Abstract

In this paper, we experimentally investigate the performance of crosstalk mitigation for 16-ary quadrature amplitude modulation orthogonal frequency division multiplexing (16QAM-OFDM) signals carrying orbital angular momentum (OAM) multiplexed free-space-optical communication (FSO) links using the pilot assisted Least Square (LS) algorithm. At the demodulating spatial light modulators (SLMs), we launch the distorted phase holograms which have the information of atmospheric turbulence obeying the modified Hill spectrum. And crosstalk can be introduced by these holograms with the experimental verification. The pilot assisted LS algorithm can efficiently improve the quality of system performance, the points of constellations get closer to the reference points and around two orders of magnitude improvement of bit-error rate (BER) is obtained.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Turbulence heterodyne coherent mitigation of orbital angular momentum multiplexing in a free space optical link by auxiliary light

Chunyong Yang, Chuang Xu, Wenjun Ni, Yu Gan, Jin Hou, and Shaoping Chen
Opt. Express 25(21) 25612-25624 (2017)

Demonstration of data-carrying orbital angular momentum-based underwater wireless optical multicasting link

Yifan Zhao, Jing Xu, Andong Wang, Weichao Lv, Long Zhu, Shuhui Li, and Jian Wang
Opt. Express 25(23) 28743-28751 (2017)

Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link

Jaime A. Anguita, Mark A. Neifeld, and Bane V. Vasic
Appl. Opt. 47(13) 2414-2429 (2008)

References

  • View by:
  • |
  • |
  • |

  1. J. Wang, J. 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).
  2. A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).
  3. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
    [PubMed]
  4. J. Wang, S. Li, C. Li, L. Zhu, C. Gui, D. Xie, Y. Qiu, Q. Yang, and S. Yu, “Ultra-high 230-bit/s/Hz spectral efficiency using OFDM/OQAM 64-QAM signals over pol-muxed 22 orbital angular momentum (OAM) modes,” in OFC/NFOEC (2014), paper W1H. 4.
  5. J. Wang, J. Liu, S. Li, L. Zhu, C. Li, M. Luo, Q. Yang, and S. Yu, “Experimental Demonstration of Free-Space Optical Communications Using OFDM-QPSK/16QAM-Carrying Fractional Orbital Angular Momentum (OAM) Multiplexing,” in OFC/NFOEC (2015), paper M2F.5.
  6. B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
    [PubMed]
  7. I. B. Djordjevic, “Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation,” Opt. Express 19(15), 14277–14289 (2011).
    [PubMed]
  8. Y. Ren, Y. Zhang, Y. Yue, N. Bozinovic, G. Xie, H. Huang, M. Tur, P. Kristensen, I. B. Djordjevic, S. Ramachandran, and A. E. Willner, “Efficient Crosstalk Mitigation of OAM Based 400-Gbit/s QPSK Data Transmission in 1.1-km Vortex Fiber by Using Soft-Decision LDPC codes,” in CLEO (2013), paper CM2G.5.
  9. Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).
  10. S. M. Zhao, B. Wang, L. Y. Gong, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the Atmosphere Turbulence Tolerance in Holographic Ghost, Imaging System by Channel Coding,” J. Lightwave Technol. 31(17), 2823–2828 (2013).
  11. S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).
  12. Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
    [PubMed]
  13. Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
    [PubMed]
  14. 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).
    [PubMed]
  15. S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
    [PubMed]
  16. 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).
    [PubMed]
  17. L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).
  18. H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
    [PubMed]
  19. V. E. Zakharov, V. S. L’Vov, and G. Falkovich, “Kolmogorov Spectra of Turbulence: I Wave Turbulence,” Springer-Verlag, (1992)
  20. E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).
  21. W. B. Miller, L. C. Andrews, and J. C. Ricklin, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
  22. L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

2016 (2)

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
[PubMed]

2015 (3)

2014 (2)

2013 (1)

2012 (3)

2011 (1)

2008 (1)

1997 (1)

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

1996 (1)

E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).

1994 (1)

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

J. Wang, J. 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).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

W. B. Miller, L. C. Andrews, and J. C. Ricklin, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Bao, C.

Bao, H.

Bock, R.

Boyd, R. W.

Cao, Y.

Chen, H. W.

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

Cheng, W. W.

S. M. Zhao, B. Wang, L. Y. Gong, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the Atmosphere Turbulence Tolerance in Holographic Ghost, Imaging System by Channel Coding,” J. Lightwave Technol. 31(17), 2823–2828 (2013).

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Ding, J.

Djordjevic, I. B.

Dolinar, S.

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

J. Wang, J. 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).

Dolinar, S. J.

Dong, X. L.

Erkmen, B. I.

Fazal, I. M.

J. Wang, J. 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).

Glinou, G. L.

E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).

Gong, L. Y.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

J. Wang, J. 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).

Koulouvari, M. J.

E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).

Lavery, M. P.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
[PubMed]

Leach, J.

Li, L.

Li, S.

Liao, R.

Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

Lyke, S. D.

Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

Malik, M.

Miller, W. B.

Mirhosseini, M.

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Morfiadakis, E. E.

E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).

Neifeld, M. A.

O’Sullivan, M. N.

Padgett, M.

Padgett, M. J.

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Ren, Y.

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

J. Wang, J. 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).

Ricklin, J. C.

Robertson, D. J.

Rodenburg, B.

Roggemann, M. C.

Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

Shapiro, J. H.

Sheng, Y. B.

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Shieh, W.

Tang, Y.

Tur, M.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

J. Wang, J. 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).

Wang, B.

S. M. Zhao, B. Wang, L. Y. Gong, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the Atmosphere Turbulence Tolerance in Holographic Ghost, Imaging System by Channel Coding,” J. Lightwave Technol. 31(17), 2823–2828 (2013).

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Wang, J.

S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
[PubMed]

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

J. Wang, J. 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).

Wang, L.

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

Weeks, A. R.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

J. Wang, J. 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).

Willner, M.

Xie, G.

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

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).
[PubMed]

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

Yan, Y.

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).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

J. Wang, J. 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).

Yang, J.

J. Wang, J. 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).

Yue, Y.

J. Wang, J. 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).

Zhao, S. M.

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

S. M. Zhao, B. Wang, L. Y. Gong, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the Atmosphere Turbulence Tolerance in Holographic Ghost, Imaging System by Channel Coding,” J. Lightwave Technol. 31(17), 2823–2828 (2013).

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).
[PubMed]

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

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).
[PubMed]

Zhao, Z. J.

Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

Zheng, B. Y.

Zhou, L.

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Zou, L.

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

Adv. Opt. Photonics (1)

A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. 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).

Chin. Phys. B (1)

L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B 25(11), 316–323 (2016).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

J. Wind Eng. Ind. Aerodyn. (1)

E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. 62, 237–257 (1996).

Nat. Photonics (1)

J. Wang, J. 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).

Opt. Express (3)

Opt. Lett. (6)

B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
[PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39(10), 2845–2848 (2014).
[PubMed]

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[PubMed]

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).
[PubMed]

S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. 41(7), 1482–1485 (2016).
[PubMed]

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).

Other (6)

V. E. Zakharov, V. S. L’Vov, and G. Falkovich, “Kolmogorov Spectra of Turbulence: I Wave Turbulence,” Springer-Verlag, (1992)

S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

Y. Ren, Y. Zhang, Y. Yue, N. Bozinovic, G. Xie, H. Huang, M. Tur, P. Kristensen, I. B. Djordjevic, S. Ramachandran, and A. E. Willner, “Efficient Crosstalk Mitigation of OAM Based 400-Gbit/s QPSK Data Transmission in 1.1-km Vortex Fiber by Using Soft-Decision LDPC codes,” in CLEO (2013), paper CM2G.5.

Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

J. Wang, S. Li, C. Li, L. Zhu, C. Gui, D. Xie, Y. Qiu, Q. Yang, and S. Yu, “Ultra-high 230-bit/s/Hz spectral efficiency using OFDM/OQAM 64-QAM signals over pol-muxed 22 orbital angular momentum (OAM) modes,” in OFC/NFOEC (2014), paper W1H. 4.

J. Wang, J. Liu, S. Li, L. Zhu, C. Li, M. Luo, Q. Yang, and S. Yu, “Experimental Demonstration of Free-Space Optical Communications Using OFDM-QPSK/16QAM-Carrying Fractional Orbital Angular Momentum (OAM) Multiplexing,” in OFC/NFOEC (2015), paper M2F.5.

Cited By

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

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Simulation of OAM beam propagation through atmospheric turbulence with the multi-phase screen method: (a) phase hologram of OAM + 1; (b) random turbulent phase screen obeying the modified Hill spectrum generated by MATLAB; (c) phase hologram of OAM + 1 with turbulent information.
Fig. 2
Fig. 2 Experimental setup (AWG, arbitrary waveform generator; LPF, low-pass filter; EA, electrical amplifier; DFB, distributed feedback laser; PMF, polarization-maintenance fiber; Col, collimator; SLM, spatial light modulator; BS, beam splitter; SMF, single mode fiber; DSO, digital storage oscilloscope): (a) block type pilot arrangement; (b)(c) the fork holograms when l=+1 and l=1 loaded at SLM1 and SLM2 for OAM beams generation respectively; (d)(e) the demodulating phase holograms with turbulent information at SLM3 and SLM4; (f) the process diagram of OFDM signal generation; (g) the process diagram of offline DSP.
Fig. 3
Fig. 3 The intensity profiles of OAM + 1 and OAM-1 multiplexed beams’ demodulation by the phase holograms of OAM-1: (a) The phase hologram with different turbulence; (b) the demodulation intensity profile when OAM + 1 and OAM-1 are on; (c) the demodulation intensity profile when OAM + 1 is on and OAM-1 is off; (d) the demodulation intensity profile when OAM-1 is on and OAM + 1 is off.
Fig. 4
Fig. 4 The average normalized received power of each collimator as a function of C n 2 : (a) Col3 (channel of OAM + 1); (b) Col4 (channel of OAM-1).
Fig. 5
Fig. 5 The decomposition of demodulated OAM beam with turbulence.
Fig. 6
Fig. 6 The bit-error rate (BER) performance of 16QAM-OFDM signals over channel OAM + 1 and OAM-1 with different turbulence: (a) C n 2 =1.5× 10 15 m 2/3 ; (b) C n 2 =9.5× 10 15 m 2/3 ; (c) C n 2 =1× 10 13 m 2/3 .
Fig. 7
Fig. 7 Recovered constellations of channel OAM + 1 and OAM-1 with and without pilot assisted LS algorithm: (a) C n 2 =1.5× 10 15 m 2/3 ; (b) C n 2 =9.5× 10 15 m 2/3 ; (c) C n 2 =1× 10 13 m 2/3 .
Fig. 8
Fig. 8 Measured BER performance with and without pilot assisted LS algorithm under C n 2 from 5× 10 16 m 2/3 to 1× 10 13 m 2/3 : (a) OAM + 1; (b) OAM-1.

Tables (1)

Tables Icon

Table 1 Transceiver Parameters

Equations (10)

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

Φ n ( k x , k y )= 0.033 C n 2 ( k r 2 + k 0 2 ) 11 6 ( 1+ a 1 ( k r k l ) a 2 ( k r k l ) 7 6 )exp( k r 2 k l 2 )
φ( x,y )=FFT( Cσ( k x , k y ) )
L G l m 1 =IFFT( FFT( L G l m ) U prop ) e i φ 1 ( x,y )
L G l m n =IFFT( FFT( ( IFFT( FFT( L G l m ) U prop ) e i φ 1 ( x,y ) ) ) U prop ) e i φ n ( x,y )
Y p ( k )= H p ( k ) X p ( k )+ W p ( k )
H ^ p ( k )= Y p ( k ) X p ( k ) = H p ( k )+ W p ( k ) X p ( k )
H ^ p =diag ( X p ) 1 Y p
Y p =diag( X p )Fh+ W p
h ^ p =Q F diag ( X p ) Y p
H ^ p =FFT( h ^ p )

Metrics