1. Introduction

Vector beams (VBs) have garnered significant interest in recent decades owing to their unique characteristics such as spatially variant polarization [1]. They are applied extensively in fields, such as optical trapping [2], high-resolution imaging [3], ultrasensitive metrology [4], and particle acceleration [5]. Considering physical essentials of VBs, spin angular momentum (SAM) and orbital angular momentum (OAM) are associated with circular polarizations (CPs) and optical vortices, respectively, and they can be coupled into the total angular momentum (TAM) states of vector vortices [6]. Two conjugate TAM states constitute higher-order Poincaré (HOP) beams [7] and full-Poincaré beams [8], which are a fundamental category of VBs. The non-separability of these beams in space and polarization degrees of freedom is similar to the local entanglement in a bipartite system [9] that enables a range of methods and applications in quantum systems [10], such as superdense teleportations [11], asymmetric optical quantum networks [12], and quantum key distribution [13]. Accordingly, VBs provide a novel resource of protocols for encoding rotationally invariant qubits in alignment-free communication over a particular distance [14–16]. The focus of most studies is on the generation of VBs for which various methods have been developed using different interferometers, such as Michelson, Mach–Zehnder, and Sagnac [17–19] with optical elements, including Dammann gratings [20], q-plates [17], spatial light modulators [21], and spiral-phase plates [22]. However, cascaded bulky devices limit the flexibility of VB manipulation and are not suitable for integration into compact optical systems.

Metasurfaces are composed of metallic or dielectric subwavelength meta-atoms with variable geometric parameters (size, shape, and orientation) and provide a powerful and flexible platform for manipulating the light field by modifying multiple degrees of freedom, including amplitudes, phases, and polarizations. These metasurfaces have been widely applied in miniaturized and multifunctional optical devices and systems, such as broadband achromatic metalenses [23,24], OAM multiplexed holograms [25–27], VB generators [28–31], and polarization beam splitters [32,33]. They have also been used to implement multichannel transforms and develop elements for entangled photons [34,35]. Metasurfaces are powerful tools for manipulating VBs in subwavelength dimension owing to the strong spin–orbit interactions of light with the constituent meta-atoms. Particularly, VBs in frequency spectrum of microwaves [36–38], terahertz waves [39,40] and light-waves have been generated using metasurfaces with meta-atoms of sophisticated multilayer structures [36–38] and anisotropic resonators [41]. In these categories, metasurfaces for generating light-wave VBs, with the advantages of miniaturization and integration over the conventional optical systems, have attracted extensive interest and have been applied in wide range of applications such as quantum entanglement [42,43], on-chip manipulation of VBs [44], vector fiber laser [45], and full-color vectorized hologram [46]. Furthermore, in the entities of light waves, metasurfaces have been used to generate versatile VBs with radial and azimuthal polarizations [47–49], and complicated HOP [50,51] and hybrid HOP beams [31,52]. Initially, VB generation using metasurfaces were realized by directly controlling the polarization and phase of the output wave of the meta-atom [48,53–55]. Subsequently, Yue et al. [29] introduced the method of superposing two optical vortices of opposite CPs [7] into the metasurface design, and HOP beams in multiple channels were generated [56]. Devlin et al. [52] manipulated arbitrary HOP beams using the spin–orbit conversion of meta-atoms to create two circularly polarized vortices. The superposition of vortices of cross-circular polarizations (CPCs) has been used to generate enriched vector fields, including multichannel and perfect HOP beams [31,57–60]. Recently, Wang et al. proposed the solid Poincaré sphere and designed the metasurface to manipulate degree of polarization of light waves [61]. However, the VBs generated by these methods are unfocused and usually have comparatively large beam sizes; even for perfect VBs, the radii of the doughnuts are as large as several tens of micrometers, which hinders certain specific but important applications.

With the beam size reduced to the subwavelength scale [62], tightly focused VBs exhibit fascinating properties and unusual abilities, including strong manipulations [63] and super-resolutions [3]. Furthermore, they have also contributed to the discovery of exotic optical phenomena in focal regions, such as topological knots [64], Möbius strips [65], and lasing emission chirality [66]. In recent years, the generation of focused VBs with metasurfaces has attracted considerable research attention; however, most studies have been implemented on simple VBs with radial and azimuthal polarizations using either simulation designs [57,62] or experimental realizations [39,40,66]. In addition, multichannel-focused un-coaxial HOP beams have been investigated [67,68] in which the cores of multiple VBs deviating from the optical axis. However, few studies have focused on the generation of coaxially focused HOP beams [50]. The difficulties encountered in this research field are due to the fact that the dark cores of the VBs would arrive at the optical axis and coincide with the probable maximum spot of the remaining co-polarization components owing to the focusing effect of the metasurface, thus resulting in deteriorated beam quality. Even in common metadevice performances, the remaining co-polarization components may also cause opposite effects. Essentially, the components are originated from such factors as errors of the meta-atoms deviating from the nano-half-waveplate in metasurface fabrication, misalignments in optical setup and mismatches of illuminating wavelengths. To decrease the influences of the co-polarization components, different schemes have been developed in recent years. Firstly, the components can be filtered out using the polarization filter composed of quarter-wave plate and polarizer [69,70]; however, this method is inapplicable in VB generation because two CPs in illuminating light are used simultaniously [56]. Then, components are also eliminated by using the nano-polarizers as the meta-atoms of the metasurfaces based on Malus law [71,72]. Besides, they can be eliminated by the destructive interference of the light waves transmitting meta-atom-pairs, which was demonstrated in the focused VB generations [50]. Yet, the last two schemes require two meta-atoms to constitute atom-pairs, leading to inconvenience for metasurface designs and fabrications.

In this study, we propose a metal metasurface comprising two sets of nanoslits with identical dimensions to generate HOP beams by overall elimination of co-polarization components based on the principle of Fresnel zones. The initial rings of the two sets of nanoslits start at two adjacent Fresnel zones [41,50,73] to provide opposite propagation phases for their wavefields. By alternately interleaving the subsequent equispaced rings and optimizing the number of rings, the overall superposition of the light field of the two nanoslits effectively eliminate the co-polarization components. The orientation angles of the nanoslits are rotated in opposite directions along the radius and identically in the azimuthal direction to form the overall hyperbolic and helical profiles of the geometric phases for the corresponding CPs; subsequently, two vortices of the opposite CPCs can be generated and selectively focused. The initial phases and amplitudes of the two CPC vortices are controlled by the initial orientation angles of the two-set nanoslits and elliptical polarization states of the incident light. Consequently, the HOP beams are realized at different points on the HOP sphere. In the practical performace, the finite-difference time-domain (FDTD) method was used to simulate HOP beams of different orders and optimize the metasurfaces. Experimentally, the samples producing different HOP beams were fabricated, and the measurements of the generated beams were conducted. The results demonstrated the feasibility of the proposed method for generating focused HOP beams of satisfactory quality. This study is of significance for compact metasurface devices and integrated optical systems. It has widespread applications in both classical and quantum physics.

2. Principles of metasurface design

2.1 Outline of the design

Figure 1 shows a schematic of the design of a gold-film metasurface for the generation of focused coaxial HOP beams. The metasurface comprises two sets of nanoslits interleaved on alternate rings with equal increments in radius. In Fig. 1(a), the metasurface on plane OXY is illuminated by an incident light of elliptical polarization |E_in> . It is the linear combination of left-handed circular polarization (LCP) |L > and right-handed circular polarization (RCP) |R > with weights ε₁ and ε₂, respectively; the focused VB is observed on focal plane oxy with focal length f behind the object plane. The two sets of nanoslits in the metasurface are specified as sets A and B, respectively, and the radii of the two adjacent rings in each set have the same increment δ_R as demonstrated in Fig. 1(b). A nanoslit of the metasurface with length L and width W is shown in green-squared inset in Fig. 1(a). It acts as a linear polarizer [50,70] with a transmitted light field perpendicular to the longer side and converts the illuminating light of the CPs into the co-polarization and cross-polarization components. With φ’ denoting the angle of the longer side of the slit with respect to x-axis, the slit orientation angles are defined as φ = φ’ + π/2. They are designed to form the geometrical phase profiles of the helical and hyperbolic phase profiles ϕ_h (θ) and ϕ_f (R), which vary with the azimuth θ and radius R to realize vortices and focusing, respectively. Furthermore, as shown in Fig. 1(a), under illuminating RCP and LCP lights, the two helical wavefronts of the corresponding cross- LCP and RCP are focused for the two sets of slits, respectively. To reduce the total co-polarization components in the paraxial area by overall destructive interference, the annulus area of nanoslit set B is increased by a Fresnel zone in the innermost and outermost radii with respect to set A as demonstrated by the schematics and bottom-line slits in Fig. 1(b). The color backgrounds depict the Fresnel zones. By controlling the elliptical polarization states of the incident light in match with metasurface design, the VBs corresponding to a point with certain latitude and longitude on the HOP sphere are obtained as shown in purple-squared inset in Fig. 1(a).

 

Fig. 1. (a) Schematic for design of metasurface to generate HOP beams under the illumination of an elliptically polarized light. The inset in green square is the enlarged view of a nanoslits. The inset in purple square schematically depicts the vector beam images of the total and component intensities under incident light of different polarizations. (b) Schematic of the nanoslit arrangements and the enlarged views. The upper panel is the arrangement of nanoslit set A, and the middle panel is for nanoslit set B, and the bottom panel depicts the enlarged view of sets A and B along the horizontal radius. The background color represents the Fresnel zones. The dot circle in the middle panel labels the first circle of set A to show the difference in the initial zones in sets A and B. In the enlarged view, n_1o, n_lo, n_1e, and n_le denote the initial and the last Fresnel zone numbers of sets A and B, respectively, and the area of each set covers (N_fz-1) of Fresnel zones. The nanoslits in sets A and B (the bottom panel) rotate oppositely along the radius; they are in response to incident LCP and RCP (labeled by red and blue circle arrows at upper-left corners in upper and middle panels) and create identical hyperbolic phase profiles on the cross- RCP and LCP components(labeled by red and blue circle arrows at upper-right corners in the upper and middle panels) to achieve selective focusing. The color maps in the lower-right corners in the upper and middle panels denote the corresponding helical phase profiles (left column) and hyperbolic phase profiles (right column), respectively. The helical phase profiles are oppositely varied in azimuth for the two vortices to have opposite chirality and helicity. (c) Schematic geometry of the optical system with the nanoslits for calculating the vector beam (VB) generation. The parameters of the nanoslits are: length L = 250 nm, width W = 80 nm, gold film thickness H = 200 nm, azimuthal period P_θ= 500 nm, and the radial period P_r/2 = 300 nm.

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Suppose that the metasurface contains nanoslits of 2N rings and that each set A and B occupies N rings. The first rings of sets A and B are located at the Fresnel zones of first even and odd numbers n_1o and n_1e= n_1o + 1, and the last rings at the last zones n_lo and n_le = n_1o+ 1, respectively. Thus, the areas of sets A and B are overlapped except for the innermost and outermost zones. The metasurface covers N_fz= n_le- n_1o Fresnel zones altogether, while each set covers N_fz-1 zones. For a nanoslit s_l at point (R, θ) on the metasurface, its orientation angle is set as φ (R, θ) = φ_h (θ) + φ_f (R), and the geometric phase for the CPC is ϕ (R, θ) = 2σφ (R, θ) = ϕ_h (θ) + ϕ_f (R). Meanwhile, ϕ_h (θ) and ϕ_f (R) are given as

2.2 Transmitted field of nanoslits

As depicted in Fig. 1(c), the unit vectors parallel and perpendicular to the longer side of the nanoslit s_la are represented by $\hat{e}_{ra}^s = \textrm{cos}\varphi {^{\prime}_a}{\hat{{\boldsymbol e}}_x} + \textrm{sin}\varphi {^{\prime}_a}{\hat{{\boldsymbol e}}_y}$ and $\hat{e}_{\varphi a}^s ={-} \textrm{sin}\varphi {^{\prime}_a}{\hat{{\boldsymbol e}}_x} + \textrm{cos}\varphi {^{\prime}_a}{\hat{{\boldsymbol e}}_y}$, respectively. Here, the subscript a specifies nanoslit in set A; and ${\hat{{\boldsymbol e}}_x}$ and ${\hat{{\boldsymbol e}}_y}$ are unit vectors in the x and y directions, respectively. In addition, $\varphi {^{\prime}_a}{\boldsymbol = }{\varphi _a} - \pi /2$ is the angle of the longer side of s_la with respect to x-axis as defined earlier for φ. In polar coordinates, the radial and azimuthal unit vectors are ${\hat{{\boldsymbol e}}_r} = \textrm{cos}{\theta _a}{\hat{{\boldsymbol e}}_x} + \textrm{sin}{\theta _a}{\hat{{\boldsymbol e}}_y}$ and ${\hat{{\boldsymbol e}}_\theta } ={-} \textrm{sin}{\theta _a}{\hat{{\boldsymbol e}}_x} + \textrm{cos}{\theta _a}{\hat{{\boldsymbol e}}_y}$, respectively. Then, the unit vectors $\hat{e}_{ra}^s$ and $\hat{e}_{\varphi a}^s$ are expressed as follows:

When the illuminating light is circularly polarized with chirality σ, it is represented as

From Eqs. (3) and (5), we obtain ${\hat{{\boldsymbol e}}_r} \cdot \hat{\boldsymbol e}_\varphi ^s = \textrm{cos(}{\varphi _a}\textrm{ - }{\theta _a}\textrm{)}$ and ${\hat{{\boldsymbol e}}_\theta } \cdot \hat{\boldsymbol e}_\varphi ^s = \textrm{sin(}{\varphi _a}\textrm{ - }{\theta _a}\textrm{)}$, and it follows that ${\boldsymbol E}_{in}^\sigma \cdot \hat{\boldsymbol e}_\varphi ^s = {e^{i\sigma {\varphi _a}}}/\sqrt 2$. Substituting Eqs. (3) and (4) in (5), the transmitted field of s_la is expressed as

2.3 Destructive and constructive interferences of co-polarization components and CPCs under CP illuminations

When the illuminating light is circularly polarized with chirality σ, the light field U (r, α; σ) generated by the metasurface at point q (r, α) on the observation plane are expressed as ${{\boldsymbol U}_{a\textrm{, }b}}\textrm{(}r\textrm{,}\alpha \textrm{;}\sigma \textrm{) = }{\boldsymbol U}_{a\textrm{, }b}^\sigma \textrm{(}r\textrm{,}\alpha \textrm{) + }{\boldsymbol U}_{a\textrm{, }b}^{ - \sigma }\textrm{(}r\textrm{,}\alpha \textrm{)}$ with subscripts a and b referring to sets A and B, and the superscripts σ and -σ represent the co-polarization component and the CPC, respectively. The light field U (r, α; σ) produced by the metasurface is expressed as

We primarily analyzed the light field U_a (r, α; σ) of set A. Based on the Huygens–Fresnel principle for surface plasmon polariton [74] and considering that the element area occupied by the slit s_la at point p (R_a, θ) is dA = R_adRdθ = R_aδ_Rdθ, U_a (r, α; σ) in Eq. (7) can be written as:

The co-polarization component ${\boldsymbol U}_a^\sigma \textrm{(}r\textrm{,}\alpha \textrm{)}$ in U_a (r, α; σ) as given in Eq. (11) is expressed as follows:

To elucidate the elimination of the co-polarization component by destructive interference, we divide the summation in Eq. (12) into two layers. The inner layer is implemented on the rings included in two adjacent Fresnel zones as a group, and the outer layer is conducted on groups of the adjacent zones of set A. As the nanoslits of set A start at zone n_1o and cover N_fz-1 zones, we assumed that the number of nanoslit rings of set A in two adjacent Fresnel zones of 2 M + 1 and 2 M + 2 (M = 0, 1, ……., (N_fz - 1)/2) is 2N_Mf for convenience, with N_Mf being an odd number. Then, the radius of the ja-th ring in set A is ${R_{ja}} = {R_M} + \textrm{(}ja - {M_f}\textrm{)}{\delta _R}$, where R_M is the radius of the first slit ring in the (2 M + 1)-th zone with its ring number in set A denoted by M_f, and ja = M_f, …, M_f+ 2N_Mf - 1 within the two adjacent zones. Equation (12) is then written as follows:

The analysis was similar for the co-polarization component ${\boldsymbol U}_b^\sigma \textrm{(}r\textrm{,}\alpha \textrm{)}$ of light field U_b (r, α) of set B. The nanoslits started at n_1o+ 1 zone and also covered N_fz-1 zones. In the adjacent two Fresnel zones, 2N_Mf rings are also included, and the radius of each ring is ${R_{jb}} = {R_M} + \textrm{(}jb - {M_f} + {N_{Mf}} + 1/2\textrm{)}{\delta _R}$. Thus, ${\boldsymbol U}_b^\sigma \textrm{(}r\textrm{,}\alpha \textrm{)}$ can be expressed as follows:

Although the summations over M in Eqs. (13) and (12) appear to be the same, the increased value of R_jb compared to R_ja indicates that the summation in Eq. (12) is implemented on the annular area enlarged by a Fresnel zone. By calculating the integrals over θ in Eqs. (13) and (14), the co-polarized fields for sets A and B were obtained as follows:

When ja = jb in Eqs. (13) and (14) for sets A and B, respectively, the difference in radius between R_ja and R_jb is $\delta R_{ja}^{\textrm{(}b - a\textrm{)}} = {\textrm{(}{R_{jb}} - {R_{ja}}\textrm{)}_{jb = ja}} = \textrm{(}{N_{Mf}} + 1/2\textrm{)}{\delta _R}$, which is also the difference in radius between two adjacent Fresnel zones. Correspondingly, the optical path difference $\delta s_{ja}^{\textrm{(}b - a\textrm{)}} = {\textrm{(}{s_{jb}} - {s_{ja}}\textrm{)}_{jb = ja}}$ between the rings of ja and jb to the central point o of observation plane is $\delta s_{ja}^{\textrm{(}b - a\textrm{)}} = \lambda /2$, and resultantly, ${s_{jb}} = {s_{ja}} + \delta s_{ja}^{\textrm{(}b - a\textrm{)}} = {s_{ja}} + \lambda /2$ for ja = jb. In Eqs. (15) and (16), the diffraction spots of ${J_0}\textrm{(}kr{R_{ja}}/f\textrm{) }$ and ${J_0}\textrm{(}kr{R_{jb}}/f\textrm{) }$ are produced by the slit rings with ${R_{ja}}$ and ${R_{jb}}$, respectively. For the rings within the two adjacent zones, the spot does not vary significantly in dimension, and the approximation ${R_{ja}}{J_0}\textrm{(}kr{R_{ja}}/f\textrm{) } \approx {R_{jb}}{J_0}\textrm{(}kr{R_{jb}}/f\textrm{) }$ is adopted. Therefore, when the metasurface is illuminated by CP light, the co-polarized field ${{\boldsymbol U}^\sigma }\textrm{(}r\textrm{,}\alpha \textrm{)}{ = }{\boldsymbol U}_a^\sigma \textrm{(}r\textrm{,}\alpha \textrm{)} + {\boldsymbol U}_b^\sigma \textrm{(}r\textrm{,}\alpha \textrm{)}$ is calculated as follows:

The derivation of Eq. (17) indicates that the difference in radius between the nanoslit ring sets A and B was the width of a Fresnel zone. The derivation of the above equation indicates that the difference in radius of a nanoslit ring in set A with the corresponding ring in set B for ja = jb was the width of a Fresnel zone. Accordingly, the destructive interference approximately eliminated the overall co-polarization component ${{\boldsymbol U}^\sigma }\textrm{(}r\textrm{,}\alpha \textrm{)}$ near the center point of the observation plane.

We now consider the light fields of CPCs on the observation plane. For the orientation angle φ_a (R_a, θ) = φ_h (θ) + φ_fa (R) of slit s_la in set A, the transmitted CPC was imposed with the geometric phase ϕ_a (R, θ) given in Eqs. (1) and (2). Notably, φ_fa (R) takes plus sign in Eq. (2) for set A, and the corresponding illumination is RCP. By substituting φ_a (R_a, θ) in Eq. (11), the CPC of the light field of set A can be obtained as follows:

In the calculations, the geometric phase ϕ_fa (R) for RCP (σ = -1) illumination cancels the hyperbolic phase profile implicated in ${e^{ik{s_{ja}}}}$ in Eq. (11); therefore, the CPC of set A is focused as expressed by Eq. (18). Although this equation appears simple, the summation therein is practically the same as in Eq. (14). In addition, the integral over θ in Eq. (18) is calculated as ${J_l}\textrm{(}k{R_{ja}}r/f\textrm{)}{e^{ - il\alpha }}$, which is the focused Bessel vortex beam with topological charge of -l = 2σm = -2 m. The superposition of the Bessel function ${J_l}\textrm{(}k{R_{ja}}r/f\textrm{)}$ over R_ja is evaluated as an integral over the radial coordinate R. Then, the ${\boldsymbol U}_a^{ - \sigma }\textrm{(}r\textrm{,}\alpha \textrm{)}$ is derived as follows:

This equation indicates that the focused circularly polarized vortex beam of CPC has an initial phase 2σφ₀ and that its doughnut profile of amplitude is expressed as [75]

While the LCP light with σ = 1 illuminates set A, the geometric phase is reversed by the opposite chirality σ, resulting in an opposite hyperbolic phase profile that diverges rather than focuses the light field. The light field of the CPC is insignificant and negligible owing to divergence; as a result,

Similarly, when the light of LCP with σ = 1 illuminates the nanoslits in set B, owing to the radial variation of orientation φ_br (R_b) opposite to φ_ar (R_a) for set A, the vortex of CPC in RCP is focused; therefore,

When the light of RCP (σ = -1) is used to illuminate nanoslits in set B for the same reason as in the calculation of Eq. (21), we obtain

Thus, selective control of the CPCs in LCP and RCP was realized by the focusing effect using the geometric phases.

2.4 Realization of HOP beams under illumination of elliptical polarizations

We now consider a general case in which the metasurface is illuminated by an elliptically polarized light u_in represented as follows:

Because each CP component in u_in produces the corresponding co-polarization component and CPC in the transmitted field, the co-polarization components on the observation plane are ${{\boldsymbol U}^{{\sigma _1}}}{ = }{\boldsymbol U}_a^{{\sigma _1}} + {\boldsymbol U}_b^{{\sigma _1}} \approx \textrm{0}$ and ${{\boldsymbol U}^{{\sigma _2}}}{ = }{\boldsymbol U}_a^{{\sigma _2}} + {\boldsymbol U}_b^{{\sigma _2}} \approx \textrm{0}$ based on Eq. (18), and according to Eqs. (21) and (23), the divergent CPC terms are ${\boldsymbol U}_a^{ - {\sigma _1}}\textrm{(}r\textrm{,}\alpha \textrm{) } \approx \textrm{0}$ and ${\boldsymbol U}_b^{ - {\sigma _2}}\textrm{(}r\textrm{,}\alpha \textrm{) } \approx \textrm{0}$. These results, along with those obtained using Eqs. (19) and (22) yield the focused vector field on the observation plane as follows:

3. Simulations

In this study, we first performed the 3D FDTD simulations to demonstrate VB generation, with the mesh size of 0.25 nm. The metasurface was designed using rectangular nanoslits with optimized dimensions of 250 × 80 nm, which were etched into a 200 nm-thick gold film. Identical nanoslits were arranged in a period of 500 nm on the perimeters of the circular rings and the increment in radius of the rings was constant. Each metasurface was designed to contain a total of 24 rings, which were for the nanoslits of sets A and B to be interleaved alternately. A perfectly matched layer was employed as the absorption boundary condition to avoid the influence of adjacent periodic elements. A planar light wave of wavelength 632.8 nm was incident on the 400 nm-thick substrate side of the metasurface. The monitors were set on the air side and focused vector fields were generated on the plane at z = f = 20 μm where their components and total intensities were observed and analyzed.

In Fig. 2(a), Panel (i) shows the image of the designed sample S1. The upper-right inset is the locally enlarged view of the sample. The sample diameter is 24 μm. Nanoslit sets A and B of the sample have the same rotation order of m = 0.5; they focus the vortices of the cross- LCP and RCP with topological charges of -1 and 1, respectively, and their superposition generates VBs of order l = 1. The lower-right inset is the designed phase map φ_a (R, θ) for the focused VB. Therein the dotted circle denotes the initial ring of set A, and the area is filled with faded colors, instead of the original blank space, for convenient comparison with the familiar focused vortex phase map. Panel (ii) in Fig. 2(a) shows the x-z cross-section image of the total intensity I = |E_x|² +|E_y|² +|E_z|² of the radially polarized VB generated by S1 under illumination of horizontal linear polarization (LP). The unneglectable z-component intensity |E_z|² resulted in a nonzero distribution in the image around the z-axis, which is the basic characteristic of focusing radially polarized VBs near the focus [50]. Panel (iii) in Fig. 2(a) shows the curves of the total and component intensities versus x. From the results, we see that an obvious longitudinal component field E_z was obtained, and its curve of I_z= |E_z|² exhibited a focused profile with full width at half-maximum (FWHM) of 0.91λ. Meanwhile, the intensity curve of the radial component I_r has a hollow-core profile with FWHM 0.70λ. Thus, the intensity I = |E_x|²+ |E_y|²+ |E_z|² exhibited a double-peak curve. Though it does not have the single sharp peak as the tightly focused radially polarized VBs, whose I_z curve usually has a peak value several times higher than that of the I_r curve, the peak value of its I_z curve reaches half that of I_r curve, demonstrating that the generated radially polarized VB is well focused. Besides, the tightness for focusing the VB may be reasonably enhanced by increase the size of the metasurface. Panels (iv–vii) in Fig. 2(a) from left to right indicate the simulated images of I_A= |E_x|² + |E_y|², I_x= |E_x|² and I_y= |E_y|² for the four VBs generated by S1 when illuminating light are horizontally, 45°, vertically, and 135°-linearly polarized from top to bottom, respectively. The purple double arrows in the title column indicate the direction of polarization of incident light, and the black arrows overlaid on the intensity doughnuts in the first column indicate the polarization state of the VBs. The light intensity images of x and y-component fields exhibit the familiar petal-like distributions, and the two lobes in the x-component images of different VBs are oriented and rotate in the same direction as the corresponding incident polarization.

 

Fig. 2. Simulation results for VB generated by samples S1 and S2. Panel (i) is the sample image and the designed phase profile of (a) S1 and (b) S2. Panel (ii) is the x-z cross-section image of the total intensity I of generated VB by (a) S1 and (b) S2. Panel (iii) is the curves of the total and component intensities in x direction of generated VB by (a) S1 and (b) S2. Panels iv-vii are the images of I_A, I_x, and I_y produced by (a) S1 and (b) S2 when illuminating light is horizontally, 45°-, vertically, and 135°-linearly polarized. The scale bar is 1μm. The purple and white arrows denote the incident polarizations and the direction of analyzing polarizer.

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Panel (i) in Fig. 2(b) is the image of sample S2 in which the two sets of nanoslits have the same rotation order m = -0.5. The topological charges of the corresponding focused vortices of RCP and LCP are l = -1 and 1, respectively, thus generating the π-phased VBs with l = -1. Panel (ii) in Fig. 2(b) shows the x-z cross-sectional image of the total intensity of the π-phased VB of radial polarization under the illumination of horizontally linear polarization. The hollow intensity distribution indicates the zero value of |E_z|² ∼ 0 near the vortex core for this VB. As shown in panel (iii) in Fig. 2(b), for the π-radially polarized VB, the intensity curve of I_φ has a profile with a hollow core with FWHM of 0.79λ, and I_z curve exhibits the hollow core and does not have focused central spot. This demonstrating the basic characteristics of the π-radially polarized VBs. Panels (iv–vii) in Fig. 2(b) show the intensity images of VBs of order l = -1; i.e., the π-radial, π-135°, π-azimuthal, and π-45° polarizations, respectively in the focal plane. Noticeably, the orientations of the two lobes in the x-component images were observed to rotate in the direction opposite to the corresponding incident polarizations, which differs from the previously discussed VBs. In addition to the numerical simulation of the VBs of order l = 1 and -1 (i.e., π-phase VBs) generated by the two samples S1 and S2, the light fields of all the samples in this study were simulated and optimized by FDTD before sample fabrications, although no simulation results are presented.

4. Experiment

4.1 Optical measurement setup and sample fabrications

Figure 3(a) shows a schematic of the experimental setup. A He-Ne laser with a wavelength of 632.8 nm is used as the light source. The incident light is passed through an attenuator (A) for power adjustment, which illuminates the sample from the side of the fused silica substrate, and the focused VBs are observed on the plane at a distance of 20 μm from the sample surface. The focused VBs are enlarged and imaged using a microscope objective (MO) and the enlarged images are recorded on a scientific complementary metal–oxide semiconductor (sCMOS). The analyzing polarizer (P) is placed in front of the sCMOS to obtain x-, 45°-, y-, and 135°- component intensity images. A half-wave plate (HP) and quarter-wave plate (QWP) are placed in front of the sample and combined to adjust the major-axis direction and ellipticity of the incident light of elliptical polarization corresponding to an arbitrary point on the traditional Poincaré sphere.

 

Fig. 3. (a) Schematic of the experimental setup. HP: half-wave plate, QWP: quarter-wave plate, A: attenuator, MO: microscope objective (NA = 0.9∕100×), and P: analyzing linear polarizer. (b) Scanning electron microscopy (SEM) images and enlarged view of the metasurface sample S1.

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In the sample fabrication, a 200 nm-thick gold film was evaporated on the 1 mm-thick fused silica substrate using magnetron sputtering and nanoslits were etched on the gold film using the focused ion beam (FIB) method. Seven samples were fabricated and denoted as samples S1–S7. The parameters of the fabricated samples are presented in Table 1. Figure 3(b) shows images of sample S1 generated through scanning electron microscopy (SEM), whose rotation order is m = 0.5 for generating VBs of order l = 1. Inset shows a locally enlarged view. The rotation orders of samples S2, S3, and S4 are m = -0.5, 1, and 1.5, respectively and their initial orientation angles φ₀ = 0. Correspondingly, they generate VBs of orders l = -1, 2, and 3 on the observation plane. Samples S5, S6, and S7 have the same rotation order as sample S1, i.e., m = 0.5; however, they have different initial orientation angles φ₀. Under the illuminating light of certain polarizations, they can generate VBs of the order l = l with different polarization states.

Table 1. Parameters of the metasurface samples

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4.2 VBs of different orders with linear polarized states

First, using the illumination of LP and changing the polarization direction (i.e., the polarization angle Φ/2 of the LP), the VBs were generated on the equator of the HOP sphere. Figure 4(a) shows a Poincaré sphere. For simplicity and convenience, we used this sphere to discuss the illuminating light on the traditional Poincaré sphere and the generated VBs on the HOP spheres of different orders. Figures 4(b)–(e) present the images of VBs of orders l = 1, -1, 2, and 3 on the equators of the HOP spheres generated by samples S1-S4, respectively. The purple double arrows to the left of the images indicate the polarizations of the incident light corresponding to the four equidistant points A, B, C, and D moving right-handedly on the equator of the conventional Poincaré sphere. The green double arrows in the dotted circles in the top title row of the figures indicate the transmission direction of the analyzed polarizer. The images in the first column are theoretical doughnuts, the overlaid line segments are schematics of the polarization distributions of the corresponding VBs, and images of the component intensities of the VBs generated by each sample are presented in the second to fifth columns.

 

Fig. 4. (a) Schematic sphere representing either the conventional or HOP spheres. (b)-(e) Theoretical images of total intensity I_A= I_x+ I_x and experimental component images of the VBs at point A′, B′, C′, and D′ in the equators on the HOP spheres with orders l = 1, -1, 2, and 3 produced by samples S1-S4 under linearly polarized illuminations, respectively. The purple double arrows in the left title column indicate the polarizations of incident light, corresponding to the four equidistant points A, B, C, and D in the equator on conventional Poincaré sphere, respectively; the green double arrows in the dotted circle in the top title row indicate the transmitted direction of the analyzing polarizer.

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Comparing the component images of samples S1 (m = 0.5) in Fig. 4(b) and S2 (m = -0.5) in Fig. 4(c), we observed that the intensity lobes of the π-phase VBs in the latter rotate with the analyzing polarizer opposite to the ordinary VBs in the former. In addition, the polarization distributions in Figs. 4(b) and (c) indicate that, for both the VBs of orders l = 1 and l = -1, the four points A’, B’, C’, and D’ representing the polarization states on their HOP spheres, respectively, move oppositely to the right-hand direction of points A, D, C, and B on the conventional Poincaré sphere. Particularly, the intensity images in the first and third rows in both Fig. 4(b) and (c) are the radial, azimuthal, π-radial, and π-azimuthal VBs of the Bell-like states [76], respectively, which correspond to the simulation results shown in Figs. 2(a) and (b). The measured results for all VBs were in good consistency.

Figures 4(d) and (e) show the focused VBs of l = 2 and 3 generated by samples S3 and S4 with rotation orders m = 1 and 1.5, respectively. The intensity images from rows 2 to 5 in each figure for the VBs of radial, 135°-slanted, azimuthal, and 45°-slanted polarizations also corresponded to points A’, B’, C’, and D’ on the equators of the corresponding HOP spheres, respectively. They exhibit properties similar to those of the VBs produced by sample S1 and are consistent with the familiar theoretical results.

4.3 HOP beams of different orders in different longitudes

For demonstrating the focused HOP beams by superposing the two CP vortices of non-equal weights, samples S1 and S4 were used as examples to generate VBs at nine points on the longitudes Φ'/2 = 135°, and Φ'/2 = 45°, on the corresponding HOP spheres of l = 1 and l = 3 as depicted in the left panels of Figs. 5(a) and (b), respectively. The corresponding theoretical, simulation, and experimental results are presented in the right panels of Figs. 5(a) and (b). On the right panel of each figure, the elliptical arrows in top title row denote the elliptical polarizations of the illuminating light, corresponding to the latitude Θ with sin Θ varying from 1 to -1 by an increment of -0.25. The images in the first and third rows are doughnuts of the theoretical and experimental total intensities of I_A, respectively, and the schematic of the polarization states is overlaid on the first row. The second and fourth rows show the simulated and experimental images of x-component intensities, respectively, and the corresponding values of latitudes on the HOP spheres are shown in the second row.

 

Fig. 5. Experimental intensity images of the focused HOP beams. The HOP sphere (left part) and theoretical, simulation, and experimental images of HOP beams (right part) corresponding to points I to A in the longitude shown on the HOP sphere of (a) order l = 1 produced by sample S1 and (b) order l = 1 produced by sample S4. The elliptical arrows in the top title row denote the elliptical polarizations of the illuminating light with latitude Θ determined by sinΘ varying from 1 to -1 by an increment of -0.25. The polarization states of the HOP beams are overlaid on the images in the first row and the latitude values of the HOP beams are given in the second row. The white double arrows indicate the transmitted direction of the analyzing polarizer.

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These images show the evolution of the HOP beams at points from the south-pole point I to the north-pole point A on the longitudes of the corresponding HOP spheres. Starting from the LCP vortex beam at south-pole point I, the VBs successively underwent elliptical polarizations of left-handedness in the southern hemisphere, LP at the equator point, elliptical polarizations of right-handedness in the northern hemisphere, and ended at the north-pole point A of the RCP vortex beam. According to the transformation relation (Θ, Φ) → (Θ’ = π-Θ, Φ’ = 2φ₀-Φ) in the above process, incident light transited in a direction opposite to RCP at the north pole to LCP at the south pole along the longitudes Θ = π-Θ’ on the conventional Poincaré sphere. As shown in the experimental images in Fig. 5(a), the lobes in the x-component images have the maximum contrast at the equator point E; however, they gradually blurred with decreasing contrast when the polarization state points moved toward the two pole points A and I. Similarly, the component intensity images of the HOP beams with l = 3 in Fig. 5(b) evolve in the same manner, but the x-component intensity images of the VBs contain 2 l = 6 lobes.

4.4 VBs generated by controlling initial orientations of nanoslits

To demonstrate manipulating polarization states of the VBs by the initial orientation angle of the nanoslits, we designed and fabricated samples S1, S5, S6, and S7 as a group with initial angles of 0, π/8, π/4, and 3π/8. According to Eq. (26), when the incident light kept horizontally polarized, the four samples generated the VBs of radial, 135°-slanted, azimuthal, and 45°-slanted polarizations, respectively. Figure 6 shows the SEM images of the four samples and experimental intensity images of the corresponding VBs. The upper-left panels in Figs. 6(a)–(d) show the SEM images and enlarged views of the four samples, respectively, in which the initial orientation of the first nanoslit in set A is marked. The upper-right panels of Figs. 6(a)–(d) depict images of the total intensity I_A with overlaid schematics of the polarization states. The lower-left and right panels of Figs. 6(a)–(d) depict images of the component intensities I_x and I_y, respectively. These images show that samples S1, S5, S6, and S7 generated radial, 135°-slanted, azimuthal, and 45°-slanted VBs, respectively, which are consistent with the theoretical results. These images demonstrate that the initial orientation angles of the samples can control the polarization states of the generated VBs. Thus, using the metasurfaces designed in this study, focused VBs were experimentally generated with satisfactory beam quality.

 

Fig. 6. Intensity images of the vector beams experimentally generated by samples S1, S5, S6, and S7 under horizontally polarized illuminations, respectively. (a-d) Images of the radial, 135°-slanted, azimuthal, 45°-slanted polarizations for generating VBs. The upper left parts show the SEM images and enlarged views of the samples. The other parts show images of the total and component intensities I_A, I_x, and I_y.

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5. Discussion

In the metasurface design, we assumed that in Eq. (17), in a pair of Fresnel zones, the sum of the co-polarization components of the nanoslits in sets A and B is approximately zero. This was realized by the destructive interference arising from the increase of optical path of nanoslit set B by a half-wavelength. Strictly speaking, the increment in radius of the Fresnel zones varied with the radius such that the number of rings 2N_Mf of sets A and B in certain adjacent zones may not be equal. However, their co-polarization components can be considered along with those of the other two similar adjacent zones. Thus, based on the overall summation, the co-polarization components in Eq. (17) can be eliminated. In the practical design of metasurfaces, this was realized by using FDTD simulations to determine the ring numbers of sets A and B and the total ring number 2N of the metasurfaces. The diameter of the metasurfaces is 24 μm, and each metasurface include 2352 nanoslits. Additionally, other factors that adversely affect VB generation, including practical errors in the amplitude and small defocusing of the two vortices, were finely optimized by the simulation. The metasurface parameters were finally determined as R_1a= 3.5 μm, R_1b= 4.95 μm, δ_R= 0.6 μm, N_A= N_B= 12. It is noted that under the preset focal length of 20 μm, the radii of the first and second Fresnel zone were calculated as 3.572 μm and 5.071 μm, respectively, and they were the theoretical values we set for R_1a and R_1b. In fact, the nanoslit would impose an inherent phase retardation on the transmitted waves [77]. To compensate this retardation, R_1a and R_1b were finely adjusted, and the optimization were performed using the FDTD simulations of the VBs. Finally, to further check the consistency of the experimental results with the metasurface designs, we measured the focal length of the metasurfaces. The measurement was performed using a piezoelectric nano-translation stage (PI E-516) with precision of 10 nm, which could finely move the metasurface longitudinally. The focal length of the sample S1 was measured 18.52 ± 0.21 μm, demonstrating a satisfactory consistency.

6. Conclusion

This study demonstrated the generation of focused coaxial HOP beams based on the proposed metasurface design with theoretical and experimental implementation. Metasurfaces were fabricated on gold films etched with nanoslits on equidistant circular rings. Using the geometric phases of two interleaved sets of nanoslits on a metasurface, two vortices with opposite CPs were generated and the focusing were realized simultaneously. By controlling the initial radii of the two sets of nanoslits to produce an optical path difference of half-wavelength, the opposite phase of the overall co-polarization fields of the two nanoslit sets was achieved. By the optimized designs with FDTD simulations, the co-polarization components were eliminated and focused coaxial HOP beams were generated. The proposed metasurface design provided a flexible and feasible approach for the manipulation of focused HOP beams. This study has significance in integrated and miniaturized optical devices and has potential applications in versatile areas in classical and quantum sciences.