Abstract

An approximate expression of a Bessel-Gaussian beam (BGB) with desired topological charge is introduced using a coherence superposition of decentered Gaussian beams (dGBs). And based on such an expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in turbulent atmosphere are explored. An analytical expression of the average intensity of a BGB with phase singularity propagating through turbulent atmosphere is obtained and analyzed numerically. It is found that intensity profiles of BGBs experienced successive variations and the phase singularity rapidly fades away during propagating in turbulent atmosphere.

©2008 Optical Society of America

1. Introduction

It is well known that the propagation of laser beams through a turbulent atmosphere has many applications in optical communication [1]. The turbulent atmosphere is an inhomogeneous medium in which the refractive index is a function of position and time. When laser beams pass through such a medium, it produces random variations in the amplitude and phase of the laser beams and induces intensity fluctuations and spread of the radiation pattern at the observation plane. In recent years, with the application of laser beams in atmosphere, for example, free space optical links and the propagation of high energy laser beams in atmosphere, turbulent atmosphere has become a very lively area of scientific research and application. The propagation properties of various types of laser beams in turbulent atmosphere have been investigated [212]. However, these studies above cited have been mainly restricted to the beams without optical vortices.

Singular optics is a new branch of modern physical optics deals with a wide class of effects associated with phase singularities in wave fields [13]. The concept of phase singularities or optical vortices in wave fields was emphasized by Nye and Berry [1416]. Higher-order Bessel Gaussian beams (BGBs) are the typical example of singularity beams and have attracted a lot of attention because of their nondiffracting property [17,18]. Although there are a few works to play attention on the propagation of vortex beams in turbulent atmosphere, only the integral expression of the intensity distribution for BGBs in the received plane is provided and the analysis mainly has to be conducted with the numerical evaluations [1924].

In this paper, we introduce the new combination beam through decentered Gaussian beams(dGBs) with different constant phases. It is found that such new combination beam can be made as a quite well approximate of a BGB with desired topological charge. And based on the approximate BGB expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in a turbulent atmosphere are explored. Nowadays beam combination has been a subject of current interest for some practical applications, where laser beams with high power or special beam profiles are required. A variety of laser beams, e.g., laser array beams, dark hollow beams, flat-topped beams and general-type beams have been developed through beam combination, and have found application in atomic physics, high-power laser systems, free-space optical communications and inertial confinement fusion [2530]. However, roughly speaking, in the studies listed above the optical beams entering into the combination beam are assumed to have the same phase.

The paper is organized as follows: Section 2 introduces the dGB combination that can also be expressed a superposition of BGB with change n. And the relative weight coefficient of the mode components with the desired topological charge is calculated analytically and numerically. It is found that, carefully adjusting the parameters, a BGB can be represented as the combination of these dGBs. Then, based on the combination beam, the intensity profiles of BGBs with phase singularity propagating through turbulent atmosphere are investigated in detail in Section 3. Some numerical examples are presented to illustrate the influence of atmospheric turbulence and the source beam parameters on the propagating beam properties. Finally, some conclusions are summed in Section 4.

2 dGB expansion of BGBs

In this paper we consider the beam combination

E(x,y,0)=m=0M1Em(x,y,0)

where

Em(x,y,0)=exp[(xi2Rcosθm)2+(yi2Rsinθm)2w02+iφm]

is a dGB with a constant phase φ m=nmα 0(m=0,1,2,…,M-1 and α 0 2π/M), w 0 is its beam waist size, θ m= 0, while n is a integer termed as a spiral number or topological charge. As the combination beam of M Gaussian beams can represent a hollow beam [2730], the combination beam (1) with an appropriate choice of parameters can mimic an optical vortex beam with desired charge n. Without loss of generality, an unimportant amplitude constant factor has been omitted.

In fact, transforming into the polar coordinates (ρ,θ) and using the Bessel function expansion exp(ixcosφ)=p=inJp(x)exp(ipφ) [31], Eq. (1) is reduced to

E(ρ,φ,0)=exp(ρ214R2w02)p=ipJp(Rρw02)exp(ipφ)m=0M1exp[im(np)α0]

Considering

m=0M1exp[im(np)α0]=Mδpn,Mq(q=0,±1,)

as well as performing the sum in Eq. (2) on m, finally, Eq. (3) is turned to

E(ρ,φ,0)=Mexp(ρ214R2w02)q=in+MqJn+Mq(Rρw02)exp[i(n+Mq)φ]

The relative weight of the beam component with charge n+Mq in the combination beam (5) is defined by [16, 31]

an+Mq=0exp(2ρ2)Jn+Mq2(Rρw0)ρdρq=0exp(2ρ2)Jn+Mq2(Rρw0)ρdρ
=In+Mq(R24w02)q=In+Mq(R24w02)

Numerical calculations demonstrate that in the sum (5) only the term with q=0 is important when M is sufficient large or R is appropriate small. Therefore, a BGB with charge n can be quite well approximated as a combination of M dGBs, that is,

E(x,y,0)=Jn(Rρw02)exp(ρ2w02+inφ)
1Minexp(R24w02)m=0M1exp[(xi2Rcosθm)2+(yi2Rsinθm)2w02+iφm]

This is one of the main results obtained in this paper. And with the aid of the approximate expansion (7) of nth-order BGB, to obtain an analytical expression for the vortex beam with charge n propagating through turbulent atmosphere is possible.

Carefully adjusting the parameters n, M, and R, the results evaluated by Eq. (7) and directly by the BGB may be quite well consistent. For example, for M=12, R=4w 0 and n=1, 2, 3 and 4, the numerical calculations show that, from the point of view of energy, the relative weight of the component with charge n is over 99%, as seen from the Table 1, which means that, in the combination beam, over 99% of the optical energy is converted into a BGB with charge n. In Fig. 1 these intensity profiles and corresponding phase distributions are drawn, which clearly represent optical vortex characteristics with desired charge values.

 figure: Fig. 1.

Fig. 1. Intensity profile (Upper row) and corresponding phase distributions (Lower row) for M=12, R=4w 0 and several different charges n=1, 2, 3, and 4.

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Tables Icon

Table 1. Dependence of an on the charge n for R=4w0 or 6w0.

3. Propagation of BGBs with topological charges in turbulent atmosphere

In this section we will consider the BGB with change n propagating along the z-axis in a turbulent atmosphere using the approximate combination beam (7). Based on the extended Huygens-Fresnel principle, the average intensity at the z-plane can be expressed as

I(x,y,z)=k2(2πz)2E(p,q,0)E*(ξ,η,0)exp[ψ(p,q,x,y)+ψ*(ξ,η,x,y)]m
·exp(ik2z[(xp)2+(yq)2(xξ)2(yη)2])dpdqdξdη

here k=2π/λ(λ is the wavelength), the asterisk ‘*‘ denotes the complex conjugation and the 〈〉m denotes the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The ensemble average term is [10]

exp[ψ(p,q,x,y)+ψ*(ξ,η,,x,y)]=exp[0.5Dψ(pξ,qη)]=exp[(pξ)2+(qη)2ρ02]

where ψ(p,q, x, y) represents the random part of the complex phase of a spherical wave that propagates from the source point to the receiver point, D ψ(p-ξ,q-η) is the wave structure function. ρ 0 is the spatial coherence radius of a spherical wave and can be expressed as ρ 0=(0.545C 2 n k 2 z)-3/5, where C 2 n is the refractive index structure constant that describes how strong the turbulence is[6]. It is worth pointing out that in order to obtain simpler and viewable analytical results, here we also have employed a quadratic approximation of the authentic Rytov’s phase structure function widely and usually accepted to be valid not only for the case of “weak fluctuations”, but for the case of “strong fluctuations” as well [3,4,32]. From Eqs. (8) and (9), the average intensity at the receiver plane can be represented as

I(x,y,z)=k2(2πz)2E(p,q,0)E*(ξ,η,0)
·exp(ik2z[(xp)2+(yq)2(xξ)2(yη)2]1ρ02[(pξ)2+(qη)2])dpdqdξdη

By inserting Eq. (7) into Eq. (10) and performing the integral[31], the average intensity at the receiver plane can be analytically expressed as

I(x,y,z)=Nw2M2Ωexp{R2(1+w02ρ02)2Ωw022Nw2Ωx2+y2w02}m,JM1exp[i(φmφl)+R2cos(θmθl)2Ωρ02]
·exp{Nw(1+iNw)RΩw02(xcosθm+ysinθm)+Nw(1iNw)RΩw02(xcosθl+ysinθl)}

where N w=πw 2 0/λz is the Fresnel number of dGBs in free space propagation and Ω=1+N 2 w+2w 2 0/ρ 2 0. This is another main result obtained in this paper. Based on the expression, some numerical evaluations can be conducted. In the following calculations λ=632.8nm, M=12, R=2w 0 and y=0 are always used.

Figure 2 shows the intensity profile of BGBs during propagations in turbulent atmospheres. For n=0, the combination (7) approximates to a 0th order BGB with n=0. For the case discussed in this paper, the intensity profile almost retains unchangeable except spreading out for larger distance propagation. However, for the beam with non-zero charge value, the intensity profile changes during the propagation. At different z-planes along propagation direction, the beam profile successively changes from an annular structure to a flattened-top profile and finally to a single-peak profile, which distinguishes from the propagation in free space because the BGB is well-known non-diffracting beams. For example, approximately, for n=2 the propagating beam loses its phase singularity at z=0.4km and becomes a flattened top profile at z=5km. In fact, at z=10km all the cases have the extremely similar intensity profiles[4].

The zero intensity located at the center characterizing the phase singularity is missed after a sufficient propagation distance. From Fig. 3(a) it is seen that the propagation distance relies on the charge value of the beam and C 2 n characterizing the turbulent atmospheric structure. And the larger of the charge n, the farther of the propagation distance before the phase singularity loses. But the bigger of C 2 n, the faster the phase singularity dies away. Additionally, the beam width notably affects the beam spreading during the propagation yet, which can be seen from Fig. 3(b). In particular, the smaller the beam width, the faster the beam spreads.

 figure: Fig. 2.

Fig. 2. Cross line (y=0) of the normalized intensity distribution at several selected distances in a turbulent atmosphere for charges n=2 or 3, C 2 n=10-14 m -2/3 and w 0=0.01m.

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 figure: Fig. 3.

Fig. 3. Cross line (y=0) of the normalized intensity distribution at z=0.4km (a) for w 0=0.01m and C 2 n:10-15 (Dotted line), 10-14 (Solid line) and 10-13 m -2/3 (Dash-dot line) and w 0=0.01m; (b) for C 2 n=10-14 m -2/3 and w 0=0.015m(Dotted line), 0.01m(Solid line) and 0.007m (Dash-dot line), while n=2.

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4. Conclusions

An approximate expression of a BGB with desired charge is derived with a coherence superposition of dGBs. And based on the expression and the extended Huygens-Fresnel principle, the propagation properties of BGBs traveling in a turbulent atmosphere are explored. An analytical average intensity expression of a BGB with phase singularity propagating through atmospheric turbulences is obtained. The average intensity distributions are studied analytically and numerically. It is found that the intensity profile of BGBs experienced successive changes during the propagation and phase singularity wholly fades out after propagating a small distance. The change speed of the intensity profile and phase singularity sensitively depends on the beam parameters as well as the atmospheric turbulence structure.

Acknowledgments

One of the authors (X. J. Zheng) thanks the postdoctorial Fond of Central South University.

References and links

1. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).

2. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006). [CrossRef]  

3. E. T. Eyyuboglu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006). [CrossRef]   [PubMed]  

4. H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006). [CrossRef]  

5. J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991). [CrossRef]  

6. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 38, 671–684 (1991).

7. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002). [CrossRef]  

8. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003). [CrossRef]   [PubMed]  

9. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). [CrossRef]  

10. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 0411171–3 (2006). [CrossRef]  

11. H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008). [CrossRef]  

12. G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]  

13. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001). [CrossRef]  

14. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974). [CrossRef]  

15. M. V. Berry, “Singularities in waves and rays,” in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453–543.

16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994). [CrossRef]  

17. J. Durnin, “Exact solutions for nondiffracting beams: I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]  

18. A. A. Tovar, “Propagation of Laguerre-Bessel-Gaussian beams,” J. Opt. Soc. Am. A 17, 2010–2108 (2000). [CrossRef]  

19. H. T. Eyyuboglu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007). [CrossRef]  

20. P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

21. V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008). [CrossRef]  

22. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A. 25, 225–229 (2008). [CrossRef]  

23. B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008). [CrossRef]  

24. T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008). [CrossRef]  

25. J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998). [CrossRef]  

26. B. Lu and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000). [CrossRef]  

27. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204. [CrossRef]  

28. Y. J. Cai, Z. Y. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16, 15254–15267 (2008). [CrossRef]   [PubMed]  

29. K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002). [CrossRef]  

30. K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002). [CrossRef]  

31. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.

32. A. Ishimaru, “Phase fluctuations in a turbulent medium,” Appl. Opt. 16, 3190–3192 (1977). [CrossRef]   [PubMed]  

References

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  1. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).
  2. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
    [Crossref]
  3. E. T. Eyyuboglu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006).
    [Crossref] [PubMed]
  4. H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
    [Crossref]
  5. J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
    [Crossref]
  6. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 38, 671–684 (1991).
  7. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [Crossref]
  8. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [Crossref] [PubMed]
  9. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [Crossref]
  10. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 0411171–3 (2006).
    [Crossref]
  11. H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
    [Crossref]
  12. G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007).
    [Crossref]
  13. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001).
    [Crossref]
  14. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [Crossref]
  15. M. V. Berry, “Singularities in waves and rays,” in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453–543.
  16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
    [Crossref]
  17. J. Durnin, “Exact solutions for nondiffracting beams: I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  18. A. A. Tovar, “Propagation of Laguerre-Bessel-Gaussian beams,” J. Opt. Soc. Am. A 17, 2010–2108 (2000).
    [Crossref]
  19. H. T. Eyyuboglu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007).
    [Crossref]
  20. P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).
  21. V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008).
    [Crossref]
  22. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A. 25, 225–229 (2008).
    [Crossref]
  23. B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
    [Crossref]
  24. T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
    [Crossref]
  25. J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045–1048 (1998).
    [Crossref]
  26. B. Lu and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
    [Crossref]
  27. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204.
    [Crossref]
  28. Y. J. Cai, Z. Y. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16, 15254–15267 (2008).
    [Crossref] [PubMed]
  29. K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
    [Crossref]
  30. K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
    [Crossref]
  31. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.
  32. A. Ishimaru, “Phase fluctuations in a turbulent medium,” Appl. Opt. 16, 3190–3192 (1977).
    [Crossref] [PubMed]

2008 (6)

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[Crossref]

V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008).
[Crossref]

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A. 25, 225–229 (2008).
[Crossref]

B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[Crossref]

T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Y. J. Cai, Z. Y. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16, 15254–15267 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (5)

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[Crossref]

E. T. Eyyuboglu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006).
[Crossref] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
[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, 0411171–3 (2006).
[Crossref]

2003 (2)

2002 (3)

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
[Crossref]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001).
[Crossref]

2000 (2)

1998 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

1991 (2)

J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 38, 671–684 (1991).

1987 (1)

1977 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[Crossref]

Aksenov, V. P.

V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008).
[Crossref]

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).

Arpali, C.

Baykal, Y.

E. T. Eyyuboglu, C. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006).
[Crossref] [PubMed]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[Crossref]

M. V. Berry, “Singularities in waves and rays,” in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453–543.

Boradman, A. D.

J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

Cai, Y.

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[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, 0411171–3 (2006).
[Crossref]

Cai, Y. J.

Capjack, C. E.

Chen, B. S.

B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[Crossref]

Chen, Z. Y.

B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[Crossref]

T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Dogariu, A.

Durnin, J.

Eyyuboglu, E. T.

Eyyuboglu, H. T.

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[Crossref]

H. T. Eyyuboglu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007).
[Crossref]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
[Crossref]

Gao, W.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204.
[Crossref]

Gao, Y. Y.

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
[Crossref]

Gbur, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 0411171–3 (2006).
[Crossref]

Ishimaru, A.

Konyaev, P. A.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

Korotkova, O.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Lin, Q.

Liu, T. N.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

Lu, B.

Lukin, V. P.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

Ma, H.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).

Pogutsa, Ch. E.

V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008).
[Crossref]

Pu, J. X.

T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.

Seguin, H. J. J.

Sennikov, V. A.

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

Sermutlu, E.

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
[Crossref]

Shirai, T.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001).
[Crossref]

Strohschein, J. D.

Tang, H. Q.

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
[Crossref]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

Tovar, A. A.

Tyson, R. K.

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A. 25, 225–229 (2008).
[Crossref]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001).
[Crossref]

Wang, T.

T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Wang, X. W.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

Wang, Z. Y.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Wolf, E.

Wu, J.

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 38, 671–684 (1991).

J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

Yin, J.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204.
[Crossref]

Zhu, K. C.

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
[Crossref]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204.
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

H. T. Eyyuboglu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88, 259–265 (2007).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 0411171–3 (2006).
[Crossref]

Atmos. Oceanic Opt. (1)

P. A. Konyaev, V. P. Lukin, and V. A. Sennikov, “Effect of phase fluctuations on propagation of the vortex beams,” Atmos. Oceanic Opt. 19, 924–927 (2006).

J. Mod. Opt. (2)

J. Wu and A. D. Boradman, “Coherence length of a Gaussian-Schell beam and atmosphere turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[Crossref]

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 38, 671–684 (1991).

J. Opt. A (1)

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[Crossref]

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

K. C. Zhu, H. Q. Tang, and Y. Y. Gao, “A new set of flattened light beams,” J. Opt. A: Pure Appl. Opt. 4, 33–36 (2002).
[Crossref]

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

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

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A. 25, 225–229 (2008).
[Crossref]

Opt. Commun. (2)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, “Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere,” Opt. Commun. 265, 399–405(2006).
[Crossref]

Opt. Eng. (1)

T. Wang, J. X. Pu, and Z. Y. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (2)

H. T. Eyyuboglu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[Crossref]

B. S. Chen, Z. Y. Chen, and J. X. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[Crossref]

Opt. Lett. (1)

Optik (1)

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik 113, 222–226 (2002).
[Crossref]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[Crossref]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42. 219–276 (2001).
[Crossref]

Quantum Electron. (1)

V. P. Aksenov and Ch. E. Pogutsa, “Fluctuations of the orbital angular momentum of a laser beam, carrying an optical vortex, in the turbulent atmosphere,” Quantum Electron. 38, 343–348 (2008).
[Crossref]

Other (4)

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, p.119–204.
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 718.

M. V. Berry, “Singularities in waves and rays,” in Les Houches Session XXV-Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds., (North-Holland; 1981), p. 453–543.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (Bellingham, Washington: SPIE Optical Engineering Press; 1998).

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Figures (3)

Fig. 1.
Fig. 1. Intensity profile (Upper row) and corresponding phase distributions (Lower row) for M=12, R=4w 0 and several different charges n=1, 2, 3, and 4.
Fig. 2.
Fig. 2. Cross line (y=0) of the normalized intensity distribution at several selected distances in a turbulent atmosphere for charges n=2 or 3, C 2 n =10-14 m -2/3 and w 0=0.01m.
Fig. 3.
Fig. 3. Cross line (y=0) of the normalized intensity distribution at z=0.4km (a) for w 0=0.01m and C 2 n :10-15 (Dotted line), 10-14 (Solid line) and 10-13 m -2/3 (Dash-dot line) and w 0=0.01m; (b) for C 2 n =10-14 m -2/3 and w 0=0.015m(Dotted line), 0.01m(Solid line) and 0.007m (Dash-dot line), while n=2.

Tables (1)

Tables Icon

Table 1 Dependence of a n on the charge n for R=4w 0 or 6w 0.

Equations (16)

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

E ( x , y , 0 ) = m = 0 M 1 E m ( x , y , 0 )
E m ( x , y , 0 ) = exp [ ( x i 2 R cos θ m ) 2 + ( y i 2 R sin θ m ) 2 w 0 2 + i φ m ]
E ( ρ , φ , 0 ) = exp ( ρ 2 1 4 R 2 w 0 2 ) p = i p J p ( R ρ w 0 2 ) exp ( i p φ ) m = 0 M 1 exp [ im ( n p ) α 0 ]
m = 0 M 1 exp [ im ( n p ) α 0 ] = M δ p n , Mq ( q = 0 , ± 1 , )
E ( ρ , φ , 0 ) = M exp ( ρ 2 1 4 R 2 w 0 2 ) q = i n + Mq J n + Mq ( R ρ w 0 2 ) exp [ i ( n + Mq ) φ ]
a n + Mq = 0 exp ( 2 ρ 2 ) J n + Mq 2 ( R ρ w 0 ) ρ d ρ q = 0 exp ( 2 ρ 2 ) J n + Mq 2 ( R ρ w 0 ) ρ d ρ
= I n + Mq ( R 2 4 w 0 2 ) q = I n + Mq ( R 2 4 w 0 2 )
E ( x , y , 0 ) = J n ( R ρ w 0 2 ) exp ( ρ 2 w 0 2 + in φ )
1 M i n exp ( R 2 4 w 0 2 ) m = 0 M 1 exp [ ( x i 2 R cos θ m ) 2 + ( y i 2 R sin θ m ) 2 w 0 2 + i φ m ]
I ( x , y , z ) = k 2 ( 2 π z ) 2 E ( p , q , 0 ) E * ( ξ , η , 0 ) exp [ ψ ( p , q , x , y ) + ψ * ( ξ , η , x , y ) ] m
· exp ( ik 2 z [ ( x p ) 2 + ( y q ) 2 ( x ξ ) 2 ( y η ) 2 ] ) dpdqd ξ d η
exp [ ψ ( p , q , x , y ) + ψ * ( ξ , η , , x , y ) ] = exp [ 0.5 D ψ ( p ξ , q η ) ] = exp [ ( p ξ ) 2 + ( q η ) 2 ρ 0 2 ]
I ( x , y , z ) = k 2 ( 2 π z ) 2 E ( p , q , 0 ) E * ( ξ , η , 0 )
· exp ( ik 2 z [ ( x p ) 2 + ( y q ) 2 ( x ξ ) 2 ( y η ) 2 ] 1 ρ 0 2 [ ( p ξ ) 2 + ( q η ) 2 ] ) dpdqd ξ d η
I ( x , y , z ) = N w 2 M 2 Ω exp { R 2 ( 1 + w 0 2 ρ 0 2 ) 2 Ω w 0 2 2 N w 2 Ω x 2 + y 2 w 0 2 } m , J M 1 exp [ i ( φ m φ l ) + R 2 cos ( θ m θ l ) 2 Ω ρ 0 2 ]
· exp { N w ( 1 + i N w ) R Ω w 0 2 ( x cos θ m + y sin θ m ) + N w ( 1 i N w ) R Ω w 0 2 ( x cos θ l + y sin θ l ) }

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