Abstract

Atmospheric transmission distortion is one of the main challenges hampering the practical application of a vortex beam (VB) which carries orbital angular momentum (OAM). In this work, we propose and investigate a deep learning based atmospheric turbulence compensation method for correcting the distorted VB and improving the performance of OAM multiplexing communication. A deep convolutional neural network (CNN) model, which can automatically learn the mapping relationship of the intensity distributions of input and the turbulent phase, is well designed. After trained with loads of studying samples, the CNN model possesses a good generalization ability in quickly and accurately predicting equivalent turbulent phase screen, including the untrained turbulent phase screens. The results show that through correction, the mode purity of the distorted VB improves from 39.52% to 98.34% under the turbulence intensity of Cn2 = 1 × 10−13. Constructing an OAM multiplexing communication link, the bit-error-rate (BER) of the transmitted signals in each OAM channel is reduced by almost two orders of magnitude under moderate-strong turbulence, and the demodulated constellation diagram also converges well after compensated by the CNN model.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

S. Lohani and R. T. Glasser, “Turbulence correction with artificial neural networks,” Opt. Lett. 43(11), 2611–2614 (2018).
[Crossref] [PubMed]

S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. 408, 68–81 (2018).
[Crossref]

2017 (5)

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]

D. Wang, M. Zhang, Z. Li, C. Song, M. Fu, J. Li, and X. Chen, “System impairment compensation in coherent optical communications by using a bio-inspired detector based on artificial neural network and genetic algorithm,” Opt. Commun. 399, 1–12 (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]

A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos Trans A Math Phys Eng Sci 375(2087), 20150439 (2017).
[Crossref] [PubMed]

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).
[Crossref]

2016 (4)

2015 (4)

G. Xie, Y. Ren, H. Huang, M. P. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015).
[Crossref] [PubMed]

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

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

M. Erhard, H. Qassim, H. Mand, E. Karimi, and R. W. Boyd, “Real-time imaging of spin-to-orbital angular momentum hybrid remote state preparation,” Phys. Rev. A 92(2), 022321 (2015).
[Crossref]

2014 (3)

2013 (2)

Y. Jiang, X. Li, and M. Gu, “Generation of sub-diffraction-limited pure longitudinal magnetization by the inverse Faraday effect by tightly focusing an azimuthally polarized vortex beam,” Opt. Lett. 38(16), 2957–2960 (2013).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

2012 (4)

D. S. Simon and A. V. Sergienko, “Two-photon spiral imaging with correlated orbital angular momentum states,” Phys. Rev. A 85(4), 43825 (2012).
[Crossref]

J. Wang, J. Y. Yang, I. M. 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]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012).
[Crossref] [PubMed]

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]

2011 (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

2010 (1)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

2006 (1)

2002 (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

2000 (1)

1996 (1)

1941 (1)

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Cr Acad. Sci. URSS 30, 301–305 (1941).

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

G. Xie, Y. Ren, H. Huang, M. P. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. 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]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. E. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “25.6-bit/s/Hz spectral efficiency using 16-QAM signals over pol-muxed multiple orbital-angular-momentum modes,” in Proceedings of IEEE Photonic Society 24th Annual Meeting (IEEE, 2011), pp. 587–588.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Anguita, J. A.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos Trans A Math Phys Eng Sci 375(2087), 20150439 (2017).
[Crossref] [PubMed]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bao, C.

Barnett, S. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

Bengio, Y.

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

Bernet, S.

Birnbaum, K.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. E. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “25.6-bit/s/Hz spectral efficiency using 16-QAM signals over pol-muxed multiple orbital-angular-momentum modes,” in Proceedings of IEEE Photonic Society 24th Annual Meeting (IEEE, 2011), pp. 587–588.

Birnbaum, K. M.

Bowman, R.

M. J. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Boyd, R. W.

M. Erhard, H. Qassim, H. Mand, E. Karimi, and R. W. Boyd, “Real-time imaging of spin-to-orbital angular momentum hybrid remote state preparation,” Phys. Rev. A 92(2), 022321 (2015).
[Crossref]

G. Xie, Y. Ren, H. Huang, M. P. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

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]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
[Crossref] [PubMed]

Cai, Y.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).
[Crossref]

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

G. Xie, Y. Ren, H. Huang, M. P. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. 40(7), 1197–1200 (2015).
[Crossref] [PubMed]

Chen, L.

X. Zhang, Y. He, Y. Cai, M. Su, X. Zhou, Y. Chen, S. Chen, Y. Xiang, L. Chen, C. Su, Y. Li, and D. Fan, “Coherent separation detection for orbital angular momentum multiplexing in free-space optical communications,” IEEE Photonics J. 9(3), 1–11 (2017).
[Crossref]

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A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos Trans A Math Phys Eng Sci 375(2087), 20150439 (2017).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014).
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A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos Trans A Math Phys Eng Sci 375(2087), 20150439 (2017).
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A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos Trans A Math Phys Eng Sci 375(2087), 20150439 (2017).
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J. Wang, J. Y. Yang, I. M. 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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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. E. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “25.6-bit/s/Hz spectral efficiency using 16-QAM signals over pol-muxed multiple orbital-angular-momentum modes,” in Proceedings of IEEE Photonic Society 24th Annual Meeting (IEEE, 2011), pp. 587–588.

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. 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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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. E. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “25.6-bit/s/Hz spectral efficiency using 16-QAM signals over pol-muxed multiple orbital-angular-momentum modes,” in Proceedings of IEEE Photonic Society 24th Annual Meeting (IEEE, 2011), pp. 587–588.

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
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J. Wang, J. Y. Yang, I. M. 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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Zhang, M.

D. Wang, M. Zhang, Z. Li, C. Song, M. Fu, J. Li, and X. Chen, “System impairment compensation in coherent optical communications by using a bio-inspired detector based on artificial neural network and genetic algorithm,” Opt. Commun. 399, 1–12 (2017).
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Zhao, Z.

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Figures (16)
Fig. 1
Fig. 1 The intensity and phase distributions of VBs with different topological charges ( l = 0 , 1 , 3 ) under the influence of AT. (a) C n 2 = 0 (without turbulence); (b) C n 2 = 1 × 10 − 14 ; (c) C n 2 = 5 × 10 − 14 ; (d) C n 2 = 1 × 10 − 13 .

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Fig. 2
Fig. 2 The mode purity of the VB ( l = 3 ) under the influence of AT. (a) C n 2 = 1 × 10 − 14 ; (b) C n 2 = 5 × 10 − 14 ; (c) C n 2 = 1 × 10 − 13 .

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Fig. 3
Fig. 3 The architecture of the CNN model used for AT compensation. Conv: convolution; Deconv: deconvolution.

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Fig. 4
Fig. 4 (a) Loss function curves. (b) The iteration numbers and loss values corresponding to A, B, C, D, and E, respectively. (c) AT phase diagrams.

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Fig. 5
Fig. 5 The phase compensation effect of the trained CNN under different AT intensities at the distance of Δ z = 20 m . (a) C n 2 =1 × 10 -14 ; (b) C n 2 =5 × 10 -14 ; (c) C n 2 =1 × 10 -13 .

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Fig. 6
Fig. 6 The beam profiles of the VB ( l = 3 ) with and without compensation corresponding to different turbulence intensities, Δ z = 20   m .

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Fig. 7
Fig. 7 The mode purity of the VB ( l = 3 ) under the influence of different AT intensities with and without CNN compensation, Δ z = 20   m . (a) The mode purity curves as the function of turbulence intensity C n 2 . (b) The mode purity before and after compensation for different turbulence intensities.

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Fig. 8
Fig. 8 The mode purity of the VB ( l = 3 ) under the AT intensity of C n 2 =5 × 10 -14 at different distance Δ z . (a) The mode purity curves as the function of distance Δ z . (b) The mode purity before and after compensation for different distances.

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Fig. 9
Fig. 9 The system diagram of OAM multiplexing FSO communication link with AT compensation. PBS: polarization beam splitter; AT: AT; MR: mirror; SLM: spatial light modulator.

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Fig. 10
Fig. 10 The BER curves as the function of SNR for different turbulence intensities C n 2 with and without compensation at the distance of Δ z = 20 m . (a) l = 3. (b) l = 1 .

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Fig. 11
Fig. 11 (a) The BER curves as the function of turbulence intensity C n 2 for each OAM channel with and without compensation at the distance of Δ z = 20 m . (b) The constellations of the channel l = 3 at C n 2 = 1 × 10 − 14 , C n 2 = 5 × 10 − 14 , and C n 2 = 1 × 10 − 13 without (top) and with (bottom) AT compensation.

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Fig. 12
Fig. 12 (a) The BER curves as the function of transmission distance of Δ z for each OAM channel with and without AT compensation at C n 2 = 5 × 10 − 14 . (b) The constellations of the channel l = 3 at Δ z = 15 m , 20 m , 25 m without (top) and with (bottom) atmospheric compensation.

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Fig. 13
Fig. 13 The constellations of the channel l = 3 with multiplexed OAM modes l = 2 , 3 , l = 4 , 3 , l = 5 , 3 with and without AT compensation.

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Fig. 14
Fig. 14 (a) The BER curves as the function of turbulence intensity C n 2 for each OAM channel with and without compensation at the transmission distance of Δ z = 1000 m . (b) The constellations of the channel l = 3 at C n 2 = 1 × 10 − 15 , C n 2 = 5 × 10 − 15 , and C n 2 = 1 × 10 − 14 without (top) and with (bottom) AT compensation.

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Fig. 15
Fig. 15 Loss function curve and mean test loss value for the weak AT of C n 2 ∈ [ 1 × 10 - 15 , 1 × 10 - 14 ] and strong AT of C n 2 ∈ [ 1 × 10 - 13 , 1 × 10 - 12 ] , Δ z ∈ [ 15 m , 25 m ] .

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Fig. 16
Fig. 16 The compensation effect after turbulence has changed to varying degrees.

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Tables (2)

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Table 1 The BER of the OAM channel when the other multiplexed OAM mode is different

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Table 2 The mean test loss for different numbers of training data

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Equations (13)

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D ( r ) = C n 2 r 2 / 3 ,                                           l 0 ≪ r ≪ L 0 ,
Φ n ( k ) = 0.033 C n 2 k − 11 / 3 ,
Φ n ( k x , k y ) = 0.033 C n 2 [ 1 + 1.802 k x 2 + k y 2 k l 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 ,
σ 2 ( k x , k y ) = ( 2 π N Δ L ) 2 Φ ( k x , k y ) ,
ϕ ( x , y ) = F F T [ C ⋅ σ ( k x , k y ) ] ,
U ( z + Δ z , x , y ) ≈ F F T − 1 [ exp ( i A Δ z ) ⋅ F F T { exp ( i ϕ ( x , y ) ) × U ( z , x , y ) } ] .
l 1 ( f ( x i , θ ) , y i ) = ( y i − f ( x i , θ ) ) 2 ,
l 2 ( f ( x i , θ ) , y i ) = R e l u ( y i − f ( x i , θ ) − a ) ,
l 3 ( f ( x i , θ ) , y i ) = R e l u ( f ( x i , θ ) − y i − a ) ,
L ( f ( X , θ ) , Y ) = ∑ i = 1 N [ l 1 ( f ( x i , θ ) , y i ) + l 2 ( f ( x i , θ ) , y i ) + l 3 ( f ( x i , θ ) , y i ) ] / N = ∑ i = 1 N [ ( y i − f ( x i , θ ) ) 2 + R e l u ( y i − f ( x i , θ ) − a ) + R e l u ( f ( x i , θ ) − y i − a ) ] / N ,
θ o p t = min θ ( L ( f ( X , θ ) , Y ) .
y ^ = f ( x , θ o p t ) .
y c o m = y − G ( y ^ ) ,

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