Monolayer Chiral Metasurface for Generation of Arbitrary Cylindrical Vector Beams

Abstract

The cylindrical vector beam (CVB) has been widely studied and applied in recent years. However, many CVB generation methods suffer from complex systems, and large-size devices are required. Here, we propose a monolayer chiral metasurface composed of spin-sensitive unit cells which can generate different holograms for left- and right-circular polarization based on the combined modulation of geometric phase and detour phase. With a linearly polarized incident beam, the metasurface can generate CVBs with controllable polarization angles and orders, and even more complex vector beams. This work provides a new idea for the design of miniaturized optical devices for generating arbitrary vector beams.

1. Introduction

A light beam exhibiting a non-uniform polarization state across its cross-section is commonly referred to as a vector beam. A cylindrical vector beam, as one of the most representative examples, is the vector eigen solution of the wave equation under the paraxial approximation [1,2]. Alongside the well-known radially and azimuthally polarized beams [3], CVBs contain more complex polarization angles and orders. They exhibit axial symmetry in both amplitude and phase. Due to this unique feature, CVBs have found extensive utility in diverse fields, such as optical tight focusing [4], optical tweezers [5], and optical communications [6,7]. In particular, their polarization singularity enables tight focusing beyond the diffraction limit and the manipulation of microparticles with lower optical power. Additionally, the orthogonality between their vector modes lays the foundation for harnessing mode-division multiplexing techniques to greatly improve optical communication capacity and spectral efficiency. Crucially, the generation of various CVBs, including radial and azimuthally polarized CVBs as well as more complex polarization profiles, is a prerequisite for many practical applications of CVBs. Currently, many schemes, including spatial light modulator [8], Q-plate [9], etc., have been proposed to generate and modulate CVBs. Although these methods can effectively generate CVBs, complex optical configurations are often required to produce CVBs with specific polarization angles and orders, leading to issues of system complexity and low robustness. There remains a need for methods that can generate arbitrary CVBs in a simple, efficient, and compact manner.

In recent years, metasurfaces [10,11], consisting of two-dimensional arrays of nanoscale structures, have demonstrated their ability to regulate multi-dimensional properties of the light field, and have therefore become an ideal choice for manipulating complex CVBs with inhomogeneous polarization states. Recently, many metasurface structures have been proposed for CVB generation, such as nano-holes metasurfaces [6], concentric-ring metasurfaces [12,13], reflective-mode metallic metasurfaces [14], and double-layer metasurfaces [15]. However, many metasurfaces based on the geometric-phase principle are only suitable for circularly polarized incident beams; thus, they require the additional pre-processing of polarization conversion from initial linear polarization to circular polarization, which increases the volume and complexity of the optical system. For instance, concentric-ring metasurfaces can be used to generate CVBs with any polarization state, but they are only suitable for circularly polarized incident beams, and the output beam always carries a helical phase. In addition, although some multi-layer metasurface structures, such as the double-layer structure [15] and the reflective-mode sandwich structures [16], can generate CVBs and realize polarization control, their manufacturing difficulties are much larger than those of the conventional single-layer metasurface structure, resulting in low manufacturing efficiency and high production cost. There are still challenges to generating CVBs with an arbitrary polarization state from linearly polarized incidence using a compact, single-layer metasurface.

In this article, a monolayer chiral metasurface structure is proposed and numerically studied for generating CVBs with arbitrary polarized angles and orders from linearly polarized incidence. Benefiting from the monolayer structure, the proposed metasurface is easily fabricated, and it works with the linearly polarized incidence and thus avoids the liner-to-circular polarization conversion required in conventional geometric-phase metasurfaces. To achieve these functions, both geometric-phase and detour-phase principles are used in the metasurface. Based on the geometric-phase principle, two types of chiral unit cells are designed with multiple rectangular nano-slits on a gold film, enabling independent modulation for the left-handed and right-handed circularly polarized (LCP and RCP) parts of the incident linearly polarized beam. The distributions of the two types of unit cells on the gold surface are arranged according to detour-phase principle [17]; thus, two different holographic phases can be loaded onto the LCP and RCP components of transmitted beam, respectively. As a result, CVBs with arbitrary polarized angles and orders, and even more complex vector beams like optical skyrmion, can be generated through the metasurface from a linearly polarized incident beam. This work offers a simple and efficient approach to the generation of CVBs and other more complex vector beams and facilitates their applications in very compact and integrated optical systems.

2. Principle of CVB Generation

Theoretically, a light beam with any polarization state can be considered as a superposition of LCP and RCP components, whose Jones vector can be expressed as follows:

$$\left[\begin{array}{c}{E}_x\\ E_y\end{array}\right]=\frac{1}{2}(E_x+i{E}_y)\left[\begin{array}{c}1\\ -i\end{array}\right]+\frac{1}{2}(E_x-i{E}_y)\left[\begin{array}{c}1\\ i\end{array}\right].$$

(1)

Thus, if the LCP and RCP components can be loaded with complex amplitude distributions of $(E_x+i{E}_y)$ and $(E_x-i{E}_y)$, respectively, in theory, an arbitrary vector beam can be generated. Similarly, an m-order CVB can also be divided into an LCP optical vortex (OV) beam with a topological charge of m and an RCP OV beam with a topological charge of -m, whose Jones vector can be expressed as follows:

$$\left[\begin{array}{c}\cos (m\varphi +{\varphi }_0)\\ \sin (m\varphi +{\varphi }_0)\end{array}\right]=\frac{1}{2}{e}^{i(m\varphi +{\varphi }_0)}\left[\begin{array}{c}1\\ -i\end{array}\right]+\frac{1}{2}{e}^{-i(m\varphi +{\varphi }_0)}\left[\begin{array}{c}1\\ i\end{array}\right],$$

(2)

where m is the polarization order of CVB, $\varphi$ is the azimuthal angle, and ${\varphi }_0$ is the initial polarized angle [1]. As illustrated by Equation (2), once the LCP and RCP parts are loaded with helical phases of $e^{i(m\varphi +{\varphi }_0)}$ and $e^{-i(m\varphi +{\varphi }_0)}$, respectively, an m-order CVB can be generated.

As an example, the principle of generating a radially polarized beam (1-order CVB with ${\varphi }_0=0$) through the proposed metasurface is shown in Figure 1. For an incident LCP beam, by loading a spiral phase to the beam with a metasurface structure S₁ that is only in response to LCP (Figure 1b), the transmitted beam becomes an RCP OV beam with a topological charge of −1. In the same way, for the RCP incident beam, an LCP OV beam with a topological charge of +1 can be generated by another metasurface structure S₂ (Figure 1c). As shown in Figure 1d, when the LCP and RCP beams are combined into a linearly polarized (LP) incident beam, and the metasurface structures S₁ and S₂ are alternately superimposed to form the structure S₃, a radially polarized beam can finally be obtained by the structure S₃ according to Equation (2).

Although the above principle of CVB generation is very simple, it is difficult to achieve in a single-layer metasurface, mainly due to two problems: First, most-conventional single-layer metasurfaces can only modulate one type of circularly polarized beam, rather than achieve independent modulation on LCP and RCP beams at the same time. To solve this problem, we designed two types of chiral unit cells based on the geometric phase-principle for achieving independent modulation on LCP and RCP beams with high modulation efficiency and robustness. The second problem is that in many polarization-multiplexing structures, the output orthogonal polarization components are not in complete overlap [17,18,19]. Here, we make the output of LCP and RCP holographic beams perfectly overlap in the same position based on the detour-phase principle. Based on the combined modulation of geometric phase and detour phase, the proposed monolayer chiral metasurface can be used to generate CVBs with arbitrary polarized angles and orders.

3. Design of the Metasurface

According to the principle shown in Figure 1, it is a key point to design a metasurface which can independently regulate LCP and RCP components. Our designed metasurface is shown in Figure 2a, where two types of chiral unit cell structures, composed of multiple nano-slits on a gold film, are presented in blue and yellow colors for the independent modulation of LCP and RCP beams, respectively. The structural parameters of each nano-slit include the following: length, $l$; width, $w$; x-direction period, Px; y-direction period, Py; and thickness of gold film, h. As shown in the inset of Figure 2a, when the single nano-slit with principal axes of u and v can be considered as a single linear polarizer [20], its Jones matrix $J$ can be expressed as follows:

$$J=\left[\begin{array}{cc}{t}_u& 0\\ 0& t_v\end{array}\right],$$

(3)

where $t_u$ and $t_v$ are the transmission coefficients of incident light along the two principal axes, respectively, and θ is angle between the $u$-axis and the x-axis. Thus, the Jones matrix of the nano-slit in the x-o-y coordinate system can be expressed as follows [20]:

$$J_{\theta }=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]J\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]=\left[\begin{array}{cc}{t}_u{\cos \theta }^2+t_v{\sin \theta }^2& (t_u-t_v)\sin \theta \cos \theta \\ (t_u-t_v)\sin \theta \cos \theta & t_u{\sin \theta }^2+t_v{\cos \theta }^2\end{array}\right].$$

(4)

When LCP and RCP beams illuminate the nano-slit, the transmission light field can be expressed as follows:

$$\left[\begin{array}{c}{E}_{x-out}\\ E_{y-out}\end{array}\right]=\frac{J_{\theta }}{\sqrt{2}}\left[\begin{array}{c}1\\ i\end{array}\right]=\frac{1}{2\sqrt{2}}\left(\left(t_u+t_v\right)\left[\begin{array}{c}1\\ i\end{array}\right]+\left(t_u-t_v\right)e^{2i\theta }\left[\begin{array}{c}1\\ -i\end{array}\right]\right),$$

(5)

$$\left[\begin{array}{c}{E}_{x-out}\\ E_{y-out}\end{array}\right]=\frac{J_{\theta }}{\sqrt{2}}\left[\begin{array}{c}1\\ -i\end{array}\right]=\frac{1}{2\sqrt{2}}\left(\left(t_u+t_v\right)\left[\begin{array}{c}1\\ -i\end{array}\right]+\left(t_u-t_v\right)e^{-2i\theta }\left[\begin{array}{c}1\\ i\end{array}\right]\right).$$

(6)

As shown in Equations (5) and (6) above, irrespective of whether the incident circular polarization state is LCP or RCP, the transmitted light always consists of two components, including the co-polarized part without phase modulation and the cross-polarized part with an additional geometric phase of 2θ. Given that the angle θ ranges from 0 to π, the corresponding range of geometric-phase modulation on the cross-polarized part spans from 0 to 2π.

To achieve spin-sensitive responses of the two unit cell structures, we arranged N numbers of nano-slits with different orientations to form a single unit cell; hence, the total length of a unit cell along x-direction is PX = N × P_x. As shown in the blue-color unit cell in Figure 2a, the rotation angle of the i-th nano-slit in unit cell is chosen as ${\theta }_i={\varphi }_0+\frac{(i-1)\times {180}^{°}}{\mathrm{N}}(i=\mathrm{1,2},\cdots N)$ with respect to the x-direction, where ${\varphi }_0$ is the initial polarized angle in Equation (2). According to Equations (5) and (6) above, each nano-slit can generate a geometric-phase modulation of 2θ only on the cross-polarized part of the transmitted light; hence, the rotation of N nano-slits leads to a gradient-phase modulation on the cross-polarized part [21], and such a gradient phase makes the cross-polarized part propagate to a different direction to the co-polarized part and thus avoids their interaction. In Figure 2a, two chiral unit cell structures are designed based on the opposite rotation directions of the N nano-slits, including the anticlockwise unit cell (blue color) and the clockwise unit cell (yellow color), for the opposite gradient-phase modulation on incident LCP and RCP beams, respectively. As shown in Figure 2c, the blue-color unit cells make the output RCP part (cross-polarized part of the incident LCP beam) propagate to the right side, while the yellow-color unit cells make the output LCP part (cross-polarized part of the incident RCP beam) propagate to the right side; thus, the output LCP and RCP parts at the right side (indicated by dashed boxes) can be combined to further generate an arbitrary polarization state according to Equation (1). As a result, when a LP beam is incident, its LCP and RCP parts can be modulated by the blue and yellow unit cells, respectively, making their cross-polarized parts propagate to the right side and superposed to form the desired CVB, as in the principle shown in Figure 1d. It is noted that here, only the output beams at the right side (dashed boxes in Figure 2c) are used, and the output beams at the left side usually give the conjugation of right-side results.

In order to optimize the performance of the nano-slit, the cross-polarized part should be enhanced for higher polarization conversion efficiency. Here, the polarization conversion efficiency (PCE) is defined as the ratio of the cross-polarized part of transmitted light to the total transmitted light. In Figure 2b, two key parameters of slit length ($l$) and width ($w$) are studied, and the other parameters are chosen as $P_x=0.2 \mathsf{\mu }\mathrm{m}$, $P_y=0.2 \mathsf{\mu }\mathrm{m}$, and $h=0.075 \mathsf{\mu }\mathrm{m}$. The depth of the nano-slits (thickness of gold film h) only influences the transmittance of the structure. The 3D finite-difference time-domain (FDTD, Lumerical FDTD Solutions) simulation method was used to investigate the influence of the parameters on polarization conversion efficiency at the incident wavelength $\lambda =0.633 \mathsf{\mu }\mathrm{m}$. All boundaries in FDTD were set to periodic boundary conditions, and the minimum grid size was 10 nm. Since all nano-slits in our metasurface have exactly the same length and width, we only need to simulate a single nano-slit structure with periodic boundary conditions and scan its length and width to obtain the optimal results. The substrate used in FDTD is glass with a refractive index of 1.515, and the dielectric constant of gold film was calculated using the Drude model [22]. Figure 2b shows that the PCE of single nano-slit can reach more than 90% in some regions. We finally select the slit parameters as $l=0.167 \mathsf{\mu }\mathrm{m}$ and $w=0.02 \mathsf{\mu }\mathrm{m}$, with the corresponding optimized PCE of ~91%.

Next, to optimize the performance of unit cell composed of N nano-slits, we investigate the effect of number (N) on the polarization ellipticity [23] of the modulated light. In the FDTD simulation, an LCP beam is incident on the blue-color unit cell under periodic boundary conditions, and its cross-polarized part (RCP) in the transmitted beam should be modulated by the gradient phase induced by the rotation of N nano-slits. However, due to the imperfection of structures in FDTD and the inevitable scattering of the non-modulated co-polarized part (LCP), the polarization ellipticity of the modulated part of the transmitted beam is usually less than 1 (not a pure RCP beam), which affects the quality of the generated CVB. The influence of N on the ellipticity is shown in Figure 2d. When N increases, the ellipticity of the modulated beam also increases, and it does not vary significantly when N ≥ 8. Thus, we finally chose N = 8 for all unit cells. Moreover, we studied the working wavelength of the unit cells at N = 8, as shown in Figure 2e, where the ellipticity reaches its peak around the incident wavelength $\lambda =0.633 \mathsf{\mu }\mathrm{m}$.

As mentioned above, the geometric-phase principle is used to design the two types of chiral unit cells for the respective responses of the LCP and RCP parts of the incident beam, while the detour-phase principle is used to achieve different holographic phase modulations on the LCP and RCP parts by controlling the unit cell distributions in the metasurface. Figure 2f shows the detour-phase principle, where the light is incident onto the gold film with two slits (actually representing two unit cells) from the bottom; then, their diffraction light from each slit produces secondary waves like two point sources. When receiving the far-field diffracted light at a fixed angle θ_D, the phase difference produced by the two slits is $∆\phi =\left(2\pi D\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_D\right)/\lambda$. By adjusting the distance D, $∆\phi$ can be changed from −π to π [24]. Through expanding the two slits to a two-dimensional array of unit cells, we can achieve a two-dimensional holographic phase modulation on the diffracted light at the angle (θ_D) by arranging the positions of all unit cells to obtain the desired two-dimensional distribution of distance D. Here, we choose the period of the two-dimensional array as 1.6 μm, which is also the value of D at $∆\phi =2\mathsf{\pi }$ for both distributions of the two unit cells. According to the formula $∆\phi =\left(2\mathsf{\pi }D\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_D\right)/\lambda$ mentioned above, the fixed angle (θ_D)of modulated light is ~23.3$°$ for both output phase-modulated LCP and RCP parts.

Based on the combination of geometric phase and detour phase in our design, two different phase holograms can, respectively, be loaded onto the LCP and RCP components by arranging the positions of two types of chiral unit cells in the metasurface, and the output LCP and RCP holographic beams perfectly overlap in the same direction. As a result, the principle of arbitrary CVB generation (as shown in Figure 1 and Equation (2)) can be realized by the designed metasurface.

4. Verification of the Designed Metasurface

To verify the CVB generation function of the designed metasurface as expected in Figure 1, we build the metasurface in FDTD with the alternately superimposed two types of unit cells, which can load two different phase holograms, as shown in Figure 3a, according to the detour-phase principle. Theoretically, the two holograms can load two spiral phases with topological charge of +1 and −1 onto the LCP and RCP components of the modulated light, respectively, and finally generate the CVB according to Equation (2). In the FDTD simulation, the metasurface has, in total, 13 × 100 unit cells in the x–y plane, including 8 × 13 × 100 nano-slits, with half for blue-color unit cells and another half for yellow-color unit cells (Figure 2a). The nano-slits in the metasurface have different directions, so we need to simulate the whole metasurface structure with absorbing boundary conditions. The absorption boundary condition is set as a perfectly matched layer (PML).

We first consider the case of an LCP beam illuminating the metasurface and simulate the electric intensity distribution in the far field at the fixed angle with FDTD, as shown in Figure 3b. The generated far-field beam presents a doughnut-shaped intensity with an RCP state, and its phase is a clockwise spiral distribution covering 0~2π, verifying that an RCP OV beam with a topological charge of −1 is generated by the metasurface, as in the prediction in Figure 1b. Similarly, when the metasurface is illuminated by a RCP beam, as shown in Figure 3c, the obtained far-field distribution is also doughnut-shaped but with am LCP state, and the phase shows an anticlockwise spiral distribution covering 0~2π, which means an LCP OV beam with a topological charge of +1 is generated by the metasurface, as in the prediction in Figure 1c. For the case of LP incidence, as illustrated in Figure 3d, a typical radially polarized beam (1-order CVB) in the far field is generated with a nearly plane-wave phase, verifying that the LCP and RCP parts can, respectively, be modulated by the two types of chiral unit cells of the metasurface, as in the prediction in Figure 1d. These FDTD-simulated results demonstrate the capability of our designed metasurface in separate modulations on LCP and RCP beams, and its potential for generating more complex CVBs and other vector beams under the LP incidence.

5. Generation of CVBs and Other Vector Beams

It is well-known that as well as the radially polarized beam in Figure 3d, the 1-order CVB also includes an azimuthally polarized beam and other cases depending on the initial polarized angle ${\varphi }_0$ in Equation (2), which is also used for designing the rotation angles (${\theta }_1$~${\theta }_8$) of nano-slits in unit cells (Figure 2a). Thus, in order to generate other polarization states of 1-order CVB by the metasurface, we change the value of ${\varphi }_0$ to modulate the angles (${\theta }_1$~${\theta }_8)$ of nano-slits in unit cells and finally control the linear polarization state at each point of the generated CVB. As shown in Figure 4a, we separately set the initial angle $({\varphi }_0=0°, 45°, 90°, \mathrm{a}\mathrm{n}\mathrm{d} 135°$) and obtain the corresponding four one-order CVBs with different polarization states but the same doughnut-shaped intensity distribution and phase singularity in the center. As the angle $\left({\varphi }_0\right)$ increases from 0$°$ to 90$°$, the polarization state at each point of the generated CVB is always linear, but its linear polarization direction gradually shifts from radial to azimuthal. Then, it continues to vary from azimuthal to radial polarization when the angle (${\varphi }_0)$ changes from 90$°$ to 180$°$.

As well as the the one-order CVBs, the metasurface we designed can, in theory, generate all other order CVBs. To verify this, we generate CVBs with order m = −1 and ${\varphi }_0=0°, 45°, 90°, \mathrm{a}\mathrm{n}\mathrm{d} 135°$, as shown in Figure 4b. According to Equation (2), we load two spiral phases with topological charges of −1 and +1 onto the LCP and RCP components by the two types of chiral unit cells of the metasurface, respectively. Under LP incidence, the designed metasurface can generate −1 order CVB as expectation, and its polarization distribution can be rotated by changing the angle ${\varphi }_0$, as presented in Figure 4b. The absorption boundary condition is also set as PML.

Furthermore, according to Equation (1), more complex vector beams can be generated when the LCP and RCP components can be loaded with the desired phase distributions. Here, we try to load a spiral phase with a topological charge of +1 onto the RCP component and a plane-wave phase (topological charge of 0) onto the LCP component, and the generated vector beam is shown in Figure 4c. It can be seen that its polarization state gradually changes from LCP to LP and finally to RCP along the radial direction, and such polarization distribution can also be rotated by changing the angle $\left({\varphi }_0\right)$. Actually, such a vector beam has recently been known as the optical Stokes skyrmion [25].

These results in Figure 4 indicate that the designed metasurface can generate CVBs with controllable polarization orders and rotation angles, as well as even more complex vector beams like optical Stokes skyrmions.

6. Discussion and Conclusions

According to Equations (1) and (2), the basic principle for generating CVBs or more complex polarized beams is simple; however, it is not easy to achieve by an easy-to-use single-layer metasurface device. Most conventional single-layer metasurfaces were designed based on only one phase modulation principle, such as the geometric phase, and thus are limited to work for only LCP or RCP beams. As a result, these conventional single-layer metasurfaces require the pre-processing of polarization conversion from initial linear polarization to circular polarization, and there is a lack of independent modulation capabilities for LCP and RCP beams. In this work, both geometric-phase and detour-phase principles are used in the design of metasurface, therefore achieving independent modulation for LCP and RCP beams in a single device and generating various CVBs based on the principle in Equations (1) and (2). Moreover, the proposed metasurface can work for the linearly polarized incident light and avoids the liner-to-circular polarization conversion and makes the optical system more compact. Compared to other multi-layer metasurfaces for complex vector beams generation, the proposed metasurface provides the same function but is more easily fabricated due to the single-layer structure.

In conclusion, we have proposed a simple monolayer chiral metasurface structure composed of two types of chiral unit cells and numerically studied its ability to generate CVBs with arbitrary polarized angles and orders. To improve the performance of the metasurface, its structural parameters have been optimized and have successfully achieved different holographic phase modulations on the LCP and RCP components of light. As a result, under LP incidence, +1 order CVB and −1 order CVB can be generated, and their polarization directions can be further rotated by changing the initial polarized angle (${\varphi }_0)$. Finally, we prove that the proposed metasurface can not only generate arbitrary CVBs but also more complex vector beams like optical Stokes skyrmions by loading the desired holographic phases onto the LCP and RCP components. The proposed metasurface device has the advantages of monolayer structure, compact size, and strong ability to generate various vector beams; thus, it has great potential in the miniaturization and integration of vector polarization optical systems.