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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 24699–24711
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Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere

Guoquan Zhou  »View Author Affiliations


Optics Express, Vol. 19, Issue 24, pp. 24699-24711 (2011)
http://dx.doi.org/10.1364/OE.19.024699


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Abstract

A radial phased-locked (PL) Lorentz beam array provides an appropriate theoretical model to describe a coherent diode laser array, which is an efficient radiation source for high-power beaming use. The propagation of a radial PL Lorentz beam array in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral and some mathematical techniques, analytical formulae for the average intensity and the effective beam size of a radial PL Lorentz beam array are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a radial PL Lorentz beam array in turbulent atmosphere are numerically calculated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a radial PL Lorentz beam array in turbulent atmosphere are discussed in detail.

© 2011 OSA

1. Introduction

However, the output of low beam power of a single diode laser hinders its applications. Accordingly, coherent diode laser arrays have been proposed to provide the efficient high-power output [20

20. J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55(6), 531–533 (1989). [CrossRef]

24

24. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009). [CrossRef]

]. Naturally, the Lorentz beam array represents an appropriate theoretical model to describe a coherent diode laser array. Formulae for the intensity distributions of coherent and incoherent combined one-dimensional Lorentz beam array have been derived by using the representation of the cross-spectral density of the far-field [25

25. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010). [CrossRef]

]. When extended to the two-dimensional case, the Lorentz beam array in Ref [25

25. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010). [CrossRef]

]. is a rectangular arrangement. The propagation properties of a P × Q rectangular Lorentz beam array have been investigated in free space [26

26. P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010). [CrossRef]

]. The reported researches on the Lorentz beam array are all confined to a rectangular arrangement. As the important Gaussian beam array is a radial arrangement [27

27. P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010). [CrossRef]

, 28

28. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011). [CrossRef] [PubMed]

] and the Gaussian beam is replaced by the Lorentz beam in the theoretical model of diode lasers, one maybe more interested in a radial Lorentz beam array. The researches on the propagation of beam arrays in a turbulent atmosphere are useful to optical communications and remote sensing [29

29. 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]

32

32. 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]

], therefore, the propagation of a radial Lorentz beam array in turbulent atmosphere is examined in the remainder of this paper. The average intensity and the spreading properties of a radial Lorentz beam array in turbulent atmosphere are mainly concerned.

2. Propagation of a radial phase-locked Lorentz beam array in turbulent atmosphere

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The scheme of a radial Lorentz beam array in the source plane is shown in Fig. 1
Fig. 1 Schematic diagram of a radial PL Lorentz beam array in the source plane.
. The radial Lorentz beam array considered here is phase-locked (PL).

E(r0,0)=n=1NEn(r0,0).
(3)

By directly using Eqs. (1) and (3), one cannot obtain the analytical expression of a beam profile for a radial PL Lorentz beam array propagating in turbulent atmosphere. Therefore, we have to expand the Lorentz distribution, which is the best choice. As the Lorentz distribution is expanded into the linear superposition of Hermite-Gaussian functions [33

33. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976). [CrossRef]

], the optical field of the radial PL Lorentz beam array in the source plane can be rewritten as follows:
E(r0,0)=π2w0xw0yn=1Nm1=0Mm2=0Mσ2m1σ2m2H2m1(x0w0x)H2m2(y0w0y)exp(x022w0x2y022w0y2+iφn),
(4)
where

x0=x0anx,y0=y0any.
(5)

M is the term number of the expansion. H2m1() and H2m2() are the 2m1-th and 2m2-th order Hermite polynomials, respectively. σ2m1 and σ2m2 are the weight coefficients and are listed in Table 1

Table 1. Value of the weight coefficient σ2m.

table-icon
View This Table
[33

33. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976). [CrossRef]

]. With increasing the integer m, however, the value of the weight coefficient σ2m dramatically decays. In the practical calculations, therefore, M takes a small integer. For an example, the original expression of the Lorentz beam in the source plane well coincides with the expansion in the Hermite-Gaussian functions with M = 5.

The propagation of the radial PL Lorentz beam array in a turbulent atmosphere can be investigated by using the following extended Huygens-Fresnel integral [34

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

36

36. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008). [CrossRef] [PubMed]

]:
E(r,z)=ik2πzE(r0,0)exp[ik2z(r0r)2+ψ(r0,r)]dx0dy0,
(6)
where r = xex + yey. (r, z) is the observation plane. ψ(r0, r) is the solution to the Rytov method that represents the random part of the complex phase. k = 2π/λ is the wave number. λ is the optical wavelength. The average intensity of the radial PL Lorentz beam array in the observation plane is given by
<I(r,z)>=k24π2z2E(r01,0)E(r02,0)exp[ik2z(r01r)2+ik2z(r02r)2]×<exp[ψ(r01,r)+ψ(r02,r)]>dr01dr02,
(7)
where the angle brackets indicate the ensemble average over the medium statistics, and the asterisk denotes the complex conjugation. The last ensemble average term in the above equation can be expressed as follows [34

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

]:
<exp[ψ(r01,r)+ψ(r02,r)]>=exp[(r01r02)2ρ02],
(8)
where ρ0=(0.545Cn2k2z)3/5 is the spherical wave lateral coherence length. Cn2 is the structure constant of the atmospheric turbulence. In Eq. (8), the quadratic structure functions are employed. Substituting Eqs. (4) and (8) into Eq. (7) and using the following mathematical formulae [37

37. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, 1980).

]:
H2m1(x)exp[(xy)2/Ω]dx=πΩ(1Ω)m1H2m1[y(1Ω)1/2],
(9)
H2m1(x)=l=0m1(1)l(2m1)!l!(2m12l)!(2x)2m12l,
(10)
(x+y)2m1=u=02m1(2m1u)x2m1uyu,
(11)
x2texp(bx2+2cx)dx=(2t)!πb(cb)2texp(c2b)s=0t1s!(2t2s)!(b4c2)s,
(12)
we can obtain the analytical average intensity of the radial PL Lorentz beam array in the observation plane:
<I(r,z)>=n1=1Nn2=1NIn1,n2(x,z)In1,n2(y,z)exp[i(φn1φn2)],
(13)
with In1,n2(x,z) and In1,n2(y,z)being given by
In1,n2(j,z)=kπ4z1α1jα2jm1=0Mm1=0Mexp(β1j24α1j+β2j24α2jan2j22w0j2an1j2ρ02+ikan1j22zikan1jjz)σ2m1σ2m1×(11α1j)m1l1=0m1(1)l1(2m1)!l1!(2m12l1)!l2=02m12l1(2m12l1l2)γj2m12l1l2δjl2l3=0m1(1)l3(2m1)!l3!(2m12l3)!×l4=02m12l3(2m12l3l4)(2an2jw0j)2m12l3l4(l2+l4)!s=0[(l2+l4)/2]α2jsl2l4β2jl2+l42ss!(l2+l42s)!,
(14)
where j = x or y in the all locations including the subscript. [(l2 + l4)/2] gives the greatest integer less than or equal to (l2 + l4)/2, and the auxiliary parameters are defined as follows:

α1j=12+w0j2ρ02ikw0j22z,α2j=12+w0j2ρ02+ikw0j22zw0j4α1jρ04,
(15)
β1j=ikw0jan1jz2w0jan1jρ02ikw0jjz,β2j=an2jw0j+βw1j0j2α1jρ02+2w0jan1jρ02+ikw0jjz,
(16)
γj=β1j(α1j2α1j)1/2,δj=w0j2(α1j2α1j)1/2ρ02.
(17)

The effective beam size of the radial PL Lorentz beam array in the x- and y-directions of the observation plane is defined as [38

38. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 (1980). [CrossRef] [PubMed]

]:

Wjz=2j2<I(r,z)>dxdy<I(r,z)>dxdy.
(18)

Substituting Eq. (13) into Eq. (18), the analytical effective beam sizes of the radial PL Lorentz beam array yield
Wxz=2n1=1Nn2=1NAn1,n2(x,z;2)An1,n2(y,z;0)exp[i(φn1φn2)]n1=1Nn2=1NAn1,n2(x,z;0)An1,n2(y,z;0)exp[i(φn1φn2)],
(19)
Wyz=2n1=1Nn2=1NAn1,n2(x,z;0)An1,n2(y,z;2)exp[i(φn1φn2)]n1=1Nn2=1NAn1,n2(x,z;0)An1,n2(y,z;0)exp[i(φn1φn2)],
(20)
with An1,n2(j,z;0) and An1,n2(j,z;2)being given by
An1,n2(j,z;v)=m1=0Mm1=0Mexp(ξj24α1j+ηj24α2jan2j22w0j2an1j2ρ02+ikan1j22z)σ2m1σ2m1(11α1j)m1×l1=0m1(1)l1(2m1)!l1!(2m12l1)!l2=02m12l1(2m12l1l2)t1=02m12l1l2(2m12l1l2t1)pj2m12l1l2t1×qjt1δjl2l3=0m1(1)l3(2m1)!l3!(2m12l3)!l4=02m12l3(2m12l3l4)(2an2jw0j)2m12l3l4(l2+l4)!×s=0[(l2+l4)/2]α2jsl2l4s!(l2+l42s)!t2=0l2+l42s(l2+l42st2)ηjl2+l42st2μjt22t1t21-v×exp(τ2j24τ1j)τ1j1(t1+t2+v)/22t1+t2+v{τ1j[1+(1)t1+t2]Γ(t1+t2+1+v2)×F11(t1+t2+v2;12;τ2j24τ1j)τ2j[(1)t1+t21]Γ(t1+t2+2+v2)×F11(t1+t2+v12;32;τ2j24τ1j)},v=0or2,
(21)
where 1F1(⋅)is a Kummer function and Γ(⋅) is a Gamma function. The auxiliary parameters are defined as

ξj=ikw0jan1jz2w0jan1jρ02,ηj=an2jw0j+ξjw0j2α1jρ02+2w0jan1jρ02,pj=ξj(α1j2α1j)1/2,
(22)
qj=ikw0jz(α1j2α1j)1/2,μj=ikw0jz(1w0j2α1jρ02),
(23)
τ1j=k2w0j24z2α1j+k2w0j24z2α2j(1w0j2α1jρ02)2,τ2j=ikw0jηj2zα2j(1w0j2α1jρ02)ikw0jξj2zα1jikan1jz.
(24)

Though the analytical expressions of the average intensity and the effective beam size of the radial PL Lorentz beam array derived here seem very complicated, the term number in the expressions of the average intensity and the effective beam size will not be too large. Moreover, the Kummer function and the Gamma function can be directly used in software Mathmatica. Compared with the direct integrals that the Hermite-Gaussian expansion is not adopted, therefore, the calculations of the average intensity and the effective beam size are convenient and fast by using the analytical formulae derived here.

3. Numerical calculations and analyses

Now, the average intensity and the spreading of a radial PL Lorentz beam array in turbulent atmosphere are calculated by using the formulae derived above. In the following numerical calculations, the beam propagation we consider is with horizontal path. Figures 2
Fig. 2 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 350zr.
-6
Fig. 6 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 8, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 350zr.
represent the contour graph of normalized intensity distribution of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. The optical wavelength is set to be λ = 0.8μm. w0x = w0y results in the symmetry of the beam spot, which is the familiar case. Therefore, we consider w0x being equal to w0y. Cn2 = 10−14m-2/3, w0x = w0y = 2mm, R = 3mm, and N = 6 in Fig. 2. In the near-field such as z=zr=kw0x2, there is a central dark region in the contour graph of normalized intensity distribution. With the increase of the propagation distance z, the central dark region disappears, and the on-axis intensity increases from zero to the maximum value. Moreover, the normalized intensity distribution in the central region tends to be uniform. In Fig. 3
Fig. 3 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 3mm, and Cn2 = 10−15m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 1500zr.
, Cn2 = 10−15m-2/3 and the rest of parameters are same as those in Fig. 2. When the structure constant of the atmospheric turbulence decreases, the propagation distance where the central dark region disappears apparently increases, which can be interpreted as follow. The coherence of the radial PL Lorentz beam array will be damaged by the atmospheric turbulence. If the structure constant of the atmospheric turbulence decreases, the extent of damage of the coherence of the radial PL Lorentz beam array also decreases. In the source plane, the beam spot of the radial PL Lorentz beam array takes on a central dark distribution. The better the coherence is, the longer the axial propagation distance within which the beam spot doesn’t distort is. With decreasing the structure constant of the atmospheric turbulence, therefore, the propagation distance where the central dark region disappears increases. w0x = w0y = 1mm in Fig. 4
Fig. 4 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 1mm, R = 3mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 1500zr.
, and the other parameters are same as those in Fig. 2. In the near-field such as z = zr, we can distinctly distinguish six petals from the whole pattern. However, the six petals are conjoint. When z = 5zr, the outer distribution in the contour graph of normalized intensity takes on a gear shape. When z = 100zr, the on-axis intensity is still zero and the two isolated side lobes are located at the y-axis. When z = 1500zr, the on-axis intensity is no longer zero and two side lobes are connected to the central dominating spot. R = 5mm in Fig. 5
Fig. 5 Contour graphs of a radial PL Lorentz beam array at several different propagation distances in turbulent atmosphere. N = 6, w0x = w0y = 2mm, R = 5mm, and Cn2 = 10−14m-2/3. (a) z = zr. (b) z = 5zr. (c) z = 100zr. (d) z = 200zr.
, and the rest of parameters are same as those in Fig. 2. Comparing Fig. 4 with Fig. 5, the effect of the decrease of w0x on the beam pattern is equivalent to that of the increase of R on the beam pattern, which leads to the similar beam pattern in the different observation planes. Moreover, When z = 100zr, the on-axis intensity is nonzero. When z = 200zr, the on-axis intensity is nearly the maximum. N = 8 in Fig. 6, and the other parameters are same as those in Fig. 2. Comparing Fig. 6 with Fig. 2, we nearly cannot distinguish one from the other. When the number of the beamlet is large enough, the effect of the further increase of the beamlet is saturated. To sum up, the fill-factor in the near-field pattern is low in the case of w0x being small or R being large. If one wants to get a dark hollow uniform beam in the near-field, the parameters should be appropriate, e.g., w0x is close to R and N is large enough. As optical communications and remote sensing is involved in the far distance and a radial PL Lorentz beam array propagating in turbulent atmosphere finally turns out to be a solid beam, a radial PL Lorentz beam array is applicable to optical communications and remote sensing.

To further reveal the spreading properties of a radial PL Lorentz beam array in turbulent atmosphere, the effective beam sizes of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere are depicted in Figs. 7
Fig. 7 The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. N = 6 and R = 3mm (a) Cn2 = 10−14m-2/3. w0x and w0y take different values. (b) w0x = w0y = 2mm, and Cn2 takes different value.
and 8
Fig. 8 The effective beam size in the x-direction of a radial PL Lorentz beam array versus the propagation distance z in turbulent atmosphere. w0x = w0y = 2mm and Cn2 = 10−14m-2/3. (a) N = 6, and R takes different values. (b) R = 3mm, and N takes different value.
. As the effective beam sizes in the x- and y-directions have the similar variational rule, only Wxz is shown in Figs. 7 and 8. The radial Lorentz beam array with the larger w0x and w0y spreads more rapidly, which seems to contradict the spreading of a single laser source. The combination of the different initial phase and the dark hollow distribution in the source plane maybe result in the difference. When reducing to R = 0 and N = 1, the result obtained here is consistent with the behavior of a single laser source. The radial Lorentz beam array spreads more rapidly in turbulent atmosphere for a larger structure constant of the atmospheric turbulence. The structure constant of the refractive index fluctuations of the turbulence increasing denotes that the turbulence strengthens, which results in the large effective beam size. The radial PL Lorentz beam array with the larger radial radius spreads more rapidly. As to the effective beam size, the effect of the increase of w0x is similar to that of the increase of R. When the radius of the beam array is not far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet under the condition of the initial number of beamlet being large. When the radius of the beam array is far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is first sensitive to the increase of the number of beamlet under the condition of the initial number of beamlet being not large enough. If the initial number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. The variational law revealed in Figs. 7 and 8 is consistent with that denoted by Figs. 2-6.

4. Conclusions

Based on the extended Huygens-Fresnel integral and some mathematical techniques, the analytical average intensity and the analytical effective beam size of a radial PL Lorentz beam array are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a radial PL Lorentz beam array are numerically examined in turbulent atmosphere. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a radial PL Lorentz beam array in turbulent atmosphere are analyzed. Upon the propagation in turbulent atmosphere, the radial PL Lorentz beam array first takes on a dark hollow distribution and finally evolves to be a solid brilliant spot. Anyone of w0x, w0y, R, N, and Cn2 will affect the propagation distance within which the radial PL Lorentz beam array propagating in turbulent atmosphere keeps a dark hollow distribution. The radial PL Lorentz beam array in turbulent atmosphere with the larger w0x and w0y or with the larger radial radius or for a larger structure constant of the atmospheric turbulence spreads more rapidly. When the number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. This research is useful to the practical application of a coherent diode laser array in turbulent atmosphere, which is used for optical communications, remote sensing, and optical imaging. When the parameters are not appropriate, the near-field beam pattern of the radial PL Loretnz beam array is not ideal due to the low fill-factor. If one wants to achieve a higher fill-factor, the P × Q rectangular PL Lorentz beam array maybe alternative, which can be handled in the similar analytical approach as here.

Acknowledgments

This research was supported by National Natural Science Foundation of China under Grant No. 10974179 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y1090073. The author is indebted to the reviewers for valuable comments.

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J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE 7749, 77491Z, 77491Z-6 (2010). [CrossRef]

26.

P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010). [CrossRef]

27.

P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE 7822, 78220J, 78220J-6 (2010). [CrossRef]

28.

H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011). [CrossRef] [PubMed]

29.

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]

30.

L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009). [CrossRef]

31.

X. Li and X. Ji, “Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence,” Appl. Opt. 48(22), 4338–4347 (2009). [CrossRef] [PubMed]

32.

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]

33.

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976). [CrossRef]

34.

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

35.

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22(8), 1527–1535 (2005). [CrossRef] [PubMed]

36.

K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008). [CrossRef] [PubMed]

37.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, 1980).

38.

W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19(7), 1027–1029 (1980). [CrossRef] [PubMed]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(140.2010) Lasers and laser optics : Diode laser arrays
(140.3290) Lasers and laser optics : Laser arrays

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: August 16, 2011
Revised Manuscript: October 17, 2011
Manuscript Accepted: November 10, 2011
Published: November 17, 2011

Citation
Guoquan Zhou, "Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere," Opt. Express 19, 24699-24711 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24699


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  27. P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE7822, 78220J, 78220J-6 (2010). [CrossRef]
  28. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A28(6), 1016–1021 (2011). [CrossRef] [PubMed]
  29. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express16(22), 18437–18442 (2008). [CrossRef] [PubMed]
  30. L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt.11(6), 065703 (2009). [CrossRef]
  31. X. Li and X. Ji, “Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence,” Appl. Opt.48(22), 4338–4347 (2009). [CrossRef] [PubMed]
  32. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express18(7), 6922–6928 (2010). [CrossRef] [PubMed]
  33. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B9(13), 2331–2339 (1976). [CrossRef]
  34. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am.69(9), 1297–1304 (1979). [CrossRef]
  35. H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A22(8), 1527–1535 (2005). [CrossRef] [PubMed]
  36. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express16(26), 21315–21320 (2008). [CrossRef] [PubMed]
  37. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, 1980).
  38. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt.19(7), 1027–1029 (1980). [CrossRef] [PubMed]

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