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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 22 — Oct. 27, 2008
  • pp: 18437–18442
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Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere

Yingbin Zhu, Daomu Zhao, and Xinyue Du  »View Author Affiliations


Optics Express, Vol. 16, Issue 22, pp. 18437-18442 (2008)
http://dx.doi.org/10.1364/OE.16.018437


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Abstract

Analytical formulas for the elements of the 2×2 cross-spectral density matrix of a kind of stochastic electromagnetic array beam propagating through the turbulent atmosphere are derived with the help of vector integration. Two types of superposition (i.e. the correlated superposition and the uncorrelated superposition) are considered. The changes in the spectral density and in the spectral degree of polarization of such an array beam generated by isotropic or anisotropic electromagnetic Gaussian Schell-model sources on propagation are determined by the use of the analytical formulas. It is shown by numerical calculations that for the array beam composed by isotropic Gaussian-Schell model sources, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic sources, the spectral degree of polarization in the far field approaches a fixed value that is different from the source.

© 2008 Optical Society of America

1. Introduction

The propagation properties of various laser beams through the turbulent atmosphere have been extensively investigated. It is well known that partially coherent beams are less influenced by turbulent atmosphere. Many studies have been done on the characterization of partially coherent beams propagating through turbulent atmosphere [1

1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef] [PubMed]

6

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

]. Quite recently a class of stochastic electromagnetic beams became of great interest. With the help of the unified theory of coherence and polarization [7

7. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003). [CrossRef]

], the propagation properties of stochastic electromagnetic beams through turbulent atmosphere have been investigated [8

8. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003). [CrossRef] [PubMed]

13

13. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

].

On the other hand, a variety of linear, rectangular and radial laser arrays have been developed to achieve high system powers, high-energy weapons, etc. The propagation of various laser arrays in atmosphere has been investigated [14

14. X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825–833 (2008). [CrossRef]

16

16. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157–167 (2007). [CrossRef]

]. In this paper, we investigate a two-dimensional (2D) rectangular array beam, composed by electromagnetic Gaussian Schell-model sources, propagating through turbulent atmosphere. Two kinds of superposition, i.e., the correlated superposition and the uncorrelated superposition are considered. The analytical propagation formulas for the array beam are obtained with the help of vector integration. The changes in the spectral density and in the spectral degree of polarization of this class of beams on propagation through turbulent atmosphere are investigated in details.

2. Theory

Suppose that the beam array consists of M×N stochastic electromagnetic Gaussian Schell-model sources, positioned at the plane of z=0 (see Fig. 1). We assume that M and N are odd numbers, the extension to the even numbers is straightforward.

Fig. 1. 2D model of the M×N stochastic electromagnetic Gaussian Schell-model sources array.

2.1. Uncorrelated superposition

Suppose that the beamlets of the array are uncorrelated with each other, then the intensity distribution in the source plane can be considered as the simple superposition of intensity of each beamlet. The elements of the cross-spectral density matrix of the beamlet with subscription m and n centered at the point ρ 0mn can be expressed as

Wijmn(0)(ρ1',ρ2',ω)=AiAjBijexp[(ρ1'ρ0mn)24σi2(ρ2'ρ0mn)24σj2]exp(ρ2'ρ1'22δij2),
(1)

where ρ1'=(x1'y1'),ρ2'=(x2'y2'),ρ0mn=(mx0ny0), x 0 and y 0 are the separation distances of the beamlets. The coefficients Ai, Aj, Bij and the variances σi, σj, σij are independent of position but may depend on frequency [9

9. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]

]. Written in the tensor form, Eq. (1) can be expressed as:

Wijmn(0)(ρ¯12',ω)=AiAjBijexp[ik2(ρ¯12'ρ¯0mn)TMij'1(ρ¯12'ρ¯0mn)],
(2)

where ρ¯12'=(ρ1'ρ2'),ρ¯0mn=(ρ0mnρ0mn), and M-1 ij is a 4×4 matrix of the following form:

Mij'1=[(i2kσi2ikδij2)I(ikδij2)I(ikδij2)I(i2kσj2ikδij2)I],
(3)

where I is a 2×2 unitary matrix. Then the elements of the cross-spectral density matrix of the laser array source in the plane z=0 can be expressed as:

Wij(0)(ρ¯12',ω)=m=M12M12n=N12N12Wijmn(0)(ρ¯12',ω)
=m=M12M12n=N12N12{AiAjBijexp[ik2(ρ¯12'ρ¯0mn)TMij'1(ρ¯12'ρ¯0mn)]}.
(4)

The propagation formula for the elements of the cross-spectral density matrix in the turbulent atmosphere has the following form [3

3. 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, 1794–1802 (2002). [CrossRef]

, 13

13. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

]:

Wij(ρ1,ρ2,z,ω)=k24π2z2Wij(0)(ρ1',ρ2',ω)exp[ik2z(ρ1ρ1')2+ik2z(ρ2ρ2')2]
×exp[ψ*(ρ1,ρ1',z,ω)+ψ(ρ2,ρ2',z,ω)]md2ρ1'd2ρ2'.
(5)

Consider the Kolmogorov power spectrum, the last term in Eq. (5) can be written as [1

1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef] [PubMed]

, 3

3. 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, 1794–1802 (2002). [CrossRef]

, 13

13. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

]:

exp[ψ*(ρ1,ρ1',z,ω)+ψ(ρ2,ρ2',z,ω)]m=exp[(12)Dψ(ρd',ρd)]
exp[(1ρ02)(ρd'2+ρd'·ρd+ρd2)],
(6)

with ρd=ρ1-ρ2, ρ d=ρ 1 ρ 2°ρ 0(0.545C 2 nk 2 z)-3/5 is the coherence length of a spherical wave propagating through the turbulent medium and C 2 n is the structure parameter of the refractive index. We have employed a quadratic approximation [1

1. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef] [PubMed]

] for the Rytov’s phase structure function in order to obtain simple and viewable analytical result.

On substituting from Eq. (6) into Eq. (5) and using the tensor method, we obtain the integral formula for the elements of cross-spectral density matrix in the half-space z>0 as follows [13

13. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

]:

Wij(ρ¯12,z,ω)=k24π2[Det(B¯)]12Wij(0)(ρ¯12',ω)exp[ik2(ρ¯12TB¯1ρ¯12'2ρ¯12'TB¯1ρ¯12+ρ¯12TB¯1ρ¯12)]
exp[ik2(ρ¯12'TP¯ρ¯12'+ρ¯12'TP¯ρ¯12+ρ¯12TP¯ρ¯12)]d4ρ¯12'
(7)

where ρ̄′T 12=(ρT 1,ρT 2)=(x1,y1,x2,y2) and ρ̄′T 12=(ρT 1,ρT 2)=(x1,y1,x2,y2) are fourdimensional vectors, T stands for the matrix transposition, Det is the determinant, and

B¯=[zI00zI],P¯=2ikρ02[IIII].
(8)

On substituting from Eq. (4) into Eq. (7), and after performing vector integration, the elements of the cross-spectral density matrix of the laser array beam in the half-space z>0 turn out to be

Wij(ρ¯12,ω)=m=M12M12n=N12N12Wijmn(ρ¯12,ω)
=m=M12M12n=N12N12AiAjBij[Det(I¯+B¯P¯+B¯Mij'1)]12exp(ik2ρ¯12TM2ij1ρ¯12)
×exp{ikρ¯0mnT[I¯+(B¯1+P¯)Mij']1(B¯112P¯)ρ¯12}exp{ik2ρ¯0mnT[Mij'+(B¯1+P¯)1]1ρ¯0mn}
(9)

where

M2ij1=(B¯1+P¯)(B¯112P¯)T(B¯1+P¯+Mij'1)1(B¯112P¯).
(10)

2.2 Correlated superposition

Suppose that the beamlets of the laser array are correlated with each other. Similar to the expression for 1D array in [17

17. B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A 5, 303–307 (2003). [CrossRef]

], the elements of the cross-spectral density matrix of the laser array source can be expressed as:

Wij(0)(ρ1',ρ2',ω)=(m=M12M12n=N12N12Ei(ρ1'ρ0mn))*(p=M12M12q=N12N12Ej(ρ2'ρ0pq))
=m=M12M12n=N12N12p=M12M12q=N12N12Ei*(ρ1'ρ0mn)Ej(ρ2'ρ0pq)
=m=M12M12n=N12N12p=M12M12q=N12N12{AiAjBijexp[(ρ1'ρ0mn)24σi2(ρ1'ρ0pq)24σj2]
×exp((ρ2'ρ0mn)(ρ1'ρ0pq)22δij2)}
(11)

where ρ 0mn and ρ 0pq are the center positions of two different beamlets, Ei(ρ1-ρ 0mn) and Ej(ρ2-ρ 0pq) are the field components of two different beamlets. If 〈E * i(ρ1-ρ 0mn)Ej(ρ2-ρ 0pq)〉=0 for ρ 0mnρ 0pq, Eq. (11) reduces to the expression for uncorrelated superposition. Equation (11) can be written in the form

Wij(0)(ρ¯12',ω)=m=M12M12n=N12N12p=M12M12q=N12N12AiAjBijexp[ik2(ρ¯12'ρ¯0mnpq)TMij'1(ρ¯12'ρ¯0mnpq)],
(12)

where ρ¯0mnpq=(ρ0mnρ0pq),Mij'1 is given by Eq. (3).

On substituting from Eq. (12) into Eq. (7), one can obtain the elements of the cross-spectral density matrix of the laser array beam in the half-space z>0:

Wij(ρ¯12,ω)=m=M12M12n=N12N12p=M12M12q=N12N12AiAjBij[Det(I¯+B¯P¯+B¯Mij'1)]12exp(ik2ρ¯12TM2ij1ρ¯12)
×exp{ikρ¯0mnpqT[I¯+(B¯1+P¯)Mij]1(B¯112P¯)ρ¯12}exp{ik2ρ¯0mnpqT[Mij'+(B¯1+P¯)1]1ρ¯0mnpq}
(13)

where M -1 2ij has the same form with Eq. (10). With the choice of ρ 1=ρ 2=ρ, one can obtain the elements of the cross-spectral density matrix Wij(ρ,z,ω) at any two coincident points in the half-space z>0. The spectral density and the spectral degree of polarization of the array beam at the point (ρ,z) with ρ̄T 12=(ρ T,ρ T) are given by [7

7. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003). [CrossRef]

, 13

13. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

, 18

18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

]

S(ρ,z,ω)=TrW(ρ,ρ,z,ω),
(14)
P(ρ,z,ω)=14DetW(ρ,ρ,z,ω)[TrW(ρ,ρ,z,ω)]2.
(15)

3. Numerical calculations and analyses

Figure 2 shows the normalized spectral density of the stochastic electromagnetic Gaussian Schell-model array beam composed by uncorrelated superposition (solid curve) and correlated superposition (dashed curve) propagating in free space and through the turbulent atmosphere. Figure 2(a) shows the free space propagation, it can be seen that the normalized spectral density always has fringe pattern on propagation for the correlated superposition, this phenomenon is similar to the fringe generating by multiple-beams interference. However, the normalized spectral density always has a Gaussianlike profile for the uncorrelated superposition. Figure 2(b) shows the propagation through the turbulent atmosphere. It can be seen that the fringe disappears and becomes a Gaussianlike profile when the array beam propagates into the sufficiently far field.

Fig. 2. Normalized spectral density for the array beam composed by uncorrelated superposition (solid curve) and correlated superposition (dashed curve) propagating in (a) free space and (b) turbulent atmosphere. The source parameters for each beamlet are: λ=632.8 nm, Ax=2, Ay=1, Bxy=0.2exp(/3), σx=1 cm, σy=1 cm, σxx=σyy=3 mm, σxy=6 mm. The other parameters are: x 0=y 0=σy, M=N=3, C 2 n=10-15m-2/3.

The physical reason for the above fringe generating in the case of correlated superposition can be explained as follows: the optical path difference from each beamlet in the source plane to the observation point is stationary, which induces the stationary phase deviation. When the array beam propagates through atmosphere, the random turbulence destroys the wavefront configuration, i.e., disturbs the stationary phase deviation, and then the fringe disappears when the array beam propagates into the sufficiently far field.

Figure 3 shows the changes in the spectral degree of polarization along the z-axis (ρ̄12=0) of an electromagnetic Gaussian Schell-model array beam passing through the turbulent atmosphere. One can find that the spectral degree of polarization has the same value in the source plane or in the sufficiently far field for the two types of superposition, while it has different values on propagation. In Fig. 3(a), σi and σj are assumed to be equal, the spectral degree of polarization returns to its value in the source plane after the beam propagates a sufficiently long distance. This phenomenon can be considered as a consequence of the fact that the polarization components of the source field have different spatial coherence properties, i.e., the coherence-induced polarization change. In Fig. 3(b), we show the spectral degree of polarization of an array beam generated by the beamlets of anisotropic source, i.e., the source with σiσj. It can be seen that the spectral degree of polarization always changes on propagation even though each beamlet satisfies δxx=δyy=δxy, which can be considered as the anisotropic-induced polarization change. In Fig. 3(c), the source of each beamlet is anisotropic and has different coherent properties of the different components of the electromagnetic field. It can be seen that the spectral degree of polarization in the sufficiently far field approaches a fixed value that is different from the source plane. This phenomenon can be considered as the combinations of the coherence-induced polarization change and the anisotropic-induced polarization change.

Fig. 3. Changes in the spectral degree of polarization P along the z-axis of electromagnetic Gaussian Schell-model array beam propagating through the turbulent atmosphere. Solid curves: the uncorrelated superposition. Dashed curves: the correlated superposition. The source parameters are the same as in Fig. 2, but (a) σx=σy=1 cm, σxx=σyy=3 mm, σxy=6 mm; (b)σx=2cm, σy=1 cm, σxx=σyy=σxy=6 mm; (c) σx=2 cm, σy=1 cm, σxx=σyy=3 mm, σxy=6 mm;. The other parameters are: x 0=y 0=σy, M=N=3, C 2 n=10-15m-2/3.

4. Conclusions

In summary, generalized analytical expressions for the elements of the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model array beam composed by uncorrelated superposition and correlated superposition propagating in the homogeneous and isotropic turbulent atmosphere have been derived with the help of vector integration. By using numerical calculations we have studied the changes in the spectral density and the spectral degree of polarization of the stochastic electromagnetic array beams. We found that, for the correlated superposition, the spectral density of the array beam has fringe pattern on propagation, and the atmosphere can destroy the fringe pattern when the array beam propagates into the sufficiently far field. For the uncorrelated superposition, the spectral density of the array beam has Gaussianlike profile in the far field. For the array beam composed by isotropic Gaussian-Schell model sources that satisfy σx=σy (uniform polarized beamlet sources) and have different correlation coefficients of the x and y components of the field, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic Gaussian-Schell model sources that satisfy σxσy (non-uniform polarized beamlet sources) and have different or the same correlation coefficients of the x and y components of the field (see Ref. [18

18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

], Appendix III), the spectral degree of polarization in the sufficiently far field approaches a fixed value that is different from the source.

Acknowledgments

This work was supported by the Program for New Century Excellent Talents in University (NCET-07-0760) and the National Natural Science Foundation of China under grant 10874150.

References and links

1.

H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef] [PubMed]

2.

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

3.

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, 1794–1802 (2002). [CrossRef]

4.

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

5.

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

6.

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

7.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003). [CrossRef]

8.

T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003). [CrossRef] [PubMed]

9.

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]

10.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005). [CrossRef]

11.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves Random Media 14, 513–523 (2004). [CrossRef]

12.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B 89, 91–97 (2007). [CrossRef]

13.

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]

14.

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825–833 (2008). [CrossRef]

15.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265–271 (2008). [CrossRef]

16.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157–167 (2007). [CrossRef]

17.

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A 5, 303–307 (2003). [CrossRef]

18.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(030.1640) Coherence and statistical optics : Coherence
(260.5430) Physical optics : Polarization

ToC Category:
Atmospheric and oceanic optics

History
Original Manuscript: July 25, 2008
Revised Manuscript: September 9, 2008
Manuscript Accepted: October 16, 2008
Published: October 24, 2008

Citation
Yingbin Zhu, Daomu Zhao, and Xinyue Du, "Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere," Opt. Express 16, 18437-18442 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18437


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References

  1. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972). [CrossRef] [PubMed]
  2. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979). [CrossRef]
  3. 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, 1794-1802 (2002). [CrossRef]
  4. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002). [CrossRef]
  5. G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007). [CrossRef]
  6. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003). [CrossRef]
  7. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003). [CrossRef]
  8. T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003). [CrossRef] [PubMed]
  9. O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004). [CrossRef]
  10. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005). [CrossRef]
  11. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004). [CrossRef]
  12. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007). [CrossRef]
  13. X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007). [CrossRef] [PubMed]
  14. X. Ji, E. Zhang, and B. Lü, "Superimposed partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. B 25, 825-833 (2008). [CrossRef]
  15. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008). [CrossRef]
  16. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007). [CrossRef]
  17. B. Li and B. Lü, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A 5, 303-307 (2003). [CrossRef]
  18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

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