# Abstract

The purity of an optical vortex beam depends on the spread of its energy among different azimuthal and radial modes, also known as
*p*-modes. The smaller the spread, the higher the vortex purity and more efficient its creation and detection. There are several methods to generate vortex beams with well-defined orbital angular momentum, but only few exist allowing selection of a pure radial mode. These typically consist of many optical elements with rather complex arrangements, including active cavity resonators. Here, we show that it is possible to generate pure vortex beams using a single metasurface plate—called *p*-plate as it controls radial modes—in combination with a polarizer. We generalize an existing theory of independent phase and amplitude control with birefringent nanopillars considering arbitrary input polarization states. The high purity, sizeable creation efficiency, and impassable compactness make the presented approach a powerful complex amplitude modulation tool for pure vortex generation, even in the case of large topological charges.

## 1 Introduction

The characterizing feature of an optical vortex is a zero of intensity, which coincides with a phase singularity of the field. The phase circulates around this point of null intensity endowing the beam of light with orbital angular momentum (OAM) [1]. The OAM and the specific ring-shaped intensity distribution of these beams make them attractive for a number of applications [2], ranging from quantum information [3], [4] to super-resolution microscopy [5], [6], and have motivated the development of several techniques of optical vortex generation [7]. However, most of these methods, such as spiral phase plates [8], computer-generated holograms [9], [10], spatial light modulators [11], *q*-plates [12], [13], and *J*-plates [14], [15], rely on phase-only (PO) transformations. The azimuthal phase modulation imparted by these optical elements allows creation of an optical vortex, but the lack of amplitude modulation prevents the output beam from being a solution of the paraxial wave equation. The missing amplitude term is compensated by the spreading of the beam energy during propagation on high-order radial modes, thus leading to impure states consisting of a superposition of vortex modes [16].

There exist a few methods that can provide both the required phase and amplitude (PA) modulation for pure vortex generation, such as mode conversion in active resonators [17], [18], [19], but these require either specific input beams or rather complex cavity set-ups. Here, we introduce a method that allows conversion of an arbitrary input beam into a pure vortex mode using a single metasurface plate—called *p*-plate as it controls radial modes—which is easy to implement in practical optics experiments and could be of use in any vortex application, requiring that the beam power is contained within a specific *p*-mode.

## 2 Methods

After presenting our metasurface theory, we will prove the approach by applying it to the design of dielectric metasurfaces operating in the near-infrared region. This spectral range is only chosen for the sake of illustration, without loss of generality. We consider amorphous silicon nanopillars with a rectangular cross section lying on a silica substrate. The pillars have a height of 600 nm and are arranged in a hexagonal close packed lattice (600-nm pillar-to-pillar separation). The wavelength of the source is 1064 nm. A library of the transmission coefficients and phases imparted by the pillars as a function of their size *L*_{x,y} (in the 100- to 480-nm range) is constructed using the finite-difference time-domain (FDTD) module of Lumerical and selecting the elements with the best performance in terms of amplitude transmission and phase accuracy [20]. The average transmittance of the pillars chosen from the library to design the metasurfaces considered in this work is typically around 95%. The design code sets the sizes and orientation angles of the pillars according to our theory to produce a desired Laguerre–Gaussian (LG) beam. We validate the method by carrying out FDTD simulations of *p*-plates with 30-μm diameter, using plane wave or Gaussian sources with linear or elliptical polarizations. The beam waist of the designed LG modes is set to

## 3 Phase and amplitude control

The operation of a single birefringent nanopillar on an incident field can be represented in Jones calculus in the pillar frame of reference as

*M*is unitary, and the pillar cannot modulate the amplitude of the field. Meta-atom geometries capable of controlling directly both the phase and amplitude of the field have been demonstrated [ 22], [ 23] but suffer in general from fabrication complexities as they rely on resonances, may be limited in efficiency, or may not allow full modulation ranges, i.e. from 0 to 1 in amplitude and from 0 to

*M*allows conversion of the polarization of the incident field, thanks to the phase retardance

The operation principle of the proposed scheme of PA control is shown in Figure 1A. Given an elliptically polarized input, the pillar dimensions and orientation angle *α* are designed so that the polarization-converted beam after passing through a polarizer has the desired phase and amplitude. The phase is determined by one of the two dimensions of the pillar, which sets *ψ*, while the amplitude can be controlled via a combination of
*α*, as shown in Figure 1B.

### Figure 1:

Now, we concentrate on the specific problem of finding analytically the exact conditions to obtain the extinction. This is much more important than determining the conditions for unity transmission as the extinction is critical to mask certain parts of the input beam. On the other hand, the conditions for unity transmission only influence the efficiency of the device and will be discussed later. The problem is illustrated in Figure 1C. We consider an incident elliptical state represented by the Jones vector

*x*-axis. Our goal is to find the pillar parameters that convert the input state into a linear state oriented perpendicularly to the polarizer axis. In the following, the subscript “0” refers to parameters and operators corresponding to the extinction condition. First of all, we use the rotation matrix

*R*

_{0}to rotate the field on the pillar basis as

*θ*to

*x*-axis, which gives the condition

Figure 1D and E represent the solutions for the extinction condition for all the possible input polarization states shown on the

## 4 Pure vortex generation

The principle of arbitrary polarization conversion illustrated above for the single nanopillar is now applied to the design of optical plates for pure vortex generation. The operation scheme is shown in Figure 2A. The input beam can be any wave of known PA distribution, such as a plane or Gaussian wave, and arbitrary polarization. In the case of a plane wave, the *p*-plate is designed to impart the PA profile of an LG mode [26]

*p*are indices denoting the OAM and radial mode of the LG beam, respectively;

*r*and

*φ*are the radial and azimuthal coordinates, respectively;

*x*. The amplitude mask

*A*(

*r*) is assimilated in the angles of the pillars

### Figure 2:

The near-field intensity maps of an LG_{5,0} vortex generator obtained by FDTD simulations are shown in Figure 2B. The uniform intensity of the input plane wave is separated by the *p*-plate into a vertically polarized component, which is aligned with the polarizer axis and exhibits the characteristic ring-shaped distribution of a vortex mode, and a horizontally polarized component, which contains essentially the complementary intensity distribution and will be filtered by the polarizer. (A small fraction of light goes also into the longitudinal component, which is not shown here.) After propagating the component filtered by the polarizer to the far field, we obtain the intensity and phase distributions shown in Figure 2D, which can be compared with those obtained from a PO metasurface operating without a polarizer (Figure 2C). While in both cases, the phase shows an azimuthal modulation consisting of five sectors (
*p* = 0). The PO metasurface instead shows multiple rings corresponding to the superposition of different *p*-modes [16]. A modal purity analysis shows that 97% of the generated beam power is in the *p* = 0 mode for the PA metasurface, while this value drops to 23% for the PO metasurface. The efficiency in generating pure beams with the two types of metasurfaces will be discussed in Section 5.

The purity of the intensity profile generated by the PA metasurface all originates from a proper use of the pillar angle degree of freedom, as highlighted by the comparison of the device designs (cf. Figure 2C and D). As described in our theory, the PA control works for any input polarization state. In Figure 2E, we show that also for elliptically polarized light, where we chose as an example
*p*-modes may be appealing in other applications, such as super-resolution microscopy. In Figure 2F, we show that this is possible using a PA metasurface, demonstrating as an example the generation of an LG_{1,3} mode. Also, in this case, the purity of the beam is high as a very large fraction of the beam power transmitted through the polarizer (92%) is retained in the target mode.

## 5 Discussion

After having demonstrated that PA metasurfaces perform better than PO metasurfaces in terms of pure vortex generation, we wish to compare their efficiencies. We define the efficiency as the fraction of power of the beam incident on the metasurface that is converted into the target
*p* = 0 mode for
*p* = 0 mode; thus, no modal filtering is needed. What limits the efficiency in this case is the fraction of input power that needs to be absorbed by the polarizer. To maximize the generation efficiency, there is an optimum choice for the beam waist

*U*is the size of the unit cell of the metasurface. Thus, for any OAM charge and for a unit cell size fixed by the phase library, the phase sampling of the PA metasurface can be chosen arbitrarily large just by adjusting

### Figure 3:

The PA metasurfaces called *p*-plates introduced here also present some limitations with respect to the existing approaches for optical vortex generation. Differently from *J*-plates [14], [15], which implement PO transformations, *p*-plates are not spin–orbit converters. In particular, they can operate only on a single input polarization state. The degree of freedom that is used in *J*-plates to achieve independent OAM conversion of two orthogonal input polarization states here is used to apply a desired amplitude modulation. In comparison with *q*-plate lasers [17] and *J*-plate lasers [19], which incorporate PO transformation optics in active resonator cavities, *p*-plates can neither convert all the input energy into a pure LG state nor generate a continuously variable superposition of OAM states [19], only a fixed one. In view of these differences, we do not expect that *p*-plates will replace the existing technologies for OAM beam generation but may represent a very convenient and powerful approach in several applications, thanks to their simple and compact implementation scheme, sizeable efficiency, and high purity.

**Funding source: **European Research Council

**Award Identifier / Grant number: **817794

# Acknowledgements

This work has been financially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme “METAmorphoses”, grant agreement no. 817794.

**Author contribution**: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

**Research funding**: This work has been financially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme “METAmorphoses”, grant agreement no. 817794.

**Conflict of interest statement:** The authors declare no conflicts of interest regarding this article.

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## Supplementary material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0332).

**Received:**2020-06-17

**Accepted:**2020-07-23

**Published Online:**2020-08-10

© 2020 Marco Piccardo and Antonio Ambrosio, published by De Gruyter.

This work is licensed under the Creative Commons Attribution 4.0 International License.