1. Introduction

Amplitude, phase, and polarization are fundamental properties of light beams that have been subject to extensive studies. The progress in controlling these quantities has resulted in the introduction of beams with novel features. For instance, vortex beams with helical phase wavefronts carrying orbital angular momentum (OAM) [1] and vector beams of spatially varying states of polarization (SoP) [2,3] have been generated. A convenient description for the polarization and spatial degrees of freedom of optical beams (spin-orbit interaction [4]) is furnished by the so-called hybrid-order Poincaré sphere (HyOPS) [5], developed from a generalized Poincaré sphere (PS) [6,7] and the earlier higher-order Poincaré sphere (HOPS) [8]. The poles of HyOPS represent circularly polarized light beams of opposite handedness, and therefore opposite spin angular momentum (SAM), while simultaneously carrying different amounts of OAM. Elsewhere on the sphere, the beam states correspond to varying measures of OAM and SAM. Coherent superpositions of two (polar) eigenstates of HyOPS are called vectorial vortex beams (VVBs) [9]. Sharing characteristics of both vector and vortex beams, VVBs have led to a number of applications, for instance, in optical communications [10], microparticle manipulation [11], and quantum optics [12].

However, VVBs have a major drawback: their creation requires two beams of opposite circular polarizations and different OAM states, and the intensity profiles of such beams do not generally coincide due to different divergences. This results in undesired polarization state mixtures. In addition, the ensuing VVBs on propagation develop spatially random SoPs [13], which further leads to zeroes in the intensity distribution [9]. To overcome the limitations of VVBs, perfect optical vortex beams (POVBs), which show equal divergences for all vortex orders, have been introduced [14,15]. By applying this principle, a perfect vectorial vortex beam (PVVB) can be generated from two circularly polarized POVBs of opposite handedness. Conventional methods of realizing POVBs and PVVBs require a combination of optical elements, such as axicons, q-plates, phase masks, spatial light modulators, and Fourier transform lenses [16–18]. This in turn increases the light-path complexity, makes the system bulky, and renders it incompatible with integrated photonics platforms.

Recently, metasurfaces have provided an alternative for conventional optical elements. Metasurfaces consist of nanostructures that enable one to control the light field’s polarization state, phase, and intensity profiles within subwavelength dimensions [19]. Due to the flexible phase modulation capabilities and several degrees of freedom in design, a variety of functional planar optical elements have been fabricated on the basis of metasurfaces, including lenses [20], holograms [21], and structured beam generators [22,23]. Also, metasurfaces for POVB and PVVB implementation have been proposed with metallic or dielectric nanostructures. Due to absorption, plasmonic metasurfaces however tend to exhibit a low efficiency for beam generation [24]. On the other hand, the dielectric approaches utilizing silicon (Si) or titanium dioxide (TiO$_2$) metasurfaces either require a complicated oblique incidence scheme [25] or are limited to generating only one type of PVVB [26].

In this paper, we put forward an approach to generate a pair of PVVBs by means of a single-layer all-dielectric metasurface and demonstrate the method by direct numerical simulations. The key element in our system is a Dammann vortex grating (DVG) which, through appropriate designs, enables the tailoring of the topological OAM and SAM properties of the ensuing PVVBs. The metasurface we consider is composed of Si subwavelength nanopillars that support efficient resonant beam excitation at the operation wavelength of 1550 nm [27], allowing for simultaneous modulation of both the geometric and propagation phase [28]. The distinguishing feature of the method is that it creates, in a controlled manner, both diffraction-related and spin-dependent topological charges in the output beams. In particular, the generated PVVBs carrying OAM are shown to depend on the diffraction order, which allows for an important, additional degree of freedom for beam engineering. The simulation results indicate that the emerging PVVBs have a high beam quality of up to 99.9${\% }$ mode purity. We also show that the structure can operate within a broad wavelength range in infrared, thus leveraging potential for applications in various laser systems. The versatility, compact size, and high efficiency indicate that these metasurfaces could be used in many fields including optical communications, quantum information science, and integrated nanophotonics.

2. Perfect vectorial vortex beams

We recall first that a scalar Bessel beam (of unit amplitude) in the cylindrical coordinate system ($r$, $\varphi$, $z$) is expressed in the form

 

Fig. 1. (a) Schematic illustration of HyOPS. Through the Fourier transformation (FT) of a VVB in the Bessel-mode basis, HyOPS can represent PVVBs. The poles (points I and IV) correspond to POVBs of opposite circular polarization, while points II and III on the equator represent PVVBs of linear polarization such that the polarization direction changes with the beam’s azimuthal angle. (b) Intensity distributions related to points I–IV on HyOPS for the case $m = 2$ and $n = 6$. Left column: total intensity profiles (white circles, arrows – polarization states). Right column: intensity patterns passing through a horizontal polarizer.

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To create a PVVB as we desire, one simple way is employing a lens to perform an optical Fourier transformation of the Bessel-type VVB. Making use of Eq. (2), on the focal plane of the lens this transformation can be expressed as [29]

3. Principles of dual PVVB generation

The focal plane field given by Eq. (5) is an idealization containing a radial delta function, which cannot be realized in practise. Thus, we consider a Bessel-Gaussian (BG) beam as a feasible approximation. By loading three optical functionalities onto a single metasurface, we can convert a Gaussian beam into two different BG-type PVVBs, as illustrated in Fig. 2. We take the incident field at plane $z = 0$ to be of the form [cf., Eq. (2)]

 

Fig. 2. Concept of metasurface for dual perfect vectorial vortex beam generation. The structure can be viewed as a combination of three conventional optical elements: an axicon, a q-plate, and a Dammann vortex grating (DVG) effecting beam splitting.

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Next we analyze the complex field amplitude of the beams emerging from the trifunctional device, when the illumination is as specified in Eq. (7). We consider first a single diffraction order. In the coordinate frame of the (paraxial) diffraction order $u$, the output beam in the DVG near field reads

To obtain the far field, we again perform an optical Fourier transformation with a lens, as in Eq. (4). In the focal plane, the beam field of diffraction order $u$ becomes

Likewise, for the diffraction order $v$, the complex Fourier-plane field reads

Figure 3 illustrates PVVB beams generated by both symmetric and asymmetric DVGs under $x$-polarized incidence (the diffraction orders are $u = -1$, $v = +1$ and $u = -1$, $v = +2$, respectively, while the other parameters are set as $l = 2$ and $q = 4$). The Fourier-plane intensity and phase profiles for the symmetric and asymmetric DVGs are calculated and shown in Fig. 3. According to the intensity profiles in Fig. 3(a), the output PVVBs are symmetrically located at diffraction orders $\pm 1$, while in Fig. 3(b) the beams are refracted by the asymmetric DVG at different angles into orders $-1$ and $2$. Moreover, one can confirm that the phase profiles shown in Figs. 3(a) and (b) are consistent with the mathematical expressions of PVVBs consisting of eigenstate pairs $m = -6$, $n = -2$ and $m' = 2$, $n' = 6$, as well as $m = -6$, $n = -2$ and $m' = 6$, $n' = 10$, respectively. Hence symmetric DVGs create eigenstates whose topological charges obey $m^\prime = -n$ and $n^\prime = -m$, whereas the eigenstates of asymmetric DVGs are not bound to any specific relationship.

 

Fig. 3. Numerically calculated intensity and phase profiles of the far-field output beam states for (a) a symmetric and (b) an asymmetric DVG, under $x$-polarized incidence with $u$ and $v$ denoting the corresponding diffraction orders.

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Our unique approach allows taking advantage of diffraction orders and DVGs for generating PVVBs with desired topological OAM and polarization properties. For a Dammann grating the absolute values $|c_{u}|$ and $|c_{v}|$ are equal [34], which guarantees the uniformity of the output beams. Moreover both symmetric and asymmetric DVGs can be applied, offering extra degrees of freedom for beam engineering.

4. Dual PVVB metasurface design

In this section, we introduce the design procedure for the metasurface. We consider the total phase modulation that includes the axicon, q-plate, and DVG for the POVB or PVVB generation and implement the associated phase maps through the different design dimensions of the metasurface (e.g., nanopillar size, shape, and rotation angle). For specificity, we present the metasurface design in detail for a symmetric DVG and briefly comment on the differences in the case of an asymmetric DVG.

A paraxial beam of an arbitrary polarization state can be decomposed into two spin eigenstates $|L\rangle$ and $|R\rangle$. When considering the beam’s passage into the far field, which amounts to a Fourier transformation of the near field, our metasurface is required to perform the conversion $|L\rangle \rightarrow |{\rm POVB}, R_m\rangle$ and $|R\rangle \rightarrow |{\rm POVB}, L_n\rangle$ by providing two independent phase profiles $\varphi _{\rm R}(x, y)$ and $\varphi _{\rm L}(x, y)$. The output beams are of opposite handedness and can have different topological charges $m$ and $n$, respectively.

The transformation can be expressed as

In order to design a metasurface whose phase modulation simultaneously satisfies Eqs. (15) and (16), we first describe the metasurface with a single Jones matrix $J(x,y)$ in the circular polarization basis [23]

Making use of the phase profiles given in Eqs. (17) and (18), the phase shifts $\delta _x$ and $\delta _y$ and the rotation angle $\theta$ are calculated as

In this work we implement three different metasurfaces (Meta1, Meta2, Meta3), each generating a pair of dual PVVBs. The design wavelength is $1550$ nm. The metasurface radius $R \approx 59\,\mu$m and the spot size of the incident Gaussian beam is $w_g = 60.0\,\mu$m. In the designs, the axicon radial period $d_a = 9.75\,\mu$m and the Dammann grating period $d = 5.85\,\mu$m. The ensuing $\delta _x$ and $\theta$ phase maps for Meta2, for which the vortex phase order $q = 4$ and the q-plate topological charge $l = 2$, are illustrated as characteristic examples in Fig. 4.

 

Fig. 4. Phase shift $\delta _x$ (left) and rotation angle $\theta$ (right) illustrated in the $xy$ plane for design metasurface Meta2. For clarity, phase $\delta _x$ is shown only for the central part of the metasurface. Features related to the radial axicon grating, the spokes of the vortex grating, and the Dammann grating along the $x$ direction are observed. Angle $\theta$ ranges twice over $2\pi$. The plotting pixel size in both figures is $650\times 650$ $\textrm {nm}^2$.

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The expressions in Eqs. (20)–(22) assign values of $\delta _x$, $\delta _y$, and $\theta$ to all points $(x,y)$ on the metasurface and we design a planar nanostructure to implement the corresponding phase map. A schematic diagram of the metasurface is shown in Fig. 5(a). The metasurface is composed of elliptical silicon nanopillars periodically arranged on a fused silica substrate. Each unit has two cross-sectional parameters – the long and short radii ($R_{\rm L}$ and $R_{\rm S}$), whose variation allows us to control the phase shifts. The rotation angle $\theta$, i.e., the angle between the $x$ axis and the $R_{\rm L}$ direction, is another parameter that can be independently varied to satisfy Eq. (22). All nanopillars are of the same height. Since $\delta _x$ and $\delta _y$ have a constant $\pi$ phase difference, the nanopillars may be viewed as acting like a half-wave plate.

To characterize the nanostructure, finite-difference time-domain (FDTD) simulations (software by Lumerical) are employed at the central wavelength of 1550 nm. The periods in the $x$ and $y$ directions are optimized at $P_x = P_y = 650$ nm to reduce coupling effects between the units. The height is chosen as $H = 800$ nm to cover a $2\pi$ phase modulation. The dimensions of eight selected nanostructures, from which the metasurfaces can be constructed, are listed in Table 1 and their simulated phase shifts and transmittance are shown in Fig. 5(b). The chosen elliptical nanopillars are seen to exhibit accurate phase control combined with high transmission, which ensures efficient generation of dual PVVBs.

Table 1. Size Parameters

View Table

The rotation angle $\theta$ is illustrated in Fig. 5(c) in two selected cases. Due to the high refractive index of the material, the nanopillars exhibit strong resonances [27] resulting in excellent polarization conversion rate $PCR = T_{{\rm cross}}/(T_{{\rm cross}}+T_{{\rm co}}$). Here $T_{{\rm cross}}$ and $T_{{\rm co}}$ are the cross- and co-polarization transmittances (calculated for an $x$-polarized incident plane wave). We observe that $PCR$, demonstrated Fig. 5(d), is close to unity over a wide wavelength range, which may enable broadband operation.

 

Fig. 5. (a) Schematic illustration of the metasurface and the elliptical nanopillar. (b) Phase shifts $\delta _x$ and $\delta _y$ and transmittance $T$ of the selected nanopillars 1–8 at the 1550 nm operation wavelength. (c) Illustration of the rotation angle $\theta$ for nanopillar index 1 at two sample locations on the metasurface. (d) Polarization conversion rate $PCR$ of each selected nanopillar within a near-infrared waveband of 1300–1800 nm.

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5. Results and discussion

To characterize the generation of dual PVVBs with hybrid polarization states, three metasurfaces (Meta1, Meta2, Meta3) were designed and simulated. Besides the axicon, Meta1 is composed of DVG of charge value $q = 2$ and a 1st-order q-plate ($l = 1$ spiral phase plate). Thus, at diffraction orders $u = -1$ and $v = +1$ Meta1 generates PVVBs that are superpositions of two POVBs of topological charges $(m,n)_{+1} = (1,3)$ and $(m,n)_{-1} = (-3,-1)$. In view of Eq. (9), the PVVBs at diffraction orders $+1$ and $-1$ can therefore be expressed as: $|\mathrm {PVVB}\rangle = \cos \alpha e^{i\beta } |\mathrm {POVB},R_1\rangle + \sin \alpha e^{-i\beta } |\mathrm {POVB},L_3\rangle$ and $|\mathrm {PVVB}\rangle = \cos \alpha e^{i\beta } |\mathrm {POVB},R_{-3}\rangle + \sin \alpha e^{-i\beta } |\mathrm {POVB},L_{-1}\rangle$, respectively. Different input polarization states (governed by $\alpha$ and $\beta$) lead to different spatial polarization distributions and phase vortices for the emerging beams. Meta2 consists of DVG of charge $q = 4$ and a 2nd-order q-plate ($l = 2$), hence the POVB topological charges are $(m,n)_{+1} = (2,6)$ and $(m,n)_{-1} = (-6,-2)$. Likewise, Meta3 corresponds to DVG of charge $q = 8$ and a 4th-order q-plate ($l = 4$), whereby $(m,n)_{+1} = (4,12)$ and $(m,n)_{-1} = (-12,-4)$.

Figure 6 shows simulated far-field diffraction patterns for metasurfaces Meta1, Meta2, and Meta3 with an $x$-polarized ($2\alpha ^\prime = \pi /2$, $2\beta ^\prime = 0$) incident field. The results are plotted in the $xy$ plane (perpendicular to the beams) at a constant propagation distance of $L = 1.4$ cm. In Fig. 6(a) annular intensity distributions of PVVBs at both $u = -1$ and $v = +1$ diffraction orders are presented for Meta1 ($|H_{-3,-1}\rangle$, $|H_{1,3}\rangle$), Meta2 ($|H_{-6,-2}\rangle$, $|H_{2,6}\rangle$), and Meta3 ($|H_{-12,-4}\rangle$, $|H_{4,12}\rangle$). As expected, the polarization states of PVVBs generated by Meta1, Meta2, and Meta3 exhibit azimuthal periodicities, respectively, of $2\pi /p$ with $p =$ 1, 2, and 4, in full accordance with the relations given in Sect. 2. We can clearly see that although the spin states and the topological charges are different for these PVVBs, their intensity profiles have equal radii (of approximately 0.34 mm). To explore the beams further, the cross-sectional intensity distributions of the PVVBs are simulated and displayed in Fig. 7. The theoretical radii of PVVBs obtained from Eq. (13) are $R_r = 0.37$ mm, which are close to the simulated values. Notably, both the ring radii and the transverse intensity profiles are essentially independent of the type of PVVB.

 

Fig. 6. (a) Simulated transverse intensity distributions (white arrows – polarization states) and (b) intensity patterns passed through a linear polarizer (white arrow) of PVVBs with different states $|H_{m,n}\rangle$ at the operation wavelength of 1550 nm. The rows marked $-1$ and $+1$ represent the different diffraction orders. $|H_{-3,-1}\rangle$, $|H_{1,3}\rangle$ are beams generated by Meta1, $|H_{-6,-2}\rangle$, $|H_{2,6}\rangle$ correspond to Meta2, and $|H_{-12,-4}\rangle$, $|H_{4,12}\rangle$ are created by Meta3. Scale bar: 0.2 mm.

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Fig. 7. Cross-sectional intensity distributions of the annular intensity rings for different PVVB states at diffraction orders $-1$ and $+1$.

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To verify the performance of these metasurfaces to generate PVVBs of varying states, we choose Meta2 as an example for a detailed study. Outputs from Meta2 can be represented by points on the HyOPS, as shown in Fig. 8(a). Depending on the order of diffraction, we encounter specific HyOPSs with different polar eigenstates. At order $u = -1$ the eigenstates that determine the HyOPS are $|\mathrm {POVB}, R_{-6}\rangle$ and $|\mathrm {POVB}, L_{-2}\rangle$, while at order $v = +1$ they are $|\mathrm {POVB}, R_{2}\rangle$ and $|\mathrm {POVB}, L_{6}\rangle$. For $u = -1, v = +1$, the PVVB states located on the HyOPS equator are characterized by polarization order [9] $p_{u(v)} = (n_{u(v)}-m_{u(v)})/2 = 2$ and by topological Pancharatnam charge [26] $s_{u(v)} = (m_{u(v)}+n_{u(v)})/2 = \pm 4$, respectively. At other HyOPS points, the PVVB polarization and topological properties are different. We have selected six points on HyOPS whose coordinates are labeled by I–VI in Fig. 8(a). The corresponding polarization states of the incident light are depicted in Fig. 8(b) and we analyze by numerical simulations the intensity patterns and phase distributions of the ensuing PVVBs in the $xy$ plane at wavelength 1550 nm.

 

Fig. 8. Properties of PVVBs generated by Meta2. (a) Six selected points I–VI on HyOPS with corresponding coordinates, each point represents a PVVB produced by the metasurface. (b) Polarization states of the incident light to achieve the various PVVBs. (c) Simulated annular intensity profiles of PVVBs (white circles, ellipses, arrows – polarization states) and (d) intensity patterns passing through a linear polarizer (white arrow) corresponding to points I–VI. (e) Phase distributions of PVVBs associated with points I–VI. Note that depending on the diffraction order, the eigenstates $|\mathrm {POVB}, R_m\rangle$, $|\mathrm {POVB}, L_n\rangle$ of HyOPS are different (for Meta2, at $-1$ the values are $m = -6$, $n = -2$, while at $+1$ they are $m = 2$, $n = 6$). Scale bar: 0.2 mm.

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In particular, Figs. 8(c), (d), and (e) show the annular intensity profiles, the intensity patterns after passing through a horizontal linear polarizer, and the beam phase distributions, respectively, corresponding to HyOPS points I–VI. In Fig. 8(c), the intensities are rings of identical radii, indicative of the perfect vortex nature of the beam. Notably, cases III and IV corresponding to $y$- and $x$-polarized incident light demonstrate that the intensity patterns show anisotropic polarization distributions with 4 lobes, which is consistent with polarization order $p=2$. This means that the polarization rotates by $4\pi$ in a full circle. In Fig. 8(e), cases III and IV show the simulated phase patterns with $s_{-1} = -4$ and $s_{+1} = 4$, which further confirm the four-fold polarization winding around the perimeter of the beam. This is in full agreement with the theory above. For cases I and VI (LCP and RCP incidence) in Fig. 8(e), the phases around the azimuth undergo $2\pi$ rotations $m_{-1} = -6$, $n_{-1} = -2$ and $m_{+1} = 2$, $n_{+1} = 6$ times, respectively. The simulation results are entirely consistent with the theoretical predictions for each HyOPS and verify that the metasurface performs as expected.

We have also studied the quality of the beams generated by the metasufaces. The beam purity plays a crucial role, in particular, in the fields of optical communication and laser physics. It is defined as

 

Fig. 9. (a) Mode purity of $|\mathrm {POVB}, R_{-6}\rangle$ and $|\mathrm {POVB}, R_{+2}\rangle$ with LCP illumination for Meta2, and (b) mode purity of $|\mathrm {POVB}, L_{-2}\rangle$ and $|\mathrm {POVB}, L_{+6}\rangle$ with RCP illumination for Meta2. Blue and orange columns: diffraction orders $-1$ and $+1$, respectively.

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6. Conclusion

In summary, we have proposed and simulated all-dielectric metasurfaces that generate dual perfect vectorial vortex beams. The metasurface is composed of a single layer of elliptic silicon nanopillars and it operates in a transmission mode at the wavelength of 1550 nm. By changing the metasurface design or the state of polarization of the incident light, the output beam can assume any state on the hybrid-order Poincaré sphere (HyOPS). The vortex order and the HyOPS eigenstates are shown to depend on the diffraction order and spin-dependent topological charges. The possibility to utilize both symmetric and asymmetric diffraction modes introduced by the DVG enable unprecedented control over the output beams. These degrees of freedom allow for a convenient and meaningful control over the topological, OAM and polarization porperties of the beams. Such metasurface elements are likely to inspire a new family of functional ultra-compact flat optical devices for various applications that require flexible phase control and high efficiency. The results presented in this work may prove important in the fields of optical communications, micro-manipulation, optical data storage, and quantum information processing.