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Abstract
The multiplexing and de-multiplexing of orbital angular momentum (OAM) beams are critical issues in optical communication. Optical diffractive neural networks have been introduced to perform sorting, generation, multiplexing, and de-multiplexing of OAM beams. However, conventional diffractive neural networks cannot handle OAM modes with a varying spatial distribution of polarization directions. Herein, we propose a polarized optical deep diffractive neural network that is designed based on the concept of dielectric rectangular micro-structure meta-material. Our proposed polarized optical diffractive neural network is optimized to sort, generate, multiplex, and de-multiplex polarized OAM beams. The simulation results show that our network framework can successfully sort 14 kinds of orthogonally polarized vortex beams and de-multiplex the hybrid OAM beams into Gauss beams at two, three, and four spatial positions, respectively. Six polarized OAM beams with identical total intensity and eight cylinder vector beams with different topology charges have also been sorted effectively. Additionally, results reveal that the network can generate hybrid OAM beams with high quality and multiplex two polarized linear beams into eight kinds of cylinder vector beams.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams with helical phase front, possessing orbital angular momentum (OAM) or topological charge (l), have attracted wide attention in many pivotal fields such as optical communication [1,2], optical tweezers [3,4], quantum information processing [5,6] and holographic display [7,8]. The ring radius of vortex beam is a straightforward feature for the identification of topological charge number [9]. As a critical technology in optical communication, vortex beams with different modes or topological charge numbers have been considered as multiple independent channels to improve communication bandwidth and capacity [10–12]. For example, 16 OAM beams with two polarization directions and a single wavelength can achieve 2.56 Tbit/s free space communication [11], showing the advantage of OAM mode multiplexing. Besides, there are also other multiplexing technologies combined with OAM mode multiplexing to empower optical communication efficiency, such as wavelength-division multiplexing [13–15], space-division multiplexing [16,17], polarization-division multiplexing [18,19] and time-division multiplexing [20,21]. Various methods have been studied to generate and modulate vortex beams with different OAM modes and polarization states such as spiral phase plate [22,23], liquid crystal spatial light modulator [24], micro-structure meta-material [25,26] and deep diffractive neural network (D$^2$NN) [27,28]. Through designing various patterns of meta-surface, multiple optical vortex beams have been reported to be produced by a single nano-device [29]. Some researchers generate a specific cylinder vector beam by the interference of vortex beams [30]. In order to avoid redundant interferometers and diverse switching elements, Gouy phase has been introduced to generate arbitrary cylindrical vector beam [31].
Deep learning as the most widely used method in artificial intelligence has made a great advance in many tasks such as image classification [32], speech recognition [33], chemical component analysis [34] and language translation [35]. Optical neural network (ONN) is a combination between optics and deep learning which was firstly proposed as an optical vector-matrix multiplication model by J. W. Goodman of Stanford University in 1978 [36]. D$^2$NN consisting of multiple diffractive layers is another typical physical implementation of ONN. D$^2$NN has been widely investigated for various optical computing tasks including number digit classification [37–39], optical logic gate operation [40] and multiplexing and de-multiplexing of OAM beams [27]. For example, Z. Huang et al. designed a D$^2$NN to achieve a depth-controllable imaging technology in OAM multiplexing holography [27,41]. For improving the accuracy of D$^2$NN, many nonlinear materials and nonlinear physical processes are designed as nonlinear optical activation functions in a D$^2$NN [42]. Additionally, in order to handle different machine learning tasks for wave sensing, programmable meta-surfaces have also been proposed to achieve in-situ modulation of diffractive layers [43].
However, in most cases, the D$^2$NN cannot perform light modulation in the dimension of polarization. Namely, there is no polarization anisotropy in most previously proposed D$^2$NNs. Recently meta-surface has been widely reported to achieve anisotropic polarized light modulation [44,45]. For example, the rectangular microstructure based on the concepts of Huygens meta-surface and Berry phase is the common anisotropic meta-surface, which is favorable for us to design optical devices with polarization anisotropy. In this paper, we propose a polarized deep diffractive neural network (PD$^2$NN) to sort, generate, multiplex and de-multiplex polarized OAM beams, which is modulated by rectangular pillar meta-materials. The proposed PD$^2$NN consists of multiple anisotropic pure phase layers which are optimized by wavefront matching algorithm [46]. The results show our PD$^2$NN can accurately recognize the optimized vortex beams or cylinder vector beams. In addition, our proposed network can spatially de-multiplex hybrid polarized OAM beams into spatially separated Gauss beams. From the aspect of generation and multiplexing, our PD$^2$NN can achieve the objective of generating two kinds of high quality cylinder vector beams and multiplexing two different polarized Gauss beams into eight kinds of cylinder vector beams with multiple polarized modes.
2. Model and principles of PD$^2$NN
2.1 Mathematical model of polarized light modulation by an anisotropic neural layer
Various materials and methods proposed previously can be used as an anisotropic neural layer in a D$^2$NN. Here, we consider taking dielectric rectangular meta-material to design our anisotropic neural layers in which there are different x-polarized and y-polarized responses in horizontal and vertical directions. In a common dielectric rectangular meta-material structure, the long and short sides correspond to different propagation phase modulation of a-polarized or b-polarized incidence [47]. And anisotropic amplitude and phase modulation can be achieved by rotating the angle of rectangular micro-structure. Therefore, through rotating the rectangular pillars with different propagation phase in two polarization directions, we can implement polarized light modulation. The structure diagrams of rectangular meta-material are shown in Fig. 1. The period of the micro pillar is P, the length and width of the rectangular pillar are a and b respectively and $\theta$ is the angle between long side and x axis. Other optimized parameters like height of proposed structure can be obtained via electromagnetic simulation based on optimized phase values.
Fig. 1. (a) Structure diagram of rectangular pillar micro-structure. (b) A rectangular pillar unit cell.
In general, polarized light can be expressed by Jones vectors and the light fields in x and y coordinates can be expressed as a column vector: E = [$E_x, E_y$]$^T$ where $\textit {E}_x$ and $\textit {E}_y$ correspond to x-polarized and y-polarized components. The transmission matrix $\textit {T}_s$ of a single-pillar unit cell can be represented as:
where $\textit {t}_a$, $\textit {t}_b$, $\varphi _a$ and $\varphi _b$ are the transmission coefficients of a-polarized and b-polarized amplitudes and phases respectively. In our calculation, $\textit {t}_a=\textit {t}_b=1$ which are determined by anisotropic amplitude modulation effect. In other words, our designed structure has no modulation in amplitude but only in phase. The transmission matrix T can be obtained by Eq. (1) when (a, b) coordinates have an in-plane anti-clockwise rotation angle $\theta$ with respect to the (x, y) coordinates as shown in Fig. 1. We set $\varphi$=$\varphi _b$-$\varphi _a$, and the transmission matrix T can be expressed as:Then the final transmission R can be obtained by Eq. (2):
2.2 Free-space propagation model and optimization of diffractive neural network
Our proposed PD$^2$NN structure has several hidden layers which have 256$\times$256 neural units in every layer. Each neural unit has a size of 5 $\mu$m. The distance d between neighboring hidden layers is 5 cm. The illumination wavelength ($\lambda$) is set as 1500 nm. The network is optimized by input vortex beams or spatially separated Gauss beams which is depending on the specific task of network. The structure diagram of PD$^2$NN is shown in Fig. 2. The light field propagation in PD$^2$NN is simulated based on Rayleigh Sommerfeld diffractive theory. Every single neural unit of a given PD$^2$NN layer is regarded as a point source of a wave and the connection weight can be expressed as:
3. Sorting, generation, multiplexing, and de-multiplexing of polarized OAM beams with PD$^2$NN
3.1 Sorting and de-multiplexing of polarized vortex beams with different topological charges
To demonstrate the performance of PD$^2$NN framework, we firstly optimized 14 kinds of x polarized or y polarized vortex beams with topological charges l = −3, −2, −1, 0, 1, 2, 3, respectively. We optimized the network with 5 hidden layers which is phase-only modulated. We sort 14 kinds of polarized vortex beams with different topological charges into 14 Gauss beams in 14 different circle areas of the detecting plane as shown in Fig. 3(a). The Gauss beam in each circle area corresponds to a kind of polarized vortex beam. The intensity of x polarized incidence and y polarized incidence, phase of x polarized incidence and y polarized incidence and total intensity of x and y polarized incidence are shown in first, second, third, fourth and fifth rows of Fig. 3(b) respectively. In our calculation, the polarized beams are expressed by two matrices. Taking x polarized beam for example, the sub-figures in the first row are beam spot intensity distributions and the sub-figures in the second row denote zero intensities in y polarized direction as shown in Fig. 3(b). The sorting results are shown in the last row of Fig. 3(b) indicated by total intensity of x and y polarized outputs in the detecting plane. Obviously, as shown in the first and second rows of Fig. 3(b), the size of hollow spots becomes bigger with the absolute value of the topological charge of vortex beam increasing. Figure 3(d) is the zoom-in image of the sorting result of x polarized vortex beam with l = 1 which indicates the output Gauss beam is located accurately in the pre-defined circle area. It can be observed that every Gauss output beam corresponds exactly to the target location in the detecting plane. These results verify the capability of the network with 5 hidden layers to sort polarized vortex beams with different topological charge numbers and polarized states. The optimized phase $\varphi$ and rotation angle $\theta$ distributions of 5 hidden layers are shown in first and second rows of Fig. 3(c) respectively.
Fig. 3. Sorting polarized vortex beams with different topological charges. (a) Ideal output plane with different sorting results. (b) Input amplitude and phase distributions of vortex beams with different topological charges and the results of sorting. (c) Optimized phase and angle distributions of proposed PD$^2$NN from first to fifth layer. (d) Zoom-in image of the output of x polarized vortex beam with l = 1.
For quantitatively evaluating the quality of sorting results, we define signal to noise ratio (SNR) of target detecting regions to measure the accuracy of sorting. The definition of the SNR can be seen as follows:
where $SNR_{\textit {i},\textit {j}}$ (i, j=1, 2, 3, $\cdots$,14) represents signal-to-noise ratio of the light intensity signal of i-th Gauss beam in j-th Gauss beam’s target region, $Signal_{\textit {i},\textit {j}}$ represents the light intensity of i-th Gauss beam in j-th Gauss beam’s target region, $Noise_{\textit {i}}$ represents the light intensity of i-th Gauss beam in non-target area. The detecting regions in the output plane are divided into 14 circles as shown in Fig. 3(a), the signal in an target region is only the total light intensity within an circle area and noise is the total intensity out of an circle area. The $\textit {SNR}$ results of 14 detecting regions are shown in the confusion matrix of Fig. 4. We can see clearly that the values of diagonal elements are much higher than the other values which indicates output Gauss beams mainly focus on the corresponding target regions.Fig. 4. The confusion matrix map of signal to noise $SNR_{\textit {i},\textit {j}}$ of detecting results for the sorting of polarized vortex beams with different topological charges.
Figure 5 shows that the proposed PD$^2$NN network optimized by 14 kinds of cross-polarized vortex beams with l = −3, −2, −1, 0, 1, 2, 3 respectively can spatially de-multiplex polarized hybrid OAM beams. In Fig. 5(a), the first, second and third rows represent the de-multiplexing results of hybrid OAM modes with two (x polarized vortex beams with l = 2 and y polarized vortex beams with l = 3), three (x polarized vortex beams with l = 1, 3 and y polarized vortex beams with l = −3) and four (x polarized vortex beams with l = 1, −3 and y polarized vortex beams with l = −1, −2) kinds of modes respectively. We can see clearly that two, three and four corresponding Gauss beams in the output plane are generated. The SNR of spatially de-multiplexing two, three and four hybrid polarized OAM modes are shown in Figs. 5(a), (b), and (c), respectively. The red bars in Figs. 5(b), (c), and (d) represent the SNR of de-multiplexing results of hybrid polarized OAM modes in which the de-multiplexed signals can be distinguished from noise in non-target regions.
Fig. 5. Spatial de-multiplexing of hybrid polarized OAM beams. (a) Input polarized beams with multiple topological charges and output results. The de-multiplexing $\textit {SNR}$ of polarized OAM beams summed by (b) two (c) three (d) four polarized Gauss beams. The red bars are corresponding to the results of de-multiplexing modes.
3.2 Sorting of polarized OAM beams with identical total intensity
In order to verify the capability of our network to sort polarized OAM beams with identical total intensity, we optimized our PD$^2$NN network with 5 hidden layers to sort four kinds of x or y polarized vortex beams with l = −2 and 2 and two kinds of cylinder vector beams which have an identical total intensity in x and y polarization directions with the first four kinds of vortex beams. The intensity patterns of x and y polarized cylinder vector beams are obtained from interference of vortex beams with l = −1 and 1. The patterns of detecting plane are shown in Fig. 6(a) which is divided into six circle areas in different locations. We can see clearly that the total intensities of the six beams in the input plane in Fig. 6(b) are identical which cannot be distinguished by human eyes while the intensities of each polarization direction are different. Fortunately, our network can convert the 6 polarized beams into 6 Gauss beams in the output plane at the target locations as shown in the last column of Fig. 6(b). In the last two rows in Fig. 6(b), there are some noise signals in non-targeted locations of the detecting plane. This is because the 6 polarized beams with identical total intensity are not orthogonal in topological states which will affect the sorting performance. For non-orthogonal input light fields, the intensity differences of output light spots on the detection plane will usually be very small and therefore affect the correct judgement of classification result [48]. It is evident that the two kinds of cylinder vector beams can be interfered by the first four kinds of polarized vortex beams. In other words, the input cylinder vector beam may include modes of vortex beams in the front which makes the output plane exhibit other possible results. Although the noise signals are relatively stronger, we can still easily distinguish the target signal from the noise signal in the output plane. The final optimized rotation angle and modulation phase distributions of each diffractive layer are shown in Fig. 6(c).
Fig. 6. Sorting polarized vortex beams with identical topological charge. (a) Incidence with identical topological charge and the results of sorter. (b) The output plane with different sorting results. (c) The optimized rotation angle and phase (d) The confusion matrix map of signal to noise of sorting polarized beams with identical total intensity.
The $SNR_{\textit {i}, \textit {j}}$ (i, j=1, 2, 3, $\cdots$, 6) of sorting results of polarized beams with identical total intensity are given in Fig. 6(d) where i and j represent light with different polarization states from 1 to 6 rows in Fig. 6(d) respectively. The confusion matrix map indicates that there is high contrast between $SNR_{\textit {i}, \textit {j}}$ (i, j=1, 2, 3, 4) while there is low contrast between $SNR_{\textit {i}, \textit {j}}$ (i, j=5, 6) which indicates some disturbing signals are emerging when detecting the results of multiplexing of two kinds of cylinder vector beams possessing the same total intensity with the first four vortex beams. Although the contrast between $SNR_{\textit {i}, \textit {j}}$ (i, j=5, 6) is low, the target signals in specific detecting regions still can be distinguished from noise signals in non-target detecting regions.
3.3 Sorting of cylinder vector beams with different topological charge numbers
For sorting cylinder vector beams with different topological charge numbers and polarization states, we optimized the PD$^2$NN network with 5 layers to sort 8 kinds of cylinder vector beams which are generated by interference of vortex beams. Eight kinds of cylinder vector beams with different polarization states and topological charge numbers are assigned into 8 circle areas in output plane as shown in Fig. 7(a). The radial and angular polarization beams of 1, 2, 3 and 4 orders are the interference results of vortex beams with l = $\pm 1$, $\pm 2$, $\pm 3$ and $\pm 4$ respectively. As shown in first and second columns of Fig. 7(b), with the order of polarization beams increasing, the petals of the intensity patterns of cylinder vector beams are increasing. And the sizes of hollow spots which is the total intensities of x and y components are becoming larger as the order of cylinder vector beams increases as shown in Fig. 7(b). The results shown in the last column of Fig. 7(b) indicate that the total intensities of x and y polarized output Gauss beams are exactly located in the target detecting regions as shown in Fig. 6(a). And the final optimized phase and rotation angle distributions are shown in Fig. 7(c). The quantitative results of $SNR_{\textit {i}, \textit {j}}$ (i, j=1, 2, 3, $\cdots$, 8) of sorting cylinder vector beams with different topological charge numbers are shown in Fig. 7(d). In Fig. 7(d), the minimum of contrast ratio between diagonal elements and other elements are larger than 22.6 which is sufficient to distinguish the target signal.
Fig. 7. Sorting cylinder vector beams with different polarization states and topological charges (l = $\pm$1, $\pm$2, $\pm$3 and $\pm$4). (a) The output plane with 8 different sorting regions. (b) Input cylinder vector beams with different topological charges and the results of sorter. (c)The optimized rotation angle and phase distributions. (d) The confusion matrix map of signal to noise ratio of detecting results for sorting cylinder vector beams with different topological charges.
3.4 Generation and multiplexing of polarized OAM beams
On the other hand, we also try to use the PD$^2$NN to generate cylinder vector beams from polarized Gauss beams located in different areas in the input plane. We firstly use four 45 degrees linear polarized Gauss beams to optimize our network with 5 hidden layers to generate four kinds of cylinder vector beams which are the interference results of vortex beams with l = $\pm 3$, $\pm 4$, $\pm 5$ and $\pm 6$ respectively. The intensity of x and y polarized input beams and the total intensity of x and y polarized input light are shown in the first, second and third columns of Fig. 8(a) respectively. The results of intensity and phase distributions of generated cylinder vector beams are shown in the last five columns of Fig. 8(a). In order to achieve generated results with higher quality, we reduce the number of generated cylinder vector beams to 2. Then we consider using two 45 degrees linear polarized Gauss beams to optimize our network with 5 hidden layers to generate two kinds of cylinder vector beams whose polarization components are the interference results of vortex beams with l = $\pm 5$ and $\pm 6$ respectively. The intensity distribution of input light and output results are shown in Fig. 8(b). Apparently, the quality of generated two kinds of cylinder vector beams is better than that of generated four kinds of cylinder vector beams.
Fig. 8. Generating cylinder vector beams. (a) Generating four kinds of cylinder vector beams (which are generated by the interference of vortex beams with $l= {\pm }3,{\pm }4,{\pm }5 \;\textrm{and}\;{\pm }6$ respectively) with diffractive layers number N = 5. (b) Generating two kinds of cylinder vector beams (which are generated by the inerference of vortex beams with $l={\pm }5 \;\textrm{and}\; {\pm }6$ respectively) with N = 5. (c) Generating two kinds of cylinder vector beams with N = 10.
In order to generate two kinds of cylinder vector beams with higher quality, we optimized the network with more hidden diffractive layers. Figure 8(c) shows the results generated from a network with 10 layers have better quality than those from a network with 5 layers and are very close to the ground truth ones. For analyzing the effect of increasing the number of diffractive layers to generate two kinds of cylinder vector beams, we optimize the network with incremental layers from 3 to 10 and roughly use the mean square error (MSE) between the output intensity results and the ground truth ones to evaluate the generated modes quality. The definition of intensity MSE can be expressed by:
where R and T represent the output intensity of PD$^2$NN and the ground truth intensity respectively and (m, n) represents the horizontal and vertical extent of the light field. The results in Fig. 9 show that the MSE of intensity is roughly always on the downward trend with the number of diffractive layers increasing regardless of intensities in each polarization directions or total intensities in two polarization directions. But the results indicate that the quality will not be enhanced very significantly as the number of layers increases when the number of diffractive layers is larger than 6. This process indicates that increasing the number of layers is an effective method to enhance the performance of our proposed network to generate cylinder vector beams. But it shall be noticed that increasing the number of hidden layers in a D$^2$NN network can improve the quality of mode generation but also cause more loss of light power during the process of propagation between layers which is not favorable for increasing the output signal power.Fig. 9. The intensity MSE results of generating two kinds of cylinder vector beams (l = $\pm$5 and $\pm$6) with different numbers of diffractive layers from 3 to 10. (a) MSE curve of intensity of x component. (b) MSE curve of intensity of y component. (c) MSE curve of total intensity with increasing layers.
Additionally, a network optimized with two 45 degrees linear beams to generate two cylinder vector beams (whose components are the interference of modes with l = $\pm 5$ and $\pm 6$ respectively) also can be used to multiplex polarized linear beams at two locations in the input plane. The results of multiplexing are shown in Fig. 10 which are obtained from an optimized network with 10 hidden diffractive layers. There are 8 kinds of input modes as input Gauss beams at two locations shown in first, second and third columns of Fig. 10 and 8 different cylinder vector beams generated shown in fourth to eighth columns of Fig. 10. The total intensity of every multiplexing result is different from each other which reflects the orthogonality. The polarization state of each multiplexing result is different from each other as well. The multiplexing results indicate that this network can use linear polarized beams at two locations to generate a mixture of 8 kinds of hybrid cylinder vector beams.
4. Conclusion
In conclusion, we propose a novel phase-only modulated PD$^2$NN framework for processing polarized vortex beams and cylinder vector beams that conventional D$^2$NN cannot handle. It is implemented by designing proper rectangular micro-structures with wavefront matching algorithms. The hidden layers of the proposed network are determined by two parameters: modulated phase and rotation angle of designed rectangular micro-structure. The simulation results show that the proposed PD$^2$NN with 5 hidden layers can sort 14 kinds of polarized vortex beams with different topological charge numbers, 6 kinds of cylinder vector beams with identical topological charges and 8 kinds of cylinder vector beams into target Gauss beam spots in detecting regions. Additionally, the network with 10 layers can achieve generation and multiplexing of cylinder vector beams with a relatively high quality. Simulation results show that two linear Gauss beams can be successfully multiplexed into 8 kinds of cylinder vector beams. This work paves the way for polarized diffractive neural network which can be applied in other potential fields such as multi-channel image recognition and logic gate operation of polarized light.
Funding
Natural Science Foundation of Shenzhen City (JCYJ20200109150212515); BNU Interdisciplinary Research Foundation for the First-Year Doctoral Candidates (BNUXKJC2119).
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.
References
1. J. Liu, P. Wang, X. Zhang, Y. He, X. Zhou, H. Ye, Y. Li, S. Xu, S. Chen, and D. Fan, “Deep leaning based atmospheric turbulence compensation for orbital angular momentum beam distortion and communication,” Opt. Express 27(12), 16671–16688 (2019). [CrossRef]
2. F. Zhu, S. Huang, W. Shao, J. Zhang, M. Chen, W. Zhang, and J. Zeng, “Free-space optical communication link using perfect vortex beams carrying orbital angular momentum (oam),” Opt. Commun. 396, 50–57 (2017). [CrossRef]
3. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
4. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: Oam manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90–367 (2019). [CrossRef]
5. E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009). [CrossRef]
6. G. B. Xavier and G. Lima, “Quantum information processing with space-division multiplexing optical fibres,” Commun. Phys. 3(1), 9–11 (2020). [CrossRef]
7. X. Fang, H. Ren, and M. Gu, “Orbital angular momentum holography for high-security encryption,” Nat. Photonics 14(2), 102–108 (2020). [CrossRef]
8. H. Zhou, B. Sain, Y. Wang, C. Schlickriede, R. Zhao, X. Zhang, Q. Wei, X. Li, L. Huang, and T. Zentgraf, “Polarization-encrypted obital anular mmentum mltiplexed mtasurface holography,” ACS Nano 14(5), 5553–5559 (2020). [CrossRef]
9. S. Zheng and J. Wang, “Measuring orbital angular momentum (oam) states of vortex beams with annular gratings,” Sci. Rep. 7(1), 1–9 (2017). [CrossRef]
10. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, and P. Winzer, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 mimo processing,” J. Lightwave Technol. 30(4), 521–531 (2012). [CrossRef]
11. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. F. Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef]
12. C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 9(2), 1–10 (2017). [CrossRef]
13. J. Jia, Z. Chang, H. Yang, Q. Liu, F. Wang, H. Gao, F. Li, and P. Zhang, “Mode sorter designed for (de) multiplexing vector vortex modes,” Appl. Opt. 58(26), 7094–7099 (2019). [CrossRef]
14. A. H. Gnauck, P. J. inzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Spectrally efficient long-haul wdm transmission using 224-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol. 29(4), 373–377 (2011). [CrossRef]
15. A. Sano, H. asuda, T. Kobayashi, M. Fujiwara, K. Horikoshi, E. Yoshida, Y. Miyamoto, M. Matsui, M. Mizoguchi, H. Yamazaki, Y. Sakamaki, and H. Ishii, “Ultra-high capacity wdm transmission using spectrally-efficient pdm 16-qam modulation and c- and extended l-band wideband optical amplification,” J. Lightwave Technol. 29(4), 578–586 (2011). [CrossRef]
16. X. Liu, S. Chandrasekhar, X. Chen, P. J. Winzer, Y. Pan, T. F. Taunay, B. Zhu, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “1.12-tb/s 32-qam-ofdm superchannel with 8.6-b/s/hz intrachannel spectral efficiency and space-division multiplexed transmission with 60-b/s/hz aggregate spectral efficiency,” Opt. Express 19(26), B958–B964 (2011). [CrossRef]
17. S. Li and J. Wang, “A compact trench-assisted multi-orbital-angular-momentum multi-ring fiber for ultrahigh-density space-division multiplexing (19 rings × 22 modes),” Sci. Rep. 4(1), 3853 (2015). [CrossRef]
18. X. Zhou, J. J. Yu, M. F. Huang, Y. Shao, T. Wang, L. Nelson, P. Magill, M. Birk, P. I. Borel, D. W. Peckham, R. Lingle, and B. Y. Zhu, “64-tb/s, 8 b/s/hz, pdm-36qam transmission over 320 km using both pre- and post-transmission digital signal processing,” J. Lightwave Technol. 29(4), 571–577 (2011). [CrossRef]
19. H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” J. Lightwave Technol. 5, 1 (2015). [CrossRef]
20. A. Wang, L. Zhu, J. Liu, C. Du, Q. Mo, and J. Wang, “Demonstration of hybrid orbital angular momentum multiplexing and time-division multiplexing passive optical network,” Opt. Express 23(23), 29457 (2015). [CrossRef]
21. Y. Fang, J. Yu, N. Chi, J. Zhang, and J. Xiao, “A novel pon architecture based on oam multiplexing for efficient bandwidth utilization,” IEEE Photonics J. 7(1), 1–6 (2015). [CrossRef]
22. X. Hui, S. Zheng, Y. Hu, C. Xu, X. Jin, H. Chi, and X. Zhang, “Ultralow reflectivity spiral phase plate for generation of millimeter-wave oam beam,” Antennas Wirel. Propag. Lett. 14, 966–969 (2015). [CrossRef]
23. V. Kotlyar, A. Kovalev, A. Porfirev, and E. Kozlova, “Orbital angular momentum of a laser beam behind an off-axis spiral phase plate,” Opt. Lett. 44(15), 3673–3676 (2019). [CrossRef]
24. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological chargee,” Opt. Express 19(5), 4085–4090 (2011). [CrossRef]
25. T. Arikawa, T. Hiraoka, S. Morimoto, F. Blanchard, and K. Tanaka, “Transfer of orbital angular momentum of light to plasmonic excitations in metamaterials,” Sci. Adv. 6(24), 1–6 (2020). [CrossRef]
26. Y. Li, X. Li, L. Chen, M. Pu, J. Jin, M. Hong, and X. Luo, “Orbital angular momentum multiplexing and demultiplexing by a single metasurface,” Adv. Opt. Mater. 5(2), 1600502 (2017). [CrossRef]
27. Z. Huang, P. Wang, J. Liu, W. Xiong, Y. He, J. Xiao, H. Ye, Y. Li, S. Chen, and D. Fan, “All-optical signal processing of vortex beams with diffractive deep neural networks,” Phys. Rev. Appl. 15(1), 014037 (2021). [CrossRef]
28. P. Wang, W. Xiong, Z. Huang, Y. He, J. Liu, H. Ye, J. Xiao, Y. Li, D. Fan, and S. Chen, “Diffractive deep neural network for optical orbital angular momentum multiplexing and demultiplexing,” IEEE J. Sel. Top. Quantum Electron. 28(4), 1–11 (2022). [CrossRef]
29. Z. Jin, D. Janoschka, J. Deng, L. Ge, P. Dreher, B. Frank, G. Hu, J. Ni, Y. Yang, J. Li, C. Yu, D. Lei, G. Li, S. Xiao, S. Mei, H. Giessen, F. M. zu Heringdorf, and C.-W. Qiu, “Phyllotaxis-inspired nanosieves with multiplexed orbital angular momentum,” eLight 1(1), 5–11 (2021). [CrossRef]
30. X. L. Wang, J. Ding, W. Ni, C. Guo, and H. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef]
31. J. Jia, K. Zhang, G. Hu, M. Hu, T. Tong, Q. Mu, H. Gao, F. Li, C.-W. Qiu, and P. Zhang, “Arbitrary cylindrical vector beam generation enabled by polarization-selective gouy phase shifter,” Photonics Res. 9(6), 1048–1054 (2021). [CrossRef]
32. J. Zhang, Y. Xie, Q. Wu, and Y. Xia, “Medical image classification using synergic deep learning,” Med. Image Anal. 54, 10–19 (2019). [CrossRef]
33. K. Noda, Y. Yamaguchi, K. Nakadai, H. G. Okuno, and T. Ogata, “Audio-visual speech recognition using deep learning,” Appl. Intell. 42(4), 722–737 (2015). [CrossRef]
34. Y. Han, J. Jang, E. Cha, J. Lee, H. Chung, M. Jeong, T.-G. Kim, B. G. Chae, H. G. Kim, S. Jun, S. Hwang, E. Lee, and J. C. Ye, “Deep learning stem-edx tomography of nanocrystals,” Nat. Mach. Intell. 3(3), 267–274 (2021). [CrossRef]
35. Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015). [CrossRef]
36. X. Sui, Q. Wu, J. Liu, Q. Chen, and G. Gu, “A review of optical neural networks,” IEEE Access 8, 70773–70783 (2020). [CrossRef]
37. X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361(6406), 1004–1008 (2018). [CrossRef]
38. L. Li, L. Zhu, Q. Zhang, B. Zhu, Q. Yao, M. Yu, H. Niu, M. Dong, G. Zhong, and Z. Zhou, “Miniaturized diffraction grating design and processing for deep neural network,” IEEE Photonics Technol. Lett. 31(24), 1952–1955 (2019). [CrossRef]
39. H. Chen, J. Feng, M. Jiang, Y. Wang, J. Lin, J. Tan, and P. Jin, “Diffractive deep neural networks at visible wavelengths,” Engineering 7(10), 1483–1491 (2021). [CrossRef]
40. C. Qian, X. Lin, X. Lin, J. Xu, Y. Sun, E. Li, B. Zhang, and H. Chen, “Performing optical logic operations by a diffractive neural network,” Light: Sci. Appl. 9(1), 59 (2020). [CrossRef]
41. S. Watanabe, T. Shimobaba, T. Kakue, and T. Ito, “Hyperparameter tuning of optical neural network classifiers for high-order gaussian beams,” Opt. Express 30(7), 11079–11089 (2022). [CrossRef]
42. Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y. Chen, P. Chen, G. Jo, J. Liu, and S. Du, “All-optical neural network with nonlinear activation functions,” Optica 6(9), 1132–1137 (2019). [CrossRef]
43. C. Liu, Q. Ma, Z. J. Luo, Q. R. Hong, Q. Xiao, H. C. Zhang, L. Miao, W. M. Yu, Q. Cheng, L. Li, and T. J. Cui, “A programmable diffractive deep neural network based on a digital-coding metasurface array,” Nat. Electron. 5(2), 113–122 (2022). [CrossRef]
44. J. Zi, Y. Li, X. Feng, Q. Xu, H. Liu, X. Zhang, J. Han, and W. Zhang, “All-function terahertz waveplate based on all-dielectric metamaterial,” Phys. Rev. Appl. 13(3), 034042 (2020). [CrossRef]
45. T. Wu, X. Zhang, Q. Xu, E. Plum, K. Chen, Y. Xu, Y. Lu, H. Zhang, Z. Zhang, X. Chen, G. Ren, L. Niu, Z. Tian, J. Han, and W. Zhang, “Dielectric metasurfaces for complete control of phase, amplitude, and polarization,” Adv. Opt. Mater. 10(1), 2101223 (2022). [CrossRef]
46. Y. Gao, S. Jiao, J. Fang, T. Lei, Z. Xie, and X. Yuan, “Multiple-image encryption and hiding with an optical diffractive neural network,” Opt. Commun. 463, 125476 (2020). [CrossRef]
47. W. Zhu, Y. Fan, R. Yang, G. Geng, Q. Fu, C. Gu, J. Li, and F. Zhang, “Polarization-multiplexed silicon metasurfaces for multi-channel visible light modulation,” Adv. Funct. Mater. 32(22), 2200013 (2022). [CrossRef]
48. S. Zheng, S. Xu, and D. Fan, “Orthogonality of diffractive deep neural network,” Opt. Lett. 47(7), 1798–1801 (2022). [CrossRef]
