## Abstract

Beyond orbital angular momentum of Laguerre-Gaussian (LG) modes, the radial index can also be exploited as information channel in free-space optical (FSO) communication to extend the communication capacity, resulting in the LG- shift keying (LG-SK) FSO communications. However, the recognition of radial index is critical and tough when the superposed high-order LG modes are disturbed by the atmospheric turbulences (ATs). In this paper, the convolutional neural network (CNN) is utilized to recognize both the azimuthal and radial index of superposed LG modes. We experimentally demonstrate the application of CNN model in a 10-meter 768-ary LG-SK FSO communication system at the AT of $C_n^2$ = 1*e*-14 m^{−2/3}. Based on the high recognition accuracy of the CNN model (>95%) in the scheme, a colorful image can be transmitted and the peak signal-to-noise ratio of the received image can exceed 35 dB. We anticipate that our results can stimulate further researches on the utilization of the potential applications of LG modes with non-zero radial index based on the artificial-intelligence-enhanced optoelectronic systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

To meet the ever-increasing demand for high capacity optical devices in information optics, various physically-orthogonal dimensions of light (e.g. the amplitude, phase, wavelength, polarization) have been employed as information carriers. A special utilization is the physical dimension of orbital angular momentum (OAM), carried by the light beam featuring a helical phase term of exp(*ilφ*) (*l* represents azimuthal index, *φ* is azimuthal angle) [1]. Due to the theoretically unlimited orthogonality, it has been utilized to boost the information capacity in quantum information optics [2], holography [3,4] and optical communication [5,6].

The current OAM-based optical communication systems can be divided into two categories, including OAM division multiplexing (OAM-DM) system where separate OAM beams are treated as individual signal carriers [7–10] and OAM shift keying (OAM-SK) system where each OAM state can represent a data symbol [11–13]. Indeed, OAM is only a subspace of well-known Laguerre-Gaussian modes, which own two degrees of freedom: an azimuthal index *l* and a radial index *p* [14]. More recent works have reported that the radial degree of freedom could also increase the bandwidth capacity in optical communication, resulting in the LG modes free-space optical (FSO) communication systems [15,16].

The OAM demodulation technique plays a key role in OAM FSO communication system. The common OAM detection methods range from the modes converter combining with the modified Mach-Zehnder interferometer (MZI) [17]. inverse modes conversion based on special-designed computer generated holograms (CGHs) [18], the Surface-Plasmon-Polaritons-based OAM-sensitive nanophotonic structures [19], and the coordinates transformation scheme based on multi-plane phase modulation [20,21]. The recognition methods of the radial index of LG modes, either by more complicated CGHs [22] or bulky optical MZI systems based on fractional Fourier transformation [23,24], are similar as the previous methods [17,18] in principle. However, the modes detection range and diffraction efficiency of the above-mentioned schemes are quite limited, and the performance degrades seriously due to the misalignments of optical setups and the phase distortion caused by the environmental perturbation. As such, the ubiquitous atmospheric turbulences (ATs) become a tough issue in the practical hybrid complex spatial-mode-involved FSO communications [25].

The adaptive optics (AO) with intelligent algorithms, such as Gerchberg-Sxton (GS) [26] and stochastic-parallel-gradient-descent (SPGD) [27] algorithms, is a potential solution to cope with the effects of turbulence [28]. However, the requirements of wavefront compensating devices and probe beams increase the cost and system complexity. Moreover, multiple iterations are needed in the common GS and SPGD algorithms, resulting in a long processing time. And the lack of learning ability in these algorithms even leads to the failing of convergence in the processes of iterative calculations. In the past several years, benefiting from the capability to extract and recognize the intrinsic features of the input raw images, convolutional neural network (CNN), as a model of deep learning [29], has been applied in OAM-based optical communication systems [30–39]. The intensity patterns of integer OAM modes [38], fractional OAM modes [40], Hermite-Gaussian modes [36] and other hybrid complex spatial modes [39,41] have been successfully categorized and demodulated with high accuracies. Particularly, CNN could be used to extract and compensate AT from the received intensity patterns [42], showing great demodulation capability especially in AT-involved FSO communication systems. Up to now, such CNN-based OAM-SK FSO communication systems have developed from 8-ary [43], 16-ary [37,44,45] to 32-ary [46]. In this paper, we have experimentally demonstrated the LG shift keying (LG-SK) FSO communication to transmit a colorful image. Based on the remarkable ability of spatial mode recognition of CNN, the alphabet in our scheme can be as large as 768 consist of 24 different LG modes. Meanwhile, the issue of the AT has also been considered.

## 2. Conceptual illustration of CNN-based LG-SK FSO communication

The conceptual illustration of CNN-based LG-SK FSO communication is shown in Fig. 1. More than the azimuthal degree of freedom (*l*) in conventional OAM optical communications, the radial index (*p*) of the LG modes is also implemented as information carrier in our scheme. In particular, the information is encoded into the radial index (*p*) and azimuthal index (*l*) of coherent superposed LG modes, which transmit in free space with certain AT in time sequence. In the demodulation processes, the CNN is utilized to recognize the hybrid spatial modes. The training data are obtained from the Charge Coupled Device (CCD) camera in Fig. 1(a). As such, the efficient performance of the CNN leads to the increment of the accuracy. More notably, no rigorous requirements on the alignment in the acquisition of the data result in the improvement of the robustness compared to the detection methods based on mode conversion. Finally, as shown in Fig. 1(b), the target information can be decoded based on the radial and azimuthal index of the hybrid spatial modes acquired from the CNN.

## 3. Experimental setup for the generation of the complex-amplitude field

As introduced below, LG mode, a common type of OAM beam, is implemented as information carrier in our scheme of optical communication. In the cylindrical coordinate (*r*, *θ*, *z*), the complex field of a LG mode at the waist plane (*z*=*0*) can be described as:

*l*and

*p*is the topological charge and the radial index of LG mode, respectively.

The schematic diagram of the experimental setup is illustrated in Fig. 2(a). A 532 nm beam from a continuous-wave laser, passes through a half-wave plate (HWP) in combination with a polarization beam-splitter (PBS), which is used to coarsely adjust the laser power. With the adjustment of a beam expansion system consisting of two convex lenses with 50-mm- and 300-mm- focal lengths, the light beam with horizontal polarization direction is illuminated on the phase-only holograms imprinted on the spatial light modulator (SLM, Holoeye Pluto-2-NIR-011). To achieve the complex amplitude single or superposed LG modes, we adopted Double-Fourier transform optical setup in our scheme (Fig. 2(b)) [47] . Notably, to spatially isolate the encoded complex-amplitude field in the back focal plane of L3 (u-v plane in Fig. 2(b)), the carrier phase modulation 2π(u_{0}x + v_{0}y) with spatial frequencies (u_{0}, v_{0}) has been added on the phase-only computer-generated-hologram (CGH) loaded on SLM. This allows the separation of Fourier spectrum of the CGH. As such, a spatial filter is put in the focal plane of L_{3} to eliminate the noise from the other diffraction orders. Finally, the target complex field can be recovered from the first order of the Fourier series of the phase-only CGH. The intensity of target filed after a 10-meter propagation distance is eventually collected by the Charge Coupled Device (CCD) camera (Thorlabs DCU224C).

Notably, the impact of AT is critical in the study of optical communication. In our experiment, the modified Von Karman turbulence spectrum model, containing both large-scale and small-scale influences, is adopted to analyse the amplitude fluctuation of the spatial modes. The atmospheric refractive index power spectrum is described as:

where ${k_l} = 5.92/{k_0},$*0≤k≤∞*means this spectral model is applicable to all wavenumber scenarios. $C_n^2$, the refractive index structure parameter, has a typical value from 10

^{−17}to 10

^{−13}m

^{−2/3}, standing for week turbulence and strong turbulence. As such, the additional fluctuation phases embedded into the original hologram could represent the AT. As shown in the first column of Fig. 3, the refractive index structure parameter with larger $C_n^2$ results in a disturbed phase term with higher spatial frequency. The intensity distributions of LG modes with azimuthal index varying from 1 to 3 (

*l*=1,2,3) and radial index from 1 to 8 (

*p*=1-8) after a 10-meter- propagation distance have been shown in the third column of Fig. 3. While AT (= $C_n^2 = 1e - 16\textrm{ }{m^{ - 2/3}}$) has a negligible influence on the intensity distributions of a single LG mode, the detection of the radial and azimuthal index after AT ($C_n^2 = 1e - 14{m^{ - 2/3}}$) calls for a robust method.

Next, in order to improve the efficiency of information transmission, a superposition of multiple LG modes can be generated simultaneously to implement LG-SK FSO communications. The complex electrical field of such coherent superposition can be expressed as

To achieve the complex-amplitude field $u_{_{\textrm{LG}}}^s({r,\theta } )= {u_{\textrm{LG}}}\left( {\sqrt {{x^2} + {y^2}} ,\arctan \frac{y}{x}} \right)$ $= a({x,y} )\exp [{i\phi ({x,y} )} ]$, where $a(x,y)$ and $\phi (x,y)$ represents amplitude and phase respectively, the phase-only CGH can be expressed as

The Fourier series in the domain of $\phi $ can be expressed as

whereFinally, the complex-amplitude field can be recovered from the first-order term (*q*=1) if the identity $\textrm{c}_1^a = Aa$. Here, *A* is a positive constant.

As an example illustrated in Fig. 4(a), five LG modes (LG_{01}, LG_{10}, LG_{20}, LG_{21}, LG_{22}) are randomly selected and coherently superposed. The intensity patterns of the superposed modes influenced by AT with different refractive index structure parameters are listed here. In comparison to the negligible influence of the small AT on the transmission of a single LG mode, the intensity pattern of the superposed LG modes has an obvious distortion caused by the AT $C_n^2$ = 1*e*-16 m^{−2/3}. As illustrated in Fig. 4(b), the center of the intensity pattern remains moving with the increase of the $C_n^2$ (from 1*e*-16 m^{−2/3} to 1*e*-14 m^{−2/3}), suggesting the difficulty of modes detection in LG multiplexing optical communication based on the traditional inverse mode-conversion method.

## 4. CNN model for the recognition of superposed LG modes

Inspired by the amazing performance of CNN for image classification tasks, we have designed a CNN model to identify the hybrid spatial modes consisting of superposed coaxial LG modes in this work. Here, we set the radial index and the azimuthal index of the constituent LG modes ranging from 0-2, resulting in a whole spatial mode set (LG_{00}, LG_{01}, LG_{02}, LG_{10}, LG_{11}, LG_{12}, LG_{20}, LG_{21}, LG_{22}). Thus, the possible hybrid spatial modes have 2^{9}=512 categories. Using the experimental setup in Fig. 2(a), two data sets of intensity distributions of the hybrid spatial modes have been acquired at different ATs ($C_n^2 = 1e - 16\textrm{ }{m^{ - 2/3}}$ and $C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$), respectively. To improve the fidelity of CNN, huge amount of data should be collected to construct the data set. As such, we synchronized the loading frequency of holograms on SLM with the automatic exposure interval time of CCD camera. Finally, for each data set at a special AT, the total number is 512×120 = 61440 patterns split into ∼80% for the training and ∼20% for the validation. As a set of examples, the 120 intensity patterns of the superposed modes (LG_{01}, LG_{10}, LG_{20}, LG_{21}, LG_{22}) at ATs ($C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$) have been listed in Fig. 5(a).

The proposed CNN architecture is illustrated in Fig. 5(b), which consists of an input, output and 5 hidden layers. Firstly, the images with a size of 256×256 is input into the first layer featuring 16 convolutional filters with 5 × 5 dimension. The next four layers have a convolutional filters of 3 × 3 dimension, while the number of each convolution filters is 32, 64, 126, 256, respectively. After passing through these 5 layers of convolution filters, the size of the image reaches 256×5×5. Then, output values of the feature extraction layers are flattened out and served as input to the fully connected classification layers. ReLU activation function is used and dropout in the fully-connected layers is added to prevent overfitting. The last layer provided the output with 512 nodes, one for each element corresponds to one state of multiplexing LG modes. Finally, the Softmax activation function guarantees that only one of these nodes is activated for corresponded single input after the training of the model.

The left panel of Fig. 5(c) shows the accuracy as the function of epoch in training process. Following by the dramatical increment in the first 200 epochs, the accuracies of modes recognition remain stable and achieved approximately 100% and 95% after multiple iterations under ATs of $C_n^2 = 1e - 16\textrm{ }{m^{ - 2/3}}$ and $C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$. Here, the mean time required for testing one pattern takes only 2.7 ms by using an Intel Xeon Platinum 8160 CPU and NVIDIA Titan RTX GPU, indicating that the CNN model can accurately and rapidly identify superposed LG modes. Besides, the right panel shows that loss will continuously decline by the epoch. Under ATs of $C_n^2 = 1e - 16\textrm{ }{m^{ - 2/3}}$, the loss function tends to be stable and reaches around 0 when the training iterations reach 400. Notably, with the increase of the ATs, it needs more iterations (400) to reduce the loss value to 0 ($C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$).

## 5. CNN-based LG-SK FSO communication

To further demonstrate the availability of CNN-based method in exploiting superposed LG modes in optical communication, we have experimentally constructed a 10-m free-space communication link to transmit an 8-bit colorful image. The encoding process is illustrated in Fig. 6(a). The colorful image has been divided into three color channels (R, G and B), denoted by the radial index of the LG modes (*p*=1, 2 and 3). Additionally, the pixel values ranging from 0 to 255 have been mapped to octets, which are represented by the azimuthal index of LG modes (*l*=1-8). For example, the pixel value 125 in the red channel can be denoted by an octet (10010011), resulting in the special superposed LG modes set (LG_{11}, LG_{41}, LG_{71}, LG_{81}). Moreover, the green pixel with a value of 193 and blue pixel with a value of 52 can be represented by the LG modes set (LG_{12}, LG_{32}, LG_{52}, LG_{62}, LG_{82}) and (LG_{13}, LG_{43}, LG_{73}), respectively. Finally, the holograms have been loaded on SLM in time sequence. As such, to transmit a colorful image with a size of 128×128 in our scheme, the number of the holograms is 128×128×3 and the total transmission time is ∼13.7 min. After a distance of 10 meters in our lab, the images captured by CCD camera were input into the CNN. The colorful picture can eventually be reconstructed with the information of color and pixel value decoded by the CNN.

Notably, the hybrid spatial modes set for the current architecture have 3×2^{8}=768 categories. At a special AT ($C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$), the total number is 768×120 = 92160 patterns. As shown in Fig. 7(a), when the number of a special superposed spatial modes for training (training number) increases from 20 to 100, the decoded image gradually approaches the ground truth. Specifically, the peak signal-to-noise ratio (PSNR) [37,48] is used as a criterion to evaluate the quality of the image in Fig. 7(b). When the training number is 60, the PSNR reaches 34.16 dB, which is comparable to the typical values for the image compression [49]. Moreover, the accuracy and loss of the CNN model are also analyzed. Similar to the curve of PSNR, the accuracy rate improves dramatically (from 48.84% to 85.12%) when the training number increases from 20 to 60, and it finally reaches 95.12% for a training number of 100. Furthermore, all of the loss function can reach around 0 when the training number is larger than 60 while it needed more training iterations for the smaller training number (Fig. 7(c)). To illustrate the remarkable performance of the CNN model, the confusion matrix of the CNN model with a training number of 100 is given in Fig. 7(d).

In our experiment, the size of receiving aperture determined by the CCD camera limits the propagation distance to 10-meter [16]. To analyze the influence of propagation distance on the CNN model, we achieve the training (76800 patterns) and test data (15360 patterns) through numerically calculating the intensity patterns of the superposed LG modes at different propagation distances (from 10 m to 100 m). As shown in Fig. 8(a), the distortion of the intensity patterns increases with the propagation distances under the same AT ($C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$), resulting in the decrease of the accuracy of the CNN model from 95.26% to 70.92% (Fig. 8(b)). Then we also study the effect of AT on the accuracy at the propagation distance of 10 m. As mentioned in the previous section (Fig. 4(a)), the additional random phase distributions representing the ATs weakens the differences among the intensity distributions of the different superposed LG modes, especially for the stronger ATs. As a result, the curve in Fig. 8(c) shows that the accuracies with $C_n^2$ varies from 1e-16 m^{−2/3} to 1e-14 m^{−2/3}can achieve over 95%, but with the further increase of $C_n^2$ to 5e-14 m^{−2/3} and 1e-13 m^{−2/3}, the accuracies decrease to 87.62% and 80.50%, respectively.

## 6. Conclusion and discussion

In this paper, a CNN-based method has been proposed to recognize both the radial index and azimuthal index of the superposed LG modes. And we have experimentally demonstrated a 10-meter 768-ary LG modes multiplexing optical communication system to transmit a colorful image. Even at an AT of $C_n^2 = 1e - 14\textrm{ }{m^{ - 2/3}}$, the CNN-based method exhibits the advantages of high-accuracy (> 95%), fast speed (< 2.7 ms), and no strict requirements on alignments in the process of mode detection, resulting in a received image with high PSNR (>35 dB). In order to further improve the capacity of such system, the LG modes with negative azimuthal index should also be exploited as information channels without an increase of the intensity area. The input LG modes can be converted into Hermitian-Gaussian modes by a placing π/2-mode-converter [50] before the CCD camera, breaking the limitation caused by the same intensity distributions of the LG modes with inverse azimuthal index and the same radial index. In addition, some phase compensation methods, such as GS algorithm [51–54] and self-adaptive methods [44,55], should be adopted at a stronger AT. Last but not least, we notice that while the decoding process is fast, the processes of data collection and training for the current CNN model is time-consuming (∼ 2 days), which is caused by the slow response speed of the digital devices such as CCD camera and computer. In the future, the all-optical diffractive neuron networks (DNN) [56–58] can be introduced to solve this urgent problem in practical applications.

## Funding

Shanghai Rising-Star Program (20QA1404100); National Natural Science Foundation of China (62005164); Zhangjiang National Innovation Demonstration Zone (ZJ2019-ZD-005).

## Disclosures

The authors declare no conflicts of interest.

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