Metafiber transforming arbitrarily structured light

Structured light has proven useful for numerous photonic applications. However, the current use of structured light in optical fiber science and technology is severely limited by mode mixing or by the lack of optical elements that can be integrated onto fiber end-faces for wavefront engineering, and hence generation of structured light is still handled outside the fiber via bulky optics in free space. We report a metafiber platform capable of creating arbitrarily structured light on the hybrid-order Poincaré sphere. Polymeric metasurfaces, with unleashed height degree of freedom and a greatly expanded 3D meta-atom library, were 3D laser nanoprinted and interfaced with polarization-maintaining single-mode fibers. Multiple metasurfaces were interfaced on the fiber end-faces, transforming the fiber output into different structured-light fields, including cylindrical vector beams, circularly polarized vortex beams, and arbitrary vector field. Our work provides a paradigm for advancing optical fiber science and technology towards fiber-integrated light shaping, which may find important applications in fiber communications, fiber lasers and sensors, endoscopic imaging, fiber lithography, and lab-on-fiber technology.

Implementing structured light on optical fibers could be crucial for widespread fiber applications ranging from fiber communications 15 , fiber lasers 38 , and fiber sensors 39 to endoscopic imaging 40 , fiber lithography 41 , and lab-on-fiber technology 42,43 .However, the practical use of structured light from optical fibers is severely hampered by modal crosstalk and polarization mixing or by the lack of optical elements that can be integrated onto fiber end-faces for complex wavefront manipulation.To date, the generation of structured light is still handled outside the fiber via bulky optics in free space, which hinders the deployment of structured light for fiber science and technology and partially nullifies the advantages of optical fiber such as flexible light guidance.Although electron- 42,[44][45][46] and ion-beam lithography 47,48 , nanoimprinting 49 and hydrofluoric chemical-etching techniques 50 have been proposed to implement metasurfaces on the fiber end-faces, these fabrication methods suffer from either a complicated manufacturing process or difficulties in interfacing arbitrary 3D nanostructures for efficient wavefront engineering.3D laser nanoprinting, based on two-photon polymerization, has been introduced to interface 3D micro-optics on fiber tips [51][52][53][54][55][56] .Recently, 3D laser-nanoprinted lenses with high numerical aperture 54 , achromatic focusing 57 , multi-focus generation 58 , and inverse-design optimization 41 have been integrated on fiber end-faces to improve fiber functionalities and applications.Nevertheless, it remains elusive to realize arbitrarily structured light directly on optical fibers.

Results
Here we demonstrate the design, 3D laser nanoprinting, and characterization of structured light generating metafibers (SLGMs), capable of on-fiber transforming arbitrarily structured light fields on the HOPS (Fig. 1).For sample implementation and polarization manipulation, we experimentally used a commercial polarizationmaintaining single-mode fiber (PM-SMF, PM1550-XP, Thorlabs).To allow the output of the PM-SMF to be freely expanded to fully cover the area of a 3D metasurface with a diameter of around 100 μm, a hollow tower structure (height of 550 μm) was first laser printed onto the fiber end-face (Figs.1a and 1b).To prove our concept, we demonstrate several different SLGMs with various structured light outputs including the radial (SLGM-1) and azimuthal (SLGM-2) polarizations, circularly polarized vortex beams with topological charges of -1 (SLGM-3) and -3 (SLGM-4), as well as an arbitrary vector field on the HOPS with spatially variant localized elliptical polarizations (SLGM-5) (Fig. 1c).We show that 3D anisotropic meta-atoms with unleashed height degree of freedom offer a greatly expanded 3D meta-atom library, allowing the independent, complete, and precise polarization and phase control at the level of a single meta-atom, paving the way for the use of a single metasurface to create arbitrarily structured light on the fiber end-face.

Design principle
An arbitrarily structured light field on the HOPS can be defined mathematically as a superposition of left-and right-handed circular polarization components (|⟩ and |⟩) carrying different OAM modes 1 : where  and  (shown in Fig. 2a) represent the weighted amplitude parameter and relative phase contributions of the circular polarization components, respectively. and  denote the topological charges of the OAM modes in the right-and left-handed circular polarization components, respectively.Eq. ( 1) can be rewritten as Jones vectors in the linear polarization basis (x-linear and y-linear polarizations): where  is the azimuthal angle in the transverse cross-section plane of a structured light field.Eq. 2 can be used to define different vector beams on the HOPS with spatially variant polarization distributions, with some representative examples shown in Fig. 2a (from I to V).
We now demonstrate the design of 3D polymeric metasurfaces for implementing vector beams on the HOPS.3D laser-nanoprinted nanopillar waveguides in a polymer matrix were employed as meta-atoms (Fig. 2b).To achieve strong birefringence necessary for the polarization control, we designed anisotropic nanopillars with rectangle cross-sections that support waveguide modes with distinctive indices for the polarizations along the short and long axes.According to the Jones matrix method, for an incident linear polarization along the x-direction (x-y-z coordinates are defined in Fig. 2b), the output light after passing through an anisotropic nanopillar waveguide is given as (Supplementary Note 1): where  is the in-plane rotation angle of the nanopillar,  x and  y are the absolute transmission amplitudes of the transverse modes polarized in the x and y directions, respectively, and ∆ is their relative phase.All these parameters are spatially dependent as functions of x and y.Eq. 3 indicates that both the polarization (controlled by ∆) and phase ( x ) of an output beam can be controlled simultaneously by a single nanopillar, which forms the physical basis for implementing any arbitrarily structured light field.The key to our design is to find 3D nanopillars to be arranged in a particular distribution so that their optical responses can be precisely matched with any desired structured light field defined in Eq. 2.
We have established a 3D meta-atom library based on the rigorous coupled-wave analysis method at a telecommunication wavelength of 1550 nm (Fig. 2c).It should be mentioned that the height degree of freedom of 3D nanopillars leads to a greatly expanded meta-atom library with a 3D dataset, which provides an expanded source to precisely match any desired polarization and phase responses.We present our simulation results of the phase difference between the x-and y-polarized transverse modes and the propagation phase of the x-linear polarization.We considered 3D nanopillars made of a lossless, commercial polymer material IP-L with an index of 1.5 Therefore, our results suggest that 3D nanopillars with the unlocked height degree of freedom provide a powerful platform for implementing arbitrarily structured light.
Without loss of generality, we picked up five different vector states on the HOPS to prove our SLGM concept, including cylindrical vector beams on the equator of the firstorder HOPS (Fig. 3), circularly polarized vortex beams carrying different OAM modes sitting on the poles of the first-and third-order HOPS (Fig. 4), and an arbitrary vector state on the HOPS carrying a spatially variant elliptical polarization distribution (Fig. 5).

Cylindrical vector beams on SLGMs
Cylindrical vector beams, located at the equator of the first-order HOPS ( = − = 1), are a special group of spatially variant vector beams with localized linear polarization.
As typical examples of cylindrical vector beams, we experimentally demonstrate two SLGMs that can produce radial (SLGM-1) and azimuthal (SLGM-2) polarizations (Fig. 3).To obtain these states, the weighted amplitude parameter  in Eq. 2 are set to  The cylindrical vector beam metasurface with a diameter of 100 μm was 3D laser nanoprinted on top of a hollow tower structure (height of 550 μm) that was interfaced on the PM-SMF end-face (Methods).To realize the correct polarization manipulation, the polarization axis of the PM-SMF must be carefully calibrated and aligned with the x-axis of the metasurface.This alignment was experimentally performed under a bright-field imaging microscope in the laser nanoprinting system (Nanoscribe GT2).
The side-view scanning electron microscope (SEM) image of SLGM-1 with the radial polarization output is given in Fig. 3d, revealing a well-defined 3D nanopillar metasurface on top of the tower.To characterize the cylindrical vector beams on SLGMs, a linearly polarized 1550 nm laser beam from a supercontinuum laser source (SuperK Fianium, NKT Photonics) and an infrared wavelength selector (SuperK Select, NKT Photonics) was coupled into the unstructured ends of SLGMs (Supplementary Figure S1).The SLGM outputs were characterized using a home-built optical imaging setup with 15x magnification and recorded with a near-infrared camera (Raptor, Owl 640 M).Placing a linear polarizer in front of the camera results in two-lobe intensity patterns with respect to the axis of the linear polarizer, allowing us to identify the SLGM outputs as the radial (Fig. 3e) and azimuthal (Fig. 3f) vector beams 59 .Specifically, the lobes follow the orientation of the polarization axis of the linear polarizer, confirming the generation of cylindrically polarized beams.

Circularly polarized vortex beams on SLGMs
Circularly polarized vortex beams sitting at the poles of the HOPS represent the base vector states of the light fields generalized on the HOPS (see Eq. 1).We now demonstrate two additional SLGMs that enable the generation of right-and left-handed circularly polarized vortex beams with topological charges of -1 and -3, respectively (SLGM-3 and SLGM-4) (Fig. 4).For the circularly polarized vortex beam at the north pole of the first-order HOPS ( = − = 1), both the weighted amplitude parameter  and relative phase  in Eq. 2 are set to 0 (yielding sample SLGM-3).The corresponding circularly polarized vortex beam can be described mathematically as Similarly, for the circularly polarized vortex beam at the south pole of the third-order HOPS ( = − = 3 , for SLGM-4), the weighted amplitude parameter  and relative phase  in Eq. 2 are set to 0 and π, respectively, and the resulting beam is given as � ,−3 � = 1 √2  −3(,) � 1  � .To realize such vector states through a metasurface, independent control of both polarization and propagation phase is necessary.
We designed nanopillars to function like quarter-wave plates, having high transmission efficiency ( x ≈  y ≈ 1) as well as a phase difference of  2 between the x-and y-linear polarization modes.According to Eq. 3, the nanopillar output should be expressed as − � and   x (,) � 1  �, respectively (Fig. 4a).Moreover, the propagation phase of x-linear polarization  x can satisfy a helical phase distribution   , where  denotes the topological charge of the OAM mode.
Owing to the greatly extended 3D meta-atom library, we can find 3D nanopillars with the propagation phase response from 0 to 2 to imprint different OAM modes while maintaining a nearly constant phase difference of  2 (Fig. 4b).
We used the same fabrication process as described above to print two more SLGMs with circularly polarized vortex beam outputs.The side-view SEM images of these SLGMs are shown in Fig. 4c.To ensure precise polarization conversion, the x-axis of nanopillars were carefully aligned with respect to the polarization axis of the PM-SMFs by rotating them 45 and 135 degrees for the right-and left-handed circular polarization outputs, respectively.To characterize the helical beams of the fiber outputs, we recorded a series of intensity maps at different propagation distances from the metasurface plane up to 1 mm distance in real space (Supplementary Figure S2).Due to the divergent wavefront of the fiber output leaving the PM-SMF, the OAM intensity distributions are enlarged as the propagation distance is increased.To further verify the OAM indices, we measured the Fourier plane image of the SLGMs (Supplementary Figure S3).We found that doughnut-shaped beam patterns in the Fourier plane exhibit consistent divergence angles (SLGM3: 0.05° and SLGM4: 0.1°) with our simulation results, corroborating the nature of the transformed OAM beams (Fig. 4d).To consider only the OAM-induced beam divergence, we experimentally characterized the OAM beam divergence at a shifted Fourier plane that has the smallest beam sizes, through which we the fiber divergence is compensated.In addition, we noticed some interference effects in the intensity patterns, which we believe are mainly due to imperfect fiber cleaver such that the fiber output was not uniformly incident on the metasurfaces.Finally, we placed a linear polarizer in front of the camera to verify the circular polarization outputs.Our experimental results indicate the successful generation of circularly polarized vortex beam outputs in both SLGMs with high degree of circular polarizations (Fig. 4e).

Arbitrarily structured light on SLGMs
To further demonstrate the superiority of 3D metasurfaces, we randomly chose an arbitrary state on the first-order HOPS (  = − = 1 ) and demonstrate the transformation of such state by designing, fabricating, and employing another SLGM (SLGM-5) (Fig. 5a).The weighted amplitude parameter  and relative phase  in Eq.
2 are set as 2 tan −1 � 1 3 � and  4 , respectively, leading to a vector state written as ) ) �.It corresponds to an elliptically polarized state with an ellipticity ratio (a/b defined in Fig. 5b) of 2 and the polarization directions  (semi-major axes) are spatially related to azimuthal angle (, ) .The specific amplitude and phase distributions for the polarization states along x and y directions are displayed in Supplementary Fig. S4.Based on vectorial diffraction theory [60][61][62] , we numerically simulated the diffraction pattern of the vector beam in a Fourier plane, with the polarization components along the x and y directions (Supplementary Fig. S4).To satisfy the desired amplitude and phase distributions, we selected 3D nanopillars with matched optical responses based on Eq. 3 to fulfill the desired amplitude and phase requirements (Fig. 5c).We also calculated the intensity response of our designed meta-atoms in the Fourier plane, which shows great agreement with our theoretical results.The resultant doughnut-shaped total intensity in the Fourier plane and the rotated two-lobe polarization patterns are consistent with our desired vector state (as shown in Fig. 5a).As such, the newly fabricated metafiber, SLGM-5 (Fig. 5d), was experimentally characterized by measuring the total intensity and polarization filtered images in the Fourier plane of the metafiber (Fig. 5e), finding a good agreement with our theoretical and simulation results.Our simulation and experimental results show a consistent divergence angle of 0.05° induced by the first-order OAM beam of the SLGM-5 output.As such, we have experimentally verified that our metafiber platform is able to transform an arbitrary structured light field directly on the end face of a PM-SMF.

Conclusions
We have demonstrated an entirely new metafiber platform that can transform the output of a single-mode fiber into arbitrarily structured light on the HOPS using nanoprinted metasurfaces.We have successfully created polymeric 3D metasurfaces on commercial PM-SMFs.The unleashed height degree of freedom in 3D nanopillar meta-atoms offers a greatly expanded 3D meta-atom library, leading to independent, complete, and precise polarization and phase control at the level of individual metaatoms.Several SLGMs were designed, 3D laser nanoprinted, and characterized, allowing for the on-fiber realization of five representative structured-light fields on the HOPS.These include radial and azimuthal polarizations (SLGM-1 and SLGM-2 in Fig. 3), circularly polarized vortex beams (SLGM-3 and SLGM-4 in Fig. 4), as well as an arbitrary vector state selected from the HOPS that carries a spatially variant elliptical polarization distribution (SLGM-5 in Fig. 5).Due to its simple and integrated nature, the implementation of structured light directly on optical fibers provides a new paradigm for advancing optical fiber science and technology towards multimode light shaping and multi-dimensional light-matter interactions [63][64][65][66][67][68][69][70] .For instance, longdistance transmission and delivery of structured light modes may not require sophisticated multimode fibers, which unfortunately suffer from intrinsic modal crosstalk and polarization mixing.Alternatively, our demonstrated metafiber platform could be used to realize arbitrarily structured light transformation at the fiber end-faces and achieve well-defined structured light modes of high quality without suffering susceptibility to bending or lack of reproducibility.Therefore, we believe that our demonstrated structured-light metafibers could find important applications, such as but not limited to fiber communications 15 , fiber lasers 38 , fiber sensors 39 , endoscopic imaging 40 , fiber lithography 41 , and lab-on-fiber technology 42,43 .

Methods
Fabrication of SLGMs: 3D laser nanoprinting of polymer-based SLGMs was realized from a two-photon polymerization process via a commercial photolithography system (Photonic Professional GT, Nanoscribe GmbH).Before nanoprinting, the fibers had to be prepared as follows: First, the coating of the PM-SMF was stripped and the end face was cleaved to ensure that the surface is complete and clean.The processed fiber was fixed in a fiber holder marked with lines to indicate the polarization direction of the fiber.The holder together with a fiber were put into the sample holder of the 3D laser nanoprinting system.
To precisely find the fiber and mark its exact 3D locations (x, y, and z coordinates), we found the fiber interface firstly with a low magnification (20x) air objective, followed by a high numerical aperture, oil immersion objective (Plan-Apochromat 63x/1.40Oil DIC, Zeiss).Thereafter, the IP-L 780 photoresist resin was dropped cast onto the fiber and a high precision translational stage was used for realignment.After the fiber end face has been found with the high numerical aperture objective, the whole structure that includes a 3D hollow tower (height of 550 μm) and a 3D metasurface was sequentially printed on top of the fiber in the dip-in configuration.We created some holes on the tower structure, ensuring that unpolymerized photoresist can be washed away during the chemical development process, and therefore creating a free-space beam expansion area inside the tower.To maximize the printing performance, we have used our optimized printing parameters as: laser power of 55 mW and scanning speed of 3500 μm/s, respectively.To increase the mechanical strength of polymer nanopillars with considerably high aspect ratios, small hatching (the lateral laser movement step: 10 nm) and slicing (the longitudinal laser movement step: 20 nm) distances were employed in our 3D nanoprinting.After laser exposure, the samples were developed by immersing them in propylene glycol monomethyl ether acetate (PGMEA, Sigma-Aldrich) for 20 min, Isopropanol (IPA, Sigma-Aldrich) for 5 min, and Methoxynonafluorobutane (Novec 7100 Engineered Fluid, 3 M) for 2 min, respectively.

Figure 1 .
Figure 1.Principle of generating arbitrarily structured light on metafibers.(a) Schematic representation of a polarization-maintaining fiber interfaced with a 3D lasernanoprinted metasurface on top of a hollow tower structure.The metafiber output features an arbitrarily structured light field.(b) Schematic of the side view of the metafiber.(c) The realized structured light fields on a HOPS carry spatially variant polarization distributions, with some examples indicated by the states I to V. The arrows refer to the local polarization of the electric field.

Figure 2 .
Figure 2. Design of a 3D nanopillar used to implement metasurfaces for generating arbitrarily structured light beams.(a) Showcase of five representative structured light fields on the first-order HOPS, defined by different angles of θ and α.(b) Schematic of a 3D laser nanoprinted nanopillar waveguide in a polymer matrix (H: height, W: width, L: length, : the in-plane rotation angle of the nanopillar with respect to x-axis).(c) Simulated 3D meta-atom library consists of the phase difference between the x-and y-linear transverse modes (left) and the propagation phase of the light polarized along the x-direction (right).(d) Three exemplary data planes in the meta-atom library are highlighted, with heights of H=8, 10, 12 μm, respectively.Red and black dashed lines mark the 3D nanopillars satisfying phase differences of π /2 and π, respectively.

(
Nanoscribe, GmbH), having a fixed pitch distance of P = 2.2 μm, a length L in the range of 0.8-1.8μm, a width W of 0.5-1.1 μm, and a height H of 6-14 μm.Both the polarization eigenstates feature high transmission efficiency.3D nanopillars exhibit a wide dynamic range from 0 to π in the phase difference and hence can function as waveplates converting an incident x-linear polarization into different polarization outputs.Specifically, when the phase difference is equal to π, the corresponding nanopillars operate as half-wave plates that can rotate the angle of incident linear polarization, paving the way for generating various cylindrical vector beams on the equator of the HOPS.When the phase difference is π/2, in contrast, each nanopillar operates as quarter-wave plate and convert the linear polarization input into circular polarization, allowing access to the poles of the HOPS.When the phase difference takes any other value, the nanopillars can convert the linear polarization into different elliptical polarizations and hence can reach to any desired state on the HOPS.To better illustrate a wide coverage of 3D meta-atoms in phase difference and propagation phase responses, we highlight three individual planes in our 3D library dataset with different heights of H=8, 10, and 12 μm in Fig.2d.As an example, we can select 3D nanopillars of different heights to function like half-or quarter-wave plates, and in the meantime, they can cover the full range (0 to 2π) of the phase response.

𝜋𝜋 2 and
relative phase  to 0 and π, respectively.The resulting radial and azimuthal polarization states are given as |  ⟩ = � (, ) (, ) � and |  ⟩ = � (, ) −(, ) � , respectively, where (, ) is the azimuthal angle in the transverse cross-section of the vector beams.To satisfy the only required polarization distributions, we used a single-sized 3D nanopillar (Fig. 3a) that behaves like a half-wave plate.The nanopillar waveguide has a length L of 1.60 μm, a width W of 0.55 μm, and a height H of 11 μm.We show that this nanopillar features both high transmission efficiency and  phase difference between the x-and y-linear transverse modes across the entire S, C and L telecommunication bands, from 1.45 to 1.65 μm (Fig.3b).Eq. 3 can then be simplifiedas   (, ) = � cos [2(, )] sin [2(, )]� by assuming  x ≈  y ≈ 1 .As such, for the radial polarization output, its localized linear polarization angle (, ) can be easily controlled by the in-plane rotation angle (, ) of the nanopillar (Fig.3c).To achieve the azimuthal polarization output, we can simply rotate all the nanopillars used for the above radial polarization by 45 degrees.

Figure 3 .
Figure 3. Design and experimental characterization of SLGMs yielding cylindrical vector beam outputs.(a) Schematic representation of the used 3D nanopillar meta-atom.(b) Simulated nanopillar response across a broad spectral range across the whole S, C and L telecommunication bands, as well as parts of the E and U bands.(c) Illustration of the in-plane angle distribution of 3D nanopillars used for creating the radial vector state.(d) Example SEM image of SLGM-1 used for creating the radial vector beam (scale bar: 25 μm).(e and f) Experimentally measured intensity patterns of the SLGM outputs of the radial (e) and azimuthal (f) vector beams, respectively.The red arrows mark the polarization filtering axis of a linear polarizer inserted in front of the camera used for recording the polarization-dependent intensity profiles.Scale bars: 25 μm.

Figure 4 .
Figure 4. Design and characterization of metafibers that generate circularly polarized vortex beams.(a-b) Schematic illustration of creating circularly polarized vortex beam outputs on metafibers (a), which is achieved from arrays of 3D nanopillars with in-plane rotation angles of 45 and 135 degrees The specific propagation phase ( x ) and phase difference (∆) maps for creating circularly polarized vortex beams of � ,−1 � and � ,−3 � are shown in (b).(c) SEM images of the fabricated metafibers of SLGM-3.Scale bar: 100 μm.Bottom left and right: zoom-in areas of SLGM-3 and SLGM-4.Scale bars: 25 μm (d) Simulation (left column) and experimental (right column) results of the intensity distributions of the two SLGMs in the Fourier plane.The dashed circles mark the beam divergence angles of 0.05 and 0.1 degrees.(e) Polarization ellipticity analysis of the SLGMs measured by inserting a rotating linear polarizer before the camera.The grey curve marks a perfectly circular polarized output.

Figure 5 . 4 )
Figure 5. Design and characterization of a metafiber (SLGM-5) used for creating an arbitrarily structured light field on the first-order HOPS.(a) Schematic representation of the desired arbitrarily structured light field on the HOPS including the intensity and polarization distributions.(b) Close-up view of a local vector state of the structured light field, in which the ellipticity ratio and polarization direction angle are defined as / and , respectively.(c) The amplitude and phase distributions of both polarization components, based on optical responses of selectively matched 3D metaatoms.Scale bars: 10 µm.(d) SEM image of SLGM-5.Scale bar: 30 μm.(e) Simulation and experimentally measured intensity distributions of SLGM-5 in the Fourier plane.Total intensity (first column) and polarization filtered intensity images (second and third columns, in which red arrows label the polarization axis of the linear polarizer).The dashed circles mark the beam divergence angle of 0.05.