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 demands1,2,3. 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 a conical 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 multidimensional control over amplitude, phase, and polarization25,26. Metasurface has been applied to imaging and display, such as metalenses27,28,29, beam shaping30, holographic display31,32,33,34 and color printing35,36. Initially, PVBs and perfect Poincaré beams required cascading multiple geometric metasurfaces to mimic bulky optical components like spiral phase plates, 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 a broad spectral range (450–780 nm), showcasing its 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 phase plates for generation, thus remaining susceptible to dispersion effects. In our work, by introducing a radial momentum component (|kr | ≫|kϕ|) that dominates the transverse momentum distribution (\(|{{{{\mathbf{k}}}}}_{{|}{|}}|=\sqrt{|{{{{\mathbf{k}}}}}_{{r}}{|}^{2}+|{{{{\mathbf{k}}}}}_{{\phi }}{|}^{2}}\approx |{{{{\mathbf{k}}}}}_{{r}}|,{{{{\mathbf{k}}}}}_{{|}{|}}={{{{\mathbf{k}}}}}_{{r}}+{{{{\mathbf{k}}}}}_{{\phi }},{{{{\mathbf{k}}}}}_{{|}{|}}\approx {{{{\mathbf{k}}}}}_{{r}}\)), 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 (\(\Delta |{{{{\mathbf{k}}}}}_{{\phi }}|\propto {l}\)) 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.

Fig. 1: Schematic depiction of the generation of an achromatic non-diffracting perfect vortex beam (ANPVB) utilizing a geometric metasurface.
Fig. 1: Schematic depiction of the generation of an achromatic non-diffracting perfect vortex beam (ANPVB) utilizing a geometric metasurface.
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a Broadband light is incident upon the metasurface, which employs precise radial momentum control to generate an ultra-broadband, achromatic, non-diffractive perfect vortex beam. b The metasurface introduces a dominant radial momentum component (|kr | ≫|kϕ|), which governs the transverse momentum distribution (\(|{{{{\mathbf{k}}}}}_{{|}{|}}|=\sqrt{|{{{{\mathbf{k}}}}}_{{r}}{|}^{2}+|{{{{\mathbf{k}}}}}_{{\phi }}{|}^{2}}\approx |{{{{\mathbf{k}}}}}_{{r}}|,{{{{\mathbf{k}}}}}_{{|}{|}}={{{{\mathbf{k}}}}}_{{r}}+{{{{\mathbf{k}}}}}_{{\phi }},{{{{\mathbf{k}}}}}_{{|}{|}}\approx {{{{\mathbf{k}}}}}_{{r}}\)). This design renders changes in azimuthal momentum due to variations in topological charge (\(\Delta |{{{{\mathbf{k}}}}}_{{\phi }}|\propto {l}\)) negligible. The suppression of diffraction is achieved through a nonlinear design of the radial momentum gradient, significantly enhancing the Rayleigh range (zR).

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

$$\psi (r)=\left\{\begin{array}{cc}t{(\frac{r-\beta R}{(1-\beta )R})}^{\alpha } & \beta R\le r\le R\\ -\delta t{(\frac{\beta R-r}{\beta R})}^{\gamma } & 0\le r < \beta R\end{array} \right.,$$
(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 δt 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 = 2 R/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:

$$U=\exp (-{r}^{2}/{w}^{2})\exp (i\psi (r))\exp (il\phi ).$$
(2)

where l denotes the topological charge and ϕ the azimuthal angle. Notably, ANPVBs exhibit co-localized concentric vortices with identical diameters despite differing topological charges. This diameter invariance persists over extended propagation distances and remains wavelength-independent, as confirmed by simulations.

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 = 2 R/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).

Fig. 2: Comparison of beam size dependence on topological charge between Laguerre-Gaussian beam (LGB) and achromatic non-diffracting perfect vortex beam (ANPVB).
Fig. 2: Comparison of beam size dependence on topological charge between Laguerre-Gaussian beam (LGB) and achromatic non-diffracting perfect vortex beam (ANPVB).
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a1–c1 Intensity and (a2–c2) phase profiles at z = 18 mm of the LGBs with l = 1, 15 and 30, respectively. d1–f1 Intensity and (d2–f2) phase profiles at z = 18 mm of the proposed ANPVB of (α = 18, β = 0.8, γ = 3, t = 700, δ = 1, R = 0.54 mm, l = 1, 15 and 30) respectively. The scale bars in (a1) and (d1) represent 1 mm and 0.3 mm, respectively.

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 = 2 R/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 \(-\frac{\pi {r}^{2}}{\lambda f}+l\phi+\frac{2\pi r}{\alpha {R}^{{\hbox{'}}}}\), where the focal length f is 18 mm, l = 30, and the adjusted parameters α and the outermost radius \({R}^{{\hbox{'}}}\) 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 \(\sqrt{2}\) 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.

Fig. 3: Diffraction comparison between perfect vortex beam (PVB) and achromatic non-diffracting perfect vortex beam (ANPVB).
Fig. 3: Diffraction comparison between perfect vortex beam (PVB) and achromatic non-diffracting perfect vortex beam (ANPVB).
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a, 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 (a1–a10) expands, while the diameter of the ANPVB in (b1–b10) remains approximately constant. c Beam radius versus axial position. d Plot of the relative variation of diameters with respect to axial positions.

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.

Fig. 4: Dispersion-free and diffractive-free characteristics of achromatic non-diffracting perfect vortex beam (ANPVB).
Fig. 4: Dispersion-free and diffractive-free characteristics of achromatic non-diffracting perfect vortex beam (ANPVB).
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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.

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: \(d\,\sin \theta=k\lambda\), 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) illustrates 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.

Fig. 5: Experimental demonstration of achromatic non-diffracting perfect vortex beam (ANPVB).
Fig. 5: Experimental demonstration of achromatic non-diffracting perfect vortex beam (ANPVB).
Full size image

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. d–g Captured intensity distributions at z = 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30 mm of these three types of ANPVBs under circularly polarized incident light at the wavelengths of 450, 532, 633 and 780 nm, respectively. Corresponding captured interference intensity distributions on the far right at z = 18 mm under linearly polarized incident light. h Plot of the corresponding diameters with respect to axial positions. The scale bars in (a1–c1), (a2–c2), (a3–c3) and (d) represent 200 μm, 1 μm, 1 μm and 100 μm, respectively.

Discussions

The generation of vortex beams with different topological charges has long been recognized as a critical factor influencing the diffraction characteristics and beam radius in conventional LG beams. This is due to the inherent relationship between the OAM associated with each topological charge and the resulting diffraction pattern, where higher l values typically lead to larger beam radii. The variation in beam radius with differing topological charges poses significant challenges for applications requiring consistent beam properties across multiple OAM states, such as high-dimensional quantum information processing or multi-channel optical communication systems. In this work, we have demonstrated an approach to generate ultra-broadband ANPVBs by employing metasurfaces that precisely control radial momentum. The key innovation lies in the introduction of engineered radial momentum far exceeding the intrinsic azimuthal momentum of the vortex beams themselves. This design choice ensures that the overall transverse momentum component is dominated by the radial momentum, thereby rendering the differences caused by varying topological charges negligible. Consequently, the beam radius remains constant regardless of the topological charge, a feature not observed in traditional vortex beams. Moreover, through meticulous manipulation of the gradient in radial momentum, our method achieves an unprecedented threefold increase in the Rayleigh range compared to the conventional PVB. This enhancement significantly extends the propagation distance over which the ANPVB maintains its non-diffracting property. The underlying mechanism hinges on the dispersion-free geometric phase distribution intrinsic to the dielectric metasurface, ensuring that the generated beams also exhibit wavelength-independent behavior. Such dispersion-free characteristics are crucial for broadband applications, where maintaining beam integrity across a wide spectral range is essential. The utilization of geometric phase distributions in metasurfaces offers a versatile platform for controlling light-matter interactions at subwavelength scales. The ultrabroadband and achromatic performance of the vortex—enabled by the radial phase—is presented in Supplementary Note 10. Supplementary Note 11 outlines the mechanism for selecting parameters of the radial-phase profile. By leveraging these principles, we have not only overcome the limitations imposed by traditional vortex beams but also opened avenues for designing other types of non-diffractive, dispersion-free beams. The scalability of this approach across different wavelengths suggests its applicability beyond visible light, potentially extending into other regions of the electromagnetic spectrum, including microwaves and terahertz waves. This broad applicability underscores the fundamental significance of our findings, providing a robust framework for future innovations in structured light fields and their diverse applications.

In 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, topologically 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.

In 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. It is worth noting that the condition kr ≫ kϕ imposes a theoretical upper bound on the topological charge l for a fixed radial momentum kr. 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 kr accordingly, presenting a rich avenue for further exploration. In summary, the ability to engineer both radial and azimuthal momenta with precision using metasurfaces represents a significant advancement in the field of structured light. It enables the creation of vortex beams whose properties are decoupled from their topological charges, thus addressing a longstanding challenge in optical physics and engineering. Our results pave the way for developing more sophisticated optical devices and systems that can harness the full potential of structured light, promising breakthroughs in areas ranging from advanced imaging techniques to secure communication protocols.

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).

$$E={F}_{0}\times {{{\boldsymbol{F}}}}(U\times F),$$
(3)

where, \({F}_{0}=\frac{\exp ({ikz})\times \exp ({ik}\frac{{x}^{2}+{y}^{2}}{2z})}{i\lambda z}\) \(F=\exp \left[\frac{i\pi }{\lambda z}({u}^{2}+{v}^{2})\right]\), 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.