Abstract

Close to the ground, it is generally known that atmospheric turbulence exhibits strong anisotropy, which affects the performance of applications such as free-space optical (FSO) communication. In this paper, we establish a theoretical model for calculating the spiral spectrum, also called the orbital angular momentum (OAM) spectrum, of a Laguerre-Gaussian (LG) beam after propagation through anisotropic turbulence along a horizontal link. This model isolates the effects of anisotropy from other parameters of the turbulence. On the basis of this model, the effects of the anisotropy on the probability density of the OAM spectrum and its corresponding modal crosstalk are studied through numerical examples. Our simulation results show that the width of the OAM spectrum will increase or slightly decrease depending on the specific nature of the anisotropy. In addition, it is demonstrated that the inner scale is more likely to cause modal crosstalk than the outer scale. Some strategies to reduce modal crosstalk in anisotropic turbulence are also discussed. Our results may be useful in OAM-based FSO communication at ground level.

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

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  1. 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]
  2. 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. R. Soc., A 375(2087), 20150439 (2017).
    [Crossref]
  3. A. Wang, L. Zhu, L. Wang, J. Ai, S. Chen, and J. Wang, “Directly using 8.8-km conventional multi-mode fiber for 6-mode orbital angular momentum multiplexing transmission,” Opt. Express 26(8), 10038–10047 (2018).
    [Crossref]
  4. 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]
  5. 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]
  6. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref]
  7. N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
    [Crossref]
  8. Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26(17), 21861–21871 (2018).
    [Crossref]
  9. Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  10. P. Li, S. Liu, T. Peng, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
    [Crossref]
  11. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
    [Crossref]
  12. L. Yu, B. Hu, and Y. Zhang, “Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence,” Opt. Express 25(16), 19538–19547 (2017).
    [Crossref]
  13. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
    [Crossref]
  14. A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of Atmospheric Turbulence by Narrow Laser Beams,” Appl. Opt. 9(11), 2543–2547 (1970).
    [Crossref]
  15. R. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IRE Trans. Antennas Propag. 34(2), 258–261 (1986).
    [Crossref]
  16. M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
    [Crossref]
  17. L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
    [Crossref]
  18. L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
    [Crossref]
  19. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
    [Crossref]
  20. C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
    [Crossref]
  21. L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
    [Crossref]
  22. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
    [Crossref]
  23. Y. Ata and Y. Baykal, “Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere,” J. Mod. Opt. 66(2), 130–135 (2019).
    [Crossref]
  24. C. Chen, H. Yang, S. Tong, B. Ren, and Y. Li, “Characterization of temporal pulse broadening for horizontal propagation in strong anisotropic atmospheric turbulence,” Opt. Express 23(4), 4814–4828 (2015).
    [Crossref]
  25. Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
    [Crossref]
  26. L. Cui, “Analysis of angle of arrival fluctuations for optical waves propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
    [Crossref]
  27. L. C. Andrews, R. L. Phillips, and R. Crabbsa, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
    [Crossref]
  28. I. Toselli and O. Korotkova, “General scale-dependent anisotropic turbulence and its impact on free space optical communication system performance,” J. Opt. Soc. Am. A 32(6), 1017–1025 (2015).
    [Crossref]
  29. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
    [Crossref]
  30. X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017).
    [Crossref]
  31. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  32. L. Allen, M. W. Beijersbergen, R. J. C. 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]
  33. M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
    [Crossref]
  34. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [Crossref]
  35. H. T. Yura, “Mutual Coherence Function of a Finite Cross Section Optical Beam Propagating in a Turbulent Medium,” Appl. Opt. 11(6), 1399–1406 (1972).
    [Crossref]
  36. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
    [Crossref]
  37. V. P. Lukin, P. A. Konyaev, and V. A. Sennikov, “Beam spreading of vortex beams propagating in turbulent atmosphere,” Appl. Opt. 51(10), C84–C87 (2012).
    [Crossref]
  38. X. Ge, B. Wang, and C. Guo, “Evolution of phase singularities of vortex beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 32(5), 837–842 (2015).
    [Crossref]

2019 (1)

Y. Ata and Y. Baykal, “Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere,” J. Mod. Opt. 66(2), 130–135 (2019).
[Crossref]

2018 (3)

2017 (4)

2016 (4)

2015 (4)

2014 (5)

2013 (2)

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

2012 (1)

2011 (1)

2008 (3)

2006 (1)

R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
[Crossref]

2005 (2)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref]

1999 (1)

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. 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]

1986 (1)

R. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IRE Trans. Antennas Propag. 34(2), 258–261 (1986).
[Crossref]

1972 (1)

1970 (1)

Agrawal, B.

Ahmed, N.

Ai, J.

Allen, L.

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbsa, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

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

Anguita, J. A.

Ashrafi, N.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Ata, Y.

Y. Ata and Y. Baykal, “Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere,” J. Mod. Opt. 66(2), 130–135 (2019).
[Crossref]

Barnett, S. M.

R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
[Crossref]

Baykal, Y.

Y. Ata and Y. Baykal, “Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere,” J. Mod. Opt. 66(2), 130–135 (2019).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Belen’Kii, M. S.

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Bock, R.

Brownn II, J. M.

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Cai, Y.

Cao, X.

L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Carrasco, S.

Chen, C.

Chen, S.

Cheng, M.

Chu, X.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Conan, J. M.

Consortini, A.

Crabbs, R.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Crabbsa, R.

L. C. Andrews, R. L. Phillips, and R. Crabbsa, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

Cui, L.

L. Cui, “Analysis of angle of arrival fluctuations for optical waves propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
[Crossref]

L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Dalaudier, F.

Djordjevic, I. B.

Du, C.

Dudley, A.

Fan, C.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Feng, X.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Forbes, A.

Fugate, R. Q.

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Gan, X.

Gao, J.

Ge, X.

Guo, C.

Guo, L.

Hu, B.

Huang, H.

Huang, Q.

Jiang, Y.

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Karis, S. J.

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Konyaev, P. A.

Korotkova, O.

Lavery, M. P. J.

Leclerc, T.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Li, J.

Li, L.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Li, N.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Li, P.

Li, Y.

Liao, P.

Liu, D.

Liu, J.

Liu, S.

Liu, X.

Lukin, V. P.

Manning, R.

R. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IRE Trans. Antennas Propag. 34(2), 258–261 (1986).
[Crossref]

Michau, V.

Mo, Q.

Molisch, A. F.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

Neifeld, M. A.

Osmon, C. L.

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Peng, T.

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbsa, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

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

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Qiao, C.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Qu, J.

Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26(17), 21861–21871 (2018).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Ren, B.

Ren, Y.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Renard, J. B.

Restaino, S.

Robert, C.

Ronchi, L.

Salem, A. B.

Sennikov, V. A.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Stefanutti, L.

Sun, R.

Tang, H.

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Tong, S.

Torner, L.

Torres, J. P.

Toselli, I.

Trichili, A.

Tur, M.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Van Der Veen, H.E. L. O.

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Vasic, B. V.

Wang, A.

Wang, B.

Wang, J.

A. Wang, L. Zhu, L. Wang, J. Ai, S. Chen, and J. Wang, “Directly using 8.8-km conventional multi-mode fiber for 6-mode orbital angular momentum multiplexing transmission,” Opt. Express 26(8), 10038–10047 (2018).
[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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Wang, L.

Wang, S.

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Wang, Z.

Willner, A.

Willner, A. E.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Xie, G.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Xu, H.

Xue, B.

L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Yan, X.

Yan, Y.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Yang, H.

Yu, L.

Yuan, Y.

Yura, H. T.

Zambrini, R.

R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
[Crossref]

Zghal, M.

Zhang, J.

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Zhang, P.

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Zhang, Y.

Zhao, F.

Zhao, J.

Zhao, Z.

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

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]

Zhou, F.

L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

Zhou, Z.

Zhu, L.

Zhu, Y.

Appl. Opt. (4)

IRE Trans. Antennas Propag. (1)

R. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IRE Trans. Antennas Propag. 34(2), 258–261 (1986).
[Crossref]

J. Mod. Opt. (1)

Y. Ata and Y. Baykal, “Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere,” J. Mod. Opt. 66(2), 130–135 (2019).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

Y. Jiang, S. Wang, J. Zhang, J. Qu, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

M. W. Beijersbergen, L. Allen, H.E. L. O. Van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993).
[Crossref]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (11)

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref]

C. Chen, H. Yang, S. Tong, B. Ren, and Y. Li, “Characterization of temporal pulse broadening for horizontal propagation in strong anisotropic atmospheric turbulence,” Opt. Express 23(4), 4814–4828 (2015).
[Crossref]

L. Cui, “Analysis of angle of arrival fluctuations for optical waves propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
[Crossref]

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]

Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
[Crossref]

X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017).
[Crossref]

L. Yu, B. Hu, and Y. Zhang, “Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence,” Opt. Express 25(16), 19538–19547 (2017).
[Crossref]

A. Wang, L. Zhu, L. Wang, J. Ai, S. Chen, and J. Wang, “Directly using 8.8-km conventional multi-mode fiber for 6-mode orbital angular momentum multiplexing transmission,” Opt. Express 26(8), 10038–10047 (2018).
[Crossref]

Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26(17), 21861–21871 (2018).
[Crossref]

P. Li, S. Liu, T. Peng, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref]

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref]

Opt. Laser Technol. (2)

L. Cui, B. Xue, X. Cao, and F. Zhou, “Atmospheric turbulence MTF for optical waves’ propagation through anisotropic non-Kolmogorov atmospheric turbulence,” Opt. Laser Technol. 63, 70–75 (2014).
[Crossref]

N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018).
[Crossref]

Opt. Lett. (2)

Philos. Trans. R. Soc., A (1)

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. R. Soc., A 375(2087), 20150439 (2017).
[Crossref]

Phys. Rep. (1)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Phys. Rev. Lett. (1)

R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
[Crossref]

Proc. SPIE (3)

L. C. Andrews, R. L. Phillips, and R. Crabbsa, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

M. S. Belen’Kii, S. J. Karis, C. L. Osmon, J. M. Brownn II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Other (1)

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

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Figures (8)

Fig. 1.
Fig. 1. Schematic for the Kolmogorov cascade theory of turbulence. (a) Isotropic atmospheric turbulence. (b) Anisotropic atmospheric turbulence. $L_0$ and $l_0$ denote the outer and inner scale of turbulence, respectively. Turbulent cells (eddies) between the scale size $L_0$ and $l_0$ form the inertial subrange. $a$, $a'$, $b'$ and $c'$ denotes the semi-principal axes of turbulent cells in different directions, respectively.
Fig. 2.
Fig. 2. Schematic diagram of the effects of an anisotropic turbulent atmosphere on the OAM mode.
Fig. 3.
Fig. 3. Different order crosstalk probability of a LG beam with $l=1$ propagating in anisotropic turbulent atmosphere for different values of $\mu _x/\mu _y$.
Fig. 4.
Fig. 4. Detection probability $P_{m=l}$ and the dimensionless variance of crosstalk probability $V$ of a LG beam as a function of $\mu _x/\mu _y$ for different values of $l$.
Fig. 5.
Fig. 5. (a) Detection probability $P_{m=l}$, (b) the variance of crosstalk probability $V$, (c-d) different order crosstalk probability $P_{m=l+\Delta l}$ of a LG beam as a function of propagation distance for several cases of anisotropy.
Fig. 6.
Fig. 6. (a) Detection probability and (b) $T$ of a LG beam plotted as a function of $\alpha$ for several inner scales of turbulence $l_0$ with $\mu _x/\mu _y=3$.
Fig. 7.
Fig. 7. Detection probability of a LG beam propagating in anisotropic turbulent atmosphere for different values of the inner scales of turbulence, outer scales of turbulence, refractive-index structure parameters and beam wavelengths. The anisotropy parameter is chosen to be $\mu _x/\mu _y=3$.
Fig. 8.
Fig. 8. Detection probability of a LG beam propagating in anisotropic turbulence atmosphere along horizontal path as a function of propagation distance for different values of the beam width $w_0$ and the radius of the receiver aperture $R$ with $w_0$=2cm.

Tables (1)

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Table 1. Radius of a LG Beam Spot against Initial Beam Width for Several Propagation Distances

Equations (22)

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D n ( R ) = β C n 2 ( x 2 / μ x 2 + y 2 / μ y 2 + z 2 / μ z 2 ) γ / 2 , l 0 < | R | < L 0 ,
Φ n ( κ ) = 1 4 π 2 κ 2 0 sin ( κ R ) κ R d d R [ R 2 d D n ( R ) d R ] d R ,
Φ n ( κ ) = μ x μ y μ z A ( α ) C ~ n 2 ( μ x 2 κ x 2 + μ y 2 κ y 2 + μ z 2 κ z 2 ) α / 2 , 1 l 0 < | κ | < 1 L 0 , 3 < α < 4 ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π / 2 ) ,
Φ n ( κ , α ) = μ x μ y μ z A ( α ) C ~ n 2 exp [ ( μ x 2 κ x 2 + μ y 2 κ y 2 + μ z 2 κ z 2 ) / κ m 2 ] ( μ x 2 κ x 2 + μ y 2 κ y 2 + μ z 2 κ z 2 + κ 0 2 ) α / 2 , 3 < α < 4 , 0 < | κ | < ,
c ( α ) = { 2 π Γ [ ( 5 α ) / 2 ] A ( α ) 3 } 1 / ( α 5 ) .
U ( r , θ , z ) = 2 π | l | ! 1 w ( z ) [ 2 r w ( z ) ] l exp [ r 2 w 2 ( z ) ] exp ( i l θ ) × exp { i ( l + 1 ) tan 1 ( z / z R ) i k r 2 2 z [ 1 + ( z R / z ) 2 ] } ,
U t u r ( r , θ , z ) = U ( r , θ , z ) exp [ ψ ( r , θ , z ) ] ,
U t u r ( r , θ , z ) = 1 2 π m = a m ( r , z ) exp ( i m θ ) ,
a m ( r , z ) = 1 2 π 0 2 π U t u r ( r , θ , z ) exp ( i m θ ) d θ .
| a m ( r , z ) | 2 = 1 2 π 0 2 π 0 2 π U t u r ( r , θ 1 , z ) U t u r ( r , θ 2 , z ) exp [ i m ( θ 1 θ 2 ) ] d θ 1 d θ 2 .
| a m ( r , z ) | 2 = 1 2 π 0 2 π 0 2 π U ( r , θ 1 , z ) U ( r , θ 2 , z ) exp [ i m ( θ 1 θ 2 ) ] × exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] d θ 1 d θ 2 = 2 π | l | ! 1 2 π [ 1 w ( z ) ] 2 [ 2 r w ( z ) ] 2 l exp [ 2 r 2 w 2 ( z ) ] × 0 2 π 0 2 π exp [ i ( m l ) ( θ 1 θ 2 ) ] exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] d θ 1 d θ 2 ,
exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] = exp { 2 π k 2 z 0 1 d t 0 d 2 κ Φ n ( κ , κ z = 0 ) [ 1 exp ( t r d κ ) ] } ,
exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] = exp { 2 π k 2 z μ x μ y 0 1 d t 0 d κ x d κ y Φ n ( κ ) [ 1 exp ( t r d κ ) ] } ,
exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] = exp { 4 π 2 k 2 z μ x μ y 0 1 d t 0 κ d κ Φ n ( κ ) [ 1 J 0 ( t κ | r d | ) ] } ,
exp [ ψ ( r , θ 1 , z ) + ψ ( r , θ 2 , z ) ] = exp ( T μ z x d 2 μ x 2 ) exp ( T μ z y d 2 μ y 2 ) ,
T = π 2 k 2 z A ( α ) 6 ( α 2 ) C ~ n 2 [ η κ m 2 α exp ( κ 0 2 / κ m 2 ) Γ 1 ( 2 α / 2 , κ 0 2 / κ m 2 ) 2 κ 0 4 α ] ,
x d = r ( cos θ 1 cos θ 2 ) , y d = r ( sin θ 1 sin θ 2 ) ,
| a m ( r , z ) | 2 = 1 π 2 | l | ! [ 1 w ( z ) ] 2 [ 2 r w ( z ) ] 2 l exp [ 2 r 2 w 2 ( z ) ] × 0 2 π 0 2 π exp [ i ( m l ) ( θ 1 θ 2 ) ] exp ( T μ z x d 2 μ x 2 ) exp ( T μ z y d 2 μ y 2 ) d θ 1 d θ 2 .
P m = C m / m = C q .
V = m = P m ( m m ¯ ) 2 ,
r ( z ) = w 0 ( l / 2 ) 1 / 2 1 + ( z / z R ) 2 ,

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