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Generation of multi-channel perfect vortex beams with the controllable ring radius and the topological charge based on an all-dielectric transmission metasurface

Yue Liu, Chengxin Zhou, Kuangling Guo, Zhongchao Wei, and Hongzhan Liu

Open Access Open Access

Abstract

The perfect vortex (PV) beam, characterized by carrying orbital angular momentum and a radial electric intensity distribution independent of the topological charge, has important applications in optical communication, particle manipulation, and quantum optics. Conventional methods of generating PV beams require a series of bulky optical elements that are tightly collimated with each other, adding to the complexity of optical systems. Here, making the amplitude of transmitted co-polarized and cross-polarized components to be constant, all-dielectric transmission metasurfaces with superimposed phase profiles integrating spiral phase plate, axicon and Fourier lens are constructed based on the phase-only modulation method. Using mathematical derivation and numerical simulation, multi-channel PV beams with controllable annular ring radius and topological charge are realized for the first time under circularly polarized light incidence combining the propagation phase and geometric phase. Meanwhile, perfect vector vortex beams are produced by superposition of PV beams under the incidence of left-handed circularly polarized and right-handed circularly polarized lights, respectively. This work provides a new perspective on generating tailored PV beams, increasing design flexibility and facilitating the construction of compact, integrated, and versatile nanophotonics platforms.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Vortex beam refers to a special beam carrying orbital angular momentum (OAM), with spiral phase wavefront ${e^{il\phi }}$ and doughnut-shaped transverse intensity distribution (where l is the topological charge (TC), theoretically taking arbitrary value and $\phi$ is the azimuthal angle) [1]. Attributed to its unique optical properties and additional information-carrying degrees of freedom, vortex beam opens a new esplanade in optical communication [2], particle manipulation [3], microscopic imaging [4], and quantum entanglement [5]. However, the annular intensity profile of vortex beam is more dependent on the TC, and it is difficult to simultaneously couple multiple OAM beams into a single hollow-core fiber, adding to the information multiplexing challenge in optical communication [6,7]. Therefore, it is necessary to find an OAM beam whose optical intensity does not vary with the TC. Furthermore, vortex beam with large TC and small ring radius is also of great importance for nanoparticle control and capture.

Surprisingly, Ostrovsky et al. have proposed perfect vortex (PV) beam with TC-independent light intensity distribution [8] that solves the above problems exactly and has great potential for optical communication [9], particle capture [10], super-resolution imaging [11], and quantum information [12]. It is worth mentioning that the PV beam not only resolves the inconsistency of light fields with different TCs during vortex beams superposition, but also overcomes the influence of propagation distance on beam quality and has good resistance to TC changes and perturbations, enabling directional long-distance information transmission. Additionally, the development of integratable PV beam and perfect vector vortex beam generation techniques can provide reliable light sources for optical systems. In general, PV beam can be realized through axicon [13], spatial light modulators [8], interferometers [14], and digital microscopy devices [15], but these methods inevitably introduce numerous bulky optical elements in free space, making their application in miniaturized photonic systems and fiber communications difficult.

Alternatively, metasurfaces, two-dimensional metamaterials composed of ultrathin metallic or dielectric elements [16–18], open a new perspective in establishing miniature integrated photonic systems and have become a rather important frontier research in recent years. Benefiting from the remarkable characteristics and great application potential in light manipulation, and allowing arbitrary and extraordinary modulation on the amplitude, polarization and phase of incident light at the subwavelength scale, metasurface has stimulated a surge of research in metalenses [19,20], holography [21–23], polarization conversion [24,25], nonlinear optics [26], and optical manipulation of vortex beams [27] (including OAM generation [28–30], OAM multiplexing [31,32], and OAM superposition [33,34]). In fact, many studies on generating PV beams based on metasurface have also been reported and demonstrated. For example, PV beams can be obtained using plasmonic metasurface [35]. Considering the low efficiency due to ohmic loss, researchers have successfully acquired high-efficiency PV beams by all-dielectric metasurfaces based on geometric phase [36–38]. However, the above metadevices can only produce PV beam with fixed OAM mode distribution. A metasurface designed via the propagation phase has realized two different PV beams under x-linearly polarized (x-LP) and y-linearly polarized (y-LP) lights incidence [39]. Inspired by plant grafting, Ahmed et al. demonstrate the superposition of grafted PV beams in multiple channels with a single geometric metasurface, leading to asymmetric singularity distributions [40]. Recently, perfect vector vortex beams are also gained using transmissive and reflective metasurfaces, respectively [41,42]. Unfortunately, the OAM mode of PV beam generated based on these designs is mostly single, and the ring size and TC of the beam cannot be flexibly changed to meet the application requirements in various practical situations. In addition, all the above methods only control the transmitted (reflective) component that is the same or opposite to the incident light, and cannot regulate the co-polarized and cross-polarized output components independently. Meanwhile, the perfect vector vortex beams are limited to gain for the linearly polarized (LP) incident light. While all these hinder the further modulation and development of PV beams.

In this paper, we break the limitation of previous studies that can only generate a single OAM mode PV beam, and combine the propagation phase and geometric phase to construct a single-layer transmission metasurface. Based on simple phase-modulation, the co-polarized and cross-polarized components in the transmitted field can be controlled independently and simultaneously, resulting in multi-channel PV beams. Through mathematical derivation and numerical simulation, two PV beam generators are realized: (i) PV beams with the same radius but carrying different TCs are generated under three CP channels. Additionally, compared to previous reports, we can also produce perfect vector vortex beams via the superposition of PV beams under left-handed circularly polarized (LCP) and right circularly polarized (RCP) incident lights, respectively. (ii) Multi-channel PV beams with a certain proportional relationship of ring radius are obtained in the transmission field, as shown in Fig. 1. We provide design guidelines for generating multi-channel tailored PV beams, and also greatly advancing applications in optical communication, particle trapping and manipulation.

 figure: Fig. 1.

Fig. 1. Schematic of multi-channel PV beams based on all-dielectric transmission metasurface.

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2. Theory and design

Theoretically, the PV beam can be obtained by Fourier transformation of higher-order Bessel beam. But the Bessel beam is only an ideal mathematical physics model because of its infinitely extended light field distribution. Therefore, the Bessel-Gaussian (BG) beam is generally used as an approximation to the Bessel beam in experiments, and the complex amplitude in cylindrical coordinate systems $(\rho ,\phi ,z)$ can be expressed as [43]:

$${E_{BG}}(\rho ,\phi ,z) = {J_l}({k_r}\rho )\exp (il\phi )\exp (i{k_z}z)\exp ( - \frac{{{\rho ^2}}}{{w_0^2}})$$
where ${J_l}$ is the first kind of l-th order Bessel function, ${k_r}$ and ${k_z}$ are the radial and longitudinal wave vectors respectively, ${w_0}$ denotes the beam radius of the Gaussian beam. The BG beam is Fourier transformed through a lens with focal length f to obtain the complex amplitude distribution of PV beam, namely:
$${E_{PV}}(\gamma ,\vartheta ) = {i^{l - 1}}\frac{{{w_0}}}{{{w_g}}}\exp (il\vartheta )\exp ( - \frac{{{\gamma ^2} + \gamma _r^2}}{{w_g^2}}){I_l}(\frac{{2{\gamma _r}\gamma }}{{w_g^2}})$$
where ${w_g} = {{2f} / {k{w_0}}}$ defines the waist of the Gaussian beam at the focal plane, ${I_l}$ is the l-th order modified Bessel function of the first kind, and ${\gamma _r}$ represents the annular ring radius of PV beam. Equation (2) is approximated by:
$${E_{PV}}(\gamma ,\vartheta ) = {i^{l - 1}}\frac{{{w_0}}}{{{w_g}}}\exp (il\vartheta )\exp ( - \frac{{{{(\gamma - {\gamma _r})}^2}}}{{w_g^2}})$$

Then the light intensity distribution of PV beam can take the form:

$${I_{PV}}(\gamma ,\vartheta ) = \frac{{w_{_0}^2}}{{w_\textrm{g}^2}}\exp ( - 2\frac{{{{(\gamma - {\gamma _r})}^2}}}{{w_g^2}})$$

Obviously, Eq. (4) shows that the light intensity distribution of PV beam is independent of the TC, meaning that the beam radius remains constant no matter how the TC changes. In fact, the incident Gaussian beam can first be converted into a Laguerre-Gaussian beam using a spiral phase plate, followed by an axicon to obtain a BG beam, and finally the PV beam is generated by Fourier transformed the corresponding BG beam through a refractive lens. Considering the integration and miniaturization of photonic systems, we use a metasurface instead of the above three bulky optical elements to achieve multi-element multifunctional process, and the applied phase is the total phase profiles of a spiral phase plate, an axicon and a Fourier-transform lens specifically, as shown in Fig. 2. It can be described as:

$${\varphi _{meta}}(x,y) = {\varphi _{spiral}}(x,y) + {\varphi _{axicon}}(x,y) + {\varphi _{lens}}(x,y)$$
$${\varphi _{spiral}}(x,y) = l \cdot \arctan (\frac{y}{x})$$
$${\varphi _{axi\textrm{c}on}}(x,y) ={-} 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{d}$$
$${\varphi _{lens}}(x,y) = \frac{{ - \pi ({x^2} + {y^2})}}{{\lambda f}}$$
where (x, y) is the position coordinate of a meta-atom. Equation (5b) represents the phase of the spiral phase plate, and l is the TC carried by the generated beam. The axicon phase profile is defined in Eq. (5c), where d is the axicon period, controlling the ring radius of PV beam. Finally, Eq. (5d) is the phase distribution of the Fourier-transform lens, λ is the incident light wavelength and f is the focal length.

 figure: Fig. 2.

Fig. 2. Phase profile encoded on the metasurface by superimposing the phases of spiral phase plate, axicon and Fourier-transform lens.

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Herein, in order to obtain multi-channel PV beams based on a single-layer metasurface, the co-polarized component and cross-polarized component of output field need to be modulated simultaneously. The transmission matrix of the birefringent nanopillar is expressed as [44,45]:

$$J(\theta ) = R( - \theta )\left[ {\begin{array}{cc} {{t_x}}&0\\ 0&{{t_y}} \end{array}} \right]R(\theta )$$
where R(θ) is the rotation matrix, θ represents the rotation angle of nanopillar with respect to the reference coordinate system, ${t_x} = {T_x}{e^{i{\varphi _x}}}$ and ${t_y} = {T_y}{e^{i{\varphi _y}}}$ refer to the transmission response of the meta-atom for x-LP and y-LP lights, and ${T_x}$ (${T_y}$) and ${\varphi _x}$ (${\varphi _y}$) are the nanopillar transmission amplitude and phase delay, respectively. Calculating Eq. (6), the Jones matrix under the linear basis is obtained as follows:
$$\begin{aligned} J(\theta ) &= \frac{1}{2}({T_x}{e^{i{\varphi _x}}} + {T_y}{e^{i{\varphi _y}}})\\ & + \frac{1}{2}( {T_x}{e^{i{\varphi _x}}} - {T_y}{e^{i{\varphi _y}}}) \cdot ({e^{i2\theta }}{{\hat{\sigma }}_R} + {e^{ - i2\theta }}{{\hat{\sigma }}_L}) \end{aligned}$$
where ${\hat{\sigma }_R} = \frac{1}{2}\left( {\begin{array}{cc} 1&{ - i}\\ { - i}&{ - 1} \end{array}} \right)$ and ${\hat{\sigma }_L} = \frac{1}{2}\left( {\begin{array}{cc} 1&i\\ i&{ - 1} \end{array}} \right)$ are the handedness operators that control the polarization state of transmitted field. It is observed that the first term of Eq. (7) coincides with the polarization state of the incident light, called the co-polarized component, while the second term is opposite to the incident polarization state, namely the cross-polarized component. We make the transmission amplitudes of nanopillar satisfy ${T_x} = {T_y} = 1$, then the phase difference along x and y axis is $\Delta \varphi = {\varphi _x} - {\varphi _y}$, and the sum of the propagation phases is denoted as $\sum \varphi = {\varphi _x} + {\varphi _y}$. Thus, Eq. (7) can be further evolved as:
$${J_{co}} = \frac{1}{2}({T_x}{e^{i{\varphi _x}}} + {T_y}{e^{i{\varphi _y}}}) = \cos \frac{{\Delta \varphi }}{2} \cdot {e^{i(\frac{1}{2}\sum \varphi )}}$$
$$\begin{aligned} {J_{cross}} &= \frac{1}{2}({T_x}{e^{i{\varphi _x}}} - {T_y}{e^{i{\varphi _y}}}) \cdot ({e^{i2\theta }} \cdot {{\hat{\sigma }}_R} + {e^{ - i2\theta }} \cdot {{\hat{\sigma }}_L})\\ &\textrm{ } = \sin \frac{{\Delta \varphi }}{2} \cdot {e^{i(\frac{1}{2}\sum \varphi )}} \cdot ({e^{i2\theta }} \cdot {{\hat{\sigma }}_R} + {e^{ - i2\theta }} \cdot {{\hat{\sigma }}_L}) \end{aligned}$$

Notably, the sum of the propagation phases can be modulated by controlling ${\varphi _x}$ and ${\varphi _y}$. When the phase $\frac{1}{2}\Sigma \varphi$ is imparted to the metasurface, i.e., the geometry of nanopillar is changed to tailor the transmitted co-polarized component. The cross-polarized term in the transmitted field is regulated by rotating θ, which introducing the geometric phase. In addition, the amplitudes of two components can also be flexibly engineered, where the amplitude of co-polarized part is ${T_{co}} = \cos \left( {\frac{{\Delta \varphi }}{2}} \right)$, the amplitude of cross-polarized component refers to ${T_{cross}} = \sin \left( {\frac{{\Delta \varphi }}{2}} \right)$, and the energy ratio of two terms is defined as $\eta = {\tan ^2}(\frac{{\Delta \varphi }}{2})$. Subsequently, to further modulate the co-polarized and cross-polarized components simultaneously, $\Delta \varphi \textrm{ = }{\pi / 2}$ is ordered here to guarantee equal output energy distribution in the two orthogonal CP channels ($\eta = 1$).

In order to satisfy the above conditions, the meta-atoms constituting a metasurface are carefully and rigorously designed. A series of subwavelength nanostructures are selected to provide the required phase shift $\sum \varphi$ covering the entire range of 2π, and to realize the rotation angle θ at any point (x, y). Moreover, it is necessary to make $\Delta \varphi \textrm{ = }{\pi / 2}$ to ensure the amplitudes of transmitted co- and cross-polarized components are equal. The adopted meta-atom in Fig. 3(a) is composed of a rectangular TiO2 nanopillar with height H = 600 nm and a square substrate of fused silica with lattice constant P = 380 nm, and TiO2 is chosen because of its high refractive index and low loss in the visible range [46]. To investigate the transmission properties of the meta-atom, all simulations are based on the 3D-finite-difference-time-domain (FDTD). We calculate the transmission amplitude and phase delay of the meta-atom under x-LP and y-LP incident lights at the design wavelength λ=630 nm, respectively. The incident light is plane wave and propagates along the + z direction. Periodic boundary conditions are applied in the x and y directions, and perfectly matched layers (PMLs) are implemented in the z direction. Figures 3(b) and 3(c) describe the relationship between the transmission amplitude ${T_x}$, phase delay ${\phi _x}$ and the size of the nanopillar (L,W) for x-LP illumination, respectively. The transmission amplitude ${T_y}$ and phase delay ${\phi _y}$ with y-LP incidence in Figs. 3(d) and 3(e) can be regarded as the transposition of ${T_x}$ and ${\phi _x}$. As shown in Figs. 3(b)-(e), ${\phi _x}$ and ${\phi _y}$ cover the full 2π phase shifts. Then arbitrary phase combinations $({\phi _x},{\phi _y})$ can be achieved to modulate the phase and transmission amplitude of the output co-polarized and cross-polarized components by choosing the nanopollar size (L, W) reasonably. A set of 8 nanopillars (black pentagrams in Fig. 3(b) with an interval of π/4 are optimized, and the corresponding characterization values are demonstrated in Fig. 3(f). Obviously, the sum of these 8 units’ propagation phases cover 2π range. Meanwhile, the phase difference between the x-LP and y-LP incident lights of each unit is close to 0.5π, ensuring that the amplitudes of co-polarized and cross-polarized components in the transmitted field are around 0.7. Therefore, the above 8 units form a complete set of structural units that modulate the phase of output co-polarized and cross-polarized components independently.

 figure: Fig. 3.

Fig. 3. (a) Perspective and top views of the meta-atom which consists of a rectangular TiO2 nanopillar and a square silica substrate. (b, c)The transmission coefficients and phase shifts for x-LP illumination. (d, e) The transmission coefficients and phase shifts under y-LP incident light. (f) The transmittance coefficients, phase differences and summation of phases for x-LP and y-LP lights of all selected 8 meta-atoms, as well as the amplitudes of transmitted co-polarized and cross-polarized components for CP light incidence.

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3. Results and discussion

Based on the above theory, two all-dielectric transmission metasurfaces composed of TiO2 with dimensions of 50 µm×50 µm are constructed. We can produce multi-channel PV beams with controllable ring radius, where the TCs carried by beams under each channel are also different. Both metadevices operate at wavelength λ=633 nm, and all numerical simulations of the designed PV beam generators are performed using the FDTD method with PML boundary conditions for the x, y and z axis.

3.1 Generation of multi-channel PV beams with the same ring radius

By modulating co-polarized and cross-polarized components in transmitted field, multi-channel PV beams with the same size but carrying different TCs can be generated. Meanwhile, these PV beams can also be superimposed under CP incidence to realize perfect vector vortex beams. To satisfy the above functions, the phase imparted to a metasurface is governed by [36,45]:

$${\varphi _{LL}} = {\varphi _{RR}} = \frac{1}{2}\Delta \varphi = {l_1} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{d} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$
$${\varphi _{LR}} = \frac{1}{2}\Delta \varphi + 2\theta = {l_2} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{d} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$
$${\varphi _{RL}} = \frac{1}{2}\Delta \varphi - 2\theta = {l_3} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{d} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$
where ${\varphi _{LL}}$ (${\varphi _{RR}}$) represents the phase profile of LCP (RCP) state under transmitted electric field when LCP (RCP) is incident, which is the co-polarized component. ${\varphi _{LR}}$ (${\varphi _{RL}}$) refers to the phase distribution of RCP (LCP) output electric field part under the LCP (RCP) light incident, i.e., cross-polarized component. Specifically, the LCP and RCP channels generate PV beams carrying l1 and l2 for LCP light, respectively. When the incident light is converted into RCP, PV beams with different TCs (l1 and l3) are realized under RCP and LCP channels. It is obvious to find that Eq. (9c) depends on the mutually independent Eqs. (9a) and (9b), which gives the relationship between the TCs under three CP channels:
$${l_3} = 2{l_1} - {l_2}$$

The designed parameters are d = 6 µm, f = 80 µm, l1=-2, and l2 = 2, then ${l_3} = 2{l_1} - {l_2} ={-} 6$ is available. The electric field intensities and phase distributions of co-polarized and cross-polarized components under LCP and RCP lights incidence are given in Fig. 4. Figures 4(a1) and 4(a2) show the electric field intensity distributions of LCP component with efficiency of 34.5% (defined as the ratio of the light intensity of the generated PV beam to the incident light intensity) and RCP component with efficiency of 33.8% at the focal plane z = 80 µm under LCP illumination. In order to better analyze the generated PV beams, normalized intensity profiles along the x direction of co- and cross-polarized channels at the focal plane are obtained in Figs. 4(b1) and 4(b2), with the same radius for both PV beams. Figures 4(c1) and 4(c2) are the corresponding phase distributions, and the TC l=-2 through LCP output state can be determined from the number of spirals and the rotation direction, while a PV beam of OAM mode l = 2 is generated in RCP channel. Thus PV beams carrying different TCs are successfully motivated in co-polarized and cross-polarized channels under LCP light incidence, and the slight inhomogeneous of ring light intensity distribution result from the non-uniform amplitudes of the 8 meta-atoms constituting the metasurface. When the light polarization is switched from LCP to RCP light, the extracted intensity distribution of LCP component (cross-polarized component) with efficiency of 34.3% at the focal plane is presented in Fig. 4(a3). According to the intensity curve pattern in Fig. 4(b3), the beam radius is slightly different from that under LCP illumination mainly because the ring width changes with the increased TC [35]. Also the TC l = 6 carried by PV beam with efficiency of 33.1% is observed from the phase distribution shown in Fig. 4(c1), confirming the derivation of Eq. (10). Alternatively, the generated beam interferes with a coaxial Gaussian beam, and the TC can also be distinguished by the number and direction of the spiral branches in the interference pattern (see Fig. S1 in Supplement 1). Figures 4(a4)-(c4) exhibit the electric field intensities and phase profiles of the extracted RCP components (co-polarized components), which are completely consistent with Figs. 4(a1)-(c1). The above analysis further verifies that the intensity of PV beam at the focal plane is independent of TC when the axicon period d is fixed, reflecting the “perfect” characteristics of the beam. Therefore, it is concluded that we can use the all-dielectric metasurface combined with the propagation phase and geometric phase to produce multi-channel PV beams with the same ring radius but adjustable TC under three CP channels.

 figure: Fig. 4.

Fig. 4. When the LCP is incident, the annular intensity distributions of generated PV beams at the focal plane under LCP channel (a1) and RCP channel (a2), the normalized intensity curves of extracted PV beams for LCP component (b1) and RCP component (b2), and the phase profiles in the xz plane at the focal plane corresponding to the LCP transmission field (c1) and RCP transmission field (c2). When the incident light is converted into RCP, the annular intensities of generated PV beams at the focal plane under LCP channel (a3) and RCP channel (a4), the normalized intensity curves of extracted PV beams for LCP component (b3) and RCP component (b4), and the phase profiles in the xz plane at the focal plane corresponding to the LCP transmission field (c3) and RCP transmission field (c4).

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3.2 Generation of multi-channel perfect vector vortex beams

In addition to generating PV beams with the same size in three channels, we found that PV beams in co-polarized and cross-polarized channels will be superimposed to obtain perfect vector vortex beams for LCP and RCP incident lights, solving the problem that vortex beam cannot be superimposed when the TC differs greatly. Figures 5(a1)/(a2) and 5(c1)/(c2) show the transmission electric field intensity distribution at the focal plane z = 80 µm and the cross-section of normalized intensity curve along y = 0 when LCP/RCP light is incident, respectively. The radii of beams are approximately the same, indirectly reflecting the perfect properties of co- and cross-polarized components described above. Figures 5(b1) and 5(b2) are the electric field intensities of transmitted field in the xz plane under the LCP and RCP incident lights, respectively. It can be found that the transmitted beam will first focus at z = 50.3 µm when passing through the designed metasurface, and then the light wave diverges in free space and evolves into a perfect ring profile, which gradually increases in radius as the beam propagates further. Because we modulate the co-polarized and cross-polarized components of transmitted field simultaneously and independently, two PV beams with the co-polarized component carrying l1 and cross-polarized part carrying l2 are superimposed ($|{L,{l_1}} \rangle + |{R,{l_2}} \rangle$) under LCP illumination to generate a perfect vector vortex beam. To observe the light field distribution, the produced beam can be passed through a horizontal linear polarizer as shown in Fig. 5(d1), resulting a petal-like light intensity pattern with the total number of petals equal to $N = |{{l_1} - {l_2}} |$. Furthermore, the electric field intensity of the superimposed beam passing through a linear polarizer at different angles (45°, 90°, and 135°) with the x-axis is also studied (see Fig. S2 in Supplement 1). The electric field rotates with the rotation of the polarizer, and the rotation angle is halved in comparison with the angle between the linear polarizer and x-axis. Similarly, PV beams carrying l1 and l3 respectively will be superimposed $|{R,{l_1}} \rangle + |{L,{l_3}} \rangle$ under RCP light incidence, Fig. 5(d2) shows the petal-shaped light intensity distribution with four lobes after the resultant vector vortex beam through a horizontal linear polarizer. Compared to the LCP light incidence, the light intensity under RCP light (in Fig. 5 (d1)) rotates a small angle due to the Gouy phase when beams are superimposed [47].

 figure: Fig. 5.

Fig. 5. The electric field intensity distributions of generated beam in the xz plane at the focal plane under the illumination of LCP (a1) and RCP (a2). The electric field intensity profiles of resultant beam in the xy plane for LCP (b1) and RCP (b2) lights. The normalized intensity curves of transmitted beam with LCP (c1) and RCP (c2) incidence. The electric field intensity patterns of output beam after transmission through a horizontal linear polarizer for LCP (d1) and RCP (d2) lights.

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3.3 Generation of multi-channel PV beams with the different ring radius

To further verify the feasibility of this method and break the limitation of single-mode PV beams in previous studies, we can also control parameters to generate multi-channel PV beams with a certain relationship based on the metasurface, specifically as follows:

$${\varphi _{LL}} = {\varphi _{RR}} = \frac{1}{2}\Delta \varphi = {l_1} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{{{d_1}}} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$
$${\varphi _{LR}} = \frac{1}{2}\Delta \varphi + 2\theta = {l_2} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{{{d_2}}} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$
$${\varphi _{RL}} = \frac{1}{2}\Delta \varphi - 2\theta = {l_3} \cdot arc\tan (\frac{y}{x}) - 2\pi \frac{{\sqrt {{x^2} + {y^2}} }}{{{d_3}}} - \frac{{\pi \sqrt {{x^2} + {y^2}} }}{{\lambda f}}$$

Similar to the relationship between l1, l2, and l3 described in Eq. (10), the same result can be obtained as:

$${d_3} = \frac{{2{d_2} - {d_1}}}{{{d_1}{d_2}}}$$

In order to confirm our derivation, the parameters l1 = 3, l2 = 2, d1 = 3 µm, and d2 = 6 µm are set, so we can get l3 = 4 and d3 = 2 µm. In principle, when LCP light is incident, PV beams of l1 = 3, d1 = 3 µm and l2 = 2, d2 = 6 µm are obtained in LCP and RCP channels, respectively. Under RCP illumination, PV beams with l1 = 3, d1 = 3 µm and l3 = 4, d3 = 2 µm can be generated in RCP and LCP transmitted fields, respectively. Figure 6(a1) shows the intensity distribution of LP state at the focal plane for LCP light, observing two ring patterns with different radii. Figure 6(b1) presents the normalized cross sections of the annular intensity profile in Fig. 6(a1). It can be found that the intensity profile decreases with increasing ring radius. To further illustrate the propagation characteristics of the resultant beam, the light intensity distribution in the xz plane under LCP incidence is studied, as shown in Fig. 6(c1), two beams are generated in free space, both of which are first focused and then diverge to evolve into a circular profile for further propagation. The two beams are focused at different positions, which is mainly due to the difference in axicon period d1 and d2. Subsequently, the electric field intensity distributions of the co-polarized and cross-polarized components are extracted, respectively. Figures 6(a2) and 6(b2) demonstrate the intensity distributions of LCP component with efficiency of 31.6% at z = 80 µm corresponding to the xy plane and the cross section along the x axis, respectively. The corresponding phase distribution (in Fig. 6(d1)) indicating the OAM mode carried by beam is l = 3. Figure 6(a3) is the electric field distribution at the focal plane under cross-polarization channel with efficiency of 35.2%, and Fig. 6(b3) shows the normalized intensity curve. Compared with Figs. 6(a2) and 6(b2), it shows that PV beams have different radii under different channels, which further verify that the size of PV beam is controlled by axicon period. From the phase profile in Fig. 6(d2), the TC l = 2 carried by RCP output beam under LCP light incidence is completely consistent with the preset.

 figure: Fig. 6.

Fig. 6. When LCP is incident, the annular electric field intensity distributions of generated beams at the focal plane for LP (a1), LCP (a2), and RCP (a3) components; the normalized intensity profiles of resultant beams at the focal plane for LP (b1), LCP (b2), and RCP (b3) components; the electric field intensity profiles of transmitted beams in the xz plane (c1); the phase distributions of transmitted beams under LCP (d1) and RCP (d2) channels. When the incident light is converted into RCP, the annular electric field intensity distributions of generated beams at the focal plane for LP (a4), LCP (a5), and RCP (a5) components; the normalized intensity profiles of resultant beams at the focal plane for LP (b4),LCP (b5), and RCP (b6) components; the electric field intensity profiles of transmitted beams in the xz plane (c2); the phase distributions of transmitted beams under LCP (d3) and RCP (d4) channels.

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When the incident light is converted into RCP, similarly, two beams are produced under the control of d1 and d3 as shown in Figs. 6(a4) and 6(b4). Figure 6(c2) displays the electric field intensity distribution in the xz plane and the focus position of beam is found to change. Combined with Fig. 6(c1), it further illustrates that the focus position of beam is greatly affected by axicon period. When d is smaller, the focus position of beam is also smaller, which is in line with the previous researches [37]. When RCP light is incident, Fig. 6(a5) shows the intensity distribution at the focal plane of LCP cross-polarized component and the efficiency is 28.7%. It is also obvious from the intensity curve of Fig. 6(b5) that the size of annular ring is different from Figs. 6(b2) and 6(b3), while the phase distribution in Fig. 6(d3) illustrates the TC of PV beam under cross-polarized channel is equal to l = 6, coinciding with Eq. (10) precisely. The interference patterns that generate the beam and the coaxial Gaussian beams are shown in Fig. S3 in Supplement 1. In addition, the electric field and phase distribution of co-polarized channel for RCP light are shown in Figs. 6(a6), 6(b6), and 6(d4), which are exactly the same as co-polarized component with efficiency of 36.2% under LCP light incidence. Therefore, it is demonstrated that our derivation can combine the propagation phase and geometric phase to generate multi-channel PV beams with controllable ring radius and TC, providing a new strategy for realizing tailored PV beams based on the metasurface.

4. Conclusion

In summary, two all-dielectric transmission metasurfaces with superimposed phases of spiral phase plate, axicon and Fourier lens are constructed, and multi-channel PV beams are successfully realized in the visible light band by combining the propagation phase and geometric phase. Moreover, both the ring radius and the carrying TC under the output CP channel can be controlled flexibly by modulating the phase of transmitted co- and cross-polarized components independently. Meanwhile, perfect vector vortex beams are also generated under CP light incident. The metadevices replace the spiral phase plate, axicon and Fourier lens, solve the collimation and compatibility problems among the three elements, and also discard a series of bulky optical elements, which has a promising development in miniaturization and multifunctional integration of devices. In addition, this work provides a simple and effective method for generating multi-channel PV beams, broadening the range of PV beam applications with far-reaching implications in optical communications, particle capture, and holographic displays

Funding

National Natural Science Foundation of China (62175070, 61875057, 61774062); Natural Science Foundation of Guangdong, China (2021A1515012652); Science and Technology Program of Guangzhou (2019050001).

Disclosures

The authors declare no conflicts of interest.

Data availability

Date underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       supplement 1

Data availability

Date underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. Schematic of multi-channel PV beams based on all-dielectric transmission metasurface.

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Fig. 2.
Fig. 2. Phase profile encoded on the metasurface by superimposing the phases of spiral phase plate, axicon and Fourier-transform lens.

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Fig. 3.
Fig. 3. (a) Perspective and top views of the meta-atom which consists of a rectangular TiO2 nanopillar and a square silica substrate. (b, c)The transmission coefficients and phase shifts for x-LP illumination. (d, e) The transmission coefficients and phase shifts under y-LP incident light. (f) The transmittance coefficients, phase differences and summation of phases for x-LP and y-LP lights of all selected 8 meta-atoms, as well as the amplitudes of transmitted co-polarized and cross-polarized components for CP light incidence.

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Fig. 4.
Fig. 4. When the LCP is incident, the annular intensity distributions of generated PV beams at the focal plane under LCP channel (a1) and RCP channel (a2), the normalized intensity curves of extracted PV beams for LCP component (b1) and RCP component (b2), and the phase profiles in the xz plane at the focal plane corresponding to the LCP transmission field (c1) and RCP transmission field (c2). When the incident light is converted into RCP, the annular intensities of generated PV beams at the focal plane under LCP channel (a3) and RCP channel (a4), the normalized intensity curves of extracted PV beams for LCP component (b3) and RCP component (b4), and the phase profiles in the xz plane at the focal plane corresponding to the LCP transmission field (c3) and RCP transmission field (c4).

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Fig. 5.
Fig. 5. The electric field intensity distributions of generated beam in the xz plane at the focal plane under the illumination of LCP (a1) and RCP (a2). The electric field intensity profiles of resultant beam in the xy plane for LCP (b1) and RCP (b2) lights. The normalized intensity curves of transmitted beam with LCP (c1) and RCP (c2) incidence. The electric field intensity patterns of output beam after transmission through a horizontal linear polarizer for LCP (d1) and RCP (d2) lights.

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Fig. 6.
Fig. 6. When LCP is incident, the annular electric field intensity distributions of generated beams at the focal plane for LP (a1), LCP (a2), and RCP (a3) components; the normalized intensity profiles of resultant beams at the focal plane for LP (b1), LCP (b2), and RCP (b3) components; the electric field intensity profiles of transmitted beams in the xz plane (c1); the phase distributions of transmitted beams under LCP (d1) and RCP (d2) channels. When the incident light is converted into RCP, the annular electric field intensity distributions of generated beams at the focal plane for LP (a4), LCP (a5), and RCP (a5) components; the normalized intensity profiles of resultant beams at the focal plane for LP (b4),LCP (b5), and RCP (b6) components; the electric field intensity profiles of transmitted beams in the xz plane (c2); the phase distributions of transmitted beams under LCP (d3) and RCP (d4) channels.

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Equations (20)

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E B G ( ρ , ϕ , z ) = J l ( k r ρ ) exp ⁡ ( i l ϕ ) exp ⁡ ( i k z z ) exp ⁡ ( − ρ 2 w 0 2 )
E P V ( γ , ϑ ) = i l − 1 w 0 w g exp ⁡ ( i l ϑ ) exp ⁡ ( − γ 2 + γ r 2 w g 2 ) I l ( 2 γ r γ w g 2 )
E P V ( γ , ϑ ) = i l − 1 w 0 w g exp ⁡ ( i l ϑ ) exp ⁡ ( − ( γ − γ r ) 2 w g 2 )
I P V ( γ , ϑ ) = w 0 2 w g 2 exp ⁡ ( − 2 ( γ − γ r ) 2 w g 2 )
φ m e t a ( x , y ) = φ s p i r a l ( x , y ) + φ a x i c o n ( x , y ) + φ l e n s ( x , y )
φ s p i r a l ( x , y ) = l ⋅ arctan ⁡ ( y x )
φ a x i c o n ( x , y ) = − 2 π x 2 + y 2 d
φ l e n s ( x , y ) = − π ( x 2 + y 2 ) λ f
J ( θ ) = R ( − θ ) [ t x 0 0 t y ] R ( θ )
J ( θ ) = 1 2 ( T x e i φ x + T y e i φ y ) + 1 2 ( T x e i φ x − T y e i φ y ) ⋅ ( e i 2 θ σ ^ R + e − i 2 θ σ ^ L )
J c o = 1 2 ( T x e i φ x + T y e i φ y ) = cos ⁡ Δ φ 2 ⋅ e i ( 1 2 ∑ φ )
J c r o s s = 1 2 ( T x e i φ x − T y e i φ y ) ⋅ ( e i 2 θ ⋅ σ ^ R + e − i 2 θ ⋅ σ ^ L )   = sin ⁡ Δ φ 2 ⋅ e i ( 1 2 ∑ φ ) ⋅ ( e i 2 θ ⋅ σ ^ R + e − i 2 θ ⋅ σ ^ L )
φ L L = φ R R = 1 2 Δ φ = l 1 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d − π x 2 + y 2 λ f
φ L R = 1 2 Δ φ + 2 θ = l 2 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d − π x 2 + y 2 λ f
φ R L = 1 2 Δ φ − 2 θ = l 3 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d − π x 2 + y 2 λ f
l 3 = 2 l 1 − l 2
φ L L = φ R R = 1 2 Δ φ = l 1 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d 1 − π x 2 + y 2 λ f
φ L R = 1 2 Δ φ + 2 θ = l 2 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d 2 − π x 2 + y 2 λ f
φ R L = 1 2 Δ φ − 2 θ = l 3 ⋅ a r c tan ⁡ ( y x ) − 2 π x 2 + y 2 d 3 − π x 2 + y 2 λ f
d 3 = 2 d 2 − d 1 d 1 d 2
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