Radially and Azimuthally Pure Vortex Beams from Phase-Amplitude Metasurfaces

To exploit the full potential of the transverse spatial structure of light using the Laguerre–Gaussian basis, it is necessary to control the azimuthal and radial components of the photons. Vortex phase elements are commonly used to generate these modes of light, offering precise control over the azimuthal index but neglecting the radially dependent amplitude term, which defines their associated corresponding transverse profile. Here, we experimentally demonstrate the generation of high-purity Laguerre–Gaussian beams with a single-step on-axis transformation implemented with a dielectric phase-amplitude metasurface. By vectorially structuring the input beam and projecting it onto an orthogonal polarization basis, we can sculpt any vortex beam in phase and amplitude. We characterize the azimuthal and radial purities of the generated vortex beams, reaching a purity of 98% for a vortex beam with l =50 and p = 0. Furthermore, we comparatively show that the purity of the generated vortex beams outperforms those generated with other well-established phase-only metasurface approaches. In addition, we highlight the formation of “ghost” orbital angular momentum orders from azimuthal gratings (analogous to ghost orders in ruled gratings), which have not been widely studied to date. Our work brings higher-order vortex beams and their unlimited potential within reach of wide adoption.

'Ghost' OAM orders generated by J-plates and q-plates In Figure 4a of the main text, we presented and discussed the azimuthal ℓ-mode purity of the vortex modes generated by the p-plate devices. Here, we extend those results to include the azimuthal ℓ-mode decomposition results for the phase-only metasurface devices, namely the J-plates and q-plates, shown in Figure S1. Recall that the form birefringence of metasurfaces allows to impart two different phase profiles, i.e., ℓ 1 , ℓ 2 on two orthogonal polarization states, |λ + ⟩ and |λ − ⟩, respectively. This allowed us to design and fabricate only two J-plate devices that act on circularly polarized light: the first with ℓ 1 = 1, ℓ 2 = 3 and the second with ℓ 1 = 5, ℓ 2 = 10. On the other hand, the q-plate design is restricted to only impart conjugate OAM, ℓ 1 = −ℓ 2 , to orthogonal circular polarization states, such that three q-plate devices were fabricated, one for each charge ℓ = 3, 5, 10. In the case of J-plates, the decomposition reveals peaks at the desired charges ℓ = 3, 5, 10 respectively, with 'ghost' OAM contributions of a few percent at the conjugate of the designed charge. For q-plates nearly all power is in the desired ℓ mode, with no auxiliary OAM contributions. To understand the origin of these OAM diffraction orders, or the lack there of, we expound the description in the main text to include the effect of a change in phase depth in azimuthal gratings for the case of J-plates and q-plates.  Figure S1: Corresponding azimuthal ℓ-mode spectra of vortex beams generated using phaseonly control in (a) J-plates and (b) q-plates, obtained via a modal decomposition in the LG ℓ,p basis. For each type of device, three devices were designed to impart the topological charge ℓ = 3, 5, 10. We note that no spatial filtering was used in the generation or detection of the vortex beams.
Recall that the p-plate devices were designed such that ℓ 1 = ℓ 2 = ℓ. Unwrapping the required azimuthal phase profile reveals a blazed grating with ℓ phase jumps for each orthogonal polarization, as seen in Figure S2a, with a total of 2ℓ phase jumps. These desired azimuthal phase delays are imparted via a propagation phase that is related to the dimensions of the nano pillar. This manifest visually as 2ℓ wedge sectors in the device design. We note that the rotation angle for the nano pillars is kept constant in the azimuthal direction and only varies in the radial direction, so as to structure the beam vectorially and sculpt the desired amplitude profile. Since the imparted phase depends on the dimensions of the pillars, any small deviation from the designed pillar size will affect the grating depth. As discussed in the main text, such a small deviation in the grating depth leads to a cross-coupling of unwanted 'ghost' OAM modes, with the first OAM diffraction order being at −ℓ. More specifically, the 'ghost' orders are formed at multiples of |ℓ 1 | + |ℓ 2 | = 2|ℓ|, centered about the desired charge.  Figure S2: Optical images of the fabricated metasurfaces for (a) p-plate, (b) J-plates and (c) q-plates, with the OAM charge ℓ 1 , ℓ 2 imparted on each orthogonal polarization state, respectively. Th middle row shows the unwrapped phase delays δ for both designed orthogonal polarization states as a function of azimuthal angle ϕ. The bottom row shows the rotation angle θ of the nano pillar for a fixed radial coordinate as a function of azimuthal angle ϕ.
Next, we extend this to the generalized case of the J-plates, which combine both prop-agation phase and geometric phase control to impart orthogonal circular polarization states with different amounts of OAM, ℓ 1 and ℓ 2 , while also performing a polarization conversion.
Lastly, we turn our attention to the azimuthal decomposition of the q-plate devices, in which we see no 'ghost' OAM orders. In contrast to the p-plate and J-plate devices, the q-plate device imparts the desired azimuthal phase using only geometric phase. Essentially, the device is designed using identical pillars, whose orientation angle varies in azimuth to impart the conjugate azimuthal phases ℓ 1 = −ℓ 2 to orthogonal circular polarization states.
The corresponding phase delay and rotation angles are shown in Figure S2c. As opposed to the case of p-plates and J-plates, fabrication errors in the size of the pillars affect the efficiency of the polarisation conversion and not the azimuthal grating depth, as the latter is geometrically defined by the rotation angles of the pillars. As a result, the generation of 'ghost' OAM orders is less susceptible in geometric phase elements, although the effect is still possible if the applied azimuthal phase grating is changed in some other way. Nevertheless, we reiterate that q-plates are not suitable in the context of this paper, as they do not structure the beam amplitude (they are phase-only devices) and thus generate vortex modes that unravel during propagation into a superposition with many radial modes.

p-Plate conversion efficiency
When applying a single-step amplitude shaping (regardless of the device used), the power conversion efficiency depends on the radial distribution of both the source and target LG ℓ,p mode. It is then important to consider the incident wave, in which case we choose a Gaussian beam as it is a readily available source in the laboratory. We can calculate the maximum modal power content η of the generated beam by taking the overlap between the Gaussian beam incident on the metasurface (or other device that implements amplitude shaping, such as an SLM) and that of the target LG ℓ,0 mode. This is given by 2 where ω s is the beam waist of the Gaussian source, ω 0 is the beam waist of the embedded Gaussian in the LG ℓ,0 mode and Γ(...) is the gamma function. There is an optimum choice for the beam waist ω s of the Gaussian source that maximizes the overlap with the mode, which occurs when ω s /ω 0 = |ℓ| + 1.This sets an upper bound in conversion efficiency that is independent of the technique used to apply the amplitude and phase modulation. The maximum modal power achievable as a function of the ℓ index is shown in Figure S3 (bars), reaching a maximum of 29% for ℓ = 3, 23% for ℓ = 5 and 16% for ℓ = 10. We see that a higher ℓ index results in lower conversion efficiency, as there is less overlap with the incident Gaussian beam and more light is discarded. The measured power conversion efficiency for each of the fabricated p-plate devices, are shown in Figure S3 Figure S4: A simulation of a Gaussian beam modulated by an azimuthal phase aperture produces a vortex mode with many concentric intensity rings. During its propagation, the power of the desired azimuthal mode is spread over many higher-order radial modes. Simulation showing the reduction in the writing area of the p-plate device as a function of the OAM it imparts as compared to that of a phase-only metasurface. This is calculated as the complement of the area of the p-plate to that of the phase-only device. The active area of the p-plate is selected where the intensity transmission efficiency is above 5%. This results in a characteristic ring shape of p-plate devices and allows to fabricate larger devices with higher OAM, while reducing the electron beam lithography writing time.   Figure S8: Schematic of the generation and detection of vortex beams from a p-plate metasurface. A focused, linearly polarized Gaussian beam propagates through the p-plate and exits the polarizer as a pure vortex beam. The purity of the vortex mode is characterized by performing optical overlap measurements. In the near-field (NF) of the beam, complexamplitude holograms are displayed on a spatial light modulator (SLM) and the resulting on-axis intensity in the Fourier plane is measured using a camera. The SLM operates in reflection but for simplicity is shown in transmission.  Figure S9: The azimuthal ℓ-mode spectrum of the vortex beam generated by a p-plate with ℓ = 50 and p = 0. The peak power contribution is at the designed charge. The inset shows a small contribution of a 'ghost' OAM order at the opposite charge −ℓ. This small contribution results in the azimuthal intensity undulations, so called 'pearls', in far-field intensity image shown in the inset of Figure 2d.