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Abstract
The orbital angular momentum (OAM) of vortex beams has great potential in optical communications due to its communication confidentiality and low crosstalk. It is necessary to design a plausible OAM pattern recognition mechanism. Abandoning AI models that require large datasets, a single passive all-dielectric metasurface consisting of TiO2 nanopillars on a SiO2 substrate is used to recognize high-order optical vortexes. In this configuration, the proposed device is capable of simultaneously encoding the wavefront and the transmission paths in different incident OAM beams. Due to the presence of spin angular momentum (SAM), the vortex beam to be identified is spatially separated after passing through the metasurface. As a proof of concept, 14 signal channels are considered in the constructed metasurface, 12 of them can be encoded at will for the detection of any vortex beam with a predefined topological charge. These results make use of metasurfaces to enable OAM pattern recognition in an effective way, which may open avenues for the ultimate miniaturization of optical vortex communication and advanced OAM detection technologies.
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1. Introduction
Data science is evolving rapidly today [1–6]. By pre-processing and dynamic-management signals, communication capacity can be effectively increased. Furthermore, expanding the physical degrees of freedom is considered to be the most constructive and effective solution to meet growing communication transmission capacity requirements [7,8]. The capacity is in direct proportion to the number of beam channels. When the beams with independent channels are mutually orthogonal, the different beams can then propagate together with a low level of inherent cross-talk [9,10]. Beams with the phase term of exp(ilψ) are referred to as vortex beams [11,12]. In 1992, Allen et al. demonstrated that the orbital angular momentum (OAM) of the optical vortex beam was associated with a helical phase front [13,14]. In theory, each photon of vortex beams carries OAM of $l{\hbar}$, where l is the topological charge [15–17]. As is known, OAM beams can have infinite orthogonal states on propagation, forming an infinite-dimensional Hilbert space [18]. It has therefore emerged as a potential candidate for increasing communications bandwidth and information capacity. Optical communication with optical vortex relies on the detection of OAM states. To encode information in a scalar field with multiple OAM states, it is necessary to use plausible pattern recognition mechanisms.
Artificial intelligence (AI) promises to improve the performance of OAM recognition systems [12,13]. Nevertheless, to make an authentic prediction, the deep learning model requires a large number of training datasets for OAM decoupling in free-space optical (FSO) communication, and the model converges slowly [19–23]. For this reason, various techniques have been developed to measure OAM states, including interferometry [24], OAM sorting [25,26], vortex diffraction grating [27], multifocal arrays [28], digital hologram technology [29], and measuring transmission matrix [30]. However, the above approaches rely heavily on a well-defined OAM mode intensity pattern. Both the growth of the phase singularity and the diffraction effect are known to have a major influence on the intensity distribution of a vortex beam. To surpass this limit, high-order optical vortex detection over the physical dimension of space makes great sense. Metasurfaces are a collection of subwavelength elements that are spatially distributed over a surface [5,31]. Due to resonant scattering, generalized Huygens’ metasurfaces can be readily designed and fabricated to act as converters between arbitrary spatial modes [32]. Importantly, a single metasurface enables the wavefront of light to be encoded in new ways, which has stimulated research on spatial encoding and detection of optical vortex. It is possible to exploit the spatial dimension by sending information along specific spatial pathways. This concept is widely used in other areas of communications and has been applied to the transmission of data in free space on the basis of orbital angular momentum [7,9,32].
Herein, we present a single passive all-dielectric metasurface for high-order optical vortex recognition. It is capable of simultaneous encoding of the wavefront and the transmission paths in different incident OAM beams. Rather than encoding OAM states onto a single well-defined scalar field, we utilize the optical vortex beams as the carriers of OAM states where phase, topological charge, spin angular momentum (SAM), and wavelength are the basic properties of beams. When the metasurface is illuminated by LP plane waves or vortex beams of the selected orders, the transmitted beams will be transformed into independent off-axis OAM beams pointing in predefined directions. The SAM direction and the topological charge of the incident vortex beam can be accurately identified by distinguishing between fundamental (solid spot) and OAM modes (ring-shaped intensity profile with a central null) for the separated spot. This allows any vortex beam with a predefined topological charge to be detected separately by encoding independent wavefronts in a single metasurface. As a proof of concept, 14 signal channels are considered in the constructed metasurface, 12 of them can be encoded at will for the detection of any vortex beam with a predefined topological charge. These results make use of metasurfaces to enable OAM pattern recognition in an effective way, which suggests a novel approach for the ultimate miniaturization of optical vortex communication and advanced OAM detection technologies.
2. Design and numerical simulation of metasurface
Figure 1 illustrates the schematic diagram of the proposed coupling mechanism. The OAM mode is characterized by the topological charge (l), which describes the azimuthal eigenmodes of vortex beams with a phase singularity. When linear polarization (LP) plane waves pass through the metasurface along the z-axis direction, they will be transformed into independent off-axis OAM beams with specific topological charges. Taking SAM (s) into consideration, the target detection field #1 appears at two different diffraction radii R1 and R2, which are associated with both the left-handed and the right-handed states of circular polarisation. The s can take only two values, 1 and -1, representing the state of the left circular polarisation (LCP) and the state of the right circular polarisation (RCP), respectively. When an LCP beam carrying topological charges lLCP passes through the metasurface, the target detection field #2 appears with diffraction radius R1. After passing through the metasurface, the incident light with topological charge lLCP focuses on only one solid point that replaces the position of one of the OAM modes in # 1. Importantly, the topological charge of the replaced OAM mode in #1 is opposite to the incident beam (lLCP). For example, in comparison with the target detection field #1, the position of the solid point in #2 replaces the position of one of the OAM modes (l = 3) in #1, then the topological charge of the incident light is -3. Similarly, when an RCP beam carrying topological charges lRCP passes through the metasurface, the target detection field #3 appears with diffraction radius R2. In comparison with the target detection field #1, the position of the solid point in #3 replaces the position of one of the OAM modes (l = -1) in #1, then the topological charge of the incident light is 1. In this way, the SAM direction and the topological charge of the incident vortex beam can be accurately identified by distinguishing between fundamental (solid spot) and OAM modes (ring-shaped intensity profile with a central null) for the separated spot.
Fig. 1. Schematic of the optical vortex recognition device. Illustrations display the target detection field for different incident beams. The target detection fields (#1), (#2), and (#3) correspond to the cases of incident beam with LP plane waves, incident beam with s = 1, and incident beam with s = -1, respectively. Both the diameter of the metasurface and the edge length of the target detection field are set to 78 µm.
The approach can be readily extended to cases of higher-order OAM modes. As is known, the OAM of the optical vortex beam is related to a helical phase front. The OAM mode is characterized by a transverse phase structure of exp(ilψ), with ψ being the azimuthal angle and l being the topological charge. In principle, the transmission function of a series of vortex beams can be defined as [33,34]:
where An(r) and kn denote the intensity and wavevector of the nth beam carrying a topological charge ln. For metasurface structures that can generate vortex beams as described above, the output light field at the Fourier plane can be calculated by Fraunhofer diffraction. Herein, normal incidence and propagation along the z-axis are considered for a vortex beam with a topological charge lin. The general relation between incident light and metasurface can be described as [34]:The direction of propagation of the nth beam is determined jointly by kxn and kyn, and each vortex beam forms an independent information channel that is physically separated from the others. Obviously, when the incident vortex beam satisfies lin = -ln, the transmitted vortex beam carries no OAM. Therefore, only one beam can focused into a solid spot with a vanishing singularity. From the degeneration of the vortex beam channel to the fundamental Gaussian mode, it is possible to identify the input OAM mode. To further increase the number of channels from n to 2n, it is also possible to include the SAM state in the OAM recognition process described above. To further increase the recognition dimension, the transmission function is modified with both SAM and OAM information:
Obviously, when the incident vortex beam satisfies lmL = -lnL (lmR = -lnR), the transmitted vortex beam carries no OAM. At the same time, if the incident light satisfies the topological charge relationship described above, the transmitted light still has a transverse wave vector kmρL (kmρR) in the positive direction along the z-axis. Consequently, one beam with a specific topological charge can be degenerated to a fundamental Gaussian mode. Furthermore, it is also possible to compute the position of this basic Gauss mode as:
The unit cell of the metasurface should be rigorously designed for efficient manipulation of the incident vortex light for OAM mode identification. Titanium dioxide (TiO2) is an excellent candidate for effectively manipulating both the propagation phase and the PB phase of the incident light [5,6,35]. In our design, the metasurface is made up of arrays of TiO2 nanopillars arranged in a periodic pattern on a SiO2 substrate. A unit cell of the metasurface is shown in Fig. 2. The wavelength was fixed at 1.55 µm which is usually used for optical communication. Figure 2(a) depicts a TiO2 nanopillar (blue square) with a given length L and width W, and the thickness of the substrate S is set to 800 nm. The height of the TiO2 nanopillar H = 1 µm and the lattice constant P = 820 nm. A non-correlated phase modulator Φσ+ (x, y) and Φσ- (x, y) is required for the designed multifunctional metasurface. When incident light with linear polarization interacts with the aforementioned metasurface, the LCP and RCP component is modulated with phase factor eiΦσ+ (x, y) and eiΦσ- (x, y), respectively. The corresponding Jones Matrix can be described as [36]:
Being azimuthally asymmetric, the nanopillar can exhibit birefringence similar to a rectangular waveguide. The calculations show that the orientation angle $\theta$ in Fig. 2(b) is determined by $\theta$ (x, y) = 1/4[Φσ+ (x, y) − Φσ- (x, y)], which can be independently adjusted the PB phase from 0 to 2$\pi$ in nanopillars. The phase delays φx (parallel to the optical axis) and φy (perpendicular to the optical axis) satisfy φx (x, y) = 1/2[Φσ+ (x, y) − Φσ- (x, y)] and φy (x, y) = 1/2[Φσ+ (x, y) − Φσ- (x, y)] − π, respectively. Nanopillar in-plane lateral dimensions (W, L) are optimized to achieve desired phase shifts φx and φy.
Fig. 2. (a) A unit cell of the metasurface consisting of a TiO2 nanopillar on a SiO2 substrate. (b) The top view of (a).
The unit cell shown in Fig. 2 was constructed as the model for the simulations. The transmissions and phase shifts were obtained by commercial simulation software (FDTD Solutions, Lumerical) based on the finite-difference time-domain method. Periodic boundary conditions were applied in both the x and y directions, while perfectly matched layers are used as the boundary condition in the propagation direction. When x-LP and y-LP plane waves pass through the unit cell of the metasurface, the transmittance maps (Tx, Ty) can be calculated as functions of the nanopillar dimensions (W, L), and are shown in Fig. 3(a) and 3(b), respectively. The corresponding phase shifts (φx, φy) are shown in Fig. 3(c) and 3(d). To cover phase delays φx and φy from 0 to 2$\pi$ at a central wavelength of 1.55 µm, the continuous phases are approximated into eight discrete phase levels. Eight different nanopillars (labeled as blue triangles in Fig. 3(c)) are selected as the basic cells for constructing the metasurface, whose phase shifts and transmittances are shown in Fig. 3(e). It can be seen that these nanopillars possess a transmittance of over 80% when the working wavelength is selected as 1.55 µm. Based on eight discrete phase levels labeled in Fig. 3(c), the relationship between the phase levels and the phase shifts is illustrated in Fig. 3(f). As seen in Fig. 3(e) and 3(f), the phase spacing between adjacent nanofins is approximately π/4, and the phase differences between φx and φy are close to π. It can be calculated that the polarisation conversion efficiencies of all the nanopillars in the selection will be in excess of 80%. The polarization conversion efficiency can defined as:
Fig. 3. (a) and (b) Calculated transmittance maps for the x-LP and y-LP plane waves pass through the unit cell of the metasurface, respectively. (c) and (d) Calculated phase shift maps for the x-LP and y-LP plane waves pass through the unit cell of the metasurface, respectively. (e) Transmittance (Tx, Ty) and normalized phase shifts (φx/2π, φy/2π) of eight selected unit structures of the metasurface. (f) Normalized Phase shifts (φx/2π, φy/2π) as a function of eight phase levels.
3. Results and discussion
Independent control of (W, L) and $\theta$ distribution can therefore guarantee the combination of the propagation phase and PB phase. Passing through the metasurface, LCP and RCP light have the same propagation phase and opposite PB phase. For the two different SAM states of left and right-handedness circular polarization presented in the incident light, the propagation phase φP and PB phase φPB can be determined by the following equations:
According to the design theory in Fig. 1, to realize optical vortex recognition, the optical vortex should be decoupled with different SAMs to propagate along different conical surfaces, as shown in Fig. 4. When linear polarization (LP) plane waves pass through the metasurface along the z-axis direction, they will be transformed into independent off-axis OAM beams with specific topological charges. The off-axis OAM beams and the center of the metasurface structure form two different diffraction azimuths ϕ1 and ϕ2, which are associated with the left-handed and the right-handed states of circular polarisation respectively. When the incident light is an OAM beam, the diffraction angle of the outgoing beam corresponds to ϕ1 if the incident OAM beam contains a SAM value of s = 1, and to ϕ2 if the incident OAM beam contains a SAM value of s = -1. The azimuth angle ϕ determines the diffraction radius R of the circle formed by the center of each off-axis OAM mode. The larger the azimuth angle, the larger the radius of the circle. The distance from the metasurface to the target detection screen is defined as f, then R1 = f·tan(ϕ1) and R2 = f·tan(ϕ2).
Fig. 4. The incident OAM beams with s = 1 and s = −1 are intended to propagate with two different diffraction azimuths ϕ1 and ϕ2, respectively.
As depicted in Fig. 1 and Fig. 4, the incident beams with s = 1 and s = −1 are intended to propagate along the inner and outer surfaces of the cone, respectively. In this case, 14 signal channels ($|{s,l} \rangle$, where s = ±1, l = 0, ± 1, ± 2, ± 3) are considered in the constructed metasurface. The transverse wave vectors in the transmission function are set as:
Fig. 5. (a) and (b) The theoretical and real component (LCP) of the metasurface phase profile, respectively. (c) and (d) The theoretical and real component (RCP) of the metasurface phase profile, respectively. (e) and (f) Profiles of the propagation phase and PB phase of the metasurface, respectively.
According to the theoretical construction above, a schematic diagram of part of the experimental setup is shown in Fig. 6. In the measurement setup illustrated in Fig. 6(a) and 6(b), the light incident on the metasurface is LP plane waves and OAM beams, respectively. For the realization of optical vortex recognition, the plane waves with LP should be perpendicularly incident on the metasurface. As can be seen from the normalized field intensity distributions in Fig. 7(a), the LP plane waves will be transformed into independent off-axis OAM beams with specific topological charges. The field intensity distribution on target detection fields is split into a small and a large torus, as illustrated in Fig. 7(a). This corresponds to the decoupling of the LCP and RCP components of the incident light, respectively. Identically, the distribution of topological charges l for OAM patterns in the counterclockwise direction is -1, 1, -2, 2, -3, 3, using the fundamental mode position as the default position. Figure 7(b)-7(f) describes the normalized field intensity distributions when the metasurface is under vertical incidence optical vortex with $|{s,l} \rangle$, where s = -1. These results are consistent with our theoretical expectations, suggesting that the encoded independent wavefronts in a single metasurface can be employed for high-order optical vortex recognition.
Fig. 6. Schematic diagram of part of the experimental setup for high-order optical vortex recognition. (a) and (b) the light incident on the metasurface is LP plane waves and OAM beams, respectively. Optical elements: linear polarizer (LP), quarter-wave plate (QWP), and Liquid Crystal Q-Plate (Q-plate).
Fig. 7. (a) The LP plane waves pass through the metasurface and are transformed into independent off-axis OAM beams with specific topological charges. (b)-(f) The normalized field intensity distributions when the metasurface is under vertical incidence optical vortex with $|{s,l} \rangle$, where s = -1.
A new approach to detecting OAM is provided here that does not require analyzing patterns using traditional algorithms with large datasets. Take Fig. 7 as an example. The azimuth angle ϕ2 is first obtained from the diffraction radius R2 on the target detection field shown in Fig. 7(b)-7(f). This determines the SAM in the incident OAM beam is RCP. Then taking the position of the fundamental mode (solid spot) in Fig. 7(a) as a reference, with the anti-clockwise direction as the observation direction, observe how many OAM modes (ring-shaped intensity profile with a central null) are spaced between the fundamental mode in Fig. 7(b)-7(f) and in Fig. 7(a). The topological charge of the incident OAM beam can be obtained from the interval number between two fundamental modes and the ordering of the preset topological charges in the metasurface. The details are given in Table 1. The case where the interval number is 0 has been shown in Fig. 1.
Table 1. Using observation data to recognize incident vortex light states (original ordering principle).
It is noted that the non-optical vortex in Fig. 7 is perfectly symmetric and has a uniform donut-shaped intensity distribution. This originates from the conservation of optical momentum. A similar metasurface component has been fabricated experimentally and the experimental results are analyzed in detail in Fig. 2 of Ref. [33]. In Ref. [33], when the component is operated at a preset wavelength, the experimentally detected OAM modes have a uniform donut-shaped intensity distribution. However, changing the wavelength of the incident light breaks this perfectly symmetrical property. This suggests that the dispersion of the component is critical to the ability to maintain the uniform donut-shaped intensity distribution of the detected OAM modes.
The conclusions in Table 1 reflect only one ordering of topological charges through the metasurface. Actually, the topological charges can be arranged in any principle when designing metasurface phase profiles. Figure 8 illustrates the OAM beam incident on the metasurface with s = 1. The normalized field intensity distribution for the incident light of $|{1,\textrm{ - }3} \rangle$ in the metasurface described in Table 1 is shown in Fig. 8(a). The inset around the mode pattern is the phase distribution corresponding to the OAM modes. The situation under the new ordering principles is detected in Fig. 8(b) to 8(i). As can be seen from Fig. 8(b), when the LP plane waves pass through the metasurface, the mode spot distribution of the off-axis OAM beams is significantly different from that shown in Fig. 7(a). Figure 8(c) describes the normalized field intensity distribution at diffraction radius R1 from Fig. 8(b). Using the fundamental mode position as the default position, the distribution of topological charges l for each spot in the counterclockwise direction is -1, -2, -3, 3, 2, 1. Similar to Table 1, Table 2 shows the scheme for identifying the incident OAM beam states according to the new ordering rule. As a proof of concept, 14 signal channels ($|{s,l} \rangle$, where s = ±1) are considered in the constructed metasurface. Apart from two channels which are reserved for the generation of the fundamental mode in the case of l = 0, the remaining 12 channels can be encoded at will. If one has already predicted which states are about to be detected, or which OAM modes will definitely not appear in the following detections, then the phase profile of the metasurface can be designed accordingly. This is of great importance for the efficient use of existing recognition channels on metasurface. For an OAM beam, its far-field radiation angle θr is $\tan {\theta _\textrm{r}} = \sqrt {\frac{l}{2}} \frac{\lambda }{{\pi {\omega _0}}}$, where ω0 is the beam waist. The maximum k-space that can be controlled by the phase modulation unit is 1/P, where P is the lattice constant. Using the off-axis method, the maximum number of OAM channels Nmax can be estimated as ${N_{\textrm{max}}} = \sqrt[3]{{\frac{{{\pi ^2}{D_{\textrm{all}}}^2}}{{{P^2}}}}}$, where Dall is the diameter of the component. For experiments, the number of OAM channels available can be increased by an increase in sample size or a reduction in unit cell size.
Fig. 8. (a) The normalized field intensity and phase distribution for the incident light of $|{1,\textrm{ - }3} \rangle$ in the metasurface which is described in Table 1. (b) The LP plane waves pass through the metasurface under the new ordering principle. (c)-(i) The normalized field intensity and phase distributions when the metasurface is under vertical incidence optical vortex with $|{s,l} \rangle$, where s = 1.
Table 2. Using observation data to recognize incident vortex light states (new ordering principle).
4. Conclusions
In conclusion, a general scheme is proposed to implement effective high-order optical vortex recognition. With the advantage of encoding independent wavefronts, the proposed metasurface can easily identify the SAM (s) and topological charge (l) of OAM patterns. The proposed optical vortex recognition device consists of TiO2 nanopillars on a SiO2 substrate. This kind of recognition can be achieved without the use of traditional artificial intelligence technology. A single passive all-dielectric metasurface is designed to simultaneously encode of wavefront and the transmission paths in different incident OAM beams. We utilize the optical vortex beams as the carriers of OAM states where phase, topological charge, spin angular momentum (SAM), and wavelength are the basic properties of beams. When the metasurface is illuminated by LP plane waves or vortex beams of the selected orders, the transmitted beams will be transformed into independent off-axis OAM beams pointing in predefined directions. Due to the presence of SAM, the vortex beam to be identified is spatially separated after passing through the metasurface. The SAM direction and the topological charge of the incident vortex beam can be accurately identified by distinguishing between fundamental (solid spot) and OAM modes (ring-shaped intensity profile with a central null) for the separated spot. As a proof of concept, 14 signal channels are considered in the constructed metasurface, 12 of them can be encoded at will for the detection of any vortex beam with a predefined topological charge. These results merge two seemingly disparate fields, OAM pattern recognition and metasurface, which may open avenues for the ultimate miniaturization of optical vortex communication and advanced OAM detection technologies.
Funding
National Natural Science Foundation of China (61827817, 62221001, 62235003); State Key Laboratory of Rail Traffic Control and Safety (RCS2019ZZ007).
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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