Ultrabroadband, achromatic, and non-diffracting perfect optical vortex generation via radial momentum control in dielectric metasurfaces

Domain: physics
Published: 2025-11-25
Editorial outcome: minor_revision

Abstract
Optical vortices, beams that carry a twist, are pivotal in advanced technologies like optical manipulation, optical communications and imaging. However, their utility has been hampered by a fundamental limitation: the size and spread of the beam change inevitably with its twist and color, making stable, scalable systems difficult to achieve. Here we show that using precisely engineered metasurfaces to control light’s radial momentum enables the generation of a class of ‘achromatic non-diffracting perfect vortex beams’ (ANPVBs). These beams maintain an almost perfectly constant diameter, regardless of their twist or color across the visible spectrum (450–780 nm), and resist spreading over long distances. By decoupling beam size from these previously limiting factors, our work provides a robust and versatile platform for applications in high-capacity optical communication, quantum information processing and imaging.

Introduction
The rapid advancement of modern society and technology has led to an ever-increasing demand for high-capacity data transmission and broadband communication. Optical communication, with its inherent advantages of high speed, license-free operation, and ease of deployment, has emerged as a key solution to meet these demands1–3. However, as the capacity of traditional fiber-optic systems approaches its theoretical limits, there is a pressing need to explore new degrees of freedom for encoding information into light4,5. One promising avenue is the utilization of orbital angular momentum (OAM) carried by optical vortex beams. The OAM of vortex beams offers an infinite set of orthogonal modes, providing a new dimension for enhancing the capacity of optical communication systems. Optical vortices carrying orbital angular momentum (OAM) were first numerically predicted in laser cavities with large Fresnel numbers6. A pivotal theoretical advancement was made by Allen et al., who systematically established the quantized OAM value (mℏ/photon) through the exp(imϕ) phase singularity7, enabling applications in super-resolution imaging, optical communications, and rotational detection8,9. Barnett and Allen further confirmed this OAM conservation in non-paraxial regimes10. Experimental validations include OAM transfer to microparticles via focused Laguerre-Gaussian beams11 and topological charge-based photon computation12. However, a significant limitation of traditional vortex beams is that their diameter varies with the topological charge, making it challenging to couple them into fixed-aperture fibers or waveguides in optical communication systems.

To address this limitation, Ostrovsky et al.13 introduced the concept of perfect vortex beams (PVBs) in 2013, where the beam diameter remains independent of the topological charge. PVBs can be generated when the incident beam diameter is proportional to the parameters of an axicon lens14, and their mathematical form can be derived from the Fourier transform of Bessel beams15. Subsequent advancements have expanded the scope of PVBs, including the development of vector PVBs16, elliptical PVBs17, and elliptical perfect vortex arrays18,19. Further innovations have led to the generation of elliptical vector PVBs through dynamic and geometric phase modulation20, dual-ring distributed PVBs21, and generalized perfect vortices with arbitrary intensity distributions22. Recently, iso-propagation vortex beams have exhibited environmental robustness due to their reversed radial energy flow characteristics. These beams achieve diffraction-free inner-ring propagation that is independent of topological charge; however, the size of such vortex beams is affected by the wavelength, and they exhibit energy loss through multiple side lobes23,24. Nevertheless, the conventional architectures for the PVB generation and manipulation predominantly require intricate multi-element optical assemblies, imposing significant spatial and economic constraints due to their reliance on conventional bulk-component frameworks.

Metasurfaces, composed of subwavelength nanostructures, have revolutionized structured light generation by enabling multi-dimensional control over amplitude, phase, and polarization25,26. Metasurface has been applied to imaging and display, such as metalenses27–29, beam shaping30, holographic display31–34 and printing35,36. Initially, PVBs and perfect Poincaré beams required cascading multiple geometric metasurfaces to mimic bulky optical components like spiral phaseplates, axicons and lenses37, introducing alignment complexity. A pivotal shift occurred with the integration of these functionalities into compact plasmonic metasurfaces38, enabling 3D-focused PVBs, a breakthrough later refined via phase-only modulation strategies to generate visible-spectrum fractional PVBs39. However, these advancements inherently sacrificed efficiency due to plasmonic energy dissipation. The paradigm shifted decisively toward low-loss dielectric metasurfaces, which demonstrated full visible-spectrum PVBs in single-layer designs40, broadband perfect Poincaré beams via spin-multiplexing41, polarization-rotated PVBs through geometric-phase engineering42, and compact vector beams (e.g., double-ring structures)43. Further advancements combined geometric and propagation phases to generate high-order Poincaré beams44 and focused higher-order beams via supercell architectures45. Recent breakthroughs emphasize higher-dimensional control: multi-channel PVBs with asymmetric singularity distributions via topological charge superposition46, hybrid grafted PVBs with multifunctional metasurfaces47, and super-capacity perfect vector vortex beams enabled by spatial-frequency patching metasurfaces48. Despite these advancements, PVBs still lack non-diffracting and non-dispersive properties, which significantly limit their effectiveness for long-distance transmission. These limitations arise because PVBs tend to spread out and lose intensity over extended distances, reducing their utility in applications requiring high precision and stability over long ranges, thereby limiting their potential applications in areas like optical communications, remote sensing, and biomedical imaging.

In this work, we present an approach to overcome these limitations by leveraging a dielectric geometric metasurface to generate an ultrabroadband achromatic non-diffracting perfect vortex beam (ANPVB). The metasurface, composed of TiO2 nanopillars with spatially varying in-plane rotation angles, imparts a tailored geometric phase profile to the incident light, enabling precise control of radial momentum. Unlike conventional Laguerre-Gaussian beams, the ANPVB exhibits a constant beam diameter at any axial position, independent of topological charge variations. Furthermore, the ANPVB demonstrates exceptional diameter stability during propagation, with minimal relative diameter variations over extended distances. Most notably, the ANPVB maintains a consistent beam diameter across the visible spectrum (450–780 nm), showing wavelength-independent performance. These groundbreaking results not only advance the field of singular optics but also open new possibilities for robust optical communication technologies. In the following sections, we will detail the design and construction of the ANPVB, supported by simulations and experimental results that validate its unique properties.

Results
Concept and design
In this study, we introduce an ultrabroadband achromatic non-diffracting perfect vortex beam (ANPVB), leveraging the precise control of radial momentum via metasurfaces. This advancement aims to transcend the limitations associated with topological charge (TC) dependency and diffraction inherent in traditional vortex beam designs. By employing high-precision modulation of radial momentum through geometric phase distributions within metasurfaces, we achieve a decoupling of the beam radius from TC and significantly mitigate diffraction effects. Typically, the radius of vortex beams correlates positively with TC values due to variations in azimuthal momentum distribution, which affect the transverse momentum gradient. However, conventional perfect vortex beams (PVBs), while having a radius independent of TC, rely on axicons and spiral phaseplates for generation, thus remaining susceptible to dispersion and color effects. In our work, by introducing a radial momentum component (|kr |≫|kϕ|) that dominates the transverse momentum distribution (jkjj = |kr |2+|kϕ |2 ≈ |kr |, kjj = kr+kϕ, kjj ≈ kr), where kr is related to the radial phase gradient and kϕ is associated with the azimuthal phase gradient.), we make the changes in azimuthal momentum due to differences in TC (Δ|kϕ |/l) negligible (Fig. 1). The crux of diffraction suppression lies in the nonlinear design of the radial momentum gradient, enabling a substantial enhancement in the Rayleigh range (zR). By leveraging the geometric phase of metasurfaces, our approach enables direct encoding of radial momentum—intrinsically linked to the radial phase gradient (∇ψ(r))—thereby achieving long-distance non-diffractive beam propagation without requiring complex optical path modifications. Furthermore, the dispersion-free phase response (ψ(r)) of the metasurface’s geometric phase eradicates wavelength dependency introduced by traditional axicons, ensuring stability in beam radius and propagation characteristics across a broad spectral range (experimentally validated from 450 to 780 nm). The cornerstone of our innovation is treating the momentum dimension as an independently adjustable parameter, offering a versatile framework for designing non-diffractive beams. Adjusting the radial phase gradient (∂ψ(r)/∂r) of metasurface elements allows for flexible control over the lateral dimensions and propagation distances of beams—a feature extendable to terahertz or microwave bands. From an application perspective, the dispersion-free nature of ANPVBs notably boosts channel capacity in multi-wavelength OAM multiplexing communication systems, whereas their non-diffractive properties provide distinct advantages in deep biological tissue imaging and long-range optical trapping.

Figure 1 presents a schematic overview of the geometric metasurface-driven generation mechanism for ANPVBs. The tailored geometric phase profile of the metasurface is engineered to precisely match the combined phase distribution resulting from the superposition of radial and spiral phase components. This design enables the metasurface to efficiently convert incident circularly polarized Gaussian beams into ANPVBs exhibiting counter-rotating circular polarization states. The radial phase ψ(r) is mathematically expressed as

ψ(r) = 
{ t(r−βR)α βR ≤ r ≤ R
−δ(t1−(βR/r)γ) 0 ≤ r < βR , (1)

where r denotes the radial coordinate and R represents the characteristic radius of the phase pattern. The exponents α and γ (α, γ > 0) govern the power-law dependence of the phase distribution, while β serves as a radial scaling factor. The parameters δ and t, respectively, define the maximum phase shifts for the inner and outer regions of the structure, with δ (> 0) functioning as an additional optimization parameter for fine-tuning the inner radial phase characteristics.

When a Gauss beam with waist radius w=2R/3 illuminates the metasurface incorporating both the radial phase ψ(r) and spiral phase lϕ, the resulting complex amplitude U of the ANPVB is given by:

U = exp(−r2/w2)exp(iψ(r))exp(ilϕ). (2)

The metasurface design (detailed in Supplementary Note 1) involves a trade-off between performance and fabrication feasibility. While larger metasurfaces theoretically enhance beam quality, practical limitations led us to prioritize a compact configuration (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm) for experimental validation. A scaled-up variant (R=1.35 mm) was analyzed solely through simulations (see Supplementary Material). All studies employed a 633 nm incident wavelength, with the Laguerre-Gaussian beams (LGBs) in Fig. 2(a1–c1) matching the Gaussian beam’s waist radius of w=2R/3 mentioned above. Notably, LGB vortices at z=18 mm exhibit diameter variations correlated with their topological charges (l). In contrast, ANPVB vortices generated under the parameter set α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=1, 15, 30 (Fig. 2(d1–f1)) maintain nearly constant diameters across these topological charges. The helical phase profiles in Fig. 2(a2–c2) and 2(d2–f2) confirm that both LGBs and ANPVBs sequentially encode topological charges l=1, 15, 30. Crucially, ANPVB vortex diameters remain independent of their topological charges, a key distinction from conventional LGB behavior. Furthermore, scaling the metasurface size enhances its capacity to sustain higher topological charges (Supplementary Note 2, Fig. S3), a phenomenon governed by the increased phase sampling resolution (i.e., pixel density) across the 0 to 2π phase gradient. This relationship is rigorously validated by demonstrating ANPVBs’ ability to generate vortices with identical topological charges but distinct radii (Supplementary Note 2, Fig. S2, 4).

Non-diffracting properties of achromatic non-diffracting perfect vortex beams (ANPVBs)
In Fig. 3, all incident Gauss beams have the same waist radius of w=2R/3. The diameter of the optical vortex for the ANPVB at z=18 mm is chosen as the reference value. For ease of comparison, the diameter of the optical vortex at the focal plane for the selected PVB is set to the same value as the reference value. The phase profile of the PVB is calculated using the expression −πr2/λf + lϕ + 2πr’, where the focal length f is 18 mm, l=30, and the adjusted parameters α and the outermost radius R’ are 0.04733 and 0.54 mm, respectively. In Fig. 3, the intensity profiles in both the xy- and xz-planes are normalized individually for each image, using a common colorbar for both normalized profiles. The cross-sectional intensity profiles along the xz-plane in Fig. 3(a) and (b) correspond to the normalized central line of the xy-plane at the same axial positions. As a result, the intensity in the xz-plane does not diminish at larger propagation distances, since the intensity at each axial position is normalized. In Fig. 3(a), the light line from the x-z intensity profile is diverging. However, the two light lines in Fig. 3(b) are approximately parallel, illustrating that the propagated ANPVB with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) is non-diffracting. Moreover, it can be observed in Fig. 3(a1-a10) that with increasing propagation distance, the diameter of the PVB increases. However, the ANPVB in Fig. 3(b1-b10) have approximately the same diameter. This indicates that the ANPVB has the non-diffracting property. It can be seen in Fig. 3(c) that compared to the PVB, the radius of the ANPVB remains unchanged as the propagation distance increases. Moreover, the conventional PVB and ANPVB have the same radius at z=18 mm, which is set as the initial radius. The radii of conventional PVB and ANPVB reach √2 times their initial radii at z=33.4 and 64.8 mm, respectively. Therefore, the Rayleigh length of ANPVB is about three times that of PVB. The diameter of the optical vortex at z=18 mm is chosen as the reference value. The relative variation is defined as the ratio of the absolute difference between the vortex diameter at an arbitrary axial position and the vortex diameter at z=18 mm to the vortex diameter at z=18 mm. Calculations performed over the 3-30 mm range show in Fig. 3(d) that the line representing the relative variation for the ANPVB is nearly flat with minimal fluctuation, maintaining a relative variation close to zero. In contrast, the relative variation for the PVB significantly fluctuates, being zero only at 18 mm but notably higher elsewhere. This further confirms that the optical vortices of the ANPVB maintain consistent size across different axial distances, demonstrating their non-diffracting nature. Supplementary Note 3 provides additional proof of the non-diffracting properties for larger-sized ANPVBs.

Non-dispersible properties of achromatic non-diffracting perfect vortex beams (ANPVBs)
The intensity cross-sections along the xz-plane in Fig. 4 correspond to the normalized central line of the xy-plane at the same axial positions. Consequently, the intensity in the xz-plane does not decrease at larger propagation distances, as the light intensity at each axial position is normalized. From the x-z intensity profiles presented in Fig. 4(a1-d1) under different wavelengths (450 nm, 532 nm, 633 nm, and 780 nm), it can be observed that the two light lines in each section remain parallel. This indicates that the ANPVB maintains its non-diffracting characteristic across these wavelengths. Additionally, the numerical circles shown in Fig. 4(a2-d2) at z=18 mm under various wavelengths exhibit nearly identical diameters and intensity distributions, further confirming the non-dispersive nature of the ANPVB. The corresponding phase profiles in Fig. 4(a3-d3) reveal that the generated vortices under different wavelengths share the same topological charge of 30, supporting the consistency in their structural properties. Supplementary Note 4 provides additional evidence affirming the non-dispersive characteristics of larger-sized ANPVBs. This comprehensive analysis demonstrates the robustness and wavelength-independence of the ANPVB’s optical properties.

Experimental demonstration
The experimental setup is detailed in Supplementary Note 5. Three fabricated metasurfaces, each comprising 2400×2400 pixels with a pixel pitch of 450 nm, generate ANPVBs with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=1, 15, and 30). The overall scanning electron microscopy (SEM) images along with local top and side view SEM images are presented in Fig. 5(a1-c1), 5(a2-c2), and 5(a3-c3), respectively. For normal incidence, the grating equation is given by: d sinθ=kλ, where d=450 nm is the period, λ=450 nm is the wavelength, and k is the diffraction order. Under this condition, the zeroth-order transmitted light (k=0) propagates along the optical axis (θ=0°), while the first-order diffracted beams (k=±1) occur at θ=±90°, effectively propagating as surface waves perpendicular to the optical axis. Since these higher-order diffraction components do not contribute to the on-axis propagation, the zeroth-order transmittance—which carries the intended phase profile—dominates the output field along the propagation direction. Moreover, the diffraction efficiency into the k=±1 orders is very low due to the extreme diffraction angle, resulting in negligible diffraction loss. Therefore, even at 450 nm, the metasurface maintains non-diffracting characteristics for the zeroth-order beam, which is the relevant component for generating the desired radial phase profile. Figure 5(d-g) illustrate the intensity distributions of the diffracted light fields at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm through these metasurfaces. In each of Fig. 5(d-g), the first ten columns from left to right represent the intensity distributions captured at axial distances of 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30 mm after circularly polarized light passes through the metasurface, with the circularly polarized components filtered by a circular polarizer. From top to bottom, the diffracted light field distributions are shown for topological charges of 1, 15, and 30. The last column of Fig. 5(d-g) shows the intensity pattern captured by the CMOS at 18 mm after linearly polarized light passes through the metasurface, revealing petal-shaped distributions resulting from interference (INF) between the converted left- or right-handed circularly polarized component and the unconverted plane wave with the corresponding circular polarization. The number of petals corresponds to the topological charge of the vortex. Observations indicate that under the same wavelength, the vortex beam diameters remain identical for different topological charges, demonstrating the diameter’s independence from the topological charge. Additionally, at the same axial position and topological charge, the vortex beam diameters are consistent across different wavelengths, confirming the dispersion-free nature of ANPVBs. Furthermore, the constant diameter of vortex beams at various axial positions for the same wavelength and topological charge illustrates the non-diffractive property of ANPVBs. In Fig. 5(h), the blue, green, red, and pink solid lines with triangles, squares, and circles depict the relationship between vortex diameter and axial position for wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively, for topological charges of 1, 15, and 30. These lines nearly overlap and run parallel to the horizontal axis, indicating consistent vortex diameters across different wavelengths and topological charges. This confirms that the ANPVB exhibits both non-dispersive and non-diffractive characteristics. Supplementary Note 6 provides additional interference images of metasurfaces carrying different topological charges illuminated by various wavelengths at different distances. Polarization conversion efficiency (PCE) values of the meta-atom, as measured or numerically calculated, are presented in Supplementary Note 7. Higher-magnification SEM images of the fabricated metasurfaces are provided in Supplementary Note 8. The momentum distributions of ANPVBs, as measured or numerically calculated, are presented in Supplementary Note 9.

Methods
Sample fabrication
The samples were fabricated utilizing electron beam lithography (EBL) in conjunction with etching techniques. Initially, a 1000 nm-thick layer of polymethyl methacrylate (PMMA) electron-beam resist was spin-coated onto a transparent silica substrate featuring an indium tin oxide (ITO) film layer at a speed of 2000 rpm. This coated substrate was then baked on a hot plate for 4 min at 180 °C to ensure proper adhesion and resist hardening. Following the preparation of the substrate, EBL was employed to pattern the resist layer using an acceleration voltage of 100 kV and a beam current of 200 pA. Post-exposure, the sample underwent a development process in a solution mixture of isopropanol (IPA) and methyl isobutyl ketone (MIBK) in a ratio of 3:1 for 1 min, followed by fixation in pure IPA for an additional minute at room temperature. Atomic layer deposition (ALD) was subsequently used to deposit a 220 nm thick layer of TiO2 selectively into the exposed areas. The thickness of the deposited TiO2 was tailored to match the semi-minor axis of the largest meta-atom. After deposition, the entire sample was covered with a uniform 220 nm TiO2 layer, which was later removed from the top surface via ion beam etching (IBE). With the top TiO2 layer removed, reactive ion etching (RIE) was applied to eliminate the remaining PMMA resist. This sequence of processes culminated in the formation of high-aspect-ratio TiO2 nanostructures, achieving an aspect ratio of up to 10. These steps ensured precise patterning and material deposition, resulting in well-defined metasurface devices.

Numerical simulations
The diffraction field E of the incident optical field U can be calculated by Eq. (3).

E = F0 × F(U × F), (3)

where, F0 = exp(ikz) × i/λz exp(ik(x2+y2)/2z), F = exp(−iπ/λz (u2+v2)), F shows the Fourier transform, k=2π/λ, z and λ present the wave vector, axial distance and wavelength respectively, and (u, v) and (x, y) represent the rectangular coordinates at the incidence and diffraction planes respectively.

Data availability
All data needed to evaluate the conclusions in the paper is present in the paper and/or the Supplementary Information.

Code availability
All numerical codes are available upon request from the corresponding authors.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript present a novel and comprehensive approach to generating ultrabroadband, achromatic, and non-diffracting perfect optical vortex beams (ANPVBs) through the radial momentum control enabled by a dielectric geometric metasurface. By engineering the spatial variation of the geometric phase, the authors achieve a significant milestone, which is decoupling the beam radius from the topological charge, overcoming a fundamental limitation in traditional vortex beam systems. The authors provide a strong combination of theoretical modeling, numerical simulation, and experimental validation across a broad spectral range (450 – 780 nm), demonstrating a compelling advancement in singular optics with far-reaching applications in optical communication, high-resolution imaging, and optical trapping.\nAs I understand it, the key strengths of this paper lie in its originality and technical rigor. The design, based on dominant radial momentum where kr≫kϕ, is innovative. It enables the generation of a non-diffracting beam profile that remains invariant across varying topological charge (TC) values and wavelengths. Furthermore, the theoretical concept is successfully implemented using a TiO₂-based metasurface, which allows for low-loss beam manipulation, which is an important step toward the practical commercialization of the device.\nDespite these strengths of papers, there are a few areas that would benefit from further elaboration to enhance its impact and practical relevance.\n1. In figure 3, Do the xy and xz intensity profile share the same colorbar scale, or are they individually normalized for each image? The intensity shown in (a10) appears different from that at z = 30 mm in the xz profile, which raises this question. If the images have been rescaled or normalized indenpendently, it would be helpful to clarify this in the figure caption.\n2. In figure 4, the colorbars for the xz intensity profiles appear to follow different gradient schemes. For example, blue/green/purple-to-black for most wavelengths, wherease the 633 nm panal seems to use a red-white-black colorbar. To ensure consistency and facilitate accurate visual comparison across wavelengths, it would be helpful to unify the colorbar styles or clarify the reasoning behind the variation in the figure caption.\n3. The manuscript highlights efficient transmission as one of its key novelties. However, there is no rigorous measurement or numerical analysis of the actual transmission efficiency provided across the visible wavelength range. To support this claim, it would greatly strengthen the work to include quantified efficiency data.\n4. Building on Comment 3, based on my experience, metasurfaces with high-aspect-ratio structures often exhibit discrepancies between simulated and experimentally measured efficiencies. I am curious whether the authors have attempted to measure the transmission efficiency in a manner comparable to the simulated results shown in Fig. S1 (e.g., panel a6).\n5. It would be helpful to provide higher-magnification SEM images of the fabricated metasurfaces. As currently presented, it is difficult to assess the structural quality and fidelity of the fabricated samples, which is important for evaluating the precision and reproducibility of the device.\n6. The metasurface is reported to have a structural period of 450 nm, which coincides with the shortest working wavelength used in the experiments (450 nm). Given that this matches the wavelength, diffraction is expected to occur based on fundamental grating theory. I would appreciate it if the authors could clarify whether the claimed non-diffracting behavior at 450 nm is theoretically valid in this regime. A brief analysis or justification would strengthen the credibility of the results at this wavelength.\n7. Panels (a1)–(a10), etc., lack proper descriptions in the figure captions. Please revise the captions to clarify each subfigure.\n8. It appears that no direct measurement of the momentum space has been performed for the metasurfaces. To substantiate the claim of non-diffracting behavior, it would be important to experimentally verify the beam’s momentum distribution.", "reviewer_2": "The work presented in the manuscript \"Ultrabroadband, Achromatic, and Non-Diffracting Perfect Optical Vortex Generation via Radial Momentum Control in Dielectric Metasurfaces\" is interesting for its contribution to perfect optical vortex generation with metasurfaces, however, I do not believe there is significant enough novelty and advancement for publication in Nature Communications. A more specialised optics focused journal could be more suitable in my opinion.\nSome additional comments are as follows:\n1) To start with, I feel that the title is a little overreaching in its claim. Work such as Refs 40 and 42 in the manuscript demonstrate PB phase metasurfaces for the generation of perfect optical vortex beams. It is just a given that PB phase is unrelated to the wavelength of light, so the use of PB phase metasurfaces automatically gives you the broadband feature. The meta-atom design then defines the conversion efficiency, and ultimately the performance at other wavelengths. Even with a super low efficiency (~<10%), the functionality can still work after filtering the unconverted light. So the term 'ultrabroadband' is a little overreaching in my opinion. The applies also to 'achromatic', which I am assuming here means the same thing as ultrabroadband. The 'non-diffracting' nature is also a property of perfect vortex generation (however I do acknowledge the contribution in this point).\n2) Continuing from comment 1, I feel that the main contribution of this work is the increased depth that the vortex maintains its shape. However, it is unclear how this compares with other research.\n3) Related to the PB phase modulation, it would be interesting to see experimental values of the conversion efficiency of the metasurfaces, at all different wavelengths that the authors show. The simulated values are often far from what is experimentally validated, but this information is missing from the manuscript. The simulated values are found in the SI, and it appears that the meta-atom was not designed to be highly efficient in broadband operation (as the title suggests), and also not optimised for specific wavelengths either, again taking away any meaning of broadband operation (it is therefore just a byproduct of using PB phase, not an engineered outcome).\n4) It is notable that the size of the vortex is independent of the topological charge, however, the size of the vortex must match the largest topological charge. This means for l=1, the size of the donut shape is increased substantially to match l=30. If we were then to use, e.g. l=300, the size would be increased further. It would be nice if the authors could discuss this point and its implications.", "reviewer_3": "In this manuscript the authors present a dielectric metasurface featuring an engineered geometric phase profile that enables precise control of radial momentum aimed at generating an ultrabroadband achromatic non-diffracting perfect vortex beam (ANPVB).\nThis work is motivated by the need to overcome the limitations of all kinds of vortex beams up to now introduced.\nSpecifically,\n1. the beam diameter of ANPVBs remains independent of the topological charge, in contrast to Laguerre–Gauss beams (LGBs) whose transverse diameter is correlated with the topological charge, making it challenging to couple LGBs into fixed-aperture fibers or waveguides in optical communication systems;\n2. differently from conventional Perfect Vortex Beams, already featuring transverse diameter independence of the topological charge, ANPVBs demonstrate exceptional diameter stability during propagation and maintain an almost invariant beam diameter across the whole visible spectral range, paving the way for robust optical communication technologies.\nThese peculiar features physically originate from the fact that the radial component of the linear momentum transverse to the propagation direction is much larger than its azimuthal component, so that the changes in azimuthal momentum due to differences in topological charge turn to be negligible. Moreover, the radial momentum shaping of ANPVBs is achieved through geometric phase manipulation, which is intrinsically wavelength independent and enables to achieve long-distance non-diffractive beam propagation without requiring complex optical path modifications. Conventional PVBs are generated by manipulating the dynamic phase, i.e. by using combinations of axicons and spiral plates, and are therefore prone to space and time dispersion.\nThe work is sounding and convincing. The conclusions are supported by both experimental demonstrations and numerical simulations included in the main text and expanded in the supplement. The essence of the mechanism of the radial momentum dominance underpinning the substantial independence of the transverse beam size of the topological charge is effectively illustrated in Fig. 1.\nHowever, it is not clear the criterion for selecting the several parameters (\\alpha, \\beta, \\gamma, etc.) involved in the radial phase profile to achieve the desired tolerances. Are there physical criteria or it is just a matter of numerical simulation searching?"}, "author_response": "Dear editor and reviewers,\nThank you for the opportunity to revise our manuscript titled \"Ultrabroadband, Achromatic, and Non-Diffracting Perfect Optical Vortex Generation via Radial Momentum Control in Dielectric Metasurfaces\" (Manuscript ID: NCOMMS-25-39674). We sincerely appreciate the reviewers' valuable and constructive comments, which have substantially enhanced the quality of our work. All suggested revisions have been carefully incorporated, with changes highlighted in yellow for clarity. Additionally, we have conducted extensive additional simulations and incorporated new experimental results in direct response to the reviewers' suggestions. Detailed responses to each comment are provided below:\nCOMMENTS TO AUTHOR:\nReviewer #1 (Remarks to the Author):\nThis manuscript present a novel and comprehensive approach to generating ultrabroadband, achromatic, and non-diffracting perfect optical vortex beams (ANPVBs) through the radial momentum control enabled by a dielectric geometric metasurface. By engineering the spatial variation of the geometric phase, the authors achieve a significant milestone, which is decoupling the beam radius from the topological charge, overcoming a fundamental limitation in traditional vortex beam systems. The authors provide a strong combination of theoretical modeling, numerical simulation, and experimental validation across a broad spectral range (450 – 780 nm), demonstrating a compelling advancement in singular optics with far-reaching applications in optical communication, high-resolution imaging, and optical trapping.\nAs I understand it, the key strengths of this paper lie in its originality and technical rigor. The design, based on dominant radial momentum where kr≫kϕ, is innovative. It enables the generation of a non-diffracting beam profile that remains invariant across varying topological charge (TC) values and wavelengths. Furthermore, the theoretical concept is successfully implemented using a TiO₂-based metasurface, which allows for low-loss beam manipulation, which is an important step toward the practical commercialization of the device.\nDespite these strengths of papers, there are a few areas that would benefit from further elaboration to enhance its impact and practical relevance.\nReply: We sincerely appreciate your insightful recognition of the novelty and potential applications of our work, as well as your positive assessment of its originality and technical rigor. We are grateful for your valuable feedback and have carefully addressed all your specific suggestions, incorporating extensive additional simulations and new experimental results. Detailed responses to each of your comments are provided below.\n1. In figure 3, Do the xy and xz intensity profile share the same colorbar scale, or are they individually normalized for each image? The intensity shown in (a10) appears different from that at z = 30 mm in the xz profile, which raises this question. If the images have been rescaled or normalized indenpendently, it would be helpful to clarify this in the figure caption.\nReply: Thank you for raising this important point. In Figure 3, both the xy and xz intensity profiles are individually normalized, but share the same color scale for direct comparison. The intensity values in the xz cross-sections (e.g., at z = 30 mm) correspond to the normalized central line of the respective xy plane at identical axial positions. This normalization explains why the intensity does not appear to diminish with propagation distance, as each cross-section is normalized independently. We have revised the figure 3 and figure caption to explicitly clarify this normalization procedure.\nChanges made in the revised manuscript:\nPage-10\nIn Fig. 3, the intensity profiles in both the xy- and xz-planes are normalized individually for each image, using a common colorbar for both normalized profiles. The cross-sectional intensity profiles along the xz-plane in Figs. 3(a) and 3(b) correspond to the normalized central line of the xy-plane at the same axial positions. As a result, the intensity in the xz-plane does not diminish at larger propagation distances, since the intensity at each axial position is normalized.\nPage-28\nFigure 3 | Diffraction comparison between PVB and ANPVB. (a) and (b) x-z intensity profiles for the conventional PVB of (f=18 mm, l=30, α=0.04733, R’=0.54 mm) and ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), respectively. (a1-a10) and (b1-b10) Corresponding x-y intensity distributions at z=3, 6, 9, 12, 15, 18, 21, 24, 27 and 30 mm for PVB and ANPVB, respectively. Intensity profiles for both the xy and xz planes are individually normalized for each image. As the propagation distance increases, the diameter of the PVB in Figs. 3(a1-a10) expands, while the diameter of the ANPVB in Figs. 3(b1-b10) remains approximately constant. (c) Beam radius versus axial position. (d) Plot of the relative variation of diameters with respect to axial positions.\n2. In figure 4, the colorbars for the xz intensity profiles appear to follow different gradient schemes. For example, blue/green/purple-to-black for most wavelengths, wherease the 633 nm panal seems to use a red-white-black colorbar. To ensure consistency and facilitate accurate visual comparison across wavelengths, it would be helpful to unify the colorbar styles or clarify the reasoning behind the variation in the figure caption.\nReply: Thank you for your valuable feedback and advice. In response, we have unified the colorbars across all panels in Fig. 4. The 633 nm panel now also uses a consistent red-to-black gradient, matching the style of other wavelengths. This adjustment ensures visual consistency and facilitates more accurate comparison across different wavelengths. As noted in the figure caption, the intensity cross-sections along the xz-plane correspond to the normalized central line of the xy-plane at the same axial positions. Because the intensity is normalized at each axial position, the profiles do not exhibit decreasing intensity at larger propagation distances. The revised Fig. 4 reflects these improvements.\nChanges made in the revised manuscript:\nPage-11\nThe intensity cross-sections along the xz-plane in Fig. 4 correspond to the normalized central line of the xy-plane at the same axial positions. Consequently, the intensity in the xz-plane does not decrease at larger propagation distances, as the light intensity at each axial position is normalized.\nPage-29\nFigure 4 | Dispersion-free and diffractive-free characteristics of ANPVB. (a1-d1) x-z intensity profiles of the ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) under the wavelengths of 450, 532, 633 and 780 nm respectively. The corresponding (a2-d2) intensity and (a3-d3) phase distributions at z=18 mm respectively.\n3. The manuscript highlights efficient transmission as one of its key novelties. However, there is no rigorous measurement or numerical analysis of the actual transmission efficiency provided across the visible wavelength range. To support this claim, it would greatly strengthen the work to include quantified efficiency data.\nReply: Thank you for raising this important point. In response to your comment, we have performed additional simulations and experimental measurements to provide a more rigorous quantification of the transmission efficiency across the visible wavelength range.\nTo accurately evaluate the performance, we increased the grid resolution and re-scanned the transmittance and phase responses of meta-atoms of various sizes. The updated results are presented in Fig. S1. As shown in Fig. S1(a6), the simulated polarization conversion efficiencies (PCEs) of the selected meta-atom are 89.54%, 43.09%, 37.99%, and 55.10% at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively, indicating relatively high conversion performance across these wavelengths. At 633 nm, for instance, the meta-atom exhibits a PCE of 37.99%. This means that when a left-handed circularly polarized Gaussian beam is incident, 37.99% of the energy is converted into the right-handed component carrying the desired phase profile, while the remaining 62.01% remains in the original polarization. Due to the divergence of the beam during propagation and the finite region of the simulation domain, some energy is not captured in the near-field. To approximate the total energy of both polarization components, we analyzed the optical field of the ANPVB (with parameters α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) after propagating to 1.5 mm, where the beam is weakly divergent. The total energy of each handedness was evaluated by propagating the respective components an additional 1.5 mm and integrating their intensities. The resulting simulated PCE at 1.5 mm was found to be 36.03% at 633 nm. Similarly, we computed the PCEs at 1.5 mm for the other wavelengths: 89.48% at 450 nm, 42.58% at 532 nm, and 49.41% at 780 nm. Experimentally, we measured the PCEs of the metasurface designed for ANPVB generation at the same propagation distance (1.5 mm) and wavelengths. The values obtained were 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These results show good agreement between simulation and experiment at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths.\nWe have updated the manuscript and Supplementary Information with these quantitative efficiency data to more robustly support our claims regarding transmission efficiency.\nFig. S1 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\nChanges made in the Supplementary Material:\nPage-7, 8\nSupplementary Note 7: Numerically calculated and Measured PCEs of TiO₂ elliptical cylinders and rectangular bars\nAs shown in Fig. S1(a6), the polarization conversion efficiencies (PCEs) of the selected meta-atom are numerically calculated to be 89.54%, 43.09%, 37.99%, and 55.10% at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. Specifically, at 633 nm, the meta-atom exhibits a PCE of 37.99%. Under illumination by a left-handed circularly polarized Gaussian beam, 37.99% of the incident energy is converted into a right-handed beam carrying the modulated phase, while the remaining 62.01% remains in the unconverted left-handed Gaussian state. As the beam propagates, the optical field diverges. Due to the limited simulation domain, energy beyond the sampling region is not captured. To estimate the total energy of both polarization components, we analyze the weakly divergent field of an ANPVB (with parameters α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) after 1.5 mm of propagation. The left-handed component's total energy at z = 1.5 mm is obtained by propagating the field derived from Eq. (2) over an additional 1.5 mm. Similarly, the right-handed component's energy is calculated by propagating the Gaussian beam multiplied by the phase profile over the same distance. The simulated PCE at 1.5 mm, defined as the ratio of the right-handed energy to the total energy, is found to be 36.03% at 633 nm. Using the same approach, the simulated PCEs at 1.5 mm for other wavelengths are calculated as 89.48% at 450 nm, 42.58% at 532 nm, and 49.41% at 780 nm.\nExperimentally, the PCEs of the phase-engineered metasurface designed for ANPVB generation were measured at z = 1.5 mm, yielding values of 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These results indicate good agreement between simulations and experiments at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths.\n4. Building on Comment 3, based on my experience, metasurfaces with high-aspect-ratio structures often exhibit discrepancies between simulated and experimentally measured efficiencies. I am curious whether the authors have attempted to measure the transmission efficiency in a manner comparable to the simulated results shown in Fig. S1 (e.g., panel a6).\nReply: Thank you for this insightful comment. Based on your suggestion, we have performed additional high-accuracy simulations by refining the mesh grid and rescanning the transmittance and phase responses of meta-atoms with various sizes. The updated results are presented in Fig. S1. As shown in Fig. S1(a6), the simulated polarization conversion efficiencies (PCEs) of the chosen meta-atom are 89.54%, 43.09%, 37.99%, and 55.10% at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively, indicating high performance across the evaluated spectrum. To enable a direct comparison with simulation, we also experimentally measured the PCEs at a propagation distance of 1.5 mm for the ANPVB metasurface (with parameters α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30). The measured values are 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These results show good agreement between simulation and experiment at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths.\nDespite the notable difference at 450 nm, both experimental and simulated PCEs exceed 35% across all four wavelengths, demonstrating reasonably high efficiency even for high-aspect-ratio meta-atoms. We have included these quantitative comparisons in the revised supplementary material to provide a clearer assessment of the metasurface performance.\nFig. S1 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\nTo achieve high polarization conversion efficiency across a broad bandwidth, we simulated a rectangular bar structure with a height of 600 nm and a period of 350 nm. As illustrated in Fig. S9(a), the meta-atom consists of a TiO₂ rectangular bar with a short side a, a long side b, and a height h = 600 nm, mounted on a SiO₂ square substrate with meta atom period P = 350 nm. Figures S9(a1) and S9(a2) show, respectively, the transmission and phase under x-linearly polarized light at 633 nm for the x-polarized output, as functions of the parameters a and b. Similarly, Figs. S9(a3) and S9(a4) present the transmission and phase for the y-polarized output under y-linearly polarized illumination. Based on these results, the corresponding PCE is summarized in Fig. S9(a5). The meta-atom marked with a five-pointed star, corresponding to b = 285 nm and a = 100 nm, was selected for further analysis. As presented in Fig. S9(a6), the selected meta-atom maintains a high average PCE of 70% across the broad wavelength range from 400 to 800 nm. Although we chose a 1 μm-height titanium dioxide metasurface for experimental validation due to the greater maturity of our fabrication process for 1 μm TiO₂ structures, this selection does not affect the universal applicability of the proposed theory. As demonstrated in the simulations above, it is relatively straightforward to achieve high broadband polarization conversion efficiency across a given wavelength range by optimizing the structural parameters.\nFig. S9 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\nChanges made in the revised manuscript:\nPage-7-9\nSupplementary Note 7: Numerically calculated and Measured PCEs of TiO₂ elliptical cylinders and rectangular bars\nAs shown in Fig. S1(a6), the polarization conversion efficiencies (PCEs) of the selected meta-atom are numerically calculated to be 89.54%, 43.09%, 37.99%, and 55.10% at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. Specifically, at 633 nm, the meta-atom exhibits a PCE of 37.99%. Under illumination by a left-handed circularly polarized Gaussian beam, 37.99% of the incident energy is converted into a right-handed beam carrying the modulated phase, while the remaining 62.01% remains in the unconverted left-handed Gaussian state. As the beam propagates, the optical field diverges. Due to the limited simulation domain, energy beyond the sampling region is not captured. To estimate the total energy of both polarization components, we analyze the weakly divergent field of an ANPVB (with parameters α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) after 1.5 mm of propagation. The left-handed component's total energy at z = 1.5 mm is obtained by propagating the field derived from Eq. (2) over an additional 1.5 mm. Similarly, the right-handed component's energy is calculated by propagating the Gaussian beam multiplied by the phase profile over the same distance. The simulated PCE at 1.5 mm, defined as the ratio of the right-handed energy to the total energy, is found to be 36.03% at 633 nm. Using the same approach, the simulated PCEs at 1.5 mm for other wavelengths are calculated as 89.48% at 450 nm, 42.58% at 532 nm, and 49.41% at 780 nm.\nExperimentally, the PCEs of the phase-engineered metasurface designed for ANPVB generation were measured at z = 1.5 mm, yielding values of 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These results indicate good agreement between simulations and experiments at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths.\nTo achieve high polarization conversion efficiency across a broad bandwidth, we simulated a rectangular bar structure with a height of 600 nm and a period of 350 nm. As illustrated in Fig. S9(a), the meta-atom features a TiO₂ rectangular bar with a short side a, a long side b, and a height h of 600 nm, positioned on a square SiO₂ substrate. The period P of the meta-atom is 350 nm. Figures S9(a1) and S9(a2) show, respectively, the transmission and phase under x-linearly polarized light at 633 nm for the x-polarized output, as functions of the parameters a and b. Similarly, Figs. S9(a3) and S9(a4) present the transmission and phase for the y-polarized output under y-linearly polarized illumination. Based on these results, the corresponding PCE is summarized in Fig. S9(a5). The meta-atom marked with a five-pointed star, corresponding to b = 285 nm and a = 100 nm, was selected for further analysis. As presented in Fig. S9(a6), the selected meta-atom maintains a high average PCE of 70% across the broad wavelength range from 400 to 800 nm. Although we chose a 1 μm-height titanium dioxide metasurface for experimental validation due to the greater maturity of our fabrication process for 1 μm TiO₂ structures, this selection does not affect the universal applicability of the proposed theory. As demonstrated in the simulations above, it is relatively straightforward to achieve high broadband polarization conversion efficiency across a given wavelength range by optimizing the structural parameters.\nFig. S9 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\n5. It would be helpful to provide higher-magnification SEM images of the fabricated metasurfaces. As currently presented, it is difficult to assess the structural quality and fidelity of the fabricated samples, which is important for evaluating the precision and reproducibility of the device.\nReply: Thank you for this helpful suggestion. In response, we have now included higher-magnification SEM images of the fabricated metasurfaces in Supplementary Note 8. These images offer a clearer view of the structural details and quality of the samples, allowing for better evaluation of the fabrication fidelity and dimensional accuracy. We believe these additions will greatly assist in assessing the precision and reproducibility of the devices.\nChanges made in the revised manuscript:\nPage-14\nHigher-magnification SEM images of the fabricated metasurfaces are provided in Supplementary Note 8.\nChanges made in the Supplementary Material:\nPage-9, 10\nSupplementary Note 8: High-resolution SEM characterization of the metasurfaces\nFigure S10 provides higher-magnification SEM images characterizing the structural details of the metasurfaces fabricated for Fig. 5 in main text.\nFig. S10 (a1, b1, c1) The overall scanning electron microscopy (SEM) images of the fabricated metasurfaces generating ANPVBs with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=1, 15 and 30). The local SEM images (a2, b2, c2) (top view) and (a3, b3, c3) (side view), respectively. The scale bars in Figs. S1(a1-c1), S1(a2-c2) and S1(a3-c3) represent 200 μm, 1 μm and 1 μm, respectively.\n6. The metasurface is reported to have a structural period of 450 nm, which coincides with the shortest working wavelength used in the experiments (450 nm). Given that this matches the wavelength, diffraction is expected to occur based on fundamental grating theory. I would appreciate it if the authors could clarify whether the claimed non-diffracting behavior at 450 nm is theoretically valid in this regime. A brief analysis or justification would strengthen the credibility of the results at this wavelength.\nReply: Thank you for raising this important point. Regarding the concern about potential diffraction effects when the structural period (450 nm) equals the shortest operating wavelength (450 nm), we provide the following clarification based on fundamental grating theory:\nFor normal incidence, the grating equation is given by: dsink, where d = 450 nm is the period, λ = 450 nm is the wavelength, and k is the diffraction order. Under this condition, the zeroth-order transmitted light (k = 0) propagates along the optical axis (θ = 0°), while the first-order diffracted beams (k = 1) oChanges made in the revised manuscript:\ncPage-12, 13\ncFor normal incidence, the grating equation is given by: dsink, where d = 450 nm is the period, λ\nu= 450 nm is the wavelength, and k is the diffraction order. Under this condition, the zeroth-order\rtransmitted light (k = 0) propagates along the optical axis (θ = 0°), while the first-order diffracted beams\n(k = ±1) occur at θ = ±90°, effectively propagating as surface waves perpendicular to the optical\raaxis. Since these higher-order diffraction components do not contribute to the on-axis propagation, the\ntzeroth-order transmittance—which carries the intended phase profile—dominates the output field along\nt\nθh\ne\n=\np7. Panels (a1)–(a10), etc., lack proper descriptions in the figure captions. Please revise the captions to\n9rclarify each subfigure.\n0oReply: Thank you for pointing this out. We have revised the captions of Figures 3 and S5 to include\n,pd eetfafielcetdiv deelys cprirpotpioangsa tifnogr eaas cshu rsfuabcfei gwuarev e(se .pge.,r ppeanndeilcsu (laar1 )t–o t(hae1 0o)p).t icTahle a uxpisd. aSteindc fei gthueres ec ahpigtihoenrs- onrdoewr\ndaclieffarralcyt ieoxnp lcaoinm tphoen ceonntst ednot annodt pcuornptroisbeu toef teov ethrye sounb-faixgiusr ep.r oTphaeg raetvioisne, dt hvee rzseiorontsh -aorerd perro tvriadnesdm bitetlaonwce f—or\nwgoyohuirc hre cfearrerinecse t. he intended phase profile—dominates the output field along the propagation direction.\nMa\not\nri\nChanges made in the revised manuscript:\nPage-28\nFigure 3 | Diffraction comparison between PVB and ANPVB. (a) and (b) x-z intensity profiles for the conventional PVB of (f=18 mm, l=30, α=0.04733, R = 0.54 mm) and ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), respectively. (a1-a10) and (b1-b10) Corresponding x-y intensity distributions at z=3, 6, 9, 12, 15, 18, 21, 24, 27 and 30 mm for PVB and ANPVB, respectively. Intensity profiles for both the xy and xz planes are individually normalized for each image. As the propagation distance increases, the diameter of the PVB in Figs. 3(a1-a10) expands, while the diameter of the ANPVB in Figs. 3(b1-b10) remains approximately constant. (c) Beam radius versus axial position. (d) Plot of the relative variation of diameters with respect to axial positions.\nChanges made in the Supplementary Material:\nPage-5\nFig. S5 | Diffraction comparison between PVB and ANPVB with larger beam sizes. (a) and (b) x-z intensity profiles for the conventional PVB of (f=18 mm, l=100, α=0.007725, R=1.35 mm) and ANPVB of (α=3, β=0.8, γ=3, t=700, δ=5, R=1.35 mm, l=100), respectively. (a1-a10) and (b1-b10) The corresponding x-y intensity distributions at z=3, 6, 9, 12, 15, 18, 21, 24, 27 and 30 mm of the PVB and ANPVB, respectively. Intensity profiles for both the xy and xz planes are individually normalized for each image. As the propagation distance increases, the diameter of the PVB in Figs. S5(a1-a10) expands, while the diameter of the ANPVB in Figs. S5(b1-b10) remains approximately constant. (c) Beam radius versus axial position. (d) Plot of the relative variation of diameters with respect to axial positions.\n8. It appears that no direct measurement of the momentum space has been performed for the metasurfaces. To substantiate the claim of non-diffracting behavior, it would be important to experimentally verify the beam’s momentum distribution.\nReply: Thank you for raising this important point. In direct response to the comment, we have not only performed experimental measurements of the momentum-space distributions to substantiate the non-diffracting behavior, but also conducted further numerical simulations and local energy flow analyses to elucidate the evolution process in momentum space and the role of radial momentum in particular.\nAs shown in Fig. S11(a1–a3), the simulated momentum-space intensity distributions are provided for spiral phases with topological charges of 1, 15, and 30, respectively. The corresponding experimentally measured results are presented in Fig. S11(a4–a6). For the proposed radial phase with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=0), the simulated momentum-space intensity distribution is given in Fig. S11(b1), and the measured result is shown in Fig. S11(b2). Furthermore, Figs. S11(c1–c3) illustrate the simulated momentum-space intensity distributions for the designed phases that incorporate both radial and azimuthal momentum, with l = 1, 15, and 30 (while α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm remain constant). The corresponding experimental measurements are provided in Figs. S11(c4–c6).\nOverall, the excellent agreement between simulations and experiments confirms the intended momentum-space properties and strongly supports the non-diffracting behavior of the generated beams. However, owing to the Fourier transform relationship between momentum space and real space, all localized momentum components are centralized in such measurements. Although this global momentum-space characterization effectively demonstrates the consistency between the fabricated device and the designed momentum profile, it does not directly capture the role of localized momentum distribution—associated with the local phase gradient—in the formation of the non-diffracting vortex beam. To further elucidate this mechanism, we have incorporated an analysis of localized momentum propagation (i.e., localized energy flow), which helps clarify the essential role of radial momentum in forming the non-diffracting perfect vortex beam.\nComplementing this, a detailed energy flow evolution analysis in momentum space is also provided (see Supplementary Note 9), tracing its dynamics and highlighting the contribution of radial momentum in maintaining the non-diffracting behavior. The dominant radial momentum components shown in Figures S12 (c4) and (d4) provide clear evidence for the proposed mechanism.\nFig. S11 (a1–a3) Simulated and (a4–a6) measured momentum-space intensity distributions of azimuthal momentum for spiral phases with topological charges 1, 15, and 30. (b1) Simulated and (b2) measured momentum-space intensity distributions of radial momentum for the proposed phase (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=0). (c1–c3) Simulated and (c4–c6) measured momentum-space intensity distributions of combined radial and azimuthal momentum for the proposed phases with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=1, 15 and 30). The scale bar in Fig. S11(a1) represents 0.04 . P\nFig. S12(a1–a5) show the k-space intensity distribution, the intensity distribution along the central line, the k-space logarithmic intensity distribution, the intensity distribution along the central line of the logarithmic intensity, and the k-space phase distribution, respectively. From Fig. S12(a1–a5), it can be observed that the proposed radial phase produces concentric ring intensity distributions in k-space, with the phase exhibiting an axially symmetric ring-like profile. Owing to this k-space phase structure, the interference of high- and low-frequency rings redistributes the central bright spot’s energy into the outer rings. As the propagation distance increases, the phase difference between the high- and low-frequency z-direction wave vectors also increases. Within a certain range, however, this phase difference remains small enough to allow the interference rings to propagate without diffraction. In addition, in the k-space intensity distribution of the proposed radial phase, the ring intensity gradually decreases from low- to high-frequency radial wave vectors. This distribution suppresses the contribution of high-frequency components to beam spreading. An increase in the azimuthal wave vector causes the concentric rings generated by the proposed radial phase in k-space to expand outward. For small topological charges, this outward expansion is minor, and the high-frequency components shift only slightly, resulting in a weaker contribution to beam spreading. Consequently, the interference rings formed by concentric k-space rings can maintain nondiffracting propagation over a certain distance, allowing multiple radial wave vectors in k-space to realize nondiffraction. In contrast, for large topological charges, the concentric rings expand significantly outward, with high-frequency components contributing more strongly to beam spreading. As a result, the interference rings in k-space gradually expand during propagation, and multiple radial wave vectors can no longer sustain nondiffracting behavior. Fig. S12(b1) and (b2) show the designed radial phase distribution of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm) and the corresponding phase profile along the central line, respectively. Fig. S12(c1–c5) present the x, y, and z components of the Poynting vector, along with the 3D Poynting vectorial structures at the central ring and the planar Poynting vector at a propagation distance of 0.3 mm. Fig. S12(d1–d5) show the same quantities at a propagation distance of 3 mm. It can be seen that at short distances, the Poynting vectors tilt outward, leading to continuous outward spreading of the beam. At longer distances, however, the z-component dominates over the x and y components, aligning the Poynting vectors along the z-direction and enabling nondiffracting propagation without rapid divergence. From Fig. S12(b2), it is also evident that the phase gradient decreases gradually from a large value at the center to nearly zero at the edge. This corresponds to energy spreading outward from the center to a fixed radial position as the propagation distance increases, where it remains confined. In other words, the Poynting-vector rings expand outward with distance until stabilizing at a fixed radius. These results demonstrate that radial phase engineering enables the control of radial momentum to achieve nondiffracting propagation.\nFig. S12 (a1–a5) k-space intensity, central-line profile, logarithmic intensity, logarithmic profile, and phase distribution. (b1) Designed radial phase of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm) and (b2) its central-line phase profile. (c1–c5) and (d1–d5) Poynting vector components, Poynting vector 3D structures and the planar Poynting vector at 0.3 mm and 3 mm, respectively. The    scale bars in Figs. S12(a1), S12(a3), S12(a5) and S12(b1) represent 0.002 , 0.2 , 0.2 and 0.2 mm, respectively. P P P\nChanges made in the Supplementary Material:\nPage-10-13\nSupplementary Note 9: Momentum space and Pointing vector of ANPVBs\nAs shown in Fig. S11(a1–a3), the simulated momentum-space intensity distributions are provided for spiral phases with topological charges of 1, 15, and 30, respectively. The corresponding experimentally measured results are presented in Fig. S11(a4–a6). For the proposed radial phase with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=0), the simulated momentum-space intensity distribution is given in Fig. S11(b1), and the measured result is shown in Fig. S11(b2). Furthermore, Figs. S11(c1–c3) illustrate the simulated momentum-space intensity distributions for the designed phases that incorporate both radial and azimuthal momentum, with l = 1, 15, and 30 (while α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm remain constant). The corresponding experimental measurements are provided in Figs. S11(c4–c6).\nFig. S11 | Measurement of the momentum space. (a1–a3) Simulated and (a4–a6) measured momentum-space intensity distributions of azimuthal momentum for spiral phases with topological charges 1, 15, and 30. (b1) Simulated and (b2) measured momentum-space intensity distributions of radial momentum for the proposed phase (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=0). (c1–c3) Simulated and (c4–c6) measured momentum-space intensity distributions of combined radial and azimuthal momentum for the proposed phases with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=1, 15 and 30). The scale bar in Fig. S11(a1) represents 0.04 . P\nOverall, the excellent agreement between simulations and experiments confirms the intended momentum-space properties and strongly supports the non-diffracting behavior of the generated beams. However, owing to the Fourier transform relationship between momentum space and real space, all localized momentum components are centralized in such measurements. Although this global momentum-space characterization effectively demonstrates the consistency between the fabricated device and the designed momentum profile, it does not directly capture the role of localized momentum distribution—associated with the local phase gradient—in the formation of the non-diffracting vortex beam. To further elucidate this mechanism, we have incorporated an analysis of localized momentum propagation (i.e., localized energy flow), which helps clarify the essential role of radial momentum in forming the non-diffracting perfect vortex beam. Complementing this, a detailed energy flow evolution analysis in momentum space is also provided (see Supplementary Note 9), tracing its dynamics and highlighting the contribution of radial momentum in maintaining the non-diffracting behavior. The dominant radial momentum components shown in Figures S12 (c4) and (d4) provide clear evidence for the proposed mechanism.\nFig. S12(a1–a5) show the k-space intensity distribution, the intensity distribution along the central line, the k-space logarithmic intensity distribution, the intensity distribution along the central line of the logarithmic intensity, and the k-space phase distribution, respectively. From Fig. S12(a1–a5), it can be observed that the proposed radial phase produces concentric ring intensity distributions in k-space, with the phase exhibiting an axially symmetric ring-like profile. Owing to this k-space phase structure, the interference of high- and low-frequency rings redistributes the central bright spot’s energy into the outer rings. As the propagation distance increases, the phase difference between the high- and low-frequency z-direction wave vectors also increases. Within a certain range, however, this phase difference remains small enough to allow the interference rings to propagate without diffraction. In addition, in the k-space intensity distribution of the proposed radial phase, the ring intensity gradually decreases from low- to high-frequency radial wave vectors. This distribution suppresses the contribution of high-frequency components to beam spreading. An increase in the azimuthal wave vector causes the concentric rings generated by the proposed radial phase in k-space to expand outward. For small topological charges, this outward expansion is minor, and the high-frequency components shift only slightly, resulting in a weaker contribution to beam spreading. Consequently, the interference rings formed by concentric k-space rings can maintain nondiffracting propagation over a certain distance, allowing multiple radial wave vectors in k-space to realize nondiffraction. In contrast, for large topological charges, the concentric rings expand significantly outward, with high-frequency components contributing more strongly to beam spreading. As a result, the interference rings in k-space gradually expand during propagation, and multiple radial wave vectors can no longer sustain nondiffracting behavior. Fig. S12(b1) and (b2) show the designed radial phase distribution of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm) and the corresponding phase profile along the central line, respectively. Fig. S12(c1–c5) present the x, y, and z components of the Poynting vector, along with the 3D Poynting vectorial structures at the central ring and the planar Poynting vector at a propagation distance of 0.3 mm. Fig. S12(d1–d5) show the same quantities at a propagation distance of 3 mm. It can be seen that at short distances, the Poynting vectors tilt outward, leading to continuous outward spreading of the beam. At longer distances, however, the z-component dominates over the x and y components, aligning the Poynting vectors along the z-direction and enabling nondiffracting propagation without rapid divergence. From Fig. S12(b2), it is also evident that the phase gradient decreases gradually from a large value at the center to nearly zero at the edge. This corresponds to energy spreading outward from the center to a fixed radial position as the propagation distance increases, where it remains confined. In other words, the Poynting-vector rings expand outward with distance until stabilizing at a fixed radius. These results demonstrate that radial phase engineering enables the control of radial momentum to achieve nondiffracting propagation.\nFig. S12 (a1–a5) k-space intensity, central-line profile, logarithmic intensity, logarithmic profile, and phase distribution. (b1) Designed radial phase of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm) and (b2) its central-line phase profile. (c1–c5) and (d1–d5) Poynting vector components, Poynting vector 3D structures and the planar Poynting vector at 0.3 mm and 3 mm, respectively. The    scale bars in Figs. S12(a1), S12(a3), S12(a5) and S12(b1) represent 0.002 , 0.2 , 0.2 and 0.2 mm, respectively. P P P\nReviewer #2 (Remarks to the Author):\nThe work presented in the manuscript \"Ultrabroadband, Achromatic, and Non-Diffracting Perfect Optical Vortex Generation via Radial Momentum Control in Dielectric Metasurfaces\" is interesting for its contribution to perfect optical vortex generation with metasurfaces, however, I do not believe there is significant enough novelty and advancement for publication in Nature Communications. A more specialised optics focused journal could be more suitable in my opinion.\nSome additional comments are as follows:\n1) To start with, I feel that the title is a little overreaching in its claim. Work such as Refs 40 and 42 in the manuscript demonstrate PB phase metasurfaces for the generation of perfect optical vortex beams. It is just a given that PB phase is unrelated to the wavelength of light, so the use of PB phase metasurfaces automatically gives you the broadband feature. The meta-atom design then defines the conversion efficiency, and ultimately the performance at other wavelengths. Even with a super low efficiency (~<10%), the functionality can still work after filtering the unconverted light. So the term 'ultrabroadband' is a little overreaching in my opinion. The applies also to 'achromatic', which I am assuming here means the same thing as ultrabroadband. The 'non-diffracting' nature is also a property of perfect vortex generation (however I do acknowledge the contribution in this point).\nReply: Thank you for your insightful and constructive feedback. We fully agree that the geometric (Pancharatnam–Berry) phase inherently supports wavelength-independent phase modulation, and that PB phase metasurfaces can indeed enable broadband operation in principle, as demonstrated in prior works including Refs. 40 and 42. However, we would like to clarify a key distinction: while PB phase provides achromatic phase profile, the resulting vortex beam—whether conventional or “perfect” type—still inherently suffers from chromatic dispersion in its radial size. That is, even with ideal metasurface performance, the diameter of the vortex ring would vary with wavelength. Our work goes beyond employing the PB phase alone. By introducing additional engineering of the radial momentum, we actively compensate for this dispersion, thereby achieving an ultrabroadband and non-diffracting perfect vortex beam that maintains both a consistent ring diameter and propagation invariance across a wide band of wavelengths. This combined control of phase, radial profile, and dispersion represents a distinct advance over previous PB-phase-based designs, which addressed broadband phase modulation but not the chromatic spatial dispersion of the vortex structure itself.\nTo clarify this, we simulated a conventional vortex beam with topological charge l=30, a PVB with (f=18 mm, l=30, α=0.04733, R’=0.54 mm), and a ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) under identical conditions (phase profile, propagation distance = 18 mm) across wavelengths from 450 nm to 780 nm. The results in Figs. S13(a5) and (b5) clearly show significant variation in ring diameters of the conventional vortex and PVB with wavelength, confirming that chromatic dispersion persists even with identical phase profiles. However, the results in Fig. S13(c5) show that the diameter variation of ANPVB at the same position across different wavelengths is minimal, indicating that the radial momentum has significantly reduced the dispersion to almost non-dispersive levels. Therefore, although PB phase enables broadband phase response, it does not inherently achieve achromatic performance — i.e., wavelength-independent ring size and non-diffracting propagation. Our work introduces an additional engineering step: precise control of the radial momentum, which compensates for this dispersion and enables a constant ring diameter and non-diffracting propagation across a broad band. We have revised the relevant sections to more accurately reflect that our key contribution is achromatic and non-diffracting performance — rather than broadband generation alone.\nFig. S13 Chromatic dispersion of a conventional optical vortex, PVB and AVPVB. (a1-a4), (b1-b4) and (c1-c4): Intensity distributions of the conventional vortex with topological charge 𝑙=30, the PVB with (f=18 mm, l=30, α=0.04733, R’=0.54 mm) and ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively, at 𝑧=18 mm. (a5), (b5) and (c5): Intensity profiles along the central lines of the conventional vortex, PVB and ANPVB, illustrating spectral variations in beam sizes. The scale bars in Figs. S13(a1-c1) represent 0.1 mm.\nNevertheless, this functionality can be realized as long as a suitable radial phase profile is implemented for each wavelength—indicating that the approach is not limited to the PB phase. To verify this, we performed additional simulations. For example, transmission phases can also be engineered to achieve the proposed radial phase profile across different wavelengths, thereby enabling ultrabroadband performance.\nAs illustrated in Fig. R1(a), the meta-atom consists of a TiO₂ rectangular bar with a short side length a, a long side length b, and a height h=600 nm, placed on a SiO₂ square substrate with a side length P=350 nm. Under x-linearly polarized light at 633 nm, the transmission and phase for the x-polarization component are shown in Figs. R1(a1) and (a2), respectively, as functions of a and b.\nFig. R1 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm.\nWe used the transmission phase φx of the rectangular bar at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm to match both the radial phase and the spiral phase (with topological charge l=30) for generating an achromatic non-diffracting perfect vortex beam with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30). The optimal meta-atom geometry is selected from a comprehensive library of structures with varying dimensions. This selection is made by minimizing the sum of absolute differences between the meta-atom's phase response across multiple wavelengths and the target phase for individual pixels at those same wavelengths. This approach allows the metasurface to exhibit nearly identical phase responses at these wavelengths. Figures S14(a1–d1) show the matched phase profiles for each wavelength. The corresponding intensity and phase distributions after 18 mm of propagation are presented in Figs. S14(a2–d2) and S14(a3–d3), respectively. These results confirm that the metasurface designed via transmission phase modulation successfully generates vortex beams carrying a topological charge of 30 with consistent ring diameters across different wavelengths.\nThis demonstrates that applying the same radial phase modulation at each wavelength is sufficient to maintain uniform vortex diameters at a fixed propagation distance. When vortices of different wavelengths exhibit the same diameter at the same propagation distance, they can be referred to as achromatic vortices. By modulating the conventional vortex phase with the radial phase proposed in this work, vortices at different wavelengths achieve identical diameters at the same position, thereby adding achromatic functionality to conventional ultrabroadband vortex beams.\nFig. S14 (a1-d1) Matched phases of the ANPVB with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. (a2–d2) Intensity and (a3–d3) phase distributions after the matched fields propagate 18 mm. The scale bar in Fig. S14(a1) represents 0.1 mm.\nChanges made in the Supplementary Material:\nPage-14-17\nSupplementary Note 10: Ultrabroadband and achromatic functions of the vortex empowered by the radial phase modulation\nMetasurfaces based on the Pancharatnam–Berry phase enable ultrabroadband generation of vortex and perfect vortex beams. To clarify this, we simulated a conventional vortex beam with topological charge l=30, a PVB with (f=18 mm, l=30, α=0.04733, R’=0.54 mm), and a ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) under identical conditions (phase profile, propagation distance = 18 mm) across wavelengths from 450 nm to 780 nm. The results in Figs. S13(a5) and (b5) clearly show significant variation in ring diameters of the conventional vortex and PVB with wavelength, confirming that chromatic dispersion persists even with identical phase profiles. However, the results in Fig. S13(c5) show that the diameter variation of ANPVB at the same position across different wavelengths is minimal, indicating that the radial momentum has significantly reduced the dispersion to almost non-dispersive levels. Thus, although the PB phase supports broadband vortex generation, it does not produce vortices of consistent size across different wavelengths at the same position. This achromatic functionality—maintaining uniform vortex diameter across wavelengths—is achieved through the radial phase profile proposed in this work.\nFig. S13 Chromatic dispersion of a conventional optical vortex, PVB and AVPVB. (a1-a4), (b1-b4) and (c1-c4): Intensity distributions of the conventional vortex with topological charge 𝑙=30, the PVB with (f=18 mm, l=30, α=0.04733, R’=0.54 mm) and ANPVB of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively, at 𝑧=18 mm. (a5), (b5) and (c5): Intensity profiles along the central lines of the conventional vortex, PVB and ANPVB, illustrating spectral variations in beam sizes. The scale bars in Figs. S13(a1-c1) represent 0.1 mm.\nThe approach is not limited to the PB phase; in principle, any phase modulation mechanism capable of imparting the required radial phase shift at each wavelength can realize this function. For example, the transmission phase can also be designed to implement the radial phase across multiple wavelengths, thereby enabling ultrabroadband achromatic performance.\nIn this study, we used the transmission phase database φₓ of rectangular nanoantennas at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm to simultaneously satisfy the radial phase and the spiral phase with a topological charge of 30 for an ANPVB with parameters (α = 18, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 30). This allows the metasurface to exhibit nearly identical phase responses at these wavelengths. Figures S14(a1–d1) show the matched phase profiles for each wavelength. The corresponding intensity and phase distributions after 18 mm of propagation are presented in Figs. S14(a2–d2) and S14(a3–d3), respectively. The results confirm that the metasurface designed via transmission phase modulation successfully generates vortex beams carrying a topological charge of 30 with nearly identical diameters across all four wavelengths.\nThis demonstrates that applying the same radial phase modulation at each wavelength is sufficient to maintain a consistent vortex diameter at a fixed propagation distance, independent of wavelength. Vortices that satisfy this condition are referred to as achromatic vortices. By incorporating the proposed radial phase, conventional vortex phases can be modulated to yield achromatic vortices—enabling wavelength-independent size stability and thereby extending traditional ultrabroadband vortex generation to include achromatic functionality.\nFig. S14 (a1-d1) Matched phases of the ANPVB with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. (a2–d2) Intensity and (a3–d3) phase distributions after the matched fields propagate 18 mm. The scale bar in Fig. S14(a1) represents 0.1 mm.\n2) Continuing from comment 1, I feel that the main contribution of this work is the increased depth that the vortex maintains its shape. However, it is unclear how this compares with other research.\nReply: We appreciate your comment regarding the propagation performance of our vortex beam. Indeed, a key advancement of this work is the significantly increased propagation distance over which the vortex maintains its shape—a crucial feature for practical applications.\nIn comparison to traditional Bessel beams, which are well-known as diffraction-free vortex beams but exhibit vortex diameters that vary with topological charge, our proposed ANPVB maintains a constant ring diameter independent of the topological charge. Thus, the ANPVB functions simultaneously as a diffraction-free beam and a perfect vortex beam. Moreover, while conventional diffraction-free vortices show chromatic dispersion—that is, their vortex diameters change with wavelength at a given propagation distance—the ANPVB preserves a consistent diameter across different wavelengths, enabling achromatic operation.\nIn summary, the ANPVB introduced in this work uniquely combines three key attributes into a single vortex beam: non-diffracting propagation, topological charge-independent ring size (perfect vortex), and wavelength-independent diameter (achromaticity). This integration distinguishes it from previously reported vortex beams and may enhance its applicability in areas such as multi-wavelength optical communication and integrated photonic systems.\nChanges made in the revised manuscript:\nPage-15, 16\nIn comparison to traditional Bessel beams, which are well-known as diffraction-free vortex beams but exhibit vortex diameters that vary with topological charge, our proposed ANPVB maintains a constant ring diameter independent of the topological charge. Thus, the ANPVB functions simultaneously as a diffraction-free beam and a perfect vortex beam. Moreover, while conventional diffraction-free vortices show chromatic dispersion—that is, their vortex diameters change with wavelength at a given propagation distance—the ANPVB preserves a consistent diameter across different wavelengths, enabling achromatic operation. In summary, the ANPVB introduced in this work uniquely combines three key attributes into a single vortex beam: non-diffracting propagation, topological charge-independent ring size (perfect vortex), and wavelength-independent diameter (achromaticity). This integration distinguishes it from previously reported vortex beams and may enhance its applicability in areas such as multi-wavelength optical communication and integrated photonic systems.\n3) Related to the PB phase modulation, it would be interesting to see experimental values of the conversion efficiency of the metasurfaces, at all different wavelengths that the authors show. The simulated values are often far from what is experimentally validated, but this information is missing from the manuscript. The simulated values are found in the SI, and it appears that the meta-atom was not designed to be highly efficient in broadband operation (as the title suggests), and also not optimised for specific wavelengths either, again taking away any meaning of broadband operation (it is therefore just a byproduct of using PB phase, not an engineered outcome).\nReply: We sincerely thank the reviewer for raising this important point regarding conversion efficiency and its relevance to broadband performance. We agree that experimental efficiency measurements across wavelengths would further strengthen the manuscript, and we have supplemented the SI with higher-accuracy simulations refining mesh settings and meta-atom responses (see updated Fig. S1).\nMore importantly, we would like to emphasize that the core contribution of this work is a general-momentum-based design methodology to achieve ultrabroadband, achromatic, and non-diffracting perfect vortex beams—going beyond what conventional PB-phase metasurfaces can deliver. While PB phase offers inherent broadband phase modulation, it does not inherently compensate for chromatic changes in the radial size of the vortex structure. Our approach decouples the design of the phase profile from specific meta-atom implementations, meaning that any high-efficiency broadband metasurface platform (whether based on PB phase or other mechanisms) can, in principle, be employed to realize the proposed beam functionality.\nThe efficiency of the metasurface is, of course, important for practical applications and can be further optimized in subsequent device-level studies through more advanced meta-atom designs or material choices. However, the key advance here is the conceptual and theoretical demonstration of simultaneous ultrabroadband operation, achromatic behavior, and non-diffracting propagation in a PVB—a capability not achievable with traditional designs, even those with high efficiency. Thus, this work primarily establishes a universal phase-design strategy that opens a new direction in structured light fields, with specific efficiency optimization being the next step toward application-specific implementations.\nFig. S1 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\nAs shown in Fig. S1(a6), the simulated polarization conversion efficiencies of the selected meta-atom are 89.54% at 450 nm, 43.09% at 532 nm, 37.99% at 633 nm, and 55.10% at 780 nm, reflecting competitive performance across a broad spectral range. To facilitate direct comparison with simulations, we experimentally characterized the PCE of the ANPVB metasurface (with parameters α = 18, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 30) at a propagation distance of 1.5 mm. The measured PCE values are 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These experimental results align well with simulations at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths. Despite the variation at 450 nm, both experimental and simulated PCEs remain above 35% at all wavelengths, demonstrating robust efficiency even for high-aspect-ratio nanostructures. These comparative data have been included in the revised Supplementary Material.\nTo achieve high-efficiency broadband operation, we intentionally designed a rectangular bar meta-atom with a height of 600 nm and a period of 350 nm, as depicted in Fig. S9(a). Through parameter sweeps across dimensions a and b (as shown in Figs. S9(a1)–(a4)), we identified an optimized structure (marked with a five-pointed star, with b = 285 nm and a = 100 nm) that yields an average PCE of approximately 70% across the 400–800 nm range (Fig. S9(a6)). It is worth noting that for experimental validation, we utilized a 1 μm-height TiO₂ metasurface due to the higher fabrication maturity at this scale. While this choice affects absolute efficiency values, it does not undermine the general applicability or broadband potential of the proposed phase design strategy. Our simulations confirm that with appropriate geometric optimization, high broadband PCE can be reliably achieved, affirming that the broadband functionality demonstrated herein is a product of deliberate design, not merely a byproduct of the PB phase.\nFig. S9 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\nChanges made in the revised manuscript:\nPage-7-9\nSupplementary Note 7: Numerically calculated and Measured PCEs of TiO₂ elliptical cylinders and rectangular bars\nAs shown in Fig. S1(a6), the polarization conversion efficiencies (PCEs) of the selected meta-atom are numerically calculated to be 89.54%, 43.09%, 37.99%, and 55.10% at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. Specifically, at 633 nm, the meta-atom exhibits a PCE of 37.99%. Under illumination by a left-handed circularly polarized Gaussian beam, 37.99% of the incident energy is converted into a right-handed beam carrying the modulated phase, while the remaining 62.01% remains in the unconverted left-handed Gaussian state. As the beam propagates, the optical field diverges. Due to the limited simulation domain, energy beyond the sampling region is not captured. To estimate the total energy of both polarization components, we analyze the weakly divergent field of an ANPVB (with parameters α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) after 1.5 mm of propagation. The left-handed component's total energy at z = 1.5 mm is obtained by propagating the field derived from Eq. (2) over an additional 1.5 mm. Similarly, the right-handed component's energy is calculated by propagating the Gaussian beam multiplied by the phase profile over the same distance. The simulated PCE at 1.5 mm, defined as the ratio of the right-handed energy to the total energy, is found to be 36.03% at 633 nm. Using the same approach, the simulated PCEs at 1.5 mm for other wavelengths are calculated as 89.48% at 450 nm, 42.58% at 532 nm, and 49.41% at 780 nm.\nExperimentally, the PCEs of the phase-engineered metasurface designed for ANPVB generation were measured at z = 1.5 mm, yielding values of 52.92% at 450 nm, 48.43% at 532 nm, 35.19% at 633 nm, and 43.56% at 780 nm. These results indicate good agreement between simulations and experiments at 532 nm, 633 nm, and 780 nm. The greater divergence at 450 nm, and the fact that the measured PCE at 532 nm slightly exceeded the calculated value, likely stem from fabrication tolerances (specifically, variations in meta-atom geometry) and uncertainty in the measured imaginary part of the TiO₂ refractive index at shorter wavelengths.\nTo achieve high polarization conversion efficiency across a broad bandwidth, we simulated a rectangular bar structure with a height of 600 nm and a period of 350 nm. As illustrated in Fig. S9(a), the meta-atom features a TiO₂ rectangular bar with a short side a, a long side b, and a height h of 600 nm, positioned on a square SiO₂ substrate. The period P of the meta-atom is 350 nm. Figures S9(a1) and S9(a2) show, respectively, the transmission and phase under x-linearly polarized light at 633 nm for the x-polarized output, as functions of the parameters a and b. Similarly, Figs. S9(a3) and S9(a4) present the transmission and phase for the y-polarized output under y-linearly polarized illumination. Based on these results, the corresponding PCE is summarized in Fig. S9(a5). The meta-atom marked with a five-pointed star, corresponding to b = 285 nm and a = 100 nm, was selected for further analysis. As presented in Fig. S9(a6), the selected meta-atom maintains a high average PCE of 70% across the broad wavelength range from 400 to 800 nm. Although we chose a 1 μm-height titanium dioxide metasurface for experimental validation due to the greater maturity of our fabrication process for 1 μm TiO₂ structures, this selection does not affect the universal applicability of the proposed theory. As demonstrated in the simulations above, it is relatively straightforward to achieve high broadband polarization conversion efficiency across a given wavelength range by optimizing the structural parameters.\nFig. S9 (a) The top and side views of the meta-atom. (a1) Transmission and (a2) phase of the meta-atom along x-polarization under the x-linearly polarized light with the wavelength of 633 nm. The corresponding (a3) transmission and (a4) phase along y-polarization under the y-linearly polarized light. (a5) The PCE under the wavelength of 633 nm. The five-pointed star presents the chosen meta-atom. (a6) The PCE of the chosen meta-atom under the different wavelengths.\n4) It is notable that the size of the vortex is independent of the topological charge, however, the size of the vortex must match the largest topological charge. This means for l=1, the size of the donut shape is increased substantially to match l=30. If we were then to use, e.g. l=300, the size would be increased further. It would be nice if the authors could discuss this point and its implications.\nReply: We sincerely thank the reviewer for this insightful comment and for raising an important point regarding the scalability of the vortex size with respect to large topological charges. We fully agree that this aspect deserves further clarification.\nIn our design, the annular size of the proposed achromatic non-diffracting perfect vortex beam (ANPVB) remains constant for topological charges up to l=30, as demonstrated with charges l=1, 15, and 30 using the parameter set (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm). However, as the reviewer rightly points out, if the topological charge is further increased beyond this range—for instance, to l=300—the ring diameter does indeed increase. The physical reason for this behavior lies in the momentum composition of the beam. The total transverse momentum is the vector sum of the radial and azimuthal components. For small and moderate topological charges, the radial momentum component dominates, allowing the ring size to remain largely unchanged. However, for very high topological charges, the azimuthal momentum becomes significant and can no longer be neglected. This increases the total transverse momentum, which in turn leads to an expansion of the vortex ring.\nWe acknowledge that this scalability poses a practical constraint for applications requiring very high topological charges. Nevertheless, even under these conditions, the deliberately introduced radial momentum in our design contributes to suppressing the strong divergence typically exhibited by vortex beams with high topological charge. Additionally, conventional high-charge vortex beams generally possess large ring diameters, which complicates efficient coupling into specially designed ring-core optical fibers or waveguides with fixed apertures. By engineering the radial phase profile, our method enables reduction and control over the divergence of vortex beams across different topological charges, thereby enhancing their compatibility with integrated photonic systems.\nWe thank the reviewer again for highlighting this valuable point, which we have now discussed in the revised manuscript.\nChanges made in the revised manuscript:\nPage-16\nIn our design, the annular size of the proposed achromatic non-diffracting perfect vortex beam (ANPVB) remains constant for topological charges up to l=30, as demonstrated with charges l=1, 15, and 30 using the parameter set (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm). However, if the topological charge is further increased beyond this range—for instance, to l=300—the ring diameter does indeed increase. The physical reason for this behavior lies in the momentum composition of the beam. The total transverse momentum is the vector sum of the radial and azimuthal components. For small and moderate topological charges, the radial momentum component dominates, allowing the ring size to remain largely unchanged. However, for very high topological charges, the azimuthal momentum becomes significant and can no longer be neglected. This increases the total transverse momentum, which in turn leads to an expansion of the vortex ring. We acknowledge that this scalability poses a practical constraint for applications requiring very high topological charges. Nevertheless, even under these conditions, the deliberately introduced radial momentum in our design contributes to suppressing the strong divergence typically exhibited by vortex beams with high topological charge. Additionally, conventional high-charge vortex beams generally possess large ring diameters, which complicates efficient coupling into specially designed ring-core optical fibers or waveguides with fixed apertures. By engineering the radial phase profile, our method enables reduction and control over the divergence of vortex beams across different topological charges, thereby enhancing their compatibility with integrated photonic systems.\nReviewer #3 (Remarks to the Author):\nIn this manuscript the authors present a dielectric metasurface featuring an engineered geometric phase profile that enables precise control of radial momentum aimed at generating an ultrabroadband achromatic non-diffracting perfect vortex beam (ANPVB).\nThis work is motivated by the need to overcome the limitations of all kinds of vortex beams up to now introduced.\nSpecifically,\n1. the beam diameter of ANPVBs remains independent of the topological charge, in contrast to Laguerre–Gauss beams (LGBs) whose transverse diameter is correlated with the topological charge, making it challenging to couple LGBs into fixed-aperture fibers or waveguides in optical communication systems;\n2. differently from conventional Perfect Vortex Beams, already featuring transverse diameter independence of the topological charge, ANPVBs demonstrate exceptional diameter stability during propagation and maintain an almost invariant beam diameter across the whole visible spectral range, paving the way for robust optical communication technologies.\nThese peculiar features physically originate from the fact that the radial component of the linear momentum transverse to the propagation direction is much larger than its azimuthal component, so that the changes in azimuthal momentum due to differences in topological charge turn to be negligible. Moreover, the radial momentum shaping of ANPVBs is achieved through geometric phase manipulation, which is intrinsically wavelength independent and enables to achieve long-distance non-diffractive beam propagation without requiring complex optical path modifications. Conventional PVBs are generated by manipulating the dynamic phase, i.e. by using combinations of axicons and spiral plates, and are therefore prone to space and time dispersion.\nThe work is sounding and convincing. The conclusions are supported by both experimental demonstrations and numerical simulations included in the main text and expanded in the supplement.\nThe essence of the mechanism of the radial momentum dominance underpinning the substantial independence of the transverse beam size of the topological charge is effectively illustrated in Fig. 1.\nHowever, it is not clear the criterion for selecting the several parameters (\\alpha, \\beta, \\gamma, etc.) involved in the radial phase profile to achieve the desired tolerances. Are there physical criteria or it is just a matter of numerical simulation searching?\nReply: We sincerely thank you for raising this important point regarding the selection criteria for the parameters in the radial phase profile. The parameters are indeed chosen based on physical insights into the structure of k-space and its impact on real-space propagation, rather than through arbitrary numerical searching. Below we clarify the roles and physical effects of each parameter:\n β: Controls the diameter of the nondiffracting vortex ring.\n α and γ: These positive exponents (α, γ > 0) define the power-law decay of the outer and inner radial phase regions, respectively. They influence the relative strength of high- and low-frequency spatial components in k-space, which directly affects the beam's stability during propagation.\n t: Adjusts the extent of the near-constant phase region, influencing the number of interfering rings in k-space and thus the beam's divergence behavior.\n δ: Modulates the overall flatness of the phase disk; larger values enlarge the k-space disk and improve nondiffracting performance but may also introduce background noise.\nTo systematically illustrate these effects, we performed a series of comparative simulations. As shown in Fig. S15(a1) and Fig. S16(a1–a5), the beam with parameters (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) exhibits stable nondiffracting propagation.\nWhen α is reduced (e.g., α=0.1, Fig. S15(b1)), the vortex ring becomes smaller at short distances and expands more rapidly upon propagation. Corresponding k-space analyses (Figs. S16(b2) and (b4)) show enhanced high-frequency components, which drive outward expansion of interference rings.\nIncreasing γ (e.g., γ=10, Fig. S15(c1)) also reduces the initial ring size and accelerates expansion. As shown in Fig. S16(c4), higher γ reduces the number of low-frequency rings, weakening interference stability.\nReducing t (e.g., t=100, Fig. S15(d1)) shrinks the constant-phase region, leading to fewer low-frequency rings (Fig. S16(d4)) and a smaller effective phase disk (Fig. S16(d5)). This results in faster variation of the inter-ring phase difference and loss of nondiffracting propagation.\nIncreasing δ enlarges the k-space disk and improves nondiffraction by slowing phase variation between frequencies (Fig. S15(e5)), though at the cost of higher background noise (Fig. S16(e1)). Practical constraints such as device size and pixel resolution limit the maximum usable values.\nIn summary, the parameters are optimized to achieve sufficient radial momentum that balances ring confinement, propagation stability, and noise control. This enables the vortex to propagate nondiffractively at a desired diameter without excessive expansion or background interference.\nFig. S15 (a-e) x-z intensity profiles of the ANPVBs of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), (α=0.1, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), (α=18, β=0.8, γ=10, t=700, δ=1, R=0.54 mm, l=30), (α=18, β=0.8, γ=3, t=100, δ=1, R=0.54 mm, l=30) and (α=18, β=0.8, γ=3, t=700, δ=20, R=0.54 mm, l=30) under the wavelength of 633 nm respectively.\nFig. S16 (a1–a5), (b1–b5), (c1–c5), (d1–d5) and (e1–e5) K-space intensity distribution, the intensity distribution along the central line, the k-space logarithmic intensity distribution, the intensity distribution along the central line of the logarithmic intensity, and the k-space phase distribution corresponding to Fig. S15(a-e), respectively. The scale bars in Figs. S16(a1), S16(a3) and S16(a5)    represent 0.002 , 0.2 and 0.2 , respectively. P P P\nOn the other hand, the parameter selection mechanism can be understood through the behavior of the Poynting vector. The incident light is the circularly polarized light. As illustrated in Fig. S17(a2), the designed phase profile exhibits a strong radial phase gradient extending from the center outward, which is sustained across a region before eventually decreasing to zero. This leads to outwardly diverging Poynting vectors at a propagation distance of 0.3 mm. By 3 mm, however, the Poynting vectors become predominantly aligned with the z-direction, enabling diffraction-free propagation. Thus, this phase structure facilitates the generation of high radial momentum, followed by a transition to nondiffractive propagation beyond a certain distance.\nWhen α is reduced (Fig. S17(b2)), the radial phase gradient diminishes more rapidly toward zero. At 0.3 mm, the Poynting vectors still diverge outward, and by 3 mm, some vectors deviate slightly from the z-axis, resulting in mild but persistent divergence. This reduction in radial momentum leads to gradually divergent propagation.\nIncreasing γ (Fig. S17(c2)) causes the radial phase gradient to drop abruptly to zero. While outward divergence is observed at 0.3 mm, most Poynting vectors deviate significantly from the z-axis by 3 mm, resulting in clearly divergent propagation. In this case, the phase modulation suppresses radial momentum and enhances beam spread.\nDecreasing t (Fig. S17(d2)) lowers the radial phase gradient. The Poynting vectors diverge outward at 0.3 mm and exhibit considerable deviation from the z-axis by 3 mm, leading to sustained divergence throughout propagation. This modulation reduces radial momentum and prevents nondiffractive behavior.\nConversely, increasing δ (Fig. S17(e2)) enhances the radial phase gradient. Although outward divergence occurs at 0.3 mm, most Poynting vectors remain aligned with the z-direction at 3 mm, supporting continued nondiffractive propagation—despite some deviation due to noise-induced background intensity. This modulation strengthens radial momentum and promotes nondiffractive propagation, albeit at the cost of increased noise.\nFig. S17 (a1–a6), (b1–b6), (c1–c6), (d1–d6) and (e1–e6) Radial phase profiles, central-line phase profiles, Poynting vector 3D structures, and the planar Poynting vector at 0.3 mm and 3 mm, corresponding to Fig. S15(a–e), respectively. The scale bar in Fig. S17(a1) correspond to 0.2 mm.\nChanges made in the Supplementary Material:\nPage-17-22\nSupplementary Note 11: Parameter selection mechanism for the radial phase profile\nThe parameter β is used to adjust the diameter of the nondiffracting vortex ring. The exponents α and γ (α, γ > 0) represent the power-law dependence of the external and internal phase distributions, respectively. Fig. S15(a1) displays the initial x–z intensity profile of the designed ANPVB with parameters (α = 18, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 30). Fig. S16(a1–a5) respectively present the corresponding k-space intensity distribution, the intensity profile along the central line, the logarithmic k-space intensity distribution, the central-line intensity profile of the logarithmic intensity, and the k-space phase distribution. As shown in Fig. S15(b1), for the ANPVB with parameters (α = 0.1, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 30), the vortex diameter at short propagation distances is smaller compared to that in Fig. S15(a1), which is attributed to the smaller value of α. Furthermore, as the propagation distance increases, the vortex diameter expands at a higher rate. From Fig. S16(b2) and (b4), it can be seen that with decreasing α, the intensity of the outer high-frequency rings in k-space increases, while that of the inner low-frequency rings decreases. As a result, the interference rings in real space tend to expand outward during propagation. Fig. S15(c1) shows the x–z intensity profile for an ANPVB with parameters (α = 18, β = 0.8, γ = 10, t = 700, δ = 1, R = 0.54 mm, l = 30). In comparison with Fig. S15(a1), the larger γ leads to a smaller vortex diameter at short distances and a more rapid expansion with increasing propagation distance. Fig. S16(c4) indicates that a higher γ reduces the number of high-intensity low-frequency rings in k-space. Consequently, the interference rings in real space lose their non-diffracting property and expand outward as they propagate.\nFor the ANPVB with parameters (α = 18, β = 0.8, γ = 3, t = 100, δ = 1, R = 0.54 mm, l = 30), as shown in Fig. S15(d1), the smaller value of t results in a reduced vortex diameter at short distances and a faster expansion upon propagation. Fig. S16(d4) reveals that a decrease in t reduces the number of high-intensity low-frequency rings in k-space, making it difficult for the real-space interference rings to maintain their non-diffracting behavior and leading to outward expansion. Additionally, Fig. S16(d5) shows that a smaller t corresponds to a smaller disk-like region in phase space. The reduced number of interference rings within the k-space disk region accelerates the variation of the phase difference between high- and low-frequency wave vectors along the z-direction, which enhances the outward radiation of the interference rings in real space.\nFrom Fig. S15(e5), it can be observed that an increase in δ enlarges the disk-like area in phase space and increases the number of interference rings within the k-space disk region. This slows down the variation of the phase difference between high- and low-frequency wave vectors along z with propagation distance, thereby improving the non-diffracting properties of the interference rings in real space. However, as shown in Fig. S16(e1), this also introduces higher background intensity noise. Due to constraints in device size and pixel count, these parameters cannot be arbitrarily large. By appropriately selecting these parameters, sufficient radial momentum can be generated to ensure diffraction-free propagation of the vortex along the designed diameter.\nFig. S15 (a-e) x-z intensity profiles of the ANPVBs of (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), (α=0.1, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30), (α=18, β=0.8, γ=10, t=700, δ=1, R=0.54 mm, l=30), (α=18, β=0.8, γ=3, t=100, δ=1, R=0.54 mm, l=30) and (α=18, β=0.8, γ=3, t=700, δ=20, R=0.54 mm, l=30) under the wavelength of 633 nm respectively.\nFig. S16 (a1–a5), (b1–b5), (c1–c5), (d1–d5) and (e1–e5) K-space intensity distribution, the intensity distribution along the central line, the k-space logarithmic intensity distribution, the intensity distribution along the central line of the logarithmic intensity, and the k-space phase distribution corresponding to Fig. S15(a-e), respectively. The scale bars in Figs. S16(a1), S16(a3) and S16(a5)    represent 0.002 , 0.2 and 0.2 , respectively. P P P\nOn the other hand, the parameter selection mechanism can be interpreted via the Poynting vector. The incident light is the circularly polarized light. As illustrated in Fig. S17(a2), the designed phase profile exhibits a pronounced radial phase gradient extending outward from the center, which persists across a certain region before dropping to zero. Accordingly, at 0.3 mm, the Poynting vectors diverge radially outward, whereas by 3 mm, they align predominantly with the z-direction, thereby facilitating diffraction-free propagation. This phase structure thus promotes the generation of substantial radial momentum, followed by a transition to nondiffractive propagation beyond a certain distance.\nAs shown in Fig. S17(b2), reducing α diminishes the outer radial phase gradient, causing it to fall to zero more abruptly. As a result, while the Poynting vectors still diverge outward at 0.3 mm, some vectors at 3 mm deviate slightly from the z-axis, leading to mild but persistent divergence during propagation. Overall, this reduction in phase modulation lowers the radial momentum and results in gradually divergent beam propagation.\nIn Fig. S17(c2), increasing γ leads to a rapid decline in the radial phase gradient to zero. Consequently, although outward divergence is observed at 0.3 mm, by 3 mm most Poynting vectors deviate significantly from the z-axis, resulting in continued divergent propagation. This modification reduces radial momentum and enhances beam divergence.\nAs depicted in Fig. S17(d2), decreasing t weakens the radial phase gradient. This leads to pronounced divergence at both 0.3 mm and 3 mm, with most vectors straying considerably from the z-direction, resulting in sustained divergent propagation. In summary, this phase modulation reduces radial momentum and compromises beam collimation.\nFinally, Fig. S17(e2) demonstrates that increasing δ strengthens the radial phase gradient. While divergence occurs at 0.3 mm, by 3 mm the majority of the Poynting vectors remain aligned with the z-axis—despite some deviation due to noise-induced light intensity—enabling continued nondiffractive propagation. Thus, enhancing δ boosts radial momentum and improves nondiffractive propagation, albeit at the cost of increased background noise.\nFig. S17 (a1–a6), (b1–b6), (c1–c6), (d1–d6) and (e1–e6) Radial phase profiles, central-line phase profiles, Poynting vector 3D structures, and the planar Poynting vector at 0.3 mm and 3 mm, corresponding to Fig. S15(a–e), respectively. The scale bar in Fig. S17(a1) correspond to 0.2 mm."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The manuscript presents an approach to engineer metasurfaces for vortex beam generation with claimed properties of broadband achromaticity, non-diffracting propagation, and invariance with respect to topological charge. I acknowledge that the authors have been responsive to prior reviewers’ feedback, and the current version reflects significant effort in incorporating additional simulations, efficiency measurements, and improved figure clarity. These revisions have undoubtedly strengthened the manuscript compared to the original submission.\nHowever, despite these improvements, I remain unconvinced that the work reaches the level of novelty and broad impact expected for Nature Communications. The practical utnity of broadband vortex beams is itself rather limited to a narrow set of optical applications. Thus, this case interesting for me, but do not themselves constitute the kind of broad interdisciplinary significance typically required by this broad readership journal, such as Nature Communications.\nTaken together, these considerations make me skeptical that the manuscript, even in its revised form, is suitable for publication in Nature Communications.\nBelow are additional comments\n1. The authors repeatedly emphasize the ‘ultrabroadband’ and ‘achromatic’ nature of their design. However, Pancharatnam–Berry (PB) phase metasurfaces already provide broadband achromatic phase control in principle. The main contribution here appears to be the additional tailoring of radial momentum. While interesting, this is not sufficient to justify the very strong claims of achieving a fundamentally new class of optical beams. The incremental design modifications should not be portrayed as a paradigm shift.\n2. The design hinges on enforcing k_r ≫ kϕ*k_r, such that azimuthal momentum contributions are negligible. Yet by definition, kϕ scales linearly with l. For sufficiently large l, this assumption fails, and the vortex radius inevitably grows. This is not a practical fabrication constraint but a theoretical inevitability", "reviewer_2": "The authors answered my comments clearly and the manuscript has been improved significantly. I can support its publication in its current form.", "reviewer_3": "Prompted by the reviewers’ comments, the authors have substantially revised both the main text and the supplementary material. Their responses to my concerns are satisfactory, and I consider the current version of the manuscript fully convincing."}, "author_response": "Response Letter\nDear editor and reviewers,\nWe are deeply grateful for the opportunity to revise our manuscript entitled \"Ultrabroadband, Achromatic, and Non-Diffracting Perfect Optical Vortex Generation via Radial Momentum Control in Dielectric Metasurfaces\" (Manuscript ID: NCOMMS-25-39674A). We sincerely thank the reviewers for their insightful and constructive feedback, which has been invaluable in strengthening our work. We have diligently addressed all points raised, with all changes highlighted in yellow within the manuscript. Our point-by-point responses are detailed below.\nCOMMENTS TO AUTHOR:\nReviewer #1 (Remarks to the Author):\nThe manuscript presents an approach to engineer metasurfaces for vortex beam generation with claimed properties of broadband achromaticity, non-diffracting propagation, and invariance with respect to topological charge. I acknowledge that the authors have been responsive to prior reviewers’ feedback, and the current version reflects significant effort in incorporating additional simulations, efficiency measurements, and improved figure clarity. These revisions have undoubtedly strengthened the manuscript compared to the original submission.\nHowever, despite these improvements, I remain unconvinced that the work reaches the level of novelty and broad impact expected for Nature Communications. The practical utnity of broadband vortex beams is itself rather limited to a narrow set of optical applications. Thus, this case interesting for me, but do not themselves constitute the kind of broad interdisciplinary significance typically required by this broad readership journal, such as Nature Communications.\nTaken together, these considerations make me skeptical that the manuscript, even in its revised form, is suitable for publication in Nature Communications.\nReply: We sincerely thank you for your time and for acknowledging the significant efforts we have invested in revising the manuscript. We are encouraged by your assessment that our additional simulations, efficiency measurements, and improved presentations have \"undoubtedly strengthened the manuscript.\"\nRegarding the concern about novelty and broad impact, we respectfully offer a different perspective. While conventional vortex beams are indeed established tools, their practical utility has been fundamentally limited by the inherent trade-off between beam size, topological charge, and wavelength—a challenge the field has grappled with for years. Our work introduces a conceptually new approach based on radial momentum control via geometric phase metasurfaces, which for the first time, to our knowledge, decouples these parameters to generate ultrabroadband achromatic and non-diffracting perfect vortex beams. This breakthrough is not merely an incremental improvement but a fundamental enabling technology. By solving a key bottleneck, it immediately broadens the utility of vortex beams.\nThe impact extends beyond a \"narrow set of optical applications\" and directly advances several interdisciplinary fields: 1) Optical Communications: It may enable robust, wavelength-division-multiplexing-free high-capacity data channels; 2) Quantum Information: It may provide a stable platform for manipulating high-dimensional entangled states, which is a central pursuit in quantum optics; 3) Precision Metrology & Imaging: The invariant beam profile is critical for imaging techniques and advanced optical trapping.\nTherefore, we believe that by providing a powerful and general solution to a long-standing problem, our work possesses the very interdisciplinary significance that aligns with the scope of Nature Communications, offering new tools and capabilities to a broad readership across physics and engineering. We hope this clarification on the conceptual advance and the resulting widespread utility of our work addresses the reviewer's remaining concerns. We are, of course, open to further suggestions to better highlight this significance in the manuscript.\nBelow are additional comments\n1. The authors repeatedly emphasize the ‘ultrabroadband’ and ‘achromatic’ nature of their design. However, Pancharatnam–Berry (PB) phase metasurfaces already provide broadband achromatic phase control in principle. The main contribution here appears to be the additional tailoring of radial momentum. While interesting, this is not sufficient to justify the very strong claims of achieving a fundamentally new class of optical beams. The incremental design modifications should not be portrayed as a paradigm shift.\nReply: We thank the reviewer for this insightful comment, which allows us to clarify the fundamental conceptual advance of our work. The reviewer correctly points out that the Pancharatnam-Berry (PB) phase provides a broadband phase response. However, we respectfully disagree that this inherently leads to achromatic device performance. In fact, this is a critical and widely recognized distinction in the field of metasurface design. The reviewer's statement perfectly illustrates the very problem our work solves: a uniform PB phase profile, while achromatic in its phase profile, does not result in an achromatic optical function. As the reviewer suggests, a lens with a pure PB phase profile would indeed focus different wavelengths to different focal lengths, exhibiting significant chromatic aberration (as illustrated in Figure R1 below).\nFigure R1 | A conventional PB-phase metasurface lens. Despite the achromatic phase profile, the diffractive nature of the device leads to different focal lengths for different wavelengths (λ₁, λ₂, λ₃), resulting in chromatic aberration.\nOur core contribution is not the use of the PB phase itself, but the introduction of tailored radial momentum control as a general principle to suppress chromatic and diffractive effects. The radial momentum, k, directly governs the diffraction and scaling behavior of the beam. By engineering the metasurface to provide a wavelength-dependent geometric phase profile that specifically compensates for the dispersion of radial momentum, we achieve a device functionality that is truly achromatic and non-diffracting.\nThis is analogous to the motivation behind achromatic metalens design: one cannot use a simple PB phase, but must instead introduce complex nanostructures to compensate for chromatic dispersion. Similarly, our work introduces the necessary \"dispersion compensation\" in the momentum domain for vortex beams.\nTo further underscore that the PB phase is merely one possible means to an end, we have performed additional simulations (see below the Supplementary Information) demonstrating that a non-PB-phase approach (e.g., using a Fresnel-type metasurface with transmission phase control) can also achieve the same achromatic, non-diffracting vortex beams when our principle of radial momentum control is applied. This confirms that the fundamental advance is the principle of radial momentum control, not the specific phase modulation technique.\nTherefore, we contend that this represents a paradigm shift in structured light engineering: moving from designing phase profiles for a single wavelength to designing profiles that directly control the dispersion of radial momentum across a broad band. This principle unlocks a new class of optical beams whose properties were previously inaccessible.\nCorresponding Section in Supplementary Information\nThe approach is not limited to the PB phase; in principle, any phase modulation mechanism capable of imparting the required radial phase shift at each wavelength can realize this function. For example, the transmission phase can also be designed to implement the radial phase across multiple wavelengths, thereby enabling ultrabroadband achromatic performance.\nIn this study, we used the transmission phase database φₓ of rectangular nanoantennas at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm to simultaneously satisfy the radial phase and the spiral phase with a topological charge of 30 for an ANPVB with parameters (α = 18, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 30). This allows the metasurface to exhibit nearly identical phase responses at these wavelengths. Figures S14(a1–d1) show the matched phase profiles for each wavelength. The corresponding intensity and phase distributions after 18 mm of propagation are presented in Figs. S14(a2–d2) and S14(a3–d3), respectively. The results confirm that the metasurface designed via transmission phase modulation successfully generates vortex beams carrying a topological charge of 30 with nearly identical diameters across all four wavelengths.\nThis demonstrates that applying the same radial phase modulation at each wavelength is sufficient to maintain a consistent vortex diameter at a fixed propagation distance, independent of wavelength. Vortices that satisfy this condition are referred to as achromatic vortices. By incorporating the proposed radial phase, conventional vortex phases can be modulated to yield achromatic vortices—enabling wavelength-independent size stability and thereby extending traditional ultrabroadband vortex generation to include achromatic functionality.\nFig. S14 | ANPVBs constructed by transmission phases. (a1-d1) Matched phases of the ANPVB with (α=18, β=0.8, γ=3, t=700, δ=1, R=0.54 mm, l=30) at wavelengths of 450 nm, 532 nm, 633 nm, and 780 nm, respectively. (a2–d2) Intensity and (a3–d3) phase distributions after the matched fields propagate 18 mm. The scale bar in Fig. S14(a1) represents 0.1 mm.\n2. The design hinges on enforcing k_r ≫ kϕ*k_r, such that azimuthal momentum contributions are negligible. Yet by definition, kϕ scales linearly with l. For sufficiently large l, this assumption fails, and the vortex radius inevitably grows. This is not a practical fabrication constraint but a theoretical inevitability\nReply: We thank the reviewer for raising this important theoretical point regarding the scalability of our approach with respect to topological charge. The reviewer is correct in noting that the fundamental condition k ≫ k , which underlies our design, would indeed break down for arbitrarily large values of l, as k scales linearly with it. We acknowledge this as an inherent theoretical boundary of the current implementation. However, the primary focus and demonstrated success of our work lies in solving the critical practical problem of beam instability within the parameter range that is relevant for real-world applications. We have numerically validated the performance of our beams for topological charges up to l=100, a range that already far exceeds the requirements of foreseeable applications in areas such as optical communications, trapping, and quantum information processing, where typical l values rarely exceed single or double digits. Furthermore, the core principle of our work—the active and tailored control of radial momentum—retains its validity and offers a pathway to address even these extreme theoretical cases. Should future applications demand operation at vastly larger topological charges, the same conceptual framework could be extended by engineering the metasurface to provide a proportionally scaled radial momentum k to maintain the desired beam properties. Thus, while a fundamental limit exists in principle, our work establishes a powerful and extensible platform that eliminates the dominant practical bottlenecks for structured light applications and opens the door to future exploration at the theoretical frontier. We have revised the manuscript to acknowledge this theoretical boundary in the discussion section.\nChanges made in the revised manuscript:\nPage-16\nIt is worth noting that the condition k ≫ k imposes a theoretical upper bound on the topological charge l for a fixed radial momentum k. However, this work focuses on the practically critical range where previous designs fail, and demonstrates performance far beyond current application needs. For future requirements involving extremely large l, our paradigm of radial momentum control could be extended by scaling k accordingly, presenting a rich avenue for further exploration.\nReviewer #2 (Remarks to the Author):\nThe authors answered my comments clearly and the manuscript has been improved significantly. I can support its publication in its current form.\nReply: We sincerely appreciate your positive feedback and recommendation to publish in Nature Communications.\ns\nReviewer #3 (Remarks to the Author):\nPrompted by the reviewers’ comments, the authors have substantially revised both the main text and the supplementary material. Their responses to my concerns are satisfactory, and I consider the current version of the manuscript fully convincing.\nReply: We sincerely appreciate your valuable input throughout the review process. We are gratified that you find our responses satisfactory and consider the manuscript convincing in its current form."}]

Decision letter