Field correlations of laser arrays in atmospheric turbulence Yahya Baykal Çankaya University, Department of Electronic and Communication Engineering, Eskişehir Yolu 29.km, 06810, Yenimahalle, Ankara, Turkey ([email protected]) Received 8 November 2013; revised 7 January 2014; accepted 18 January 2014; posted 22 January 2014 (Doc. ID 201046); published 24 February 2014

Correlations of the fields at the receiver plane are evaluated after a symmetrical radial laser array beam incident field propagates in a turbulent atmosphere. The laser array configuration is composed of a number of the same size laser beamlets symmetrically located around a ring having a radius that determines the distance of the ring from the origin. The variations of the correlations of the received field originating from such laser array incidence versus the diagonal length starting from a receiver point are examined for various laser array parameters, turbulence parameters, and the locations of the reception points. Laser array parameters consist of the ring radius and the number and size of the beamlets. Structure constant, link length, and wavelength are the turbulence parameters whose effects on the field correlation of the laser arrays are also investigated. © 2014 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (010.3310) Laser beam transmission. http://dx.doi.org/10.1364/AO.53.001284

1. Introduction

The correlations of optical entities play an important role in understanding the behavior of physical quantities such as intensity, scintillations, coherence, and noise in an optical system. Most of the time, these entities are directly evaluated from the special cases of correlation formulations. For example, the average received intensity profile in a random medium, which is a special case of field correlations, is usually evaluated by employing a direct formulation without going into the detailed correlation analysis. However, in some cases, the detailed analysis of correlations is needed for a better understanding in the design of an optical system. Optical heterodyne telecommunication system design is one such fields where a good analysis of the correlations of the received field and the received intensity is crucial. Field correlations have been studied in a weakly turbulent medium for flat-topped Gaussian, annular [1], and sinusoidal 1559-128X/14/071284-06$15.00/0 © 2014 Optical Society of America 1284

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Gaussian [2] and in a extremely strong turbulent medium for annular beams [3]. The evaluations of the intensity correlation in weak turbulence are reported for the general type beams [4]. The correlations in turbulence of the log amplitude function [5], received power-signal fluctuations [6], partially coherent beams [7–9], multi-input multi-output (MIMO) optical wireless links [10], scintillations [11–14], and multiple beams [15] are investigated. We have also reported the formulation of the correlations for general-type beams [16]. On the other hand, recently the performances of the laser arrays in atmospheric turbulence have been scrutinized in detail. In this respect, various characteristics in turbulence of radial [17–24], flat-topped [25–27], rectangular [17,28,29], Lorentz [20], Airy [30], and partially coherent [26,28,31] beam arrays have been analyzed. Among the numerous physical entities examined for array beams propagating in atmospheric turbulence are the beam propagation [20,32], propagation factor [18,26], spectrum [19,29], scintillation [21,27,28,30,31], radius of curvature [23], reliability [25], beam quality factor [22], average

spreading [33], Rayleigh range [34], and directionality [35] parameters. In this paper, we have formulated the received field correlations of radial laser beam arrays after the array beam horizontally propagates in a turbulent atmosphere. Evaluations of the field correlations are needed in determining the performance parameters such as the average intensity, scintillation index, and bit error rate (BER) of heterodyne and MIMO optical wireless telecommunication systems. The effects of the number of beamlets forming the laser array, radius of the ring on which beamlets are located, size of the beamlets, structure constant, link length, wavelength, and locations of the reception points on the field correlations are investigated versus the diagonal distance at the receiver plane. We intend to apply the results obtained in this paper in our heterodyne detection study in which the laser arrays will be employed. 2. Formulation

The field originating from the nth single beamlet of the radial laser array beam at the source plane (i.e., single incident beam) is given by [17]:   rx − r0 2 r2y inc (1) − 2 : un rx ; ry   exp − 2α2s 2αs Here rx ; ry  denote the x and y coordinates of the radial direction at the source plane; r0 is the radius of the ring on which all the beamlets composing the array beam are symmetrically located. The origin of this ring coincides with the origin of the source plane. αs is the size of the beamlet where all the beamlets forming the array are assumed to have the same size, making the following coordinate transformation: rx → sx cos θn  sy sin θn ; ry → sy cos θn − sx sin θn ;

(2)

where θn  2πn − 1∕N, n  1; 2; …N is the angle at which the beamlets are symmetrically distributed on the ring of radius r0 and sx ; sy  is the source rectangular coordinate. Here, N represents the number of beamlets composing the laser array. Substituting Eq. (2) in Eq. (1), 

sx cos θn  sy sin θn − r0 2 uinc n sx ; sy   exp − 2α2s  sy cos θn − sx sin θn 2 − 2α2s  1  exp − 2 s2x  s2y − 2r0 sx cos θn 2αs  2 −2r0 sy sin θn  r0  :

uinc N sx ; sy  

N X n1

uinc n sx ; sy :

(4)

In our formulation, the focal length for all the laser beams in the array is taken to be infinity, which means that the beams forming the array are all perfectly collimated. As the result, the correlations found reflect the diffraction effects in turbulence of the collimated beams of finite sizes where, in application, the relevant values of source sizes of the array beams are several centimeters. Using the extended Huygens Fresnel principle [36], the received field after the field of the laser array beam propagates in the turbulent atmosphere is found to be Z Z k expikL ∞ ∞ dsx dsy uinc upx ; py   N sx ; sy  2πiL −∞ −∞   ik sx − px 2  sy − py 2  × exp 2L × expψsx ; sy ; px ; py ;

(5)

where i  −10.5 , k  2π∕λ, λ is the wavelength, px ; py  is the transverse receiver coordinate, L is the horizontal link length, and the ψsx ; sy ; px ; py  term represents the turbulence, which is the Rytov solution of the random part of the complex phase of the spherical wave. We define the field correlations at two receiver points as hupx1 ; py1 u px2 ; py2 i, where  is the complex conjugate, h·i is the ensemble average, and px1 ; py1  and px2 ; py2  are the coordinates of the first and the second receiver point. Using Eqs. (4) and (5), hupx1 ; py1 u px2 ; py2 i  2 r Z ∞Z ∞ exp − α02 Z ∞ Z ∞ s dsx1 dsy1 dsx2 dsy2  λL2 −∞ −∞ −∞ −∞  N X 1 × exp − 2 s2x1  s2y1 − 2r0 sx1 2αs n1  × cos θn − 2r0 sy1 sin θn   1 × exp − 2 s2x2  s2y2 − 2r0 sx2 2α s m1  × cos θm − 2r0 sy2 sin θm   ik s − p 2  sy1 − py1 2 × exp 2L x1 x1  2 2 −sx2 − px2  − sy2 − py2   N X

(3)

The total field of the radial laser array at the source plane is obtained by superposing the fields arising from each beamlet as

× hexpψsx1 ;sy1 ; px1 ;py1   ψ  sx2 ;sy2 ; px2 ;py2 i;

(6)

where the last ensemble average term representing the effect of turbulence is given by [37] 1 March 2014 / Vol. 53, No. 7 / APPLIED OPTICS

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2 2 hexpψ  ψ  i  expf−ρ−2 0 s1x − s2x   s1y − s2y 

3. Results

 s1x − s2x p1x − p2x   s1y − s2y p1y − p2y   p1x − p2x 2  p1y − p2y 2 g;

(7)

where ρ0  0.546C2n k2 L−3∕5 is the coherence length of turbulence, C2n being the structure constant. For simplicity, we have taken p2x ; p2y   p1x  rx ; p1y  ry  and defined the diagonal length as r2x  r2y 0.5. In Eq. (7), the quadratic form of the exponent is utilized to replace the exact 5∕3 power, which is known to be a good approximation in the evaluations of the secondorder field moments to cover both weak and strong turbulence regimes. In our figures, however, we have chosen the range of C2n such that in combination with the chosen horizontal link length and the wavelength, all the results presented in the figures fall in the weak turbulence condition under the definition that the spherical wave scintillation index is less than unity. Taking the integrations in Eq. (6), the field correlation function of a symmetrical radial laser array at the receiver plane in a turbulent atmosphere is found to be hupx1 ; py1 u p1x  rx ; p1y  ry i  2  2   2 rx  r2y r π  exp − 02 exp − λL αs ρ20   ik 2 2 × exp − 2p1x rx  rx  2p1y ry  ry  2L    N X N X 1 0.25 0.125 2 2 exp  B1x  B1y  × A A A1 A2 A21 ρ20 n1 m1 1 2   0.5 0.25 2 2 × exp B1x B2x  B1y B2y   B2x  B2y  ; A2 A2 A1 ρ20

In all the figures, except Figs. 1(a) and 7(b), presented in this section, the absolute values of the received field correlations of the laser array beams in turbulence versus the diagonal length from the receiver point px1 ; py1  are presented by evaluating Eq. (8). In each figure, one of the parameters such as the ring radius, number and the size of the beamlets, structure constant, link length, wavelength, and locations of the reception points is varied, and the effects of these parameters on the variation of the absolute value of the received field correlations are observed. One common observation which is valid for all the figures is that as the diagonal length increases, the absolute field correlations in general decrease and eventually vanish, as expected. Again, being valid for all the figures, the explanations individually provided below for each figure are true for all the diagonal lengths. In Fig. 1(a), a schematic diagram of the radial laser array is given. Figures 1(b), 2, and 3 reflect the effects of the laser array parameters on the absolute field correlations. In Fig. 1(b), it is seen that the laser array with larger ring radius will exhibit smaller absolute field correlations. Within the given parameters of αs  3 cm, N  4, and L  3 km in Fig. 1(b), for the curve of r0  10 cm, the beam widths at the receiver plane of the individual array beams are small compared to r0 . This means that even in the presence of turbulence, the individual beams are separated at the receiver plane, which in turn (a)

(8) with the parameters defined as ik 1 1   ; 2L 2α2s ρ20

A2 

ik 1 1 1  2 2− ; 2L 2αs ρ0 A1 ρ40

B1x 

r0 cos θn ikp1x rx  2; − L α2s ρ0

B1y 

r0 sin θn ikp1y ry  2; − L α2s ρ0

B2x B2y

r cos θ ik r  0 2 m  p1x  rx  − x2 ; L αs ρ0 ry r sin θ ik  0 2 m  p1y  ry  − 2 : L αs ρ0

Equation (8) is the main result of this paper from which we have evaluated the laser array field correlations as represented in Section 3. 1286

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(b)

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

A1  −

0.8

Cn2 = 5 x 10 -15 m-2/3, L = 3 km, λ = 1.55 µ m, αs = 3 cm Array Beam, N = 4, p1x = p1y = 0

0.6

r0 = 5 cm r0 = 6 cm r0 = 7 cm

0.4

r0 = 8 cm r0 = 9 cm

0.2

r0 = 10 cm 0 0

1

2

3

4

5

6

7

8

Diagonal length (cm) from the point (p1x , p1y )

Fig. 1. (a) Schematic diagram of radial laser array. (b) Field correlations of array beams in turbulence for various r0 .

1.4

Cn2 = 5 x 10 -15 m-2/3, L = 3 km, λ = 1.55 µ m, αs = 3 cm 2

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

2.5

Array Beam, r0 = 7 cm, p1x = p1y = 0 N=1 N=3 N=5 N=7 N=9 N = 11

1.5 1 0.5

L = 3 km, αs = 4 cm, λ = 1.55 µ m, N = 4

1.2

Array Beam, r0 = 7 cm, p1x = p1y = 0

Cn2 = 1 x 10 -17 m-2/3

1 0.8 0.6 0.4

Cn = 3 x 10

2

-16

m

Cn2 = Cn2 = Cn2 = Cn2 =

5 x 10

-16

-2/3

m-2/3

1 x 10

-15

m-2/3

2 x 10

-15

m-2/3

5 x 10 -15 m-2/3

0.2 0

0 0

1

2

3

4

5

6

7

0

8

5

10

15

20

25

Diagonal length (cm) from the point (p1x , p1y)

Diagonal length (cm) from the point (p1x , p1y)

Fig. 2. Field correlations of array beams in turbulence for various N.

Fig. 4. Field correlations of array beams in turbulence for various C2n .

results in almost zero field at the center of the receiver plane. Thus, as seen in Fig. 1(b), when r0  10 cm, the physical values of the field correlations are close to zero even at very small diagonal lengths. Figure 2 shows that the absolute field correlations become smaller as the number of beamlets forming the laser array is reduced. The physical reason behind this is that the increase in the number of beamlets in a laser array results in an increase in the total received field, which in turn increases the absolute field correlations. As observed from Fig. 3, the effect of the size of the beamlets in the laser array on the absolute field correlations is that large-sized beamlets yield larger absolute correlations. This result is physically justified when the diffraction in turbulence is considered for large-sized sources. Here we note that each curve in Fig. 3 reflects a laser array with four equal-sized beamlets. In Fig. 3, for very narrow beams, the curves first decrease, then increase and again decrease. Specifically, secondary peaks are observed at diagonal lengths of around 4 cm. As is well known, the coherent radiation originating from pinholes, namely spherical waves, forms fringe patterns at the receiver plane. This means, as the source sizes of the laser array elements become small, the fringes formed at the receiver plane become apparent, which in turn causes minimum and maximum field

correlation values to occur along the diagonal distance. We note that the fringes become more distinct in the absence of turbulence, and they are smoothed out; that is, the curves monotonically decrease for larger sized sources due to diffraction and turbulence effects. In all the field correlation figures presented in this paper, except in Fig. 3, relatively large source sizes are used for the laser array elements so the secondary peaks are only seen in Fig. 3 for very narrow beams. In the next three figures, the effects of the turbulence parameters on the absolute field correlations are separately investigated. The conclusion we derive from Figs. 4 –6 is that the increase in the structure constant and the link length and decrease in the wavelength result in lower-value absolute received field correlations of the laser arrays in turbulence. Finally, the effect of the locations of the reception points on the absolute field correlations of laser arrays is examined in Fig. 7(a), where it is observed that as the diagonal distance starts from a more distant receiver location (i.e., from a larger px1  py1 location), the absolute value of the laser array correlations decreases. Here, for simplicity, only px1  py1 cases are considered. In some comparisons it could be more convenient to observe the normalized correlation. For this purpose, in Fig. 7(b) we have repeated Fig. 7(a) by normalizing each curve of Fig. 7(a) by its 5

Cn2 = 5 x 10 -15 m-2/3, L = 3 km, λ = 1.55 µ m, N = 4

0.3

αs = 0.5 cm

Array Beam, r0 = 7 cm, p1x = p1y = 0

0.25

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

0.35

αs = 1 cm αs = 1.5 cm

0.2

αs = 2 cm

0.15

αs = 2.5 cm αs = 3 cm

0.1 0.05 0

Cn2 = 5 x 10 -15 m-2/3, αs = 6 cm, λ = 1.55 µ m, N = 4 4

Array Beam, r0 = 7 cm, p1x = p1y = 0 L = 0.5 km L = 1 km L = 2 km L = 3 km L = 4 km L = 5 km

3 2 1

0 0

1

2

3

4

5

6

7

8

Diagonal length (cm) from the point (p1x , p1y)

Fig. 3. Field correlations of array beams in turbulence for various αs .

0

5

10

15

Diagonal length (cm) from the point (p1x , p1y)

Fig. 5. Field correlations of array beams in turbulence for various L. 1 March 2014 / Vol. 53, No. 7 / APPLIED OPTICS

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4

2.5

m

-2/3

, αs = 6 cm, L = 3 km, N = 4

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

-15

Cn2 = 5 x 10

3.5

Array Beam, r0 = 7 cm, p1x = p1y = 0

3

λ = 1.55 µ m λ = 1.31 µ m λ = 0.90 µ m λ = 0.80 µ m λ = 0.63 µ m λ = 0.51 µ m

2.5 2 1.5 1 0.5

Cn2 = 5 x 10 -15 m-2/3, L = 3 km, λ = 1.55 µ m, αs = 3 cm 2

Array Beam, p1x = p1y = 0

N = 1 , r0 = 0.0 cm N = 3 , r0 = 6.0 cm

1.5

N = 5 , r0 = 5.4 cm N = 7 , r0 = 5.8 cm

1

N = 9 , r0 = 7.0 cm N = 11, r0 = 7.0 cm

0.5

0

0 0

1

2

3

4

5

6

7

Diagonal length (cm) from the point (p1x , p1y)

0

1

2

3

4

5

6

7

8

Diagonal length (cm) from the point (p1x , p1y)

Fig. 8. Equal-correlation curves of array beams in turbulence using two parameters, N and r0 .

own largest value, that is, by that curve’s value at zero diagonal length. Thus, the title of the vertical axis |. |N of Fig. 7(b) denotes jhup1 ; Lu p1  r; Lij∕jhup1 ; Lu p1 ; Lij. In Fig. 7(b), the normalized curves naturally meet at r  0, and the general behavior of the curves follows a similar trend as in Fig. 7(a). Figure 8 is presented in an attempt to find equalcorrelation curves by using two parameters, N and r0 . It is seen that for all the diagonal distances, the arrays of N  1 and r0  0 cm (which actually represents a single Gaussian beam at the origin), N  5 and r0  5.4 cm, and N  7 and r0  5.8 cm

have almost the same field correlations when compared to the arrays of N  3 and r0  6 cm, N  9 and r0  7 cm, and N  11 and r0  7 cm, respectively. The formulation in this paper provides the selffield correlations, that is, the correlations between the same field are evaluated at two different receiver points. In heterodyne and MIMO optical wireless communication systems which perform coherent detection, self correlations are required to be large within the area of reception. The cross correlations of the fields, that is, the correlations between different fields at two different receiver points, which are not the topic of this paper, are required to be small. In the design of a free space optics (FSO) link, the performance is found by measures such as the intensity fluctuations (scintillations) and the resulting BER. The field correlations form part of the scintillation index and BER formulations in heterodyne and MIMO FSO links. The results presented in this paper are intended to be used in the evaluation of BER in heterodyne and MIMO FSO links where a laser array incidence is used. To our knowledge, there exists no experimental work on the field correlations of laser arrays in atmospheric turbulence so we were not able to compare our results with the corresponding experimental outcomes.

(a)

4

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐

Fig. 6. Field correlations of array beams in turbulence for various λ.

3.5

Cn2 = 5 x 10 -15 m-2/3, αs = 6 cm, L = 3 km, N = 4 r0 = 7 cm, λ = 1.55 µm

3

Array Beam

2.5

p1x = p1y = 0 cm p1x = p1y = 1 cm p1x = p1y = 2 cm

2

p1x = p1y = 3 cm

1.5

p1x = p1y = 4 cm

1

p1x = p1y = 5 cm

0.5 0 0

1

2

3

4

5

6

7

8

Diagonal length (cm) from the point (p1x , p1y)

⏐〈 u (p1, L) u*(p1+ r, L) 〉⏐N

(b)

1 2

Cn = 5 x 10

0.8

-15

-2/3

m

, αs = 6 cm, L = 3 km, N = 4

r0 = 7 cm, λ = 1.55 µ m Array Beam

0.6

4. Conclusions

p1x = p1y = 0 cm p1x = p1y = 1 cm p1x = p1y = 2 cm p1x = p1y = 3 cm

0.4

p1x = p1y = 4 cm p1x = p1y = 5 cm

0.2 0 0

1

2

3

4

5

6

7

8

Diagonal length (cm) from the point (p1x , p1y)

Fig. 7. (a) Field correlations of array beams in turbulence for various p1x ; p1y . (b) Normalized field correlations of array beams in turbulence for various p1x ; p1y . 1288

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The correlation function of the received field is derived for a symmetrical radial laser array excitation in atmospheric turbulence. Our evaluations show that the absolute field correlations decrease as the ring radius, structure constant, link length, and diagonal distance starting from a more distant receiver location increase and as the number of beamlets, the size of the beamlet, and the wavelength decrease. Valid for any parameter variation, an increase in the diagonal length at the receiver plane results in a decrease in the absolute field correlations of the laser array in turbulence, and as the diagonal length is further increased, the absolute field correlations eventually approach zero. The

results presented in this paper will be useful in heterodyne detection analysis of wireless atmospheric optics links that employ laser arrays. Yahya Baykal gratefully acknowledges the support provided by the Çankaya University and the ICT COST Action IC1101 entitled “Optical Wireless Communications—An Emerging Technology.” References 1. Y. Baykal, “Field correlations of flat-topped Gaussian and annular beams in turbulence,” Opt. Lasers Eng. 49, 647–651 (2011). 2. Y. Baykal, “Sinusoidal Gaussian beam field correlations,” J. Opt. 14, 075707 (2012). 3. Y. Baykal, Y. Cai, and X. Ji, “Field correlations of annular beams in extremely strong turbulence,” Opt. Commun. 285, 4171–4174 (2012). 4. Y. Baykal, “Intensity correlations of general type beam in weakly turbulent atmosphere,” Opt. Laser Technol. 43, 1237–1242 (2011). 5. V. S. R. Gudimetla, R. B. Holmes, C. Smith, and G. Needham, “Analytical expressions for the log-amplitude correlation function of a plane wave through anisotropic atmospheric refractive turbulence,” J. Opt. Soc. Am. A 29, 832–841 (2012). 6. J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013). 7. X. Ji and G. Ji, “Spatial correlation properties of apertured partially coherent beams propagating through atmospheric turbulence,” Appl. Phys. B 92, 111–118 (2008). 8. X. Ji, X. Li, S. Chen, E. Zhang, and B. Lü, “Spatial correlation properties of Gaussian–Schell model beams propagating through atmospheric turbulence,” J. Mod. Opt. 55, 877–891 (2008). 9. 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, 3554–3563 (2007). 10. Z. Chen, S. Yu, T. Wang, G. Wu, H. Guo, and W. Gu, “Spatial correlation for transmitters in spatial MIMO optical wireless links with Gaussian-beam waves and aperture effects,” Opt. Commun. 287, 12–18 (2013). 11. H. Wang and X. Li, “Correlations between intensity fluctuations in apertured stochastic electromagnetic twist anisotropic Gaussian-Schell model beam propagating in turbulent atmosphere,” Optik 124, 1711–1715 (2013). 12. V. Kornilov, “Correlation of stellar scintillation in different photometric bands,” Appl. Opt. 50, 3717–3724 (2011). 13. J. A. Anguita and J. E. Cisternas, “Influence of turbulence strength on temporal correlation of scintillation,” Opt. Lett. 36, 1725–1727 (2011). 14. A. Jurado-Navas and A. Puerta-Notario, “Generation of correlated scintillations on atmospheric optical communications,” J. Opt. Commun. Netw. 1, 452–462 (2009). 15. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space,” Appl. Opt. 46, 6561–6571 (2007). 16. Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006). 17. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).

18. H. Tang and B. Ou, “Beam propagation factor of radial Gaussian–Schell model beam array in non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 1442–1447 (2011). 19. P. Pan and Y. Dan, “Changes in the spectrum of radial array beams through turbulent atmosphere,” J. Mod. Opt. 60, 177–184 (2013). 20. G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19, 24699–24711 (2011). 21. H. Tang, B. Wang, B. Luo, A. Dang, and H. Guo, “Scintillation optimization of radial Gaussian beam array propagating through Kolmogorov turbulence,” Appl. Phys. B 111, 149–154 (2013). 22. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol 42, 604–609 (2010). 23. 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, 6922–6928 (2010). 24. J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12, 065401 (2010). 25. S. Golmohammady, M. Yousefi, F. D. Kashani, and B. Ghafary, “Reliability analysis of the flat-topped array beam FSO communication link,” J. Mod. Opt. 60, 696–703 (2013). 26. Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012). 27. Y. Yuan and Y. Cai, “Scintillation index of a flat-topped beam array in a weakly turbulent atmosphere,” J. Opt. 13, 125701 (2011). 28. P. Pan, B. Zhang, N. Qiao, and Y. Dan, “Characteristics of scintillations and bit error rate of partially coherent rectangular array beams in turbulence,” Opt. Commun. 284, 1019–1025 (2011). 29. P. Pan, “Spectrum changes of rectangular array beams through turbulent atmosphere,” Opt. Commun. 293, 95–101 (2013). 30. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35, 3456–3458 (2010). 31. Ç. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboğlu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011). 32. P. Zhou, X. Wang, Y. Ma, H. Ma, H. Xu, and Z. Liu, “Propagation of Gaussian beam array through an optical system in turbulent atmosphere,” Appl. Phys. B 103, 1009–1012 (2011). 33. H. Tang and B. Ou, “Average spreading of a linear Gaussian– Schell model beam array in non-Kolmogorov turbulence,” Appl. Phys. B 104, 1007–1012 (2011). 34. X. Ji and Z. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010). 35. X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26, 236–243 (2009). 36. Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967). 37. S. J. Wang, Y. Baykal, and M. A. Plonus, “Receiver aperture averaging effects for the intensity fluctuation of a beam wave in the turbulent atmosphere,” J. Opt. Soc. Am. 73, 831–837 (1983).

1 March 2014 / Vol. 53, No. 7 / APPLIED OPTICS

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Field correlations of laser arrays in atmospheric turbulence.

Correlations of the fields at the receiver plane are evaluated after a symmetrical radial laser array beam incident field propagates in a turbulent at...
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