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

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References

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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, 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]

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, 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]

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.

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]

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, 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, 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, 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.

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[Crossref] [PubMed]

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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).
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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).
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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).
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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).
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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).

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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).
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[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).
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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]

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Opt. Lett. (3)

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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).
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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)

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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 )),

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