Vortex beams are characterized by a helical wavefront and a phase singularity point on the propagation axis that results in a doughnut-like intensity profile. These beams carry orbital angular momentum proportional to the number of intertwined helices constituting the wavefront. Vortex beams have many applications in optics, such as optical trapping, quantum optics and microscopy. Although beams with such characteristics can be generated holographically, spin-to-orbital angular momentum conversion has attracted considerable interest as a tool to create vortex beams. In this process, the geometrical phase is exploited to create helical beams whose handedness is determined by the circular polarization (left/right) of the incident light, that is by its spin. Here we demonstrate high-efficiency Spin-to-Orbital angular momentum-Converters (SOCs) at visible wavelengths based on dielectric metasurfaces. With these SOCs we generate vortex beams with high and fractional topological charge and show for the first time the simultaneous generation of collinear helical beams with different and arbitrary orbital angular momentum. This versatile method of creating vortex beams, which circumvents the limitations of liquid crystal SOCs and adds new functionalities, should significantly expand the applications of these beams.
© 2017 Optical Society of America
CorrectionsRobert Charles Devlin, Antonio Ambrosio, Daniel Wintz, Stefano Luigi Oscurato, Alexander Yutong Zhu, Mohammadreza Khorasaninejad, Jaewon Oh, Pasqualino Maddalena, and Federico Capasso, "Spin-to-orbital angular momentum conversion in dielectric metasurfaces: erratum," Opt. Express 25, 4239-4239 (2017)
A helical mode of light is an optical field whose azimuthal phase evolution around the propagation axis (z) has the form , φ being the azimuthal angle and (an integer) called topological charge of the beam. The wavefront of a helical mode of charge is constituted by helical surfaces twisted together, whose handedness is set by the sign of , resulting in a topological singularity (optical vortex), along the propagation axis . Such vortex beams carry an average of orbital angular momentum (OAM) per photon [2,3]. Additionally, circularly polarized modes carry a spin angular momentum of per photon, depending on the polarization handedness. Such beams are central to the field of singular optics  and have found numerous applications such as optical trapping  where the angular momentum is a powerful manipulation tool to spin the trapped object [6,7] as well as to control its orientation .
The characteristic screw-type dislocation of a helical mode can be imposed on the wavefront of a propagating beam by means of different devices, for example, pitch-fork holograms [9,10] or cylindrical and axicons lenses and reflectors [11,12]. Additionally, helical modes can be also produced by exploiting the geometrical phase (also known as Pancharatnam-Berry (PB) phase) [13–15], to create inhomogeneous gratings for the wavefront reshaping [16,17]. In these spin-orbital angular momentum converters (SOC) the OAM of the vortex beam is coupled with the spin angular momentum of the illuminating light: switching the handedness of the illuminating beam polarization (spin angular momentum) flips the handedness of the vortex (orbital angular momentum). Locking the OAM to the spin angular momentum has unique applications in quantum computing and communications, allowing high dimensionality encoding of quantum units  and fast switching related to the modulation of the incident polarization of light [19–21].
More recently, the wavefront manipulation allowed by metasurfaces  has been used to produce a variety of PB optical elements, e.g., lenses [23,24] and vortex beam generators in the near-infrared [25,26]. Similar approaches have allowed working with visible light although with low transmission efficiency in the bluest part of the spectrum [27–32]. To date, the most versatile spin-orbital angular momentum converters for visible light are the liquid crystal devices developed by Marrucci et al. in 2006 and known as q-plates . They have found numerous applications in quantum optics although they are limited by degradation effects and resolution in defining the extent of the topological singularity region [34–38].
2. Nanostructured dielectric Spin-to-Orbital angular momentum Converter
In order to describe some general features of a SOC based on PB phase, it is useful to define the orientation angle of the optical axis (fast or slow) of each element of the device in the transverse plane (x-y plane). Regardless of the constituents, if each element imposes a π phase delay between the field transverse components, an incident uniform left-circularly polarized beam is turned into the beam that is right-circularly polarized with a geometrical phase in the transverse plane. Analogously to what reported in the first description of a q-plate , if the azimuthal variation of the angle in the PB-device follows the relation , the incident wave front is then turned into a helical wavefront composed of intertwined helical surfaces which carries an orbital angular momentum , where the sign depends on the handedness of the incident light polarization (is a constant). For instance, if and the incident light is left-circularly polarized (spin angular momentum of ), the out coming light is right-circularly polarized (spin angular momentum of ) with an OAM per photon of and zero net angular momentum transferred to the device (Fig. 1(a)). For there will be a net angular momentum exchange with the PB-device to preserve the total angular momentum of the system.
In our devices, as compared to previous work on metallic metasurface q-plates [28,29], the constitutive elements (nanofins) are subwavelength dielectric resonators [39–42] made of TiO2  (Appendix). Each nanofin is 250 nm long, 90 nm wide and 600 nm tall. The radial distance between two fins is of 325 nm (Fig. 1(b)). Figures 2(a) and 2(b) show the scanning electron microscope (SEM) images of the devices with and (and respectively). The insets of Figs. 2(a) and 2(b) show the devices as imaged in cross-polarization. The first polarizer sets the incident polarization direction. Each nanofin works as a half waveplate for the incident light: the nanofin rotates the incident polarization according to its orientation. The cross polarizer after the metasurface filters out the polarization opposite to that of the light incident on the metasurface thus creating intensity lobes in the camera image.
In order to fully characterize the vortex beams, we used a Mach-Zehnder interferometer as shown in Fig. 2(c). In this configuration, the source beam (a solid-state laser emitting at 532 nm with power lower than 2mW) is split in two linearly polarized beams by means of a 50/50 beam splitter. Half of the light (upper arm of the interferometer) passes through a quarter waveplate (QWP1) to produce a circularly polarized beam incident on the device. The vortex beam created by the device then passes through a polarization filter made of a quarter waveplate (QWP2) and a linear polarizer (LP2) in cross-polarization with respect to QWP1. This polarization filter is used to eliminate non-converted light passing through the device (Appendix). The reference beam propagates in the lower arm of the interferometer and passes through a half waveplate (HWP) to acquire the same polarization of the helical mode in port 1 (as well as in port 2). This maximizes the intensity modulation (thus the contrast) in the interference pattern.
Figure 2(d) shows the intensity distribution of a vortex beam with, generated by the device in Fig. 2(a), in a transverse plane (plane of the camera at port 1 of the setup) at about 45 cm from the device exit plane, when the reference beam is blocked. Figure 2(f) shows the intensity profile for the vortex beam with generated by the device of Fig. 2(b). The four insets of Figs. 2(e) and 2(g) show the intensity patterns produced in the plane of the camera by interfering the vortex beam with the reference beam. Such interference experiments are widely used to reveal phase singularities . The pitchfork-like interference is obtained when a vortex beam and a Gaussian beam interfere with an angle between their propagation axes, which sets the fringe spacing. If the vortex beam is collinear with the reference beam from the lower arm of the interferometer, a spiral is obtained as an interference pattern, with the number of arms equal to the topological charge of the vortex beam. The handedness of the incident circularly polarized light sets the orientation of the pitchforks and spirals.
Figure 3 shows how our approach can be used to produce optical vortices with higher values of topological charge, (Figs. 3(a)-3(d)) and (Figs. 3(e)-3(h)). Each individual device is 500μm in diameter and all devices are on the same glass substrate of 1 inch diameter (Appendix). This allows mounting the device on standard opto-mechanical components and to select the desired topological charge just by translating the corresponding device into the laser beam path.
Another important feature of our devices is related to the localization of the beam singularity. The fabrication process is based on atomic layer deposition (ALD) and electron beam lithography (EBL) (Appendix). This guarantees high resolution and reproducibility, resulting in precise definition of the singularity region and improving the vortex beam quality. For example, the device has a singularity region smaller than 3μm (Appendix).
In our devices we reached absolute efficiencies (the amount of light from the illuminating beam that is actually converted into the helical mode while also accounting for absorption and reflection from the device/substrate) of 60% (Appendix). Since TiO2 is ideally transparent at these wavelengths and the nanofins are only 600 nm in height, this measured efficiency is limited mainly due to reflections at the air-substrate and substrate-metasurface interfaces and error between the fabricated and designed nanofin dimensions. Thus this device provides a substantial improvement in efficiency as compared to previous metallic metasurface q-plate devices with conversion efficiencies of 8.4% at an operating wavelength of 780 nm . The simulated phase delay between the x- and y-component of the electric field and resulting conversion efficiency as a function of wavelength are shown in Appendix E and while the efficiencies we reported here were for 532 nm illumination, high efficiencies can be achieved at any visible wavelength simply by re-optimizing the lengths and widths of the individual TiO2 nanofins, as we showed in ref . Finally, the fact that these devices are fabricated using lithography and etching allows many devices with different topological charge to be placed on a single substrate—this is not easily-achievable with liquid crystal devices.
3. Fractional and interlaced spin-to-orbital angular momentum converters
As a further demonstration of the versatility of our approach, we designed a SOC that produces a vortex beam with fractional topological charge. This is possible when a non-integer phase discontinuity is introduced in the azimuthal evolution of the helical mode. In this case, Berry described the optical vortex as a combination of integer charge vortices with a singularity line in the transverse plane surrounded by alternating optical single charge vortices [44,45]. From a quantum optics point of view, the average angular momentum per photon has a distribution peaked around the nearest integer value of the topological charge and a spread proportional to the fractional part of the charge . We fabricated a SOC producing a 6.5 topological charge vortex beam. Figure 4(a) shows the intensity distribution of the resulting helical mode at about 55μm from the device plane (Appendix) and Fig. 4(b) shows pitchfork-like interference obtained in the Mach-Zehnder configuration of Fig. 2(c). The phase singularity line predicted for such vortices is evident. The interference pattern (Fig. 4(b)) also shows the line of alternating vortices (single line pitchforks) along the singularity line. For half odd-integer values of the OAM, two helical modes with same OAM but phase singularities lines with a relative π orientation are orthogonal . This has been used, for instance, to observe high-dimensional photon entanglement [47,48].
Our approach to SOC enables a new and unique feature, the generation of collinear beams from a single device with arbitrary and different OAM. In contrast, a detuned q-plate (i.e., a q-plate with phase delay not equal to π) can only produce a beam with OAM of 2q and a 0th order Gaussian beam with zero OAM, which is unconverted light.
To demonstrate this concept, we designed an interlaced and device (Fig. 5(a)). Two metasurfaces with different azimuthal patterns are interleaved by placing the nanofins at alternating radii. Although they have different topological charges (and), the beams emerge collinearly from the device, interfering in the plane transverse to the propagation direction. Figures 5(b) and 5(c) show the intensity patterns recorded in transverse planes (far from the device) for opposite handedness of the incident light. It is evident that the two interference patterns are flipped according to what is expected for beams with opposite topological charges. Figures 5(c) and 5(d) show the calculated interference patterns of two collinear helical modes of topological charges 5 and 10 with opposite handedness. These interference patterns are close to what we found experimentally if we assume for the charge 5 beam a Rayleigh range three times greater than for the charge 10 beam. In the calculations this accounts for the different divergence of the two experimental beams. While the interlaced designs allow for multiple values of OAM to be imprinted on a single beam, the measured efficiency for the interlaced device is 20%, which is less than the single topological charge. This drop in efficiency results from the spatial multiplexing of two devices—the period of each individual device is doubled leading to higher orders of diffraction.
It is important to note that each nanofin in our device has two interfaces, glass-TiO2 and air-TiO2. Illuminating one side or the other, as in Fig. 5(f), does not alter the phase delays imposed by the nanofins (Appendix) but only slightly affects light coupling into the latter, due to the different reflectance of the air-TiO2 and glass-TiO2 interfaces. We measured a small decrease (< 5%) in the device efficiency when illuminating from the air-side, due to the larger refractive indices difference with TiO2.
In the setup of Fig. 5(f), the beams illuminating the sample from opposite interfaces have opposite handedness. The double-face characteristic of our devices together with the illumination configuration of Fig. 5(f) allows one to simultaneously generate similar beams with opposite topological charges. This configuration was also used to obtain the intensity distributions of Fig. 5(b) and 5(c) representing the helical modes at optical ports 3 and 2 respectively.
Although we limited our interlaced design to two collinear beams, it is possible to produce three or more collinear vortices simultaneously as well as + and -collinear vortices (Appendix). This can find important applications in entanglement and quantum computing experiments. Moreover, the quantum description of a device simultaneously generating co-propagating vortex beams with different topological charges has never been investigated and represents a stimulating direction for future work. Finally, we expect good tolerance to heating since TiO2 has an intensity damage threshold of 0.5 J/cm2 in the femtosecond regime ; thus we envision using such devices for non-linear optics with pulsed lasers. We actually exposed one of our devices to a CW laser (532nm wavelength) with a power of 1W over the device area for 5 hours without observing any change in the device efficiency and beam quality.
In summary we have demonstrated that the interaction of light with designer metasurfaces can lead to the generation of complex wavefronts characterized by arbitrary integer and fractional topological charges and co-propagating beams with different orbital angular momenta. Our approach represents a major advance in design with respect to liquid crystals devices and as such has considerable potential in several areas of optics and photonics, ranging from quantum information processing to optical trapping and complex beam shaping.
Appendix A The device constitutive element (nanofin)
The individual units of the devices demonstrated in the main text are TiO2 nanofins, shown schematically in Figs. 6(a) and 6(b). These units were fabricated using electron beam lithography and atomic layer deposition of TiO2 onto the electron beam resist, as was previously described by our group in reference . The low temperature deposition yields amorphous TiO2 that has minimal surface roughness, which minimizes scattering losses. Additionally the TiO2 has a high refractive index, ranging from 2.64 at λ= 400 nm to 2.34 at λ = 700 nm and a bandgap of 3.46 eV, which lies outside of the visible portion of the spectrum. At the design wavelength (λ= 532 nm) for the devices described in the main text the measured TiO2 refractive index is 2.43. This value of refractive index is sufficiently high to confine the incident light to individual nanostructures and the bandgap occurring in the ultraviolet ensures there is no absorption at visible wavelengths.
In order to impose a geometric phase on an incident light field while maximizing the efficiency , the nanofins must possess structural birefringence so that a π phase delay can be imparted on orthogonal components of the incident electric field (the x- and y-components of the electric field in the example shown here). With our nanofins this birefringence is implemented for a fixed height (h), Fig. 6(a), by varying the length (L) and width (W), Fig. 6(b). In this way, different each component of light experiences a different effective refractive index, i.e. the nanofins are acting as waveplates with a fast and slow axis.
As stated in the main text, a waveplate with a spatially-varying fast axis (Pancharatnam-Berry phase optical elements [PBOE]), can impart a geometry-dependent phase on an incident circularly polarized light field. The imparted phase arises due to rotation of individual elements causing incident light to traverse two different paths on the Poincaré sphere. When the paths form a closed loop, the phase of the exiting light is then equal to half the solid angle of the loop. For our nanofins, which act as half waveplates, spatial rotation of each nanofin by an angle α (Fig. 6(b)) allows us to imprint the geometric phase and ultimately achieve the phase profile for producing a vortex beam, .
The geometric parameters of the fins, W and L, were determined using 3-dimensional finite difference time domain simulations (Lumerical). At a fixed height of 600 nm, W and L were allowed to vary so that for an incident wavelength of λ = 532 nm, there would be a π phase shift between the x- and y-components of the incident electric field. From this optimization, it was found that fins with W = 90 nm and L = 250 nm provided the desired phase shift as can be seen from the simulated electric field profiles in Fig. 6(c) .
Appendix B Device fabrication
All devices used were fabricated on a fused silica substrate. Resist was spun at 1750 rpm in order to achieve a thickness of 600 nm. The resist was then baked at C for 5 mins. The patterns were exposed using electron beam lithography (ELS-F125, Elionix Inc) and developed in o-Xylene for 60 s. For the ALD (Savannha, Cambridge Nanotech) of TiO2, TDMAT precursor was used to avoid chlorine contamination and the system was left under continuous 20 sccm flow of N2 carrier gas and maintained at C throughout the process. Reactive ion etching was carried out on Unaxis ICP RIE with a mixture of Cl2 and BCl3 gas (3 and 8 sccm, respectively) at a pressure of 4 mTorr, substrate bias of 150 V and ICP Power of 400 W. The samples were finally placed in 2:1 sulfuric acid:hydrogen peroxide to remove any residual electron beam resist.
Appendix C The Device
The devices described in the main text have been designed such that they can be easily integrated into manual or automatic positioning systems. Figure 7(a) shows a 1-inch fused silica substrate that has been patterned with TiO2 metasurfaces. The sample sits on a manual translation stage but automatic translation stages could be used for faster switching. Such design allows SOCs producing different topological charges to be arrayed on a single device at regular distances, (Fig. 7(b)) and easy switching between an output beam with a desired topological charge.
Appendix D Imaging at a distance from the device exit plane
In order to produce an image of the transverse light distribution at different distances from the sample plane, the upper arm of the interferometer was slightly modified. In particular the source beam was focused on the device by means of a 75mm aspherical lens. The light emerging from the device was then collected by means of a 4X infinity corrected microscope objective. The objective has 0.10 NA and works in confocal configuration with the aspheric lens. The sample is mounted on a translation stage that allows changing the distance from the objective. This configuration has been used, for instance, in the figure below that shows a fractional optical vortex at 55μm from the device exit plane. Figure 8 shows the image of the center of the device when the device plane is in focus. This image is what is sometimes referred to as near-field image although does not contain any near-field component. The image shows that the singularity localization in the device plane is smaller than 3μm. In ref  device singularities of 750 nm are reported. The physical dimension of the device singularity (region with no nanofins) in these metasurface devices is 1200 nm, however the fabrication technique used here can produce devices with controlled vacancies (i.e., lack of nanofin) on the order of the unit cell dimension of a few hundred nanometers. As pointed out in  this is an advantage of the metasurface q-plates.
Appendix E Device efficiency and conversion over the visible range
The polarization filter of Fig. 2(c) is used to eliminate the light that passes unperturbed through the device. This light has opposite handedness with respect to the helical mode and is absorbed by the polarization filter. It is known that when a waveplate that introduces a phase delay between the transverse field components is illuminated by circularly polarized light, the out coming field has two components with opposite handedness :
The term weighted by the has the same handedness () of the incident light, the term weighted by has opposite handedness () with respect to the incident field and acquires an extra phase term , where is the orientation of the plate axis (each nanofin in our devices).
In some experiments in literature, the percentage of light with handedness () is reported as the beam purity since represents the fraction of total outcoming light converted into the desired helical mode to the unconverted light . This feature, though, does not account whatsoever for the device transmittance since the purity can be close to 100% even if the transmittance is as low as a few percent. This is a particularly important point for communication and quantum optics applications where for high fidelity systems it is necessary that both the transmission efficiency and conversion efficiency of the transmitted light be maximized.
We designed our devices for a π phase delay at λ=532 nm. The transmittance of TiO2 in the visible is greater than 80%. The efficiency that we report in our paper is the amount of light that is converted in a helical mode with respect to the light incident on the device. We think that this value (experimentally measured to be up to 60%) is more useful for practical uses since it takes into account the fact that in a real device with discrete features like our TiO2 nanofins, a certain amount of light just passes through non-modulated. Furthermore, in a real experiment there are also reflections at the interfaces. These effects are all included in the figure of merit that we measured.
Actually, an inhomogeneous grating design  with densely packed grooves can in principle allow reducing the amount of light that passes through the device without acquiring the proper phase delay. This could result in even higher efficiency values at the designed wavelength. However, such design is limited to cylindrically symmetric structures like those necessary to generate a topological charge 2 vortex and cannot be applied to generate arbitrary vortex beams.
The wavelength dependence of the phase delay Г imposed by the constitutive elements of spin-orbital momentum converters based on metasurfaces limits the efficiency bandwidth of such devices. The actual phase delay is equal to π only at the designing wavelength. The farther the wavelength is from the designed wavelength the larger the amount of unperturbed light, according to Eq. 1. However, the Pancharatman-Berry phase is path-length independent and does not change with the wavelength of the incident light. Figure 9 shows a single charge helical mode obtained at different wavelengths from a supercontinuum laser with our 532nm optimized device.
The simulated phase shift between the x- and y- components of the electric field after passing through the nanofin and resulting as a function of wavelength is shown in Fig. 10. As can be seen from the figure, the conversion peaks around the design wavelength (532 nm) and is lower away from the design wavelength since the nanofin no longer acts as a half wave plate. Changing the design parameters of the nanofins would allow optimizing the device operation at other wavelengths .
Appendix F Fractional Vortex in far-field
Figure 11 shows the light distribution of the fractional 6.5 vortex beam at the camera plane, about 45cm from the device exit plane. The singularity line visible at 55μm from the device (Fig. 4 (a)) is no longer visible .
Appendix G Plus/Minus topological charge interlaced device
In addition to the interlaced +5,+10 device, we have also produced a device with the same magnitude but opposite sign of topological charge (Fig. 12).
R.C.D is supported by a Draper Laboratory Fellowship. We also acknowledge financial support from Air Force Office of Scientific Research (AFOSR) contract FA9550-14-1-0389 and FA9550-16-1-0156); Charles Stark Draper Laboratory Contract SC001-0000000959 and from Thorlabs Inc. Fabrication was performed at the Harvard University Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is a part of Harvard University.
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