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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1871–1883
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Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

Rong Chen, Lin Liu, Shijun Zhu, Gaofeng Wu, Fei Wang, and Yangjian Cai  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 1871-1883 (2014)
http://dx.doi.org/10.1364/OE.22.001871


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Abstract

Laguerre-Gaussian Schell-model (LGSM) beam was proposed in theory [Opt. Lett. 38, 91 (2013 Opt. Lett. 38, 1814 (2013)] just recently. In this paper, we study the propagation of a LGSM beam in turbulent atmosphere. Analytical expressions for the cross-spectral density and the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere are derived. The statistical properties, such as the degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere are studied in detail. It is found that a LGSM beam with larger mode order n is less affected by turbulence than a LGSM beam with smaller mode order n or a GSM beam under certain condition, which will be useful in free-space optical communications.

© 2014 Optical Society of America

1. Introduction

Propagation characteristics of different types of beams propagating in the turbulent atmosphere are of interest for free-space optical communications and remote sensing applications [7

7. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

, 11

11. G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]

, 16

16. Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

, 18

18. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]

, 20

20. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]

22

22. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]

, 24

24. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

, 28

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]

, 30

30. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge Univeristy, 2007).

52

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]

]. It has been found that one can overcome or reduce the negative influence of turbulence by use of a laser beam with special beam profile, phase and polarization or a partially coherent beam. Propagation properties of partially coherent beams whose degrees of coherence satisfy Gaussian distributions in turbulent atmosphere have been studied in detail. Up to now, only few papers were devoted to the propagation of partially coherent beams whose degrees of coherence do not satisfy Gaussian distributions [16

16. Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

, 18

18. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]

, 20

20. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]

22

22. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]

, 24

24. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

, 28

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]

]. In [28

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]

], Cang et al. derived the analytical expressions for the average intensity and the beam width of a LGSM beam in turbulent atmosphere and studied the evolution properties of the average intensity and the beam width of such beam in turbulent atmosphere, while they did not derive the analytical expression for the cross-spectral density of the LGSM beam in turbulent atmosphere. To study the statistical properties, such as degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere, one should know the expressions for cross-spectral density and the second-order moments of the Wigner distribution function. In this paper, our aim is to derive the analytical expressions for the cross-spectral density and the second-order moments of the Wigner distribution function of a LGSM beam in turbulent atmosphere, and study its statistical properties. Some useful results are found.

2. Cross-spectral density of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

The cross-spectral density (CSD) of a LGSM beam at z = 0 is defined as [26

26. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

]
W(r1,r2)=exp[r12+r224σ02(r1r2)22σg2]Ln0[(r1r2)22σg2],
(1)
where r1(x1,y1) and r2(x2,y2) are two arbitrary transverse position vectors at z = 0, σ0 and σg are the transverse beam width and the transverse coherence width of the LGSM beam, respectively, Ln0 denotes the Laguerre polynomial of mode order n and 0. The degree of coherence of the LGSM beam at z = 0 is given as
μ(r1,r2)=W(r1,r2)W(r1,r1)W(r2,r2)=exp[(r1r2)22σg2]Ln0[(r1r2)22σg2].
(2)
One finds from Eq. (2) that the degree of coherence of a LGSM beam has a non-Gaussian distribution, which induces unique propagation properties of such beam [26

26. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

28

28. J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]

]. Under the condition of n = 0, Eq. (1) reduces to the expression for the CSD of a GSM beam.

Within the validity of the paraxial approximation, propagation of the CSD of a partially coherent beam in turbulent atmosphere can be studied with the help of the following extended Huygens-Fresnel integral [7

7. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

, 31

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

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]

, 52

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]

]
W(ρ1,ρ2)=1λ2z2W(r1,r2)×exp[ik2z(r1ρ1)2+ik2z(r2ρ2)2]×exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]d2r1d2r2,
(3)
where the asterisk denotes the complex conjugate and the angular brackets denote ensemble average, ρ1(ρ1x,ρ1y) and ρ2(ρ2x,ρ2y) are two arbitrary transverse position vectors at the receiver plane, dr1dr2=dx1dy1dx2dy2, k=2π/λ is the wave number with λ being the wavelength. The expression in the angular brackets in Eq. (3) can be expressed as [7

7. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

, 31

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

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]

, 52

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]

]
exp[Ψ(r1,ρ1)+Ψ*(r2,ρ2)]=exp[(r1r2)2ρ02(r1r2)(ρ1ρ2)ρ02(ρ1ρ2)2ρ02],
(4)
where ρ0=(0.545Cn2k2z)3/5is the coherence length of a spherical wave propagating in the turbulent medium with Cn2 being the structure constant. Following [7

7. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

, 31

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

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]

, 52

52. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]

], we have applied the Kolmogorov turbulence spectrum and a quadratic approximation for wave structure function.

Substituting Eqs. (1) and (4) into Eq. (3), we obtain

W(ρ1,ρ2)=1λ2z2exp[ik2z(ρ12ρ22)(ρ1ρ2)2ρ02]×exp[(ik2z14σ02)r12+(ik2z14σ02)r22+r1(ikzρ1ρ1ρ2ρ02)]×exp[r2(ikzρ2ρ1ρ2ρ02)(12σg2+1ρ02)(r1r2)2]Ln0[(r1r2)22σg2]d2r1d2r2.
(5)

If we set
ikzρ1'=ikzρ1ρ1ρ2ρ02,ikzρ2'=ikzρ2ρ1ρ2ρ02,
(6)
A(ρ1,ρ2)=exp[ik2z(ρ12ρ22)(ρ1ρ2)2ρ02],
(7)
Equation (5) is simplified as

W(ρ1',ρ2')=A(ρ1,ρ2)λ2z2exp[(12σg2+1ρ02)(r1r2)2]Ln0[(r1r2)22σg2]×exp[(ik2z14σ02)r12+(ik2z14σ02)r22+ikz(r1ρ1'r2ρ2')]d2r1d2r2.
(8)

For the convenience of integration, we introduce the following “sum” and “difference” coordinates
r=r1+r22,rd=r1r2,ρ'=ρ1'+ρ2'2,ρd'=ρ1'ρ2',
(9)
then Eq. (8) can be expressed as

W(ρ',ρd')=A(ρ1,ρ2)λ2z2exp[(12σg2+1ρ02+18σ02)rd2+ikzrdρ']Ln0(rd22σg2)d2rd×exp[12σ02r2+r(ikzrd+ikzρd')]d2r.
(10)

After integration over r, Eq. (10) reduces to

W(ρ',ρd')=A(ρ1,ρ2)λ2z22πσ02exp(k2σ022z2ρd'2)Ln0(rd22σg2)×exp[(12σg2+1ρ02+18σ02+k2σ022z2)rd2+rd(ikzρ'+k2σ02z2ρd')]d2rd.
(11)

If we set
a=12σg2+1ρ02+18σ02+k2σ022z2,b=ikz,c=k2σ02z2,
(12)
Equation (11) reduces to

W(ρ',ρd')=A(ρ1,ρ2)λ2z22πσ02exp(k2σ022z2ρd'2)×exp[ard2+rd(bρ'+cρd')]Ln0(rd22σg2)d2rd.
(13)

By using the following expansion formulae [53

53. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

]
Ln0(x)=p=0n(np)(1)pp!xp,
(14)
(x2+y2)p=m=0p(pm)x2(pm)y2m,
(15)
Equation (13) can be expressed in the following alternative form

W(ρx',ρy',ρdx',ρdy')=2πσ02λ2z2A(ρ1,ρ2)exp[k2σ022z2(ρdx'2+ρdy'2)]p=0nm=0p(np)(pm)(1)p2pp!σg2p×exp[ardx2+rdx(bρx'+cρdx')ardy2+rdy(bρy'+cρdy')]rdx2(pm)rdy2mdrdxdrdy.
(16)

With the help of the following integral formulae [54

54. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

]
xnexp[(xβ)2]dx=(2i)nπHn(iβ),
(17)
where Hn denotes the Hermite polynomial of mode order n, after integration over rdx and rdy, we obtain the following expression for the CSD of a LGSM beam in the output plane
W(ρ1x,ρ1y,ρ2x,ρ2y)=p=0nm=0p(np)(pm)123pp!σg2pap+12π2σ02λ2z2exp{k2σ022z2[(ρ1x'ρ2x')2+(ρ1y'ρ2y')2]}×exp[ik2z(ρ1x2+ρ1y2ρ2x2ρ2y2)(ρ1xρ2x)2+(ρ1yρ2y)2ρ02]×exp{14a[b(ρ1x'+ρ2x'2)+c(ρ1x'ρ2x')]2}H2(pm)[ib(ρ1x'+ρ2x'2)+c(ρ1x'ρ2x')2a]×exp{14a[b(ρ1y'+ρ2y'2)+c(ρ1y'ρ2y')]2}H2m[ib(ρ1y'+ρ2y'2)+c(ρ1y'ρ2y')2a],
(18)
where

ρ1x'=(1zikρ02)ρ1x+zikρ02ρ2x,ρ1y'=(1zikρ02)ρ1y+zikρ02ρ2y,
(19)
ρ2x'=(1+zikρ02)ρ2xzikρ02ρ1x,ρ2y'=(1+zikρ02)ρ2yzikρ02ρ1y.
(20)

The average intensity of the LGSM beam in the output plane is obtained as I(ρx,ρy)=W(ρx,ρy,ρx,ρy), and the degree of coherence of the LGSM beam in the output plane is obtained as
μ(ρ1x,ρ1y,ρ2x,ρ2y)=W(ρ1x,ρ1y,ρ2x,ρ2y)W(ρ1x,ρ1y,ρ1x,ρ1y)W(ρ2x,ρ2y,ρ2x,ρ2y).
(21)
Applying Eqs. (18) and (21), one can study the evolution properties of the degree of coherence of a LGSM beam in turbulent atmosphere numerically.

3. Second-order moments of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

Applying the following “sum” and “difference” coordinates,
r=r1+r22,rd=r1r2,ρ=ρ1+ρ22,ρd=ρ1ρ2,
(22)
Equation (3) can be expressed as
W(ρ,ρd)=(k2πz)2W(r,rd)exp[ikz(ρr)(ρdrd)]×exp(rd2ρ02rdρdρ02ρd2ρ02)d2rd2rd,
(23)
where

W(r,rd)=W(r1,r2)=W(r+rd2,rrd2).
(24)

After some operations as shown in [39

39. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

], i.e., by expressing W(r,rd)in term of its Fourier transform and applying the properties of the Dirac delta function, Eq. (23) can be expressed in the following alternative form
W(ρ,ρd)=(12π)2W(r',ρd+zkκd)d2r'd2κd×exp(iρκdir'κd3ρ02ρd2z2ρ02k2κd23zρ02kρdκd),
(25)
whereκd(κdxκdy)is the position vector in the spatial-frequency domain.

For a LGSM beam, we can express its CSD W(r',ρd+zkκd) as follows

W(r',ρd+zkκd)=exp[12σ02r'2(18σ02+12σg2)(ρd+zkκd)2]Ln0[12σg2(ρd+zkκd)2].
(26)

The Wigner distribution of a partially coherent beam can be expressed in terms of the CSD by the formula [39

39. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

]
h(ρ,θ)=(k2π)2W(ρ,ρd)exp(ikθρd)d2ρd,
(27)
whereθ(θx,θy)denotes an angle which the vector of interest makes with the z-direction, kθx and kθy are the wave vector components along the x-axis and y-axis, respectively.

Applying Eqs. (25)-(27), we obtain the following expression for the Wigner distribution of a LGSM beam in turbulent atmosphere
h(ρ,θ)=k216π42πσ02Ln0[12σg2(ρd+zkκd)2]×exp(aρd2bκd2cρdκdikθρd+iρκd)d2κdd2ρd.
(28)
where

a=18σ02+12σg2+3ρ02,b=z28k2σ02+z22k2σg2+σ022+z2k2ρ02,c=z4kσ02+zkσg2+3zkρ02.
(29)

The moments of order n1+n2+m1+m2of the Wigner distribution function of a beam is defined as
<xn1yn2θxm1θym2>=1Pxn1yn2θxm1θym2h(ρ,θ)d2ρd2θ,
(30)
where

P=h(ρ,θ,z)d2ρd2θ.
(31)

Substituting Eq. (28) into Eqs. (30) and (31), we obtain (after integration) the following expressions for the second-order moments of the Wigner distribution function of a LGSM beam in a turbulent atmosphere

ρ2=z2k2[12σ02+2σg2(1+n)]+2σ02+4z2k2ρ02,
(32)
θ2=1k2[12σ02+2σg2(1+n)]+12k2ρ02,
(33)
ρθ=zk2[12σ02+2σg2(1+n)]6zk2ρ02.
(34)

The propagation factor of a partially coherent beam in a turbulent atmosphere is defined in terms of the second-order moments as follows [39

39. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

]
M2(z)=k(ρ2θ2ρθ2)1/2.
(35)
Substituting Eqs. (32)-(34) into Eq. (35), we obtain the following expression for the propagation factor of a LGSM beam in turbulent atmosphere

M2(z)={[z22k2σ02+2z2k2σg2(1+n)+2σ02+4z2k2ρ02][12σ02+2σg2(1+n)+12ρ02][z2σ02+2zσg2(1+n)+6zρ02]2}
(36)

Under the condition of ρ0=, Eq. (36) reduces to the following expression for the propagation factor of a LGSM beam in free space
M2=[1+4σ02σg2(1+n)]1/2.
(37)
From Eq. (37), we see that the propagation factor of a LGSM beam in free space is independent of the propagation distance as expected, and its value increases as the beam order increases.

Under the condition of ρ0= and n = 0, Eq. (37) reduces to the following expression for the propagation factor of a GSM beam in free space [55

55. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

]

M2=(1+4σ02σg2)1/2.
(38)

4. Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere

In this section, we study the statistical properties of a LGSM beam in turbulent atmosphere numerically by applying the formulae derived in above sections.

Figure 1
Fig. 1 Modulus of the degree of coherence of a LGSM beam at several propagation distances in free space.
shows the modulus of the degree of coherence of a LGSM beam at several propagation distances in free space with λ=632.8nm,σg=5mm,σ0=10mmand n = 2. One finds from Fig. 1 that the degree of coherence of the LGSM beam in the source plane has non-Gaussian distribution, and there are side robes around the main peak. The side robes in the degree of coherence disappear gradually on propagation in free space. In the far field, only the main peak exists which also has non-Gaussian distribution. Figure 2
Fig. 2 Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant Cn2.
shows the modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the structure constant Cn2 with λ=632.8nm, σg=5mm, σ0=10mm and n = 1. From Fig. 2, one finds that the evolution properties of the degree of coherence at short propagation distance in turbulent atmosphere are similar to the corresponding evolution properties in free space, i.e., the side robes disappear gradually on propagation. While at long propagation distance, the distribution of the degree of coherence in turbulent atmosphere is much different from that in free space. In turbulent atmosphere, the degree of coherence becomes of Gaussian distribution in the far field. We may explain this phenomenon by the fact that at the short propagation distance, the influence of the turbulence can be neglected and the role of free-space diffraction plays a dominant role. At long propagation distance, the influence turbulence plays a dominant role, and the degree of coherence takes a Gaussian distribution due to the isotropic influence of the turbulence. Figure 3
Fig. 3 Modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order n.
shows modulus of the degree of coherence of a LGSM beam at several propagation distances in turbulent atmosphere for different values of the mode order n. One finds from Fig. 3 that the evolution properties of the degree of coherence of the LGSM beam in turbulent atmosphere are also affected by the mode order n. The conversion from the non-Gaussian distribution to Gaussian distribution becomes slower as the mode order n increases, which means that a LGSM beam with larger n is less affected by turbulence.

Figure 4
Fig. 4 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the structure constantCn2.
shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the structure constant Cn2 with λ=632.8nm,σ0=10mm,σg=10mm. Fig. 5
Fig. 5 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the coherence widthσg.
shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the coherence width σg withλ=632.8nm,σ0=10mm,Cn2=5×1015m2/3. Figure 6
Fig. 6 Normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the wavelengthλ.
shows the normalized propagation factor of a LGSM beam versus the propagation distance in turbulent atmosphere for different values of the mode order n and the wavelengthλwith Cn2=5×1015m2/3,σ0=10mm,σg=10mm.. One finds from Figs. 4-6 that the normalized propagation factor of a LGSM beam increases on propagation in turbulent atmosphere, which is much different from its properties in free space, where the propagation factor is independent of the propagation distance z. Thus, the turbulence degrades the beam quality of the LGSM beam on propagation. We find that the normalized propagation factor of a LGSM beam with larger n increases slower than a LGSM beam with smaller n or a GSM beam (n = 0) on propagation, which means that the LGSM beam with larger n is less affected by turbulence. Furthermore, we note that the advantage of a LGSM beam with larger n over a LGSM beam with smaller n or a GSM beam is enhanced for larger structure constantCn2, larger coherence width σg and smaller wavelengthλ. Thus, it is necessary for us to take these parameters into consideration in practical applications.

5. Summary

We have derived the analytical expressions for the CSD and the second-order moments of a LGSM beam in turbulent atmosphere, and we have studied the statistical properties, such as the degree of coherence and the propagation factor, of a LGSM beam in turbulent atmosphere with the help of the derived formulae. We have found that a LGSM beam with larger mode order n is less affected by turbulence than a LGSM beam with smaller mode order n or a GSM beam by choosing suitable beam parameters. In [7

7. 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(9), 1794–1802 (2002). [CrossRef] [PubMed]

], it is shown that a GSM beam has advantage over a coherent Gaussian beam for reducing the turbulence-induced degradation, thus it is useful in free-space optical communication. The results in our manuscript have shown that a LGSM beam has advantage over a GSM beam for reducing the turbulence-induced degradation, thus we can expect that the LGSM beam will be useful in free-space optical communication.

Acknowledgments

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Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]

6.

A. Belendez, L. Carretero, and A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993). [CrossRef]

7.

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(9), 1794–1802 (2002). [CrossRef] [PubMed]

8.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). [CrossRef] [PubMed]

9.

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012). [CrossRef]

10.

C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

11.

G. Wu and Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]

12.

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]

13.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

14.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]

15.

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

16.

Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]

17.

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]

18.

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]

19.

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

20.

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]

21.

S. Du, Y. Yuan, C. Liang, and Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]

22.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]

23.

Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]

24.

Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]

25.

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. Doc. ID 202151 (2014).

26.

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

27.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]

28.

J. Cang, P. Xiu, and X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]

29.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]

30.

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge Univeristy, 2007).

31.

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

32.

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]

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.

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]

35.

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45(16), 3793–3797 (2006). [CrossRef] [PubMed]

36.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]

37.

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]

38.

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]

39.

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]

40.

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]

41.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]

42.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]

43.

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]

44.

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef] [PubMed]

45.

Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010). [CrossRef] [PubMed]

46.

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]

47.

F. Wang and Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]

48.

F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012). [CrossRef] [PubMed]

49.

Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37(9), 1553–1555 (2012). [CrossRef] [PubMed]

50.

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]

51.

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef] [PubMed]

52.

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]

53.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

54.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

55.

M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(030.0030) Coherence and statistical optics : Coherence and statistical optics

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: November 25, 2013
Revised Manuscript: January 11, 2014
Manuscript Accepted: January 13, 2014
Published: January 21, 2014

Citation
Rong Chen, Lin Liu, Shijun Zhu, Gaofeng Wu, Fei Wang, and Yangjian Cai, "Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere," Opt. Express 22, 1871-1883 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1871


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. Y. Cai, S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005). [CrossRef] [PubMed]
  3. F. Wang, Y. Cai, S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006). [CrossRef] [PubMed]
  4. D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am. 65(8), 887–891 (1975). [CrossRef]
  5. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53(11), 1057–1060 (1984). [CrossRef]
  6. A. Belendez, L. Carretero, A. Fimia, “The use of partially coherent light to reduce the efficiency of silve-halide noise gratings,” Opt. Commun. 98(4-6), 236–240 (1993). [CrossRef]
  7. J. C. Ricklin, F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef] [PubMed]
  8. T. van Dijk, D. G. Fischer, T. D. Visser, E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104(17), 173902 (2010). [CrossRef] [PubMed]
  9. Y. Dong, F. Wang, C. Zhao, Y. Cai, “Effect of spatial coherence on propagation, tight focusing and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012). [CrossRef]
  10. C. Zhao, Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]
  11. G. Wu, Y. Cai, “Detection of a semirough target in turbulent atmosphere by a partially coherent beam,” Opt. Lett. 36(10), 1939–1941 (2011). [CrossRef] [PubMed]
  12. Y. Cai, U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
  13. F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]
  14. F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]
  15. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]
  16. Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef] [PubMed]
  17. Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]
  18. Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013). [CrossRef] [PubMed]
  19. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]
  20. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]
  21. S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013). [CrossRef]
  22. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013). [CrossRef]
  23. Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]
  24. Z. Mei, E. Schchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef] [PubMed]
  25. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. Doc. ID 202151 (2014).
  26. Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]
  27. F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]
  28. J. Cang, P. Xiu, X. Liu, “Propagation of Laguerre-Gaussian and Bessel-Gaussian Schell-model beams through paraxial optical system in turbulent atmosphere,” Opt. Laser Technol. 54, 35–41 (2013). [CrossRef]
  29. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]
  30. E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge Univeristy, 2007).
  31. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Washington, 2001).
  32. H. T. Eyyuboğlu, 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]
  33. T. Shirai, A. Dogariu, 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. Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
  35. Y. Baykal, H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45(16), 3793–3797 (2006). [CrossRef] [PubMed]
  36. Y. Cai, S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]
  37. Y. Cai, Y. Chen, H. T. Eyyuboğlu, Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32(16), 2405–2407 (2007). [CrossRef] [PubMed]
  38. R. J. Noriega-Manez, J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007). [CrossRef] [PubMed]
  39. Y. Dan, B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef] [PubMed]
  40. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, 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]
  41. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008). [CrossRef] [PubMed]
  42. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]
  43. Y. Cai, Q. Lin, H. T. Eyyuboğlu, 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]
  44. W. Cheng, J. W. Haus, Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009). [CrossRef] [PubMed]
  45. Y. Gu, G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010). [CrossRef] [PubMed]
  46. P. Zhou, Y. Ma, X. Wang, H. Zhao, Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35(7), 1043–1045 (2010). [CrossRef] [PubMed]
  47. F. Wang, Y. Cai, “Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 18(24), 24661–24672 (2010). [CrossRef] [PubMed]
  48. F. Wang, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012). [CrossRef] [PubMed]
  49. Y. Gu, G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37(9), 1553–1555 (2012). [CrossRef] [PubMed]
  50. F. Wang, X. Liu, L. Liu, Y. Yuan, Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]
  51. X. Liu, Y. Shen, L. Liu, F. Wang, Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef] [PubMed]
  52. S. C. H. Wang, M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]
  53. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  54. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  55. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999). [CrossRef]

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