## Abstract

The analytical formula for the effective radius of curvature of radial Gaussian array beams propagating through atmospheric turbulence is derived, where coherent and incoherent beam combinations are considered. The influence of turbulence on the effective radius of curvature of radial Gaussian array beams is studied by using numerical calculation examples.

## 2. Analytical Formulae

#### 2.1 Mean squared beam width in free space

Assume that a radial array beam consists of N equal elements, which are Gaussian beams and located symmetrically on a ring with radius $r0$, and separation angle $α0=2π/N$. For the coherent combination, the cross-spectral density function of the radial Gaussian array beam in the plane z = 0 is expressed as

$W(x1′,x2′,y1′,y2′,z=0)=∑p=0N−1∑q=0N−1exp[−(x1′cosαp+y1′sinαp−r0)2+(y1′cosαp−x1′sinαp)2w02]$$×exp[−(x2′cosαq+y2′sinαq−r0)2+(y2′cosαq−x2′sinαq)2w02],$
where $αj=jα0$ (j = p, q = 0, 1, 2, …N-1), $N≥2$ and $w0$ is the waist width of Gaussian beams.

Based on the Huygens-Fresnel principle, the intensity of the radial Gaussian array beam represented by Eq. (1) propagating in free space reads as

where $r′≡(x′,y′),$ $r≡(x,y),$ k is the wave number related to the wavelength λ by $k=2π/λ.$ The mean squared beam width $$ is defined as [18]

$=∬r2I(r,z)d2r/∬I(r,z)d2r.$

Upon substituting from Eq. (2) into Eq. (3), and making use of the integral transform technique [20], after very tedious integral calculations we obtain

$=G+(Q/k2)z2,$
where
$G=∑p=0N−1∑q=0N−1w02{r0′2[1+cos(αp−αq)]+1}S/2∑p=0N−1∑q=0N−1S,$
$Q=2w02∑p=0N−1∑q=0N−1{1−r0′2[1−cos(αp−αq)]}S/∑p=0N−1∑q=0N−1S,$
$S=exp{r0′2[1+cos(αp−αq)]},$
with $r0′=r0/w0$ being the inverse radial fill-factor [21].

#### 2.2 Effective radius of curvature in turbulence

The transformation of the mean squared beam width of an arbitrary field by the optical ABCD system can be expressed as [18]

$=<(Ar1+Bθ1)2>,$
where A, B and D are elements of the transfer matrix $(ABCD)$, $θ≡(θx,θy)$, $kθx$ and $kθy$ are the wave vector components along the x-axis and y-axis respectively. The parameters with the subscripts “1” and “2” denote those before and after the optical ABCD system respectively.

Comparing the propagation equation of the mean squared beam width of an arbitrary field (i.e., Eq. (8)) with that of an ideal Gaussian beam, the effective radius of curvature is defined by [18]

$R=/.$

When the beam waist is located in the plane z = 0, from Eq. (9) we obtain the free-space propagation equation of the effective radius of curvature of an arbitrary field, i.e.,

$R=z+zR2z,$
where $zR2=0/<θ2>0$ is the Rayleigh range of an arbitrary field in free space [18], the angle brackets with the subscript 0 denote the second moments in the plane z = 0. The Rayleigh range is used in the theory of lasers to characterize the distance over which a beam may be considered effectively non-spreading. It is clear that the effective radius of curvature of an arbitrary field defined by Eq. (9) obeys the same free-space propagation equation as does the wavefront curvature of an ideal Gaussian beam.

On the other hand, the general formulae of the second moments $$ and $$ of partially coherent beams propagating through atmospheric turbulence can be expressed as [19]

$=0+20z+<θ2>0z2+(4/3)Tz3,$
$=0+<θ2>0z+2Tz2,$
where
$T=π2∫0∞κ3Φn(κ)dκ.$
with $Φn$ being the spatial power spectrum of the refractive index fluctuations of the turbulent atmosphere.

Upon substituting from Eqs. (11)-(12) into Eq. (9), and considering $0=A,$ $0=0$ and $<θ2>0=B/k2$ which can be derived from comparing Eq. (4) with Eq. (11) for T = 0, we obtain the effective radius of curvature of radial Gaussian array beams propagating through atmospheric turbulence for the coherent combination case, i.e.

$R=A+(B/k2)z2+(4/3)Tz3(B/k2)z+2Tz2.$

Similarly, for the case of incoherent combination, the effective radius of curvature of radial Gaussian array beams in turbulence can be obtained as

$R=w02(1/2+r0′2)+(2/k2w02)z2+(4/3)Tz3(2/k2w02)z+2Tz2.$
From Eqs. (14) and (15), we have $R∞=limz→∞R=z$ in free space. It implies that the wavefront can be regarded as the spherical surface when the free-space propagation distance is large enough. Equation (15) indicates that for the incoherent combination, R increases with increasing $r0′$, but R is independent of N.

## 3. Numerical calculation results and analysis

The numerical calculation results are given in Figs. 1 -3 to show the influence of turbulence on the effective radius of curvature R, where $λ=1.06μm$ and $w0=0.01m$ are kept fixed. In the numerical calculations, Tatarskii spectrum is adopted, i.e., $Φn(κ)=0.033Cn2κ−11/3exp(−κ2/κm2),$ where $κm=5.92/l0,$ $l0$ is the turbulence inner scale, $Cn2$ is the refraction index structure constant. If the typical value of $l0=0.01m$ is taken, from Eq. (13) yields $T=7.6113Cn2.$

Fig. 1 Effective radius of curvature R versus the propagation distance z, $r0′=2$, N = 15

Fig. 3 Effective radius of curvature R versus the beam number N, z = 4km, $r0′=2$.

It is noted that in Figs. 1-3, two combinations of the sources, i.e., coherent and incoherent beam arrays are considered. From Figs. 1-3 it can be seen that for the two types of beam combination R decreases due to turbulence. But the decrease of R is generally larger for the coherent combination than that for the incoherent combination, i.e., for the incoherent combination R is less sensitive to turbulence than that for the coherent combination. Figure 1 shows that in free space for the coherent combination R is always larger than that for the incoherent combination, while for the coherent combination R may be smaller than that for the incoherent combination as the propagation distance z increases in turbulence. Figure 1 also indicates that, there exists the minimum R min as z increases, and position z max of R min is further away from the source plane for the coherent combination than that for the incoherent combination. From Fig. 2 it can be seen that there may exist a minimum of R as $r0′$ varies for the coherent combination in turbulence. It has been shown that this result is valid only when the strength of turbulence is strong enough, and the propagation distance is long enough. In addition, for the two types of beam combination R approaches the same value when $r0′$ is large enough. Figure 3 shows that for the coherent combination, R increases with increasing N in free space, but R decreases with increasing N in turbulence. Furthermore, R tends to its asymptotical value when N is large enough, and for the two types of beam combination R approaches the same value when N is small enough.

Fig. 2 Effective radius of curvature R versus the inverse radial fill-factor $r0′$, z = 4km.

We note that a radial array beam is quite different from an annulus beam even when N is large enough since it is composed of multi-beams that are combined coherently or incoherently. Thus, propagation of radial array and annulus beams are different.

## 4. Conclusions

In summary, the simple and analytical formula for the effective radius of curvature R of radial Gaussian array beams in turbulence has been derived in this paper. It has been shown that for the two types of beam combination R decreases due to turbulence. However, for the incoherent combination R is less sensitive to turbulence than that for the coherent combination. In free space R for the coherent combination is always larger than that for the incoherent combination, but this situation may reverse as the propagation distance increases in turbulence. For the coherent combination, R increases with increasing the beam number N in free space, but R decreases with increasing N in turbulence. In addition, there may exist a minimum of R as the inverse radial fill-factor $r0′$ varies when the strength of turbulence is strong enough and the propagation distance is long enough. The method used in this paper would be useful for studying the effective radius of curvature of an arbitrary single laser beam or an arbitrary array laser beam in free space or in turbulence.

## Acknowledgment

The authors are very thankful to the reviewers for valuable comments. Xiaoling Ji acknowledges the support provided by the National Natural Science Foundation of China under grant 60778048.

1. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.

2. L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

3. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003). [CrossRef]   [PubMed]

4. 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]

5. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009). [CrossRef]

6. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007). [CrossRef]

7. H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008). [CrossRef]

8. 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(25), 16909–16915 (2007). [CrossRef]   [PubMed]

9. M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992). [CrossRef]   [PubMed]

10. 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]

11. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996). [CrossRef]

12. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).

13. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008). [CrossRef]   [PubMed]

14. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007). [CrossRef]

15. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008). [CrossRef]

16. X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26(2), 236–243 (2009). [CrossRef]

17. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010). [CrossRef]

18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992). [CrossRef]

19. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef]   [PubMed]

20. To derive Eq. (4), we first introduce the new variables of integration by setting $u=(r2′+r1′)/2$ and $v=r2′−r1′$. Then we use the formulae$∫exp(−i2πxs)dx=δ(s)$, $∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2$, $∫f(x)δ″(x)dx=f″(0)$ and $∫f(x)δ(x)dx=f(0)$, where δ denotes the Dirac delta function and $δ″$ is its second derivative, and f is an arbitrary function and $f″$ is its second derivative.

21. J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37(6), 1045–1048 (1998). [CrossRef]

### References

• View by:

1. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.
2. L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
3. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
[Crossref] [PubMed]
4. 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]
5. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]
6. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24(11), 3554–3563 (2007).
[Crossref]
7. H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]
8. 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(25), 16909–16915 (2007).
[Crossref] [PubMed]
9. M. A. Porras, J. Alda and E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31(30), 6389–6402 (1992).
[Crossref] [PubMed]
10. 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]
11. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]
12. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).
13. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]
14. Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]
15. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]
16. X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26(2), 236–243 (2009).
[Crossref]
17. X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]
18. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[Crossref]
19. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009).
[Crossref] [PubMed]
20. To derive Eq. (4), we first introduce the new variables of integration by setting u=(r2′+r1′)/2 and v=r2′−r1′. Then we use the formulae∫exp(−i2πxs)dx=δ(s), ∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2, ∫f(x)δ″(x)dx=f″(0) and ∫f(x)δ(x)dx=f(0), where δ denotes the Dirac delta function and δ″ is its second derivative, and f is an arbitrary function and f″ is its second derivative.
21. J. D. Strohschein, H. J. Seguin, and C. E. Capjack, “Beam propagation constants for a radial laser array,” Appl. Opt. 37(6), 1045–1048 (1998).
[Crossref]

#### 2010 (1)

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

#### 2009 (3)

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

#### 2008 (3)

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]

#### 2007 (3)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

#### 2006 (1)

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]

#### 1996 (1)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### 1992 (2)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[Crossref]

#### Baker, H. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### Baykal, Y.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

#### Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

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]

#### Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

#### Eyyuboglu, H. T.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

#### Hall, D. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### He, S.

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]

#### Hornby, A. M.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### Ji, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

#### Korotkova, O.

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

#### Li, X.

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

#### Morley, R. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### Pu, J.

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

#### Strohschein, J. D.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[Crossref]

#### Yelden, E. F.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### Appl. Phys. B (3)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007).
[Crossref]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91(2), 265–271 (2008).
[Crossref]

X. Li, X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Turbulence distance of radial Gaussian Schell-model array beams,” Appl. Phys. B 98(2-3), 557–565 (2010).
[Crossref]

#### Appl. Phys. Lett. (1)

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]

#### IEEE J. Quantum Electron. (1)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996).
[Crossref]

#### J. Opt. A, Pure Appl. Opt. (1)

H. T. Eyyuboglu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10(1), 015005 (2008).
[Crossref]

#### Opt. Commun. (1)

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere,” Opt. Commun. 282(9), 1691–1698 (2009).
[Crossref]

#### Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–1049 (1992).
[Crossref]

#### Other (4)

W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser,” in Gas and Chemical Lasers and Applications, H. R. C. Sze and E. A. Dorko, eds., Proc. SPIE 2987, 13–21 (1997).

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics 22, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Chap. 6.

L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

To derive Eq. (4), we first introduce the new variables of integration by setting u=(r2′+r1′)/2 and v=r2′−r1′. Then we use the formulae∫exp(−i2πxs)dx=δ(s), ∫x2exp(−i2πxs)dx=−δ″(s)/(2π)2, ∫f(x)δ″(x)dx=f″(0) and ∫f(x)δ(x)dx=f(0), where δ denotes the Dirac delta function and δ″ is its second derivative, and f is an arbitrary function and f″ is its second derivative.

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### Figures (3)

Fig. 1 Effective radius of curvature R versus the propagation distance z, $r 0 ′ = 2$ , N = 15
Fig. 3 Effective radius of curvature R versus the beam number N, z = 4km, $r 0 ′ = 2$ .
Fig. 2 Effective radius of curvature R versus the inverse radial fill-factor $r 0 ′$ , z = 4km.

### Equations (16)

$W ( x 1 ′ , x 2 ′ , y 1 ′ , y 2 ′ , z = 0 ) = ∑ p = 0 N − 1 ∑ q = 0 N − 1 exp [ − ( x 1 ′ cos α p + y 1 ′ sin α p − r 0 ) 2 + ( y 1 ′ cos α p − x 1 ′ sin α p ) 2 w 0 2 ]$
$× exp [ − ( x 2 ′ cos α q + y 2 ′ sin α q − r 0 ) 2 + ( y 2 ′ cos α q − x 2 ′ sin α q ) 2 w 0 2 ] ,$
$< r 2 > = ∬ r 2 I ( r , z ) d 2 r / ∬ I ( r , z ) d 2 r .$
$< r 2 > = G + ( Q / k 2 ) z 2 ,$
$G = ∑ p = 0 N − 1 ∑ q = 0 N − 1 w 0 2 { r 0 ′2 [ 1 + cos ( α p − α q ) ] + 1 } S / 2 ∑ p = 0 N − 1 ∑ q = 0 N − 1 S ,$
$Q = 2 w 0 2 ∑ p = 0 N − 1 ∑ q = 0 N − 1 { 1 − r 0 ′2 [ 1 − cos ( α p − α q ) ] } S / ∑ p = 0 N − 1 ∑ q = 0 N − 1 S ,$
$S = exp { r 0 ′2 [ 1 + cos ( α p − α q ) ] } ,$
$< r 2 2 > = < ( A r 1 + B θ 1 ) 2 > ,$
$R = < r 2 > / < r ⋅ θ > .$
$R = z + z R 2 z ,$
$< r 2 > = < r 2 > 0 + 2 < r ⋅ θ > 0 z + < θ 2 > 0 z 2 + ( 4 / 3 ) T z 3 ,$
$< r ⋅ θ > = < r ⋅ θ > 0 + < θ 2 > 0 z + 2 T z 2 ,$
$T = π 2 ∫ 0 ∞ κ 3 Φ n ( κ ) d κ .$
$R = A + ( B / k 2 ) z 2 + ( 4 / 3 ) T z 3 ( B / k 2 ) z + 2 T z 2 .$
$R = w 0 2 ( 1 / 2 + r 0 ′2 ) + ( 2 / k 2 w 0 2 ) z 2 + ( 4 / 3 ) T z 3 ( 2 / k 2 w 0 2 ) z + 2 T z 2 .$