Figure 1A schematically illustrates the working principle of the proposed GSP metasurface for orthogonal linear-polarization conversion and focused vortex-beam generation. Once an *x*-polarized beam is incident on the metasurface at normal incidence, the reflected waves will become cross-polarized (i.e. *y*-polarized), gain additional phase shifts, and then interfere constructively in the far field, thereby forming a focused vortex beam in cross-polarization. Specifically, the basic GSP meta-atom consists of gold (Au) nanoantennas tilted by 45° with respect to the *x*-axis, a middle silicon dioxide (SiO_{2}) spacer layer, and a bottom continuous Au film, which functions as a nanoscale HWP (nano-HWP) to, simultaneously and independently, engineer the phase and polarization of the reflected light (Figure 1B) [23]. When an *x*-polarized wave is normally impinging on the meta-atoms, electric-dipole oscillations along the long and short axis of the Au nanoantennas are excited, resulting in the cross-polarized scattering. The cross-polarized scattering is further enhanced by the constructive interference in the multireflection process within the GSP cavity, consequently leading to highly efficient orthogonal linear-polarization conversion [24], [25], [26].

Figure 1: Schematic of the metasurface.

(A) Artistic illustration of the metasurface for linear-polarization conversion and focused vortex-beam generation. (B) Schematic of the basic meta-atom that consists of an Au nanoantenna on top of a spacer and Au substrate with dimensions of *p*=550 nm, *d*=130 nm, *t*_{s}=110 nm, and *t*_{m}=80 nm.

The design principle of such a GSP metasurface is based on the use of meta-atoms that enable the HWP functionality, i.e. efficient orthogonal linear-polarization conversion, along with the complete phase control over the polarization-converted reflected fields. Taking into account the fact that the phase of the reflected cross-polarized field can be changed by *π* radian simply by rotating the corresponding meta-atom over 90°, designing two meta-atoms that produce the reflected cross-polarized fields with the phases *φ*_{cr1} and *φ*_{cr2} being different by *π*/2 (elements 1 and 2 in Figure 2A) is sufficient to construct the four-element-based phase gradient metasurface by rotating elements 1 and 2 over 90° so as to cover the whole 2*π* range, because *φ*_{cr3}=*φ*_{cr1}+*π* and *φ*_{cr4}=*φ*_{cr2}+*π*. By tailoring the shapes and dimensions (*a* and *b*) of the topmost Au nanoantennas, we can independently control the reflection phase of the cross-polarized reflected light, achieving eventually the desired phase relationship *φ*_{cr2}=*φ*_{cr1}+*π*/2 at the design wavelength of *λ*_{d}=850 nm and enabling the design of the four-element supercell (the details of simulation is shown in Supplementary Section S1). The simulated cross-polarized reflectivity *R*_{cr}, co-polarized reflectivity *R*_{co}, orthogonal linear-polarization conversion ratio *PCR* [*PCR*=*R*_{cr}/(*R*_{cr}+*R*_{co})], and cross-polarized reflection phase *φ*_{cr} of the four selected meta-atoms indicate that each of the four meta-atoms functions as a reflective nano-HWP with high efficiency at *λ*_{d}=850 nm (Figure 2B). Specifically, the linear-polarization conversion efficiency is over 95% with the averaged cross-polarized reflectivity of ~85%. At the same time, a linear phase gradient with the phase increment of π/2 between adjacent meta-atoms has been realized for the cross-polarized reflected light. When the field distributions of the four meta-atoms are plotted side by side, one can see that the wavefronts of the cross-polarized reflected light experience significantly different phase shifts upon interaction with the meta-atoms, forming four phase values with the phase step of π/2 (Figure 2C). Consequently, the cross-polarized reflected light would be steered into the perpendicular direction to the titled wavefronts marked with the black dashed line. As a final comment, it should be noted that the designed metasurfaces maintain high-efficiency linear-polarization conversion over a broad wavelength range of 800–900 nm as the phase gradient for all four meta-atoms remains approximately linear, thus enabling the broadband manipulation of the cross-polarized reflected light (Figure 2D and E).

Figure 2: Reflection amplitude and phase responses of the selected meta-atoms.

(A) Schematic of the four meta-atoms. The dimensions of element 1–4 are (1) *a*=228 nm, *b*=90 nm; (2) *a*=155 nm, *b*=360 nm; (3) *a*=90 nm, *b*=228 nm; (4) *a*=360 nm, *b*=155 nm, indicating that elements 3 and 4 are simply rotated by 90° with respect to elements 1 and 2, respectively. (B) Simulated cross-polarized reflection phase *φ*_{cr}, cross-polarized reflectivity *R*_{cr}, co-polarized reflectivity *R*_{co}, and linear-polarization conversion ratio *PCR* of the four elements at *λ*_{d}=850 nm. (C) Simulated electric field distribution (*E*_{y}) in each meta-atom at *λ*_{d}=850 nm when an *x*-polarized plane wave is incident from the top. (D, E) Simulated cross-polarized reflectivity *R*_{cr}, co-polarized reflectivity *R*_{co}, and cross-polarized reflection phase *φ*_{cr}, respectively, for four meta-atoms as a function of incident wavelength.

To generate a focused vortex beam with orthogonal linear-polarization conversion, the GSP metasurface should first incorporate the phase profiles of a lens and a q-plate together. Therefore, the phase distribution imposed on the metasurface is calculated via the following formula:

$$\phi \mathrm{(}x,\text{\hspace{0.17em}}y\mathrm{)}=\frac{2\pi}{{\lambda}_{\text{d}}}\mathrm{(}\sqrt{{x}^{2}+{y}^{2}+{f}^{2}}-f\mathrm{)}+l\cdot \mathrm{arctan}\mathrm{(}\raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$x$}\right.\mathrm{)}$$(1)

where *λ*_{d} is the design wavelength in free space, *f* is the focal length, and *l* is the topological charge. Figure 3A displays the phase profile of the designed focused vortex-beam generator with a diameter of *D*=50 μm, a focal length of *f*=60 μm, and the topological charge of *l*=2, indicating the twofold spiral distribution. The phase profile is then discretized in steps on the *x*–*y* plane and implemented using the meta-atoms (Figure 3B). Within the metasurface, all meta-atoms are rotated by 45° with respect to the *x*-axis, ensuring that the *x*-polarized incident light is efficiently converted into the cross-polarized reflected beam. Figure 3C presents the scanning electron microscope (SEM) images of the sample fabricated with standard electron-beam lithography (Supplementary Section S2), displaying the twofold spiral pattern consistent with the twofold spiral phase distribution. Following the fabrication, we characterized the metasurface sample using a home-built optical system that includes a couple of optical components mounted on a three-dimensional translation stage (Supplementary Section S3). By moving all the mounted components together along the axis of light propagation (*z*-direction) step by step, we could record the intensity profiles at different *x*–*y* planes in sequence and reconstruct the three-dimensional intensity distribution.

Figure 3: Design and fabrication of the focused vortex-beam generator with a diameter of *D*=50 μm, a focal length of *f*=60 μm, and the topological charge of *l*=2 at *λ*_{d}=850 nm.

(A, B) Calculate phase profile and designed geometry of the metasurface. (C) SEM images of the fabricated sample.

We plot in Figure 4A–C the measured cross-polarized intensity distributions at different wavelengths when the fabricated sample is illuminated with an *x*-polarized Gaussian beam at normal incidence. As shown in the left panels of Figure 4A–C, the two-dimensional intensity slices along the *z*-direction reveal the evolution of the generated vortex beam, which is slightly focused at the corresponding focal plane of each wavelength. All the intensity patterns on the *x*–*y* planes have the doughnut-shaped intensity distributions with the intensity minimum at the center, which is the main characteristic of the vortex beams, that is, intensity singularity. To prove the helicity wavefront of the focused vortex beam, we performed the interference experiment by using a home-built Michelson interferometer, where the generated vortex beam and the tilted Gaussian beam are arranged in the sample and reference arms, respectively (Supplementary Figure S2). The fork dislocations are observable in the interference patterns (Figure 4A–C), verifying the phase singularity in a broadband spectrum. If the topological charge *l* is increased from 1 to 3, the corresponding focal spot expands, and the ring radius of the vortex intensity profile becomes larger accordingly, while the doughnut shape is preserved (Supplementary Figure S3). Additionally, the number of dislocations would be equal to the increased topological charge.

Since the focused vortex beam is solely assembled by the cross-polarized light, *PCR* can quantitatively gauge the conversion efficiency of the metasurface, which is defined as the ratio of the power within the vortex beam in cross-polarization to the overall reflected power [13]. As depicted in Figure 4D, the fabricated sample exhibits efficient (*PCR*>80% on average) focused vortex-beam generation in a wide wavelength spectrum ranging from 800 to 950 nm, superior to some of the reported vortex-beam generators based on plasmonic metasurfaces in visible and near-infrared wavelengths [12], [13]. Here it should be mentioned that even though the conversion efficiency has been increased, it is still inferior to the dielectric metasurface platforms [17], [18]. Although the *PCR* drops to ~35% at the short wavelength of *λ*=775 nm, the fabricated metasurface sustains the good performance of generating a focused vortex beam with linear-polarization conversion (Supplementary Figure S4). Subsequently, we measured the absolute focusing efficiency of the total reflected beam and the cross-polarized component (Figure 4E). Predictably, the vortex beam is predominantly focused in cross-polarization with the averaged efficiency exceeding 27%, exhibiting at least 10-fold enhancement compared with that from V-shaped antennas [12], while the remaining co-polarized component is strongly suppressed over the wavelength range from 800 to 950 nm. At the design wavelength of *λ*_{d}=850 nm, the measured focusing efficiency of the polarization-converted vortex beam is found to be ~26%, which is considerably lower than the averaged cross-polarized reflectivity of the four meta-atoms (~85%). We believe that the discrepancies are related to the imperfections and the surface roughness of the fabricated sample, the slight variations in the optical constants of the deposited Au, the increased damping related to the titanium adhesion layer between the Au and SiO_{2} layers, different excitation conditions in the simulation and experiment, and the near-field coupling between adjacent meta-atoms that is not considered in the simulations [27]. As a final comment, it should be emphasized that the focused vortex-beam generator shows the similar performance of orthogonal linear-polarization conversion and vortex-beam focusing when the incident polarization is switched to *y*-polarization, in accord with the symmetry of the meta-atoms (Supplementary Figure S5).

Figure 4: Optical characterization of the fabricated vortex-beam generator with the topological charge of *l*=2.

Left (A–C): focal spots evolution of the generated vortex beam along the optical axis (*z*-axis) in cross-polarization. Right top (A–C): measured focal spot profiles at the corresponding focal planes in cross-polarization. Right down (A–C): interference pattern of the focused vortex beam and the copropagating Gaussian beam when the beam axes are tilted with respect to the other. (D) Measured *PCR* as a function of wavelength. (E) Measured efficiency as a function of wavelength. The total efficiency is defined as the ratio of the light intensity from the corresponding focal spot to the incident intensity, while the converted efficiency only considers the cross-polarized reflected beam. In all measurement, the *x*-polarized Gaussian beam is normally incident on the sample.

In addition to the aforementioned single-focal metasurfaces capable of generating focused vortex beams in cross-polarization with different topological charges, we further extended our approach to realize a dual-focal metasurface that transforms the incident Gaussian beam into two slightly focused spots with distinct wavefronts while maintaining the capability of orthogonal linear-polarization conversion. In our design, the dual-focal metasurface is segmented into two sub-metasurfaces, which spatially focus a linearly polarized incident beam to an orthogonal-polarized beam and a vortex beam with intensity singularity, respectively, at different locations in the same focal plane. Therefore, the dual-focal metasurface should impose different phase profiles on the left and right to convert the incident planar wavefronts into two spherical wavefronts in reflection:

$$\begin{array}{l}\phi \mathrm{(}x,\text{\hspace{0.17em}}y\mathrm{)}=\\ \text{\hspace{0.17em}}\{\begin{array}{c}\frac{2\pi}{{\lambda}_{\text{d}}}\mathrm{(}\sqrt{{\mathrm{(}x+\frac{s}{2}\mathrm{)}}^{2}+{y}^{2}+{f}^{2}}-\sqrt{{\mathrm{(}\frac{s}{2}\mathrm{)}}^{2}+{f}^{2}}\mathrm{)}\hspace{0.17em}\mathrm{(}x\ll 0\mathrm{)}\\ \frac{2\pi}{{\lambda}_{\text{d}}}\mathrm{(}\sqrt{{\mathrm{(}x-\frac{s}{2}\mathrm{)}}^{2}+{y}^{2}+{f}^{2}}-\sqrt{{\mathrm{(}\frac{s}{2}\mathrm{)}}^{2}+{f}^{2}}\mathrm{)}+l\cdot \mathrm{arctan}\left(\frac{y}{x-\frac{s}{2}}\right)\hspace{0.17em}\mathrm{(}x>0\mathrm{)}\end{array}\end{array}$$(2)

where *λ*_{d} is the wavelength in free space, *f* is the focal length along the axial direction, *l* is the topological charge, and *s* is the distance between two foci. Figure 5A plots the phase profile of the dual-focal metasurface with a diameter of *D*=50 μm, a focal length of *f*=60 μm, the topological charge of *l*=1, and the separation between the two foci of *s*=20 μm. Similarly, the corresponding meta-atoms are properly selected and arranged with a uniform rotation of 45° with respect to the *x*-axis after discretizing the phase distribution on the *x*–*y* grid, as shown in Figure 5B. Figure 5C presents the SEM image of the fabricated dual-focal metasurface, where the spiral pattern can be clearly seen on the right side. Subsequently, the dual-focal sample was measured using the same setup, and the intensity distributions at the corresponding focal planes of different wavelengths are displayed in Figure 5D–I. Generally, two focal spots are formed with negligible backgrounds, indicating the excellent phase realization. The diffraction-limited focal spot on the left has a Gaussian-distributed intensity profile, while the right spot exhibits the intensity singularity with a minimum in the central part. When a linear polarizer oriented 90° with respect to the illumination polarization was placed in front of the CCD camera to filter out any co-polarized light, the intensity profiles remain largely unaffected and the moderate noise is removed, qualifiedly indicating the high performance of our metasurface design. Here, the non-negligible noise is most likely associated with the imperfections in fabrication and the gap between two sub-metasurfaces, leading to increased co-polarized reflection.

Figure 5: Design, fabrication, and experimental demonstration of the dual-focal metasurface.

(A) Calculated phase profile, (B) designed pattern, and (C) SEM image of the dual-focal metasurface that simultaneously generates dual-focal spots with different intensity profiles for the cross-polarized reflected light, where the left sub-metasurface is to create the convention focused Gaussian beam and the right sub-metasurface is responsible for generating a focused vortex beam. The dual-focal metasurface has a diameter of *D*=50 μm, a focal length of *f*=60 μm, the topological charge of *l*=1, and the distance between two foci of *s*=20 μm. (D–F) Measured intensity profiles at the corresponding focal planes of different wavelengths without polarization analyzer. (G–I) Measured intensity profiles at the corresponding focal planes of different wavelengths when a linear polarizer oriented 90° with respect to the illumination polarization was placed in front of the camera. In (D–I), an *x*-polarized Gaussian beam is impinging on the sample at normal incidence.

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