoretical bar diagram is in accordance with the distribution of the amplified edge mode plotted in Fig. 3(a). The experimental distribution is similar with that of calculated edge mode. However, the intensity at the bulk part of the measured profile is stronger than that of theoretical edge mode. The deviation can be attributed to two main reasons. First, the inevitable spreading of injected current in the lateral direction, along with lateral diffusion of carriers [83], may result in increasing intensity distribution outside of the pumped waveguides by spontaneous emission. Especially for the output pattern collected near threshold, the light intensity of spontaneous emission is not negligible. Second, with the current injection and the carrier density variation, the refractive index fluctuates due to free carrier plasma effects [84]. The light profile tends to expand into the unpumped region because the variation of refractive index is negatively associated with carrier density. In addition, the lateral far-field pattern of this laser array, which functions as an indicator of output mode, is also measured (blue circles) and compared with the calculated edge mode (red line), as plotted in Fig. 8(a). The calculated pattern is obtained by FEM. The positions of the lobes in the theoretical calculation and measured result are almost identical, with a deviation around 1°. We also present the calculated far-field pattern of one typical bulk mode in Fig. 8(b). The far-field distribution of bulk mode with sharper peaks is obviously different from that of the measured result in Fig. 8(a), revealing that the amplified edge mode lases dominantly near threshold as calculated.
Fig. 8. The measured and calculated far-field patterns in lateral direction. (a) Blue circles present the measured pattern of the laser array whose cavity length is 2.0 mm (obtained at 280mA). The red line is the calculated pattern of the amplified edge mode. (b) The calculated far-field pattern of one bulk mode.
Figure 7(b) and (e) show the intensity profile of the stripe laser array that is 1.0 mm in length. The output profile of this laser array distributes more uniformly compared to that shown in Fig. 7(a) and (d), while the intensity peak still locates at the edge waveguide. It corresponds to the phase where γ is set to be 5.99 cm-1, as plotted in Fig. 2(e). Two more modes except the amplified edge mode possess modal gain and the intensity distributions of amplified bulk modes in this partially broken PT phase II have been discussed and plotted in Fig. 3(c) and (d). Therefore, the light intensity at the bulk of array increases in this phase. The stripe laser array whose cavity length is 0.6mm does not hold edge-location and the maximum of the intensity mainly locates around the center of the stripe laser array, as presented in Fig. 7(c) and (f). In this phase with γ = 9.79 cm-1, half of the eigenmodes in this phase possess modal gain and contribute to the lasing pattern, as shown in Fig. 2(f). The topological edge state is not observable in this phase. This behavior of peak shift may be attributed to two factors. First, the number of amplified bulk modes increases. Second, modal gain difference between amplified bulk modes and edge mode is smaller compared to that in partially broken PT phase II.
According to results above, with the increasing value of γ, we observe output profiles in different quantum phases among these three stripe laser arrays. Along with the increasing degree of non-Hermicity, the edge mode becomes more obscure due to the existence of more amplified bulk modes. Finally in completely broken PT phase III, the intensity peak does not locate at the edge.
5. Conclusion
In summary, we have proposed and fabricated topological semiconductor stripe laser arrays as the platform to analyze and observe their mode characteristics in different non-Hermitian phases. With fixed dimerization and the increasing degree of non-Hermicity, the peaks of intensity distributions move from the edge to the bulk of stripe laser arrays, on account of more lasing bulk modes and decreasing gain differences among bulk modes and edge mode. It demonstrates that for applications of topological edge state in non-Hermitian laser or waveguide systems, the device should be deliberately designed and work in unbroken PT phase I to exclude the appearance of other bulk modes. On the one hand, we tune the radiation loss in the semiconductor stripe laser arrays by altering the cavity lengths, manipulating them working in different quantum phases. The tunable loss in proposed model is much more feasible than that in other platforms, which yields highly tunable non-Hermicity. On the other hand, PT symmetry in proposed semiconductor stripe laser array is realized by introducing patterned electrode, which is compatible with normal optoelectronic chip fabrication process. Meanwhile, the test system is rather attainable since the stripe laser array is electrically injected by one single current source, which simplifies test process in experiments compared to those optically-pumped systems. The proposed semiconductor stripe laser array has the potential to be applied as an effective non-Hermitian topological experimental platform, which can be combined with other paradigms of topological chains, to further explore non-Hermitian topology. In the future, further studies on multimode lasing and quasi-PT-symmetry in the non-Hermitian SSH model should be conducted.
Funding
National Key Research and Development Program of China (2021YFA1400604, 2021YFB2801400); National Natural Science Foundation of China (62075213, 62135001, 62205328, 91850206).
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.
References
1. F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008). [CrossRef]
2. S. Iwamoto, Y. Ota, and Y. Arakawa, “Recent progress in topological waveguides and nanocavities in a semiconductor photonic crystal platform [Invited],” Opt. Mater. Express 11(2), 319–337 (2021). [CrossRef]
3. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014). [CrossRef]
4. Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, and S. Iwamoto, “Active topological photonics,” Nanophotonics 9(3), 547–567 (2020). [CrossRef]
5. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019). [CrossRef]
6. M. Segev and M. A. Bandres, “Topological photonics: Where do we go from here?” Nanophotonics 10(1), 425–434 (2020). [CrossRef]
7. B.-Y. Xie, H.-F. Wang, X.-Y. Zhu, M.-H. Lu, Z. D. Wang, and Y.-F. Chen, “Photonics meets topology,” Opt. Express 26(19), 24531–24550 (2018). [CrossRef]
8. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013). [CrossRef]
9. B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358(6363), 636–640 (2017). [CrossRef]
10. F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”,” Phys. Rev. Lett. 61(18), 2015–2018 (1988). [CrossRef]
11. Y. G. N. Liu, P. S. Jung, M. Parto, D. N. Christodoulides, and M. Khajavikhan, “Gain-induced topological response via tailored long-range interactions,” Nat. Phys. 17(6), 704–709 (2021). [CrossRef]
12. C. L. Kane and E. J. Mele, “Quantum Spin Hall Effect in Graphene,” Phys. Rev. Lett. 95(22), 226801 (2005). [CrossRef]
13. S. Barik, A. Karasahin, C. Flower, T. Cai, H. Miyake, W. DeGottardi, M. Hafezi, and E. Waks, “A topological quantum optics interface,” Science 359(6376), 666–668 (2018). [CrossRef]
14. A. Dikopoltsev, H. Harder Tristan, E. Lustig, A. Egorov Oleg, J. Beierlein, A. Wolf, Y. Lumer, M. Emmerling, C. Schneider, S. Höfling, M. Segev, and S. Klembt, “Topological insulator vertical-cavity laser array,” Science 373(6562), 1514–1517 (2021). [CrossRef]
15. Z.-K. Shao, H.-Z. Chen, S. Wang, X.-R. Mao, Z.-Q. Yang, S.-L. Wang, X.-X. Wang, X. Hu, and R.-M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15(1), 67–72 (2020). [CrossRef]
16. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013). [CrossRef]
17. A. Bandres Miguel, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, N. Christodoulides Demetrios, and M. Khajavikhan, “Topological insulator laser: Experiments,” Science 359(6381), eaar4005 (2018). [CrossRef]
18. G. Harari, A. Bandres Miguel, Y. Lumer, C. Rechtsman Mikael, Y. D. Chong, M. Khajavikhan, N. Christodoulides Demetrios, and M. Segev, “Topological insulator laser: Theory,” Science 359(6381), eaar4003 (2018). [CrossRef]
19. X.-T. He, E.-T. Liang, J.-J. Yuan, H.-Y. Qiu, X.-D. Chen, F.-L. Zhao, and J.-W. Dong, “A silicon-on-insulator slab for topological valley transport,” Nat. Commun. 10(1), 872 (2019). [CrossRef]
20. M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, “Robust topologically protected transport in photonic crystals at telecommunication wavelengths,” Nat. Nanotechnol. 14(1), 31–34 (2019). [CrossRef]
21. Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong, and Q. J. Wang, “Electrically pumped topological laser with valley edge modes,” Nature 578(7794), 246–250 (2020). [CrossRef]
22. I. Martin, Y. M. Blanter, and A. F. Morpurgo, “Topological Confinement in Bilayer Graphene,” Phys. Rev. Lett. 100(3), 036804 (2008). [CrossRef]
23. Z. Qiao, J. Jung, Q. Niu, and A. H. MacDonald, “Electronic Highways in Bilayer Graphene,” Nano Lett. 11(8), 3453–3459 (2011). [CrossRef]
24. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological States and Adiabatic Pumping in Quasicrystals,” Phys. Rev. Lett. 109(10), 106402 (2012). [CrossRef]
25. F. Alpeggiani and L. Kuipers, “Topological edge states in bichromatic photonic crystals,” Optica 6(1), 96–103 (2019). [CrossRef]
26. Y. Xing, L. Qi, J. Cao, D.-Y. Wang, C.-H. Bai, W.-X. Cui, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Controllable photonic and phononic edge localization via optomechanically induced Kitaev phase,” Opt. Express 26(13), 16250–16264 (2018). [CrossRef]
27. Y. E. Kraus and O. Zilberberg, “Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model,” Phys. Rev. Lett. 109(11), 116404 (2012). [CrossRef]
28. M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silberberg, “Topological pumping over a photonic Fibonacci quasicrystal,” Phys. Rev. B 91(6), 064201 (2015). [CrossRef]
29. Z. Guo, H. Jiang, Y. Sun, Y. Li, and H. Chen, “Asymmetric topological edge states in a quasiperiodic Harper chain composed of split-ring resonators,” Opt. Lett. 43(20), 5142–5145 (2018). [CrossRef]
30. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in Polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979). [CrossRef]
31. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015). [CrossRef]
32. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity–time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017). [CrossRef]
33. P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, “Lasing in topological edge states of a one-dimensional lattice,” Nat. Photonics 11(10), 651–656 (2017). [CrossRef]
34. Y. Ota, R. Katsumi, K. Watanabe, S. Iwamoto, and Y. Arakawa, “Topological photonic crystal nanocavity laser,” Commun. Phys. 1(1), 86 (2018). [CrossRef]
35. M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Edge-Mode Lasing in 1D Topological Active Arrays,” Phys. Rev. Lett. 120(11), 113901 (2018). [CrossRef]
36. H. Zhao, P. Miao, M. H. Teimourpour, S. Malzard, R. El-Ganainy, H. Schomerus, and L. Feng, “Topological hybrid silicon microlasers,” Nat. Commun. 9(1), 981 (2018). [CrossRef]
37. C. Han, M. Lee, S. Callard, C. Seassal, and H. Jeon, “Lasing at topological edge states in a photonic crystal L3 nanocavity dimer array,” Light: Sci. Appl. 8(1), 40 (2019). [CrossRef]
38. C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]
39. C. M. Bender, M. V. Berry, and A. Mandilara, “Generalized PT symmetry and real spectra,” J. Phys. A: Math. Gen. 35(31), L467–L471 (2002). [CrossRef]
40. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5(1), 4034 (2014). [CrossRef]
41. B. Peng, K. Özdemir Ş, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346(6207), 328–332 (2014). [CrossRef]
42. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
43. H. Hodaei, M.-A. Miri, M. Heinrich, N. Christodoulides Demetrios, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]
44. L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
45. M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]
46. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]
47. T. Fu, Y.-F. Wang, X.-Y. Wang, X.-Y. Zhou, and W.-H. Zheng, “Mode Control of Quasi-PT Symmetry in Laterally Multi-Mode Double Ridge Semiconductor Laser *,” Chin. Phys. Lett. 37(4), 044207 (2020). [CrossRef]
48. J. Chen, Y. Wang, T. Fu, X. Wang, Y. Dai, and W. Zheng, “Mode degeneracy and enhanced sensitivity in electrically injected PT-symmetric semiconductor laser with a quasi-high-order exceptional point,” Appl. Phys. Express 14(12), 122005 (2021). [CrossRef]
49. T. Fu, Y. Wang, X. Zhou, F. Du, J. Fan, X. Wang, J. Chen, A. Qi, and W. Zheng, “Approaches to tuning the exceptional point of PT-symmetric double ridge stripe lasers,” Opt. Express 29(13), 20440–20448 (2021). [CrossRef]
50. X. Wang, J. Li, T. Fu, J. Chen, Y. Dai, R. Han, Y. Wang, and W. Zheng, “Electrically injected double-taper semiconductor laser based on parity-time symmetry,” Electron. Lett. 58(5), 216–217 (2022). [CrossRef]
51. S.-D. Liang and G.-Y. Huang, “Topological invariance and global Berry phase in non-Hermitian systems,” Phys. Rev. A 87(1), 012118 (2013). [CrossRef]
52. T. E. Lee, “Anomalous Edge State in a Non-Hermitian Lattice,” Phys. Rev. Lett. 116(13), 133903 (2016). [CrossRef]
53. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems,” Phys. Rev. Lett. 118(4), 040401 (2017). [CrossRef]
54. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, “Topological Phases of Non-Hermitian Systems,” Phys. Rev. X 8(3), 031079 (2018). [CrossRef]
55. H. Shen, B. Zhen, and L. Fu, “Topological Band Theory for Non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120(14), 146402 (2018). [CrossRef]
56. S. Yao, F. Song, and Z. Wang, “Non-Hermitian Chern Bands,” Phys. Rev. Lett. 121(13), 136802 (2018). [CrossRef]
57. S. Yao and Z. Wang, “Edge States and Topological Invariants of Non-Hermitian Systems,” Phys. Rev. Lett. 121(8), 086803 (2018). [CrossRef]
58. K. Yokomizo and S. Murakami, “Non-Bloch Band Theory of Non-Hermitian Systems,” Phys. Rev. Lett. 123(6), 066404 (2019). [CrossRef]
59. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, “Symmetry and Topology in Non-Hermitian Physics,” Phys. Rev. X 9(4), 041015 (2019). [CrossRef]
60. T. Liu, Y.-R. Zhang, Q. Ai, Z. Gong, K. Kawabata, M. Ueda, and F. Nori, “Second-Order Topological Phases in Non-Hermitian Systems,” Phys. Rev. Lett. 122(7), 076801 (2019). [CrossRef]
61. L. Qi, G.-L. Wang, S. Liu, S. Zhang, and H.-F. Wang, “Robust Interface-State Laser in Non-Hermitian Microresonator Arrays,” Phys. Rev. Appl. 13(6), 064015 (2020). [CrossRef]
62. D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, “Non-Hermitian Boundary Modes and Topology,” Phys. Rev. Lett. 124(5), 056802 (2020). [CrossRef]
63. H. Schomerus, “Topologically protected midgap states in complex photonic lattices,” Opt. Lett. 38(11), 1912–1914 (2013). [CrossRef]
64. H. Zhao, S. Longhi, and L. Feng, “Robust Light State by Quantum Phase Transition in Non-Hermitian Optical Materials,” Sci. Rep. 5(1), 17022 (2015). [CrossRef]
65. H. Jiang, C. Yang, and S. Chen, “Topological invariants and phase diagrams for one-dimensional two-band non-Hermitian systems without chiral symmetry,” Phys. Rev. A 98(5), 052116 (2018). [CrossRef]
66. S. Lieu, “Topological phases in the non-Hermitian Su-Schrieffer-Heeger model,” Phys. Rev. B 97(4), 045106 (2018). [CrossRef]
67. B. Midya, “Non-Hermitian tuned topological band gap,” Ann. Phys. 421, 168280 (2020). [CrossRef]
68. C. Yuce, “Edge states at the interface of non-Hermitian systems,” Phys. Rev. A 97(4), 042118 (2018). [CrossRef]
69. S. Ke, D. Zhao, J. Liu, Q. Liu, Q. Liao, B. Wang, and P. Lu, “Topological bound modes in anti-PT-symmetric optical waveguide arrays,” Opt. Express 27(10), 13858–13870 (2019). [CrossRef]
70. M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng, “Photonic zero mode in a non-Hermitian photonic lattice,” Nat. Commun. 9(1), 1308 (2018). [CrossRef]
71. W. Song, W. Sun, C. Chen, Q. Song, S. Xiao, S. Zhu, and T. Li, “Breakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical Lattices,” Phys. Rev. Lett. 123(16), 165701 (2019). [CrossRef]
72. J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a Topological Transition in the Bulk of a Non-Hermitian System,” Phys. Rev. Lett. 115(4), 040402 (2015). [CrossRef]
73. F. Hentinger, M. Hedir, B. Garbin, M. Marconi, L. Ge, F. Raineri, J. A. Levenson, and A. M. Yacomotti, “Direct observation of zero modes in a non-Hermitian optical nanocavity array,” Photonics Res. 10(2), 574–586 (2022). [CrossRef]
74. A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, “Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial,” Proc. Natl. Acad. Sci. U. S. A. 117(47), 29561–29568 (2020). [CrossRef]
75. C. H. Lee and R. Thomale, “Anatomy of skin modes and topology in non-Hermitian systems,” Phys. Rev. B 99(20), 201103 (2019). [CrossRef]
76. N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological Origin of Non-Hermitian Skin Effects,” Phys. Rev. Lett. 124(8), 086801 (2020). [CrossRef]
77. P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98(2), 023838 (2018). [CrossRef]
78. S. Zhao, A. Qi, M. Wang, H. Qu, Y. Lin, F. Dong, and W. Zheng, “High-power high-brightness 980 nm lasers with >50% wall-plug efficiency based on asymmetric super large optical cavity,” Opt. Express 26(3), 3518–3526 (2018). [CrossRef]
79. W. S. C. Chang, “Guided-wave interactions,” in Fundamentals of Guided-Wave Optoelectronic Devices (Cambridge, 2009), pp. 39–68.
80. A. U. Hassan, H. Hodaei, M.-A. Miri, M. Khajavikhan, and D. N. Christodoulides, “Nonlinear reversal of the PT-symmetric phase transition in a system of coupled semiconductor microring resonators,” Phys. Rev. A 92(6), 063807 (2015). [CrossRef]
81. L. A. Coldren, S. W. Corzine, and M. L. Mašanović, “A Phenomenological Approach to Diode Lasers,” in Diode Lasers and Photonic Integrated Circuits (Wiley, 2012), pp. 45–90.
82. D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014). [CrossRef]
83. L. Rahman and H. G. Winful, “Nonlinear dynamics of semiconductor laser arrays: a mean field model,” IEEE J. Quantum Electron. 30(6), 1405–1416 (1994). [CrossRef]
84. T. Numai, “Fundamentals of Semiconductor Lasers,” in Fundamentals of Semiconductor Lasers (Springer, 2015), pp. 89–186.
