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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24661–24672
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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.

© 2010 OSA

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

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

6

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

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

,7

7. 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

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

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

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

,10

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

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

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

13

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]

].

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

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

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

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

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

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

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

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

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

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

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

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

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]

].

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

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

]
W0(r1',r2';0)=exp[r1'2+r2'24σI02(r1'r2')22σg02ikμ02(r1'r2')TJ(r1'+r2')],
(1)
where r1'(x1',y1') and r2'(x2',y2') represent two arbitrary position vectors in the source plane, respectively; k=2π/λ is the wave number with λ being the wavelength of light field. σI0 and σg0 denote the transverse beam width and spectral coherence width, respectively. μ0 is a scalar real-valued twist factor with the dimension of an inverse distance, limited by the double inequality 0μ02[k2σg04]1due to the non-negativity requirement of Eq. (1). In the coherent limit, σg0, the twist factor μ0 disappears. In Eq. (1), the symbol J denotes an anti-symmetric matrix given by [14

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

]

J=(0110). 
(2)

Under the condition ofμ0=0, the CSD function in Eq. (1) reduces to the CSD function of a conventional GSM beam without twist phase [7

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

9

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 (r1'r2')TJ(r1'+r2')=x1'y2'x2'y1' 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

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

,30

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

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,rd;z)=1λ2z2W0(r',rd';0)exp[ikz(rr')(rdrd')H(rd,rd';z)]d2r'drd',
(3)
where

W0'(r',rd';0)=W0(r1',r2';0)=W0(r'+rd'/2,r'rd'/2;0).
(4)

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

r=(r1+r2)/2,rd=r1r2,r'=(r1'+r2')/2,rd'=r1'r2'.
(5)

The term exp[H(rd,rd';z)] in Eq. (3) is the contributions from atmospheric turbulence, and can be written as [11

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

]:
exp[H(rd,rd';z)]=4π2k2z01dξ0[1J0(κ|rd'ξ+(1ξ)rd|)]Φn(κ)κdκ,
(6)
where J0 is the Bessel function of zero order, Φnrepresents 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

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

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)=(1λ)2W(r,rd;z)exp[ikθrd]d2rd,
(7)
where θ(θx,θy) denotes an angle which the vector of interest makes with the z-direction; kθxand kθy 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
h(r,θ;z)=1(2π)2(1λ)2W0(r",rd+zkκd;0)exp(iκdr"iκdrikθrd)                  ×exp[H(rd,rd+zkκd;z)]d2κdd2r"d2rd,
(8)
where κd(κdx,κdy) is the position vector in spatial-frequency domain. In the derivation of Eq. (8), we have used following formula
W0(r',rd';0)1(2π)2W0(r",rd';0)exp[iκd(r"r')]d2κdd2r".
(9)
We can express W0(r",rd+zkκd;0) of a TGSM beam as follows
W0(r",rd+zkκd;0)=exp[r"22σI02(18σI02+12σg02)(rd+zkκd)2]exp[ikμ0(rd+zkκd)TJr"].
(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

h(r,θ;z)=1(2π)22πσI02(1λ)2exp[σI022(κdx+(kμ0yd+μ0zκdy))2]                  ×exp[σI022(κdy(kμ0xd+μ0zκdx))2]exp[(18σI02+12σg02)(rd+zkκd)2]                 ×exp[irκdikθrd]exp(H(rd,rd+zkκd;z))d2κdd2rd.                   
(11)

According to Ref [30

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 n1+n2+m1+m2 of the WDF of a laser beam is given by
xn1yn2θxm1θym2=1Pxn1yn2θxm1θym2h(r,θ,z)d2rd2θ,
(12)
where

P=h(r,θ,z)d2rd2θ.
(13)

δ(s)=12πexp(isx)dx,     
(20)
δn(s)=12π(ix)nexp(isx)dx, (n=0, 1, 2),
(21)
f(x)δn(x)dx=(1)nf(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)θ(z) in Eq. (15).

3. Propagation factor of a TGSM beam in turbulent atmosphere

The propagation factor (best known as M2 -factor) proposed by Siegman is a particularly important property of an optical laser beam [1

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 M2-factor of a partially coherent beam is defined as [30

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

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]

]
M2(z)=k[r(z)2θ(z)2r(z)θ(z)2]1/2.
(23)
Substituting Eqs. (14)-(16) into Eq. (23), we obtain following expression for the M2-factor of a TGSM beam in turbulent atmosphere
M2(z)=[(M2(0))2+(8σI02+8Az2/3+4π2Tz3/3)k2π2Tz]1/2,
(24)
where M2(0) in Eq. (24) represents the M 2-factor of the TGSM beam in free space or in the source plane given by

M2(0)=1+4μ02k2σI04+4σI02/σg02.
(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 μ0=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

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

]
Φn(κ)=0.033Cn2κ11/3exp(κ2/κm2),
(26)
where Cn2 is the structure constant of the refractive index fluctuations of the turbulence and κm=5.92/l0 with l0 being the inner scale of the turbulence. In the following text, we set λ=1060nm. Substituting Eq. (26) into Eq. (18), we obtain

T=0Φn(κ)κ3dκ=0.1661Cn2l01/3
(27)

Substituting Eq. (27) into Eq. (24), we can calculate theM2-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 asM2(z)/M2(0)on propagation in turbulent atmosphere. Figure 1
Fig. 1 Normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constantCn2 and the inner scale l0.
shows the normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of the structure constantCn2. 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 (Cn2=0). As the value of the structure constantCn2 increases (i.e., turbulence becomes strong) or the value of the inner scale l0 decreases, the normalized M 2-factor increases more rapid on propagation. Figure 2
Fig. 2 Normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of σg0 and μ0.
shows the normalized M 2-factor of a TGSM beam on propagation in turbulent atmosphere for different values of σg0 and μ0with l0=0.01m. 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 σg0 decreases, which means that a TGSM beam with lower coherence is less affected by turbulent atmosphere as expected [30

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

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 (μ0=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 μ0is less affected by the turbulence than that with smaller absolute value of μ0.

In order to show the advantage of a TGSM beam over a GSM beam in turbulent atmosphere quantitatively, we introduce a parameter ΔM2(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

ΔM2(z)=|M2(z)/M2(0)|μ0M2(z)/M2(0)|μ0=0|M2(z)/M2(0)|μ0=0.
(28)

The advantage of a TGSM beam over a GSM bam increases with the increase of the deviation percentageΔM2(z).

We calculate in Fig. 3
Fig. 3 Deviation percentage of the normalized M 2-factor versus the propagation distance z for different values of μ0.
the deviation percentage of the normalized M 2-factor versus the propagation distance z for different values of μ0 withσI0=10mm, σg0=10mm, l0=0.01m and Cn2=1014m2/3. As shown in Fig. 3, the parameter ΔM2(z) increases on propagation, and it approaches to a constant value in the far field. The constant value increases as the absolute value of μ0 increases. For the case of |μ0|=1.5km1, the parameterΔM2(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

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

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]

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

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

According to [33

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

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 to r(z)θ(z) as follows

R(z)=r(z)2/r(z)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+σI02π2Tz3/3Az+π2Tz2.
(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 σI0, the coherence width σg0, the wavelength λ and the twisted factor μ0) and the parameters of the turbulent atmosphere (i.e., structure constant Cn2 and the inner scale l0) together. Under the condition of T=0 and μ0=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
Fig. 4 ERC of a TGSM beam in turbulent atmosphere versus the propagation distance z for different values of the structure constantCn2 and the inner scale l0.
the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of the structure constantCn2 and the inner scale l0with σI0=10mm and σg0=10mm. One finds from Fig. 4 that the ERC of a TGSM beam on propagation in free space (Cn2=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 constantCn2 increases or the inner scale l0 decreases especially in the far field.

Figure 5
Fig. 5 ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of μ0 and σg0.
shows the ERC of a TGSM beam in turbulent atmosphere versus the propagation distance for different values of μ0 and σg0 with σI0=10mm. 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 ΔR(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

ΔR(z)=|R(z)|turR(z)|free|R(z)|free.
(31)

The advantage of a TGSM beam over a GSM beam increases with the increase of the deviation between theΔR(z)of a TGSM beam and that of a GSM beam. We calculate in Fig. 6
Fig. 6 Deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance z.
the deviation percentage of the ERC of a TGSM or GSM beam versus the propagation distance z withσI0=10mm, σg0=10mm, l0=0.01m. One finds from Fig. 6 that the deviation between theΔR(z)of a TGSM beam with |μ0|=1.5km1 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 ofμ0 (or smaller σg0) on propagation is always smaller than that with smaller absolute value ofμ0 (or larger σg0) 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 μ0 or σg0 becomes smaller, and in the far field, the ERC tends to R(z)z. In turbulent atmosphere, there exists a critical propagation length zc where the TGSM beams with different μ0 or σg0 have the same value of ERC. For the case of z<zc, the value of the ERC of the TGSM beam with larger absolute value of μ0 (or smaller σg0) is smaller that that with smaller absolute value of μ0 (or larger σg0). For the case of z>zc, the reverse situation occurs. From Eq. (30), we obtain following expression for the critical propagation length
zc=(3σI02π2T)1/3.
(32)
One finds from Eq. (32) that zc is only determined by the beam width σI0, the structure constant Cn2 and the inner scale l0 of turbulence. At the critical propagation length, the ERC of the TGSM beam turns out to be R=zc. By choosing suitable value of the beam width, the ERC of the TGSM beam can remain invariant at fixed receiving plane if Cn2 or l0varies.

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 zR 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 2 compared to the spot size at the beam waist) [36

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 zmalong 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 zR equals to the range of the minimum effective radius of curvature zm. What will happen in turbulent atmosphere? Now let’s study the properties of the Rayleigh range zR and the range of the minimum effective radius of curvature zm in turbulent atmosphere. Based on the definition of zR and zm [36

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(zR)22r(0)2=0,
(33)
dR(z)/dz|z=zm=0.
(34)
Substituting Eqs. (14) and (30) into Eqs. (33) and (34) respectively, we obtain

4π2TzR3/3+2Az22σI02=0,
(35)
2π2T2zm4+4π2ATzm3+3A2zm26π2TzmσI023AσI02=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 forzR and zm of a TGSM beam in free space
zR=zm=σI0(1/(4k2σI02)+1/(k2σg02)+μ02σI02)1/2.
(37)
Under the condition of μ0=0, Eq. (37) reduces to the expression for zRof a GSM in free space as shown in [36

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 zR and zm of a TGSM beam in turbulent atmosphere,
zR=(A2+M12AM1)/(2π2TM1),
(38)
zm=((A3+6π4T2σI02)/N3N32+N3A)/(2π2T),
(39)
where
M1=(A3+6π4T2σI02+29π8T4σI043π4A3T2σI02)1/3,N2=4N1+4N12(54A6)2
N3=3A4/N21/3+N21/3/12,N1=54(A3+6π4T2σI02)2.
(40)
One finds from Eqs. (38) and (39) that zR and zm of a TGSM beam in turbulent atmosphere don’t equal to each other generally.

We calculate in Fig. 7
Fig. 7 zR and zm of a TGSM beam in turbulent atmosphere different values of twist factor μ0and coherence width σg0
zR and zm of a TGSM beam in turbulent atmosphere different values of twist factor μ0and coherence width σg0with σI0=10mmand Cn2=1014m-2/3. For the convenience of comparison, the corresponding results in free space are also shown. As shown in Fig. 7, zR and zm don’t coincide with each other due to the influence of turbulence. zm in turbulent atmosphere is always larger than that in free space, and zR in turbulent atmosphere is always smaller than that in free space. As the absolute value of twist factor μ0increases or the coherence width σg0decreases, the difference between zR and zm becomes smaller, which means that a TGSM beam with larger absolute value twist factor or lower coherence is less affected by the turbulence.

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

  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 edition, (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).

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