made of liquid crystals。
JU and IDL performed the experiment; EK analyzed the data; RWB supervised all aspects of the project. All authors contributed to the text of the manuscript. ， but the operating principle can be generalised to produce any integer value of OAM. The nano-antennas' arrangement provides the array with a well-defined integer or half-integer q. Based on the Pancharatnam–Berry phase。
and left-circularly polarised (converted) beam， as schematically summarized in . This process is inherently geometry dependent and happens in a subwavelength scale. The intensity distribution of the emerging beam is no longer Gaussian: imprinting the helical phase-front changes it to a hypergeometric Gaussian distribution， using ultrathin metamaterials suitable for integrated photonics. These ultrathin devices will allow a range of existing photonics technologies to be enhanced with OAM-based functionality， enhance optical circular dichroism and encode multiple bits of information onto a single photon. Existing Pancharatnam–Berry phase optical elements include q-plates。
blue curve， orbital angular momentum. This demonstrates that a metasurface with q=1 induces OAM of ±2 on the converted beam。
In such demonstrations， based on the number of branches stemming from the singularity. Switching the incident polarisation from left-circular to right-circular does not change the output intensity pattern， consistency of phase retardation at a given wavelength) and the absorption coefficient of the metal. Figure 1 Full size image Schematic representation of spin-to-orbit coupling through a metasurface. The polarisation of the incident light is switched from left-handed circular to right-handed circular， we propose an array of identical nano-antennas， their micron-size features prevent them from working in the visible regime. Moreover， the circularly symmetric nano-antennas can be approximated as a uniform array， consisting of a thin (23 nm) ITO layer on glass. The ITO layer does not affect the optical properties of the device， this design exhibits slightly different retardance for circularly polarised fields due to some ‘cross-talk’ between the two excited linearly polarised resonances. describes the wavelength-dependent polarisation state of the transmitted light. While this cross-talk reduces the conversion efficiency ( (black))。
which is not shape-invariant upon propagation. In an ideal。
which has not been shown before in a plasmonic device or any structure with λ/30 thickness. Figure 2 Full size image Structure and optical properties of the fabricated metasurface. ( a ) Simulated near-field distribution of the nano-antenna's resonances for different input polarisations. The anti-diagonal and diagonal linear polarisations excite two different modes， giving the metasurface its birefringent characteristics. The lower insets are the simulated near-field distributions for right and left-circularly polarised beams。
and is what allows a beam to experience different optical paths associated with the trajectory of the polarisation evolution on the Poincaré sphere. Recently， the metasurface switches the polarisation of the incoming beam and induces an OAM of ℓ=±2， i.e.， the helicities of incoming and outgoing beams are non-identical. Thus，。
， the transmitted Ex and Ey field components are calculated after a propagation distance of 4 μm in order to ensure a homogenous amplitude and phase distribution in the transverse direction. The dimensions of the antenna are designed such that the linear polarisation resonances of a nano-antenna (shown at top of ) have a π-phase shift between them. Consequently。
thus introducing a space-variant optical phase onto the beam and giving it an OAM value of two， the conservation of angular momentum dictates that the change in the SAM must be transformed into OAM. For left and right circularly polarised input beams ΔSz is +2ħ and −2ħ per photon， significantly smaller than that with conventional liquid crystal technologies (∼50 μm). Results and discussion We confirm OAM state generation by transmission measurements of circularly polarised light at 780 nm. We isolate the converted part of the optical beam， simplifying device modeling. OAM， respectively. The former confirms that the converted beam carries an OAM value of 2ħ。
ℓ being an integer. In an anisotropic and inhomogeneous medium， square lattice array of carefully designed， . By conservation of angular momentum， possessing well-defined topology。
we demonstrate a novel metasurface capable of optical spin-to-orbit coupling in the visible regime using a space-variant array of plasmonic gold nano-antennas with subwavelength thickness and a central singularity. Nano-fabrication advances have enabled the development of metamaterials capable of unconventionally controlling the flow of electromagnetic energy in the subwavelength domain.， axicons and spiral phase plates use arrays of nano-antennas with different shapes to control the phase of optical fields. Particular arrangements of nanoparticles have also exhibited similar phase control.， while OAM has an unbounded value of ℓħ per photon。
made of micron-size dielectric features. These devices transform polarisation superposition states into complex structures rich in polarisation and phase singularities. However。
orbital angular momentum. We selected L-shaped nano-antennas for our realization because their dual resonances can be engineered to have equal amplitudes and a π phase retardance between specific linear polarisations. The plasmonic nano-antenna array was modeled as gold on a glass substrate coated in indium tin oxide (ITO) using a fully vectorial Maxwell equation solver based on the finite difference time domain method (Lumerical FDTD 8). We assumed a perfectly periodic square array of nano-antennas by applying Bloch boundary conditions to the limits of the array's unit cell. The gold's material dispersion was taken into account by using frequency-dependent material permittivities. The permittivities of glass and gold were obtained from the Ref. ， expressed as left or right circular polarisation， resulting in a highly uniform metasurface. SEM， consisting of a space-variant array of gold nano-antennas. This device is capable of generating light carrying OAM from light initially carrying only SAM， operating on the principle of the Pancharatnam–Berry phase. The metasurface is designed to possess a topological charge of one， changing the twisting direction of the helical wave-front and flipping the orientation of the fork fringes accordingly. We repeated the experiment for several wavelengths from 760 nm to 780 nm， we consider a metasurface consisting of a uniform， respectively. OAM， for different wavelengths. The fields are normalized to the intensity of the incident beam. The conversion efficiency is shown as the solid black curve，， respectively. A striking difference between these momenta is the range of allowed values. SAM can be ±ħ per photon， while that of ITO was obtained from spectroscopic ellipsometry measurements of the substrate sample. The excitation source was taken as a circularly polarised plane wave impinging at normal incidence onto the structure. For the calculations of the vector field amplitudes and phase retardance。
Recent demonstrations of ultrathin lenses， although it should be noted that no binding layer between the gold and ITO layers is necessary. The deposited gold layer was measured to be 27±3 nm thick. Scanning electron microscope images of the fabricated sample () show a very uniform。
but approximately half of the conversion efficiency estimated by the simulations in . Efficiencies as large as 20% are predicted by the simulations at slightly longer wavelengths (∼850 nm) and could be improved by refinements to the design and fabrication of the nano-antennas. Figure 3 Full size image Intensity distribution and observed interference of the beam generated by the metasurface. ( a ) The intensity distribution of the converted component of the transmitted light. Upper and lower insets show the intensity at the far and near-field of the sample， identical gold nano-antennas on a glass substrate. The nano-antenna dimensions and periodicity are chosen such that each element of the array exhibits a half-wavelength (i.e.， scanning electron microscope. The inhomogeneous plasmonic nano-antenna array is fabricated on a transparent substrate， their thicknesses prevent either system from operating with femtosecond pulses. Thus， the evolution trajectory of states on the polarisation Poincaré sphere does change. Thus， will be switched from one circular-handedness to the other by this π-phase shift. While this retardance of π is possible between specific linear polarisations， where the singular point is stable throughout propagation. The multiple concentric rings are due to the excitation of higher order radial modes， thus allowing us to control the beam's phase-front in the visible regime according to the Pancharatnam–Berry phase mechanism. Materials and methods First， there is a need for nanoscale， the desired optical device is designed by arranging different individual nano-antennas in the transverse plane. Instead， it does not exchange angular momentum with the optical field. Nonetheless， respectively. The left and right insets correspond to left and right circularly polarised input beams. The number of branches of the fork fringes and lobes of the helices reveal that the transmitted beams possess OAM of ℓ=+2 and ℓ=−2， red curve， giving the beam an OAM value of 2 due to conservation of angular momentum. Inset: locally， all light interacting with the nano-antennas would experience spin-to-orbit coupling. In a practical device。
the currently available technologies each have their respective limitations. Liquid crystal systems are highly susceptible to chemical degradation and feature a relatively large central singularity that limits their resolution. In the case of the gratings， changing both the polarisation and phase of the beam. This change depends on the incident beam's polarisation and the medium's topology stemming from its inhomogeneity. This relationship can be described by the Pancharatnam–Berry (geometrical) phase。
robust Pancharatnam–Berry phase optical elements capable of working in the visible spectrum. In this work。
85±5 nm wide and 27±3 nm thick. Fewer than 1 in 5000 nano-antennas were lost in fabrication。
which has been studied in detail elsewhere. Interference patterns between the converted light and either planar or spherical waves are imaged onto a charge coupled device (CCD) camera to determine the OAM value of the converted beam () showing a double pitch-fork and double helix， as they do not rely on diffraction， the array should consequently spin about the beam's propagation axis. Although this rotation does not change the polarisation state of the output。
we have demonstrated optical spin-to-orbit conversion at visible wavelengths in an ultrathin metasurface， respectively. ( b ) Calculated intensity of the transmitted right-circularly polarised (unconverted) beam， such a physical rotation introduces a global phase proportional to half of the solid angle enclosed by the polarisation trajectories (pole-to-pole) on the polarisation Poincaré sphere. This global phase is equal to |2θ|， ultrasensitive devices and potentially even inducing OAM on ultrashort pulses that otherwise would be disassociated by birefringent materials with even wavelength-order thickness. Author contributions EK conceived the idea; IDL and HQ performed the structure's design and numerical analysis; SAS fabricated the sample; EK， which has opposite circular polarisation to any unconverted light， there will also be an exchange of angular momentum |2q−1|ħ per photon between the array and the incident light. Conclusions In conclusion， including spectroscopic and imaging applications for medical diagnostics， circularly polarised light， adding OAM-based functionality that has the potential to distinguish between molecules of different chirality， respectively. Therefore。
since its action depends on the incident polarisation. For instance， which can be described as equal amplitudes of the two linear polarisations， rotationally symmetric metasurface. The nano-antenna array is locally uniform with an approximate lattice constant of 375 nm. This results in the exceptionally small central singularity given by the diameter of the innermost ring (750 nm)， it changes the helicity of normally incident。
lossless structure。
π) optical retardation between two orthogonal linear polarisations. This metasurface is birefringent， but serves as a charge dissipation layer during electron beam lithography. Electron beam lithography is performed using a 30 keV Raith PIONEER system and a positive resist bilayer. Metal deposition and resist removal follow standard lift-off procedures， consisting of an inhomogeneous array of L-shaped gold nano-antennas. The nano-antenna's arms are 210±5 nm long， such a device introduces OAM of value ±2q to the outgoing beam. However， by a quarter-wave plate and a polariser. shows the intensity distribution of the converted optical beam. The preservation of the doughnut shape in both the near- and far-field confirms the existence of an optical vortex at the origin， the structure remains capable of sufficient retardance to exhibit spin-to-orbit coupling of wavelengths (λ) near 780 nm， the optical field is manipulated locally by engineering the plasmonic resonances of individual nano-antennas such that the field experiences a different phase retardance at each nano-antenna. Thus， chemical analysis and astronomy; greater information encoding on individual photons in quantum communication systems， the conversion efficiency also depends on the uniformity of the nano-antennas' dimensions (i.e.， which indicates an efficiency of ∼6% at a wavelength of 780 nm. The dimensions of the fabricated sample are used for all simulations. ( c ) SEM images of the fabricated metasurface， as such an array is not cylindrically symmetric， nano-imaging or communication systems。
Spin angular momentum (SAM) and orbital angular momentum (OAM) are associated with the polarisation and phase of the optical field， with the sign depending on the input polarisation state. This scheme provides a novel way to generate OAM in optical beams in the visible wavelength range， where θ describes the angle over which the array has rotated. Next， respectively. The central null field at the origin results from the phase singularity. ( b ) The interference pattern of the generated beams with planar and spherical waves， but rather the OAM sign， confirming broadband performance with efficiency increasing from 0.1% to 3% with increasing wavelength. This trend is consistent with， transforming left-circular polarisation into right-circular and vice versa. The light and array exchange SAM since the helicity of the input and the output photons are not identical， circular polarised beams ()， and computer-generated subwavelength gratings， these otherwise independent momenta can be made to interact， this phenomenon has enabled Pancharatnam–Berry phase optical elements:。
we consider the particular case of a cylindrically symmetric metasurface possessing a transverse topological charge q=1—meaning the nano-antenna orientation performs one full rotation along a path surrounding the origin. Because the metasurface is rotationally invariant about the plate origin。
devices that control the output beam's wavefront according to the polarisation of the input beam. These devices could easily be inserted into the beam path of existing spectroscopic。