Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere

doi:10.1364/OE.18.024661

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Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere

Fei Wang and Yangjian Cai »View Author Affiliations

Optics Express, Vol. 18, Issue 24, pp. 24661-24672 (2010)
http://dx.doi.org/10.1364/OE.18.024661

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Abstract

We present a detailed investigation of the second-order statistics of a twisted Gaussian Schell-model (TGSM) beam propagating in turbulent atmosphere. Based on the extended Huygens-Fresnel integral, analytical expressions for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere are derived. Evolution properties of the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of a TGSM beam in turbulent atmosphere are explored in detail. Our results show that a TGSM beam is less affected by the turbulence than a GSM beam without twist phase. In turbulent atmosphere the Rayleigh range doesn’t equal to the distance where the ERC takes a minimum value, which is much different from the result in free space. The second-order statistics are closely determined by the parameters of the turbulent atmosphere and the initial beam parameters. Our results will be useful in long-distance free-space optical communications.

1. Introduction

In the past decades, partially coherent beams have been widely investigated and applied in free space optical communication, optical imaging, nonlinear optics, optical trapping, inertial confinement fusion, optical projection and laser scanning [1

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

–6

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

]. Gaussian Schell-model (GSM) beam is a typical and commonly encountered partially coherent beam, whose spectral density and spectral degree of coherence have Gaussian shapes [1

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

]. By scattering a coherent laser beam from a rotating grounded glass, then transforming the spectral density distribution of the scattered light into Gaussian profile with a Gaussian amplitude filter, a GSM beam can be generated [8

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef]

]. GSM beams can also be generated with specially synthesized rough surfaces, spatial light modulators and synthetic acousto-optic holograms (c.f [9

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

].). Propagation properties of a GSM beam have been studied widely [1

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

,10

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]

–13

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]

]. It has been found that a GSM beam is less affected by the turbulent atmosphere compared to a coherent Gaussian beam, thus have important applications in free space optical communication, remote sensing and radar system [11

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).

–13

A more general partially coherent beam can possess a twist phase, which differs in many respects from the customary quadratic phase factor. In 1993, Simon and Mukunda first introduced the twisted Gaussian Schell-model (TGSM) beam [14

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]

]. Unlike the usual phase curvature, the twist phase is bounded in strength due to the fact that the cross-spectral density function must be nonnegative and it is absent in a coherent Gaussian beam. The twist phase has an intrinsic chiral property and is responsible for the rotation of the beam spot on propagation. Friberg et al. first carried out experimental demonstration of TGSM beams [15

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]

]. Superposition, coherent-mode decomposition and the analysis of the transfer of radiance of the TGSM beam have been investigated in [16

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]

,17

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996). [CrossRef]

]. Dependence of the orbital angular momentum of a partially coherent beam on its twist phase was revealed in Ref [18

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [CrossRef]

]. The conventional method for treating the propagation of TGSM beams is the Wigner-distribution function [14

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]

]. Lin and Cai have introduced a convenient alternative tensor method for treating the propagation of TGSM beams [19

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]

]. With the help of the tensor method, the propagation properties of a TGSM beam through paraxial ABCD optical system, dispersive media and nonlinear media were studied in [20

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef]

–23

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

]. More recently, Ghost imaging with a TGSM beam was explored in [24

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009). [CrossRef] [PubMed]

]. Zhao et al. studied the radiation force of a TGSM beam on a Rayleigh particle [25

C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]

]. Twist phase-induced polarization changes in electromagnetic GSM beam were studied in [26

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]

Investigations of the propagation properties of laser beams in a turbulent atmosphere become more and more important because of their wide applications in e.g. free-space optical communications and remote sensing [2

Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]

,11

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).

–13

,27

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]

–29

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]

]. Average intensity and spreading properties of a TGSM beam have been studied in [29

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]

]. Recently, more and more attention is being paid to the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of laser beams in turbulent atmosphere [30

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]

–34

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010). [CrossRef]

]. To our knowledge no results have been reported up until now on the second-order statistics of a TGSM beam in turbulent atmosphere. The purpose of this paper is to investigate the propagation factor, the ERC and the Rayleigh range of a TGSM beam in turbulent atmosphere, and to explore the advantage of a TGSM beam over a GSM beam for overcoming or reducing the turbulence-induced degradation. Analytical expressions are derived for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere, and some useful and interesting results are found.

2. Second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere

A partially coherent beam is generally characterized by the cross-spectral density (CSD) function, and the CSD function of a TGSM beam in the source plane (z = 0) is expressed as [14

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]

]

W_{0} (r_{1}^{'}, r_{2}^{'}; 0) = \exp [- \frac{r_{1}^{' 2} + r_{2}^{' 2}}{4 σ_{I 0}^{2}} - \frac{{(r_{1}^{'} - r_{2}^{'})}^{2}}{2 σ_{g 0}^{2}} - \frac{i k μ_{0}}{2} {(r_{1}^{'} - r_{2}^{'})}^{T} J (r_{1}^{'} + r_{2}^{'})],

(1)

where

r_{1}^{'} \equiv (x_{1}^{'}, y_{1}^{'})

and

r_{2}^{'} \equiv (x_{2}^{'}, y_{2}^{'})

represent two arbitrary position vectors in the source plane, respectively;

k = 2 π / λ

is the wave number with λ being the wavelength of light field.

σ_{I 0} and σ_{g 0}

denote the transverse beam width and spectral coherence width, respectively.

μ_{0}

is a scalar real-valued twist factor with the dimension of an inverse distance, limited by the double inequality

0 \leq μ_{0}^{2} \leq {[k^{2} σ_{g 0}^{4}]}^{- 1}

due to the non-negativity requirement of Eq. (1). In the coherent limit,

σ_{g 0} \to \infty

, the twist factor

μ_{0}

disappears. In Eq. (1), the symbol J denotes an anti-symmetric matrix given by [14

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]

]

J = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) .

(2)

Under the condition of

μ_{0} = 0

, the CSD function in Eq. (1) reduces to the CSD function of a conventional GSM beam without twist phase [7

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]

–9

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

]. Due to the existence of the term

{(r_{1}^{'} - r_{2}^{'})}^{T} J (r_{1}^{'} + r_{2}^{'}) = x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}

in the right side of Eq. (1), the two-dimensional CSD function cannot be split in a product of two one-dimensional CSD functions.

Within the validity of the paraxial approximation, based on the extended Huygens-Fresnel integral, the CSD function of a partially coherent beam propagating in turbulent atmosphere at z is expressed as [11

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).

,30

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]

–34

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010). [CrossRef]

]

W (r, r_{d}; z) = \frac{1}{λ^{2} z^{2}} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} W_{0} (r^{'}, r_{d}^{'}; 0) \exp [\frac{i k}{z} (r - r^{'}) \cdot (r_{d} - r_{d}^{'}) - H (r_{d}, r_{d}^{'}; z)] d^{2} r^{'} d r_{d}^{'}

(3)

where

W_{0}' (r^{'}, r_{d}^{'}; 0) = W_{0} (r_{1}^{'}, r_{2}^{'}; 0) = W_{0} (r^{'} + r_{d}^{'} / 2, r^{'} - r_{d}^{'} / 2; 0)

(4)

In the derivation of Eq. (3), we have used following sum and difference notations

r = (r_{1} + r_{2}) / 2, r_{d} = r_{1} - r_{2}, r^{'} = (r_{1}^{'} + r_{2}^{'}) / 2, r_{d}^{'} = r_{1}^{'} - r_{2}^{'} .

(5)

The term

\exp [- H (r_{d}, r_{d}^{'}; z)]

in Eq. (3) is the contributions from atmospheric turbulence, and can be written as [11

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).

\exp [- H (r_{d}, r_{d}^{'}; z)] = 4 π^{2} k^{2} z \int_{0}^{1} d ξ \int_{0}^{\infty} [1 - J_{0} (κ | r_{d}^{'} ξ + (1 - ξ) r_{d} |)] Φ_{n} (κ) κ d κ,

(6)

where

J_{0}

is the Bessel function of zero order,

Φ_{n}

represents the spectral density for the index-of-refraction fluctuations in turbulent atmosphere, and κ is the magnitude of the spatial wave-number.

The Wigner distribution function (WDF) of a partially coherent beam on propagation in turbulent atmosphere can be expressed in terms of the CSD function by the formula [30

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]

,31

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]

]

h (r, θ; z) = {(\frac{1}{λ})}^{2} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} W (r, r_{d}; z) \exp [- i k θ \cdot r_{d}] d^{2} r_{d},

(7)

where

θ \equiv (θ_{x}, θ_{y})

denotes an angle which the vector of interest makes with the z-direction;

k θ_{x}

and

k θ_{y}

are the wave vector components along the x-axis and y-axis, respectively.

Substituting Eq. (3) into Eq. (7), we obtain (after some operation) following expression for the WDF of a partially coherent in turbulent atmosphere

\begin{array}{l} h (r, θ; z) = \frac{1}{{(2 π)}^{2}} {(\frac{1}{λ})}^{2} \int_{- \infty}^{\infty} \int_{\infty}^{\infty} \int_{- \infty}^{\infty} W_{0} (r^{"}, r_{d} + \frac{z}{k} κ_{d}; 0) \exp (i κ_{d} \cdot r^{"} - i κ_{d} \cdot r - i k θ \cdot r_{d}) \\ \times \exp [- H (r_{d}, r_{d} + \frac{z}{k} κ_{d}; z)] d^{2} κ_{d} d^{2} r^{"} d^{2} r_{d}, \end{array}

(8)

where

κ_{d} \equiv (κ_{d x}, κ_{d y})

is the position vector in spatial-frequency domain. In the derivation of Eq. (8), we have used following formula

W_{0} (r^{'}, r_{d}^{'}; 0) \equiv \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} W_{0} (r^{"}, r_{d}^{'}; 0) \exp [i κ_{d} \cdot (r^{"} - r^{'})] d^{2} κ_{d} d^{2} r^{"} .

(9)

We can express

W_{0} (r^{"}, r_{d} + \frac{z}{k} κ_{d}; 0)

of a TGSM beam as follows

W_{0} (r^{"}, r_{d} + \frac{z}{k} κ_{d}; 0) = \exp [- \frac{r^{"}^{2}}{2 σ_{I 0}^{2}} - (\frac{1}{8 σ_{I 0}^{2}} + \frac{1}{2 σ_{g 0}^{2}}) {(r_{d} + \frac{z}{k} κ_{d})}^{2}] \exp [- i k μ_{0} {(r_{d} + \frac{z}{k} κ_{d})}^{T} J r^{"}] .

(10)

Substituting Eq. (10) into Eq. (8), we obtain (after integration over

r^{"}

) the following expression for the WDF of a TGSM beam in turbulent atmosphere

\begin{array}{l} h (r, θ; z) = \frac{1}{{(2 π)}^{2}} 2 π σ_{I 0}^{2} {(\frac{1}{λ})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \exp [- \frac{σ_{I 0}^{2}}{2} {(κ_{d x} + (k μ_{0} y_{d} + μ_{0} z κ_{d y}))}^{2}] \\ \times \exp [- \frac{σ_{I 0}^{2}}{2} {(κ_{d y} - (k μ_{0} x_{d} + μ_{0} z κ_{d x}))}^{2}] \exp [- (\frac{1}{8 σ_{I 0}^{2}} + \frac{1}{2 σ_{g 0}^{2}}) {(r_{d} + \frac{z}{k} κ_{d})}^{2}] \\ \times \exp [- i r \cdot κ_{d} - i k θ \cdot r_{d}] \exp (- H (r_{d}, r_{d} + \frac{z}{k} κ_{d}; z)) d^{2} κ_{d} d^{2} r_{d} . \end{array}

(11)

According to Ref [30

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]

], the moments of order

n_{1} + n_{2} + m_{1} + m_{2}

of the WDF of a laser beam is given by

〈 x^{n 1} y^{n 2} θ_{x}^{m 1} θ_{y}^{m 2} 〉 = \frac{1}{P} \int \int_{- \infty}^{\infty} x^{n 1} y^{n 2} θ_{x}^{m 1} θ_{y}^{m 2} h (r, θ, z) d^{2} r d^{2} θ,

(12)

where

P = \int \int_{- \infty}^{\infty} h (r, θ, z) d^{2} r d^{2} θ .

(13)

The second-order statistics of a laser beam, such as the propagation factor, the ERC and the Rayleigh range, are closely related with the second-order moments of the WDF. Substituting Eq. (11) into Eq. (12), we obtain (after tedious integration and operation) following expressions for the second-order moments of the WDF of a TGSM beam propagating in turbulent atmosphere

〈 r {(z)}^{2} 〉 = 〈 x {(z)}^{2} 〉 + 〈 y {(z)}^{2} 〉 = 2 σ_{I 0}^{2} + 2 A z^{2} + 4 π^{2} T z^{3} / 3,

(14)

〈 r (z) \cdot θ (z) 〉 = 〈 x (z) θ_{x} (z) 〉 + 〈 y (z) θ_{y} (z) 〉 = 2 A z + 2 π^{2} T z^{2},

(15)

〈 θ {(z)}^{2} 〉 = 〈 θ_{x}^{} {(z)}^{2} 〉 + 〈 θ_{y}^{} {(z)}^{2} 〉 = 2 A + 4 π^{2} T z,

(16)

where

A = 1 / (4 k^{2} σ_{I 0}^{2}) + 1 / (k^{2} σ_{g 0}^{2}) + μ_{0}^{2} σ_{I 0}^{2},

(17)

T = \int_{0}^{\infty} Φ_{n} (κ) κ^{3} d κ,

(18)

P = 2 π σ_{I 0}^{2} .

(19)

In the above derivations, we have used following integral formula [35

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

δ (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} \exp (- i s x) d x,

(20)

δ^{n} (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} {(- i x)}^{n} \exp (- i s x) d x, (n = 0, 1, 2),

(21)

\int_{- \infty}^{\infty} f (x) δ^{n} (x) d x = {(- 1)}^{n} f^{(n)} (0), (n = 1, 2) .

(22)

In Eqs. (14) and (16), the symbols

〈 r {(z)}^{2} 〉

and

〈 θ {(z)}^{2} 〉

represent the squared beam width and the squared far-field divergence of the TGSM beam in turbulent atmosphere, respectively. The ERC of the TGSM beam is closely determined by

〈 r (z) \cdot θ (z) 〉

in Eq. (15).

3. Propagation factor of a TGSM beam in turbulent atmosphere

The propagation factor (best known as

M^{2}

-factor) proposed by Siegman is a particularly important property of an optical laser beam [1

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

] being regarded as a beam quality factor in many practical applications. Based on the second-order moments of the Wigner distribution function, the

M^{2}

-factor of a partially coherent beam is defined as [30

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]

–32

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

]

M^{2} (z) = k {[〈 r {(z)}^{2} 〉 〈 θ {(z)}^{2} 〉 - {〈 r (z) \cdot θ (z) 〉}^{2}]}^{1 / 2} .

(23)

Substituting Eqs. (14)-(16) into Eq. (23), we obtain following expression for the

M^{2}

-factor of a TGSM beam in turbulent atmosphere

M^{2} (z) = {[{(M^{2} (0))}^{2} + (8 σ_{I 0}^{2} + 8 A z^{2} / 3 + 4 π^{2} T z^{3} / 3) k^{2} π^{2} T z]}^{1 / 2},

(24)

where

M^{2} (0)

in Eq. (24) represents the M ²-factor of the TGSM beam in free space or in the source plane given by

M^{2} (0) = \sqrt{1 + 4 μ_{0}^{2} k^{2} σ_{I 0}^{4} + 4 σ_{I 0}^{2} / σ_{g 0}^{2}} .

(25)

Under the condition of

T = 0

(without turbulence), Eq. (24) reduces to the expression for the M ²-factor of a TGSM beam in free space. Under the condition of

μ_{0} = 0

, Eq. (24) reduces to the expression for the M ²-factor of a GSM beam without twist phase in turbulent atmosphere. From Eq. (25), it is clear that the M ²-factor of a TGSM beam in free space is independent of the propagation distance, and increases with the increase of the absolute value of the twist factor. This phenomenon is caused by the fact that the twist factor cause more rapid spreading of a TGSM beam on propagation.

Now we study the evolution properties of the M ²-factor of a TGSM beam in turbulent atmosphere. In the following numerical examples, we adopt the Tatarskii spectrum for the spectral density of the index-of-refraction fluctuations, which is expressed as [11

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).

]

Φ_{n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3} \exp (- κ^{2} / κ_{m}^{2}),

(26)

where

C_{n}^{2}

is the structure constant of the refractive index fluctuations of the turbulence and

κ_{m}^{} = 5.92 / l_{0}

with

l_{0}

being the inner scale of the turbulence. In the following text, we set

λ = 1060 nm

. Substituting Eq. (26) into Eq. (18), we obtain

T = \int_{0}^{\infty} Φ_{n} (κ) κ^{3} d κ = 0.1661 C_{n}^{2} l_{0}^{- 1 / 3} .

(27)

Substituting Eq. (27) into Eq. (24), we can calculate the

M^{2}

-factor of a TGSM beam in turbulent atmosphere numerically.

For the convenience of comparison, we now study the normalized M ²-factor of a TGSM beam defined as

M^{2} (z) / M^{2} (0)

on propagation in turbulent atmosphere. Figure 1 shows the normalized M ²-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constant

C_{n}^{2}

. As illustrated by Fig. 1, the normalized M ²-factor of a TGSM beam in turbulent atmosphere increases on propagation, which is much different from its propagation-invariant properties in free space (

C_{n}^{2} = 0

). As the value of the structure constant

C_{n}^{2}

increases (i.e., turbulence becomes strong) or the value of the inner scale

l_{0}

decreases, the normalized M ²-factor increases more rapid on propagation. Figure 2 shows the normalized M ²-factor of a TGSM beam on propagation in turbulent atmosphere for different values of

σ_{g 0}^{}

and

μ_{0}

with

l_{0} = 0.01 m

. One finds from Fig. 2(a) that the normalized M ²-factor of a TGSM beam increases slower on propagation as its initial coherence width

σ_{g 0}^{}

decreases, which means that a TGSM beam with lower coherence is less affected by turbulent atmosphere as expected [30

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]

–32

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

]. One finds from Fig. 2(b) that the normalized M ²-factor of a TGSM beam increases slower than that of a GSM beam without twist phase (

μ_{0} = 0

) on propagation in turbulent atmosphere, which means that a TGSM beam is less affected by atmospheric turbulence than a GSM beam. Furthermore, as shown by Fig. 2(b), the TGSM beam with larger absolute value of

μ_{0}

is less affected by the turbulence than that with smaller absolute value of

μ_{0}

Fig. 1 Normalized M ²-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constant

C_{n}^{2}

and the inner scale

l_{0}

Fig. 2 Normalized M ²-factor of a TGSM beam on propagation in turbulent atmosphere for different values of

σ_{g 0}^{}

and

μ_{0}

In order to show the advantage of a TGSM beam over a GSM beam in turbulent atmosphere quantitatively, we introduce a parameter

Δ M^{2} (z)

named the deviation percentage of the normalized M ²-factor to show the difference between the normalized M ²-factor of a TGSM beam and that of a GSM beam. The deviation percentage of the normalized M ²-factor is defined as

Δ M^{2} (z) = \frac{| M^{2} (z) / M^{2} (0) |_{μ_{0}} - M^{2} (z) / M^{2} (0) |_{μ_{0} = 0} |}{M^{2} (z) / M^{2} (0) |_{μ_{0} = 0}} .

(28)

The advantage of a TGSM beam over a GSM bam increases with the increase of the deviation percentage

Δ M^{2} (z)

We calculate in Fig. 3 the deviation percentage of the normalized M ²-factor versus the propagation distance z for different values of

μ_{0}

with

σ_{I 0} = 10 m m

σ_{g 0} = 10 m m

l_{0} = 0.01 m

and

C_{n}^{2} = 10^{- 14} m^{- 2/3}

. As shown in Fig. 3, the parameter

Δ M^{2} (z)

increases on propagation, and it approaches to a constant value in the far field. The constant value increases as the absolute value of

μ_{0}

increases. For the case of

| μ_{0} | = 1.5 {km}^{- 1}

, the parameter

Δ M^{2} (z)

approaches to 5%, which is quite significant. In practical experiment, we can convert a GSM beam into a TGSM beam with a six-element astigmatic lens system as shown in [15

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]

], and control the twist phase by controlling the astigmatic lens. A GSM beam can be generated with the help of a rotating grounded glass and a Gaussian amplitude filter conveniently [8

]. Thus it is economic and realizable to generate a TGSM beam for application in free-space optical communications.

Fig. 3 Deviation percentage of the normalized M ²-factor versus the propagation distance z for different values of

μ_{0}

4. Effective radius of curvature of a TGSM beam in turbulent atmosphere

According to [33

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010). [CrossRef] [PubMed]

,34

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010). [CrossRef]

], the ERC of a laser beam at z is defined in terms of the ratio of

〈 r {(z)}^{2} 〉

〈 r (z) \cdot θ (z) 〉

as follows

R (z) = 〈 r {(z)}^{2} 〉 / 〈 r (z) \cdot θ (z) 〉 .

(29)

Substituting Eqs. (14) and (15) into Eq. (29), we obtain following expression for the ERC of a TGSM beam in turbulent atmosphere

R (z) = z + \frac{σ_{I 0}^{2} - π^{2} T z^{3} / 3}{A z + π^{2} T z^{2}} .

(30)

From Eq. (30), one finds that the ERC of a TGSM beam on propagation are determined by the beam parameters (i.e., beam width

σ_{I 0}

, the coherence width

σ_{g 0}

, the wavelength λ and the twisted factor

μ_{0}

) and the parameters of the turbulent atmosphere (i.e., structure constant

C_{n}^{2}

and the inner scale

l_{0}

) together. Under the condition of

T = 0

and

μ_{0} = 0

, Eq. (30) reduces to the ERC of a GSM beam in free space. Equation (30) provides a convenient way for studying the evolution properties of a GSM beam with or without twist phase in turbulent atmosphere.

We calculate in Fig. 4 the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of the structure constant

C_{n}^{2}

and the inner scale

l_{0}

with

σ_{I 0} = 10 m m

and

σ_{g 0} = 10 m m

. One finds from Fig. 4 that the ERC of a TGSM beam on propagation in free space (

C_{n}^{2} = 0

) or in turbulent atmosphere will initially display a downward trend in the near field, but after reaching a dip, will star to increase. The value of the ERC on propagation decreases as the structure constant

C_{n}^{2}

increases or the inner scale

l_{0}

decreases especially in the far field.

Fig. 4 ERC of a TGSM beam in turbulent atmosphere versus the propagation distance z for different values of the structure constant

C_{n}^{2}

and the inner scale

l_{0}

Figure 5 shows the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of

μ_{0}

and

σ_{g 0}^{}

with

σ_{I 0} = 10 m m

. One finds from Fig. 5 (a) that the difference between the ERC of a TGSM beam in free space and that in turbulence is smaller than the difference between the ERC of a GSM beam in free space and that in turbulent atmosphere, which means a TGSM beam is less affected by the turbulent atmosphere than a GSM beam from the aspect of ERC. From Fig. 5 (b), it is also clear that the TGSM beam with lower coherence is less affected by the turbulence than that with higher coherence. To show the advantage of a TGSM beam over a GSM beam quantitatively, we introduce a parameter

Δ R (z)

named the deviation percentage of the ERC to show the difference between the ERC of a TGSM or GSM beam in turbulent atmosphere and that of a TGSM or GSM beam in free space. The deviation percentage of the ERC is defined as

Fig. 5 ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of

μ_{0}

and

σ_{g 0}^{}

Δ R (z) = \frac{| R (z) |_{t u r} - R (z) |_{f r e e} |}{R (z) |_{f r e e}} .

(31)

The advantage of a TGSM beam over a GSM beam increases with the increase of the deviation between the

Δ R (z)

of a TGSM beam and that of a GSM beam. We calculate in Fig. 6 the deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance z with

σ_{I 0} = 10 m m

σ_{g 0} = 10 m m

l_{0} = 0.01 m

. One finds from Fig. 6 that the deviation between the

Δ R (z)

of a TGSM beam with

| μ_{0} | = 1.5 k m^{- 1}

and that of a GSM beam increases on propagation, and it approaches to a constant value (about 5%) in far field. This result agrees well with the result shown in Fig. 3. One also finds from Fig. 5 that evolution of the ERC of a TGSM beam is little different from that in free space. In free space, the value of the ERC of a TGSM beam with larger absolute value of

μ_{0}

(or smaller

σ_{g 0}^{}

) on propagation is always smaller than that with smaller absolute value of

μ_{0}

(or larger

σ_{g 0}^{}

) in the near field or in the intermediate propagation distance. With the increase of propagation distance, the difference between the ERC of TGSM beams with different

μ_{0}

σ_{g 0}^{}

becomes smaller, and in the far field, the ERC tends to

R (z) \to z

. In turbulent atmosphere, there exists a critical propagation length

z_{c}

where the TGSM beams with different

μ_{0}

σ_{g 0}^{}

have the same value of ERC. For the case of

z < z_{c}

, the value of the ERC of the TGSM beam with larger absolute value of

μ_{0}

(or smaller

σ_{g 0}^{}

) is smaller that that with smaller absolute value of

μ_{0}

(or larger

σ_{g 0}^{}

). For the case of

z > z_{c}

, the reverse situation occurs. From Eq. (30), we obtain following expression for the critical propagation length

z_{c} = {(\frac{3 σ_{I 0}^{2}}{π^{2} T})}^{1 / 3} .

(32)

One finds from Eq. (32) that

z_{c}

is only determined by the beam width

σ_{I 0}^{}

, the structure constant

C_{n}^{2}

and the inner scale

l_{0}

of turbulence. At the critical propagation length, the ERC of the TGSM beam turns out to be

R = z_{c}

. By choosing suitable value of the beam width, the ERC of the TGSM beam can remain invariant at fixed receiving plane if

C_{n}^{2}

l_{0}

varies.

Fig. 6 Deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance z.

5. Rayleigh range of a TGSM beam in turbulent atmosphere

The Rayleigh range is an important beam parameter for characterizing the distance within which the laser beam can be considered effectively non-spreading. The Rayleigh range is defined as the distance

z_{R}

along the propagation direction of a beam from the beam waist to the place where the area of the cross section is doubled (i.e., the diameter of the spot size increases by a factor

\sqrt{2}

compared to the spot size at the beam waist) [36

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

]. The range of the minimum effective radius of curvature is defined as the distance

z_{m}

along the propagation direction of a beam from the beam waist to the place where the ERC of the beam takes the minimum vale. In free space, the Rayleigh range

z_{R}

equals to the range of the minimum effective radius of curvature

z_{m}

. What will happen in turbulent atmosphere? Now let’s study the properties of the Rayleigh range

z_{R}

and the range of the minimum effective radius of curvature

z_{m}

in turbulent atmosphere. Based on the definition of

z_{R}

and

z_{m}

[36

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

], they can be obtained by solving following equations

〈 r {(z_{R})}^{2} 〉 - 2 〈 r {(0)}^{2} 〉 = 0,

(33)

d R (z) / d z |_{z = z_{m}} = 0.

(34)

Substituting Eqs. (14) and (30) into Eqs. (33) and (34) respectively, we obtain

4 π^{2} T z_{R}^{3} / 3 + 2 A z^{2} - 2 σ_{I 0}^{2} = 0,

(35)

2 π^{2} T^{2} z_{m}^{4} + 4 π^{2} A T z_{m}^{3} + 3 A^{2} z_{m}^{2} - 6 π^{2} T z_{m}^{} σ_{I 0}^{2} - 3 A σ_{I 0}^{2} = 0.

(36)

Under the condition of T = 0 (free space), Eqs. (35) and (36) reduce to the same quadratic equation. After some calculation, we obtain following analytical expression for

z_{R}

and

z_{m}

of a TGSM beam in free space

z_{R} = z_{m} = σ_{I 0}^{} {(1 / (4 k^{2} σ_{I 0}^{2}) + 1 / (k^{2} σ_{g 0}^{2}) + μ_{0}^{2} σ_{I 0}^{2})}^{- 1 / 2} .

(37)

Under the condition of

μ_{0} = 0

, Eq. (37) reduces to the expression for

z_{R}

of a GSM in free space as shown in [36

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

]. By solving Eqs. (35) and (36), we obtain (after tedious operation and calculation) following expressions for

z_{R}

and

z_{m}

of a TGSM beam in turbulent atmosphere,

z_{R} = (A^{2} + M_{1}^{2} - A M_{1}) / (2 π^{2} T M_{1}),

(38)

z_{m} = (\sqrt{(A^{3} + 6 π^{4} T^{2} σ_{I 0}^{2}) / N_{3} - N_{3}^{2}} + N_{3} - A) / (2 π^{2} T),

(39)

where

M_{1} = {(- A^{3} + 6 π^{4} T^{2} σ_{I 0}^{2} + 2 \sqrt{9 π^{8} T^{4} σ_{I 0}^{4} - 3 π^{4} A^{3} T^{2} σ_{I 0}^{2}})}^{1 / 3}, N_{2} = 4 N_{1} + 4 \sqrt{N_{1}^{2} - {(54 A^{6})}^{2}}

N_{3} = \sqrt{3 A^{4} / N_{2}^{1 / 3} + N_{2}^{1 / 3} / 12}, N_{1} = 54 {(A^{3} + 6 π^{4} T^{2} σ_{I 0}^{2})}^{2} .

(40)

One finds from Eqs. (38) and (39) that

z_{R}

and

z_{m}

of a TGSM beam in turbulent atmosphere don’t equal to each other generally.

We calculate in Fig. 7

z_{R}

and

z_{m}

of a TGSM beam in turbulent atmosphere different values of twist factor

μ_{0}

and coherence width

σ_{g 0}

with

σ_{I 0} = 10 m m

and

C_{n}^{2} = 10^{- 14} m^{-2/3}

. For the convenience of comparison, the corresponding results in free space are also shown. As shown in Fig. 7,

z_{R}

and

z_{m}

don’t coincide with each other due to the influence of turbulence.

z_{m}

in turbulent atmosphere is always larger than that in free space, and

z_{R}

in turbulent atmosphere is always smaller than that in free space. As the absolute value of twist factor

μ_{0}

increases or the coherence width

σ_{g 0}

decreases, the difference between

z_{R}

and

z_{m}

becomes smaller, which means that a TGSM beam with larger absolute value twist factor or lower coherence is less affected by the turbulence.

Fig. 7

z_{R}

and

z_{m}

of a TGSM beam in turbulent atmosphere different values of twist factor

μ_{0}

and coherence width

σ_{g 0}

6. Conclusion

In conclusion, we have derived the analytical expressions for the second-order moments of the WDF of a TGSM beam in turbulent atmosphere based on the extended Huygens-Fresnel integral. The second-order statistics, such as the propagation factor, the ERC and the Rayleigh range, of a TGSM beam propagating in turbulent atmosphere have been studied and compared with the results in free space. Our numerical results show that a TGSM beam is less affected by the turbulence than a GSM beam, and a TGSM beam with larger absolute value of twist factor or lower coherence is less affected by the turbulence than that with smaller twist factor or higher coherence. Our results will be useful in long-distance free-space optical communications.

Acknowledgments

Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No. 210081.

References and links

1.	L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
2.	Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]
3.	M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]
4.	Y. Cai and 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]
5.	Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
6.	C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
7.	A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
8.	F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef]
9.	E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]
10.	M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]
11.	L. C. Andrews, and R. L. Phillips, Laser Beam Propagation in the Turbulent Atmosphere , 2nd ed. (SPIE Press, Bellington, 2005).
12.	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]
13.	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]
14.	R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]
15.	A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]
16.	D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]
17.	R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996). [CrossRef]
18.	J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [CrossRef]
19.	Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]
20.	Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef]
21.	Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [CrossRef]
22.	Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]
23.	Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
24.	Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009). [CrossRef] [PubMed]
25.	C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]
26.	Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]
27.	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]
28.	Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
29.	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]
30.	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]
31.	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]
32.	S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]
33.	X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010). [CrossRef] [PubMed]
34.	H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010). [CrossRef]
35.	A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
36.	G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(030.1670) Coherence and statistical optics : Coherent optical effects
(350.5500) Other areas of optics : Propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: September 13, 2010
Revised Manuscript: October 13, 2010
Manuscript Accepted: October 28, 2010
Published: November 10, 2010

Citation
Fei Wang and Yangjian Cai, "Second-order statistics of a twisted Gaussian Schell-model beam in turbulent atmosphere," Opt. Express 18, 24661-24672 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24661

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References

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
Y. Baykal, “Average transmittance in turbulence for partially coherent sources,” Opt. Commun. 231(1-6), 129–136 (2004). [CrossRef]
M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46(5), 357–362 (2008). [CrossRef]
Y. Cai and 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]
Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982). [CrossRef]
F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007). [CrossRef]
E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]
M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef] [PubMed]
L. C. Andrews, and R. L. Phillips, Laser beam propagation in the turbulent atmosphere, 2nd edition, (SPIE Press, Bellington, 2005).
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]
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]
R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]
A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]
D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994). [CrossRef]
R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996). [CrossRef]
J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [CrossRef]
Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]
Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002). [CrossRef]
Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002). [CrossRef]
Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006). [CrossRef] [PubMed]
Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007). [CrossRef] [PubMed]
Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009). [CrossRef] [PubMed]
C. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17(24), 21472–21487 (2009). [CrossRef] [PubMed]
Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009). [CrossRef]
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]
Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
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]
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]
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]
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