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Abstract
In light of pending capacity crunch in information era, orbital-angular-momenta-carrying vortex beams are gaining traction thanks to enlarged transmission capability. However, high-order beams are confronted with fundamental limits of nontrivial divergence or distortion, which consequently intensifies research on new optical states like low-order fractional vortex beams. Here, we experimentally demonstrate an alternative mean to increase the capacity by simultaneously utilizing multiple non-orthogonal states of structured light, challenging a prevailing view of using orthogonal states as information carriers. Specifically, six categories of beams are jointly recognized with accuracy of >99% by harnessing an adapted deep neural network, thus providing the targeted wide bandwidth. We then manifest the efficiency by sending/receiving a grayscale image in 256-ary mode encoding and shift keying schemes, respectively. Moreover, the well-trained model is able to realize high fidelity recognition (accuracy >0.8) onto structured beams under unknown turbulence and restricted receiver aperture size. To gain insights of the framework, we further interpret the network by revealing the contributions of intensity signals from different positions. This work holds potential in intelligence-assisted large-capacity and secure communications, meeting ever growing demand of daily information bandwidth.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There is a large number of ongoing research in the free-space optical (FSO) communication community on the exploration of utilizing various physically distinguishable degrees of freedom of light, e.g. amplitude, phase, frequency, polarization, aiming to ease the bandwidth crunch due to exponentially growing data traffic in the information age [1,2]. As a special member of structured light, vortex beams carrying orbital angular momenta (OAM) are intensively investigated due to the unlimited Hilbert space they provide [3]. Such OAM modes are usually characterized by their topological charge $\ell $, a single quantum number that can be any integer number in theory [4]. Till now, vortex beams have expedited the advances from optical trapping [5], imaging [6], metrology [7] to quantum information processing [8].
Although arbitrary orders of OAM modes are available in principle, some challenging issues remains in terms of generation, propagation and detection. In practical communication scenarios, one can only generate finite amount of OAM modes regardless of using spatial light modulator [9], or laser cavity [10]. Higher order OAM modes are endowed with a larger effective beam width and they diverge faster and are more prone to be affected by turbulence than lower order ones, leading to nontrivial channel cross-talk and power loss [2]. Besides, the comparison results show that only OAM-based multiplexing may not outperform both conventional line-of-sight multi-input multi-output transmission and spatial-mode multiplexing [11,12]. Therefore, other than only focusing on eigen modes with different topological charges, to further expand the available state space, recent developments highlight communication systems leveraging low-order fractional OAM beams [11,13–17], Laguerre-Gaussian (LG) modes with different radial indices [18–20], multi-vortex geometric modes [21] and novel broad bandwidth OAM beams [22], to name a few. In particular, fractional vortex beams with edge dislocations can greatly increase the communication capacity, manifested in a recent work by transferring information via OAM modes with minimum interval $\mathrm{\Delta }\ell $ = 0.01 [17]. However, a bottleneck of fractional OAM modes falls into propagation instability, which poses another challenge for communications [13,23]. Additionally, LG modes with nonzero radial indices and other crafted OAM modes only offer limited improvement of the available state space.
Recently, it is noteworthy that data-driven deep learning algorithms have emerged as an exceptional tool to solve difficult problems in photonics such as holography [24], microscopy [25], device design [26] and sensing [27]. They also advance in recognizing orthogonal OAM modes [28], HG modes [29], multi-singularities modes [30] and non-orthogonal fractional OAM modes [15], as a novel and effective demodulation technique compared to traditional counterparts (e.g. interferometer [31], customized computer generated holograms [32], geometric transformation [33,34]).
In this paper, we seek to utilize non-orthogonal structured beams from different categories as information carriers. We jointly use Laguerre-Gaussian (LG) modes [35], Hermite-Gaussian (HG) modes [36], Bessel-Gaussian (BG) modes [37], Airy vortex (AV) modes [38], SU(2) vortex modes [39] and Mathieu-Gaussian (MG) modes [40] altogether to transmit messages. Embracing their non-orthogonality not only unleashes tremendous state space in hand, but also enables us to avoid disadvantages brought by high-order modes. The key to harness them is the capability of resolving these states [41], in which we resort to end-to-end neural networks. We apply an adapted ResNet [42] to successfully classify 288 different single modes and 256 superposition modes using transfer learning technique. Two structured-beam-based communication schemes including mode encoding and mode shift keying (SK) are demonstrated as proof of principle, where a 256-grayscale Richard Feynmann portrait is successfully reconstructed with satisfactory image quality. To alleviate the ‘black-box’ nature of neural networks, we interpret the model by figuring out which and how the spatial intensity signals influence the final predictions. More importantly, we investigate more practical scenarios which incorporate: (i) atmosphere turbulence (AT) and (ii) limited receiver aperture into the system, where the proposed method is still feasible under unknown turbulence and with partial received signals due to the improved model generalization ability. This study liberates us from conventional limited spatial optical fields and paves the way towards next-generation high-capacity optical communications.
2. Methods
In addition to vortex beams carrying well-known OAM which are extensively investigated over the years, two-dimensional structured beams are rich in members. According to the Helmholtz equation (HE) or paraxial wave equation (PWE) at different coordinate systems, one can obtain many groups of beams propagating in free space such as HG beam (Cartesian coordinate), LG beam (polar coordinate) [1]. In the following, to validate our proposed FSO data transmission scheme, we experimentally utilize six categories of structured beams including LG, HG, BG, AV, SU(2), MG beams altogether as information carriers. We focus on the case of scalar optics without considering spatial polarization states of the beams. Generally, one can express the electric field component as $\psi ({{q_x},{q_y},z} )$ where $({{q_x},{q_y}} )\; $ are the transverse coordinates that vary with coordinate systems and z is the propagation axis. For two-dimensional HE, $\psi ({{q_x},{q_y},z} )$ satisfies [43]:
To increase the modal diversity, we solve Eq. (1) and Eq. (2) under different conditions (see analytical expressions for each category of beams in Supplement 1). For sake of a vivid illustration of these exotic beams, we simulate six examples in Fig. 1. For each beam, the three-dimensional intensity shows its corresponding evolution along the propagation direction from $z = 0$ to $z = L$ where $L$ equals to the Rayleigh distance for LG, HG, SU(2), MG beams and half the Rayleigh distance for BG, AV beams given the same beam width and optical frequency. The right panel for each subfigure displays the intensity and phase patterns at plane $z = 0$.
Fig. 1. The beam families that are exploited in this work. (a) LG mode with $p = 5$, $\ell = 6$. (b) HG mode with $m = 6$, $n = 5$. (c) BG mode with ${k_t} = 100$, $\ell = 6$. (d) AV mode with ${r_0} = 3.0 \times {10^{ - 5}}$, $\alpha = 0.4$. (e) SU(2) vortex mode with $Q = 6$, ${n_0} = 5$ and $M = 7$. (f) MG mode with ${k_t} = 3.0 \times {10^4}$, $m = 4$. Refer to Supplement 1 for the definitions of these parameters. For each subfigure, the left column displays the evolution of three-dimensional intensity (with opacity of 0.1) along propagation axis z. The right column depicts the intensity and phase at initial plane ($z = 0$).
Intuitively, the different categories endow us with rich available state spaces even only counting on low-order beams, where the key to harness them is how to decode the data from these non-orthogonal states.
The conceptual illustration of our system is illustrated in Fig. 2. We first collect the states via a conventional free space setup in Fig. 2(a). In particular, a continuous-wave light source (CNI Laser, MGL-III-532) is firstly expanded $8$-times in diameter. Then a half wave plate is used to tune the polarization of the incoming beam to match with the working polarization of the phase-only spatial light modulator (SLM, Hamamatsu, X13138-04, resolution of $1280 \times 1024$, pixel size of $12.5\mathrm{\;\ \mu m}$). Multiple computer-generated-holograms (CGHs) are loaded on the SLM in succession to modulate the fundamental Gaussian beam into different structures. The inset showcases a hologram corresponding to a superposition state under turbulence generated using a method from [44,45]. The modulated beam is then projected to the Complementary-Metal-Oxide-Semiconductor (CMOS) camera (AVT Mako G-131B, resolution of 1280 ${\times} $ 1024, pixel size of $5.3\mathrm{\;\ \mu m}$) plane through a 4-f system. Note the inserted polarizer is to adjust beam intensity to avoid the camera from overexposure. The iris selects the target first-order signals at the spatial spectrum plane. This experimental setup allows us to acquire large datasets efficiently via LabVIEW program control. As elucidated in Fig. 2(b), the obtained beam sets are preprocessed and fed into the decoding DNN. In the following sections, we thoroughly validate our method by considering various situations, including distinctive beam categories, either single or multiplexed, unknown AT and partial receiver aperture. The core to drive this work is the brute-force neural network, whose detailed architecture is shown in Fig. 2(c). The end-to-end network is called ResNet composing of 35 layers (adapted from ResNet34), which is well-known for the contribution in preventing gradient vanishing problem [42]. Given an intensity image with size of $224 \times 224$ pixels, a series of convolutional (${\times} 33$), pooling and nonlinear layers with skip connections are applied to extract the hierarchical high-dimensional features, which are then gathered by fully connected layers (${\times} 2$) to make a prediction. We identify the recognition process as a classification problem and optimize the DNN through standard error backpropagation, where the cross-entropy loss is calculated by
Fig. 2. Schematic of the experimental setup and workflow. (a) Beam preparation setup. BE, beam expander; HWP, half-wave plate; BS, beam splitter; SLM, spatial light modulator; P, polarizer; L1, lens with $f = 150\,\textrm{mm}$; L2, lens with $f = 75\,\textrm{mm}$; CMOS, camera. Inset, details of a hologram. The SLM and camera are monitored by the computer to work synchronously. (b) Once the datasets containing training, validation and test samples are prepared, they are fed into the deep neural network for further decoding. (c) The network architecture composed of 35 effective layers. Conv, convolutional layer; Pool, max or average pooling operation; ReLu, rectified linear units; Flatten, flatten operation; FCL, fully connected layer (with dropout of 0.5); Softmax, data transformation. Note the output classes may vary with different tasks (288 or 256).
3. Results
3.1 Single mode results
We first investigate the recognition capability of the DNN with respect to single modes. Note here “single” means that we do not superpose these structured light from different categories, therefore SU(2) modes are also treated as single modes. We record the model’s performance during updating via the visualization toolkit TensorBoard as shown in Fig. 3(a). For single modes, we classify 288 different structured beams, where there are 48 elements for each category (see Supplement 1 for detailed modal parameters). We then augment the dataset with different initial phases. Specifically, for each structured beam, we change the initial phase from 0 to 1.96$\pi $ with a step of 0.04$\pi $ and 42 of them are for training, 5 for validation and 3 for test. Consequently, there are overall 12096 (1440, 864) data samples within the training (validation, test) set and the structured beams in test set are never seen by the model during training and validation stages. To improve the model robustness and generalization ability, random rotation and horizontal flip are also introduced into the input intensity images [50]. After hundreds of iterations, the training loss and validation loss are both low enough (∼ 1 ${\times} {10^{ - 2}}$) and close to each other, showing that our model is not overfitted. The blue line in Fig. 3(a) denotes the validation accuracy evolution as the training proceeds. Here accuracy $\mathrm{{\cal A}}$ is defined as the proportion of correctly classified beams over the whole set. We then test the obtained model and summarize the results in the confusion matrix of Fig. 3(b), where $\mathrm{{\cal A}} = 99.7\mathrm{\%}$ is achieved. Only $0.02\mathrm{\%}$ of the LG beams are incorrectly recognized as MG beams, indicating the model’s performance is encouraging for further applications.
Fig. 3. (a) Convergence plots of the model from single mode dataset. Left vertical axis, training and validation loss. Right vertical axis, validation accuracy. Inset, enlarged drawing of model log after convergence. (b) Confusion matrix for the recognition of experimental unseen structured modes. The diagonal entries represent the modes are correctly predicted.
Although we can attain impressive recognition results of optical beams through DNN, we are actually not aware how it can happen. That is often called “the black-box nature of neural networks” [51]. This problem is of relevance especially for medical image analysis, since a wrong diagnosis from an untrustworthy model can be detrimental to a patient [52,53]. In light of this, substantial efforts have been paid to the reasonable explanation and visualization of neural network models, which leads to some profound tools such as t-SNE [54], CAM [55] and Grad-CAM [56]. Similarly, unlocking how the model identifies different structured beams can benefit relevant applications such as communications and encryption. Here we conduct such an analysis through the help of Captum library [57]. In particular, we calculate the contributions of input intensities to the model’s final decisions. Figure 4(a) depicts six typical results, one for each category. Based on a well-trained model, the tagged numbers, e.g. $99.97\mathrm{\%}$, at the first and fourth columns of Fig. 4(a) represents the probabilities that predicted classes match with the corresponding labels. Then the integrals of the gradients of the model’s predictions with respect to the inputs are evaluated as shown in the second and fifth columns of Fig. 4(a). The greater integrated gradients are, the more important these pixels are to the correct outputs. Moreover, we use a sliding window with the size of $15 \times 15$ pixels and a stride of 5 along both intensity image dimensions to occlude the corresponding image pixels, so as to gauge the attributions of different spatial signals. Quantitative results are presented in the third and sixth columns of Fig. 4(a). As expected, the spatial regions with nontrivial intensities are more critical for the final results. The combination of integrated gradients and occlusion-based attribution could help us in structured-beam-based applications since in real-world scenarios, the receiver’s aperture is always limited in size and the information may not be fully captured especially considering beams’ divergence upon propagation [59]. These model interpretation analyses can remind us which parts of the signals are more important, whether the input modes are proper for the model and whether the predictions are convincing.
Fig. 4. (a) Attributions visualization results by interpreting the well-trained model. Two different methods are respectively used including integrated gradients and occlusion. The attributions in the 2nd, 3rd, 5th and 6th columns highlight the decisive intensity regions used for classification. (b) The conceptual illustration of structured-beam encoding scheme used for transmitting a grayscale image. Richard Feynman portrait reuse is officially permitted by Mark Roesler (CMGWorldwide.com) [58].
We then demonstrate a structured-beam encoding scheme as shown in Fig. 4(b). Specially, we transform the data into independent beams, and then transmit the beams to the receiver in succession followed by decoding via the DNN. For an 8-bit-depth Richard Feynman portrait, 256 distinct beams from six categories (48 of LG, 48 of HG, 48 of BG, 48 of AV, 48 of SU(2), 16 of MG beams, detailed in Supplement 1) are used to represent 256 gray levels. In our experiments, the resolution of the image is $100 \times 100$, which corresponds to a structured light sequence with the length of 10000. After about 30 minutes of transmission and 75 seconds of decoding, the reconstructed image is displayed in Fig. 4(b). The structural similarity index measure (SSIM) as image quality metric is calculated as 0.99, indicating a decent match with the original image and also primitively demonstrating the feasibility of communication using non-orthogonal structured beams.
3.2 Multiplexed mode results
Although the mode encoding scheme in Fig. 4 works, it is obviously less efficient due to necessitating substantial well-defined modes. Therefore, it is interesting to explore whether mode shift keying (SK) system is feasible, where each mode is identified as a data symbol [60]. More specifically, we use 8 structured beams (2 of LG, 2 of HG, 1 of BG, 1 of AV, 1 of SU(2), 1 of MG beams, detailed in Supplement 1) to represent 8 bits as shown in Fig. 5(a). For instance, given a gray value of 92, one can treat it as a binary vector as (0, 1, 0, 1, 1, 1, 0, 0), then each binary number is distributed to its corresponding structured beam as a superposition coefficient. The eight beams are superposed coherently which leads to the multiplexed state. Mathematically, the final state ${E_f}$ is defined as
where ${b_i}$ is the $i$th binary coefficient and ${E_i}$ is the $i$th mode base. As shown in the subfigures of Fig. 5(a), the intensity and phase patterns of a multiplexed state can be exotic, thereby difficult to be decoded using conventional methods.Fig. 5. (a) The encoding principle of structured-beam-based shift keying scheme. Eight beams from six categories are adopted as 8 bits for 256 gray levels $({{2^8} = 256} )$. (b) The intensity pattern evolutions of four different modes at different atmospheric turbulences. These modes represent gray values of (b1) 8, (b2) 29, (b3) 69 and (b4) 455. At around AT5, one can hardly recognize these modes via naked eye.
To make the system more practical and the results convincing, we first investigate how the AT affects the decoding performance. AT is ubiquitous caused by temperature and pressure’s inhomogeneities and is notoriously harmful to the free-space optical communication schemes thus a decoding technique robust against various and unknown AT is of significance. Here, we digitally simulate AT phase screens according to the modified von Karman model [61]. Specifically, the phase power spectral density (PSD) of each screen can be described as
Here we classify 256 different multiplexed beams and the dataset at each degree of AT is composed of $256 \times 50 = 12800$ samples, reserving $84\%$ ($10\%$, $6\%$) of which for training (validation, test). The model’s performance is evaluated based on accuracy metric $\mathrm{{\cal A}}$ and the results are summarized in Fig. 6. First, we train a model for each AT environment (therefore 8 neural network models in total) and assess the model over all blind test sets. In Fig. 6(a), the diagonal values represent the accuracies of the models on corresponding test set with the same turbulence while the nondiagonal terms are accuracies on test sets with different turbulences. Ideally, we hope the accuracy matrix including diagonal and nondiagonal elements is composed of all ‘1.00’ values, which represents the perfect generalization ability on various turbulences. It can be seen that a trained model performs remarkably well on data samples from equal $C_n^2$ but languishes on samples from different turbulences. In Fig. 6(b), blue bars are calculated based on 7 test sets with unknown turbulences while green bars are calculated based on all 8 test sets. From the results, we can conclude that the generalization ability of models trained on weak AT environments are better than models obtained from strong AT environments. Even so, the recognition accuracies are not outstanding. To improve model’s generalization ability on unknown data samples with other turbulences, we adopt dataset scaling technique, which is proven to be effective in advancing model’s performance in neural network studies. In other words, we now construct datasets with samples under two or more AT environments to retrain the neural networks. Indeed, Figs. 6(c) and (d) reveal that the accuracies are greatly enhanced. In particular, four new models are obtained under 2 (AT1 + AT5), 3 (AT1 + AT4 + AT6), 4 (AT0 + AT2 + AT4 + AT7) and 5 (AT0 + AT2 + AT4 + AT6 + AT7) types of ATs. The comparison of Figs. 6(a)-(b) and Figs. 6(c)-(d) indicates that, as the diversity of data samples increases, the model’s generalization ability on various environments also increases. With the enhancement of generalization ability, it is more favorable for our model to be utilized in real-world scenes.
Fig. 6. Experimental performance of the trained models on test sets with various turbulences. (a) Accuracy matrix of models trained on samples with single turbulence. (b) Statistical results based on (a). (c) Accuracy matrix of models trained on samples with multiple turbulences. (d) Statistical results based on (c).
Thanks to the powerful recognition capability of DNN, we then demonstrate a structured-beam-based shift keying system as illustrated in Fig. 7(a). We also transmit an 8-bit Richard Feynman gray portrait like the case in Fig. 4(b). Each gray value, e.g. 152, is encoded as a superposition state of 8 various structured beams, which are the pseudo-bases, and then the multiplexed state propagates 500 m under turbulence to the receiver. The distorted signals are injected into the trained DNN for decoding. All 10000 multiplexed states are received in a time-varying manner. Similarly, we also investigate the image transmission accuracy under unknown AT environments. In particular, the transmitted beams carrying information are distorted by AT3 but the evaluated models only see samples from AT0, AT1 + AT5, AT1 + AT4 + AT6, AT0 + AT2 + AT4 + AT7 and AT0 + AT2 + AT4 + AT6 + AT7 respectively when training. We quantitatively assess the 5 models. The SSIM values are analyzed in Fig. 7(b). Not surprisingly, as the training dataset scales, the image fidelity with respect to the ground truth increases. The lowest bit error rate (BER) in this case can be $32\%$. Notably, the SSIM value tends to approach saturation at around 0.75, we argue that one can further improve it by adding more data into the training set or adding network model parameters. Recently some Foundation models like GPT-3 [63] with at-scale parameters (hundred million level) have achieved state-of-the-art performances for various image and language tasks. Thus it is believed that as the model enlarges, the inner complexity as well as model expression ability will get better.
Fig. 7. Experimental results of structured-beam-based shift keying scheme. (a) The conceptual workflow of the data transmission system under unknown turbulences. (b) The reconstructed image quality versus training dataset. Here the structured beams carrying information are distorted by AT3 while the evaluated models are trained over beams distorted by other ATs. The image quality is enhanced as the diversity of data samples in terms of turbulence increases.
Apart from AT, another pronounced challenge for optical communication is the mismatch between the receiver aperture and the beam size [2,64]. Due to the diffraction limit or the geometric divergence nature of a transmitted beam, the required whole-beam receiver aperture can be extremely large, e.g. meter-scale in [65]. To tackle this problem, in the following, we examine the model’s recognition performance for input modes with limited aperture. As depicted in Fig. 8(a), the transmitted beam diverges upon propagation. At the receiver, we suppose the size of an ideal aperture that collects all beam’s information is constantly D and the size of a cut-off aperture that filters out accessible signals is d. Ten different cases are quantitatively considered from $d/D = 0.1$ to $d/D = 1$. When $d/D = 1$, the system degenerates to the ideal case that is generally assumed in recent works [11,14–18,22,28,29,65–67]. Notably, Fig. 8(b) denotes the results from a model trained with data samples under $d/D = 1$. In general, the decoding model provides superior performance on samples under larger apertures. When $d/D < 0.4$, the recognition accuracies $\mathrm{{\cal A}} < 0.5$. When $d/D > 0.6$, the accuracies $\mathrm{{\cal A}}$ saturate at around $1.0$. Recalling the interpretation results reported in Fig. 4(a), the intensity signals at differently locations contribute differently to the decision, which suggests that one may not need to capture every part of the signals to recognize beams correctly. Inspired by dataset scaling experiments in Fig. 6 and Fig. 7, we retrain a model using data samples under $d/D = 0.5$ and $d/D = 1$ and present the blind test results in Fig. 8(c). Indeed, now the accuracies $\mathrm{{\cal A}} > 0.9$ when $d/D > 0.5$. We continue to add data $({d/D = 0.1} )$ into the previous training set and the corresponding results are shown in Fig. 8(d). Although the accuracy $\mathrm{{\cal A}} > 0.9$ at $d/D = 0.1$, the generalization ability to other small apertures e.g. $d/D = 0.15$ is still poor. It is reasonable because as shown in Fig. 8(a), when $d/D\sim 0.1$, too much intensity information is lost so that the performance of DNN collapses. The results in Fig. 8 remind us that in practice when the receiver collects very little information, the DNN’s predictions are generally not credible. While a proper aperture that takes sufficient signals can turn the model reliable again.
Fig. 8. Influences of receiver aperture sizes on the performances of the trained models. (a) Illustration of the mismatch between beam size and receiver size. Red circle denotes full receiver aperture size that collects all intensity signals. Orange circle denotes partial (effective) receiver aperture size that loses part of intensity signals. (b)-(d) The classification accuracies on different partial receivers using different models. The performance is moderately improved as the diversity of data samples in terms of receiver size increases.
4. Discussions and conclusions
The above results and analyses exhibit the potential of deep neural networks in unlocking non-orthogonal spatial beams for communications (also see Supplement 1 for intensity-based modal cross-talk analyses). To improve this work, in future research, it is meaningful to investigate how different modes diverge differently and elaborately pick out those low order beams with less divergence to constitute a structured beam library, which one can use to extend the communication distance [68]. Second, when one uses bigger model architecture to improve the generalization ability for higher accuracies, the decoding speed and energy cost could be a sacrifice. To avoid this, model pruning [69] and knowledge distillation [70] can be applied to compress the network at the software level. Besides, the novel all-optical or optoelectronic neural network hardwares can be exploited as optimal decoding platforms [71]. Moreover, in our setup, SLM is used for beam preparation which is low in refresh rate resulting in ${\sim} \; 30$ minutes to send an image, therefore replacing it with a digital mirror device is definitely helpful to improve the speed. As for LG modes with the same values of p and $|\ell |$, their intensity signals are the same (degenerate intensity), to effectively maintain the recognition robustness, one should break the symmetry before intensity detection through special diffraction [72], modulation [73], interference [28], etc. Instead of weak diffusing like atmospheric turbulence, what remains enticing is to explore the effectiveness and novel functionalities e.g. encryption in strong scattering environments including ground glasses and multimode fibers [74].
To conclude, in this work by revisiting the rich families of structured light, we experimentally demonstrate that, one can utilize multiple categories of (low-order) structured light simultaneously as information carriers for large-capacity optical communications. A well-trained model can identify these non-orthogonal beams with an accuracy of $> 99\%$ in a short time (∼ 7.5 ms), loosening the rigorous alignment need in experiments. Model interpretation analysis is conducted to identify nontrivial signals. In particular, structured-beam-based 256-ary mode encoding and SK schemes are successively verified. For the SK experiments, we investigate the model’s generalization ability from two aspects: unknown turbulences and partial signals input. At an AT of $1.0 \times {10^{ - 14}}\; {\textrm{m}^{ - 2/3}}$, the neural network trained on samples with other ATs can correctly recognize $85.3\%$ of samples in the blind test set without any adaptive optics compensations, resulting in 0.75 of SSIM value for an image transmission experiment. For a receiver aperture with finite size of $d/D \ge 0.5$, the accuracy is still as high as more than $0.8$. Our platform is highly scalable and readily be extended to other wavelengths including millimeter wave, terahertz wave and acoustic waves. The insights provided by this study may inspire further investigations on multi-channel communications, high-security encryption, novel information processing, etc.
Funding
National Natural Science Foundation of China (61975087).
Acknowledgments
H. Wang thanks Jia Guo and Zhensong Wan for useful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data and codes underlying the results presented in this paper can be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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