Abstract

A novel joint atmospheric turbulence (AT) detection and adaptive demodulation technique based on convolutional neural network (CNN) are proposed for the OAM-based free-space optical (FSO) communication. The AT detecting accuracy (ATDA) and the adaptive demodulating accuracy (ADA) of the 4-OAM, 8-OAM, 16-OAM FSO communication systems over computer-simulated 1000-m turbulent channels with 4, 6, 10 kinds of classic ATs are investigated, respectively. Compared to previous approaches using the self-organizing mapping (SOM), deep neural network (DNN) and other CNNs, the proposed CNN achieves the highest ATDA and ADA due to the advanced multi-layer representation learning without feature extractors designed carefully by numerous experts. For the AT detection, the ATDA of CNN is near 95.2% for 6 kinds of typical ATs, in cases of both weak and strong ATs. For the adaptive demodulation of optical vortices (OV) carrying OAM modes, the ADA of CNN is about 99.8% for the 8-OAM system over the computer-simulated 1000-m free-space strong turbulent link. In addition, the effects of image resolution, iteration number, activation functions and the structure of the CNN are also studied comprehensively. The proposed technique has the potential to be embedded in charge-coupled device (CCD) cameras deployed at the receiver to improve the reliability and flexibility for the OAM-FSO communication.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

To progressively improve the data transmission capacity of the free-space optical (FSO) communication, the amplitude, phase, wavelength and polarization of light fields have been fully exploited. Lately, the techniques based on the orbital angular momentum (OAM), one of the fundamental property of Laguerre Gaussian (LG) beams, have been proposed and attracted much attention [1]. Benefited from the theoretically unlimited range of available OAM states [2], the capacity of the FSO communication system is possible to be improved significantly by OAM division multiplexing (OAM-DM) [3–5] or OAM shift keying (OAM-SK) [6–8]. To be more specific, LG beams carrying diverse OAM states serve as channels to multiplex information streams in the OAM-DM. Besides, the information is modulated into LG beams carrying diverse OAM modes, which is considered as a new modulation format in the OAM-SK. The OAM-SK-FSO communication system has unique advantages in the equipment cost, transmission distance, photon efficiency and information security, which has the potential to be applied in the cost-effective emergency communication and the middle-distance FSO communication such as the deep-space and near-Earth communications and the high-dimensional quantum key distribution (QKD). Moreover, with the development of the spatial light modulator (SLM), the capacity of the OAM-SK communication system is able to be further improved.

However, the performances of the FSO-OAM communication systems expressly subject to the influence of the atmospheric turbulence (AT) [9], which randomly disturbs the wave front phase, damages the orthogonality among distinct optical vortices (OV) beams and further induces severe signal crosstalk [10]. To enhance the reliability and flexibility of the FSO-OAM communication system, it is indispensable to sense link impairments and detect the AT information of the free-space turbulent channel in order to select the appropriate modulation format, specify OAM modes interval, determine transmission distance without relay and estimate quality of transmission (QoT). However, to the best of our knowledge, few work about the AT detection for the OAM-FSO communication has been reported.

Besides the AT detection, the OAM-SK demodulation is also important. But the traditional coherent demodulation is inefficient and extremely sensitive to arriving of arriving (AOA) fluctuations and the beam wander [10]. Thus, an efficient artificial neural network (ANN) based demodulation technique is proposed, where intensity images of received LG beams are directly recognized and the corresponding OAM mode information is successfully obtained [11]. However, ANN may not be suitable to directly process original images in their raw form (pixel points of the image) and generally feature extractors need to be designed by numerous experts. Therefore, more effective and appropriate image recognition techniques should be proposed to improve the performance of the OAM-SK demodulation.

As the powerful interdisciplinary science combined with the mathematics, computer and biology science, machine learning recently has been successfully applied in computer visual [12], natural language processing [13], data mining [14] as well as the optical communication [15–18], where we have already utilized convolutional neural network (CNN) for optical performance monitoring [17, 18]. As a core member in the machine learning community, CNN has made great breakthrough in the image recognition [19, 20]. Due to the advanced multi-layer representation learning, CNN is good at directly recognizing raw images and discovering intrinsic features of input images without the careful feature engineering, which makes CNN distinct from the conventional machine learning techniques [21].

There are some attempts used the conventional machine learning techniques and the deep learning to detect the OAM modes. In the paper [11], the self-organizing mapping (SOM) network was creatively adopted to identify the intensity images of the 16 kinds LG beams transmitted over the 3-km turbulent link. Then the same group used this SOM technique to enable the 143-km free-space transmission of OAM modes of light [22]. Meanwhile, the research about transmitting more OAM modes was also reported in [23], where multi-hidden-layer deep neural network was exploited to differentiate 110 OAM modes. With the development of the deep leaning, convolutional neural networks (CNN) have also been introduced to extract the multiplexed OAM modes information of OV beams with the higher accuracy than conventional machine learning algorithms, such as ANN, K-nearest neighbor (KNN) and naive Bayes classifier (NBC) [24, 25]. To reduce the model complexity of the CNN, radon-cumulative distribution transform (R-CDT) technique was utilized to pre-process the raw images and render signal classes linearly separable in transform domain, which decreased the difficulties for classifying the intensity of OV beams carrying different OAM modes dramatically [26]. Therefore, the shallow CNN with one convolution layer can differentiate them with similar accuracy of the AlexNet-structure CNN mentioned in [25]. The researches listed above mainly focus on using advanced machine learning techniques to discovery the OAM modes by identifying the intensity images of different LG beams. However, besides the OAM modes, the AT information is also contained in the intensity images of LG beams, where different ATs cause diverse wave front perturbation and further deform their appearances. Different from the previous works, we first attempt to extract the AT information from the intensity images of LG beams with different distortions through the CNN technique. At the same time, the CNN based AT detector we proposed is also applicable for extracting the OAM mode, which can also be used in the OAM-SK demodulation.

In this paper, a joint AT detection and adaptive demodulation technique based on the CNN is proposed from the perspective of image processing for the OAM-SK-FSO communication. The CNN simultaneously discoveries AT strength and OAM modes by identifying the category of intensity images of LG beams transmitted through computer-simulated free-space channels with diverse ATs. Further, the AT detection accuracy (ATDA) and adaptive demodulation accuracy (ADA) of the CNN are investigated respectively for 4-ary, 8-ary and 16-ary 1000-m OAM-SK-FSO systems with 4, 6 and 10 kinds of classic ATs, where the structure constant of the refractive index of air Cn2  ranges from 1×1016m2/3 to 5×1014m2/3, considering both weak and strong AT cases. The numerical results show that the CNN can achieve the high accuracy. The ATDA of the CNN is ~95.2% for the 8-ary OAM-SK system in case of 6 kinds of classic ATs and the corresponding ADA is higher than 99.5% in the 8-OAM system with the strong turbulent link. Moreover, the joint AT detecting and OAM-SK demodulating technique is tested and compared with previous approaches using the SOM, DNN and CNN [24–26]. Results show that the proposed CNN can obtain better AT detecting accuracy and OAM-SK demodulating accuracy performance. The proposed joint AT detecting and OAM-SK demodulating technique has the potential to be embedded in charge-coupled device (CCD) cameras deployed at the receiver to improve the reliability and flexibility for the OAM-FSO communication. In addition, the effects of image resolution, iteration number, activation functions and the structure of the CNN are also analyzed.

2. Operational principle

2.1 Atmospheric turbulence analytical model

In the atmospheric turbulence, the random refractive index profile disturbs the wave front phase of the transmitted LG beams and further induces the signal fading and cross-talk significantly. Numerous analytical models have been proposed to simulate the AT. Here we adopt the classic and widely utilized model developed by Hill [27] and defined analytically by Andrews [28]. In this analytical model [29], the AT is emulated by random phase screens loaded with the spectrum of fluctuation in the refractive index: θn(kx,ky).

θn(kx,ky)=0.033Cn2[1+1.802(kx2+ky2kl2)120.254(kx2+ky2kl2)712]×exp(kx2+ky2kl2)(kx2+ky2+1l02)116,
where Cn2  denotes the structure constant of the refractive index of air, used to describe the strength of the AT, kx and ky is the wavenumber in the x and y direction respectively. Moreover, l0 is the inner scale of AT and kl=3.3/l0. The fluctuation of the wave-front phase is further modeled through a random distribution with variance φ2. In the random phase screens, the perturbation of the refractive index is approximately represented by the Kolmogorov spectrum.
φ2(kx,ky)=(2πNΔx)22πk02Δzθn(kx,ky),
where N and Δx respectively denotes the size and the grid interval of the random phase screens. The wavenumber k0=2π/λ and the wavelength of the light beams is λ. What’s more, Δz is the interval distance between sequential phase screens. To facilitate the calculation, the random distribution of phase perturbation is described in the Cartesian coordinate system and the phase screen is then expressed in the frequency domain through the fast Fourier transform operation.
θ(x,y)=FFT(Mφ(kx,ky)),
where M is a complex random matrix with a mean of 0 and a variance of 1, and the perturbation of the index of refraction φ(kx,ky) is described in the Equation. (2).

2.2 Concept of the convolutional neural network

Convolutional neural network is specially designed to process two-dimension (2D) data, i.e. images and audio spectrograms, and there are four key ideas making it exceed to other usual neural networks: local connections, shared weights, pooling and multi-layer structure [30]. Moreover, the multi-layer CNN generally consist of five types of layers and they are the input layer, the convolutional layer, the pooling layer, the full connected layer and the output layer respectively. The specific structure of the CNN is illustrated in the Fig. 1.

 figure: Fig. 1

Fig. 1 Schematic diagram of the specific structure of the CNN used to jointly detect atmospheric turbulence and demodulate LG beams carrying certain OAM mode. In the input layer, original intensity images of the received LG beams are resized from 1200×900 into 96×96 to accelerate the processing procedure. In the convolution layer 1 (conv1), 16 96×96 feature maps are emerged by the 5×5  convolutional kernels. In the pooling layer 1 (pool1), 16 48×48 feature maps are generated through the maximum pooling process. Similar operations are performed in the later convolution layers and pooling layers. In the full connected layer, 526 nodes are full connected with the nodes in the pooling layer 3. In the output layer, 10 nodes are also full connected with the nodes in the previous 526 nodes, then the AT class and OAM mode are output by 6 nodes and 4 nodes respectively through the softmax classifier.

Download Full Size | PPT Slide | PDF

In the input layer, the intensity images of received LG beams serve as the input images, which are captured and collected by the charge coupled device (CCD) camera deployed at the receiver. To reduce the computation complexity and accelerate the training procedure, the input images are resized from 1200×900 into 96×96. Further, the resized images are sent into the subsequent convolutional layer and the core convolution operation in the CNN is then performed.

2.2.1 Convolution operation

In the convolutional layer, the units are organized into planes called as the feature map. The neurons in the feature map only take input from partial nodes in the previous layer, which is the notion of “local connections”. The local connection strategy is adopted because that the neighbor pixels are more correlative than distant pixels in the image [30]. Moreover, all of neurons in one feature map share the same weights. Weight sharing is based on one critical conclusion in the computer vision field: features learned from one part of the image can also be applied to other parts [30]. The shared weights are exact the convolutional kernel functioning as the future extractor to perform the convolution operation with the small sub-region of the input images. An example of 2D convolution operation is shown in the Fig. 2. A 2×2 kernel convolves with a 4×4 input image and then the 3×3  convolved feature map is generated. Due to the weight sharing, the activation of feature map is capable of shifting synchronously with the shift of the input image. This enables the CNN to be more insensitive with the distortions of the input image and further improves its generalization performance. After the convolutional layer, similar futures extracted by the kernel will be merged in the pooling layer.

 figure: Fig. 2

Fig. 2 Diagram illustrating the convolution operation [17]. The 4×4 input image is convolved with a 2×2 convolution kernel and then a 3×3  feature map is generated in the convolution layer.

Download Full Size | PPT Slide | PDF

2.2.2 Pooling operation

In the polling layer, the pooling unites take inputs from a non-overlapping 2×2 sub-region in the convolved feature map and generally the maximum of these inputs is calculated as displayed in the Fig. 3. After the maximum pooling, half the number of columns and rows compared with the convolved feature maps are conserved in the pooled feature maps. On the one hand, this pooling operation dramatically decrease the parameters need to be trained in the CNN. On the other hand, the maximum of the sub-region in the convolved feature maps serves as the representation of this region, which greatly decreases the effect of numerous noise information and enables the CNN to discovery more intrinsic features of the input images.

 figure: Fig. 3

Fig. 3 Diagram illustrating the maximum pooling operation. The maximum in the non-overlapping 2×2 sub-region of the 4×4 convolved map is computed and a 2×2 pooled feature map is then emerged in the pooling layer.

Download Full Size | PPT Slide | PDF

As shown in the Fig. 1, the features of the input images are gradually distracted. The multi-layer structure transforms the representation at one level starting with raw input into the representation at a higher and more abstracter level. The higher representations magnify eccentric features of input images that are important for classification and restrain the irrelevant variations [30]. After multi convolutional layers and polling layers, the subsequent layer of the CNN is the full connected layer. Different from the local connections, the units in this layer fully connect the neurons in the last pooling layer with a typical softmax classifier for outputting the category of the input images, which contains the AT strength and OAM modes information. The schematic diagram of the softmax classifier is depicted in the Fig. 4. In this multi-class classifier, the probabilities that the input x(x1, x2 xm) belongs to every class are respectively calculated and the category corresponding to the maximum probability is further output by the CNN.

 figure: Fig. 4

Fig. 4 Schematic diagram of the softmax classifier.

Download Full Size | PPT Slide | PDF

2.2.3 Softmax classifier

The figure below illustrates the softmax classifier.

3. Numerical results and analysis

The schematic diagram of the joint AT detection and adaptive demodulation technique based on the CNN for the 4-ary OAM-SK-FSO communication system is illuminated in Fig. 5. At the transmitter, the quaternary number sequence is mapped into diverse phase masks loaded in the spatial light modulator (SLM), which then modulate Gaussian beams into corresponding LG beams carrying diverse OAMs (e.g. s = 0, 1, 2, 3l = ±1, ±3, ±5, ±7). The turbulence models adopted in the OAM-SK communication system are analytical simulated and serve as the zero order approximation of the true atmospheric turbulence. Further, the LG beams are transmitted through the computer-simulated free-space turbulence channels. To simulate the AT, we adopt the classic and widely used model developed by Hill and defined analytically by Andrews, which is specially described in the section 2. In the numerical model, we have followed system parameters: λ = 1550nm (wavelength), ω0 = 3cm (beam waist), Δz = 200m (interval between sequential phase screens), N = 500 (grid interval of phase screens), l0 = 0.0003m (the value of inner scale of AT) and l1= 50m (the value of outer scale of AT). At the receiver, intensity images of distorted LG beams are captured by the CCD camera and then they are sent to the CNN to analyze the AT of the turbulence channels and the OAM modes offline.

 figure: Fig. 5

Fig. 5 Numerical model of the OAM-SK-FSO communication system. SLM: spatial light modulator, CCD camera: charge-coupled device camera.

Download Full Size | PPT Slide | PDF

After being transmitted over different computer-simulated free-space AT channels, the wave-front phase of the received LG light beams has experienced diverse perturbations, which further changes the distortion degree of the received LG beams. Meanwhile, it is observed that the appearance of the intensity patterns of LG beams carrying distinct OAM modes are also different from each other. Therefore the corresponding intensity images of LG beams contain relevant AT and OAM mode information. Figure 6 displays the wave-front phase perturbation caused by the 6 kinds of classic AT, where the structure constant of the refractive index of air Cn2 ranges from 1×1016m2/3 to 5×1014m2/3 (weak and strong AT cases are both considered). In the Fig. 6, the maximum phase perturbation ranges from 0.21rad to 6.2rad. With the increase of AT strength, the original wave-front phase structure is distorted gradually and the appearances of the intensity images of received LG beams carrying different OAM modes become more and more indistinct as shown in the Fig. 7.

 figure: Fig. 6

Fig. 6 Wave-front phase perturbations cause by a random phase screen with Cn2 valued respectively in (a) 1×1016m2/3, (b) 5×1016m2/3, (c) 1×1015m2/3, (d) 5×1015m2/3, (e) 1×1014m2/3, (f)  5×1014m2/3.

Download Full Size | PPT Slide | PDF

 figure: Fig. 7

Fig. 7 Intensity images of the received LG beams carrying the 16 kinds of OAM modes (l = ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16) over the computer-simulated 1000-m free-space turbulent channels with Cn2 valued in (a) 1×1016m2/3, (b) 5×1016m2/3, (c) 1×1015m2/3, (d) 5×1015m2/3, (e) 1×1014m2/3, (f)  5×1014m2/3 respectively.

Download Full Size | PPT Slide | PDF

In the Fig. 7, every column displays the intensity images of the received LG beams carrying certain OAM mode through 6 kinds of computer-simulated free-space AT channels and every row shows intensity images of LG beams with 16 types of OAM modes over given AT links. For the columns, with the increase of the strength of ATs, the intensity images of LG beams carrying certain one OAM mode become more and more anamorphic, which can be regarded as different categories. For the rows, under the case of the same AT channel, the patterns of intensity images corresponding to varied OAM modes are distinct with each other and they also can be considered as diverse classes. Therefore, the intensity images of LG beams contain both AT strength and OAM mode information. For each class of AT and OAM mode, 1200 intensity images of the received LG beams are collected, where 2/3 of them are selected as training data and 1/3 of them serve as testing data. During the training stage, the CNN gradually extracts the effective features of input intensity images of LG beams over diverse kinds of AT links and it constructs the mapping relationship between input intensity images and output class of AT and OAM mode. During the testing stage, the trained CNN calculates the AT and OAM mode class of input intensity images of the received LG beams. Therefore, the CNN can analyze and discovery the AT and OAM mode information hidden behind the intensity images. Further, the AT detection performances and the OAM-SK adaptive demodulation performances of the CNN will be investigated in the later sections, where the effects of the number of training iterations, image resolution, activation functions and structures of the CNN on the performances will also be analyzed one by one.

3.1 AT detection

To select appropriate system parameters for the CNN, we firstly explore the effects of input image resolution on the AT detection accuracy (ATDA). As shown in the Fig. 8(a), the ATDA of the CNN increases from 77.8% to 95.78% when the size of input images varies from 16×16 to 96×96. With the increase of the image resolution, more detailed information of the input images can be exploited and then the convergent ATDA is improved gradually. Then the ATDA increases by a small margin and achieves the saturation when the image resolution is set as 96×96. Thus, the input images fed with the CNN is resized as 96×96.

 figure: Fig. 8

Fig. 8 (a) The AT detecting accuracy of the CNN fed with the input images respectively resized as, 16×16, 32×32, 64×64 and 96×96; (b) The ATDA of the CNN with different activation functions for the image resolution of 96×96; (c) The effects of diverse structure of the CNN on the ATDA. (d) The ATDA for LG beams carrying varied OAM modes (l = ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16).

Download Full Size | PPT Slide | PDF

Further, the type of activation functions in the CNN is determined by comparing the corresponding ATDA performance. It is observed that the performance curves of the CNN with rectified linear unit (Relu) function and hyperbolic tangent (Tanh) function almost overlap in the Fig. 8(b). The reason why they significantly exceed the performance of the CNN with the sigmoid activation function is that the vanishing gradient problem is serious in the CNN using sigmoid function. Here, we choose the simple Relu function as the activation function for the CNN to simplify the computation.

In the Fig. 8(c), the ATDA performances of five kinds of network structure of the CNN for the 8OAM system are displayed. The network structure of the CNN generally means the number of feature maps in the convolutional layers and the number of layers. Here, we adopt classic structure, where there are three convolutional layers, three polling layers and one full connected layer as shown in the Fig. 1. The effect of the number of feature maps is investigated and the ATDA curves are represented in the Fig. 8(c). With increase of the number of feature maps in the convolution layer, more feature extractors are generated enables the CNN to be more powerful to discovery intrinsic features of the intensity images of the received LG beams, which promotes the corresponding detecting accuracy. However, more parameters need to be trained at the same time. For the 6 kinds of AT, the CNN with (16, 32, 128) feature maps in three convolution layers is acceptable, where the detecting accuracy of this CNN is over than 95%. Moreover, we can see that the influence of different OAM modes on the AT detecting is small from the Fig. 8(d) when the input images is resized as 96×96, the Relu activation function is adopted and the number set of feature maps in the convolutional layers is set as (16,32,128). The average ATDA of the CNN for 16 kinds of OAM modes is about 95% with the variance valued in 0.98.

To further analyze the AT detection results of the CNN, the classification distributions of every kinds of ATs are illuminated in the Fig. 9. It is observed that the detecting accuracy for the weakest AT and strongest AT are 99.3% and 98% respectively and the classification error mainly occurs in the middle section of the AT range. As shown in the Fig. 10, the ATDA is near 100% in the 4AT, where the interval between different ATs is large. With the decrease of the interval, the ATDA deteriorates dramatically to 76% under the case of 10AT. It is more difficult for the CNN to recognize these ATs due to their similar wave-front phase perturbation which further causes analogous appearance of the intensity images of the received LG beams.

 figure: Fig. 9

Fig. 9 The classification proportion of the CNN at each AT (AT1-AT6) respectively denotes Cn2 valued in 1×1016m2/3, 5×1016m2/3, 1×1015m2/3, 5×1015m2/3, 1×1014m2/3, and  5×1014m2/3.

Download Full Size | PPT Slide | PDF

 figure: Fig. 10

Fig. 10 The ATDA performance comparisons among of the CNN, SOM, DNN, CNN2, CNN3 and CNN5 for the computer-simulated 1000-m free-space link with 4, 6 and 10 kinds of ATs respectively. The parameters in the SOM [22], DNN [23], CNN2 [25], CNN3 [26] and CNN5 [24] are adopted in [22–26].

Download Full Size | PPT Slide | PDF

After the discussion of the CNN, we further compare the CNN with other five machine learning models. The five machine learning models used in the similar works are self-organizing network (SOM) [22], AlexNet-structure convolutional neural networks (CNN) [25], one-convolution-layer shallow CNN [26], two-hidden-layer deep neural network (DNN) [23] and two-convolution-layer CNN [24]. To simplify the expression, the descriptions of these algorithms above are abbreviated to the “SOM”, “CNN2”, “CNN3”, “DNN” and “CNN5” respectively. What’s more, the “CNN” refers to the CNN model adopted in our work. We have implemented these algorithms with the specific parameters and the structure descripted in the corresponding references. Further, their AT detection accuracy (ATDA) and the OAM-SK adaptive demodulation accuracy (ADA) performances are investigated as following.

Further, the specific comparison of the AT detection accuracy is displayed in the Fig. 10. We can see that the AT detecting accuracy of CNN exceeds those of SOM, DNN, CNN2, CNN3 and CNN5. To be more specific, the ATDA of CNN is 100% for the 4AT case and the detecting accuracy is near 95.2% for 6 kinds of classic ATs. For the 6AT, the detecting error rate (DER) of CNN is ~5%, which is only about 17.4% of DER by SOM, 46.6% by CNN2, 53.0% by CNN3, 76.2% by CNN5 and 73.8% by DNN. The theory of the SOM and the multi-hidden-layer neural network is intelligible, but generally the manual designed feature extractors are needed to discovery effective features of the input images. Compared with these traditional machine learning algorithms, the CNN is specialized in processing raw images and is capable of automatically extracting eccentric features to improve the robust of the trained network and avoid over-fitting, which enable CNN to be more insensitive to the irrelevant variations of the input images and achieve the better generation performance. In the CNN2 and the CNN3, the five-convolutional-layer structure and one-convolutional-layer structure are adopted to recognize different intensity images of the received vortex beams through diverse computer-simulated ATs, the over-fitting and under-fitting might occur. As for the different CNN, appropriate structure is needed to be selected to balance the fitting performance and the generation performance.

3.2 Adaptive demodulation

Meanwhile, the OAM mode information is also output by the CNN. The OAM mode of the LG beams is then transformed into the bit stream and the OAM-SK demodulation is achieved adaptively. To study the effect of different ATs, the adaptive demodulation accuracy of the CNN for the computer-simulated 1000-m free-space turbulent channels with diverse ATs is then researched. The ADA curves for the 4OAM, 8OAM and 16OAM system with the Cn2 valued in 1×1016m2/3, 1×1015m2/3, 5×1015m2/3 and  1×1014m2/3 are respectively displayed in Fig. 11(a)-11(d). The results show that the convergent ADA of the CNN is near 100% for the 8OAM system in the strong turbulence, which means that the CNN is capable of demodulating the LG beams through identifying the corresponding intensity images with zero error.

 figure: Fig. 11

Fig. 11 Adaptive demodulating accuracy of the CNN for 4-OAM (l = ±2, ±6, ±10, ±14), 8- OAM (l = ±2,  ±4, ±6, ±8, ±10, ±12, ±14, ±16) and 16-OAM (l = ±1, ±2, ±3, ±4, ±5,±6, ±7, ±8,  ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16) systems over computer-simulated 1000-m space-free turbulent channel with Cn2 valued in (a) 5×1016m2/3, (b) 1×1015m2/3, (c) 5×1015m2/3 and (d) 1×1014m2/3 respectively.

Download Full Size | PPT Slide | PDF

Moreover, we investigate the differences of ADA performance among these models. Seen from the Fig. 12 (a)-12(c), the ADA performance curves of the CNN are close to those of the CNN2 and the CNN5 in the 4OAM and the 8OAM system, which exceed those of the SOM, DNN and CNN3. With the increase of the number of OAM modes used in the OAM-SK system, the ADA of the CNN is much better than those of other four models. In the 16OAM system, the adaptive demodulating error rate of the CNN is near 8.5% under the case of 1000-m computer-simulated strong turbulent link with Cn2 valued in 1×1014m2/3 and the input images resized as 96×96. However, the corresponding demodulating error rates of the SOM, the DNN, the CNN2, the CNN3 and the CNN5 are respectively 20.1%, 11.4%, 17.1%, 43.7% and 11.5%. The reasons why the ADA performance of the CNN outperform those of DNN and SOM are that CNN is capable of automatically discovering intrinsic features of images through multilevel representations without manually designed feature extractors, where the effects of the essential features are magnified and those of irrelevant noise are suppressed significantly. By contrast, conventional machine learning algorithms are not good at processing the raw image without handy designed feature extractors, where intrinsic features of the images cannot be exploited and overfitting is inevitable generally. Therefore, they are more sensitive with the ambiguous intensity of LG beams under stronger atmospheric turbulence. The demodulating accuracy performances of them deteriorate dramatically in the worse atmospheric turbulence channel and these approaches are no longer effective for the OAM-SK communication systems. As for the CNN3, due to the limited learning ability of one-layer shallow network, the OAM-SK adaptive demodulating performance decreases with the intensity of AT increased, where the intensity images of distorted LG beams become more and more ambiguous and the one- convolutional-layer shallow CNN is not capable of extracting the essential features to be less insensitive to the noise.

 figure: Fig. 12

Fig. 12 Adaptive demodulating accuracy of the CNN, SOM [21], DNN [22], CNN2 [24], CNN3 [25] and CNN5 [23] under the case of computer-simulated 1000-m free-space turbulent link for (a) 4-OAM (l = ±2, ±6,±10,±14), (b) 8-OAM ( l = ±2, ±4, ±6, ±8, ±10, ±12,±14,±16) and (c) 16-OAM (l = ±1,±2, ±3, ±4, ±5, ±6, ±7, ±8,±9, ±10, ±11,±12, ±13, ±14,±15,±16).

Download Full Size | PPT Slide | PDF

4. Conclusions

In this paper, a CNN based joint atmospheric turbulence detection and OAM-SK adaptive demodulation technique was proposed. The AT detecting accuracy and the OAM demodulating accuracy of the 4-OAM, 8-OAM, 16-OAM FSO systems over the computer-simulated 1000-m turbulent channels with 4, 6, 10 kinds of classic ATs were respectively investigated. Compared with previous approaches using the SOM, DNN and other CNNs, the proposed CNN achieves the highest ATDA and ADA. The ATDA of CNN is near 95.2% for 6 kinds of typical ATs, in cases of both weak and strong ATs. The adaptive demodulation of optical vortices (OV) carrying OAM modes, the ADA of CNN is about 99.8% for the 8-OAM system over the computer-simulated 1000-m free-space strong turbulent link. Moreover, the proposed CNN technique is also applicable for multiple OAM states demultiplexing. The multiplexed OAM modes of the OV beams can be recognized through differentiating the intensity images of OV beams carrying multiple OAM modes. Further, corresponding computer-generated holograms (CGHs) are loaded in the SLMs to demultiplex the OAM-DM OV beams. Thus, the CNN based mode recognition and optical performance monitoring technique for the OAM-DM systems can be explored in our future works. These implantable technologies also have the potential to be embedded in the CCD camera deployed at the receiver to improve the reliability and the intelligent processing ability for the OAM-FSO communication.

Funding

NSFC Project No.61705016, the Open Research Fund of Key Laboratory of Space Utilization, Chinese Academy of Sciences (No.LSU-DZXX-2017-01) and the China Postdoctoral Science Foundation (CPSF) No.2017M620697.

References and links

1. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]   [PubMed]  

2. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]  

3. T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015). [CrossRef]  

4. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016). [CrossRef]   [PubMed]  

5. J. Wang, S. Li, M. Luo, J. Liu, L. Zhu, C. Li, D. Xie, Q. Yang, S. Yu, J. Sun, X. Zhang, W. Shieh, and A. E. Willner, “N-Dimentional multiplexing link with 1.036-Pbit/s transmission capacity and 112.6-bit/s/Hz spectral efficiency using OFDM-8QAM signals over 368 WDM pol-muxed 26 OAM modes,” in Proc. European Conference and Exhibition on Optical Communication (ECOC, 2014), paper Mo.4.5.1. [CrossRef]  

6. A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41(13), 3086–3089 (2016). [CrossRef]   [PubMed]  

7. L. Zhu, J. Liu, Q. Mo, C. Du, and J. Wang, “Encoding/decoding using superpositions of spatial modes for image transfer in km-scale few-mode fiber,” Opt. Express 24(15), 16934–16944 (2016). [CrossRef]   [PubMed]  

8. X. Wang and Y. Song, “Encoding and decoding by the states of vector modes for vortex beams propagating in air-core fiber,” Opt. Express 25(23), 29342–29355 (2017). [CrossRef]  

9. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007). [CrossRef]  

10. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). [CrossRef]   [PubMed]  

11. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014). [CrossRef]  

12. K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

13. R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

14. M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015). [CrossRef]  

15. D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016). [CrossRef]  

16. D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016). [CrossRef]  

17. D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017). [CrossRef]   [PubMed]  

18. D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017). [CrossRef]  

19. A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

20. K. Simonyan and A. Zisserman, “Very Deep Convolutional Networks for Large-Scale Image Recognition,” in Proc. Computer Vision and Pattern Recognition (CVPR, 2014).

21. Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015). [CrossRef]   [PubMed]  

22. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” in the Proceedings of the National Academy of Sciences, 113(48), 13648–13653. (2016). [CrossRef]  

23. T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56(12), 3386–3396 (2017). [CrossRef]   [PubMed]  

24. S. R. Park, L. Cattell, J. M. Nichols, A. Watnik, T. Doster, and G. K. Rohde, “De-multiplexing vortex modes in optical communications using transport-based pattern recognition,” Opt. Express 26(4), 4004–4022 (2018). [CrossRef]   [PubMed]  

25. E. M. Knutson, S. Lohani, O. Danaci, S. D Huver, and R. T Glasser, “Deep learning as a tool to distinguish between high orbital angular momentum optical modes,” in Optics and Photonics for Information Processing, 9970, (2016).

26. J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017). [CrossRef]  

27. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978). [CrossRef]  

28. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

29. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012). [CrossRef]   [PubMed]  

30. C. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

References

  • View by:

  1. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  2. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  3. T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
    [Crossref]
  4. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
    [Crossref] [PubMed]
  5. J. Wang, S. Li, M. Luo, J. Liu, L. Zhu, C. Li, D. Xie, Q. Yang, S. Yu, J. Sun, X. Zhang, W. Shieh, and A. E. Willner, “N-Dimentional multiplexing link with 1.036-Pbit/s transmission capacity and 112.6-bit/s/Hz spectral efficiency using OFDM-8QAM signals over 368 WDM pol-muxed 26 OAM modes,” in Proc. European Conference and Exhibition on Optical Communication (ECOC, 2014), paper Mo.4.5.1.
    [Crossref]
  6. A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41(13), 3086–3089 (2016).
    [Crossref] [PubMed]
  7. L. Zhu, J. Liu, Q. Mo, C. Du, and J. Wang, “Encoding/decoding using superpositions of spatial modes for image transfer in km-scale few-mode fiber,” Opt. Express 24(15), 16934–16944 (2016).
    [Crossref] [PubMed]
  8. X. Wang and Y. Song, “Encoding and decoding by the states of vector modes for vortex beams propagating in air-core fiber,” Opt. Express 25(23), 29342–29355 (2017).
    [Crossref]
  9. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007).
    [Crossref]
  10. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
    [Crossref] [PubMed]
  11. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
    [Crossref]
  12. K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.
  13. R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).
  14. M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
    [Crossref]
  15. D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
    [Crossref]
  16. D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
    [Crossref]
  17. D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
    [Crossref] [PubMed]
  18. D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
    [Crossref]
  19. A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).
  20. K. Simonyan and A. Zisserman, “Very Deep Convolutional Networks for Large-Scale Image Recognition,” in Proc. Computer Vision and Pattern Recognition (CVPR, 2014).
  21. Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
    [Crossref] [PubMed]
  22. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” in the Proceedings of the National Academy of Sciences, 113(48), 13648–13653. (2016).
    [Crossref]
  23. T. Doster and A. T. Watnik, “Machine learning approach to OAM beam demultiplexing via convolutional neural networks,” Appl. Opt. 56(12), 3386–3396 (2017).
    [Crossref] [PubMed]
  24. S. R. Park, L. Cattell, J. M. Nichols, A. Watnik, T. Doster, and G. K. Rohde, “De-multiplexing vortex modes in optical communications using transport-based pattern recognition,” Opt. Express 26(4), 4004–4022 (2018).
    [Crossref] [PubMed]
  25. E. M. Knutson, S. Lohani, O. Danaci, S. D Huver, and R. T Glasser, “Deep learning as a tool to distinguish between high orbital angular momentum optical modes,” in Optics and Photonics for Information Processing, 9970, (2016).
  26. J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
    [Crossref]
  27. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978).
    [Crossref]
  28. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  29. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012).
    [Crossref] [PubMed]
  30. C. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

2018 (1)

2017 (5)

2016 (5)

2015 (3)

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

2014 (2)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

2012 (3)

A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012).
[Crossref] [PubMed]

2009 (1)

2007 (1)

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1978 (1)

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978).
[Crossref]

Ahmed, N.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Andrews, R.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

Bock, R.

Boyd, R. W.

Cai, Z.

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Cattell, L.

Chen, X.

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

Cui, Y.

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

Deoras, A.

R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

Ding, J.

Djordjevic, I. B.

Dolinar, S.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Doster, T.

Du, C.

Dudley, A.

Fazal, I.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Forbes, A.

Fu, M.

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Gong, L. Y.

Gopaul, C.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007).
[Crossref]

Han, H.

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

He, K.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

Hill, R.

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978).
[Crossref]

Hinton, G.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

Hinton, G. E.

R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

Huang, H.

Jia, P.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Khoshgoftaar, T. M.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Krizhevsky, A.

A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

Lavery, M. P. J.

Leach, J.

LeCun, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

Lei, T.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Li, J.

J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

Li, L.

Li, Y.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Li, Z.

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Liao, P.

Lin, J.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Liu, G.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Liu, J.

Luo, B.

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Malik, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Min, C.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Mo, Q.

Muharemagic, E.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Najafabadi, M. M.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Neifeld, M. A.

Nichols, J. M.

Niu, H.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Park, S. R.

Ren, S.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

Ren, Y.

Rohde, G. K.

Salem, A. B.

Sarikaya, R.

R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

Scheidl, T.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Seliya, N.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Song, C.

Song, Y.

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sun, J.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

Sutskever, I.

A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

Trichili, A.

Tur, M.

Tyler, G. A.

Ursin, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Villanustre, F.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Wald, R.

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

Wang, D.

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Wang, J.

L. Zhu, J. Liu, Q. Mo, C. Du, and J. Wang, “Encoding/decoding using superpositions of spatial modes for image transfer in km-scale few-mode fiber,” Opt. Express 24(15), 16934–16944 (2016).
[Crossref] [PubMed]

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, X.

Wang, Z.

Watnik, A.

Watnik, A. T.

Willner, A.

Willner, A. E.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Xie, G.

Xu, X.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Yan, Y.

Yang, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yu, C.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Yuan, X.

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Yue, Y.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Zghal, M.

Zhang, M.

J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, J. Li, Z. Li, J. Li, C. Song, and X. Chen, “Intelligent constellation diagram analyzer using convolutional neural network-based deep learning,” Opt. Express 25(15), 17150–17166 (2017).
[Crossref] [PubMed]

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Zhang, X.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

Zhao, S. M.

Zhao, Z.

Zheng, B. Y.

Zhu, L.

Adv. Neural Inf. Process. (1)

A. Krizhevsky, I. Sutskever, and G. Hinton, “Imagenet classification with deep convolutional neural networks,” Adv. Neural Inf. Process. 25, 1106–1114 (2012).

Appl. Opt. (1)

IEEE Photonics Technol. Lett. (3)

J. Li, M. Zhang, and D. Wang, “Adaptive Demodulator Using Machine Learning for Orbital Angular Momentum Shift Keying,” IEEE Photonics Technol. Lett. 29(17), 1455–1458 (2017).
[Crossref]

D. Wang, M. Zhang, M. Fu, Z. Cai, Z. Li, H. Han, Y. Cui, and B. Luo, “Nonlinearity Mitigation Using a Machine Learning Detector Based on k-Nearest Neighbors,” IEEE Photonics Technol. Lett. 28(19), 2102–2105 (2016).
[Crossref]

D. Wang, M. Zhang, Z. Li, J. Li, M. Fu, Y. Cui, and X. Chen, “Modulation Format Recognition and OSNR Estimation Using CNN-Based Deep Learning,” IEEE Photonics Technol. Lett. 29(19), 1455–1458 (2017).
[Crossref]

IEEE/ACM Trans. Audio, Speech, Lang. Process. (1)

R. Sarikaya, G. E. Hinton, and A. Deoras, “Application of Deep Belief Networks for Natural Language Understanding,” IEEE/ACM Trans. Audio, Speech, Lang. Process. 22(4), 778–784 (2014).

J. Big Data (1)

M. M. Najafabadi, F. Villanustre, T. M. Khoshgoftaar, N. Seliya, R. Wald, and E. Muharemagic, “Deep learning applications and challenges in big data analytics,” J. Big Data 2(1), 1–21 (2015).
[Crossref]

J. Fluid Mech. (1)

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(03), 541–562 (1978).
[Crossref]

Light Sci. Appl. (1)

T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

Nat. Photonics (1)

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Nature (1)

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref] [PubMed]

New J. Phys. (2)

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9(4), 94 (2007).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatial modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Opt. Commun. (1)

D. Wang, M. Zhang, Z. Cai, Y. Cui, Z. Li, H. Han, M. Fu, and B. Luo, “Combatting nonlinear phase noise in coherent optical systems with an optimized decision processor based on machine learning,” Opt. Commun. 369, 199–208 (2016).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Other (7)

J. Wang, S. Li, M. Luo, J. Liu, L. Zhu, C. Li, D. Xie, Q. Yang, S. Yu, J. Sun, X. Zhang, W. Shieh, and A. E. Willner, “N-Dimentional multiplexing link with 1.036-Pbit/s transmission capacity and 112.6-bit/s/Hz spectral efficiency using OFDM-8QAM signals over 368 WDM pol-muxed 26 OAM modes,” in Proc. European Conference and Exhibition on Optical Communication (ECOC, 2014), paper Mo.4.5.1.
[Crossref]

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR, 2016), pp. 770–778.

C. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

E. M. Knutson, S. Lohani, O. Danaci, S. D Huver, and R. T Glasser, “Deep learning as a tool to distinguish between high orbital angular momentum optical modes,” in Optics and Photonics for Information Processing, 9970, (2016).

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” in the Proceedings of the National Academy of Sciences, 113(48), 13648–13653. (2016).
[Crossref]

K. Simonyan and A. Zisserman, “Very Deep Convolutional Networks for Large-Scale Image Recognition,” in Proc. Computer Vision and Pattern Recognition (CVPR, 2014).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1 Schematic diagram of the specific structure of the CNN used to jointly detect atmospheric turbulence and demodulate LG beams carrying certain OAM mode. In the input layer, original intensity images of the received LG beams are resized from 1200×900 into 96×96 to accelerate the processing procedure. In the convolution layer 1 (conv1), 16 96×96 feature maps are emerged by the 5×5  convolutional kernels. In the pooling layer 1 (pool1), 16 48×48 feature maps are generated through the maximum pooling process. Similar operations are performed in the later convolution layers and pooling layers. In the full connected layer, 526 nodes are full connected with the nodes in the pooling layer 3. In the output layer, 10 nodes are also full connected with the nodes in the previous 526 nodes, then the AT class and OAM mode are output by 6 nodes and 4 nodes respectively through the softmax classifier.
Fig. 2
Fig. 2 Diagram illustrating the convolution operation [17]. The 4×4 input image is convolved with a 2×2 convolution kernel and then a 3×3  feature map is generated in the convolution layer.
Fig. 3
Fig. 3 Diagram illustrating the maximum pooling operation. The maximum in the non-overlapping 2×2 sub-region of the 4×4 convolved map is computed and a 2×2 pooled feature map is then emerged in the pooling layer.
Fig. 4
Fig. 4 Schematic diagram of the softmax classifier.
Fig. 5
Fig. 5 Numerical model of the OAM-SK-FSO communication system. SLM: spatial light modulator, CCD camera: charge-coupled device camera.
Fig. 6
Fig. 6 Wave-front phase perturbations cause by a random phase screen with C n 2 valued respectively in (a) 1× 10 16 m 2/3 , (b) 5× 10 16 m 2/3 , (c) 1× 10 15 m 2/3 , (d) 5× 10 15 m 2/3 , (e) 1× 10 14 m 2/3 , (f)  5× 10 14 m 2/3 .
Fig. 7
Fig. 7 Intensity images of the received LG beams carrying the 16 kinds of OAM modes ( l = ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16 ) over the computer-simulated 1000-m free-space turbulent channels with C n 2 valued in (a) 1× 10 16 m 2/3 , (b) 5× 10 16 m 2/3 , (c) 1× 10 15 m 2/3 , (d) 5× 10 15 m 2/3 , (e) 1× 10 14 m 2/3 , (f)  5× 10 14 m 2/3 respectively.
Fig. 8
Fig. 8 (a) The AT detecting accuracy of the CNN fed with the input images respectively resized as, 16×16, 32×32, 64×64 and 96×96; (b) The ATDA of the CNN with different activation functions for the image resolution of 96×96; (c) The effects of diverse structure of the CNN on the ATDA. (d) The ATDA for LG beams carrying varied OAM modes ( l = ±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8, ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16 ).
Fig. 9
Fig. 9 The classification proportion of the CNN at each AT (AT1-AT6) respectively denotes C n 2 valued in 1× 10 16 m 2/3 , 5× 10 16 m 2/3 , 1× 10 15 m 2/3 , 5× 10 15 m 2/3 , 1× 10 14 m 2/3 , and  5× 10 14 m 2/3 .
Fig. 10
Fig. 10 The ATDA performance comparisons among of the CNN, SOM, DNN, CNN2, CNN3 and CNN5 for the computer-simulated 1000-m free-space link with 4, 6 and 10 kinds of ATs respectively. The parameters in the SOM [22], DNN [23], CNN2 [25], CNN3 [26] and CNN5 [24] are adopted in [22–26].
Fig. 11
Fig. 11 Adaptive demodulating accuracy of the CNN for 4-OAM (l = ±2, ±6, ±10, ±14), 8- OAM (l = ±2,  ±4, ±6, ±8, ±10, ±12, ±14, ±16) and 16-OAM (l = ±1, ±2, ±3, ±4, ±5,±6, ±7, ±8,  ±9, ±10, ±11, ±12, ±13, ±14, ±15, ±16) systems over computer-simulated 1000-m space-free turbulent channel with C n 2 valued in (a) 5× 10 16 m 2/3 , (b) 1× 10 15 m 2/3 , (c) 5× 10 15 m 2/3 and (d) 1× 10 14 m 2/3 respectively.
Fig. 12
Fig. 12 Adaptive demodulating accuracy of the CNN, SOM [21], DNN [22], CNN2 [24], CNN3 [25] and CNN5 [23] under the case of computer-simulated 1000-m free-space turbulent link for (a) 4-OAM (l = ±2, ±6,±10,±14), (b) 8-OAM ( l = ±2, ±4, ±6, ±8, ±10, ±12,±14,±16) and (c) 16-OAM (l = ±1,±2, ±3, ±4, ±5, ±6, ±7, ±8,±9, ±10, ±11,±12, ±13, ±14,±15,±16).

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

θ n ( k x , k y )=0.033 C n 2 [1+1.802 ( k x 2 + k y 2 k l 2 ) 1 2 0.254 ( k x 2 + k y 2 k l 2 ) 7 12 ] ×exp( k x 2 + k y 2 k l 2 ) ( k x 2 + k y 2 + 1 l 0 2 ) 11 6 ,
φ 2 ( k x , k y )= ( 2π NΔx ) 2 2π k 0 2 Δz θ n ( k x , k y ),
θ(x,y)=FFT(Mφ( k x , k y )),

Metrics