Realization of a Dirac-vortex topological photonic crystal fiber

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

Abstract
Photonic crystal fibers (PCFs) that trap and guide light using photonic bandgaps have revolutionized modern optics with enormous scientific innovations and technological applications spanning many disciplines. Recently, inspired by the discovery of topological phases of matter, Dirac-vortex topological PCFs have been theoretically proposed with intriguing topological properties and unprecedented opportunities in optical fiber communications. However, due to the substantial challenges of fabrication and characterization, experimental demonstration of Dirac-vortex topological PCFs has thus far remained elusive. Here, we report the experimental realization of a Dirac-vortex topological PCF using the standard stack-and-draw fabrication process with silica glass capillaries. Moreover, we experimentally observe that the Dirac-vortex single-polarization single-mode is bound to and propagates along the fiber core in the full communication window (1260-1675 nm). Our study pushes the research frontier of PCFs and provides a new avenue to enhance their performance and functionality.

Introduction
Since their discovery in 1991, photonic crystal fibers (PCFs) that use photonic bandgaps to confine and guide light have attracted great attention across many disciplines due to their outstanding ability to overcome the limitations of conventional optical fibers, such as bending-insensitive low-loss guidance of light, design flexibility of fiber cross-sections, and high-power light delivery. Recently, inspired by the advancements in topological photonics, the concept of topological PCFs with robust topological protection has been theoretically proposed based on different physical mechanisms and experimentally demonstrated in a multicore PCF whose cores are arranged in a Su–Schrieffer–Heeger chain with topological end states, significantly improving the performance and functionality of conventional PCFs. The unique properties of topological PCFs, such as the single-polarization single-mode, azimuthally and radially polarized orbital angular momentum (OAM) and near-zero flattened dispersion, exhibit promising potential applications in next-generation robust quantum networks and optical communications. In particular, Dirac-vortex topological PCFs, originating from Dirac-vortex states in a Kekulé-distorted honeycomb lattice, have been theoretically proposed with ultra-broadband single-polarization single-mode light transport or arbitrary degenerate fiber modes. However, due to the challenge of sample fabrication and experimental characterization, Dirac-vortex topological PCFs have thus far remained out of experimental reach.

Here, we fabricated and experimentally demonstrated, for the first time to our best knowledge, the Dirac-vortex topological PCF using the standard stack-and-draw process with silica glass tubes. In particular, we directly observed that the Dirac-vortex fiber mode bounds to and propagates along the fiber core, with its unique intensity profile and polarization distribution consistent with the theoretical results in a wide wavelength range (1260–1675 nm). Moreover, we experimentally explored the influence of different incident light beams on the fiber output, exhibiting excellent and robust single-polarization single-mode property of the Dirac-vortex topological PCF.

Results
We begin with a common silica PCF with perfect triangular lattice of air holes and introduce Kekulé distortion to the thickness of the silica struts in the cross section to construct a Dirac-vortex topological PCF. Figure 1b shows a unit cell of the cross section of the perfect triangular PCF with a lattice constant of 3a (a=3 µm) and a silica strut width of t0=0.16a. The Kekulé distortion is introduced by changing the strut width with Dirac mass m=m0eiφ, where m0 is the amplitude of the Dirac mass, φ=ωθ(r) is the vortex phase angle, θ(r) is the space phase angle, and ω is the vortex winding number. After Kekulé distortion, the strut width becomes t1=t0+m0cos(φ), t2=t0+m0cos(φ+2π/3), t3=t0+m0cos(φ+4π/3), and t=t0+m0cos(φ+4π/3) (green color), where t0=0.16a and winding number ω=1. When the Dirac mass m0=0 (without Kekulé distortion), the bulk band structure of the perfect triangular PCF (blue lines) exhibits a double Dirac point with fourfold degeneracy at the Γ point with nonzero wave vector kz. When m0≠0 (m0=0.024a) and φ=π/3 (with Kekulé distortion), the double Dirac point with fourfold degeneracy will be lifted (gray lines), opening a complete topological photonic bandgap at kz=4π/a, as shown in Fig. 1d. More significantly, though the topological photonic bandgap range has a minor regular fluctuation from 186.26 to 189.08 THz with different φ, it persists for arbitrary phase of φ if m0≠0, as shown in Fig. 1e, in which the color represents the phase φ varying from 0 to 2π. We then arrange a series of Kekulé-distorted triangular photonic crystals with six different modulation phases φ angularly around the fiber core to form a Dirac-vortex topological PCF. As a result, a single Dirac-vortex topological PCF mode appears in the middle of the bandgap, being tightly localized and propagating along the fiber core with a nonzero wave vector kz.

Figure 2a shows the cross section of the Dirac-vortex topological PCF with space-dependent six different modulation phases φ and fixed m0=0.024a angularly around the fiber core. From the fiber dispersions with absolute (Fig. 2b) or relative frequency (Fig. 2c), we can observe a single Dirac-vortex topological PCF mode (red line) in a broad frequency range of 164–317 THz (946–1829 nm) located in the topological photonic bandgap induced by the Kekulé distortion. Besides the nontrivial Dirac-vortex topological PCF mode, there also exist trivial index-guided modes (blue line), which usually appear in pairs with different polarizations and are confined wherever there is a local maximum of the strut thickness (equivalent to a high effective refractive index). Here, we only focus on the Dirac-vortex fiber modes inside the topological bandgap, as shown in Fig. 2d, whose field intensity (red color) and electric vector (black arrows) distribution are tightly localized around the fiber core with a C3v symmetry, similar to an azimuthally polarized beam (APB). To facilitate the fabrication of the Dirac-vortex topological PCF, we round off the six sharp corners of the hexagon with a radius of r. As we increase the radius r, the topological bandgap (the red region represents the minimal bandgap range of φ=0) gradually decreases but always exists as long as the hexagonal air holes do not transform into circular air holes. Figure 2f shows the eigenmode spectrum of a Dirac-vortex topological PCF with a radius r=0.3a, from which we can see a Dirac-vortex topological PCF mode (red dot) exists in the topological bandgap, whose field intensity is almost the same as that in Fig. 2d.

Next, we fabricate the Dirac-vortex topological PCF using the standard stack-and-draw technique. We first stack four kinds of silica glass capillaries with the same outer diameter d and different inner diameters to form the fiber preform. The combination of silica glass capillaries with inner diameters d1 and d2 (d3 and d4) can be used to construct the unit cell with modulation phase φ=0, 2π/3, 4π/3 (φ=π/3, π, 5π/3), respectively, which consist of struts with thickness t0±m0/2 (t0±m0), as illustrated in Fig. 1f. Then we draw the stacked silica glass capillary structure using a fiber drawing tower to produce a uniformly sized Dirac-vortex topological PCF. The difference between the actual and theoretical geometrical parameters is less than 0.1 μm, and the fabricated fiber exhibits the unique properties of Dirac-vortex topological PCF, indicating the stack-and-draw technique is stable and suitable to fabricate the Dirac-vortex topological PCF.

We then experimentally characterize the fabricated Dirac-vortex topological PCF. The simplified diagram of the experimental setup is illustrated in Fig. 3b. The light manipulating module consists of a Sagnac interferometer with a spatial light modulator to generate arbitrarily shaped scalar and vectorial modes as the incident beam. The imaging module, composed of an illuminating source and an imaging system, is placed along the optical field coupling path to monitor the fiber input facet. The observing module is adopted to map the polarization-intensity patterns of the output beam of the Dirac-vortex topological PCF. To match the topological mode supported by the Dirac-vortex fiber, we first use an APB beam created by the light manipulating module as the input beam at 1550 nm, as shown in the left panel of Fig. 3c, which is carefully aligned to the fiber core with beam size reduction by a long focal lens combined with an objective lens to match the effective mode field area of the Dirac-vortex topological PCF. By rotating the polarizer (white arrows), the intensity patterns of the APB turn into two symmetric petals whose orientation is altered with different polarizations, as shown in the right panel of Fig. 3c. The measured intensity pattern of the output Dirac-vortex topological PCF mode is shown in the left panel of Fig. 3d, matching well with the simulation results shown in Fig. 2d, f. When the output Dirac-vortex topological PCF modes are passed through different oriented polarizers (white arrows), the intensity patterns of the Dirac-vortex fiber modes exhibit similar distributions with those of the APB, verifying the unique vectorial nature of the Dirac-vortex topological PCF modes.

To investigate the broadband single-polarization single-mode property of the Dirac-vortex topological PCF, three different input beams, including linearly polarized (LP) Gaussian beam, circularly polarized (CP) Gaussian beam, and the first-order OAM+1 mode, are coupled into the Dirac-vortex topological PCF, respectively, as shown in the left panel of Fig. 4a. The corresponding measured output intensity patterns without and with different oriented polarizer (white arrows) are presented in the right panel of Fig. 4a. It can be seen that the output fields remain the Dirac-vortex fiber mode even under different incident beams. Figure 4b shows the measured relative output power at various wavelengths under different incident beams. For each incident beam, the power variation is about 1 dB, and the measured transmission loss and bending loss of APB remain relatively stable, indicating that the Dirac-vortex topological PCF exhibits a broadband and robust performance across the full communication window (1260–1675 nm). Moreover, the output powers of the Dirac-vortex fiber under APB (blue line) and OAM+1 (red line) beams are much higher than those under CP (brown line) and LP (green line) Gaussian beams, indicating that the more closely the input beam aligns with the topological Dirac-vortex fiber mode, the higher the output power. It is worth mentioning that, although the APB and OAM+1 beams both have similar doughnut intensity patterns, the output power of the Dirac-vortex fiber under APB (blue line) is higher than that under OAM+1 (red line). This can be explained by the fact that the APB beam with azimuthal polarization distribution is closer to the topological Dirac-vortex fiber mode. To further demonstrate the broadband single-polarization single-mode characteristic of the Dirac-vortex topological PCF, we also plot the measured output optical fields without and with different oriented polarizer (white arrows) at different wavelengths, as shown in Fig. 4c. We observe that the output topological fiber modes keep almost the same across the full communication window (1260–1675 nm), verifying that the Dirac-vortex fiber maintains its topological properties over a broad wavelength range. It is worth noting that the measured 415 nm wavelength range (1260–1675 nm) is mainly based on the current available lab conditions. From the trend of the curves in Fig. 4b, there should be a larger bandwidth of the Dirac-vortex topological PCF, as predicted by theory, indicating its superior performance in an ultra-broad wavelength range.

Methods
Numerical simulations
All numerical results presented in this work are simulated using the RF module of COMSOL Multiphysics. In the simulation, since the Dirac-vortex topological photonic crystal is continuous in the z-direction and has no periodicity, all simulations are performed using a two-dimensional structure. By setting a nonzero kz, the transmission characteristics of its Dirac-vortex mode in the z-direction are obtained. The bulk band structures are calculated using a unit cell with periodic boundary conditions in the x and y directions. The Dirac-vortex mode in the Dirac-vortex topological photonic crystal fiber (Dirac-vortex topological PCF) is calculated using a supercell with perfectly matched layers that are applied around the supercell. Meanwhile, by varying the kz, the relationship between the frequency of the Dirac-vortex mode and the propagation constant kz can be derived.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this paper, topological photonic crystal fiber which support, Dirac-vortex single-polarization single-mode, is proposed.\nThe structure is simple and the proposed PCF is actually fabricated.\nThe output field for three different incidence mode is measured\nand the results show that the stable vortex beam is observed irrespective to the incidence mode.\nTherefore, Dirac-vortex single-polarization single-mode transmission seems to be achieved in the proposed fiber.\nThe results is very interesting to a lot of researchers in this field. I have some comments as follows:\n1) First of all, the quality of figures (resolution) is very low and is improved before publication.\nEspecially, Figure 2 should be drawn in more high resolution.\nGray branch exit in this figure. Dose gray branch mean m_0 = 0 ?\nThe structure is difficult to be recognized.\n2) The dispersion relations are written in Fig. 1D for with/without distortion.\nHowever, the parameter is just written m_0 ≠ 0.\nThe actual value should be given. Is the value of m_0 = 0.024a ?\nIn addition, this figure may be drawn for φ = π based on Fig. 1F.\nHowever, it is written that φ = π/3 in the main text. Which is correct ?\n3) Dispersion relations for other phase, φ= 0, π/3, 2π/3 should be provided.\n4) Please explain Fig. 1E in more detail. Does the bandgap width remain unchanged?\nIf so, please explain briefly why the bandgap width remain unchanged.", "reviewer_2": "In this article, the authors have experimentally demonstrated Dirac-vortex topological photonic crystal fiber (PCF) for the first\ntime. Using stack-and-draw technique with silica glass capillaries, a Dirac-vortex topological PCF has been fabricated,\nwhich exhibits Dirac-vortex single mode guiding in the full communication window from 1260 nm to 1675 nm. This work has\nthe land-marking significance in topological PCF technique, so that the results are very interesting and meet the requirement\nfor the journal. However, the following issues should be resolved for acceptance in line with high standard of the journal."}, "author_response": "Manuscript ID: NCOMMS-25-27489\nTitle: Realization of a Dirac-vortex topological photonic crystal fiber\nAuthor: Quanhao Niu, Bei Yan, Lei Shen, Hao Lin, Xi Zhang, Zhenyu Wan, Mutian Xu, Hui\nZhang, Jie Luo, Lei Zhang, Perry Ping Shum, Zhen Gao, Jian Wang\nDear reviewers of Nature Communications,\nWe have carefully revised our paper (Manuscript ID: NCOMMS-25-27489) according to the\ncomments of editors and reviewers. Enclosed please find the detailed response. We would\nlike to thank you for the valuable comments and helpful suggestions. In the text below, all\nreviewers’ comments are quoted in italics and are followed by our corresponding detailed\nresponse, and the revised edition in manuscript is marked in blue. We also thoroughly revise\nthe manuscript and the supplementary information according to all Reviewers' comments, and\nthese updates are highlighted in yellow in those files.\nSincerely yours,\nQuanhao Niu, Bei Yan, Lei Shen, Hao Lin, Xi Zhang, Zhenyu Wan, Mutian Xu, Hui Zhang, Jie\nLuo, Lei Zhang, Perry Ping Shum, Zhen Gao, Jian Wang\n******************************************************************************************\nResponse to the comments of the reviewers\n******************************************************************************************\nReviewer 1– Comment 1\nIn this paper, topological photonic crystal fiber which support, Dirac-vortex single-polarization single-\nmode, is proposed.\nThe structure is simple and the proposed PCF is actually fabricated.\nThe output field for three different incidence mode is measured\nand the results show that the stable vortex beam is observed irrespective to the incidence mode.\nTherefore, Dirac-vortex single-polarization single-mode transmission seems to be achieved in the\nproposed fiber.\nThe results is very interesting to a lot of researchers in this field. I have some comments as follows:\n1) First of all, the quality of figures (resolution) is very low and is improved before publication.\nEspecially, Figure 2 should be drawn in more high resolution.\nGray branch exit in this figure. Dose gray branch mean m_0 = 0 ?\nThe structure is difficult to be recognized.\nResponse from Authors:\nWe thank the reviewer for the valuable comments and helpful suggestions. In the revised manuscript,\nwe completely replace the figures with high-definition version and ensure a clear vision for a detailed\nview. We also adjust the color illustration in Fig. 1 and Fig. 2. Now gray lines are unified to illustrate\nthe bulk band with Kekulé distortion (in Fig. 1d, Fig. 2b&2c&2f), and the topological photonic bandgap\ncolor in Fig. 2e uses the minimal bandgap’s color which are mentioned in Fig. 1e so that the difference\nof described objects can be clearly distinguished via figure review.\nFig. 2 | Dirac-vortex topological PCF. a Schematic of the Dirac-vortex topological PCF cross-section.\nThe inset color bar represents the phase distribution of the Kekulé modulation. b, c Simulated fiber\ndispersion with absolute (b) and relative (c) frequencies, and the red (blue) line represents the Dirac-\nvortex fiber (index-guided) modes. The intensity (red color) and electric vector (black arrows)\ndistributions of the index-guided modes are shown in the inset of b. d Simulated intensity (red color)\nand electric vector (black arrows) distributions of the topological Dirac-vortex fiber modes at k = 4π/a\nz\nwith the refractive index of silica equal to 1.4491 (ε = 2.1). e Topological Dirac-vortex bandgap as a\nfunction of the radius r of the rounded corners at k = 4π/a. Inset: a unit cell with r = 0.3a and t = 0.16a.\nz 0\nf Simulated eigenstate spectrum of the Dirac-vortex topological PCF with r = 0.3a. Inset: simulated\nintensity (red color) and electric vector (black arrows) distributions of the topological Dirac-vortex fiber\nmodes with r = 0.3a at k = 4π/a.\nz\n(Please see pages 12, line 345-348 in the revision)\nThe above discussions are appropriately added in the revised manuscript.\nAs we increase the radius r, the topological bandgap (the red region represents the minimal bandgap\nrange of φ=0) gradually decreases but always exists as long as the hexagonal air holes do not transform\ninto circular air holes.\n(Please see pages 4, line 106-108 in the revision)\nComment 2\n2) The dispersion relations are written in Fig. 1D for with/without distortion.\nHowever, the parameter is just written m_0 ≠ 0.\nThe actual value should be given. Is the value of m_0 = 0.024a ?\nIn addition, this figure may be drawn for φ = π based on Fig. 1F.\nHowever, it is written that φ = π/3 in the main text. Which is correct ?\nResponse from Authors:\nWe thank the reviewer for the professional comments. Here we unify the example illustration on Fig.1d\nwith the definite legend of m =0.024a instead of vague image of m ≠0 and accurate unit cell figure at φ\n0 0\n= π/3.\nThe above discussions are appropriately added in the revised manuscript.\nWhen the Dirac mass m = 0 (without Kekulé distortion), the bulk band structure of the the perfect\n0\ntriangular PCF (blue lines) exhibits a double Dirac point with fourfold degeneracy at the Γ point with\nnonzero wave vector k . When m ≠ 0 (m = 0.024) and φ = π/3 (with Kekulé distortion), the degeneracy\nz 0 0\nwill be lifted (grey lines), opening a complete topological photonic bandgap at k = 4π/a (see the band\nz\ninversion and Wilson loop in the Supplementary Note 1), as shown in Fig. 1d.\n(Please see pages 3, line 78-84 in the revision)\nFig. 1 | Designing a Dirac-vortex topological PCF by introducing Kekulé distortion in a triangular\nPCF. a Schematic of the Dirac-vortex topological PCF. b, c The unit cell of the cross-section of a\ntriangular PCF without (b) and with (c) Kekulé distortion. The Kekulé distortion is introduced by\nchanging the strut width with m and phase φ, where t = t + m cos(φ), t = t + m cos(φ+2π/3), t = t +\n0 1 0 0 2 0 0 3 0\nm cos(φ+4π/3), t = 0.16a, and a = 3µm. d Simulated bulk band structures of the cross section of the\n0 0\ntriangular PCF without (blue lines) and with (grey lines) φ=π/3 Kekulé distortion at k = 4π/a,\nz\nrespectively. e Vortex bandgaps with fixed Dirac mass m = 0.024a and different phases φ at k = 4π/a.\n0 z\nThe color represents the phase φ varying from 0 to 2π. f Schematic of the Dirac-vortex topological PCF\nconsisting of six aperiodic Kekulé-distorted triangular photonic crystals with a winding number of +1.\nThe color of the struts represents their widths.\n(Please see page 11, line 332-334 in the revision)\nComment 3\n3) Dispersion relations for other phase, φ= 0, π/3, 2π/3 should be provided.\nResponse from Authors:\nThe authors thank the reviewer for the professional comments. We print all six unit cells (φ= 0, π/3,\n2π/3, π, 4π/3 and 5π/3) in Fig. R1, respectively. The range of topological photonic bandgap is marked\nin the middle of figures. It can be seen that different unit cells have different bandgap ranges, when φ =\n0, 2π/3 and 4π/3, the topological photonic bandgap range is within [186.27, 189.03]THz, and when φ =\nπ/3, π and 5π/3, the range is within [186.26, 189.08]THz. Each unit cell’s topological photonic bandgap,\ntogether with its different phase, is colored with the relative rainbow colorbar shown in Fig. 1e.\nFig. R1. Topological photonic bandgap with φ= 0, π/3, 2π/3, π, 4π/3 and 5π/3.\nComment 4\n4) Please explain Fig. 1E in more detail. Does the bandgap width remain unchanged?\nIf so, please explain briefly why the bandgap width remain unchanged\nResponse from Authors:\nWe thank the reviewer for the helpful suggestions. As the previous reply of discussion of the relationship\nbetween the phase and topological photonic bandgap, different unit cells have different bandgap ranges\nso that there are range fluctuation in Fig.1e’s bandgap. The updated high-quality version of the figure\ncan also help to clearly see the slight fluctuation in Fig.1e.\nThe above discussions about comment 3&4 are appropriately added in the revised manuscript.\nMore significantly, though the topological photonic bandgap range has a minor regular fluctuation from\n186.26 to 189.08 THz with different φ, it persists for arbitrary phase of φ if m ≠0, as shown in Fig. 1e,\n0\nin which the color represents the phase φ varying from 0 to 2π.\n(Please see page 3, line 84-87 in the revision)\nReviewer 2 – Comment 1\nIn this article, the authors have experimentally demonstrated Dirac-vortex topological photonic crystal\nfiber (PCF) for the first time. Using stack-and-draw technique with silica glass capillaries, a Dirac-\nvortex topological PCF has been fabricated, which exhibits Dirac-vortex single mode guiding in the\nfull communication window from 1260 nm to 1675 nm. This work has the land-marking significance in\ntopological PCF technique, so that the results are very interesting and meet the requirement for the\njournal. However, the following issues should be resolved for acceptance in line with high standard of the\njournal.\n- My main concern is the comparison of experimental data with theoretical prediction. In the initial part\nof the Results section, they presented dispersion and showed the existence of topological Dirac-vortex\nguiding mode. Theoretical results are, however, presented separately from the experimental ones to\nsome extent.\nCorrespondence between the designed and fabricated fibers is not clear. The topological property of\nfabricated fiber shown in Fig. 3 should be discussed theoretically: the best way is to numerically\nevaluate the topological invariant (here, winding number) and the topological Dirac-vortex mode\ndirectly by using the cross-section of fabricated fiber, but not with schematic shown in extended data\nFig. 3. In such a way, the readers can be convinced that the finally obtained fiber is genuinely\nDirac-vortex topological PCF with winding number 1, as principally shown in the first part of Method section.\nIn Fig. 3a, scale bars should also be clearly shown. Refractive index of silica taken in the calculation\nshould be indicated in the main text.\nResponse from Authors:\nWe would like to thank the reviewer for the professional comments and valuable suggestions. The fiber\ndisplay theoretically in Supplementary Fig. S3 and fiber perform are constructed as the same parameter\nwith the winding number of 1 (depends on the Dirac-vortex structure’s phase modulation). Undeniably,\nthe practically fabricated fiber is certain to have the mismatching with ideal structure. Practically\nfabricated fiber has parameter variation and structural deformation (mainly on the curvature of hexagon\ncorner changing). We scan the parameters of fabricated fiber structure and the average data are shown\nin the following Table. R1. Simulation based on fabricated fiber cross-section is built on these\nparameters and the hexagon corner radius r=0.3a. The band invasion and Wilson loop of fabricated\nfiber are simulated in Fig. R2a-R2d, indicating the correspondence of nontrivial topological insulator\nand guarantee of Dirac-vortex mode existence under the disturbance of parameters in manufacturing\nprocess. The simulated Dirac-vortex mode spectrum and intensity distribution are shown in Fig.\nR2e&R2f, and the light field in the center of fiber definitely displays the polarization feature of\ntopological mode (APB), the same as the theoretical and experimental research. The results in fabricated\nDirac-vortex topological PCF are consistent with the theoretical results, which proves the unity of theory\nand experiment.\nTo prevent such fragmentation, we have also added a section in the Supplementary Information to\ndiscuss this issue.\nTable. R1. The parameters of topological photonic crystal fiber in designed and practically fabricated structures\nand their absolute difference.\nThe above discussions are appropriately added in the revised manuscript and supplementary\ninformation.\nThe difference between the actual and theoretical geometrical parameters is less than 0.1 μm, and the\nfabricated fiber exhibits the unique properties of Dirac-vortex topological PCF, indicating the stack-\nand-draw technique is stable and suitable to fabricate the Dirac-vortex topological PCF (see detailed\ntheoretical analysis of the fabricated Dirac-vortex topological PCF in Supplementary Note 4).\n(Please see pages 4&5, line 124-129 in the revision)\nSupplementary Note 4. Simulation on fabricated Dirac-vortex topological PCF\nIn the fabrication process of the Dirac-vortex topological PCF, the geometrical parameters of the\ndesigned Dirac-vortex topological PCF are unavoidably changed under the inhomogeneous temperature\nand pressure control, which may degrade the fiber's performance. To make sure that the fabrication error\nwill not change the topological properties of the Dirac-vortex topological PCF, we simulate the bulk\nband structures and the Wilson loop of the cross section of the practically fabricated Dirac-vortex\ntopological PCF with the geometrical parameters obtained from the scanning of the electronic telescope,\nas shown in Supplementary Figs. S4a-S4d, demonstrating that the fabricated Dirac-vortex topological\nPCF exhibits similar nontrivial topological properties to the designed one. Supplementary Fig. S4e\nshows the simulated eigenstate spectrum of the fabricated Dirac-vortex topological PCF, in which we\ncan see a Dirac-vortex fiber mode (red dot) emerges within the photonic bandgap, whose simulated\nintensity and electric vector distributions are shown in Supplementary Fig. S4e.\n(Please see Supplementary Note 4 in Supplementary Information)\nFig. R2. Simulated results of the fabricated Dirac-vortex topological PCF. a, b Simulated bulk\nband structures of the cross section of the fabricated Dirac-vortex topological PCF with φ = 0 (a) and\nφ = π (b) at k = 4π/a. The insets show the simulated electric field distributions of the p and d modes\nz\nat the Γ point. c, d Wilson loop of the Kekulé-distorted triangular photonic crystals at k = 4π/a for φ =\nz\n0 (c) and φ = π (d), respectively. The inset shows the discretization of the first BZ. e Simulated\neigenstate spectrum of the fabricated Dirac-vortex topological PCF. f Simulated intensity (red color)\nand electric vector (black arrows) distributions of the topological topological Dirac-vortex fiber modes\nat k = 4π/a, corresponding to the red dots in e.\nz\n(Please see Supplementary Note 4 in Supplementary Information)\nComment in comment 1\nIn Fig. 3a, scale bars should also be clearly shown.\nResponse from Authors:\nAll the fiber structures pictured in electron microscope scanning are attached to a standard scale.\nThe above discussions are appropriately added in the revised manuscript and supplementary\ninformation.\nFig. 3 | Fabrication and characterization of Dirac-vortex topological PCF. a Photograph and magnified image\nof the cross-section of the Dirac-vortex topological PCF. b Schematic diagram of the measurement of Dirac-\nvortex topological PCF. c The input APB and different polarization states detected in the light manipulating\nmodule. d The output Dirac-vortex topological PCF beam and different polarization states detected in the\nobserving module.\n(Please see pages 13 in the revision)\nComment in comment 1\nRefractive index of silica taken in the calculation should be indicated in the main text.\nResponse from Authors:\nIn the simulation, the dielectric constant ε of silica is about 2.1, therefore, the refractive index of silica\nis set as the fix number of 1.4491.\nThe above discussions are appropriately added in the revised manuscript and supplementary\ninformation.\nd Simulated intensity (red color) and electric vector (black arrows) distributions of the topological\nDirac-vortex fiber modes at k = 4π/a in Dirac-vortex topological PCF with its refractive index of silica\nz\nas 1.4491 (ε = 2.1).\n(Please see pages 12, line 351-353 in the revision)\nComment 2\n- From the basic point of view, the presented results are satisfactory, while from the practical one fiber\nparameters, including propagation loss and banding loss, are very important. Thus, they have to show\nexperimentally measured basic parameters of fabricated PCF.\nResponse from Authors:\nWe thank the reviewer for the professional comments. Following the reviewer’s kind suggestions, we\nmeasure the transmission loss and bend loss in a wide bandwidth (1260-1675 nm, the same range as in\nthe main manuscript). The measured results of transmission loss and bending loss are shown in Fig. R6.\nWe choose the bending radii R=25cm, 15cm and 5cm in the experiment. The transmission loss is 0.64\ndB/m at 1550nm and the loss range in the whole test band is [0.21, 1.23] dB/m. The bending loss are\n0.16, 0.62, and 1.42 dB/10 turn with bending radii R=5cm, 15cm and 25cm respectively at 1550 nm,\nand the loss ranges in the whole test band are [0.14, 0.16], [0.58, 0.86] and [1.38, 1.74] dB/10 turn.\nBecause of premature manufacturing process of the new type of fiber, the loss of fabricated Dirac-vortex\ntopological PCF is in a high level. To overcome this problem, following the reviewer’s valuable\nsuggestions, we discuss the improvement orientation of next step of manufacturing process.\nEmployment of dynamic gradient etching technology, pressure-assisted secondary drawing technology\ninvolving initial high-temperature drawing, capillaries zonal gradient pressure application, and real-\ntime closed-loop pressure control system integration, etc. could take effort into mitigates elliptical\ndeformation and suppresses structural distortion for loss suppression.\nThe above discussions are appropriately added in the revised manuscript and supplementary\ninformation.\nFig. R3. Measured transmission loss and bending loss of the Dirac-vortex topological PCF. a,\nb Measured transmission loss (a) and bending loss with different bending radii R = 5cm, 15cm, and\n25cm (b) within the wavelength range of 1260-1675 nm.\n(Please see Supplementary Note 6 in Supplementary Information)\nFor each incident beam, the power variation is about 1 dB, the measured transmission loss and bending\nloss of APB remain relatively stable (see detailed loss analysis in Supplementary Note 6),\nindicating that the Dirac-vortex topological PCF exhibits a broadband and robust performance across the\nfull communication window (1260-1675 nm).\n(Please see pages 6, line 160-164 in the revision)\nSupplementary Note 6. Experimental measurement of transmission loss and bending loss of the\nDirac-vortex topological PCF and further loss reducing improvement of fabrication\nWe measure the transmission loss and bending losses of the Dirac-vortex topological PCF with incident\nAPB within the wavelength range of 1260-1675 nm, as shown in Supplementary Figs. S6a-S6b,\nrespectively. The transmission loss of the Dirac-vortex topological PCF is about [0.21, 1.23] dB/m, and\nthe bending losses under different bending radii R = 5cm, 15cm, and 25cm are [0.14, 0.16], [0.58, 0.86],\nand [1.38, 1.74] dB/10 turn, respectively.\nAlthough the transmission loss in novel topological PCF seems a little bit higher, with future\nimprovement, there are also some solutions that can take effort on loss suppression. The high loss\nprimarily stems from geometric irregularities within the air-hole structures, including size fluctuations,\npositional deviations, and Mie scattering induced by inner wall roughness. Furthermore, minute\nvariations in the fiber drawing process, such as unstable temperature gradients, tension fluctuations, or\npressure control deviations, can readily lead to air-hole collapse or the formation of elliptical\ndeformation. These geometric discontinuities compromise the periodic arrangement characteristic of\ntopological PCF, promoting mode degradation and optical field energy leakage. To enhance air-hole\nuniformity, subsequent processes could employ dynamic gradient etching technology. By axially\nmodulating the etching intensity along the capillary, this technique enables precise control over capillary\nsurface roughness and longitudinal wall thickness uniformity. Additionally, utilizing precursor tubes\nwith improved dimensional consistency, coupled with the implementation of high-precision capillary\nmeasurement platforms for selecting tubes with uniform inner/outer diameters, ensures material\nhomogeneity. Zoned pressure gradient control, integrated with inert gas-assisted drawing (e.g., Argon,\n1-15 kPa), and the co-optimization of parameters including temperature gradient, thermal sensitivity,\nand draw speed (0.5-10 m/min), can effectively maintain the morphology of variously sized air holes,\nmitigate elliptical deformation, and suppress structural distortion.\nIncreased bending loss is closely associated with insufficient modal confinement capability in the\ncladding. Deficiencies in air-hole arrangement symmetry or deviations in the fabricated air-hole\ndiameter to pitch ratio (d/Λ) readily cause modal leakage under bending conditions. This effect is\nparticularly pronounced in regions experiencing localized air-hole collapse or deformation induced by\nexcessive drawing tension or thermal field instability, where structural defects significantly amplify\nradiation loss. To optimize bending loss performance, subsequent efforts could focus on enhancing air-\nhole shape retention and structural symmetry: implement preform inverse topology optimization to\nincrease the number of cladding layers and improve geometric tolerance; employ pressure-assisted\nsecondary drawing technology, involving initial high-temperature drawing (1790–1950°C) to solidify\nthe internal structure, followed by zonal gradient pressure application within a confinement sleeve to\ncompress capillaries. This sharpens the air-hole profiles, reduces viscous flow-induced circularity\ndistortion, aligns the fabricated fiber structure more closely with the design intent, minimizes fabrication\nerrors, and strengthens core optical confinement. Concurrently, the integration of a real-time closed-\nloop pressure control system stabilizes the dynamic morphology of air holes, thereby significantly\nreducing the critical bending radius. These synergistic strategies, through systematic optimization of\nair-hole geometric regularity and cladding confinement efficacy, can concurrently achieve substantial\nsuppression of both transmission loss and bending loss.\n(Please see Supplementary Note 6 in Supplementary Information)\nComment 3\n- There are missing references relevant to topological fibers, which should also be cited and discussed:\nZ. Zhang, et al, Nanophotonics 10, 4067—4074 (2021); K.-H. O, et al, Opt. Fiber Tehcnol. 73, 103054\n(2022), etc.\nResponse from Authors:\nWe thank the reviewer for the professional comments and sharing very nice references relevant to\ntopological fibers. Innovative topological photonic crystal fiber is an important step on topological\nphotonic crystal development. Following the reviewer’s valuable suggestions, in the revised paper, we\nhave cited and discussed the relevant references in the introduction as Refs. 20 and 21, which makes\nthe research investigation contain a wider range of the knowledge.\nThe above discussions are appropriately added in the revised manuscript.\nThe unique properties of topological PCFs, such as the single-polarization single-mode [7], azimuthally\nand radially polarized orbital angular momentum [20], and near-zero flattened dispersion [21], exhibit\npromising potential applications in next-generation robust quantum networks and optical\ncommunications.\n(Please see page 2, line 48-52 in the revision)\nThe authors thank the reviewer again for the professional comments and\nvaluable suggestions.\nIn this article, the authors have experimentally demonstrated Dirac-vortex topological photonic crystal\nfiber (PCF) for the first time. Using stack-and-draw technique with silica glass capillaries, a Dirac-vortex\ntopological PCF has been fabricated, which exhibits Dirac-vortex single mode guiding in the full\ncommunication window from 1260 nm to 1675 nm. This work has the land-marking significance in\ntopological PCF technique, so that the results are very interesting and meet the requirement for the\njournal. However, the following issues should be resolved for acceptance in line with high standard of the\njournal.\n- My main concern is the comparison of experimental data with theoretical prediction. In the initial\npart of the Results section, they presented dispersion and showed the existence of topological\nDirac-vortex guiding mode. Theoretical results are, however, presented separately from the\nexperimental ones to some extent.\nCorrespondence between the designed and fabricated fibers is not clear. The topological property\nof fabricated fiber shown in Fig. 3 should be discussed theoretically: the best way is to numerically\nevaluate the topological invariant (here, winding number) and the topological Dirac-vortex mode\ndirectly by using the cross-section of fabricated fiber, but not with schematic shown in extended\ndata Fig. 3. In such a way, the readers can be convinced that the finally obtained fiber is genuinely\nDirac-vortex topological PCF with winding number 1, as principally shown in the first part of\nMethod section. In Fig. 3a, scale bars should also be clearly shown. Refractive index of silica taken\nin the calculation should be indicated in the main text.\n- From the basic point of view, the presented results are satisfactory, while from the practical one\nfiber parameters, including propagation loss and banding loss, are very important. Thus, they have\nto show experimentally measured basic parameters of fabricated PCF.\n- There are missing references relevant to topological fibers, which should also be cited and\ndiscussed: Z. Zhang, et al., Nanophotonics 10, 4067—4074 (2021); K.-H. O, et al., Opt. Fiber\nTehcnol. 73, 103054 (2022), etc."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors revised their manuscript appropriately.\nTherefore, I think this paper is ready to be published.", "reviewer_2": "Authors addressed satisfactorily in response to referee comments and I recommend to accept as it is."}, "author_response": ""}]

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