OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 11 — May. 26, 2008
  • pp: 7665–7673
« Show journal navigation

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

Yangjian Cai, Qiang Lin, Halil T. Eyyuboğlu, and Yahya Baykal  »View Author Affiliations


Optics Express, Vol. 16, Issue 11, pp. 7665-7673 (2008)
http://dx.doi.org/10.1364/OE.16.007665


View Full Text Article

Acrobat PDF (512 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Recenlty, radially and azimuthally polarized beams attract more and more attention due to their unique focusing properties and important applications in optical data storage, particle trapping and acceleration, high-resolution microscopy, laser cutting, and determination of single fluorescent molecule orientation, etc [1-8

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

]. Radially polarized beams can be generated either inside a laser resonator, e.g. by using a conical mirror or a conical Brewster element or outside a laser cavity, e.g. by using a space-invariant dielectric subwavelength gratings, a dual conical prism or an interferometric technique [6-9

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

]. The focusing properties, paraxial and nonparaxial propagation properties through paraxial optical system or free space have been widely studied [1-3

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

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

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

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

]. To the best of our knowledge, the propagation properties of a radially or azimuthally polarized beam in a turbulent atmosphere have not been studied so far.

2. Formulation

Within the framework of the paraxial approximation, the vectorial electric field of a radially PDB is expressed as the coherent superposition of a TEM01 with a polarization direction parallel to the x-axis and a TEM10 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

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]

]

Er(x,y)=E1ex+E2ey=E0[xw0exp(r2w02)ex+yw0exp(r2w02)ey],
(1)

where 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

Eθ(x,y)=E0[yw0exp(r2w02)ex+xw0exp(r2w02)ey].
(2)

Here we do not consider the longitudinal electric field component, since it is know to be negligible under the paraxial condition [11

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

, 27

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

]. The beam coherence-polarization (BCP) matrix provides the information of polarization and spatial correlation, and the BCP matrix for a vectorial electric field across a typical plane z=constant (here 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]

]

Γ̂(r1,r2,z)=(Γ11(r1,r2,z)Γ12(r1,r2,z)Γ21(r1,r2,z)Γ22(r1,r2,z)),
(3)

where

Γαβ(r1,r2,z)=Eα(r1,r2,z)Eβ*(r1,r2,z),(α,β=1,2)
(4)

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]

]

I(r,z)=Γ11(r,r,z)+Γ22(r,r,z),
(5)

and the degree of polarization is expressed as

P(r,z)=14det[Γ̂(r,r,z)]{Tr[Γ̂(r,r,z)]}2,
(6)

where det and Tr stand for determinant and trace of the BCP matrix, respectively.

By applying Eqs. (1)-(4), the BCP matrices for a radially PDB and an azimuthally PDB at source plane (z=0) are expressed as follows

Γ̂r(r1,r2,0)=E02w02exp(r12+r22w02)(x1x2x1y2y1x2y1y2),
(7)
Γ̂θ(r1,r2,0)=E02w02exp(r12+r22w02)(y1y2x2y1x1y2x1x2).
(8)

One finds from Eqs. (5)-(8) that a radially PDB and an azimuthually PDB have the same irradiance distribution and degree of polarization Pr(r, 0)=Pθ(r, 0)=1 at z=0.

Figure 1 shows the propagation geometry of a radially or azimuthally PDB in a turbulent atmosphere. The paraxial propagation of a laser beam in a turbulent atmosphere can be treated with the well-known extended Huygens-Fresnel integral formula, and the elements of BCP matrix Γαβ (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]

]

Γαβ(r,r,z)=k24π2z2Γαβ(r1,r2,0)exp[ik2z(r1r)2+ik2z(r2r)2]
×exp[1ρ02(r1r2)2]dr1dr2,
(9)

Γαβ (r1, r2, 0) is given by Eq. (4) and d r 1 d r 2=dx 1 dy 1 dx 2 dy 2. ρ 0=(0.545C 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

12. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

], k=2π/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

12. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

]. 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]

, and has been used widely (see e.g. Refs. [17]-[26]

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]

).

Fig. 1. Propagation geometry of a radially or azimuthally PDB in a turbulent atmosphere

Substituting Γαβ (r 1,r 2,0) in Eq. (7) as Γαβ (r1,r2,0) 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

Γr11(r,r,z)=E022k2ρ04w06z2A12[1ρ02+k2ρ02(k2w04+4z2)2z2A1x2]exp[2k2ρ02w02A1r2],
(10)
Γr12(r,r,z)=Γr21(r,r,z)=E02k4ρ06w06(k2w04+4z2)xyA13exp[2k2ρ02w02A1r2],
(11)
Γr22(r,r,z)=E022k2ρ04w06z2A12[1ρ02+k2ρ02(k2w04+4z2)2z2A1y2]exp[2k2ρ02w02A1r2],
(12)

with A 1=k 2 ρ 2 0 w 4 0+4z 2 (ρ 2 0+2w 2 0). In a similar way, we obtain the following expressions for the elements of BCP matrix of an azimuthally PDB

Γθ11(r,z)=Γr22(r,z),Γθ22(r,z)=Γr11(r,z),Γθ12(r,z)=Γθ21(r,z)=Γr12(r,z).
(13)

In the absence of turbulence(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

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

]. One finds from Eqs. (5), (6) and (10)-(13) that a radially PDB and an azimuthally PDB have the same irradiance distribution and degree of polarization in a turbulent atmosphere. To distinguish a radially PDB and an azimuthally PDB in a turbulent atmosphere, it is necessary to introduce some linear polarizer. Suppose that a linear polarizer is located at a position z, whose transmission axis forms an angle ϕ with the x-axis. Then the irradiances of a radially PDB and an azimuthally PDB at z becomes

Ir(r,z)=Γr11(r,z)cos2ϕ+Γr22(r,z)sin2ϕ+Γr12(r,z)sin2ϕ,
(14)
Iθ(r,z)=Γθ11(r,z)cos2ϕ+Γθ22(r,z)sin2ϕ+Γθ12(r,z)sin2ϕ.
(15)

One finds from Eqs. (10)-(15) that we can distinguish a radially PDB and an azimuthally PDB by measuring their irradiances behind the linear polarizer with proper ϕ.

3. Numerical examples

Fig. 2. Cross line (y=0) of the normalized irradiance distribution I(x,0,z)/I(x,0,0)max of a radially PDB at several propagation distances in a turbulent atmosphere for two different values of structure constant C 2 n
Fig. 3. Normalized 3D-irradiance distribution max I(x,y,z)/I(x,0,0)max of a radially PDB and the corresponding contour graph at several propagation distances in a turbulent atmosphere with w 0=2cm and C 2 n=3×10-15 m -2/3
Fig. 4. Normalized on-axis irradiance distribution I(0,0,z)/I(x,0,0)max of a radially PDB along z in a turbulent atmosphere for different values of W 0 and C 2 n
Fig. 5. Degree of polarization P(x, y, z) of a radially PDB and the corresponding cross line (y=0) at several propagation distance with w 0=2cm and C 2 n=10-15 m -2/3
Fig. 6. Cross line (y=0) 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

To learn about the influence of the atmospheric turbulence on the polarization properties of a radially PDB in a turbulent atmosphere, we calculate in Fig. 5 the degree of polarization P(x, y,z) of a radially PDB and the corresponding cross line (y=0) at several propagation distance with λ=632.8nm, w 0=2cm and C 2 n=10-15 m -2/3 by applying Eqs. (6) and (10)-(12). For comparison, the far-field (z=15km) 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=632.8nm. 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

Y. Cai gratefully acknowledges the supports from the Alexander von Humboldt Foundation. Q. Lin gratefully acknowledges the supports from the Ministry of Science and Technology of China (grant no. 2006CB921403 & 2006AA06A204).

References and links

1.

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

2.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

3.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef] [PubMed]

4.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000). [CrossRef] [PubMed]

5.

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]

6.

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

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]

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. 31, 2969–2971 (2006). [CrossRef] [PubMed]

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. 79, 1587–1589 (2001). [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]

12.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, (Academic Press, New York, 1978).

13.

T. L. Ho, “Coherence degration of Gaussian beams in a turbulent atmosphere,” J. Opt. Soc. Am 60, 667–673 (1970). [CrossRef]

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]

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. 69, 1297–1304 (1979). [CrossRef]

16.

J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–185 (1978). [CrossRef]

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]

18.

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]

19.

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]

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 24, 2891–2901 (2007). [CrossRef]

21.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89, 91–97 (2007). [CrossRef]

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. 46, 1–5 (2008). [CrossRef]

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 21, 1613–1620 (2004). [CrossRef]

26.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998). [CrossRef]

27.

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

28.

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

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  2. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]
  3. Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004). [CrossRef] [PubMed]
  4. B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000). [CrossRef] [PubMed]
  5. 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]
  6. A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2705-2711 (1998). [CrossRef]
  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]
  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. 31, 2969-2971 (2006). [CrossRef] [PubMed]
  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. 79, 1587-1589 (2001). [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]
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic Press, New York, 1978).
  13. T. L. Ho, "Coherence degration of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am 60, 667-673 (1970). [CrossRef]
  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]
  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. 69, 1297-1304 (1979). [CrossRef]
  16. J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175-185 (1978). [CrossRef]
  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]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. X. Chu, "Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere," Opt. Express 15, 17613-17618 (2007). [CrossRef] [PubMed]
  23. 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]
  24. 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]
  25. 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]
  26. 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]
  27. 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]
  28. F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 241-243 (1998). [CrossRef]
  29. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084-1086 (2003).
  30. J. Tervo, "Azimuthal polarization and partial coherence," J. Opt. Soc. Am. A 20, 1974-1980 (2003).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited