Abstract

The beam spreading and evolution behavior of the intensity profile and coherent vortices of partially coherent, four-petal elliptic Gaussian vortex beams propagating in atmospheric turbulence are studied. The analytical expressions for cross-spectral density function, as well as root mean square (rms) beam width, are derived based on the extended Huygens–Fresnel principle. Results showed that, unlike the partially coherent four-petal Gaussian vortex beams, the partially coherent four-petal elliptic Gaussian vortex beam could change its petal number into six. The dependencies of occurrence, appearance, and transition speed from four- to six-petal profile on the topological charge, the beam order, and the ellipticity factor are illustrated. The far field behaviors of partially coherent four-petal elliptic Gaussian vortex beams propagating in atmospheric turbulence and are compared in free space. Beams with larger topological charge, smaller beam order, and larger ellipticity factor were found to be less influenced by atmospheric turbulence. Further, the ellipticity factor can be used as an additional degree of freedom in controlling the conservation distance of coherence vortices’ topological charge.

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

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References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
    [Crossref] [PubMed]
  4. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55(7), 1757–1764 (2016).
    [Crossref] [PubMed]
  5. M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(11), 1982–1992 (2015).
    [Crossref] [PubMed]
  6. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref] [PubMed]
  7. M. Lavery, S. Barnett, F. Speirits, and M. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014).
    [Crossref]
  8. K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
    [Crossref]
  9. G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
    [Crossref]
  10. Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
    [Crossref]
  11. X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
    [Crossref]
  12. L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
    [Crossref]
  13. D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
    [Crossref]
  14. D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
    [Crossref]
  15. D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
    [Crossref]
  16. M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010).
    [Crossref] [PubMed]
  17. V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
    [Crossref]
  18. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
    [Crossref] [PubMed]
  19. A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
    [Crossref]
  20. H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
    [Crossref]
  21. Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
    [Crossref]
  22. Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
    [Crossref]
  23. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  24. Z. Chen and D. Zhao, “Focusing of an elliptic vortex beam by a square Fresnel zone plate,” Appl. Opt. 50(15), 2204–2210 (2011).
    [Crossref] [PubMed]
  25. H. D. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 4th ed. (Academic, 2008).
  26. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979).
    [Crossref]
  27. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28(8), 610–612 (2003).
    [Crossref] [PubMed]
  28. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
    [Crossref]
  29. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
    [Crossref] [PubMed]
  30. X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
    [Crossref]
  31. X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
    [Crossref]

2018 (2)

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

2017 (2)

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

2016 (2)

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55(7), 1757–1764 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

M. Lavery, S. Barnett, F. Speirits, and M. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014).
[Crossref]

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

2013 (1)

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

2011 (5)

2010 (3)

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010).
[Crossref] [PubMed]

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

2009 (1)

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

2008 (2)

2007 (2)

2006 (2)

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

2003 (2)

1979 (1)

Bai-Da, L.

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

Barnett, S.

Barnett, S. M.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Cai, Y.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Chen, Z.

Davis, C. C.

Dogariu, A.

Doktorov, E. V.

Du, X.

Duan, K.

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

Gbur, G.

Golmohammady, S.

Gori, F.

Guo, J.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Guo, L.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Guo-Quan, Z.

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

He, X.

Kashani, F. D.

Korotkova, O.

Kotlyar, V.

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Kovalev, A.

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Lavery, M.

Lavery, M. P.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Li, K.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Li, X.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Liu, D.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Liu, X.

Liu, Y.

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
[Crossref]

Long, X.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Lu, K.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Lü, B.

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

Luo, X.

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Luo, Y.

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

Mashal, A.

Molchan, M. A.

Nelson, W.

Padgett, M.

Padgett, M. J.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Plonus, M. A.

Porfirev, A.

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Pu, J.

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
[Crossref]

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
[Crossref] [PubMed]

Santarsiero, M.

Shirai, T.

Speirits, F.

Speirits, F. C.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Sprangle, P.

Tang, Z.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Tyson, R. K.

Visser, T. D.

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
[Crossref]

Vlasov, R. A.

Wan, W.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Wang, F.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Wang, G.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Wang, S. C. H.

Wang, Y.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Wolf, E.

Yan, F.

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

Yan, H.

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

Yin, H.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Yousefi, M.

Zeng-Hui, G.

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

Zhang, Y.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Zhao, D.

Zhong, H.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Chin. Phys. B (1)

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

Chin. Phys. Lett. (1)

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Laser Phys. (1)

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Opt. Commun. (6)

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
[Crossref]

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (2)

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Opt. Lett. (3)

Optica (1)

Optik (Stuttg.) (2)

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

Science (1)

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Other (3)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

H. D. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 4th ed. (Academic, 2008).

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Figures (8)

Fig. 1
Fig. 1 Evolution of normalized intensity distribution of a partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere for (a) z = 200m, (b) z = 300m, (c) z = 500m, (d) z = 1000m, (e) z = 2000m, (f) z = 50000m, respectively. The calculation parameters are seen in the text. All subfigures share the same coordinate axis direction. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.754, 0.564, 0.385, 0.188, 0.060, and 1.61 × 10−5 for (a)-(f), respectively.
Fig. 2
Fig. 2 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with n = 1 and different m. (a-c) m = 2, (d-f) m = 3, (g-i) m = 4, (j-l) m = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.350, 0.201, 1.73 × 10−5, 0.373, 0.183, 1.82 × 10−5, 0.389, 0.165, 1.88 × 10−5, 0.401, 0.153, and 1.92 × 10−5 for (a)-(l), respectively.
Fig. 3
Fig. 3 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1 and different n. (a-c) n = 2, (d-f) n = 3, (g-i) n = 4, (j-l) n = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.291, 0.147, 1.57 × 10−5, 0.255, 0.113, 1.55 × 10−5, 0.260, 0.0936, 1.55 × 10−5, 0.255, 0.0865, and 1.54 × 10−5 for (a)-(l), respectively.
Fig. 4
Fig. 4 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different α. (a-c) α = 0.5, (d-f) α = 0.75, (g-i) α = 1.33, (j-l) α = 2, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.708, 0.373, 5.17 × 10−5, 0.630, 0.270, 3.41 × 10−5, 0.377, 0.200, 1.84 × 10−5, 0.407, 0.138, and 1.15 × 10−5 for (a)-(l), respectively.
Fig. 5
Fig. 5 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. (a-c) m = 1, n = 1, (d-f) m = 5, n = 1, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.406, 0.218, 1.62 × 10−4, 0.414, 0.176, and 1.04 × 10−4 for (a)-(f), respectively.
Fig. 6
Fig. 6 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different σ. (a-c) σ = 20mm, (d-f) σ = 10mm, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.296, 0.139, 1.57 × 10−5, 0.191, 0.0977, 1.44 × 10−5 for (a)-(f), respectively.
Fig. 7
Fig. 7 relative normalized rms beam width of partially coherent four-petal elliptic Gaussian vortex beams with (a) n = 1, α = 1.5, and different m, (b) m = 1, α = 1.5, and different n, and (c) m = 1, n = 1, and different α. Insets show enlarged view of curves for z<6000m.
Fig. 8
Fig. 8 Curves of Re μ = 0 and Im μ = 0 of a partially coherent four-petal elliptic Gaussian vortex beam with m = 1, n = 1 and (a) z = 200m, (b) z = 500m and (c) z = 1000m.

Tables (1)

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Table 1 Conservation distance d for different values of α

Equations (37)

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W( r 1 , r 2 ,0 )= p( v ) H * ( r 1 ,v )H( r 2 ,v )dv,
W( r 1 , r 2 ,0 )= τ * ( r 1 )τ( r 2 ) p ˜ [ a( r 1 r 2 ) ],
τ( x 0 , y 0 ,0 )= ( x 0 y 0 ω 0 2 ) 2n exp( x 0 2 + α 2 y 0 2 ω 0 2 ) ( x 0 2 + α 2 y 0 2 ω 0 2 ) | m |/2 exp[ im tan 1 ( α y 0 / x 0 ) ],
W( r 01 , r 02 ,0 )= ( x 01 y 01 ω 0 2 ) 2n ( x 02 y 02 ω 0 2 ) 2n exp( x 01 2 + α 2 y 01 2 + x 02 2 + α 2 y 02 2 ω 0 2 ) ( 1 ω 0 2 ) | m | [ x 01 isgn( m )α y 01 ] | m | [ x 02 +isgn( m )α y 02 ] | m | exp[ ( r 01 r 02 ) 2 2 σ 2 ].
( x+iy ) M = l=0 M M! i l l!( Ml )! x Ml y l ,
W( r 01 , r 02 ,0 )= 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 x 01 | m | m 1 +2n y 01 m 1 +2n x 02 | m | m 2 +2n y 02 m 2 +2n exp( x 01 2 + x 02 2 ω 0 2 )exp( y 01 2 + y 02 2 ω 0 2 / α 2 )exp[ ( x 01 x 02 ) 2 + ( y 01 y 02 ) 2 2 σ 2 ],
W( r 1 , r 2 ,z )= ( k 2πz ) 2 d r 01 d r 02 W( r 01 , r 02 ,0 ) exp{ ik 2z [ ( r 1 r 01 ) 2 ( r 2 r 02 ) 2 ] } exp[ Ψ * ( r 01 , r 1 )+Ψ( r 02 , r 2 ) ] ,
exp[ Ψ * ( r 01 , r 1 )+Ψ( r 02 , r 2 ) ] =exp{ [ ( r 01 r 02 ) 2 +( r 01 r 02 )( r 1 r 2 )+ ( r 1 r 2 ) 2 ] ρ 0 2 },
x n exp( p x 2 +2qx )dx=n!exp( q 2 p ) π p ( q p ) n s=0 E[ n 2 ] 1 (n2s)!s! ( p 4 q 2 ) s = π p 2 n i n exp( q 2 p ) ( 1 p ) n/2 H n ( iq p ),
W( r 1 , r 2 ,z )= ( k 2πz ) 2 1 ω 0 8n+2| m | exp[ ik 2z ( x 1 2 x 2 2 + y 1 2 y 2 2 ) ] exp[ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ρ 0 2 ] m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 W x W y ,
W x = n x1 ! π A x π D x exp[ 1 A x ( x 2 x 1 2 ρ 0 2 + ik x 1 2z ) 2 + E x 2 D x ] s=0 E( n x1 /2 ) h=0 n x1 2s 1 ( n x1 2s )!s! 4 s A x ( n x1 s ) ( n x1 2s h ) ( x 2 x 1 2 ρ 0 2 + ik x 1 2z ) n x1 2sh ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D x ) | m | m 2 +2n+h H | m | m 2 +2n+h ( i E x D x ),
n x1 =| m | m 1 +2n,
A x = 1 ρ 0 2 + 1 2 σ 2 + 1 ω 0 2 + ik 2z ,
D x = 1 A x ( 1 ρ 0 2 + 1 2 σ 2 ) 2 + 1 ρ 0 2 + 1 2 σ 2 + 1 ω 0 2 ik 2z ,
E x =[ 1( 1 ρ 0 2 + 1 2 σ 2 ) 1 A x ] x 1 x 2 2 ρ 0 2 +( 1 ρ 0 2 + 1 2 σ 2 ) 1 A x ik x 1 2z ik x 2 2z ,
W y = n y1 ! π A y π D y exp[ 1 A y ( y 2 y 1 2 ρ 0 2 + ik y 1 2z ) 2 + E y 2 D y ] s=0 E( n y1 /2 ) h=0 n y1 2s 1 ( n y1 2s )!s! 4 s A y ( n y1 s ) ( n y1 2s h ) ( y 2 y 1 2 ρ 0 2 + ik y 1 2z ) n y1 2sh ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D y ) m 2 +2n+h H m 2 +2n+h ( i E y D y ),
n y1 = m 1 +2n,
I( r,z )=W( r,r,z ).
ω(z)= r 2 I( r,z ) d 2 r I( r,z ) d 2 r ,
ω(z)= ω 1x + ω 1y ω 2 ,
ω 2 = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R x R y ,
ω 1x = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R 2x R y ,
ω 1y = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R x R 2y ,
R x = n x1 ! π A x π D x s=0 E( n x1 /2 ) h=0 n x1 2s t=0 E[ ( | m | m 2 +2n+h )/2 ] G x [ 1 n x2 +2E( n x2 /2 ) ]Γ( n x2 +1 2 )/ ( F x ) n x2 +1 2 ,
R y = n y1 ! π A y π D y s=0 E( n y1 /2 ) h=0 n y1 2s t=0 E[ ( m 2 +2n+h )/2 ] G y [ 1 n y2 +2E( n y2 /2 ) ]Γ( n y2 +1 2 )/ ( F y ) n y2 +1 2 ,
R 2x = n x1 ! π A x π D x s=0 E( n x1 /2 ) h=0 n x1 2s t=0 E[ ( | m | m 2 +2n+h )/2 ] G x [ 1 n x2 +2E( n x2 /2 ) ]Γ( n x2 +3 2 )/ ( F x ) n x2 +3 2 ,
R 2y = n y1 ! π A y π D y s=0 E( n y1 /2 ) h=0 n y1 2s t=0 E[ ( m 2 +2n+h )/2 ] G y [ 1 n y2 +2E( n y2 /2 ) ]Γ( n y2 +3 2 )/ ( F y ) n y2 +3 2 ,
F x = k 2 4 D x z 2 ( 1 A x ρ 0 2 + 1 2 A x σ 2 1 ) 2 k 2 4 A x z 2 ,
G x = 1 ( n x1 2s )!s! 4 s A x n x1 s ( n x1 2s h ) ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D x ) | m | m 2 +2n+h ( | m | m 2 +2n+h )! ( 1 ) t ( | m | m 2 +2n+h2t )!t! [ 1 D x ( 1 A x ρ 0 2 + 1 2 A x σ 2 1 ) k z ] | m | m 2 +2n+h2t ( ik 2z ) | m | m 1 +2nh2s ,
n x2 = m 2 +4n2t+2| m | m 1 2s,
F y = k 2 4 D y z 2 ( 1 A y ρ 0 2 + 1 2 A y σ 2 1 ) 2 k 2 4 A y z 2 ,
G y = 1 ( n y1 2s )!s! 4 s A y n y1 s ( n y1 2s h ) ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D y ) m 2 +2n+h ( m 2 +2n+h )! ( 1 ) t ( m 2 +2n+h2t )!t! [ 1 D y ( 1 A y ρ 0 2 + 1 2 A y σ 2 1 ) k z ] m 2 +2n+h2t ( ik 2z ) m 1 +2nh2s ,
n y2 = m 1 + m 2 +4n2t2s.
Re[ μ( r 1 , r 2 ,z ) ]=0,
Im[ μ( r 1 , r 2 ,z ) ]=0,
μ( r 1 , r 2 ,z )= W( r 1 , r 2 ,z ) I( r 1 ,z )I( r 2 ,z ) .
ω( z )= ( 1 4 + 1 4 α 2 ) ω 0 2 +( α 2 +1 ω 0 2 + 2 σ 2 ) ( z k ) 2 + 4 ρ 0 2 ( z k ) 2 = I+II+III .