High-E ﬃ ciency All-Dielectric Metasurfaces for the Generation and Detection of Focused Optical Vortex for the Ultraviolet Domain

Featured Application: Optics and Lasers, Abstract: The optical vortex (OV) has drawn considerable attention owing to its tremendous advanced applications, such as optical communication, quantum entanglement, and on-chip detectors. However, traditional OV generators su ﬀ er from a bulky conﬁguration and limited performance, especially in the ultraviolet range. In this paper, we utilize a large bandgap dielectric material, niobium pentoxide (Nb 2 O 5 ), to construct ultra-thin and compact transmission-type metasurfaces to generate and detect the OV at a wavelength of 355 nm. The meta-atom, which operates as a miniature half-wave plate and demonstrates a large tolerance to fabrication error, manipulates the phase of an incident right-handed circular polarized wave with high cross-polarized conversion e ﬃ ciency (around 86.9%). The phase delay of π between the orthogonal electric ﬁeld component is attributed to the anti-parallel magnetic dipoles induced in the nanobar. Besides, focused vortex generation (topological charge l from 1 to 3) and multichannel detection (l from − 2 to 2) are demonstrated with high e ﬃ ciency, up to 79.2%. We envision that our devices of high ﬂexibility may have potential applications in high-performance micron-scale integrated ultraviolet nanophotonics and meta-optics. illustrate high-performance dielectric metasurfaces for the generation of a focused optical vortex in the UV range with a maximum e ﬃ ciency of over 79%. multichannel topological charge resolved devices based on multi-wave interference are demonstrated with detection of the topological charges from − 2 to 2. Larger topological charges of the optical vortex can be generated and detected if the metasurface is big enough. Our devices of high ﬂexibility may have potential applications in high-performance micron-scale integrated ultraviolet nanophotonics and meta-optics.


Introduction
The optical vortex (OV), characterized by a donut intensity distribution and a helically structured wavefront, shows a phase evolution in the form of e ilθ , where θ is the azimuthal angle and l is the topological charge. It has aroused significant scientific curiosity since its discovery in 1992 [1] and produced tremendous applications in the field of optical tweezers [2], optical communication [3], quantum entanglement [4], nanotechnology [5], biomedicine [6], on-chip detectors [7], etc. Nevertheless, conventional OV generators, such as spiral phase plates [8], axicon lenses [9], and pitch fork holograms [10], suffer from limited performance, bulky configurations, and difficulty of integration. Therefore, it is of great urgency to seek miniaturized devices for OV generation and detection, and metasurfaces seem a promising candidate.
A metasurface, composed of arrays of sub-wavelength nanostructures, is a two-dimensional (2D) analog of metamaterials. It can manipulate the phase, polarization, and amplitude of electromagnetic

Theory and Meta-Atom Design
Depending on the geometry, a high refractive indexed meta-atom with sufficient height can manipulate the phase, polarization, and amplitude of the incident electromagnetic wave. Figure 1a shows the sketch of the proposed meta-atom. A Nb 2 O 5 nanobar with an orientation-rotating angle θ is patterned on a silica substrate to obtain the desired phase manipulation. As the nanostructure in the designed meta-atom is a simple rectangular nanobar, the transmission coefficients of this structure can be represented using the Jones matrix [30]: t(θ) = t x cos 2 θ + t y sin 2 θ (t x − t y )sinθcosθ (t x − t y )sinθcosθ t y cos 2 θ + t x sin 2 θ where t x and t y are the complex transmission coefficients when the polarization of an incident wave is x-polarized and y-polarized, respectively. Given the right-handed circular-polarized (RCP) light E i = (ê x + iê y / √ 2) of incidence, the transmitted electric field E t can be written as follows [31]: Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 11 The high conversion efficiency of the meta-atom requires two conditions: a high transmission of both electric fields and a phase delay of π between the orthogonal electric field components, in other words, the meta-atom should operate as a high-performance miniature half-wave plate. It has been reported that in dielectric metasurfaces, the displacement current and its induced magnetic field produced both electric and magnetic resonance in the nanostructure, resulting in a high transmission efficiency [34]. Besides, the complex refractive index of Nb2O5 has a sufficiently larger real part with The first term in Equation (2) represents a co-polarized transmitted wave component with the same helicity as the incident light. The second term represents a cross-polarized transmitted wave with the opposite helicity and an additional Pancharatnam-Berry (PB) phase manipulation of e i2θ . Therefore, through rotating the anisotropic nanobar from 0 to π, the full 2π phase shift of the transmitted cross-polarized wave component can be achieved.
The meta-atom is under the incidence of the RCP plane wave at a wavelength at 355 nm. The simulation of the structure is implemented by using the finite-difference time-domain (FDTD) method through the commercial software Lumerical FDTD. The computational area is set to be 200 × 200 × 200 nm in the x, y, and z directions, respectively. Periodic boundary conditions are used in the x and y directions of a unit cell. In contrast, the perfectly matched layer (PML) boundaries are applied along the z direction to eliminate unwanted reflections. The optical constant of niobium pentoxide is taken from F. Lemarchand's report [32], and the optical constant of the silica substrate is taken from the work of Palik [33]. The length (L) and width (W) are swept to maintain a high cross-polarized efficiency, which is defined as the ratio of the transmitted light with cross-polarization to the total incident light. Figure 1b shows the simulated efficiency changes with the length (L) varying from 110 to 170 nm and the width (W) from 40 to 100 nm. The height (H) of the nanobar and the period (P) of the square meta-atom are fixed to the optimized values of 340 nm and 200 nm, respectively. The peak efficiency of 86.9% is achieved with L = 165 nm and W = 45 nm, and the adjacent geometries also exhibit efficiencies over 80%, thus resulting in a relatively large tolerance to fabrication error. Figure 1c shows that the transmission (T), co-polarized, and cross-polarized efficiencies remain uniform as the Nb 2 O 5 nanobar is rotated from 0 to π. Figure 1d illustrates the simulated broadband analysis of an optimized nanobar in the range of 320 nm to 500 nm. The efficiency of the cross-polarized component is over 80% around the working wavelength of 355 nm. Moreover, there is a gradual increase in the transmission of the meta-atom, which implies that this Nb 2 O 5 -based nanostructure with an optimized geometry could gain high efficiency in the visible range.
The high conversion efficiency of the meta-atom requires two conditions: a high transmission of both electric fields and a phase delay of π between the orthogonal electric field components, in other words, the meta-atom should operate as a high-performance miniature half-wave plate. It has been reported that in dielectric metasurfaces, the displacement current and its induced magnetic field produced both electric and magnetic resonance in the nanostructure, resulting in a high transmission efficiency [34]. Besides, the complex refractive index of Nb 2 O 5 has a sufficiently larger real part with a low extinction coefficient. This also contributes to the high efficiency of transmission. The phase delay of π can be attributed to the existing antiferromagnetic resonance mode in the structure [35]. For a better understanding of the physical mechanism behind the phase delay of π, the vector field orientations and cross-sectional intensities of electric and magnetic fields for an optimized nanobar are plotted in Figure 2 under the incidence of a plane wave at 355 nm. As is demonstrated in Figure 2a,b, for the x-polarized incident wave, four circle displacement currents with the alternate direction of rotation induce four AMDs vertically located along the z-axis. The induced AMDS are staggered in anti-parallel orientations. This even number of displacement currents keeps the electric vectors of the incident and transmitted wave in the same direction. Figure 2c shows the simplified sketches of this antiferromagnetic mode under an x-polarized wave incidence. In contrast, the y-polarized incident wave induces three AMDs vertically located along the z-axis, resulting in the electric vector of the transmitted light being inverse to that of incident light (see Figure 2d-f). Thus, the transmitted electric vectors of xand y-polarized components are in the opposite direction, introducing a phase delay of π.

Focused Optical Vortex Generation
A succinct view of the designed transmission-type focused optical vortex (FOV) generator is presented in Figure 3a, the device combines the functions of a lens and a spiral phase plate into a single metasurface. The phase distribution profile of the FOV generator can be divided into the optical vortex profile and lens profile . Thus, the expression for the phase profile can be written as follows: where λ is the design wavelength (here λ = 355 nm), f is the focal length, l is the topological charge, which defines the degree of helicity at the focal point, while x and y are the transverse coordinates on the metasurface plane. Then we can obtain the spatial distribution of the phase profile compensated by the metasurface through adjusting the rotation angle ( , ) of the optimized Nb2O5 nanobars discussed above, where ( , ) = ( , )/2.

Focused Optical Vortex Generation
A succinct view of the designed transmission-type focused optical vortex (FOV) generator is presented in Figure 3a, the device combines the functions of a lens and a spiral phase plate into a single metasurface. The phase distribution profile of the FOV generator can be divided into the optical vortex profile ϕ OV and lens profile ϕ lens . Thus, the expression for the phase profile can be written as follows: where λ is the design wavelength (here λ = 355 nm), f is the focal length, l is the topological charge, which defines the degree of helicity at the focal point, while x and y are the transverse coordinates on the metasurface plane. Then we can obtain the spatial distribution of the phase profile compensated by the metasurface through adjusting the rotation angle θx, y of the optimized Nb 2 O 5 nanobars discussed above, where θ(x, y) = φ m (x, y)/2.
The calculation for numerical aperture (NA) is displayed in Figure 3b and Equation (4). Here, D is the diameter of the metasurface and s is the hypotenuse of the right triangle. The diameter of the metasurface is kept at 12μm to achieve a NA of 0.514. A full-wave numerical simulation of all the designed FOV generators is calculated through FDTD methods. PML boundary conditions are applied along all the x, y, and z axes, and the incidence is the RCP plane wave. A perfect electric conductor (PEC) ring is placed between the RCP source and the metasurface to limit the area of incidence to the circular metasurface region. Figure 4a-c are the intensity profiles of x-z cross sections for l = 1,2,3, respectively, and the white dashed line marks the position of the focal-plane (z = 10.0 μm). There is a small deviation (<1%) between the simulated focal length and the designed focal length. The deviation can be attributed to the discrete phase of each meta-atom for an approximation of the continuous phase distribution. Here, the efficiency of the FOV is defined as the ratio of the transmitted power passing through the focal spot in a circle whose radius is three times the FWHM. The definition of the FOV's efficiency is the same as in [36]. The focusing efficiency decreases as the topological charge increases, mainly due to the increase in the vortex ring's diameter.
The FWHM of horizontal cuts of vortex rings and focusing efficiencies are calculated under a different NA in Figure 5. The radius of all the designed metasurfaces is 6 μm, and the topological charge l is kept at 1. It can be seen that, as NA increases, both FWHM and focusing efficiency are decreased, which can also be attributed to the decrease in the vortex ring's diameter. All the metasurfaces were diffraction limited (FWHM < λ/(2NA)). The maximum value of NA reaches 0.788, and the maximum of focusing efficiency is measured to be as high as 79.2%. As far as we are concerned, the focusing efficiency seems to be promising compared to prior work [37][38][39][40]. The calculation for numerical aperture (NA) is displayed in Figure 3b and Equation (4). Here, D is the diameter of the metasurface and s is the hypotenuse of the right triangle. The diameter of the metasurface is kept at 12 µm to achieve a NA of 0.514.
A full-wave numerical simulation of all the designed FOV generators is calculated through FDTD methods. PML boundary conditions are applied along all the x, y, and z axes, and the incidence is the RCP plane wave. A perfect electric conductor (PEC) ring is placed between the RCP source and the metasurface to limit the area of incidence to the circular metasurface region. Figure 4a-c are the intensity profiles of x-z cross sections for l = 1,2,3, respectively, and the white dashed line marks the position of the focal-plane (z = 10.0 µm). There is a small deviation (<1%) between the simulated focal length and the designed focal length. The deviation can be attributed to the discrete phase of each meta-atom for an approximation of the continuous phase distribution. Figure 4d-f demonstrate the donut-shaped focal spot on the focal plane. It can be observed that the minimum intensity of the vortex ring is at the center of the focal plane. The bottom-left insets of Figure 4d-f show corresponding phase patterns at the focal plane. It illustrates that, as the topological charge l increases, the number of spirals in the phase pattern also increases. Figure 4g-i demonstrate the corresponding horizontal cuts of focal spots. The full widths at half maximum (FWHM) of horizontal cuts of the optical vortex ring are 311 nm, 335 nm, and 355 nm, respectively. The efficiencies of the FOV are as high as 75.1% (for l = 1), 73.0% (for l = 2), and 71.0% (for l = 3). Here, the efficiency of the FOV is defined as the ratio of the transmitted power passing through the focal spot in a circle whose radius is three times the FWHM. The definition of the FOV's efficiency is the same as in [36]. The focusing efficiency decreases as the topological charge increases, mainly due to the increase in the vortex ring's diameter.
The FWHM of horizontal cuts of vortex rings and focusing efficiencies are calculated under a different NA in Figure 5. The radius of all the designed metasurfaces is 6 µm, and the topological charge l is kept at 1. It can be seen that, as NA increases, both FWHM and focusing efficiency are decreased, which can also be attributed to the decrease in the vortex ring's diameter. All the metasurfaces were diffraction limited (FWHM < λ/(2NA)). The maximum value of NA reaches 0.788, and the maximum of focusing efficiency is measured to be as high as 79.2%. As far as we are concerned, the focusing efficiency seems to be promising compared to prior work [37][38][39][40].

Multichannel FOV Generation and Detection
A transmission-mode multichannel FOV (MCFOV) generator design principle based on metasurfaces is demonstrated. The phase distribution profile φ m (x, y) of the MCFOV generator of the x-y cross section at z = 0 is the synthetic phase of the multiple FOV beam interference.
Here, E i (x, y) is the amplitude distribution of the i th targeted beam, ϕ i is the phase profile of the i th targeted beam, where these two factors determine the intensity distribution of the transmitted light beam in the focal plane, l i is the topological charge number of the i th targeted beam, f is the distance between the metasurface and the focal plane, (x i , y i ) is the i th designed coordinate of the focal spot in the focal plane, and n is the total number of the targeted beam. It is worth mentioning that if the metasurface is large enough, both n and |l| can be any arbitrary large positive integer. Here, for simplicity, we assume that n = 5, E i = √ 1/n, and l = [−2, −1, 0, 1, 2]. Through adjusting the rotation angle θ(x, y) of optimized nanobars on the metasurface, the design of the MCFOV can be obtained.
As schematically depicted in Figure 6a, the elaborately designed metasurface with a radius of 6 µm can generate five-channel FOV beams with topological charges from −2 to 2 under the RCP plane wave incidence at 355 nm. The corresponding focal spots are generated on the focal plane 10 µm away from the metasurface. The simulated result of the multichannel FOV generator is shown in Figure 6a. The topological charges change from −2 to 2, and the corresponding focal spots are located at (0, −3 µm), (−3 µm, 0), (0, 0), (3 µm, 0), (0, 3 µm) on the focal plane, respectively. This device can realize a high efficiency of over 12% in every channel, which is much higher than the previous work [39].

Conclusions
In summary, an ultra-thin, highly efficient all-dielectric metasurface is demonstrated for FOV generation and optical vortex detection. The all-dielectric metasurface comprises Nb2O5 nanobars on a silica substrate. The meta-atom operates as a miniature half-wave plate with a cross-polarized efficiency of 86.9%. This high efficiency is achieved due to the existence of both electric and magnetic Besides, the designed metasurface is capable of interacting with incident OV beams. These OV beams can be obtained by a traditional OV generator [8] and a metasurface. When the OV beam with topological charge l = l in , illustrated in the metasurface mentioned above, the transmitted beam carries the orbital angular momentum (OAM) l out = (l in + l i ) , where is the reduced Planck's constant. The annihilation of the OAM occurs when l out equals zero, resulting in a solid spot on the focal plane. One can identify the OAM of the incidence by recognizing the solid spot's position according to the metasurface design. For better understanding, the previously designed metasurface is illuminated by normal incident RCP OV beams (RCP plane wave with a phase evolution in the form of e ilθ ) with different topological charges from −2 to 2 and the corresponding simulation results are shown in Figure 6b-e, respectively. It is necessary to mention that an optical vortex with a larger topological charge can be identified with more sizeable metasurfaces.

Conclusions
In summary, an ultra-thin, highly efficient all-dielectric metasurface is demonstrated for FOV generation and optical vortex detection. The all-dielectric metasurface comprises Nb 2 O 5 nanobars on a silica substrate. The meta-atom operates as a miniature half-wave plate with a cross-polarized efficiency of 86.9%. This high efficiency is achieved due to the existence of both electric and magnetic resonance in the nanobar. A phase delay of π between the xand y-polarized electric field components is attributed to the different parity of the numbers of AMDs induced by the incident light. Based on this concept, we demonstrate the FOV generation with distinct topological charges and NA. The designed FOV generator shows a maximum efficiency of 79.2%. Moreover, a multichannel device which can simultaneously generate and detect the OV beams with varying OAMs in a single metasurface has been demonstrated in the UV range. We envision that our designed devices of high flexibility and ultra-thin thickness may have potential use in high-performance micron-scale integrated ultraviolet nanophotonics and meta-optics.