Defect-evolved quadrupole higher-order topological nanolasers

Domain: chemistry
Published: 2026-02-26
Editorial outcome: major_revision

Abstract
Topological photonics have been garnering widespread interest in engineering the flow of light with topological ideas. Strikingly, the recent introduction of higher-order topological insulators has generalized the fundamental framework of topological photonics, endowing counterintuitive strong confinement of light at lower-dimensional boundaries, thus unlocking exciting prospects for the exploration of topological phenomena in fresh routes as well as the design of topology-driven nanoscale light sources. Here, we revealed the photonic quadrupole topological phases can be activated by defect evolution and performed experimental demonstrations of associated nanoscale lasing operation under this paradigm. The quadrupole higher-order topological nanocavity is constructed by two topologically distinct photonic crystal slabs with opposite directions of defect evolution. Stable single-mode emission and low lasing threshold in telecom C-band are achieved at room temperature of the defect-evolved quadrupole topological nanolaser. This work reveals new possibilities for photonic quadrupole topological phase transition, providing an intriguing route toward light confinement and modulation under the topological framework.

Introduction
The past few decades have witnessed the accelerated development of the fascinating area of topological insulator (TI). In general, the representative features of TI are the topological properties of the momentum space and the robust boundary states which reflected by the bulk-boundary correspondence1-3. Benefiting from the similarities between photonic crystal (PhC) and solid-state physics, analogous concepts extend into photonic systems, giving rise to the discipline of topological photonics4-6. Since the demonstration of quantum Hall effect edge states7, topological photonics has emerged as a remarkable platform for exploring diverse exotic topological phenomena, such as Floquet photonic TIs8, quantum spin Hall photonic TIs9,10, valley Hall photonic TIs11-13, and phase change material induced Chern topological phase transition14. In addition to the intrinsic importance for fundamental physics of topological aspect, topological photonics also provides unprecedented avenues for manipulating and confining light, thus opening a paradigm for the design of advanced photonic devices particularly in topological lasers, including but not limited to quantum Hall lasers15,16, spin-momentum-locked edge state laser17, topological valley hall laser18, and the topological laser based on the bulk states of quantum spin Hall insulators19. Compared to the conventional laser architecture, topological lasers are prospected as a promising candidate for future on-chip high-performance coherent light sources with inherent robustness against imperfections and disorder20.

Recently, a class of topological insulator, termed higher-order topological insulator (HOTI), has attracted significant attention due to their ability to host lower-dimensional boundary states (e.g., corner state) that go beyond the conventional bulk-boundary correspondence21,22. This line of research began with the prediction of multipole TIs23,24, characterized by their quantized multipole moments. To obtain quadrupole TIs in two-dimensional (2D) systems that enable the emergence of highly localized 0D corner states, the critical approach is using the appropriate distribution of positive and negative couplings to introduce π-flux23,24. However, the practical implementation of this approach in photonic systems remains challenging and has so far been reported in limited platforms such as coupled ring resonators25 and waveguide arrays26. Consequently, significant research efforts have been devoted to alternative schemes in photonic systems for the quadrupole topology beyond the π-flux mechanism. Until yet, the potential for photonic quadrupole topological phases beyond the π-flux mechanism has not been fully explored, only a few approaches have been proposed, such as Floquet PhCs27, magneto-optical PhCs28,29, and twisted PhCs30. Additionally, the promising characterizations of topological corner state provide an innovative route to achieve lasing operation in an ultimate small scale with diffraction-limited mode volume (Vm) and ultra-low energy consumption. Nevertheless, the most extensively investigated higher-order topological corner state nanolasers are primarily based on Wannier-type HOTI with the expanded-shrunken scheme31-35, which is a different class of symmetry-protected topological phases, where the Wannier centers are localized at maximal Wyckoff positions by manipulating the difference between inter- and intracell couplings36,37. The quadrupole topological nanolaser has only been demonstrated in a twisted PhC platform at a cryogenic temperature, mainly hindered by the implementation scheme of the photonics quadrupole topological phase38. Naturally, we wonder if there are new possibilities for realizing the photonic quadrupole topological phase and whether it can be applied to high-performance topological nanolasers, which is fundamentally crucial for exploring intriguing topological physics within the nanoscale platforms.

In this work, we propose a defect evolution method for realizing photonic quadrupole topological phases and experimentally demonstrate its topological features within 2D photonic crystal nanolasers. Specifically, the different branches of quadrupole topology are obtained by exploiting the evolution of the introduced geometrical defects in opposite directions (clockwise and anticlockwise). The corresponding 0D corner state is predictably located at the corner of two topologically distinguished semiconductor PhC slabs in quadrupole topology. Using InGaAsP multi-quantum wells as gain materials, we further experimentally demonstrate its stable single-mode lasing operation in quadrupole higher-order topological nanolasers in telecommunication band. This work unveils a crucial method for realizing photonic quadrupole topological phase, establishing a new paradigm for the higher-order topological nanolaser design.

Results
Defect-evolved photonic quadrupole topological phase

Fig. 1a shows a schematic diagram of the considered defect evolution for the square lattice PhC, where the original PhC structure without introducing defects is commonly employed to describe the 2D Su-Schrieffer-Heeger model with the topological feature of Wannier-type HOTIs31. Here, we consider air-hole type PhC thin slab for lasing purposes. In this scenario, the original PhC unit cell with lattice parameter a consists of four square-shaped air holes with length s = 0.3a located at (± a/4 , ± a/4). Afterward, to obtain the photonic quadrupole topological phases, the notch structural defects at the diagonal position for each original air hole are introduced, and the defect evolution in this context refers to the systematic geometric adjustments of the geometrical defects for air holes. The configuration describing the initial state of this process is illustrated in the left panel of Fig. 1a, where the side lengths d and d of introduced defects at the diagonal positions are correlated through a fixed total length D = d1 + d2 = 0.5s and equal to d1 = d2 = D/2 in the initial state. A defect-related parameter Δd1 (in unit of s) is used to describe the change of the length d1 relative to theinitial state in the evolved configuration. Unlike the manipulative strategy of breathing-type PhC lattices31,39,40, the topological properties of this photonic system we proposed here are manipulated by the evolution of defects in different directions. When the commutative mirror symmetries Mx and My are removed by evolving the defect size either counterclockwise or clockwise, the system maintains glide symmetries Gx = (x, y) → (-x+a/2, y+a/2) and Gy = (x, y) → (x+a/2, -y+a/2), the inversion symmetry I = (x, y) → (-x, -y), and the C4 symmetry, providing the possibility of the quadrupole topological bandgap.

To explore quadrupole topological phases, we present the typical transverse-electric (TE) like band structure of PhC with representative clockwise evolution of defects at Δd1 = 0.25 as shown in Fig. 1b. For the identical magnitude of Δd1, the clockwise and counterclockwise configurations share same band structure due to the fact that they belong to mirror-symmetric partners. Nonetheless, the bandgap for unit cell with defects evolving in different directions possesses different topological properties. To elucidate this, Wannier band polarizations are utilized to describe their quadrupole topological phases. The j-th Wannier bands vjy(kx) can be obtained from the eigenvalues e2\piivy j(kx) of Wilson-loop operator Wk,y = \prod i= -0 1 Fk + i\Delta ky 24,41, where the subscript y indicates the considered path of the Wilson loop, and the matrix element for Fk is F nk m = \langle un,k | um,k + \Delta ky \rangle, with un,k being the periodic part of the Bloch wavefunction. Further, the polarizations of Wannier band can be determined by iteratively repeating the Wilson loop, i.e., nested Wilson loop along the orthogonal direction24, where un,k in Wilson loop is replaced by w jk,y n occ 1 un,k [ jk,y] n \xi = \sum = , which is constructed by wavefunction and n-th element [ jk,y] n \xi of the j-th Wilson-loop eigenvector. Fig. 1c shows the Wannier bands for clockwise configuration with Δd1 = 0.25. There are four gapped wannier bands labeled 1 to 4 that come in pairs above and below νy = 0, indicating the vanishing total dipole moment and providing a vital piece to the emergence of the quadrupole topological phase. Noteworthy, the rich Wannier bands here can yield different combinations. The combinations “1 + 3” and “2 + 4” are presented in Fig. 1c, exhibiting the gapped composite Wannier bands. The vanishing total dipole moment can also be captured within the framework of symmetry-indicator invariant36, i.e., comparing the symmetry representations of the mode profile for occupied bands at high symmetric points. Under the framework of symmetry-indicator invariant with C4 symmetry, bulk dipole moment is expressed as P(4) = 1/2 [ X(21) ] ( a1 + a2 ) , in which the C2 invariants 2 [X(2)]=#X(2) −#\Gamma(2) describes the difference in the number of occupied bands with a given symmetry eigenvalue for X and Γ points36, and a1 and a2 are lattice vectors. The C2 eigenvalues (±1) at the high-symmetry points are labeled in Fig. 1b. Since the zero-frequency optical mode is generally even-parity, P(4) = ( 0 , 0 ) , which is consistent with the results obtained from the Wannier band analysis. Distinguishing the quadrupole topology goes beyond the framework of the symmetry-indicator invariants and instead requires to be diagnosed through the nested Wannier bands. To further describe the topological feature for the quadrupole topology, Fig. 1d shows the ky-resolved polarizations p vx y ( ky ) of Wannier bands. For Wannier sectors “1 + 3” and “2 + 4” (orange lines in Fig. 1c), they exhibit nontrivial polarization of  1 / 2 due to the glide symmetries constraints on Wannier band polarizations (Supplementary Sec. S1). Moreover, the equivalent polarization in orthogonal direction pvx y = pvy x can be deduced via C4 symmetry, thus the quantized quadrupole moment is qxy =2pvx y pvy x =1/2 . Such quadrupole topology revealed by gapped composite Wannier bands is so-called anomalous quadrupole topology to differentiate it from conventional quadrupole topology30,42,43. Meanwhile, the nontrivial quadrupole topological phases also occur in counterclockwise configuration, but they are topologically distinguished from clockwise cases by the mutually flipped signs of the Wannier band polarization shown in Fig. 1e and indicate the phase transition at Δd1 =0. These results reveal that the evolution of defects in different directions in our scheme hosts distinct branches of quadrupole topology.

Experimental demonstration of defect-evolved quadrupole topological nanolasers

In photonic HOTIs, a finite-size photonic system constructed by two topologically distinguished PhCs is known to support the extremely localized 0D topological corner state. To investigate the feasibility of the proposed architecture for lasing emission, the defect-evolved clockwise and counterclockwise configuration PhCs are combined to form a quadrupole higher-order topological PhC nanocavity based on previous analyses. Figure 2a shows a schematic of the defect-evolved quadrupole topological nanolaser, in which the different topological regions are identified by green and purple, respectively. The device is fabricated using the 260-nm-thick slab with InGaAsP multiple quantum wells as gain medium44. The scanning electron microscope (SEM) image of the fabricated device with Δd1 = 0.25 is shown in Fig. 2b, which is composed of two topologically distinguished PhCs, with the left-bottom green region corresponding to the clockwise configuration, while the remaining region is composed of the counterclockwise counterpart that evolves in the opposite direction. Figure 2c further shows a zoomed-in SEM image around the central corner boundary region. Before numerically studying the corner state, we first verified the edge states at the topologically distinct domains in ribbon-like supercells. Figure 2d shows the projected band structure that reveals the 1D edge states (purple lines) inside the bulk gap. The field distribution for the typical lower frequency edge state is plotted in the top panel of Fig. 2d. The electromagnetic field is strongly localized at the topologically distinct domain. Topological 1D edge state lasing emission from the flat interface can be realized and has been experimentally verified, as shown in Supplementary Sec. S2. To demonstrate the existence of the corner state, Fig. 2e displays the calculated eigenfrequency spectra of the topological nanocavity for a typical magnitude of Δd1 = 0.25. A corner state as a single state (orange dot) emerges within the bandgap. The corresponding simulated electric field profile is presented in the inset of Fig. 2e. As expected, the defect-evolved higher-order corner state in the quadrupole topological phase is tightly localized around the corner region assembled by two topologically distinguished slabs. Additionally, this extreme case of defect evolution (Δd1 = 0.25) is similar to quadrupole topological structure in acoustic systems with complex arch-shaped geometry transformation42,43. Figure 2f shows the spatial Fourier transforms of the Hz field for corner state at the corresponding frequency, the wavevector component inside the light cone boundary (labeled by gray region) is negligible compared to the outside, which is concentrated on the extended Brillouin zone due to being built on the folded band. While the chiral symmetry for most higher-order topological insulators leads to the corner states pinned to zero energy, the chiral symmetry is always broken in photonic crystals, and crystalline symmetries generally do not inherently constrain the energy of corner states. In Supplementary Sec. S3, we show the robustness of the corner state against several defect types and disorders. To further analyze the confined corner state, the parametric dependence of the calculated Vm and quality factors (Q-factors) on the defect-related parameter Δd1 is shown in Fig. 2g, with Δd1 here defined from left-bottom region, while the rest changes in the opposite direction by the same magnitude. The Vm increase as the magnitude of Δd1 decreases can be attributed to the gradual narrowing of the bandgap (Supplementary Sec. S4), and gradually shows the trend of the increasing Q-factors at the expense of sacrificing the Vm. The Q-factors are on the order of 104 to near 106 with the out-of-plane loss significantly suppressed, in which the larger Q-factors for the case with expanded mode volumes can be attributed to the suppression of out-of-plane loss45, which is indicated by the smaller wavevector component in the light cone (Supplementary Sec. S5).

The lasing emission of the fabricated devices was characterized by the micro-photoluminescence (μ-PL) system at room temperature. The quadrupole topological nanolasers are optically pumped using a 632 nm pulsed laser with a repetition rate of 200 kHz and a duty cycle of 0.5%. Figure 3a presents the power-dependent spectra of a quadrupole topological PhC nanolaser with structural parameters a = 855 nm, s = 0.3a, and Δd1 = 0.25, measured by locating the optical excitation spot at the corner region (Supplementary Sec. S6). The corner state exhibits suppressed radiation loss, a high Q-factor with a small Vm, and robustness against defects and disorders, thereby enabling room temperature lasing emission in our scheme and naturally tend to stable single-mode operation. As illustrated in the figure, a sharp lasing peak (~ 1567 nm) springs from the background of spontaneous emission. The intensity ascends as the input power increases, exhibiting the stable single-mode operation over a broad range of input powers. And the collected topological corner state nanolaser is linearly polarized, illustrated by the polarization-angle-dependent output intensity (Supplementary Sec. S7). To further demonstrate the signatures of the lasing operation, the corresponding light in-light out (L-L) curve and the linewidth evolution of the device under average input power are illustrated in Fig. 3b, both an obvious kink in the L-L curve and the linewidth narrowing effect are observed, presenting clear evidences for the stimulated emission in the fabricated quadrupole topological nanolaser. A relatively low lasing threshold approximately 0.5 µW (peak power 100 µW, power density 3.18 kW/cm2) is estimated from the L-L curve, and the linewidth near the threshold of ~ 0.68 nm corresponds to an experimental Q-factor of 2300. In contrast to the simulated Q-factor, the lower experimental Q-factor may be attributed to the unavoidable fabrication induced imperfections. For the designed quadrupole topological nanolaser, the stimulated emission from the corner state can be further verified by collecting near-field optical profiles using an InGaAs camera. Figure 3c depicts the near-field optical profile measured of a fabricated corner state nanolaser measured above lasing thresholds. The optical mode is spatially localized in the central corner region, and speckle pattern occurs due to the coherent emission above the threshold. Furthermore, the thermal stability for the proposed defect-evolved topological corner state nanolaser is evaluated by placing the devices on a thermoelectric cooler (TEC) stage. The fabricated device presents stable single-mode emission as temperature increases from 30 to 70°C (Supplementary Fig. S11), suggesting the potential practical applications for the quadrupole topological nanolaser in relatively high temperature scenes. Meanwhile, the quadrupole topological nanolaser is optically pumped at room temperature with various pulse widths ranging from 50 to 150 ns (Supplementary Fig. S12). Higher lasing thresholds are observed at elevated temperatures and with longer pulse widths, mainly due to reduced gain and increased nonradiative recombination rates46. To excite the lasing from edge and bulk states, the optical excitation spot is selectively located at the corresponding edge and bulk region, respectively. Figure 3d shows the lasing spectra for different states (the near-field optical profiles and L-L curves of edge and bulk states are shown in Supplementary Sec. S9). The lasing behavior, including the anticipated topological mode selection and spectral trends, is consistent with the expected characteristics of the corner, edge, and bulk states. To further assess the underlying field profiles of the corner state, we examine the position-dependent PL intensities shown in Fig. 3e. The observed lasing mode is tightly localized in the corner region and exhibits a slower decay along the edge region direction compared to the opposite direction (the topologically distinct region), which is consistent with the simulated results. Moreover, the far-field pattern of corner state lasing emission exhibits a unique distribution, distinct from that of edge state (Supplementary Fig. S14). These results, together with the following tunable wavelength emission, unambiguously confirm the single-mode lasing operation origin from the designed topological corner state.

In addition to the realization of photonics quadrupole phase transition, the defect evolution also provides an opportunity for the topological nanolasers with spectrally tunable emission. Distinct from the previous prevalent wavelength manipulation strategy34,47, the tunable wavelength emission of quadrupole topological nanolasers here is realized only through manipulating the defect-related parameter Δd1 (Fig. 4a). To obtain a quantitative insight into the modulation of the output lasing wavelength upon the defect evolution, the variation of simulated wavelength (blue dots) of the corner state with different Δd1 is shown in Fig. 4b (geometric parameter s = 0.287a here to better match the experimental lasing wavelengths). It can be seen that the wavelength of the corner state increases with the magnitude of Δd1. For the fabricated nanolasers, the lasing wavelengths with the defect-related parameter Δd1 are also presented in Fig. 4b (orange dots). Despite irregularities and roughness of the devices induced by fabrication fluctuations, the lasing wavelengths exhibit a systematic redshift that strongly correlates with increasing Δd1, closely aligning with the simulation trend. Figure 4c further shows the corresponding normalized lasing spectra, exhibiting that the lasing wavelengths are approximately tuned from 1519 to 1543 nm, which offers a tunable wavelength range of 24 nm (The individual lasing spectra, L-L curve, and linewidth for the nanolasers with representative defect parameter Δd1 = 0.18 to 0.15 are shown in Supplementary Fig. S16). These results reveal that the geometrical defect evolution can be employed as an efficient method to achieve both precise lasing wavelength tuning of corner state and quadrupole topological phase. This approach establishes a versatile design strategy for wavelength-tunable topological nanolaser, particularly advantageous for high-throughput densely integrated photonic systems.

Methods
Device fabrication

A 260-nm-thick slab with six strained InGaAsP multi-quantum wells as the gain medium was used to fabricate the defect-evolved quadrupole topological nanolasers. First, plasma-enhanced chemical vapor deposition was employed to deposit 90-nm-thick SiO2 as hard mask. Then, the designed topological PhC structure was defined using electron beam lithography in photoresist of PMMA950 A4 and transferred to the SiO2 hard mask and further into the active layer by plasma dry etching. Afterwards, the residual resist was removed using O2 plasma, and the buffered oxide etching solution was used to remove the remaining SiO2 hard mask. Finally, the InP sacrificial layer is selectively removed by diluted hydrochloric acid solution, resulting in the suspended defect-evolved quadrupole topological nanolasers.

Optical measurement

The defect-evolved quadrupole topological nanolasers were optically pumped using a 632 nm nanosecond pulsed laser with a repetition rate of 200 kHz in the μ-PL system. The pumping beam, with a spot diameter of approximately 2 µm, was focused onto the sample using a 100× objective lens and spatially controlled by piezoelectric nanopositioners. The emission from the fabricated nanolaser was collected by the same objective lens and characterized by the spectrometer with an infrared InGaAs detector cooled by liquid nitrogen. The near-field optical images were captured by using an InGaAs camera and the far-field pattern is obtained through a 4f optical system .The time-resolved photoluminescence measurements were performed using a time-correlated single-photon counting system.

Numerical simulations

In numerical studies, we conduct 3D simulations to obtain the photonic bulk and projected band structure, eigenfrequency spectra, Q-factor, Vm, and field profile by the finite element method. In the 3D simulations, the refractive index of the gain slab is set to n = 3.25. To compute the Wannier band and nested Wannier bands, 2D simulations are performed with an effective refractive index of n = 2.6. The mode volume Vm of the topological nanocavity is calculated by Vm = \int\epsilon(r)E(r)2dV / max(\epsilon(r)E(r)2), where ε(r) and E(r) denote the spatially dependent dielectric constant and electric field intensity, respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript presents a method to generate quadrupole corner states via \"defect evolution\" and demonstrates a topological nanolaser based on these states. This manuscript is well-written and organized. While the concept of defect evolution for topological phase engineering is novel and theoretically sound, the experimental demonstration of the topological laser lacks sufficient novelty compared to prior experimental works (e.g., the authors' own study in ACS Photonics 9, 3824 (2022)). My main concern is the absence of a clear comparison between the defect-evolved quadrupole corner state and conventional higher-order topological corner states, which undermines the significance of the approach. Why is this method necessary and better than other higher-order topological lasers? The authors should clarify this concern first.\nBelow are some other minor comments:\n1. The authors define their method as \"defect evolution,\" but the final photonic crystal structure is periodic without obvious defects. It is unclear how the term \"defect\" is justified here, as the design involves systematic geometric adjustments rather than introducing disorder or missing elements. Please clarify the definition of \"defect\" in this context.\n2. The caption states, \"The gray line and yellow region represent the light line and bandgap, respectively,\" but the visual elements in Fig. 1a do not clearly align with this description. Please check.\n3. Quadrupole topological phases in magnetic photonic crystals (e.g., Nat. Commun. 11, 3119 (2020); Natl. Sci. Rev. 11, nwae121 (2024)) rely on magneto-optical effects, whereas this work achieves quadrupole topology without magnetism. What is the fundamental difference between those two methods? Please explicitly compare them.\n4. The manuscript states that Q-factors increase as the mode volume (Vm) expands, which contradicts the conventional understanding that tighter light confinement (smaller Vm) enhances Q-factors. A more detailed explanation is needed.\n5. The optical pumping setup lacks clarity such as what is the pump spot size and its spatial overlap with the corner state.\n6. In Fig. 3b, the light-in-light-out (L-L) curve exhibits an indistinct threshold kink when pumping power increases. Can the authors explain why?\n7. The temperature-dependent lasing spectra (Fig. 3e) shows no evidence of bulk modes at all. Can the authors explain why?", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "The development of quadrupole topological lasers is a timely and important research topic. The idea of defect evolution to realize quadrupole phases could potentially enrich the field of topology-based nanophotonics. However, while the lasing operation in experiment itself appears convincing, I find that the topological origin of the observed lasing remains insufficiently demonstrated. At present, the results could be interpreted as stemming from other conventional localized modes rather than quadrupole topological states.\nThe suggested structure in the manuscript still raises a specific concerns, such as chiral symmetry, robustness, and quadrupole-based lasing. Related to the comment above, I have some comments for clarification.\nComment 1.\nThe manuscript lacks a clear explanation of where the quadrupole topological mode emerges within the photonic band structure. Specifically, it is essential to identify whether the edge and corner modes lie inside or outside the light cone. Given that modes inside the light cone are susceptible to significant radiation losses, understanding their position is crucial, especially in the context of laser performance. The authors are encouraged to present the band structure overlaid with the light cone and to indicate the frequency and momentum location of the corner modes, preferably in relation to the gap illustrated in Fig. 1d.\nComment 2.\nTo verify a topological transition driven by the defect parameter Δd, it would be helpful to track the evolution of the photonic band structure as a fucntion of Δd. The authors need to perform such a scan and to visualize the modal evolution, particularly at high-symmetry points such as the M point. Including mode profiles and phase distributions (e.g., arg(E_{z})) for both trivial(supercell, Δd=0) and topological phases would help clarify whether band inversion or gap opening occurs. At present, the distinction between trivial and topological phases is not clearly visible in the data.\nComment 3.\nIt is not clear whether the simulations shown in Fig. 1b, Fig. 2d, and Fig. 2e were performed using a 2D approximation or full 3D finite-element calculations. Given the slab nature of the experimental system, a thickness is critical for accurate modeling of mode confinement, radiation losses, and Q factor.\nComment 4.\nWhile the authors mention the robustness of the corner state against disorder (see Supplementary Information), it remains unclear whether the structure indeed exhibits genuine topological protection. In many topological systems, such protection arises from underlying chiral or crystalline symmetries. The manuscript should address whether the corner states are pinned to midgap via symmetry constraints or are simply localized modes that are weakly perturbed by disorder. If topological protection is claimed, a clearer explanation of the protecting symmetry and the nature of the topological invariant is needed.\nComment 5.\nThe Q-factor plotted in Fig. 2e shows an inverse relationship with Δd₁, with the highest Q obtained as Δd approaches zero(=the trivial lattice limit). This observation raises the question whether the trivial design may, in fact, be more favorable for laser operation. If no protection mechanisms exist to ensure robustness or midgap pinning, and if the topological design results in lower Q, the motivation for using a quadrupole lattice in a laser becomes unclear. The authors should comment on this trade-off and explain whether the quadrupole configuration offers any definitive advantage for conceptual novelty.\nComment 6.\nThe manuscript lacks simulations or experimental discussion of edge states. In higher-order topological insulators, the edge states is often essential in confirming the hierarchy of bulk-edge-corner correspondence. The authors are encouraged to simulate or measure the edge configurations between topologically distinct domains. Simulations at 1D interfaces (edges) would help resolve this issue.\nComment 7.\nTo better understand the modal characteristics, I suggest the authors simulate the far-field patterns and analysis their polarization dependence for the simulated lasing mode. Quadrupole states might be expected to exhibit unique field symmetries and polarization features. These could help distinguish them from other possible defect-localized modes. Providing polarization-resolved near-field or far-field measurements would also be valuable.\nComment 8.\nRelated to the previous point, it would be helpful to show polarization-resolved near-field images of the lasing emission and to compare these with the simulated quadrupole mode profiles. I hope that such measurements would offer further insight into whether the lasing mode indeed arises from the expected quadrupole corner state. Additionally, further verification is needed to ensure that the observed lasing mode is not a fundamental mode(1st band) arising from non-topological band folding or structural asymmetry.\nComment 9.\nIt is unclear how temperature-dependent measurements(Fig. 3f, 3g) relate to the topological aspects of the design. Is there any reason to expect a topological invariant or corner-state localization to degrade (or persist) with temperature? To my knowledge, it is confusing why the authors showed the thermal dependence for quadrupole lasing operations.\nComment 10.\nFor further insight into the lasing behavior, the authors might consider varying the duty cycle of the pump pulses by increasing the pulse width at fixed repetition rate. This would help assess thermal effects, and potentially the CW response of the lasing mode. While not essential, such measurements could enhance the depth of characterization.\nWhile the reported results are promising in terms of device functionality and experimental feasibility for lasing, the connection between lasing and quadrupole topology is not sufficiently substantiated at this stage. I encourage the authors to address the above points(notably, concerning the topological origin of the lasing mode). Solving those clarifications, I will look forward a definitive demonstration for a next round of revision."}, "author_response": "We thank the reviewer for the positive recognition that our work is “well-written and organized” and that “the concept of defect evolution for topological phase engineering is novel and theoretically sound”. And we are appreciative of the informative comments and suggestions that help us to enhance the quality of this work. Regarding the concerns about the novelty of the experimental demonstration of the topological laser. We would like to clarify these from the following :\nIn our previous study in ACS Photonics 9, 3824 (2022), we demonstrate corner-state nanolasers grown on a CMOS-compatible Si platform. Such corner states are the signature characteristic of the Wannier-type HOTIs in real space, in which the topological property is generally characterized by 2D Zak phase (bulk polarization) and mainly relies on the difference between inter- and intracell couplings. Instead, the introduction of tailored defect in our work naturally produces a quadrupole band gap. For quadrupole TIs, they host vanishing bulk polarization but a quantized quadrupole moment acting as the topological invariant. Compared with the quadrupole topological phase in our work, the Wannier-type HOTIs belong to a different class of symmetry-protected topological phases, which can be seen as the 2D generalization of the SSH model with 1D Zak phase. The realization of Wannier-type HOTIs in photonic systems can be traced back to early research in microwave platforms [Phys. Rev. B 98, 205147 (2018), Phys. Rev. Lett. 122, 233903 (2019), Phys. Rev. Lett. 122, 233902 (2019)] and subsequently such a concept and manipulation strategy is used to construct topological nanocavities [Optica 6, 786-789 (2019)], thus motivating the applications in topological nanolasers based on Wannier-type HOTIs, e.g., low-threshold quantum dot corner-state nanolasers [Light Sci. Appl. 9, 109 (2020)], the lasing of hierarchical eigenstates [ACS Photonics 7, 2027 (2020)], the coupling between the corner states and their lasing action [Nat Commun 11, 5758 (2020)], and the corner state lasing based on coupled perovskite micropillars [Sci. Adv. 9, eadg4322 (2023)]. Compared to these studies, the key contribution of our work is to propose a defect evolution method for realizing photonic quadrupole topological phase transition, and the room temperature quadrupole higher-order topological nanolaser is realized for the first time under this method. We noticed that the previous corner state lasers with Q-factor limited to 103 to 104, the simulated Q-factor of the proposed nanocavity is as high as 104 to near 106 without further optimized the structural parameters of the corner, and the lasing threshold is much smaller than the reported quantum-well-based corner state lasers. In the revised version, we emphasize that this is based on different topological phases, and a table is shown below to compare with prior corner state lasers to more clearly highlight the distinctive contributions of our work.\n[... Table S1 omitted for brevity ...]\nMeanwhile, the proposed defect evolution method inspired the innovation strategy of the PhC laser-simulated novelty topological state and provides avenues for the further research of topological photonics and photonic devices. In addition to topological lasers, the proposed defect evolution method for topological quadrupole phase may find applications in nonlinear photonics and topological photonic crystal fibers [Light Sci. Appl. 9, 202 (2020)]. And one may use the similar method of defect evolution to explore a plethora of intriguing ideas for further research, which is promising in terms of device functionality and experimental feasibility for lasing, as commented by Reviewer #3.\nOn the other side, embracing distinct topological physics or the manipulation strategy of light confinement is non-trivial for nanophotonic devices. Fruitful examples include the various topological edge-state lasers based on different types of band topology and topological defect lasers with real-space topology [e.g., Science 358, 636 (2017)，Science 359, eaar4005 (2018)，Phys. Rev. Lett. 125, 013903 (2020), Nat. Commun. 12, 3434 (2021)，Nature 578, 246 (2020), Light Sci. Appl. 9, 127 (2020)，Nat. Photonics 11, 651 (2017)，Phys. Rev. Lett. 120, 113901 (2018), Light Sci. Appl. 8, 40 (2019), Nat. Photonics 16, 279 (2022), Light Sci. Appl. 12, 255 (2023), Nat. Commun. 14,707 (2023), Nat. Commun. 15, 4431 (2024)], ranging from cryogenic to room temperature, optical to electrical pumping, micro-rings/pillars to nanocavities, and have led to outstanding discoveries such as topological bulk lasers in the quantum spin Hall phase [Nat. Nanotechnol. 15, 67 (2020), Light Sci. Appl. 12, 145 (2023)] and topological valley polaritons [Nat. Nanotechnol. 19, 1283 (2024), Nat. Commun. 15,10563 (2024)]. Other than that, the introduction of the magic angle as a new strategy to manipulate light confinement in laser cavities [Nat. Nanotechnol. 16, 1099 (2021)] has also promoted the research of reconfigurable nanolaser arrays [Nature 624, 282 (2023)], atomic-scale field localization [Nature 632, 287 (2024)], cavity quantum electrodynamics [Nat. Commun. 16, 4634 (2025)], and exciton polariton condensation [Sci. Adv. 11, eadx2361 (2025)]. The above-mentioned works indicate the importance of different topological physics and manipulation strategies for nanocavity design.\nConsequently, we believe that our work provides valuable insights for the fields of topological photonics, nanophotonics, and nanolasers, as well as a possible reference approach for the community to develop further research. We hope the above explanations emphasize the novelty and impact of our work.\n1. The authors define their method as \"defect evolution,\" but the final photonic crystal structure is periodic without obvious defects. It is unclear how the term \"defect\" is justified here, as the design involves systematic geometric adjustments rather than introducing disorder or missing elements. Please clarify the definition of \"defect\" in this context.\nResponse: We thank the reviewer for the suggestion. We agree with the reviewer that the final photonic crystal structure is periodic, without disorder or missing elements in real space. The defects in this context refer to the square notch into the air hole, which is typically seen in metasurface design [e.g. Phys. Rev. Lett. 123, 253901 (2019]. To prevent potential confusion or misunderstandings, we have added the following sentences to page 3 in the revised manuscript: “the notch structural defects at the diagonal position for each original air hole are introduced, and the defect evolution in this context refers to the systematic geometric adjustments of the geometrical defects for air holes”.\n2. The caption states, \"The gray line and yellow region represent the light line and bandgap, respectively,\" but the visual elements in Fig. 1a do not clearly align with this description. Please check.\nResponse: We thank the reviewer for pointing out our omission. The error has been corrected in the revised manuscript.\n3. Quadrupole topological phases in magnetic photonic crystals (e.g., Nat. Commun. 11, 3119 (2020); Natl. Sci. Rev. 11, nwae121 (2024)) rely on magneto-optical effects, whereas this work achieves quadrupole topology without magnetism. What is the fundamental difference between those two methods? Please explicitly compare them.\nResponse: We would like to thank the reviewer for the comment. The symmetry constraints on the Wannier band polarizations are different between those two methods. For quadrupole topological phases in magnetic photonic crystals, the Wannier band polarizations are quantized by the combined operation of mirror and time-reversal symmetries M T and M T, x y which subsequently leads to the quantized quadrupole moment q xy. While the topological PhCs in our methods host quantized quadrupole moments and composite Wannier bands polarizations under the action of glide symmetries G and G . The symmetry constraints on Wannier bands and Wannier band polarizations can deduced by analyzing the transformation properties of the Wilson loop and nested Wilson loop. Consider the Wilson loop along x, the symmetry G x imposes constraints on the Wilson loop eigenvalues asei2\\piv x j(ky) G = x e i2\\pi(-v x j(ky)+ 1 2 ) , G x i.e.,vj(k )=-vj(k )+1/2 mod(1), indicating the Wannier bands come in pairs as ν (k ) and x y x y x y -ν x (k y )+1/2. Similarly, the Wilson loop eigenvalues satisfie ei2\\piv x j(ky) G = y e i2\\pi(v x j(-ky)+ 1 2 ) , i.e., G y 1 v x j(k y )=v x j(-k y )+ under the action of G y . Hence, the Wannier bands occur in ν x (k y ), 2 -ν (k )+1/2, -ν (-k ), and ν (-k )+1/2. Meanwhile, for the constraints from G and G on the x y x y x y x y Wannier bands polarizations pv x j , they obey pv1 x(k ) G = x pv x 2 (-k )+ 1 , pv3 x(k ) G = x pv x 4 (-k )+ 1 , y y x y x 2 y x y x 2 pv1 x(k ) G = y - pv x 3 (k )+ 1 , and pv3 x(k ) G = y - pv x 4 (k )+ 1 , thus provide the quantization of the y x y x 2 y x y x 2 composite Wannier bands polarizations constructed by Wannier bands “1+3” and “2+4”, namely, nontrivial polarization of±1/2presented in the main text. As demonstrated above, glide symmetry is important in the quantization of Wannier bands polarizations. To further characterize it, we consider a configuration with glide symmetry broken, as shown in Fig. R1. For this configuration, although the Wannier bands are gapped, the polarizations of Wannier sectors “1+3” and “2+4” without quantization due to the strong glide symmetry broken perturbations.\nIn the revised version, we have added the above discussion of the symmetry constraints on Wannier bands/Wannier band polarizations and the glide symmetry-broken case in the Supplementary Information (Fig. S1), and revised a sentence in the revised main text: For Wannier sectors “1 + 3” and “2 + 4” (orange lines in Fig. 1c), they exhibit nontrivial polarization of ±1/2 due to the glide symmetries constraints on Wannier band polarizations (Supplementary Sec. S1).\n4.The manuscript states that Q-factors increase as the mode volume (Vm) expands, which contradicts the conventional understanding that tighter light confinement (smaller Vm) enhances Q-factors.\nResponse: We thank the reviewer for the insightful comment. The weaker mode confinement may lead to the light leakage through the domain boundary, namely that the enhanced mode confinement can indeed have larger Q-factors. For the presented topological cavity, although the mode volume increases slightly with the decrease of the defect-dependent parameter Δd , 1 the mode size remains small compared to the finite-size footprint (~533 um2), providing sufficient mode confinement to suppress the leakage through the domain boundary. The larger Q-factors for the case with expanded mode volumes in our work can be attributed to the suppression of out-of-plane loss [Nature 425,944 (2003)], which is indicated by the smaller Fourier component in the light cone. To illustrate the influence of out-of-plane loss for Q-factors, Fig. R2 gives the spatial Fourier transformation of the H field, where the light z cone boundary is indicated by a white circle. It can be observed that the Fourier components inside the light cone boundary for the cases with Δd = 0.25 [Fig. R2(a)] are more prominent 1 compared with Δd = 0.05 [Fig. R2(b)], indicating the former hosts more out-of-plane 1 radiation loss, thus resulting in larger Q factors for the cases with Δd = 0.05 despite the 1 expanded mode volume.\nIn the revised version, we have added the above discussion in Sec. S5 of Supplementary Information (Fig. S8) to clarify the enhanced Q-factor as the mode volume expands, and the discussion is added on page 7 in the revised main text: The Q-factors are on the order of 104 to near 106 with the out-of-plane loss significantly suppressed, in which the larger Q-factors for the case with expanded mode volumes can be attributed to the suppression of out-of-plane loss45, which is indicated by the smaller Fourier component in the light cone (Supplementary Sec. S5).\n5.The optical pumping setup lacks clarity such as what is the pump spot size and its spatial overlap with the corner state.\nResponse: For greater clarity regarding the optical pumping setup, we added the optical image in the revised Supplementary Information, showing the pumping laser spot with a diameter of around 2 µm that is close to the size of spatial distribution for the corner state, and it is precisely positioned at the nanocavity by using piezoelectric nanopositioners. The information of pump spot size and Figure R3 is added in the Methods Section and Supplementary Information (Fig. S9), respectively.\n6.In Fig. 3b, the light-in-light-out (L-L) curve exhibits an indistinct threshold kink when pumping power increases. Can the authors explain why?\nResponse: We thank the reviewer for the comment. In Fig. R4, we plot the L-L curve over a larger range of input power and correct the aspect ratio of the image to exhibit a threshold kink. Moreover, we provide the lasing spectra, L-L curve and linewidth for the representative nanolaser with different magnitudes of defect parameter to further show the lasing behavior with distinct kink in the L-L curves (Fig. R5). In the revised manuscript, we revise Fig. 3a over a larger range of input power, replace Fig. 3b from Fig. R4, and add Fig. R5 in the Supplementary Information (Fig. S16).\n7.The temperature-dependent lasing spectra (Fig. 3e) shows no evidence of bulk modes at all. Can the authors explain why?\nResponse: We thank the reviewer for the comment. For temperature-dependent lasing spectra, we focused on the corner mode in which the pumping spot is spatially located at the corner site rather than the bulk region, thus only the corner state is expected to be excited. To clarify this issue, we revised the sentence on page9: Furthermore, the thermal stability for the proposed defect-evolved quadrupole topological corner state nanolaser is evaluated by placing the devices on a TEC stage. The fabricated device presents stable single mode emission as temperature increases from 30 to 70°C (Supplementary Fig. S11). And according to the comment of Reviewer #3, the discussion of temperature-dependent measurements is moved into the Supplementary Information (Sec. S8) in the revised version.\n[... Remaining responses to Reviewer 3 omitted for brevity ...]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I appreciate the authors' efforts during the revision. The previous technical concerns have been resolved. The authors have introduced a novel approach for constructing a photonic quadrupole insulator. This method does not rely on tight-binding models such as the 2D SSH model, nor does it require magnetic materials, making it highly promising for photonic applications such as topological lasers as demonstrated in this work. I recommend the manuscript for publication.", "reviewer_3": "This manuscript introduces an topologically quadrupolar design for topological nanolasers. However, I still have several key concerns:\n(1) The position of the mode relative to the light cone is not clearly resolved.\n(2) The connection between the simulated Wannier-like topology and the lasing mode remains indirect.\n(3) I am not sure that the experimental evidence does not yet clearly support the interpretation of a quadrupole HOTI mode.\nI appreciate the authors’ efforts for high quality experimental results. However, the origin of quadrupolar topology and their experimental lasing system remains insufficiently demonstrated. Below, I provide detailed comments for addressing my concerns.\nComment 1.\nMost central and critical issue is that the claimed quadrupole HOTI mode seems to form a frequencies in above-light cone regime. While Wannier-like photonic system may be allowed in theory or idealized simulations, but in practice, light cone and its position in photonic bands leads to substantial radiation loss, and limit the Q factor and making laser operation extremely challenging.\nMost successful demonstrations of photonic topological lasers operate near the fundamental band and their topology because it is clearly positioned below to the light cone. In contrast, the current design places the quadrupole mode well above the light cone at Gamma and X. This reason makes it difficult whether the observed lasing is truly from a topological mode, or a conventional cavity mode. The authors should clearly identify the wavevector and frequency of the quadrupole mode relative to the light cone in the band diagram\nComment 2.\nIf the mode lies inside the light cone, it becomes important to clarify which wavevector the red dashed line (corner state) in Fig. 1b corresponds to. Specifically, at which momentum point is the band topology formed?\nComment 3.\nAlthough robustness tests are presented (Fig. S3), the perturbations are too far from the corner. This does not adequately test whether the corner mode is topologically protected. Generally, it would be meaningful to introduce bulk disorder including or near the corner region.\nComment 4.\nAs the authors already addressed in their response, the existence of chiral symmetry in current design should be mentioned in the main text. While there are topological studies on systems with chiral symmetry, such investigations remain relevant even when the symmetry is not strictly preserved in photonic system. It is important to reflect this point in the manuscript as well. In my view, even a single sentence clarifying this in the main text would be sufficient for readers.\nComment 5.\nThe authors explain on Wannier band polarizations and nested Wilson loops to argue for quadrupole topology. I agree to this theoretical approaches to design a quadrupole in photonics and it already introduced in several times in other reports. However, in current version there still lack clear real-space verification that the experimental lasing mode corresponds to the predicted quadrupole corner state.\nComment 6.\nThe authors are encouraged to provide unsaturated real-space images of the lasing mode above threshold. To confirm that the observed lasing mode indeed corresponds to a Wannier-based quadrupole topological state, it is essential to analyze the spatial symmetry and vector-field distribution of the mode.\nCurrently, the mode profile images (Fig. 3c–e) do not provide sufficient information to assess the underlying field profiles, due to the saturation. It is not possible to analyze the in-plane E-field components, which may introduce evidence for identifying the mode’s symmetry and matching to theoretical quadrupole corner states model.\nComment 7.\nThe reported threshold (~0.5 µW) appears extremely low and raises concerns about how it was measured. Is the value based on average or peak pump power? This must be clarified in both the text and figure captions. If using average power, convert to peak power for comparison with other nanolaser works. on general understanding, a peak pump power of 0.5 µW is extremely low, and it is unlikely that suggested III–V semiconductors would respond at such levels.\nComment 8.\nI did not specifically intend to highlight the M point alone, but rather to suggest that the authors identify the conditions under which topologically quadrupole mode inversion occurs, and the corresponding wavevector where it takes place. I mentioned the M point simply as an example, because a mode inversion occurring within the light cone is more likely to result in a physically realizable lasing system.\nThe current analysis still lacks sufficient mode profile information of the unit cell as a function of wavevector. This makes it difficult to determine under what conditions the E-field distribution can be considered as “Wannier-like”.\nUpon reviewing the current data, however, the closest relevant bands seem to lie along the Γ–X direction. I still find it unclear whether the observed lasing mode genuinely originates from a topological Wannier-like system, and whether this system can be convincingly interpreted as a laser based on a quadrupole topological phase."}, "author_response": "We sincerely thank you for taking the time and effort to review our manuscript and for providing insightful comments that have helped us further improve its quality and clarity.\nWe also appreciate your positive remarks regarding our efforts to obtain high-quality experimental results. Regarding the concern raised about whether the observed lasing originates from the focusing system, we have carried out additional simulations and experiments and provided further clarifications to address this issue. Below, we respond to each comment point by point and have incorporated the corresponding revisions into the revised manuscript.\n[... Responses to comments 1-8 omitted for brevity ...]"}, {"round": 3, "reviewer_comments": {"reviewer_3": "The authors have carefully and substantially addressed my previous concerns and significantly improved the revised manuscript. In particular, the revision clarifies\n(1) the position of the lasing mode relative to the light cone,\n(2) the relationship between the Wannier-based description and the quadrupole topological phase,\n(3) the experimental evidence supporting the lasing interpretation, and\n(4) the symmetry considerations, including the role and limitations of chiral symmetry in photonic systems.\nOverall, I am satisfied that the central claims of the manuscript are now adequately supported, and I believe that the work has reached the level appropriate for publication in Nature Communications."}, "author_response": ""}]

Decision letter