## Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere

Optics Express, Vol. 16, Issue 11, pp. 7665-7673 (2008)

http://dx.doi.org/10.1364/OE.16.007665

Acrobat PDF (512 KB)

### Abstract

Analytical formulas are derived for the average irradiance and the degree of polarization of a radially or azimuthally polarized doughnut beam (PDB) propagating in a turbulent atmosphere by adopting a beam coherence-polarization matrix. It is found that the radial or azimuthal polarization structure of a radially or azimuthally PDB will be destroyed (i.e., a radially or azimuthally PDB is depolarized and becomes a partially polarized beam) and the doughnut beam spot becomes a circularly Gaussian beam spot during propagation in a turbulent atmosphere. The propagation properties are closely related to the parameters of the beam and the structure constant of the atmospheric turbulence.

© 2008 Optical Society of America

## 1. Introduction

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

6. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

6. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

10. E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett. **32**, 3417–3419 (2007). [CrossRef] [PubMed]

11. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B **23**, 1228–1234 (2006). [CrossRef]

## 2. Formulation

_{01}with a polarization direction parallel to the x-axis and a TEM

_{10}with a polarization direction parallel to the y-axis [1

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

7. R. Oron, S. Blit, N. Davidson, and A. A. Friesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. **77**, 3322–3324 (2000). [CrossRef]

*r*

^{2}=

*x*

^{2}+

*y*

^{2},

*w*

_{0}denotes the beam waist size of a Gaussian beam,

*E*

_{0}is a constant. In a similar way, the vectorial electric field of an azimuthally PDB is expressed as follows

11. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B **23**, 1228–1234 (2006). [CrossRef]

27. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. **28**, 1084–1086 (2003). [CrossRef] [PubMed]

*z*is the propagation axis) is defined as follows [28

28. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A **20**, 1974–1980 (2003). [CrossRef]

*E*

_{1}and

*E*

_{2}are the components of the vectorial electric field in the x and y directions, respectively, and the angle brackets denote an ensemble average over the medium statistics. The equivalent irradiance distribution of a polarized beam is given by [28

28. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A **20**, 1974–1980 (2003). [CrossRef]

_{r}(

**r**, 0)=P

_{θ}(

**r**, 0)=1 at z=0.

_{αβ}(

**r**,

**r**,

*z*) at the output plane are given as follows [14-26

14. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. **11**, 1399–1406 (1972). [CrossRef] [PubMed]

_{αβ}(

**r**′

_{1},

**r**′

_{2}, 0) is given by Eq. (4) and

*d*

**r**

_{1}

*d*

**r**

_{2}=

*dx*

_{1}

*dy*

_{1}

*dx*

_{2}

*dy*

_{2}.

*ρ*

_{0}=(0.545

*C*

^{2}

_{n}

*k*

^{2}

*z*)

^{-3/5}is the coherence length (induced by the atmospheric turbulence) of a spherical wave propagating in the turbulent medium with

*C*

^{2}

_{n}being the structure constant [12-26],

*k*=2

*π*/

*λ*is the wavenumber and

*λ*is the wavelength of the light. In the derivation of Eq. (9), we have employed Kolmogorov spectrum and a quadratic approximation for Rytov’s phase structure function [12-26]. The extended Huygens-Fresnel integral formula Eq. (9) has been approved to be reliable in e.g. Refs. [14]-[16]

14. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. **11**, 1399–1406 (1972). [CrossRef] [PubMed]

17. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A **19**, 1794–1802 (2002). [CrossRef]

_{αβ}(

**r**

_{1},

**r**

_{2},0) in Eq. (7) as Γ

_{αβ}(

**r**′

_{1},

**r**′

_{2},0) into Eq. (9), after some tedious integration, we obtain the following expressions for the elements of BCP matrix of a radially PDB in a turbulent atmosphere

*A*

_{1}=

*k*

^{2}

*ρ*

^{2}

_{0}

*w*

^{4}

_{0}+4

*z*

^{2}(

*ρ*

^{2}

_{0}+2

*w*

^{2}

_{0}). In a similar way, we obtain the following expressions for the elements of BCP matrix of an azimuthally PDB

*ρ*

_{0}→∞,i.e.,

*C*

^{2}

_{n}=0), Eqs. (10)-(13) reduce to the expressions for a radially or azimuthally PDB in free space under the paraxial condition [6

6. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

11. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B **23**, 1228–1234 (2006). [CrossRef]

*ϕ*with the x-axis. Then the irradiances of a radially PDB and an azimuthally PDB at z becomes

*ϕ*.

## 3. Numerical examples

*x*,

*y*,

*z*) of a radially PDB and the corresponding cross line (y=0) at several propagation distance with

*λ*=632.8

*nm*,

*w*

_{0}=2

*cm*and

*C*

^{2}

_{n}=10

^{-15}

*m*

^{-2/3}by applying Eqs. (6) and (10)-(12). For comparison, the far-field (z=15

*k*m) degree of polarization of a radially PDB and the corresponding cross line (y=0) are also shown in Fig. 5. What’s more, we also calculate in Fig. 6 the cross line (y=0) of the degree of polarization P(

*x*,0,

*z*) of a radially PDB at z=15km in a turbulent atmosphere for different values of

*w*

_{0}and

*C*

^{2}

_{n}with

*λ*=632.8

*nm*. From the other tests conducted and from the illustration in Fig. 5, one finds that the degree of polarization of a radially PDB in free space equals 1 for all the points across the entire transverse plane and remains invariant during propagation, which means that radial polarization structure of a radially PDB in free space will not be destroyed during propagation in free space. While in a turbulent atmosphere, one finds from Fig. 5 that the a dip appears in the distribution of the degree of polarization, in other words, the degree of polarization of the on-axis point becomes zero after propagation and the degree of polarization of the off-axis point rises gradually towards the edges of the off-axis regions. One also finds that the width of the dip increases during propagation. Thus, one comes to the conclusion that the radial polarization structure of a radially PDB is destroyed during propagation in a turbulent atmosphere (i.e., a radially PDB is depolarized during propagation), and the radially PDB becomes a partially polarized beam. What’s more, one finds from Fig. 6 that the depolarization of an off-axis point becomes larger for a larger

*w*

_{0}or a larger

*C*

^{2}

_{n}.

## 4. Conclusion

## Acknowledgments

## References and links

1. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a radially polarized light beam,” Phys. Rev. Lett. |

2. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

3. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express |

4. | B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. |

5. | L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. |

6. | A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A |

7. | R. Oron, S. Blit, N. Davidson, and A. A. Friesem, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. |

8. | J. Li, K. I. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, “Generation of radially polarized mode in Yb fiber laser by using a dual conical prism,” Opt. Lett. |

9. | Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. |

10. | E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett. |

11. | D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B |

12. | A. Ishimaru, |

13. | T. L. Ho, “Coherence degration of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am |

14. | H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. |

15. | S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. |

16. | J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. |

17. | J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A |

18. | Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. |

19. | Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. |

20. | H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A |

21. | H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B |

22. | X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express15, 17613–17618 (2007). Z. Chen, J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 9, 1123–1130 (2007). [CrossRef] [PubMed] |

23. | R.J. Noriega-Manez and J. C. Gutierrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express15, 16328–16341 (2007). Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405–2407 (2007). [CrossRef] [PubMed] |

24. | M. Alavinejad, B. Ghafary, and F. D. Kashani, “Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere,” Opt. Lasers Eng. |

25. | K. Duan and B. Lu, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A |

26. | F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. |

27. | Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. |

28. | J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(260.5430) Physical optics : Polarization

**ToC Category:**

Atmospheric and oceanic optics

**History**

Original Manuscript: March 10, 2008

Revised Manuscript: April 28, 2008

Manuscript Accepted: May 7, 2008

Published: May 12, 2008

**Citation**

Yangjian Cai, Qiang Lin, Halil T. Eyyuboglu, and Yahya Baykal, "Average irradiance and polarization properties
of a radially or azimuthally polarized beam in a
turbulent atmosphere," Opt. Express **16**, 7665-7673 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7665

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### References

- R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]
- Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004). [CrossRef] [PubMed]
- B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000). [CrossRef] [PubMed]
- L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001). [CrossRef] [PubMed]
- A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998). [CrossRef]
- R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000). [CrossRef]
- J. Li, K. I. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, "Generation of radially polarized mode in Yb fiber laser by using a dual conical prism," Opt. Lett. 31, 2969-2971 (2006). [CrossRef] [PubMed]
- Z. Bomzon, V. Kleiner and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings," Appl. Phys. Lett. 79, 1587-1589 (2001). [CrossRef]
- E. Y. S. Yew and C. J. R. Sheppard, "Tight focusing of radially polarized Gaussian and Bessel-Gauss beams," Opt. Lett. 32, 3417-3419 (2007). [CrossRef] [PubMed]
- D. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006). [CrossRef]
- A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic Press, New York, 1978).
- T. L. Ho, "Coherence degration of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673 (1970). [CrossRef]
- H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972). [CrossRef] [PubMed]
- 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, 1297-1304 (1979). [CrossRef]
- J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175-185 (1978). [CrossRef]
- J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002). [CrossRef]
- Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006). [CrossRef]
- Y. Cai and S. He, "Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere," Opt. Lett. 31, 568-570 (2006). [CrossRef] [PubMed]
- H. T. Eyyubo�?lu, Y. Baykal and Y. Cai, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891-2901 (2007). [CrossRef]
- H. T. Eyyubo�?lu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007). [CrossRef]
- X. Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007). [CrossRef] [PubMed]
- Z. Chen, J. Pu, "Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere," J. Opt. A: Pure Appl. Opt. 9, 1123-1130 (2007). [CrossRef] [PubMed]
- R. J. Noriega-Manez and J. C. Gutierrez-Vega, "Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere," Opt. Express 15, 16328-16341 (2007). [CrossRef]
- Y. Cai, Y. Chen, H. T. Eyyubo�?lu, and Y. Baykal, "Scintillation index of elliptical Gaussian beam in turbulent atmosphere," Opt. Lett. 32, 2405-2407 (2007). [CrossRef]
- M. Alavinejad, B. Ghafary, and F. D. Kashani, "Analysis of the propagation of flat-topped beam with various beam orders through turbulent atmosphere," Opt. Lasers Eng. 46, 1-5 (2008). [CrossRef]
- K. Duan and B. Lu, "Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture," J. Opt. Soc. Am. A 21, 1613-1620 (2004). [CrossRef] [PubMed]
- F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998). [CrossRef]
- Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084-1086 (2003).
- J. Tervo, "Azimuthal polarization and partial coherence," J. Opt. Soc. Am. A 20, 1974-1980 (2003).

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