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Abstract
In this paper, we propose a bilayer metasurface with dual-operating modes to realize the generation and conversion of dual-band Laguerre-Gaussian beam with different OAM. The bilayer metasurface consists of amorphous silicon (a-Si) and amorphous antimony selenide (a-Sb2Se3). In transmission mode, the metasurface can transform incident left-handed circularly polarized (LCP) Gaussian beam into one LCP Laguerre-Gaussian beam carrying topological charge l=−2 and one right-handed circularly polarized (RCP) Laguerre-Gaussian beam carrying topological charge l=−1, which enabled OAM modes multiplexing. In reflection mode, the metasurface can convert an incident LCP Gaussian beam into a LCP Laguerre-Gaussian beam carrying a topological charge l=−3 owing to the a-Sb2Se3 nanopillar that acts as a half-wave plate with high reflectivity. In addition, the OAM conversion between two arbitrary modes can be realized by the proposed metasurface. This work provides a valid method for OAM multiplexing and conversion, which holds a great applications value in high-speed optical communication systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The angular momentum possessed by photons can be divided into spin angular momentum (SAM) and orbital angular momentum (OAM). SAM is related to the circularly polarized light, and the SAM can be ${\pm }\hbar$ per photon [1]. In contrast, OAM is associated with the helically shaped wavefront which is expressed by helical phase factors $\exp (il\theta )$, where $l$ is topological charge and $\theta$ is the azimuthal angle [2]. Light beam carrying OAM, also known as vortex beam, has been extensively employed in various fields such as optical tweezers [3], quantum information processing [4], and super-resolution imaging [5], etc. In addition to the aforementioned applications, the most promising prospect provided for vortex beam is in optical communications owing to the theoretically infinite values of $l$ and the spatial orthogonality of different OAM modes [6–11]. In recent years, to further improve the channel capacity of optical communication systems and save spectrum resources, OAM multiplexing and demultiplexing have been used in free-space wireless optical communication. High-capacity communications by combing the OAM multiplexing with polarization multiplexing in the communication C-band [12], or in the millimeter-wave band [13] have been demonstrated. Unfortunately, the generation of OAM modes in these communication systems usually uses bulk optical elements such as spatial light modulators [12], spiral phase plates [13], and so on.
Metasurfaces, the two-dimensional artificial metamaterials, consist of nanoscale optical element arrays. By modifying the size or azimuth of the optical elements constituting the metasurface, the light properties including phase, amplitude, and OAM can be readily controlled. As far as we know that one of the most promising strategies for generating OAM modes is to use optical metasurface. The previous works demonstrate that reflective or transmissive metasurface can generate a single OAM mode under a type of polarized light [14–17]. The plasmonic nanopillars, constituting the reflective metasurface, will cause low efficiency due to the inevitable Ohmic loss in it. However, the transmissive metasurface composed of dielectric nanopillars shows remarkable performance in efficient modulated light fields and has been applied to implement multi-channel OAM modes [18–20]. Regrettably, the above works investigate how to generate one vortex beam, without involving OAM modes multiplexing. In the visible light band, combing the OAM multiplexing with wavelength-division multiplexing on a single metasurface has been numerically demonstrated [21]. Subsequently, H. Zhao et al. [22] experimentally demonstrated OAM multiplexing and demultiplexing by a single ultrathin metasurface, which is composed of various C-shaped slots on a gold sheet. Due to the use of the C-shaped slot on a gold sheet, the amplitude of the transmitted cross-polarized light is less than 25%, resulting in low efficiency for the metasurface. In addition, H. Tan et al. [23] also experimentally demonstrated a kind of OAM multiplexing communication system based on single-layer reflective metasurface. By combining four different OAM modes with two polarizations, the operation of a 448 Gbit $s^{-1}$ data transmission is realized in free space. The designed metasurface is divided into four regions, and each region produces an OAM mode. Therefore, only with off-axis incidence can coaxial OAM modes be generated. For previous methods, energy loss was inevitably caused due to the off-axis designing principle. More importantly, the OAM mode can only be generated in the reflective or transmissive mode. However, OAM coaxial multiplexing and conversion based on bilayer metasurface with dual working modes have not been achieved.
In this paper, we propose a bilayer metasurface with dual-operating modes to realize the generation and conversion of dual-band Laguerre-Gaussian beam with different OAM. The bilayer metasurface is composed of amorphous silicon (a-Si) [24] and amorphous antimony selenide (a-Sb$_2$Se$_3$) [25]. At the wavelength of 1550nm, the metasurface working in transmission mode can convert incident left-handed circularly polarized (LCP) Gaussian beam into one LCP Laguerre-Gaussian beam carrying topological charge $l$=−2 and one right-handed circularly polarized (RCP) Laguerre-Gaussian beam carrying topological charge $l$=−1, realizing OAM modes multiplexing. The simulated results show that the total efficiency for the transmitted two Laguerre-Gaussian beams can reach 96%, and the proportion of the main modes with $l$ of −1, −2 is 94%, 81%. It should be noted that the metasurface can also work in reflection mode. At the resonance wavelength (975nm) of the a-Sb$_2$Se$_3$ nanopillar, the metasurface can convert incident LCP Gaussian beam into reflected LCP Laguerre-Gaussian beam carrying topological charge $l$=−3. The corresponding proportion of the main mode and the corresponding efficiency is as high as 83%, 80%, respectively. Besides, the conversion between arbitrary OAM modes based on the proposed metasurface is also be investigated. The metasurface has great applications values in high-capacity optical communication, optical tweezers, and super-resolution imaging.
2. Theoretical analysis
Figure 1(a) shows the proposed bilayer Pancharatnam-Berry (PB) phase elements, which is consisted of a-Sb$_2$Se$_3$ nanopillar and a-Si nanopillar. Considering two cascaded anisotropic nanopillars with the fast ($o$ and $o^\prime$) and slow axes ($e$ and $e^\prime$), which are under normal illumination of $x$- and $y$-linear polarized ($x$- and $y$-LP) light respectively, as plotted in Fig. 1(b). $t_o$ ($t_o^\prime$) and $t_e$ ($t_e^\prime$) denote the complex transmission coefficients of LP light along the fast and slow axes of a-Sb$_2$Se$_3$ (a-Si) nanopillar, respectively. Supposing the two nanopillars are rotated by angles of $\theta _1$ and $\theta _2$ relative to the laboratory axis ($x$-axis), respectively, the rotated bilayer PB phase elements can be described by using the Jones matrix under a pair of orthogonal circularly polarized basis $({\hat e_R},{\hat e_L})$ [26]:
For LCP incidence, the transmitted waves passing through the bilayer PB phase nanostructures can be written as:
It is obvious that the resultant transmitted waves include four diffraction orders, the first two orders have opposite helicity to the incident waves, and their phase shift satisfies 2$\theta _1$ and 2$\theta _2$, respectively. However, the helicity of the last two orders keeps the same with the incident waves, and the phase change of 2($\theta _1$ - $\theta _2$) is introduced to the last order. In theory, three completely different Laguerre-Gaussian beams in transmission mode can be achieved by using these phase shifts. While the arbitrary control of three Laguerre-Gaussian beams is difficult to achieve with only two degrees of freedom $\theta _1$ and $\theta _2$. Besides, component ${T_1}T_1^\prime {\hat e_L}/4$ cannot modulate phase shifts, leading to a decrease in the utilization of the transmitted waves. Therefore, we designing a-Si nanopillar as a half-wave plate (HWP) to satisfy the condition $T_1^\prime$=0, implying that the Eq. (3) can be simplified as:
In our design, the two degrees of freedom in Eq. (4) are exploited to design two Laguerre Gaussian beams with different topological charges in transmission mode. In the lower half space of Fig. 1(c), the illustration of the bilayer metasurface working in transmission mode is indicated.
Fig. 1. (a) Schematic for the bilayer PB phase elements. (b) The rotated bilayer PB phase elements, where $x$ and $y$ represent the laboratory axis, and $o$ ($o^\prime$) and $e$ ($e^\prime$) denote the fast and slow axis. (c) The schematic diagram of the bilayer metasurface.
To generate high efficiently Laguerre-Gaussian beam in reflection mode, the a-Sb$_2$Se$_3$ nanopillar should be designed as a reflective HWP, which can be described by the Jones matrix, as follows [27]:
The cross-polarized reflection coefficients $r_{xy}$ and $r_{yx}$ are zero owing to the mirror symmetry of the nanostructure. It is worth mentioning that the phase shifts the reflected light satisfies $\pm 2\theta$, namely, the PB phase [27–29]. Utilizing this characteristic of a-Sb$_2$Se$_3$ nanopillar, the designed metasurface can realize the generation of the Laguerre-Gaussian beam in reflection mode. In the upper half space of Fig. 1(c), the illustration of the bilayer metasurface operating in reflection mode is shown.
3. Optimization of nanopillars
In order to simplify Eq. (3) to Eq. (4), the a-Si nanopillar should be designed as a HWP. Here, the finite-difference time domain (FDTD) technique is used to research the transmission characteristics of nanopillars. The mesh size is set to 30 nm$\times$30 nm$\times$50 nm. To ensure high-efficiency PB phase modulation, the lattice constant $P$ is fixed at 650 nm, the height of the lower layer a-Si nanopillar $H_2$ is chosen to be 1.3 $\mu$m. Figures 2(a) and 2(b) denote the simulated transmission amplitudes of $x$- and $y$-LP light at 1550 nm wavelength as a function of the a-Si nanopillar dimensions ($W_2$ and $L_2$), respectively. The phase difference of $x$- and $y$-LP light as a function of the a-Si nanopillar dimensions are indicated in Fig. 2(c). It is clear that the condition $T_1^\prime$=0 can be satisfied for a-Si nanopillar with $W_2$=520 nm and $L_2$=190 nm. The height of filling material $H_3$ is 1.8 $\mu$m. Figures 2(d) and 2(e) show the simulated transmission amplitudes of RCP and LCP lights at 1550 nm wavelength as a function of the a-Sb$_2$Se$_3$ nanopillar dimensions ($W_1$ and $L_1$), respectively. In order to make sure that the resultant two diffraction orders have basically the same energy, the height $H_1$, width $W_1$, and $L_1$ of a-Sb$_2$Se$_3$ nanopillar are set to 900 nm, 365 nm, and 225 nm, respectively. The simulated transmission amplitudes and phase shifts of RCP and LCP light for bilayer PB phase elements versus rotation angle $\theta _2$ of a-Si nanopillar are shown in Fig. 2(f), implying that the components of RCP light are basically the same as LCP light.
Fig. 2. The simulated transmission amplitudes (a, b) and phase difference (c) for $x$- and $y$-LP light versus the a-Si nanopillar dimensions ($W_2$ and $L_2$) at a wavelength of $\lambda _1$ = 1550 nm. The simulated transmission amplitudes of RCP (d) and LCP (e) lights as a function of the a-Sb$_2$Se$_3$ nanopillar dimensions ($W_1$ and $L_1$) at a wavelength of $\lambda _1$ = 1550 nm. The selected a-Si and a-Sb$_2$Se$_3$ nanopillars are marked with black circles. (f) The simulated transmission amplitudes and phase shifts of RCP and LCP light for bilayer PB phase elements versus the rotation angle $\theta _2$ of a-Si nanopillar.
In order to manipulate the reflected circularly polarized light, the a-Sb$_2$Se$_3$ nanopillar should be designed as a reflective HWP to achieve highly efficient modulation of the geometric phase. Figure 3(a) shows the simulated reflection amplitudes and phase difference of a-Sb$_2$Se$_3$ nanopillar upon $x$- and $y$-LP incidences. There is a phase difference of $\pi$ at the resonance wavelength of 975 nm, and the reflection amplitudes of $x$- and $y$-LP light are 0.54 and 1, respectively. In Fig. 3(b), the reflection amplitudes and phase shifts of LCP light with the same helicity as incident light versus the rotation angle $\theta _1$ of a-Sb$_2$Se$_3$ nanopillar are plotted. It is evident that a high reflectivity and a linear phase increase from 0 to 2$\pi$ with an interval of 2$\pi$/9 can be acquired for LCP light. Figures 3(c) and 3(d) denotes the simulated phase shifts and reflection amplitudes of LCP light as a function of incident wavelength and rotation angle $\theta _1$, which reveals that the designed structures with working bandwidth ranging from 970 nm to 990 nm in reflection mode.
Fig. 3. (a) The simulated reflection amplitudes and phase difference of $x$- and $y$-LP light for a-Sb$_2$Se$_3$ nanopillar. (b) The simulated reflection amplitudes and phase shifts for LCP light versus the rotation angle $\theta _1$ of a-Sb$_2$Se$_3$ nanopillar at a wavelength of $\lambda _2$ = 975 nm. The simulated phase shifts (c) and reflection amplitudes (d) of LCP light as a function of wavelength and rotation angle $\theta _1$.
4. Results and discussion
4.1 Multiplexing and generation of OAM modes
In order to obtain two completely different Laguerre-Gaussian beams in transmission mode, the spatial variation of the phase shift for the two diffraction orders is governed by [30]:
In Figs. 4(d) and 4(h), the corresponding histogram of the OAM spectrum is plotted. It is worth noting that the proportion of the main modes with $l$ of −1, −2 is 94%, 81%, implying the designed metasurface can generate the Laguerre-Gaussian beam with high OAM mode purity.
Fig. 4. The simulated intensity at the $x$-$z$ plane (a), the intensity at the $x$-$y$ plane (b), phase distribution (c), and OAM spectrum (d) of the transmitted RCP Laguerre-Gaussian beams with topological charge $l_2$=−1. The simulated intensity at the $x$-$z$ plane (e), the intensity at the $x$-$y$ plane (f), phase distribution (g), and OAM spectrum (h) of the transmitted LCP Laguerre-Gaussian beams with topological charge $l_1$=−2.
According to Eq. (6), the phase distribution for the upper layer of bilayer metasurface obey the following form:
Near the resonance wavelength of 975 nm, the a-Sb$_2$Se$_3$ nanopillar can be perceived as a HWP in reflection mode. In addition, a phase shift $\pm 2\theta$ can be added into the reflected light with the same helicity as the incident light by the a-Sb$_2$Se$_3$ nanopillar. Therefore, when the metasurface is illuminated with a collimated LCP Gaussian beam, one LCP Laguerre-Gaussian beam carrying a topological charge of $l_3$=−3 is observed in the reflected plane. As shown in Fig. 5, the intensity distributions in the $x$-$y$ plane of the reflected LCP Laguerre-Gaussian beam exhibit a "donut" shape and there is a phase singularity at the central position. Owing to the different reflection amplitude of a-Sb$_2$Se$_3$ nanopillar at various wavelengths and orientations, leading to the doughnut intensity change with the incident wavelengths. For the phase distributions, it exhibits distinct helical characteristics and is consistent with the topological charge $l_3$=−3. When the position of the sampling plane is fixed (62 $\mu$m away from the exit facet of the metasurface), different wavelengths correspond to different propagation phases, so Fig. 5 indicates a difference in phase at the same azimuthal angle. In reflection mode, the efficiency for the reflected LCP Laguerre-Gaussian beam carrying topological charge $l_3$=−3 is up to 80%. Figures 5(d), 5(h), and 5(l) illustrate the corresponding histogram of the OAM spectrum. One can see that the proportion of the main modes with $l$ of −3 at different wavelengths is above 80%.
Fig. 5. The simulated intensity at the $x$-$z$ plane, the intensity at the $x$-$y$ plane, phase distribution, and OAM spectrum of the reflected LCP Laguerre-Gaussian with topological charge $l_3$=−3 at three different wavelengths, (a)-(d) 975 nm, (e)-(h) 980 nm, (i)-(l) 985 nm.
4.2 Conversion of OAM modes
Considering a circularly polarized Laguerre-Gaussian beam as the incident light, the corresponding complex amplitude of the electric field in the cylindrical coordinates $(r,\varphi,z)$ can be described by the following form:
Fig. 6. The simulated intensity, the phase distribution, and corresponding OAM spectrum of transmitted LCP and RCP Laguerre-Gaussian beams under the illumination of (a) LCP with a topological charge $l$=−1, (b) RCP with a topological charge $l$=−1, (c) LCP with a topological charge $l$=−2, and (d) RCP with a topological charge $l$=−2.
5. Conclusion
In conclusion, we propose a strategy for the generation and conversion of dual-band Laguerre-Gaussian beam with different OAM based on bilayer metasurface. In transmission mode, the incident LCP Gaussian beam is converted into two Laguerre-Gaussian beams with different OAM modes by the metasurface, achieving OAM modes multiplexing. The simulated total efficiency of the two transmitted Laguerre-Gaussian beams is as high as 96%. In reflection mode, the metasurface can transform the incident LCP Gaussian beam into the reflected LCP Laguerre-Gaussian beam carrying topological charge $l$=−3 with an efficiency of 80%. Furthermore, the metasurface can implement the conversion between arbitrary OAM modes. With high efficiency, OAM coaxial multiplexing and conversion, and dual-working modes, the designed metasurface has excellent potential in the field of high-capacity optical communication, optical tweezers, super-resolution imaging.
Funding
National Natural Science Foundation of China (62175070, 61875057, 61774062); Natural Science Foundation of Guangdong Province (2021A1515012652); Science and Technology Program of Guangzhou (2019050001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data 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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