Topological quantum cascade laser with Dirac-scaled valley edge modes

Domain: physics
Published: 2025-12-11
Editorial outcome: major_revision

Abstract
Topological lasers, born from the fusion of topological physics and lasing theory, offer an unprecedented mechanism to achieve coherent lasing. By utilizing topologically protected boundary states, they exhibit robustness against defects and imperfections. However, this mechanism faces a fundamental limitation: while relying on the bulk of a photonic topological insulator for topological protection, the lasing modes, defined as boundary states, must be evanescently confined to the boundary, leading to severely limited mode volume. Here, using the platform of topological quantum cascade lasers previously demonstrated with valley edge states, we demonstrate a substantial expansion of the lasing mode volume under topological protection. Unlike conventional designs, we introduce an additional bulk region featuring gapless Dirac points at the interface between topologically distinct domains. This Dirac region enables the topological lasing mode to uniformly extend across the entire region, dramatically increasing the lasing mode volume. The scaled-up mode volume enhances the robustness of the laser against defects of comparable size, outperforming unscaled designs. Our results resolve a fundamental challenge in topological laser design and also contribute to the development of cutting-edge quantum cascade lasers with enhanced efficiency and robustness.

Introduction
Topological lasers represent an emerging frontier in photonics by integrating the concepts of topological insulators and coherent photon emission1–5. In condensed matter physics, topological insulators are phases of matter in which the bulk is insulating but the edges or surfaces exhibit robust conductance due to the topological boundary states6–8. In photonics, topological lasers utilize the photonic counterparts of topological insulators, known as photonic topological insulator, to realize highly stable light generation9,10; by incorporating optical gain into those photonic topological boundary modes, one can realize robust lasing that is intrinsically protected against back-scattering from defects and disorder11–20. This unique interplay between topological physics and photon emission lies at the heart of topological lasers, opening pathways toward compact, defect-tolerant light sources with potential applications in spectroscopy, communication, and integrated photonics1,2.

Despite their technological promise, the transition of topological lasers from conceptual demonstrations to practical applications remains hindered by fundamental challenges. At the core of topological physics is the bulk–boundary correspondence principle21, which dictates that the nontrivial band topology of the bulk gives rise to topologically protected boundary states. While this principle provides robustness against scattering and fabrication imperfections, it also requires the lasing mode, as a boundary state, to be evanescently confined to the boundary. As a result, a large portion of the bulk remain ‘inactive’ for laser emission, and the limited volume of topological edge modes constrains the amount of light power that can be harvested.

Quantum cascade lasers (QCLs)22–25 are promising platforms for exploiting the topological edge mode lasing due to their semiconductor nature and compactness26. Among the various topological concepts explored for topological QCLs27–29, one of the most notable for demonstrating topological properties such as robustness to defects is the valley photonic crystal design. A valley in a band structure refers to a local minimum/maximum of an upper/lower band separated by a band gap, usually occurring at a high-symmetry point of the Brillouin zone30. The valley degree of freedom (DOF) first gained prominence in condensed matter physics in the context of valleytronics31. Valley photonic crystals were later introduced into lasers, giving rise to the first electrically pumped topological laser19. However, the pumping efficiency power was limited as only edge-modes regions were pumped. As a result, despite the intrinsic advantages offered by topological photonics for robust operation, a substantial gap remains between current topological QCL implementations and the practical requirements for THz radiation sources25.

Here, we demonstrate a topological QCL with a substantially expanded lasing mode volume. This expansion is achieved by introducing an additional bulk region exhibiting Dirac point degeneracies at the interface between two photonic domains with opposite valley-Hall (VH) topologies. Building upon the celebrated valley Hall edge states32–38, here we demonstrate their transition into ‘valley Hall bulk states’ while preserving the topological properties. The topological lasing mode is no longer a conventional topological boundary mode, but a hybridized mode that uniformly extends across the entire Dirac bulk region, which we term the Dirac-scaled topological mode. Our experimental results show regularly spaced emission peaks, consistent across multiple samples, and indicative of running wave modes in the Dirac-scaled cavity. Further cover-plate measurements reveal a uniform mode distribution throughout the Dirac bulk region, highlighting the increased mode volume enabled by our design. Finally, we assess the robustness of the Dirac-scaled design, in light of recent concerns regarding the vulnerability of VH topological waveguides to common structural imperfections39–41. By deliberately introducing multiple defects of comparable size into both the scaled and unscaled devices, we observe a clear contrast in the lasing performance: while the unscaled structure undergoes serious mode localization due to the defects, the scaled configuration maintains mode circulation, demonstrating a significant enhancement in robustness. This finding underscores the potential of Dirac-scaling to mitigate the fragility often observed in valley topological systems, further strengthening the applicability of topological photonics in real-world terahertz applications.

Results
Devicestructure
The QCL active region is fabricated into a double-metal waveguide, which supports only the transverse magnetic (TM) polarized mode due to the dipole selection rule. The photonic crystal pattern etched into the QCL consists of three distinct regions, as shown in Fig. 1a, b. The core of the scaled VH waveguide is formed using lattice unit with C6 rotational symmetric hexagonal shaped air-hole. Owing to its inversion symmetry, the scaled region hosts a pair of Dirac points at K and K’ valleys at ~3.35 THz. This region is then electrically pumped to provide gain to the guided VH modes. On either side of the waveguide, electrically isolated lattices with C3 rotational symmetry hexagonal air holes exhibit opposing valley Chern numbers of ±0.5. As a result, the waveguide supports a guided VH dispersion band, as shown in Fig. 1c, due to its nonzero valley Chern number42–44. The unit cell used in our device has a lattice constant of a = 16.77 μm. The degree of asymmetry in the hexagonally shaped gapped lattice is optimized to maximize the band gap opening, with the shorter and longer sides denoted as d1 = 2.33 μm and d2 = 11.58 μm. This parameter choice enables a band gap spanning from 3.02 to 3.45 THz.

In contrast, for the gapless Dirac lattice, the hexagonal air hole has equal-length sides with d1,2 = 7.08 μm. The unit cell parameters and structure can be found in Fig. 1a, and their respective dispersion details are presented in Supplementary Fig. 1a in the Supplementary Materials. We stress that our approach differs from earlier studies focused on homogeneous photonic crystals with Dirac dispersions45–47 or transport experiments in acoustic and microwave waveguide systems48–51. Instead, our central objective is to explore whether lasing performance can be scaled while preserving the robust characteristics afforded by topological protection.

Our VH waveguide is structured into a triangular-loop-shaped optical cavity19,52, where the sharp corners serve as a stringent test to confirm the topological characteristics of cavity modes. Unlike trivial modes, topological modes are immune to scattering at these corners53. The degree of scaling is quantified by the number of Dirac lattices spanning the waveguide width, denoted as DDirac (Fig. 1b). The total number of defects within the cavity is represented by NDefect. These defects are introduced to assess the uniformity and circulation properties of the guided mode by scattering a fraction of the in-plane radiation toward the out-of-plane Fourier transform infrared spectroscopy (FTIR) detection window. All fabricated devices maintain a consistent size for the inner region enclosed by the waveguide. As the waveguide is scaled, the increasing width of the topological waveguide extends outward toward the device boundary. The pump regions are carefully designed to cover most of the mode area while remaining confined to the perimeter of the Dirac lattices, thereby minimizing excessive heat generation and the excitation of bulk modes outside the waveguides (electrical simulation shown in Supplementary Fig. 13 in the Supplementary Materials). In unscaled devices with DDirac = 0, where Dirac lattices are not used in the waveguide, simulation results show that a width of five lattice units is sufficient to excite the valley edge mode19. In the DDirac = 3 and DDirac = 6 cavities, the pump region is slightly increased, with pump widths of six and seven lattice units, respectively, to accommodate the broadened mode area in each case.

By analyzing the measured spectral and intensity characteristics, we can evaluate the robustness of cavity modes to perturbations across different scaling designs. To facilitate localized spectral measurements at individual defect sites, metal cover plates were employed to suppress surface emissions from other regions of the cavity during experiments (schematic illustrations shown in Supplementary Fig. 5 in the Supplementary Materials). The defects were strategically spaced to minimize spectral overlap, ensuring clean and independent measurements.

Mode uniformity
A distinctive feature of Dirac-scaled VH modes, compared to other non-topological guided modes, is their relatively uniform mode envelope across the waveguide width within the region of Dirac lattices (see Fig. 1d). As such, probing the mode’s field distribution within the cavity is crucial for verification. Due to the electrical, cryogenic, and vacuum conditions required for the operation of THz QCLs, conventional near-field measurements are inaccessible. Thus, we introduce rectangular, wavelength-sized defects, randomly placed along and across the waveguide in the scaled cavity, to probe the in-plane field distribution. The homogeneous and circulating nature of the VH mode in the Dirac-scaled cavity ensures that the defects interact uniformly with the in-plane field, scattering a portion of the radiation towards free space with approximately equal intensity and spectral composition, regardless of the defect’s position within the cavity.

To experimentally verify the homogeneity of the lasing mode, a scaled VH cavity with DDirac = 6, NDefect = 3, as shown in Fig. 2a, b, was designed and fabricated. The defects were patterned into the QCL simultaneously with photonic crystal air holes using lithography and reactive ion etching (RIE) processes. The simulated normalized electric field of the VH cavity mode in this device is shown in Fig. 2c, where, except in the immediate vicinity of the defects, the VH mode remains evenly distributed without any localization in other regions of the cavity. During characterization, metal cover plates were used to block emission from the other two defects in the cavity, enabling isolated measurements from a single defect. Despite the random placement of the defects, the individually measured scattered emission, shown in Fig. 2d, exhibited nearly identical spectra and intensity. This provides direct evidence of spatial homogeneity and circulating characteristics of the lasing modes within the waveguide of the cavity.

Power scaling
A direct consequence of the enlarged lasing mode volume is the improvement of lasing power. To verify this, we fabricated and characterized three different cavity designs with DDirac = 0, 3 and 6. Surface emission was found to be weaker compared to devices with fabricated defects. This is attributed to the out-of-plane confinement provided by the under-light cone positioning of the VH modes. The simulated VH mode and device structure are shown in Fig. 3a-c, while the experimentally measured light-current-voltage (LIV) curves and maximum detected intensity of each device are presented in Fig. 3d, e. These measurements reveal a clear trend of more than a 10-fold increase in surface emission as the cavity scales from DDirac = 0 to DDirac = 6. This improved performance is directly attributed to the enhanced interaction between the scaled lasing modes and the QCL active region. As the voltage/current densities increase further (near the right edge of Fig. 3d), a decrease in the output power can be found across all the lasing devices. This phenomenon, also known as the roll-over effect, arises due to intersubband band misalignment in QCLs54.

Mode robustness in Dirac-scaled cavity
Unlike systems that break time-reversal symmetry, valley photonic crystals are inherently less robust due to the possibility of intervalley scattering caused by defects39–41. While lasing modes may remain intact with a single defect inside the cavity19, this resilience diminishes when multiple wavelength-scale defects are present. In such cases, intervalley mixing can lead to standing waves or localization within confined regions of the cavity. While the increased mode volume significantly enhances the mode’s robustness against defects along the transmission pathway, as observed in our Dirac-scaled cavity presented in Fig. 2. For controlled comparison with the DDirac = 6, NDefect = 3 cavity shown in Fig. 2, we design an unscaled cavity with DDirac = 0, NDefect = 3, as shown in Fig. 4a, b. The defect positions and dimensions are similar to those in the DDirac = 6, NDefect = 3 cavity, except that they are slightly shifted to the middle of the 1D topological waveguide, where the mode’s peak intensity is located. Simulated results for the DDirac = 0, NDefect = 3 device are presented in Fig. 4c and Supplementary Fig. 4b in Supplementary Materials. In the simulation setup shown in Supplementary Fig. 4b, c in Supplementary Materials, the VH cavity mode is excited using a directional source to observe the Poynting vector flow within the cavity. In the DDirac = 6, NDefect = 3 cavity, the Poynting vector circulate around the defects, while in the unscaled DDirac = 0, NDefect = 3 cavity, much of the flow is obstructed.

Examples of the in-gap cavity modes for the DDirac = 0, NDefect = 3 configuration are presented in Fig. 4c. The results reveal that the normalized electric field is no longer uniformly distributed and can vary in intensity across different sections of the cavity. This suggests that the defects have induced sufficient perturbation to alter the optical path of the mode. Consequently, scattered emissions from individual defects should exhibit significant variations in both spectral composition and intensity. Using the same defect emission measurement method for the DDirac = 6, NDefect = 3 cavity, the experimental results for the DDirac = 0, NDefect = 3 cavity reveal inconsistencies in both spectral characteristics and emission intensity, as shown in Fig. 4d.

In addition to comparing with previous VH topological laser designs, we further test our devices against homogeneous Dirac photonic crystals45–47. To this end, we fabricated a control set of THz QCLs with the same sandwiched structure design but with the inner triangular cavity replaced by different photonic crystals (Supplementary Fig. 6). When the Dirac material is not sandwiched between regions of opposite valley Chern number (Supplementary Fig. 6a, b), VH physics is completely removed and the dominant lasing states become localized defect modes. In this case, a similar inconsistent pattern in both spectral characteristics and emission intensity is found for the cover-plate measurements on these Dirac-bulk devices (Supplementary Fig. 6b), indicating that the lasing modes are confined between defects. This set of control experiments emphasizes that topological protection plays an essential role to the robust and extended lasing observed in our design, whereas the Dirac bulk alone does not yield comparable performance.

Methods
Device fabrication
The 12.8 μm thick QCL active region is processed into a double metal Fabry-Pérot (FP) ridge laser to evaluate its dynamic range, gain profile, and effective index of the waveguide. The growth sheet of the QCL active region could be found in Supplementary Fig. 9 in Supplementary Materials. From the fabricated ridge laser, the active region is estimated to provide sufficient gain for lasing in the range of 3–3.3 THz as shown in Supplementary Fig. 10 in Supplementary Materials, with an effective index of 4.08.

To process the QCL into double metal photonic crystal devices, a bottom electrode consisting of Ti/Au with a thickness of 20 nm/700 nm was deposited onto the QCL and N+ GaAs receiving substrate. The gold layers were then bonded together through a thermal compression process at 300 °C for 1 h. The growth substrate was subsequently lapped down to a thickness of 50 μm, with the remaining substrate removed via selective wet etching. The etch mixture consisted of deionized (DI) water and anhydrous citric acid in a mass ratio of 1:1, respectively. H2O2 was then added to the citric acid mixture at a volume ratio of 1:4, respectively. Following the removal of GaAs substrate, a 300 nm etch-stop layer composed of Al50Ga50As was removed using 49% HF acid.

The pump region was defined using 200 nm SiO2 lithography followed by fluorine-base RIE. This insulating layer reduces the device’s current requirements, helping to minimize thermal issues and prevent potential damage to the cavity. Afterwards, the top electrode and hard mask were defined through lift-off lithography, consisting of Ti/Au/Ti/SiO2 with thicknesses of 20 nm/400 nm/20 nm/900 nm, respectively. After lift-off, the electrode and hard mask layer adopted the pattern of the photonic crystal design.

The photonic crystal air hole patterns were leveled by RIE, removing 200 nm of exposed SiO2 insulation above the QCL to ensure proper leveling of the air hole patterns before transferring the photonic crystal design into the QCL using fluorine-base RIE. The process gas used for etching the QCL consists of Cl2/BCl3/CH4 at a ratio of 5:100:20 in SCCM. The etch process was terminated once the bottom electrode in the etched region was exposed. The remaining hard mask was removed using RIE. Finally, before gold wire bonding, the chip was cleaved and mounted onto a copper submount using an indium solder.

Characterization
After attaching the chip to a copper submount, the device under test is electrically connected to an electrode using wedge wire bonder. All fabricated devices are characterized within the temperature range of 8.5–9.5 K in a helium-gas-cooled cryostat. Detectable surface emission is measured from the out-of-plane direction of the photonic crystal device using Bruker Vertex 80 FTIR equipped with deuterated-triglycine sulfate (DTGS) detector. The minimal spectral resolution achievable in the wavelength of interest is 0.08 cm−1.

To power the QCL device, a rectangular pulsed electrical signal at a frequency of 10 kHz with pulse duration of 200 ns is applied. Similar to previous work19, metal cover plates are used to block unwanted surface emission from the other areas of the device, allowing for the measurement of spectral emission from a particular defect in the waveguide. These plates are positioned and adjusted in close proximity using a 3-axis translator setup mounted onto the cryostat’s cold finger where the samples are inserted.

Lasing performance for trivial Dirac-bulk devices
In addition to the Dirac-scaled VH modes, there are several other modes in our cavity which could potentially contribute to lasing. For example, finite-sized Dirac-point bulk hosts band edge mode of high Q-factor, which could lase favorably. To rule out these possibilities, we perform the following control experiments by considering two topologically trivial cavity designs which are:
1. The inner triangle cavity material is replaced with the same material as that of the outer gapped material. Valley Hall physics is completely removed as materials across the interface have the same Valley Chern number.
2. The inner triangle cavity material is replaced with the same Dirac material as that of the interface. Valley Hall physics is also completely removed as now the lasing is dominated by bulk modes closer to the Dirac point of the Dirac material.

In both cases, the dominant lasing modes are now localized modes instead of the running-wave modes for Dirac-scaled VH modes, which result in distinctive lasing behaviours among them. The detailed simulation results can be found in Supplementary Materials. Here, we summarize the experiment results and their comparison with the Dirac-scaled topological design in Supplementary Fig. 6 in the Supplementary Materials.

First, we measure the emission spectrum for these three types of cavities, as shown in the second column of Supplementary Fig. 6. Three distinctive lasing behaviors are observed: the trivial sample exhibits multimode lasing with random mode spacings, as the sandwiched cavity consists of mostly standing-wave modes now. Lasing in the Dirac-bulk sample is dominated by the band-edge mode near the Dirac point, which exhibits nearly single-mode lasing. Lastly, the Dirac-scaled cavity shows regularly spaced lasing peaks, like that being found in the main text.

Second, we consider robustness against defects for the Dirac-bulk and Dirac-scaled cavities in Supplementary Fig. 6b, c. As shown by the cover-plate measurements for the Dirac-bulk sample, most intensity is concentrated close to defect 1 and 3 and the new lasing peaks emerge in addition to the defect-free sample. In contrast, lasing from the Dirac-scaled device shows minimal change, both for the dominant lasing frequency and intensity distribution. This is direct evidence differentiating Dirac-scaled VH modes from the Dirac bulk modes, as the running-mode nature of the former modes prevents it from disorder-induced scatterings.

Last, we compare the emission power from these three designs in Supplementary Fig. 6d. Consistent with the above observations, the Dirac-scaled cavity design provides the largest output power while lasing from the other two control groups delivers far less emission power.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Report on NCOMMS-25-09265:\nThe authors investigated the topological quantum cascade laser with Dirac-scaled valley edge modes. Unlike conventional\ndesigns, they introduce an additional bulk region featuring gapless Dirac points at the interface between topologically\ndistinct domains. This Dirac region enables the topological lasing mode to uniformly extend across the entire region,\ndramatically increasing the lasing mode volume. The scaled-up mode volume enhances the robustness of the laser against\ndefects of comparable size, outperforming unscaled designs. Such topological quantum cascade laser results may have\npotential applications in quasi-quantum information processing devices.\nTopological lasers, born from the fusion of topological physics and lasing theory, offer an unprecedented mechanism to\nachieve coherent lasing. By utilizing topologically protected boundary states, they exhibit robustness against defects and\nimperfections. The manuscript reports a significant advance and offers incremental improvement to existing work in the\nreferences. The similar topological defects and edge states [PRL 122, 233905 (2019); NC 11, 1902 (2020)] in atomic and\natomic-like ensemble are studied theoretically and experimentally. The present manuscript might be of interest to a specific\ncommunity and not of the general audience. The authors should discuss the broad interesting and significant progress of the\ntopology and quantum cascade laser. The authors also should clarify the overestimated “quantum effect” [Science Advances\n10, eado3199 (2024) from time-energy-entangled W state], it might be only quasi-quantum effect ? In addition, the English\nwriting (i.e. author list “Baile Zhang” mistakes in supplemental information) and figures throughout this manuscript might be\nfurther revised to meet the standard of Nature Communications. The authors should give a brief discussion on Dirac-scaled\nvalley edge modes issue in the introduction section to highlight the significant novelty and broad impact of the current work.\nThe author’s work of the topological quantum cascade laser with Dirac-scaled valley edge modes (studied experimentally) is\ninteresting and useful. Their results resolve a fundamental challenge in topological laser design and also contribute to the\ndevelopment of cutting-edge quantum cascade lasers with enhanced efficiency and robustness. After above improvement, I\ncould recommend publication of the manuscript in Nature Communications.", "reviewer_2": "Review of Topological Quantum Cascade Laser with 1 Dirac-Scaled Valley Edge Modes by Yunda Chua et al.\nThis manuscript reports on a study demonstrating a substantial increase in lasing mode volume through the introduction of a\nDirac region, which enhances the laser's robustness against defects. While the presented concept holds significant promise,\nseveral aspects of the work require further clarification and improvement before the paper is suitable for publication in Nature\nCommunications.\nMajor Concerns:\n1. The authors state in the Introduction that “…unscaled structures exhibit significant mode localization due to defects…”.\nThis statement appears to contradict the findings in Reference 19, where a similar unscaled structure maintains robust mode\ncirculation in the presence of defects. A detailed explanation of the underlying reasons for this apparent discrepancy is\nessential.\n2. Please provide a concise explanation of the physical mechanisms responsible for the QCL active region supporting only\nthe TM mode.\n3. The paper states that the Dirac cone is formed within a C6 symmetry hexagonal air-hole structure. Given that Dirac cones\ndo not arise from arbitrary mode choices in C6 lattices, a clear and concise explanation of the specific modes involved in the\nDirac cone formation and the formation process itself is necessary.\n4. A more detailed explanation of how the pump region is defined to selectively pump only the Dirac lattices is required.\nSpecifically, the role of the SiO2 layer beneath the top Au electrode in carrier injection should be addressed, as SiO2\ntypically acts as an electrical insulator. Inclusion of the electrical simulation results for the pump region is crucial.\n5. The statement in line 132 regarding the sufficiency of a five-lattice-unit pump width for effective pumping is based on\nelectrical simulation results. These results are not included in the manuscript. The inclusion of these simulation results is\nvital for understanding the carrier injection characteristics and supporting the authors' claim.\n6. For an electrically pumped device, a detailed material stack diagram, including all doped and active semiconductor\nlayers, is essential for a comprehensive understanding of the carrier injection mechanism. The authors are strongly\nencouraged to provide this information.\n7. The measurements were performed at cryogenic temperatures. For practical applications, room-temperature operation is\nhighly desirable. Please discuss the fundamental limitations that currently prevent the device from operating at room\ntemperature.\n8. For a thorough evaluation of power scaling in electrically pumped devices, the output power should be reported in SI units\n(e.g., mW). The current presentation using arbitrary units is insufficient.\n9. The LIV curve presented in Figure 3 lacks a clear transition from spontaneous emission to lasing. A more pronounced\nthreshold characteristic is necessary to unequivocally demonstrate lasing, rather than just amplified spontaneous emission.\nThe authors must provide a more rigorous justification for claiming lasing behavior.\n10. An explanation for the observed decrease in output power in the LI curve at higher current densities is required.\n11. The slope of the IV curve is typically indicative of device resistance, which would be expected to vary with the number of\nDirac lattices. The presented IV curves, however, exhibit nearly identical slopes. An explanation for this unexpected\nbehavior is needed.\n12. The LIV curves are presented as continuous lines without explicit indication of the underlying experimental data points. If\ncurve fitting was employed, the raw experimental data should also be shown to allow for a proper assessment of the fit and\nthe data quality.\n13. Given that a pulsed source was used for the measurements, please clarify whether the reported current density values\nrepresent the average current or the peak current.\n14. To adequately assess the enhanced power output efficiency claimed in line 223, the calculation and presentation of\nwall-plug efficiency and slope efficiency are essential.\n15. The manuscript refers to simulating the Q factor in Supplementary Line 44. A plot of the simulated Q factor should be\nincluded in the Supplementary Information.\n16. Figure S2 shows single-mode lasing for unscaled cavities and multimode lasing for scaled cavities. This observation\nappears to contradict the stated claim of enhanced power scaling and also disagrees with Reference 19, where unscaled\ncavities exhibit multimode lasing. A thorough explanation of this discrepancy is required.\n17. The manuscript mentions gain competition as a factor contributing to single-mode lasing in Supplementary Line 45. A\nmore detailed explanation of this phenomenon, potentially including the gain profile of the active medium, would be\nbeneficial.\n18. The mode selection in Figure S3 appears to be largely uncontrolled and somewhat random. An explanation of the\nfactors determining which mode will lase is needed.\nMinor Concerns:\n1. The sentence \"Because of the absence of … but remained detectable\" (line 171) is unclear and should be rewritten for\nimproved readability. Consider splitting it into two sentences.\n2. The difference between the top and bottom panels of Figure S1(a) should be clearly stated.\n3. The thickness of the photonic crystal region should be explicitly stated.\n4. The phrase \"back dashed line\" in the caption of Figure 2 (line 408) should be corrected to \"black dashed line.\"\n5. Correct the typographical errors in line 265 (\"direction if the photonic …\") and in the caption of Figure S2 (currently labeled\nas S1).", "reviewer_3": "In the paper titled \"Topological Quantum Cascade Laser with Dirac-Scaled Valley Edge Mode,\" Yunda Chua et al. report the\nrealization of photonic crystal quantum cascade lasers that integrate Valley Hall concepts with open Dirac singularities.\nThey propose an original design in which the interface between two crystals with opposite\nvalley Chern numbers is replaced by a finite-thickness region of gapless Dirac material, highlighting improvements in laser intensity and robustness against defects.\nThese photonic crystals exist at the intersection of two well-established model systems:\n• Valley Hall Effect: This occurs at the interface between two time-reversal symmetric gapped Dirac materials with opposite\nvalley topological invariants. At the interface, topological edge states emerge due to the discontinuity in valley topological\ninvariants. However, these states are helical rather than chiral, as in a Chern topological insulator, and thus have limited\ntopological robustness against defects.\n• Lasing in Gapless, Time-Reversal Symmetric Dirac Materials: It has been shown that when lasing occurs near the Dirac\ncone energy, the linear dispersion leads to an extended lasing mode with highly homogeneous intensity distributed\nthroughout the entire device.\nIn this paper, the authors explore an intermediate scenario between these two models. Instead of the usual atomically abrupt\ninterface found in the Valley Hall effect, they introduce a finite-thickness region of gapless Dirac material between two\ngapped materials with opposite valley Chern numbers.\nTheir study yields two key observations:\n1. The lasing mode’s intensity is homogeneously distributed within the Dirac material region and increases with the size of\nthis region.\n2. The mode does not become localized when impurities are introduced at the edges.\nAs a counter-experiment, the authors also examine a pure Valley Hall system without any Dirac material. In this case, the\nlasing edge mode exhibits strong localization in the presence of defects.\nWhile the experimental and simulated data are well-presented and properly analyzed, I do not fully agree with the authors’\ninterpretation and find their conclusions somewhat misleading. Below, I outline my concerns:\nThe authors interpret their results as evidence of a topological lasing process, asserting that Valley Hall physics remains\ndominant, with improved robustness against defects and increased homogeneity due to the presence of the Dirac material at\nthe interface.\nHowever, I believe an alternative perspective could better explain the reported data—namely, that the observed effects are\nentirely governed by lasing near the Dirac point of the Dirac material. Homogeneous intensity distribution is a defining\ncharacteristic of lasers operating at Dirac points. Furthermore, since such lasing modes extend throughout the entire bulk of\nthe Dirac crystal, they are naturally less sensitive to defects at the boundaries.\n1) Verification That Valley Hall Physics Is Still at Play\n• Could the authors confirm that the lasing mode indeed propagates around the central triangle (as expected in the Valley\nHall effect) when Ddirac is equal to 6 or 12?\n• Is there a critical value of Ddirac beyond which the characteristic bulk lasing mode of a Dirac material (as reported in Refs.\n32–34) emerges?\n• Is there a clear crossover between Valley Hall and Dirac physics?\n2) A Crucial Missing Counter-Experiment\nA key missing experiment would be one in which Valley Hall physics is entirely removed, leaving only lasing near the Dirac\npoint. The authors could simulate and fabricate a photonic crystal quantum cascade laser in which the inner triangular\ngapped material is replaced with either Dirac material or vacuum. Important questions to address include:\n• How do the results in this configuration compare to those reported in the paper?\n• What is the output power in this setup?\n• Is the intensity distribution and defect sensitivity significantly different?\nWithout such additional analysis, I believe it is impossible to conclude that Valley Hall physics and topological lasing are\ngenuinely at play in the proposed design.\nConclusion\nI remain unconvinced by the authors' interpretation of their results. They have yet to provide definitive proof that topology\nplays a crucial role in this work.\nIn its current form, I cannot recommend this paper for publication."}, "author_response": "Reply to reviewers’ comments\nIn the text below, your comments are quoted in Italics and followed by our detailed\nresponse. These updates are also highlighted in blue in the main text. We hope that\nthe revised version of the manuscript and our responses meet the expectations of the\nreviewers.\nReviewer #1 – General Comments:\n“The author’s work of the topological quantum cascade laser with Dirac-scaled valley\nedge modes (studied experimentally) is interesting and useful. Their results resolve a\nfundamental challenge in topological laser design and also contribute to the\ndevelopment of cutting-edge quantum cascade lasers with enhanced efficiency and\nrobustness. After above improvement, I could recommend publication of the manuscript\nin Nature Communications.”\nResponse from authors:\nThank you very much for the appreciation of our work and the positive comments that\n“Their results resolve a fundamental challenge”, “contribute to the development of\ncutting-edge quantum cascade lasers with enhanced efficiency and robustness”. Below\nwe will clarify your concern point by point.\nReviewer #1 – Comments 1:\n“The authors should discuss the broad interesting and significant progress of the\ntopology and quantum cascade laser.”\nResponse from authors:\nWe thank the reviewer for bringing this up. Indeed, the reviewer is correct that aside\nfrom topological insulator lasers, many other intriguing phenomena and devices have\nbeen investigated combining topology and quantum cascade laser. One recent\ndevelopment is the study of real-space topological structures like the Dirac vortex mode\n(Nat Commun 14, 707 (2023), Light Sci Appl 12, 145 (2023), Nat Commun 15,\n4431 (2024).), which possesses single-mode lasing behavior and a nontrivial\npolarization-winding emission profile. In the revised text, we have also rewritten the\nthird introductory paragraph, covering a more complete and detailed background about\ntopology and quantum cascade laser.\nRevision:\nThe following sentences are revised from line 61 to line 73 of paragraph 3 in the main\ntext under section ‘Introduction’:\n“Quantum cascade lasers (QCLs)22-25 are promising platforms for exploiting the\ntopological edge mode lasing due to their semiconductor nature and compactness26. One\nof the most notable topological designs for QCLs is valley photonic crystals. Among the\nvarious topological concepts explored for topological QCLs27-29, one of the most notable\nfor demonstrating topological properties such as robustness to defects is the valley\nphotonic crystal design. A “valley” in a bandstructure refers to a local\nminimum/maximum of an upper/lower band separated by a bandgap, usually occurring\nat a high-symmetry point of the Brillouin zone30. The “valley” degree of freedom (DOF)\nfirst gained prominence in condensed matter physics in the context of “valleytronics”31.\nValley photonic crystals were later introduced into lasers, giving rise to the first\nelectrically pumped topological laser19. However, the pumping efficiency power was\n2\nlimited as only edge-modes regions were pumped. As a result, despite the intrinsic\nadvantages offered by topological photonics for robust operation, a substantial gap\nremains between current topological QCL implementations and the practical\nrequirements for THz radiation sources32.”\nReviewer #1 – Comments 2:\n“The authors also should clarify the overestimated “quantum effect” [Science\nAdvances 10, eado3199 (2024) from time-energy-entangled W state], it might be only\nquasi-quantum effect?”\nResponse from authors:\nWe thank the reviewer for the comment. We respectfully consider this comment to be\nirrelevant to our work. This article (Science Advances 10, eado3199 (2024)) discusses\nthe correlation time function of a three-photon state, which to our understanding is\nirrelevant to the work presented here.\nReviewer #1 – Comments 3:\n“The authors should give a brief discussion on Dirac-scaled valley edge modes issue\nin the introduction section to highlight the significant novelty and broad impact of the\ncurrent work.”\nResponse from authors:\nWe thank the reviewer for this helpful suggestion. In the revised text, we have included\nnew references on the importance of valley edge modes in the photonics community,\nwhich include both seminal works (1,2,3,4,5) and recent developments (6,7) in this field.\n1. First proposal of valley photonic crystal: New J. Phys. 18, 025012 (2016)\n2. Seminal review paper on valleytronics: Nature Reviews Materials 1.11: 1-15\n(2016). Advanced Photonics Research, 2(8), p.2100013 (2021).\n3. Robust waveguiding around path bends: Phys. Rev. B 96, 201402 (2017)\n4. Topological protected refraction: Nat. Phys. 14, 140 (2018)\n5. Robust on-chip communication: Nat. Photon. 14, 446 (2020)\n6. Routing, sorting and splitting EM waves: Nature 632, 522–527 (2024)\n7. Room-temperature polaritonic valley Hall topological insulator: Nat. Nanotech\n19, 1283–1289 (2024)\n8. Room-temperature valley exciton polaritons: Nat. Comm 15, 10563 (2024)\nRevision:\nIn addition to the brief discussion on valley photonic crystals from line 61 to line 73 in\nthe revised text (shown in response to Comment 1 above), the following sentences are\nadded from line 80 to line 82 of paragraph 4 in the main text under Introduction section:\n3\n“Here we demonstrate a topological QCL with a substantially expanded lasing mode\nvolume. This expansion is achieved by introducing an additional bulk region exhibiting\nDirac point degeneracies at the interface between two photonic domains with opposite\nvalley-Hall (VH) topologies. Building upon the celebrated valley Hall edge states33-39,\nhere we demonstrate their transition into ‘valley Hall bulk states’ while preserving the\ntopological properties. The topological lasing mode is no longer a conventional\ntopological boundary mode, but a hybridized mode that uniformly extends across the\nentire Dirac bulk region, which we term the Dirac-scaled topological mode.”\nReviewer #2 – General Comments:\n“This manuscript reports on a study demonstrating a substantial increase in lasing\nmode volume through the introduction of a Dirac region, which enhances the laser's\nrobustness against defects. While the presented concept holds significant promise,\nseveral aspects of the work require further clarification and improvement before the\npaper is suitable for publication in Nature Communications..”\nResponse from authors:\nWe sincerely appreciate your endorsement of the “significant promise” of our work\nand the constructive comments on our work. Below we will address your concern point\nby point.\nReviewer #2 – Comments 1:\n“The authors state in the Introduction that “…unscaled structures exhibit significant\nmode localization due to defects…”. This statement appears to contradict the findings\nin Reference 19, where a similar unscaled structure maintains robust mode circulation\nin the presence of defects. A detailed explanation of the underlying reasons for this\napparent discrepancy is essential.”\nResponse from authors:\nWe thank the reviewer for this insightful observation. We would like to clarify that our\nfindings do not contradict those presented in Reference 19. The key distinction lies in\nthe extent of topological protection offered by valley photonic crystals (VPCs) and the\nspecific configurations studied.\nUnlike Chern insulators which break time reversal symmetry, valley photonic crystals\n(VPC) are intrinsically reciprocal. A key signature of VPCs is its valley-dependent\nBerry curvatures, which gives nonzero valley Chern number with opposite signs at\npaired valleys. However, when a defect is incorporated in the VPCs, the topological\nmode residing at individual valley will be perturbed and scattered away from their\noriginal position in the momentum space. As a result, backscattering is generally\nallowed in VPC and could lead to localizations when disorders of sufficient strength\nare present (Nat. Photonics 17, 386 (2023), Phys. Rev. Lett. 132, 116901 (2024), Nat.\nCommun. 15, 926 (2024)).\nRegarding the apparent discrepancy with the findings present in Reference 19 in the\nmain text, we would like to point out that in their paper, the authors have only explored\nthe lasing performance under one defect. In that case, as their lasing modes compose\nequal weights of CW and CCW modes and the total path length of the triangular loop\nstays nearly constant, a single defect would not spoil the running-wave character even\nin the presence of backscattering.\nedges, as we have shown explicitly from the lasing mode profiles in Fig. R1a top row\nand Poynting vector flow in Fig. R1b. Now the backscattering is sufficient to induce\nresonance that is confined between defects.\nFig. R1 Simulated mode distribution and energy flow in cavity with defects. a, Normalized\nE-field of VH cavity modes in cavity design with 𝐷 = 0 and 𝐷 = 6 with increasing\n𝐷𝑖𝑟𝑎𝑐 𝐷𝑖𝑟𝑎𝑐\ndefect count ranging from 𝑁 = 1 to 𝑁 = 3. b-c, Simulated Poynting vector flow\n𝐷𝑒𝑓𝑒𝑐𝑡 𝐷𝑒𝑓𝑒𝑐𝑡\nrepresented by the black arrows within 𝐷 = 0, 𝑁 = 3 cavity is much more affected\n𝐷𝑖𝑟𝑎𝑐 𝐷𝑒𝑓𝑒𝑐𝑡\ncompared to 𝐷 = 6, 𝑁 = 3. The red arrow highlights the position and direction of\n𝐷𝑖𝑟𝑎𝑐 𝐷𝑒𝑓𝑒𝑐𝑡\nthe excitation source.\nRevision:\nTo address this potential source of confusion, we have included new references\ndiscussing backscattering in valley photonic crystals and revised the following\nsentences from line 197 to 202 under the first paragraph of the section ‘Mode robustness\nin Dirac-scaled cavity’:\n“Unlike systems that break time-reversal symmetry, valley photonic crystals are\ninherently less robust due to the possibility of intervalley scattering caused by defects40-\n42. While lasing modes may remain intact with a single defect inside the cavity19, this\nresilience diminishes when multiple wavelength-scale defects are present. In such cases,\nintervalley mixing can lead to standing waves or localization within confined regions of\nthe cavity. While the increased mode volume significantly enhances the mode's\n6\nrobustness against defects along the transmission pathway, as observed in our Dirac-\nscaled cavity presented in Fig. 2.”\nReviewer #2 – Comments 2:\n“Please provide a concise explanation of the physical mechanisms responsible for the\nQCL active region supporting only the TM mode.”\nResponse from authors:\nQCLs support only electric field polarized normal to the epitaxial layers (or the growth\naxis, which is along z) due to the dipole selection rule. Specifically:\n1) Intersubband transition selection rules: in QCLs, lasing occurs via intersubban\nd transitions within quantum wells. The transition dipole moments vanish in th\ne x and y directions because of translational symmetry. Only the growth directi\non, which is the z-component, remains finite because of the quantum well conf\\inement, thus allowing intersubband transitions to happen (Science 264 (5158):\n553-556 (1994)).\n2) Waveguide structure: terahertz QCLs typically employ double-metal layers for\ncurrent injection and strong optical confinement in the vertical direction. The\nelectrodes act as near-perfect electrical conducting (PEC) layers. Since the\nelectrical field parallel to the PEC surfaces must be zero, TE modes whose\nelectric fields lie entirely in the plane of the layers are suppressed due to the\nelectromagnetic boundary conditions (Optical and Quantum Electronics 47, 4,\n843-849 (2014)). In contrast, TM mode with E-field normal to PECs are allowed\nand can propagate with relatively low loss (Optics Express 14, 24 (2006)).\nRevision:\nThe response above has been incorporated into a section in the Supplementary\nMaterials, titled ‘Supported mode in the QCL active region’.\nReviewer #2 – Comments 3:\n“The paper states that the Dirac cone is formed within a C6 symmetry hexagonal air-\nhole structure. Given that Dirac cones do not arise from arbitrary mode choices in C6\nlattices, a clear and concise explanation of the specific modes involved in the Dirac\ncone formation and the formation process itself is necessary.”\nResponse from authors:\nWe appreciate the reviewer's comment regarding the formation of Dirac cones in our\nC6-symmetric photonic crystal structure. We would like to clarify here that the C6\nsymmetry hexagonal air-hole structure is the exact reason behind the formation of\nDirac cones in our system, just as how the Dirac cones are found in graphene (Rev.\nMod. Phys. 81, 109 (2009)). We feel that the reviewer may have confused symmetry-\n7\ninduced Dirac points with accidental Dirac points (like the one in Opt. Lett. 39, 2072–\n2075 (2014)), and we provide the following detailed explanations for their differences.\nIn our study, the Dirac cones emerge at the K and K′ points of the Brillouin zone due\nto the inherent symmetry of the hexagonal lattice. The photonic band structure exhibits\nlinear crossings at high symmetry points and becomes gapped once the lattice symmetry\nis broken, i.e. C6 to C3. In contrast, Dirac cones at the Γ point, which are not studied in\nthis work, result from accidental degeneracies between modes of different symmetries\nand typically require fine-tuning of structural parameters to achieve the necessary\nconditions for their formation.\nTherefore, the Dirac cones in our C6-symmetric hexagonal air-hole photonic crystal are\na direct consequence of the lattice symmetry and occur naturally at the K and K′\npoints without the need for accidental degeneracy or specific mode engineering.\nRevision:\nTo clarify this potential source of confusion, we add and change the following sentences\non line 104-107 in the first paragraph of section ‘Device structure’:\n“The QCL active region is processed into a double-metal waveguide, which supports\nonly the transverse magnetic (TM) polarized mode. The photonic crystal pattern etched\ninto the QCL comprises three distinct regions, as shown in Fig. 1a,b. The core of the\nscaled VH waveguide is formed using lattice unit with C rotational symmetric\n6\nhexagonal shaped air-hole. Owing to its inversion symmetry, the scaled section hosts a\npair of Dirac points at K and K’ valleys at ~3.35 THz. This region is electrically pumped\nto provide gain to the guided VH modes. On either side of the waveguide, electrically\nisolated lattices with C3 rotational symmetry hexagonal air holes exhibit opposing\nvalley Chern numbers of ±0.5. To show the effect of inversion-symmetry breaking\nclearly, we also add the following figures in the supplementary Materials, which show\nthe time-reversed vortex profiles near K valleys for the two C3 rotational symmetric\nhexagonal lattice, as well as the energy flow in the C6 rotational symmetric lattice.”\nthe supplementary Fig. S1 (shown as Fig. R2 below) in the Supplementary Materials\nunder the section ‘Cavity design and band dispersion’, which include the time-reversed\nvortex profiles near K valleys for the two C3 rotational symmetric hexagonal lattice,\nas well as the energy flow in the C6 rotational symmetric lattice.\nb |𝐸 𝑧 |\nMin Max\nFig. R2 Waveguide scaling simulation. a, TM dispersion of lattice unit cell used in this work.\nDispersion of Dirac lattice unit cell (top). Dispersion of gapped lattice unit cell (bottom). b,\nPlots of the absolute value of the out-of-plane electric field |Ez| (colour maps) and Poynting\nvector (white arrows) within each unit cell at the K and K′ points. The first row corresponds to\nmodes residing on the upper band, and the second row corresponds to those on the lower band\nof the bulk band structure in a. c, Dispersion of waveguide with 𝐷 = 0,3,6. Grey shaded\n𝐷𝑖𝑟𝑎𝑐\nregion represents the bandgap of the guided mode. d, Normalized E-field of VH mode across\ncavity waveguide with 𝐷 = 0,3,6. The shaded colored area indicates region covered by\n𝐷𝑖𝑟𝑎𝑐\nDirac lattices.\nReviewer #2 – Comments 4:\n“A more detailed explanation of how the pump region is defined to selectively pump\nonly the Dirac lattices is required. Specifically, the role of the SiO2 layer beneath the\ntop Au electrode in carrier injection should be addressed, as SiO2 typically acts as an\nelectrical insulator. The inclusion of the electrical simulation results for the pump\nregion is crucial.”\nResponse from authors:\nThank you for your careful observations regarding this choice of electrical insulator\nmaterial. Indeed, you are correct that the SiO layer beneath the top electrode is to\n2\nrestrict the area pumped by the top electrode, thereby reducing the total current injected\n9\ninto the device. SiO is commonly used as a passive layer for quantum cascade lasers,\n2\nespecially for emissions at shorter wavelength (Comparison across different insulating\nmaterials: Opt. Express 24, 24032-24044 (2016). Previous work using SiO2 as\nelectrical insulator: Nature 578, 246-250 (2020).). It helps suppress the heat generation\nand prevent unwanted bulk modes from lasing.\nWe also fully agree that electrical simulation results are vital in determining the actual\npump region, which suffers from the current spreading effect. Unfortunately,\nsimulations were not conducted during the design stage of the device, due to the\ndifficulty in estimating carrier concentrations in the QCL active region at operating\nconditions and temperatures. To verify this assumption, we proceed to calculate the\nseverity of current spreading through effective pump area using experimentally derived\nvalues.\nThe following are the experimental steps used to determine the effective pump area:\n1. Extract the roll-over current density from the QCL ridge laser cavity. The roll-\nover current density is used because it is independent of the Q-factor/loss of the\nlaser cavity, but the quantized energy levels of the QCL itself.\n2. Under the scenario that current spreading does not occur within the laser device,\nthe area not covered by the SiO insulation layer is exactly the effective pump\n2\narea. Multiplying this area by the roll-over current density obtained in the ridge\nlaser gives us a reference value for current that is running through the QCL at\nroll-over operating condition if no current spreading is taking place.\n3. We then characterize the LIV of our valley Hall device and acquired the actual\ncurrent that is running through the QCL at roll-over operating condition. Due to\ncurrent spreading, the experimentally acquired value is larger than the ideal\nreference value. We then fit the effective pump area using the reference value\nobtained in step 2, which allows us to obtain similar roll-over current density as\nin the ridge samples. This effective pump area provides a gauge of the severity\nof current spreading in the device.\nFollowing the procedures above, the experimental results obtained are shown in Fig.\nR3. The area of the SiO window (non-insulating area) is calculated to be 7.81x10-4\n2\ncm2, while the derived effective pumped area is 2.9x10-3 cm2. This indicates that the\nexperimental pumping region is approximately 3.6 to 3.7 times larger than the SiO\n2\nwindow, which is equivalent to a current spreading area of five lattice periods on each\nside of the domain wall. We note that this is close to previous QCL work which also\nadopted selective pumping using SiO layers (Nature 578, 246-250 (2020)), where\n2\nelectrical pumping is applied to three lattice periods on each side of the domain wall.\nfabricated using active region of the same wafer. b, LIV data of the ridge laser in a. c, Valley\nHall laser cavity fabricated in this work. d, LIV data collected from valley Hall laser cavity.\nRevision:\nTo address this point, we have amended the following sentences on line 264 in the third\nparagraph of the section ‘Device fabrication’:\n“The pump region was defined using 200 nm SiO lithography followed by fluorine-\n2\nbase RIE. This insulating layer reduces the device's current requirements, helping to\nminimize thermal issues and prevent potential damage to the cavity. Afterwards, the top\nelectrode and hardmask were defined through lift-off lithography, consisting of\nTi/Au/Ti/SiO with thicknesses of 20 nm/400 nm/20 nm /900 nm, respectively.”\nsection ‘Lasing threshold in QCLs’\nReviewer #2 – Comments 5:\n“The statement in line 132 regarding the sufficiency of a five-lattice-unit pump width\nfor effective pumping is based on electrical simulation results. These results are not\nincluded in the manuscript. The inclusion of these simulation results is vital for\nunderstanding the carrier injection characteristics and supporting the authors' claim.”\nResponse from authors:\nWe appreciate the reviewer’s suggestion. The choice of a five-lattice-unit-wide\npumping region is based on electromagnetic simulations of the edge-state field\ndistribution, rather than electrical simulations. As shown in our simulated mode profiles\non Fig. R4 (enlarged Fig. S1d in the Supplementary materials), the edge states for\n𝐷 = 0 exhibit strong confinement within approximately five lattice periods from\n𝐷𝑖𝑟𝑎𝑐\nthe domain wall. Therefore, pumping within this range is sufficient to make sure that\nthe topological modes receive the largest effective gain and lase first (Science 359,\neaar4003 (2018)).\nFig. R4 Enlarged Fig. R2d. Normalized E-field of VH mode across cavity waveguide with\n𝐷 = 0,3,6. The shaded colored area indicates region covered by Dirac lattices.\n𝐷𝑖𝑟𝑎𝑐\nRevision:\nTo avoid this potential source of confusion, we have changed the following sentences\non line 139 in the third paragraph of section ‘Device structure’:\n“All fabricated devices maintain a consistent size for the inner region surrounded by\nthe waveguide. As the waveguide is scaled, the topological waveguide extends outward\ntoward the device boundary. The pump regions are designed to cover all the mode area\nbut confined to the perimeter of the Dirac lattices to avoid exciting bulk modes outside\nthe waveguides. In unscaled devices with D = 0, where Dirac lattices are not used\nDirac\nin the waveguide construction, simulation results show that a width of five lattice units\nis sufficient to excite the valley edge mode19. In the D = 3 and D = 6 cavities, the\nDirac Dirac\npump region is only slightly increased, with pump widths of six and seven lattice units,\nrespectively, to accommodate the broadened mode area.”\nReviewer #2 – Comments 6:\n“For an electrically pumped device, a detailed material stack diagram, including all\ndoped and active semiconductor layers, is essential for a comprehensive understanding\nof the carrier injection mechanism. The authors are strongly encouraged to provide\nthis information.”\nResponse from authors:\nWe thank the reviewer for raising this concern. The detailed structure of the\nsemiconductor layers in the active region is provided below in Fig. R5. The figure\npresents the full layer sequence used in epitaxial growth, including the thickness and\ndoping concentration of all semiconductor layer. Row 9 and row 28 correspond to the\ntop and bottom N+ Ohmic contact layers, respectively. During fabrication, the top\ncontact layer (Row 9) is thinned to below 100nm to minimize lateral current spreading.\nAdditionally, all layers above Row 9 are completely removed by selective wet etching\nprocess before top metal deposition.\nFig. R5 Growth sheet of THz QCL active region used in this work.\nRevision:\nThe response above has been incorporated as Fig. S9 in the Supplementary Materials\nand mentioned on line 248 in the first paragraph of section ‘Device fabrication’.\n“The 12.8 μm thick QCL active region is processed into a double metal Fabry-Pérot\n(FP) ridge laser to evaluate its dynamic range, gain profile, and effective index of the\nwaveguide. The growth sheet of the QCL active region could be found in Fig. S9 in the\nSupplementary Materials. From the fabricated ridge laser, the active region is\nestimated to provide sufficient gain for lasing in the range of 3-3.3 THz as shown in\n(Fig. S10 a,b,), with an effective index of 4.08.”\nReviewer #2 – Comments 7:\n“The measurements were performed at cryogenic temperatures. For practical\napplications, room-temperature operation is highly desirable. Please discuss the\nfundamental limitations that currently prevent the device from operating at room\ntemperature.”\nResponse from authors:\nTHz QCLs operate based on intersubband transitions between quantized energy states\nwithin cascading quantum well. At long wavelengths, the energy separation between\nthe lasing levels becomes very small (on the order of a few meV). At room temperature\n(where kT≈25 meV), the energy separation between the quantized states is comparable\nto or even smaller than the thermal energy. As a result, thermal backfilling and phonon-\nassisted scattering significantly depopulate the upper lasing state, thereby suppressing\npopulation inversion and preventing lasing at room temperature with current QCL\ntechnologies.\nRecently, the highest reported operating temperature for a THz QCL is 261 K (−12 °C),\nachieved at a lasing frequency of approximately 4 THz (Applied Physics Letters 122,\n16 (2023)). This temperature is within the cooling capacity of thermoelectric (TEC)\ncoolers, thus significantly enhances the practical application potential of THz QCLs.\nReviewer #2 – Comments 8:\n“For a thorough evaluation of power scaling in electrically pumped devices, the output\npower should be reported in SI units (e.g., mW). The current presentation using\narbitrary units is insufficient.”\nResponse from authors:\nWe thank the reviewer for raising these issues. We have conducted the output power\nmeasurement using a terahertz power meter (Gentec-EO T-Rad with detector head of\nTHZ9B-BL-DZ-D0) and converted the arbitrary units to peak power. The updated\npower measurement results are shown in Fig. R6d below.\nin devices with D = 0, 3 and 6. The fabricated device of the respective simulation is\nDirac\npositioned on the right with zoom-in diagrams showing details of a section of its waveguide.\nd,\nExperimentally measured LIV of the respective device. e, Comparison of peak power detected\nfrom the respective devices in part d of the figure with the power detected from D = 0\nDirac\nnormalized as 1.\nRevision:\nWe have updated the LIV curves in Fig. 3d in the main text with the actual measured\npower as shown in Fig. R6d.\nReviewer #2 – Comments 9:\n“The LIV curve presented in Figure 3 lacks a clear transition from spontaneous\nemission to lasing. A more pronounced threshold characteristic is necessary to\nunequivocally demonstrate lasing, rather than just amplified spontaneous emission.\nThe authors must provide a more rigorous justification for claiming lasing behavior.”\nResponse from authors:\nWe appreciate the concern raised by the reviewer. In practice, since QCL technology is\nalready well-established, it is widely accepted in the QCL community that so long as a\nthreshold is clearly visible in the I-L curve, it is recognized as a lasing device (Appl.\nPhys. Lett. 87, 211112 (2005); Appl. Phys. Lett. 90, 091113 (2007); Nature Photonics\n3, 46–49 (2009); and Scientific Reports 6, 19325 (2016)).\nTo further support our claim, we performed additional measurements on a conventional\nridge laser fabricated from the same wafer. The structure and corresponding data are\nshown in Fig. R7a–b (same as previous Fig. R3). As illustrated, a distinct “kink”\nappears in the I–V curve (position indicated by dashed circles), which is commonly\nobserved in QCLs and is indicative of successful intersubband alignment for THz lasing.\nIt also marks the transition from spontaneous emission to stimulated emission, which\ndefines the lasing threshold in QCLs (Proc. SPIE 9382, 93821R (2015)). After that\nthreshold point, the light emitted by the device rises quickly until the roll-over point is\nreached.\nIn our valley Hall cavities, although the IV characteristics may vary in detail due to\nstructural variations, the essential lasing behaviors remain present in our current work.\nBesides, we note that the FTIR system used in our setup is equipped with a DTGS\ndetector, which lacks the sensitivity to detect spontaneous emissions. Observation of\nspontaneous emission in the THz regime typically requires a helium-cooled bolometer,\nwhich was not employed in this experiment.\nFig. R7 Comparison of LIV data from ridge lasers. a, Ridge laser fabricated using active\nregion of the same wafer. b, LIV data of the ridge laser in a. c, Valley Hall laser cavity fabricated\nin this work. d, LIV data collected from valley Hall laser cavity.\nRevision:\nWe have included Fig. R7 as Fig. S11 in the supplementary materials under the section\n‘Lasing threshold in QCLs’ and the following sentence is added to line 283 in the first\nparagraph of section ‘Characterization’:\nAfter attaching the chip to a copper submount, the device under test is electrically\nconnected to an electrode using wedge wire bonder. All fabricated devices are\ncharacterized within the temperature range of 8.5–9.5 K in a helium-gas-cooled\ncryostat. Detectable surface emission is measured from the out-of-plane direction of\nthe photonic crystal device using Bruker Vertex 80 FTIR equipped with deuterated-\ntriglycine sulfate (DTGS) detector. The minimal spectral resolution achievable in the\nwavelength of interest is 0.08 cm-1.\nReviewer #2 – Comments 10:\n“An explanation for the observed decrease in output power in the LI curve at higher\ncurrent densities is required.”\nResponse from authors:\nThe phenomenon of reduced power at higher current density is known as the “roll-over”\neffect (IEEE journal of quantum electronics 44.4: 319-323 (2008)). Such phenomena\nare unique in quantum cascade lasers (QCLs) compared to interband lasers, owing to\nthe operating principle of QCL which rely on intersubband transitions band alignment.\nAfter the roll-over point, with the increase of the pumping voltage, the carrier injection\nlevel is misaligned with the upper laser level, thus the injection efficiency of the carrier\nto the upper laser level is significantly reduced which caused the optical laser power\ngreatly reduced.\nRevision:\nWe have revised the following sentence on line 191 in the first paragraph of section\n‘Power scaling’.\nThe simulated VH mode and device structure are shown in Fig. 3a-c, while the\nexperimentally measured light-current-voltage (LIV) curves and maximum detected\nintensity of each device are presented in Fig. 3d,e. These measurements reveal a clear\ntrend of more than a 10-fold increase in surface emission as the cavity scales from\nD = 0 to D = 6. This improved performance can be easily attributed to the\nDirac Dirac\nenhanced interaction between the scaled lasing modes and the QCL active region. As\nthe voltage/current densities increase further (near the right edge of Fig. 3d), a decrease\nin the output power can be found across all the lasing devices. This phenomenon,\nalso known as the roll-over effect, arises due to intersubband band misalignment in QCLs59.\nReviewer #2 – Comments 11:\n“The slope of the IV curve is typically indicative of device resistance, which would be\nexpected to vary with the number of Dirac lattices. The presented IV curves, however,\nexhibit nearly identical slopes. An explanation for this unexpected behavior is needed.”\nResponse from authors:\nWe thank the reviewer for this concern. We would like to clarify that the IV data\npresented in the manuscript is plotted as voltage-current density J (A/cm²), where the\ncurrent has been normalized by the effective pump area of each device. As such, the\nslope of the voltage versus current density reflects the intrinsic resistance characteristics\n17\nof the QCL active region, which are primarily determined by the epitaxial layer design\nand doping level, rather than the physical cavity size or the number of Dirac lattice units.\nReviewer #2 – Comments 12:\n“The LIV curves are presented as continuous lines without explicit indication of the\nunderlying experimental data points. If curve fitting was employed, the raw\nexperimental data should also be shown to allow for a proper assessment of the fit and\nthe data quality.”\nResponse from authors:\nWe appreciate the concern raised by the reviewer. We would like to clarify that the\ncurves shown are directly plotted from raw experimental measurements, and each point\non the curve corresponds to a real, experimentally acquired data point as shown in Fig.\nR8.\nFig. R8 Plotting of LIV graph. a, Data points collected from experimental measurements. b,\nline plot with no fitting created by joining the points in a.\nRevision:\nWe have included Fig. R8 as Fig. S12 in the supplementary materials.\nReviewer #2 – Comments 13:\n“Given that a pulsed source was used for the measurements, please clarify whether the\nreported current density values represent the average current or the peak current.”\nResponse from authors:\nWe thank the reviewer for this concern. The reported current density values were\nmeasured at the top of the square-wave pulse, during the positive range of the duty\ncycle. This value represents the peak current, not the average current over the entire\npulse period. This is the standard method for IV measurements for QCLs.\nFig. R9 Oscilloscope measurement of current pulse used for pumping the valley Hall QCL\nin this experiment. The white dashed line at the plateau of the pulse indicates the value\nextracted during LIV measurement.\nReviewer #2 – Comments 14:\n“To adequately assess the enhanced power output efficiency claimed in line 223, the\ncalculation and presentation of wall-plug efficiency and slope efficiency are essential.”\nResponse from authors:\nWe thank the reviewer for this detailed comment and sincerely apologize for the\nmisleading phrasing. The term “power output efficiency” was indeed inaccurate. What\nwe intended to convey is the observation of enhanced output power, rather than an\nimprovement in efficiency metrics. This has been clarified in Fig. 3d (Fig. R6d), where\nthe trend of higher output power is evident in cavities with broader waveguide mode\ndistributions.\nRevision:\nWe have revised the corresponding statement in line 240 of the second paragraph under\nsection ‘Discussion’ to:\n“These findings underscore the potential of Dirac-scaled VH cavities to drive\nadvancements across diverse application domains, including photonic integrated\ncircuits, telecommunications, sensing, precision metrology, biomedical and healthcare\ntechnologie55-57. The enhanced power output, and robustness of these cavities are\n19\nexpected to deliver transformative advancements across these fields, effectively\naddressing challenges related to power scaling and fabrication imperfections58.”\nReviewer #2 – Comments 15:\n“The manuscript refers to simulating the Q factor in Supplementary Line 44. A plot of\nthe simulated Q factor should be included in the Supplementary Information.”\nResponse from authors:\nWe apologize for the confusion caused. The photonic cavity Q-factor is available in\nsupplementary Fig. S2a (shown as Fig. R10 below). From left to right, the plot\ncompares the Valley Hall dispersion, cavity Q-factor, and experimentally measured\nlasing mode. The red dashed line highlights the eigenfrequency region of the frequency\naxis, showcasing the progression from periodic simulation to 3D full device simulation\nand eventually the experimental results.\nFig. R10 3D Simulation and experimental results. a, Simulated dispersion of waveguide (left),\nsimulated cavity modes Q factor (middle), and experimentally detected lasing modes (right) of\n𝐷 = 6 cavity. The red dashed line indicates the equifrequency value across the three plots.\n𝐷𝑖𝑟𝑎𝑐\nRevision:\nIn the revised version we have added a figure reference to direct the reader’s attention\nto the appropriate figures. The following sentence in Supplementary line 45 of the\nsection ‘Emission spectra for devices with varying cavity sizes’ is revised:\n“Fig. S2a presents the comparison between simulated and experimental results for\ndevices with D = 6. The experimentally acquired lasing spectrum plotted towards\nDirac\nthe far right shows that the lasing mode tends towards the lower frequency end of the\nbandgap. This is expected as the simulated Q-factor of VH cavity mode peaks around\nthe mentioned region. The experimentally derived FSR of lasing peaks in device of\nD = 6 is approximately 0.03 THz (Fig. S2d), while the simulated FSR ranges from\nDirac\n0.025-0.031 THz. In this case, larger FSR values within the range are expected due to\n20\nthe lower group index of the VH waveguide dispersion towards the lower frequency end\nof the bandgap.”\nReviewer #2 – Comments 16:\n“Figure S2 shows single-mode lasing for unscaled cavities and multimode lasing for\nscaled cavities. This observation appears to contradict the stated claim of enhanced\npower scaling and also disagrees with Reference 19, where unscaled cavities exhibit\nmultimode lasing. A thorough explanation of this discrepancy is required.”\nResponse from authors:\nWe would like to clarify here that there is no contradiction between our results and\nresults in Reference 19, as there are several fundamental differences between our work\nand Reference 19:\n• The active material and active region designs used in Zeng’s work are different to\nthose used in the current work.\n• The emission spectrum reported in Zeng’s work demonstrates clear multi-mode lasing\nmostly for the cavity with defects. This is due to the “poor vertical outcoupling” of the\nvalley edge modes below the light cone. In our case, since our emphasis is not on\ndemonstrating the established valley topological lasers, we have not added these defects\nto enhance the scattering of other modes in our unscaled cavity.\n• Instead, in our work we compared devices fabricated with similar active regions to\nensure a fair and meaningful comparison. As a result, all our measured emission spectra\nin Figure S2 were obtained without the inclusion of scatters or defects to ensure a fair\ncomparison.\nReviewer #2 – Comments 17:\n“The manuscript mentions gain competition as a factor contributing to single-mode\nlasing in Supplementary Line 45. A more detailed explanation of this phenomenon,\npotentially including the gain profile of the active medium, would be beneficial. The\nmode selection in Figure S3 appears to be largely uncontrolled and somewhat random.\nAn explanation of the factors determining which mode will lase is needed.”\nResponse from authors:\nWe thank you for bringing up this point, which has not been thoroughly discussed.\nThe\nresultant lasing profile in a multimode laser is often complicated and could be affected\nby several mechanisms.\nWe would first present a typical lasing spectrum of a ridge laser across its dynamic\nrange as shown in Fig. R11a. Ridge laser cavity relies on facet reflection and does not\nhave wavelength selective mechanism; hence, its performance is a good indication of\nits gain profile and lasing range. Here, two phenomena are observed. Firstly, the lasing\npeaks exhibit varying intensity as the pump current varies, in some cases (520A/cm2 in\ngain spectrum shifts with the increasing pump current, is known as stark shift, which is\ndue to shifting of the quantized states in the QCL due to variation of electric fields.\nSecondly, the lasing dynamics are subjected to gain competition and possibly some\ncontribution of material absorption. Overall, results from the ridge laser cavity provide\na good indication of its lasing range of frequency which we match them to the valley\nHall modes through simulation.\nFor the case of valley Hall laser cavities, we expect these factors which cause unequal\nlasing intensities in ridge laser cavities to play out more aggressively due to higher\nstructural complexities and non-uniform Q-factor distribution. This may cause certain\ncavity modes to appear weaker or unable to reach the lasing threshold, forming non-\nuniform lasing intensity/FSR at certain operating conditions.\nFig. R11 Operating conditions and gain profile. a, Lasing spectrum of ridge laser fabricated\nusing similar active region. The lasing range blueshift at higher pump current density. b, Lasing\nspectrum of our Dirac-scaled devices. The grey region signifies the lasing range of the ridge\nlaser shown in Fig. R11a under 450 A/cm2.\nRevision:\nWe have added Fig. R11 as Fig. S10 in the supplementary materials and we have revised\nthe following sentence on line 249 in the first paragraph of section ‘Device fabrication’:\n“The 12.8 μm thick QCL active region is processed into a double metal Fabry-Pérot\n(FP) ridge laser to evaluate its dynamic range, gain profile, and effective index of the\nwaveguide. The growth sheet of the QCL active region could be found in Fig. S9. From\nthe fabricated ridge laser, the active region is estimated to provide sufficient gain for\nlasing in the range of 3-3.3 THz as shown in Fig. S10 in the supplementary materials,\nwith an effective index of 4.08.”\nReviewer #2 – Comments 18:\n1. The sentence \"Because of the absence of … but remained detectable\" (line 171) is\nunclear and should be rewritten for improved readability. Consider splitting it into two\nsentences.\n2. The difference between the top and bottom panels of Figure S1(a) should be clearly\nstated.\n3. The thickness of the photonic crystal region should be explicitly stated.\n4. The phrase \"back dashed line\" in the caption of Figure 2 (line 408) should be\ncorrected to \"black dashed line.\"\n5. Correct the typographical errors in line 265 (\"direction if the photonic …\") and in\nthe caption of Figure S2 (currently labeled as S1).\nResponse from authors:\nWe are deeply sorry for these mistakes and thank you for your close attention to the\ndetails of our work. We have made amendments to these errors accordingly.\nResponse to Reviewer #3\nReviewer #3 – General Comments:\n23\n“In the paper titled \"Topological Quantum Cascade Laser with Dirac-Scaled Valley\nEdge Mode,\" Yunda Chua et al. report the realization of photonic crystal quantum\ncascade lasers that integrate Valley Hall concepts with open Dirac singularities. They\npropose an original design in which the interface between two crystals with opposite\nvalley Chern numbers is replaced by a finite-thickness region of gapless Dirac material,\nhighlighting improvements in laser intensity and robustness against defects.\nThese photonic crystals exist at the intersection of two well-established model systems:\n• Valley Hall Effect: This occurs at the interface between two time-reversal symmetric\ngapped Dirac materials with opposite valley topological invariants. At the interface,\ntopological edge states emerge due to the discontinuity in valley topological invariants.\nHowever, these states are helical rather than chiral, as in a Chern topological insulator,\nand thus have limited topological robustness against defects.\n• Lasing in Gapless, Time-Reversal Symmetric Dirac Materials: It has been shown that\nwhen lasing occurs near the Dirac cone energy, the linear dispersion leads to an\nextended lasing mode with highly homogeneous intensity distributed throughout the\nentire device.”\nResponse from authors:\nWe thank you for your thorough review and accurate summary of our paper. Following\nthe constructive suggestions provided, we have performed additional simulations and\nexperiments to better address the Dirac physics present in our system, which were not\nsufficiently discussed in the previous version. Below, we provide point-by-point\nresponses to your comments.\nReviewer #3 – Comments 1:\n“In this paper, the authors explore an intermediate scenario between these two models.\nInstead of the usual atomically abrupt interface found in the Valley Hall effect, they\nintroduce a finite-thickness region of gapless Dirac material between two gapped\nmaterials with opposite valley Chern numbers. Their study yields two key observations:\n1) The lasing mode’s intensity is homogeneously distributed within the Dirac material\nregion and increases with the size of this region. 2) The mode does not become localized\nwhen impurities are introduced at the edges.”\nResponse from authors:\nWe thank the reviewer for the careful reading of our manuscript, and we fully agree\nwith these two observations.\nReviewer #3 – Comments 2:\n24\n“The authors interpret their results as evidence of a topological lasing process,\nasserting that Valley Hall physics remains dominant, with improved robustness against\ndefects and increased homogeneity due to the presence of the Dirac material at the\ninterface. However, I believe an alternative perspective could better explain the\nreported data namely, that the observed effects are entirely governed by lasing near the\nDirac point of the Dirac material. Homogeneous intensity distribution is a defining\ncharacteristic of lasers operating at Dirac points. Furthermore, since such lasing\nmodes extend throughout the entire bulk of the Dirac crystal, they are naturally less\nsensitive to defects at the boundaries.”\nResponse from authors:\nWe thank the reviewer for the insightful comments regarding the potential contributions\nfrom bulk modes of the Dirac material, a point that was indeed not adequately discussed\nin the previous manuscript version. To address this question, we have conducted a series\nof new experiments and simulations. Our observations support our claims in the original\nmanuscript that the robustness of our laser comes mainly from the topological valley\nedge states. In the following, we will provide a detailed response to this comment.\nIn general, lasing modes in our Dirac-scaled cavity design can be put into two categories\ndepending on their frequencies: the topological modes which lie in the band gap region\n(shaded area in Fig. R12b) and bulk modes of high quality factors, which typically\nreside near the band edges. While both modes are extended, their transport properties\nare drastically different: the topological mode exhibits clear momentum-valley locking\nwithin the band gap, which corresponds to running-wave mode localized at the\nextended interface (Fig. R12c). As a result, such mode has regularly spaced emission\npeaks with spacing Δk=2π/L (L is the circumference of the cavity loop).\nIn contrast, as the band edge modes are localized modes, their mode spacing follows a\ndifferent trend, which scales as Δk≈2π/D (D is the total area of the cavity loop)\nand is\nmuch smaller compared with that of the running wave mode. Moreover, modes other\nthan the fundamental mode generally exhibit more localized field distributions and\nsignificantly lower Q-factors, as discussed in other Dirac bulk laser works (Nature 608,\n692–698 (2022)). Therefore, since the bulk modes in Dirac crystals have closer mode\nspacing and larger intensity variations, we expect Dirac crystals to be more prone to\ndefects compared with our design. structural complexities and non-uniform Q-factor\ndistribution. This may cause certain cavity modes to appear weaker or unable to reach\nthe lasing threshold, forming non-uniform lasing intensity/FSR at certain operating\nconditions.\n25\nb |𝐸 𝑧 |\nMin Max\nFig. R12 Waveguide scaling simulation. a, TM dispersion of lattice unit cell used in this work.\nDispersion of Dirac lattice unit cell (top). Dispersion of gapped lattice unit cell (bottom). b,\nPlots of the absolute value of the out-of-plane electric field |Ez| (colour maps) and Poynting\nvector (white arrows) within each unit cell at the K and K′ points. c, Dispersion of waveguide\nwith 𝐷 = 0,3,6. Grey shaded region represents the bandgap of the guided mode. d,\n𝐷𝑖𝑟𝑎𝑐\nNormalized E-field of VH mode across cavity waveguide with 𝐷 = 0,3,6. The shaded\n𝐷𝑖𝑟𝑎𝑐\ncolored area indicates region covered by Dirac lattices.\nTo directly validate this argument, we conduct a new set of experiments as well as finite\ncavity simulations for three distinctive cavity configurations. We consider three\ndistinctive cavity designs, which are:\n1. The inner triangle cavity material is replaced with the same material as that of the\nouter gapped material. Valley Hall physics is completely removed as materials across\nthe interface have the same Valley Chern number.\n2. The inner triangle cavity material is replaced with the same Dirac material as that of\nthe interface. Valley Hall physics is also completely removed as now the lasing is\ndominated by bulk modes closer to the Dirac point of the Dirac material.\n3. Proposed Dirac-scaled cavity design.\nThe simulation results are summarized in Fig. R13 and the corresponding experimental\nresults are shown in Fig. R14.\nillustration for the trivial, Dirac-bulk material and topological arrangements, respectively. The\nfield profiles corresponding to the mode of maximum Q-factors are shown, both for the case of\nwithout defect (middle column) and with defect (right column). d, IPR comparison for models\nwithout defects. e, same as above with defects. The dashed line indicates the frequency of mode\nwith largest quality factor for the Dirac-bulk cavity (~3.32 THz) or topological cavity (~3.38\nTHz).\nIn the trivial configuration (Fig. R13a), where the Dirac material is sandwiched between\nregions with the same valley Chern number, clear standing-wave patterns are observed\n(Fig. R13a middle column). In contrast, when one region is replaced by either an\nopposite Chern number region (Fig. R13c) or by the Dirac material itself (Fig. R13b),\nthe resulting eigenmodes exhibit nearly uniform intensity distributions within the\nconfinement region. However, here we would like to emphasize that different from\nprevious work which considered Dirac point at Γ point, our lasing mode is near K/K’\npoint. As a result, the field distribution of our fundamental mode is less uniform and\nexhibits periodic modulations at wavelengths corresponding to lattice periodicity, as\nshown in the middle panel of Fig.R13b. This is also reflected in the inverse participation\nratio (IPR) calculation for the cavities’ eigenmodes in Fig.R13d, for which we find that\n1) the bulk Dirac mode is much more spread compared with our guided mode and\ndemonstrates an IPR closer to that of the modes of trivial configuration. 2) modes in\nthe Dirac-bulk cavity are spaced much closer, just as we discussed above.\nTo further distinguish between guided modes and bulk Dirac modes, we introduce an\nidentical defect (dark rectangle) into each cavity and examine the resulting spectrum\nand mode distributions. As illustrated in the right column of Figs. R13a-c, both\ntopological and Dirac bulk modes show considerable robustness against defects due to\ntheir extended nature. However, significant differences arise in their mode spacing\npatterns (Fig. R13e). The topological cavity maintains a nearly constant free spectral\nrange (FSR) with minimal frequency shifts around the mode with the highest Q-factor.\nWhile the degenerate CW and CCW modes are now separated due to the defect, in\npractice a single steady-state lasing mode is spontaneously chosen from one of the two\npossible solutions (Nature 578, 246-250 (2020)). In contrast, the Dirac-bulk material\n27\ndisplays a more irregular mode pattern upon defect introduction. This irregularity likely\narises from the lasing mode’s interactions with higher-order bulk modes that are closer\nand more localized, thus becoming more susceptible to defects.\nFig. R14 Comparison of lasing characteristics among trivial cavity, Dirac-bulk cavity and\nDirac-scaled cavity designs. a, Trivial cavity design with D =6, random lasing peaks are\nDirac\nobserved without clear FSR. b, Dirac-bulk cavity design by replacing the inner triangle in the\nfirst row with Dirac lattices. Nearly single-mode lasing behavior is observed. However, the\nlasing modes exhibit large intensity variations inside the cavity when defects are introduced. c,\nTopological cavity design with D =6 and, D =6, N = 3. Both variations demonstrate\nDirac Dirac Defect\ncharacteristics of valley Hall topological mode such as orderly FSR or circulating mode\nproperties.\nWe confirm the simulation results in Fig. R13 by two sets of experiments onto the\nrespective cavity designs. First, we measure the emission spectrum for these three types\nof cavities, as shown in the second column of Fig. R14. Three distinctive lasing\nbehaviors are observed: the trivial sample exhibits multimode lasing with random mode\nspacings, the Dirac-bulk sample exhibits nearly single-mode lasing, and our Dirac-\nscaled cavity shows regularly-spaced lasing peaks, similar to what was shown in the\nmain text. Second, we consider robustness against defects for the second and third\ndesigns (last two columns of Fig. R14). As shown by the cover-plate measurements for\nthe Dirac-bulk sample, most intensity is concentrated close to defect 1 and 3 and the\nnew lasing peaks emerge in addition to the defect-free sample. In contrast, lasing from\nthe Dirac-scaled device shows minimal change, both for the dominant lasing frequency\nand intensity distribution. Last, we check the output power from these three designs in\nFig. R14d. Consistent with the above observations, the Dirac-scaled cavity design\nprovides the largest output power in spite of its smallest mode volume as shown Fig.\nR13.\nRevision:\nWe have added a section in the supplementary materials titled “Comparison among\ndifferent cavity designs” containing the discussion above.\nReviewer #3 – Comments 3:\n“Could the authors confirm that the lasing mode indeed propagates around the central\ntriangle (as expected in the Valley Hall effect) when Ddirac is equal to 6 or 12?”\nResponse from authors:\nWe thank the reviewer for raising this important point. A clear signature of the running-\nwave modes circulating around the triangular cavity is the regular spacing of the lasing\nmodes, analogous to whispering-gallery modes in ring cavities. Indeed, this\ncharacteristic is directly visible in our experimental emission spectra (Fig. R15), where\nwe compare the results for both defect-containing and defect-free samples. In both cases,\nwe observe regularly spaced emission peaks, and the average free spectral range (FSR)\naligns closely with predictions from our eigenmode simulations. These observations\nconfirm that the lasing mode indeed circulates around the central triangle as anticipated\nfrom the Valley Hall effect. In contrast, higher-order modes in a uniform Dirac material\nare unlikely to lase due to their lower Q-factor and would exhibit a different FSR, as\nwe have discussed.\nFig. R15. Emission spectrum of a Dirac-scaled lasing device with corner defect. a, Device\nimage of D = 6, N = 1 cavity with corner defect. b, Magnified image for a clearer view\nDirac Defect\nof the defect site within the device. c, Normalized E-field of D = 6, N = 1 corner defect\nDirac Defect\n29\ncavity compared with D = 6 cavity. d, Experimentally measured lasing peaks from both\nDirac\ndevices are found to exhibit uniform FSR, which align relatively well despite the inclusion of\nthe corner defect.\nRevision:\nWe have added a section in the supplementary materials titled “Robustness against\ndefects” containing the discussion above.\nReviewer #3 – Comments 4:\n“Is there a critical value of D beyond which the characteristic bulk lasing mode of\na Dirac material (as reported in Refs. 32–34) emerges? Is there a clear crossover\nbetween Valley Hall and Dirac physics?”\nResponse from authors:\nWe thank the reviewer for this insightful question. Indeed, the bulk Dirac mode and\nguided mode coexist in our samples, and their respective lasing behaviors depend\ncritically on their Q-factors. To explore this systematically, we perform simulations on\ntriangular cavities by gradually increasing D while keeping the circumference of\nDirac\nthe outer triangle constant, effectively transitioning our structure from a heterostructure\ninto a uniformly Dirac-bulk configuration, as illustrated in Fig. R16.\nNoticeably, a direct crossover can be found at D =7, at which the Q-factor of the\nDirac\nbulk mode takes over that of the bandgap mode. Beyond this point, as D is further\nDirac\nincreased, significant coupling via evanescent waves between adjacent cavity arms\nenhances the quality factor of the bulk Dirac mode, making it eventually dominating\nthe lasing process. Nevertheless, for practical purposes this can always be prevented by\nenlarging the inner triangle cavity’s sizes, which suppresses couplings in between the\nneighboring cavity arms.\nDirac\ntransitioning process. b, Simulated results for the maximum Q-factors found within the bandgap\nand bulk, respectively. c,d, corresponding field distribution for the bandgap mode and bulk\nmode shown in b, with D Dirac =6,7,8 from top to bottom.\nRevision:\nWe have added a section in the supplementary materials titled “Crossover from\ntopological lasing modes to band edge modes” containing the discussion above.\nReviewer #3 – Comments 5:\n“A key missing experiment would be one in which Valley Hall physics is entirely\nremoved, leaving only lasing near the Dirac point. The authors could simulate and\nfabricate a photonic crystal quantum cascade laser in which the inner triangular\ngapped material is replaced with either Dirac material or vacuum. Important questions\nto address include:\n• How do the results in this configuration compare to those reported in the paper?\n• What is the output power in this setup?\n• Is the intensity distribution and defect sensitivity significantly different?”\nResponse from authors:\nWe thank the reviewer for this helpful advice. A summary of the measurement results\nis provided in Fig. R14. We have studied key features including both the emission\nspectrum, defect sensitivity and emission power for devices in which the inner\ntriangular gapped material is replaced with either Dirac material (Fig. R14b) or gapped\nmaterial of the same valley Chern number (Fig. R14a). Both the simulation results and\nexperimental observations favor our proposed Dirac-scaled laser design (Fig. R14c).:\namong the three candidates, it shows the most stable and robust lasing performance.\nRevision:\nWe have added a section in the supplementary materials titled “Comparison among\ndifferent cavity designs” containing the discussion above.\nIn the text below, your comments are quoted in Italics and followed by our detailed\nresponse. These updates are also highlighted in blue in the main text. We hope that\nthe revised version of the manuscript and our responses meet the expectations of the\nreviewers.\nResponse to Reviewer #1\nReviewer #1 – General Comment:\n“Now, I recommend publication of the manuscript in Nature Communications.\nResponse from authors:\nWe would like to thank the reviewer for the time and effort dedicated to evaluating our\nmanuscript and for the recommendation for publication.\nResponse to Reviewer #2\nReviewer #2 – General Comments:\n“The majority of the points have been addressed by the authors. I would still ask the\nfollowing clarifications by accepting the manuscript.”\nResponse from authors:\nWe sincerely thank the reviewer for your time and effort in carefully reading our\nmanuscript and providing valuable feedback to improve it. Below, we will address your\nconcern point by point.\nReviewer #2 – Comments 1:\n“Could the authors provide at least a simplified 2D electrical stimulation of current\nspreading to strengthen their experimental estimation?”\nResponse from authors:\nWe thank the reviewer for this concern. We performed a 3D electrical simulation to\nanalyze the current spreading in our device. The simulation results are shown below,\nwhich agrees well with our experimental estimations.\nRevision:\nIn the revised text, we have included additional discussion in supplementary materials\nunder “Supplemental Figure 12” and referred it directly on line 135 under the section\n‘Device structure’ in the main text.\n“To evaluate the current spreading within the device, we performed electrical\nsimulations using the “Electric Currents” module in COMSOL Multiphysics. The QCL\nlayers were modeled as thin film structures according to their differential\nresistance/conductivity extracted from the linear region of the IV curve1.\nThe QCL stack was represented by two distinct layers with different conductivities: a\ntop 600-nm-thick n+ GaAs ohmic contact layer with a doping concentration of 3.5x1018\ncm-3 (conductivity denoted as 𝜎 ), and a 12.2-um-thick active region with\n𝑛+\nconductivity denoted as 𝜎 .\n𝐴𝑅\nTo determine 𝜎 , we performed a parametric fit between the simulated and\n𝐴𝑅\nexperimental IV characteristics of a reference double-metal ridge laser. In this\nconfiguration, both the top and bottom electrode fully cover the active region with\nminimized lateral current spreading. A parametric simulation is performed based on a\n3D ridge structure to capture the IV characteristics under various voltage bias, and the\nIV slope was matched to experimental data to extract 𝜎 . This method provides a\n𝐴𝑅\ngood estimation of 𝜎 as the magnitude of 𝜎 is expected to be much higher than\n𝐴𝑅 𝑛+\n𝜎 . As a result, when stacked in series, the total device conductivity is primarily\n𝐴𝑅\ngoverned by 𝜎 and will be insensitive to a large variation of 𝜎 . An approximate\n𝐴𝑅 𝑛+\n𝜎 value of 1250 S/cm is adopted2,3, and the simulated results of the ridge laser fits its\n𝑛+\nexperimental IV well when 𝜎 = 0.046 S/cm.\n𝐴𝑅\nUsing these parameters, we performed 3D electrical simulation in our topological laser\ndevice. The simulated current values agree well with our experimental results as shown\nin fig. S12. The gradient of the simulated IV is slightly less than the gradient of\nexperimental IV data. We attribute the slight mismatch of the simulated IV and the\nexperimental values to parasitic resistance between the metal-semiconductor interfaces.\nFig. S12 3D Electrical Simulation of laser cavity D =6 a. Simulated results\nDirac\nillustrating current density within the device at 16V. b. Experimental LIV (black solid\nlines) vs Simulated IV plot (red dashed line). c. Cutline across waveguide structure\n(black dashed line) where current density is evaluated. d. Current density evaluated\nacross the cutline in c. at various operating voltages. The spacings in the solid lines\nwhere no data is shown represent air in the photonic crystal where electrical current is\nnot present.\nReviewer #2 – Comments 2:\n“Please clarify more explicitly why the unscaled cavity shows single-mode lasing in\nyour experiment but multimode in Ref. 19. Could fabrication quality or coupling\ngeometry explain this?”\nResponse from authors:\nWe thank the reviewer for this keen observation. Indeed, this lasing difference from\nRef. 19 arises primarily from our different coupling geometries.\nIn our measurements (Fig. S2), we did not introduce outcoupling-enhancing defects, so\nthe unscaled cavity’s multiple lasing modes, which are located below the light cone\nwith poor vertical outcoupling, have very low detectable emissions. Consequently, at\nlow current (430 A/cm2) these modes are buried in the noise floor and remain\nindistinguishable. The single-mode-like behavior observed at higher currents are likely\ndue to gain competition or scattering from fabrication imperfections.\nReviewer #2 – Comments 3:\n“Even if wall-plug efficiency is low, reporting estimated values (with caveats) would\nallow readers to benchmark performance. Can you provide approximate slope\nefficiency and wall-plug efficiency?”\nResponse from authors:\nWe thank the reviewer for this insightful suggestion and for the interest in the\nquantitative aspects of our work. We calculated the wall plug efficiency 𝑛 and the\n𝑤𝑝\nslope efficiency 𝑛 in our supplemental materials under the section “Slope efficiency\n𝑠\nand wall-plug efficiency”, the results are detailed below.\nRevision:\nThe optical and electrical characteristics of the QCL devices were characterized under\npulsed operation (10 kHz repetition rate, 200 ns pulse width, 1% duty cycle) at a heat-\nsink temperature of 10 K. The output power was measured using a calibrated DTGS\ndetector.\nThe slope efficiency 𝑛 was determined from a linear fit to the L–I-V curve in the\n𝑠\nregime above the lasing threshold:\nd𝑃\no𝑢𝑡\n𝑛 =\n𝑠 d𝐼\nThe maximum wall-plug efficiency 𝑛 was calculated using the measured peak\n𝑤𝑝\ncurrent (I ), peak voltage (V ), and peak output power (P ):\npeak peak peak\n𝑃\no𝑢𝑡,𝑝𝑒𝑎𝑘\n𝑛 =\n𝑤𝑝 V ×𝐼\n𝑝𝑒𝑎𝑘 𝑝𝑒𝑎𝑘\nIn the present work, the slope efficiency of the device with D of 0, 3, 6 layers were\nDirac\nestimated to be 0.7 mW/A, 2.42 mW/A and 3.07 mW/A respectively. And the\ncorresponding peak wall-plug efficiency were calculated to be 5.09E-5 %, 8.83E-5 %\nand 1.77E-4 % respectively. We attribute the low wall-plug efficiency (WPE) to the\npoor surface-emission characteristics of guided topological edge modes located below\nthe light cone.\nReviewer #2 – Comments 4:\n“The new control experiments are valuable, but please emphasize more explicitly in\nthe Discussion why your results cannot be solely attributed to Dirac-bulk effects.”\nResponse from authors:\nWe thank the reviewer for this suggestion. We have now moved these new control\nexperiments’ results and corresponding discussions to the main text.\nRevision:\nWe have included the new control experiments as Extended Data Fig. 1 under the\nMethods section titled ‘Lasing performance for trivial Dirac-bulk devices’ from line\n303 and added a new paragraph on line 220 following the first paragraph under the\nsection ‘Mode robustness in Dirac-scaled cavity’:\n“In addition to comparing with previous VH topological laser designs, we further test\nour devices against homogeneous Dirac photonic crystals. To this end, we fabricated a\ncontrol set of THz QCLs with the same sandwiched structure design but with the inner\ntriangular cavity replaced by different photonic crystals (Extended Data Fig. 1). When\nthe Dirac material is not sandwiched between regions of opposite valley Chern number\n(Extended Data Fig. 1a,b), VH physics is completely removed and the dominant lasing\nstates become localized defect modes. In this case, a similar inconsistent pattern in\nboth spectral characteristics and emission intensity is found for the cover-plate\nmeasurements on these Dirac-bulk devices (Extended Data Fig. 1b), indicating that the\nlasing modes are confined between defects. This set of control experiments emphasizes\nthat topological protection plays an essential role to the robust and extended lasing\nobserved in our design, whereas the Dirac bulk alone does not yield comparable\nperformance.”\nResponse to Reviewer #3\nReviewer #3 – General Comments:\n“I am very satisfied with the precise and rigorous answers the authors provide to my\nquestions. Additional experiments realized by the authors now fully support the interest\nof the newly designed photonic crystal cavities and the interpretations proposed by the\nauthors. To my opinion there are crucial to assess scientific validity of the work.”\nResponse from authors:\nWe sincerely thank the reviewer for your time and effort in carefully reading our\nmanuscript and providing valuable feedback to improve it. We are grateful for your\nhigh evaluation of our work.\nReviewer #3 – Comments 1:\n“I regret that at least part of these new data were neither incorporated nor discussed\nin the main text. The novel sections the authors have added in the supplementary\nmaterial are not even cited in the main text. I think comparing the different cavity\ngeometries and also showing the different modes separated by the FSR of the ring cavity\n(they are not so nicely resolved in the figure of the main text) is of crucial importance\nto support the main claims of the paper.”\nResponse from authors:\nWe thank the reviewer for raising this important point. We have now added the control\nexperiments’ results as Extended Data Fig. 1 and mentioned it explicitly in the main\ntext.\nRevision:\nWe have included the new control experiments as Extended Data Fig. 1 under the\nMethods section titled ‘Lasing performance for trivial Dirac-bulk devices’ from line\n303 and added the following paragraph on line 220 following the first paragraph under the\nsection ‘Mode robustness in Dirac-scaled cavity’:\n“In addition to comparing with previous VH topological laser designs, we further test\nour devices against homogeneous Dirac photonic crystals. To this end, we fabricated a\ncontrol set of THz QCLs with the same sandwiched structure design but with the inner\ntriangular cavity replaced by different photonic crystals (Extended Data Fig. 1). When\nthe Dirac material is not sandwiched between regions of opposite valley Chern number\n(Extended Data Fig. 1a,b), VH physics is completely removed and the dominant lasing\nstates become localized defect modes. In this case, a similar inconsistent pattern in\nboth spectral characteristics and emission intensity is found for the cover-plate\nmeasurements on these Dirac-bulk devices (Extended Data Fig. 1b), indicating that the\nlasing modes are confined between defects. This set of control experiments emphasizes\nthat topological protection plays an essential role to the robust and extended lasing\nobserved in our design, whereas the Dirac bulk alone does not yield comparable\nperformance.”\nReviewer #3 – Comments 2:\n“I am now in favor of publication of the present work but would strongly suggest adding\nthese highly relevant features to the main text (at least refer to them explicitly with\nproper physical discussion).”\nResponse from authors:\nWe would like to express our sincere gratitude to the reviewer for the recommendation.\nThese control experiments’ results have now been incorporated and referred directly\nwithin the main text.\nReferences\n1. Yousefvand, Hossein Reza. \"Equivalent circuit-level model of quantum cascade\nlasers with integrated hot-electron and hot-phonon effects.\" Quantum Electronics vol.\n47, no. 11: 992 (2017).\n2. Schlecht, Erich, et al. \"Design considerations for heavily-doped cryogenic\nSchottky diode varactor multipliers.\" Proceedings of the Twenlfth International\nSymposium on Space Terahertz Technology. (2001).\n3. Sapteka, Anak Agung Ngurah Gde, Anak Agung Ngurah Made Narottama, and\nKadek Amerta Yasa. \"Modelling of Phosphorus and Boron Doping Concentration on\nSOI Wafer Based Diffusion Process.\" Logic: Jurnal Rancang Bangun dan Teknologi\n20.1: 53-58 (2020)."}]

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