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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21163–21173
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Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence

Huafeng Xu, Zhifeng Cui, and Jun Qu  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21163-21173 (2011)
http://dx.doi.org/10.1364/OE.19.021163


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Abstract

The analytical formulas for the average intensity and the beam width of the elegant Laguerre–Gaussian beam (ELGB) in non-Kolmogorov turbulence have been derived based on the extended Huygens–Fresnel principle. Numerical results reveal that the ELGB converts to Gaussian form quicker for smaller values of beam order and for smaller wavelengths. The root-mean-square (rms) beam width of ELGB increases markedly with the propagation distance for higher beam order, smaller waist width, and larger wavelength. Furthermore, discussions of the influence of ELGB by the non-Kolmogorov turbulence reveal that the normalized intensity distribution of ELGB converts into Gaussian form more quickly and that the rms beam width of ELGB increases more rapidly in non-Kolmogorov turbulence with smaller parameter α, larger outer scale, smaller inner scale and larger structure constant.

© 2011 OSA

1. Introduction

For a long time, Kolmogorov’s power spectrum of refractive-index fluctuations has been widely accepted and has shown good agreement with experiment results [1

1. X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. 50(21), 3871–3878 (2011). [CrossRef] [PubMed]

,2

2. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011). [CrossRef] [PubMed]

]. However, recent experiments have indicated extensive deviations from the Kolmogorov’s model in some portions of the atmosphere [3

3. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).

7

7. M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999). [CrossRef]

], and in the case of a laser beam propagation along the vertical direction in an extremely stable atmosphere (non-Kolmogorov turbulence), some anomalistic behavior has been reported [8

8. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. 17(24), 1737–1739 (1992). [CrossRef] [PubMed]

]. This behavior results from the vertical component being suppressed and the turbulence no longer being homogeneous in three dimensions. By using a generalized exponent and a generalized amplitude factor, Toselli et al. [9

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E–1­ 65510E-12 (2007). [CrossRef]

,10

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

] introduced a non-Kolmogorov power spectrum that reduces to the conventional Kolmogorov spectrum when the power-law exponent value α=11/3. Nevertheless, based on this non-Kolmogorov power spectrum, to the best of our knowledge, a great deal of work has been investigated: the spreading and direction of Gaussian–Schell model beam [11

11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

], the average spreading of a Gaussian beam array [12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

], the second-order statistics of stochastic electromagnetic beams [13

13. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

], and so on. Up to now the propagations of ELGB in non-Kolmogorov turbulence have not yet been reported, even though propagations of this type of laser beam in Kolmogorov turbulence have been widely studied [14

14. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011). [CrossRef]

18

18. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010). [CrossRef]

] and play an important role in connection with practical applications, such as remote sensing and optical communications [19

19. T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256(4-6), 197–209 (2005). [CrossRef]

].

To date, the propagation properties of ELGB in Kolmogorov turbulence have been well discussed [14

14. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011). [CrossRef]

,17

17. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]

,18

18. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010). [CrossRef]

], However, further study of the propagation properties of ELGB in non-Kolmogorov turbulence, which is our goal in this paper, is required. The analytical expressions of the average intensity and the rms beam width in non-Kolmogorov turbulence are derived by use of the extended Huygens–Fresnel principle. Some numerical results illustrate the influence by both the beam parameters and the non-Kolmogorov spectrum.

2. Theoretical model

2.1 Average intensity of ELGB in non-Kolmogorov turbulence

The electric field distribution of ELGB at the source plane (z=0) is expressed in cylindrical coordinates as follows [20

20. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]

23

23. D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003). [CrossRef]

]
Enm(r,θ,0)=(rm)mLnm(r2w02)exp(r2w02)exp(imθ),
(1)
where r and θ are the radial and azimuthal (angle) coordinates, Lnm denotes the Laguerre polynomial with mode orders n and m, and w0 is the beam width of the fundamental Gaussian mode. By use of the following relation [24

24. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

]
eimθρmLnm(ρ2)=(1)n22n+mn!t=0ns=0mis(nt)(ms)H2t+ms(x)H2n2t+s(y),
(2)
with Hn(·)being the Hermite polynomial of ordern and (nt) and(ms) being binomial coefficients, Eq. (1) can be expressed in following alternative form in Cartesian coordinates:

E(x,y,0)=(1)n22n+mn!t=0ns=0mis(nt)(ms)H2t+ms(xw0)×H2n2t+s(yw0)exp[x2+y2w02].
(3)

With the extended Huygens–Fresnel principle, the average intensity of the ELGB at the z plane after propagating in non-Kolmogorov turbulence can be expressed [25

25. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, SPIE Press, Washington, 2001.

31

31. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005). [CrossRef] [PubMed]

]
I(px,py,z)=1(λz)2E(x1,y1,0)E*(x2,y2,0)exp[ik2z(x1px)2ik2z(y1py)2]×exp[ik2z(x2px)2+ik2z(y2py)2]exp[x12+y12w02]exp[x22+y22w02]×exp[ψ(x1,y1,px,py)+ψ*(x2,y2,px,py)]dx1dx2dy1dy2,
(4)
wheredenotes ensemble average over the turbulent media. Here (x,y)and(px,py)are the coordinates at the input plane(z=0)and the output plane(z).

The term exp[ψ(ρ,ρ1,z)+ψ*(ρ,ρ2,z)]can expressed [11

11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

,12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

,32

32. 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]

]
exp[ψ(ρ,ρ1,z)+ψ*(ρ,ρ2,z)]=exp{4π2k2z010κΦn(κ)[1J0(κξ|ρ1ρ2|)]dκdξ},
(5)
where J0 is the Bessel function of zero order and has the approximation [33

33. 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(6), 1094–1102 (2003). [CrossRef] [PubMed]

]
J0(κξ|ρ1ρ2|)~114(κξ|ρ1ρ2|)2,
(6)
Φn(κ) denotes the spatial power spectrum of the refractive-index fluctuations of the atmosphere turbulence, and κis the magnitude of two-dimensional spatial frequency. To model atmospheric turbulence, the form of the non-Kolmogorov spectrum used is [9

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E–1­ 65510E-12 (2007). [CrossRef]

,12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

,34

34. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]

,35

35. G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011). [CrossRef]

]
Φn(κ,α)=A(α)C˜n2exp[(κ2/κm2)](κ2+κ02),0κ<,3<α<4,
(7)
where κ0=2π/L0,with L0being the outer scale parameter; κm=c(α)/l0, with l0 being the inner scale parameter; parameter “α” is the power-law exponent that assumes all values between the range from 3 to 4; and [11

11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

,12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

]
c(α)=[Γ(5α2)A(α)2π/3][1/(α5)].
(8)
The term C˜n2 in Eq. (7) is the generalized structure parameter with units m3α [11

11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

,12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

] and
A(α)=Γ(α1)cos(απ/2)/4π2.
(9)
Note that the spectrum expressed in Eq. (7) reduces to conventional Kolmogorov spectrum whenα=11/3, A(α)=0.033,L0=, l0=0, and C˜n2=Cn2.

On substituting Eq. (6) into Eq. (5), one obtains
exp[ψ(ρ,ρ1,z)+ψ*(ρ,ρ2,z)]=exp[13π2k2z|ρ1ρ2|2Tdκ],
(10)
where the expression of Tcan be written
T=0κ3Φn(κ)dκ=A(α)C˜n22κm2αβexp(κ02κm2)Γ(2α/2,κ02κm2)2κ04αα2,
(11)
β=2κ022κm2+ακm2, and Γdenotes the incomplete Gamma function.

On substituting Eqs. (3) and (10) into Eq. (4) and after careful integration, the average intensity of an ELGB in non-Kolmogorov turbulence at the z plane can be expressed as
I(px,py,z)=π2p1p21(λz)2(1)2n24n+2m(n!)2t1=0ns1=0mt2=0ns2=0mk1=0[2t1+ms12]k2=0[2n2t1+s12]v1=0[2t2+ms22]×v2=0[2n2t2+s22]u1=02t2+ms22v1u2=02n2t2+s22v2g1=0[2t1+2t2+2m2k12v1s1s2u12]g2=0[4n2t12t2+s1+s22k22v2u22]×is1(i)s2(nt1)(ms1)(nt2)(ms2)(2t2+ms22v1u1)(2n2t2+s22v2u2)×(1)k1(2t1+ms1)!k1!(2t1+ms12k1)!(1)k2(2n2t1+s1)!k2!(2n2t1+s12k2)!(1)v1(2t2+ms2)!v1!(2t2+ms22v1)!×(1)v2(2n2t2+s2)!v2!(2n2t2+s22v2)!(2t1+2t2+2m2k12v1s1s2u1)!(2t1+2t2+2m2k12v1s1s2u12g1)!g1!×(4n2t12t2+s1+s22k22v2u2)!(4n2t12t2+s1+s22k22v2u22g2)!g2!(11w02p1)v1+v2×(b1)u1+u2×(b2)4n+2m2k12k22v12v2u1u22g12g2(w0)4n2m+2k1+2k2+2v1+2v2×24n+2m2k22k12v12v22g12g2(p1)2nm+2v1+2v2(p2)4n2m+2k1+2k2+2v1+2v2+u1+u2+g1+g2×(Ai)2n+m2v12v2u1u2(px)2t1+2t2+2m2k12v1s1s22g1(py)4n2t12t2+s1+s22k22v22g2×exp[(b12p1+b22p2)(px2+py2)],
(12)
whereAi=13π2k2zT, b1=ik2z, b2=b1(1Aip1), p1=(ik2z+Ai+1w02), and

p2=AiAi2p1+ik2z+1w02.

2.2 Rms beam width of ELGB in non-Kolmogorov turbulence

To learn more about the propagation properties of ELGB in non-Kolmogorov turbulence, we further studied the rms beam width to examine the directionality of ELGB propagating through non-Kolmogorov turbulence. When the variancesx oryare used twice, the rms beam width of an ELGB atz can be defined [11

11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

,12

12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

,31

31. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005). [CrossRef] [PubMed]

,32

32. 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]

]

Wρ(z)=ρ2I(x,y,z)dxdyI(x,y,z)dxdy,(ρ=x,y).
(13)

HereWx(z) and Wy(z)are the effective beam sizes of an ELGB in the x and they directions, respectively. Substituting Eq. (12) into Eq. (13), we obtain the following expression for the rms beam width of an ELGB in non-Kolmogorov turbulence:
Wx(z)=Wy(z)=P1(z)P2(z).
(14)
Here

P1(z)=π2p1p21(λz)2(1)2n24n+2m(n!)2t1=0ns1=0mt2=0ns2=0mk1=0[2t1+ms12]v1=0[2t2+ms22]u1=02t2+ms22v1×g1=0[2t1+2t2+2m2k12v1s1s2u12]k2=0[2n2t1+s12]v2=0[2n2t2+s22]u2=02n2t2+s22v2g2=0[4n2t12t2+s1+s22k22v2u22]is1(i)s2×(nt1)(ms1)(nt2)(ms2)(2t2+ms22v1u1)(2n2t2+s22v2u2)×(1)k1(2t1+ms1)!k1!(2t1+ms12k1)!(1)k2(2n2t1+s1)!k2!(2n2t1+s12k2)!(1)v1(2t2+ms2)!v1!(2t2+ms22v1)!×(1)v2(2n2t2+s2)!v2!(2n2t2+s22v2)!(2t1+2t2+2m2k12v1s1s2u1)!(2t1+2t2+2m2k12v1s1s2u12g1)!g1!×(4n2t12t2+s1+s22k22v2u2)!(4n2t12t2+s1+s22k22v2u22g2)!g2!×(Ai)2n+m2v12v2u1u2×(11w02p1)v1+v2×(w0)4n2m+2k1+2k2+2v1+2v2(b1)u1+u2(b2)4n+2m2k12k22v12v2u1u22g12g2×24n+2m2k22k12v12v22g12g2(p1)2nm+2v1+2v2(p2)4n2m+2k1+2k2+2v1+2v2+u1+u2+g1+g2×Γ(2t1+2t2+2m2k12v1s1s22g1+32)(b12p1b22p2)2t1+2t2+2m2k12v1s1s22g1+3×Γ(4n2t12t2+s1+s22k22v22g2+12)(b12p1b22p2)4n2t12t2+s1+s22k22v22g2+1,
(15)
P2(z)=π2p1p21(λz)2(1)2n24n+2m(n!)2t1=0ns1=0mt2=0ns2=0mk1=0[2t1+ms12]v1=0[2t2+ms22]u1=02t2+ms22v1×g1=0[2t1+2t2+2m2k12v1s1s2u12]k2=0[2n2t1+s12]v2=0[2n2t2+s22]u2=02n2t2+s22v2g2=0[4n2t12t2+s1+s22k22v2u22]is1(i)s2×(nt1)(ms1)(nt2)(ms2)(2t2+ms22v1u1)(2n2t2+s22v2u2)×(1)k1(2t1+ms1)!k1!(2t1+ms12k1)!(1)k2(2n2t1+s1)!k2!(2n2t1+s12k2)!(1)v1(2t2+ms2)!v1!(2t2+ms22v1)!×(1)v2(2n2t2+s2)!v2!(2n2t2+s22v2)!(2t1+2t2+2m2k12v1s1s2u1)!(2t1+2t2+2m2k12v1s1s2u12g1)!g1!×(4n2t12t2+s1+s22k22v2u2)!(4n2t12t2+s1+s22k22v2u22g2)!g2!×(Ai)2n+m2v12v2u1u2×(11w02p1)v1+v2×(w0)4n2m+2k1+2k2+2v1+2v2(b1)u1+u2(b2)4n+2m2k12k22v12v2u1u22g12g2×24n+2m2k22k12v12v22g12g2(p1)2nm+2v1+2v2(p2)4n2m+2k1+2k2+2v1+2v2+u1+u2+g1+g2×Γ(2t1+2t2+2m2k12v1s1s22g1+12)(b12p1b22p2)2t1+2t2+2m2k12v1s1s22g1+1×Γ(4n2t12t2+s1+s22k22v22g2+12)(b12p1b22p2)4n2t12t2+s1+s22k22v22g2+1.
(16)

Equations (12) and (14) are the main results of this manuscript, which presents a tool to study the average intensity and the rms beam width of ELGB and its dependence on various parameters of the non-Kolmogorov turbulence.

3. Numerical results

Figure 1
Fig. 1 Normalized intensity distribution of ELGB through non-Kolmogorov turbulence for different propagation distances zwithm=2,n=1,λ=850 nm,w0=0.02 m, and C˜n2=1×1014m3α.
shows the cross line normalized intensity distribution of ELGB propagating in non-Kolmogorov turbulence for different propagation distances zwith m=2, n=1,λ=850 nm, and w0=0.02 m. Note that the non-Kolmogorov spectrum parameters are set as α=3.8,L0=1 m,, l0=0.01 m,and C˜n2=1014m3α. From Fig. 1, one finds that the hollow part of the ELGB profile gradually disappears as the propagation distances z increase and become Gaussian in form after propagating over a sufficiently long distance.

Figure 2
Fig. 2 Normalized intensity distribution of ELGB through non-Kolmogorov turbulence for different values of the wavelength and the beam order, respectively. (a)z=5 km, m=2, and n=1. (b)z=6 km and λ=850 nm.
plots the cross line normalized intensity distribution of ELGB propagating in non-Kolmogorov turbulence [z=5 km,m=2, andn=1in Fig. 2(a) andz=6 km andλ=850 nmin Fig. 2(b)] for different values of the wavelength and the beam order, respectively; other parameters are the same as Fig. 1. We can deduce from Fig. 2 that the ELGB profile becomes Gaussian in form quicker with smaller wavelength λ[see Fig. 2(a)] and smaller beam ordermand n[see Fig. 2(b)], that is to say, the ELGB with larger wavelength and larger beam orderm and n is less affected by turbulence.

Figure 3 plots the cross line normalized intensity distribution of ELGB propagating in non-Kolmogorov turbulence at a fixed propagation distance z=5 km for different values of parameter α, outer scale L0, inner scalel0, and structure constant C˜n2. Note that the other parameters are set as m=2,n=1, λ=850 nm,,w0=0.02 m.. It can be seen from Fig. 3
Fig. 3 Normalized intensity distribution of ELGB through non-Kolmogorov turbulence at a fixed propagation distancez=5 km for different values of parameterα, outer scaleL0, inner scalel0, and structure constant c˜n2.
that the ELGB profile converts into a Gaussian beam more rapidly with smaller exponent values α, larger outer scaleL0, smaller inner scalel0,and larger structure constant C˜n2, indicating less influence on the beam by the turbulence with higher α, smaller outer scaleL0, larger inner scalel0,and smaller structure constant C˜n2. The inner scale l0, which forms the lower limit of the inertial range, has a smaller value for strong turbulence and a larger value for weak turbulence. The outer scale L0 forms the upper limit of the inertial range and increases with the strength of turbulence. The decreasing of inner scale l0 or the increasing of outer scale L0 is equivalent to increasing the strength of the turbulence. In these cases, the laser beam will meet more turbulence cells along its propagation paths, and as a result, ELGB beam will be affected more.

To further study the ELGB propagation properties in non-Kolmogorov turbulence, based on Eqs. (14), (15), and (16), numerical calculations for the rms beam width of ELGB through non-Kolmogorov turbulence were performed. Some typical examples are compiled in Figs. 4
Fig. 4 Rms beam width of ELGB versus through non-Kolmogorov turbulence for different values of the beam order, waist size, and wavelength.
-6
Fig. 6 Rms beam width of ELGB versus through non-Kolmogorov turbulence for different parameter α, outer scale L0, inner scalel0,and structure constantC˜n2. The calculation parameters are m = 2, n = 1, λ = 850 nm, and w0 = 0.02 m. (a) L0 = 1 m, l0 = 0.01 m, and C˜n2 = C˜n210−14 m3-α. (b) α = 3.8, l0 = 0.01 m, and C˜n2 = 10−14 m3-α. (c) α = 3.8, L0 = 1 m, and C˜n2 = 10−14 m3-α. (d) α = 3.8, L0 = 1 m, and l0 = 0.01 m.
.

Figure 4 shows the rms beam width of ELGB for different values of beam orderm and n, waist width, and wavelength versus the propagating distance in non-Kolmogorov turbulence. As indicated by Figs. 4(a) and 4(b), the calculation parameters areα=3.8, L0=1 m,l0=0.01 m, C˜n2=1014m3α,w0=0.02 m, and λ=850 nm, revealing that the rms beam width w(z) of ELGB in non-Kolmogorov apparently increases with the propagation distance for higher beam order. It follows from Figs. 4(c) and 4(d) that the rms beam width increase is slower with the propagation distance for higher waist width and smaller wavelength and that the beam order is chosen to bem=2 and n=1.The other parameters are the same as those in Figs. 3(a) and 3(b).

Figure 5
Fig. 5 Rms beam width of ELGB through non-Kolmogorov turbulence as a function of α for different propagation distances. The calculation parameters are:m=2, n=1, λ=850 nm,,w0=0.02 m,,L0=1 m,,l0=0.01 m,, and C˜n2=1014m3α.
plots the rms beam width of an ELGB through non-Kolmogorov turbulence as a function of α for different propagation distances; the calculation parameters are w0=0.02 m,, C˜n2=1014m3α,andλ=850 nm.. As indicated by Fig. 5, the rms beam widthw(z) of ELGB increases with the increasing of α until it reaches the maximum point (approximatelyα=3.1). After the maximum point, the rms beam width w(z) decreases with the increasingα. This phenomenon corresponds to the case of the Gaussian–Schell model beam through non-Kolmogorov turbulence. We assume that as α3,A(α)0 and the turbulence power spectrum vanishes. So the rms beam width first grows and then decreases with the increasing of the parameterα. However, when the values of α are fixed, the rms beam width has larger values for longer propagation distance.

Figure 6 plots the rms beam width of ELGB versus that through non-Kolmogorov turbulence for different values of parameter α, outer scale L0, inner scale l0, and C˜n2, which represent the characteristics of the non-Kolmogorov spectrum. It reveals that the ELGB spreads more rapidly in non-Kolmogorov turbulence for smaller α, larger outer scaleL0, smaller inner scale l0, and larger structure constant C˜n2, i.e., the influence on the ELGB is less affected by the non-Kolmogorov turbulence with higher α,, smaller L0, larger l0, and smaller C˜n2.

4. Conclusion

In summary, the analytical formulas for the average intensity and the rms beam width of ELGB in non-Kolmogorov turbulence have been derived based on the extended Huygens–Fresnel principle, the normalized intensity, and the rms beam width of ELGB through non-Kolmogorov turbulence and have been discussed with numerical examples. Results show that the ELGB evolves into a circular Gaussian beam after propagating a sufficiently long distance. The intensity distribution of the ELGB is less affected by the turbulence with larger wavelength, larger beam order m and n, higher α, smaller outer scale L0, larger inner scale l0,and smaller structure constant C˜n2. The spreading of ELGB occurs more rapidly with larger beam order, smaller waist width, larger wavelength, larger exponent value α, larger outer scale L0, smaller inner scale l0, and larger structure constant C˜n2 through non-Kolmogorov turbulence. These results may be useful in the practical beam propagation.

Acknowledgments

Jun Qu acknowledges the support by AnHui Provincial Natural Science Foundation of China under grant 11040606M154 and Foundation of AnHui Educational Committee under grant KJ2010A155.

References and links

1.

X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt. 50(21), 3871–3878 (2011). [CrossRef] [PubMed]

2.

X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011). [CrossRef] [PubMed]

3.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375A, 1111–1126 (1995).

4.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000). [CrossRef]

5.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702, 598702-12 (2005). [CrossRef]

6.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 2471–22 (1995).

7.

M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE 3749, 50–51 (1999). [CrossRef]

8.

D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett. 17(24), 1737–1739 (1992). [CrossRef] [PubMed]

9.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E–1­ 65510E-12 (2007). [CrossRef]

10.

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

11.

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef] [PubMed]

12.

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]

13.

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

14.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011). [CrossRef]

15.

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef] [PubMed]

16.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98(4), 857–863 (2010). [CrossRef]

17.

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]

18.

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–56 (2010). [CrossRef]

19.

T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256(4-6), 197–209 (2005). [CrossRef]

20.

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]

21.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009). [CrossRef] [PubMed]

22.

S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67(5), 699–700 (1977). [CrossRef]

23.

D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt. 5(5), 489–494 (2003). [CrossRef]

24.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2563–2567 (1993).

25.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, SPIE Press, Washington, 2001.

26.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd edition SPIE Press, Bellington, (2005)

27.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]

28.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008). [CrossRef] [PubMed]

29.

Y. Cai, H.T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25(7), 1497–1503 (2008). [CrossRef]

30.

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12(20), 4659–4674 (2004). [CrossRef] [PubMed]

31.

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44(6), 976–983 (2005). [CrossRef] [PubMed]

32.

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]

33.

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(6), 1094–1102 (2003). [CrossRef] [PubMed]

34.

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]

35.

G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(7), 1225–1228 (2011). [CrossRef]

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: July 12, 2011
Revised Manuscript: September 8, 2011
Manuscript Accepted: September 13, 2011
Published: October 10, 2011

Citation
Huafeng Xu, Zhifeng Cui, and Jun Qu, "Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence," Opt. Express 19, 21163-21173 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21163


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References

  1. X. Chu, C. Qiao, X. Feng, and R. Chen, “Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model,” Appl. Opt.50(21), 3871–3878 (2011). [CrossRef] [PubMed]
  2. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett.36(14), 2701–2703 (2011). [CrossRef] [PubMed]
  3. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE2375A, 1111–1126 (1995).
  4. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt.47(6), 1111–1126 (2000). [CrossRef]
  5. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE5987, 598702, 598702-12 (2005). [CrossRef]
  6. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE2471, 2471–22 (1995).
  7. M. S. Belen’kii, S. J. Kars, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence,” Proc. SPIE3749, 50–51 (1999). [CrossRef]
  8. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, “Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor,” Opt. Lett.17(24), 1737–1739 (1992). [CrossRef] [PubMed]
  9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE6551, 65510E–1­ 65510E-12 (2007). [CrossRef]
  10. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng.47(2), 026003 (2008). [CrossRef]
  11. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett.35(5), 715–717 (2010). [CrossRef] [PubMed]
  12. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett.35(7), 1043–1045 (2010). [CrossRef] [PubMed]
  13. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express18(10), 10650–10658 (2010). [CrossRef] [PubMed]
  14. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere,” Appl. Phys. B102(4), 937–944 (2011). [CrossRef]
  15. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009). [CrossRef] [PubMed]
  16. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B98(4), 857–863 (2010). [CrossRef]
  17. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun.283(14), 2772–2781 (2010). [CrossRef]
  18. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and Elegant Laguerre-Gaussian beam in turbulent atmosphere,” Prog. Electromagn. Res.103, 33–56 (2010). [CrossRef]
  19. T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun.256(4-6), 197–209 (2005). [CrossRef]
  20. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A3(4), 465–469 (1986). [CrossRef]
  21. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express17(13), 11130–11139 (2009). [CrossRef] [PubMed]
  22. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am.67(5), 699–700 (1977). [CrossRef]
  23. D. Deng, H. Guo, X. Chen, and H. Kong, “Characteristics of coherent and incoherent off-axis elegant Hermite–Gaussian beam combinations,” J. Opt. A, Pure Appl. Opt.5(5), 489–494 (2003). [CrossRef]
  24. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron.29, 2563–2567 (1993).
  25. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, SPIE Press, Washington, 2001.
  26. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd edition SPIE Press, Bellington, (2005)
  27. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express17(20), 17344–17356 (2009). [CrossRef] [PubMed]
  28. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express16(11), 7665–7673 (2008). [CrossRef] [PubMed]
  29. Y. Cai, H.T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A25(7), 1497–1503 (2008). [CrossRef]
  30. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere,” Opt. Express12(20), 4659–4674 (2004). [CrossRef] [PubMed]
  31. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt.44(6), 976–983 (2005). [CrossRef] [PubMed]
  32. 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]
  33. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A20(6), 1094–1102 (2003). [CrossRef] [PubMed]
  34. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express18(20), 21269–21283 (2010). [CrossRef] [PubMed]
  35. G. Wu, T. Zhao, J. Ren, J. Zhang, X. Zhang, and W. Li, “Beam propagation factor of partially coherent Hermite-Gassian beams through non-Kolmogorov turbulence,” Opt. Laser Technol.43(7), 1225–1228 (2011). [CrossRef]

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