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Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere
G Model IJLEO-53359; No. of Pages 5
ARTICLE IN PRESS Optik xxx (2013) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere Huiyun Wu a,∗ , Xin Li a , Shen Sheng a , Zhisong Huang a , Shuhai Huang a , Siqing Zhao a , Hua Wang a , Zhenhai Sun a,∗∗ , Xiegu Xu a , Rui Xiao b,∗∗ a b
Department of Science and Technology, Academy of Military Medical Sciences, Beijing 100071, PR China Beijing Institute of Radiation Medicine, Academy of Military Medical Sciences, Beijing 100850, PR China
a r t i c l e
i n f o
Article history: Received 8 December 2012 Accepted 1 May 2013 Available online xxx Keywords: Optical system Coherent laser array Phase optimization Turbulent atmosphere
a b s t r a c t Model of an optical system with coherent laser array source and the piston phase optimized by the stochastic parallel gradient descent algorithm is established. With this model, theory of beam propagation through the optical system in turbulent atmosphere is analyzed, and the analytical formulas of the beam average intensity along the propagation path are derived. Strehl ratio of the received beam induced by intensity disorderly distribution and power efficiency of the received beam are introduced to evaluate performance of the optical system. Under the H-V 5/7 atmospheric turbulent model, performance of an optical system with determinate parameters was calculated, and the influences of the propagation distance and the laser wavelength were numerically analyzed, respectively. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Considerable interest has been exhibited in coherent laser array due to their importance in basic science and some attractive applications [1–5]. The propagation properties of coherent laser array in turbulent atmosphere have been widely investigated, Chu et al. have studied the propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path and analyzed the average intensity along the propagation path [6], Wang has studied performance of the active phase locking of coherently combined beam array by using the optimization algorithm based adaptive optics and suggested that the optimization algorithm based adaptive optics can be an effective way in phase locking of coherently combined beam array [7], Zhou et al. have numerically studied the propagation performance of adaptive phase-locked fiber laser array in turbulent atmosphere and suggested that the adaptive phaselocked fiber laser array can improve the power centrality in far field [8]. In recent years, the optical systems in which beam propagates long-distance through the turbulent atmosphere and is then received as a new source for further propagation, such as the laser relay mirror system and the free-space optical communication system, have attracted much attention [9–14]. And the potential
∗ Corresponding author. Tel.: +86 13366069770. ∗∗ Corresponding authors. E-mail addresses: [email protected] (H. Wu), [email protected] (Z. Sun), [email protected] (R. Xiao).
application of phase-locked coherent laser array in the aforementioned optical systems has become an attractive area of scientific research. As a new area of scientific research, study on performance of the optical system with phase-locked coherent laser array source is a key work. In the present paper, we theoretically analyze the optical system with coherent laser array source and the piston phases optimized by the stochastic parallel gradient descent algorithm, and calculate the optical system performance under the H-V 5/7 turbulent model. 2. Models and theoretical analysis 2.1. Models Schematic diagram of beam propagation in an optical system with coherent laser array source and phase optimization is shown in Fig. 1. The optical system is composed of the laser array source, the transmitter, the receiver, two adaptive optics installations (AO1 at the source and AO2 at the receiver) [15,16]. AO1 optimizes the piston phase of each beam before entering into the transmitter, which aims to improve performance of the received beam. The laser array propagates from the transmitter to the receiver through the turbulence on the way up and is then received by the Cassegrain receiver. The received beam is cleaned up by AO2 and used as a new source for further propagation. Shown in Refs. [16,17], when using a high-precision adaptive optics system, phase of the received beam can be effectively corrected. Thus, in the present paper, we assume that phase of the received beam can be ideally compensated by AO2.
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Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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Fig. 1. Schematic diagram of beam propagation in an optical system with coherent laser array source.
The optical field at the source plane U0 (x, y, 0) can be expressed as U0 (x, y, 0) =
M
Um (x, y, 0) × tm (x, y, 0)
2
Um (x, y, 0) = exp −
tm (x, y, 0) =
(x − am ) + (y − bm ) ω02
1 0
= exp
(1)
m 2
× exp(
(2)
m)
x 2 + y2 ≤
(3)
−
k
−D (x − w, y − v) w 2
[(x − w)2 + (y − v)2 ]
(6)
constant of the refractive index. By inserting Eqs. (1)–(4) and Eq. (6) into Eq. (5), we can express the average intensity at the receiver as I(p, q, z) =
mnjl (p, q, z) =
1 r02
(w, v, p, q)] = exp
where Dw (x − w, y − v) is the wave structure function, r0 = −3/5 is the coherence length of spherical wave propa(0.545C¯ n2 k2 L) gating in the turbulent medium with C¯ n2 being averaged structure
else
where Um (x, y, 0) denotes the mth Gaussian beam with beam width ω0 and central coordinate (am , bm , 0), m denotes the piston phase
∗
exp[ (x, y, p, q) +
M N N M
(7)
mjnl (p, q, z)
m=1 j=1 n=1 l=1
where
2
(2z)
Bj Bl
2
∞
∞
∞
∞
× −∞
× exp
−∞
−∞
2
2
−
exp
ω02
−∞
ik 2 2 2 2 [(p − x) + (q − y) − (p − w) − (q − v) ] 2z
exp
−
1 r02
2
2
2
2
[(x − am ) + (y − bm ) + (w − an ) + (v − bn )]
−
2
2
Cj [(x − am ) + (y − bm ) ] + Cl [(w − an ) + (v − bn ) ]
2
(8)
2
[(x − w) + (y − v) ]
exp[i(
m
−
n )]dxdydwd
v
After calculation, we can obtain of the mth beam, tm (x, y, 0) is the hard-aperture truncation function of the sub-transmitter, M denotes the number of beams in the array, denotes the sub-aperture radius. By using the Gaussian functions, the truncation function of the sub-transmitter can be expressed as
tm (x, y, 0) =
N
Cj
−
Bj exp
2
j=1
H(p, z) =
2
mnjl (p, q, z) =
2
[(x − am ) + (y − bm ) ]
k2 (2z) exp
2
ik 2z
∞
−∞
∞
−∞
∞
−∞
∞
U(x, y, 0)U ∗ (w, v, 0)
−∞
[(p − x)2 + (q − y)2 − (p − w)2 − (q − v)2 ]
× exp[ (x, y, p, q) +
∗
(w, v, p, q)]dxdydwdv
−
exp
m
(1 + Cj )a2m ω02
−
−
n ))
(9)
(1 + Cl )a2n ω02
ˇl
× exp
(2am ˇl Kj r02 + 2p ˇl K3 r02 − 2an Kl − 2p K3 )
2
4(ˇj + 4/ˇl r04 )ˇl2 r04 (10)
ˇj ˇl + (4/r04 )
exp
b K + p K 2 n l 3
+ (5)
where k is the wave number, the asterisk denotes the complex conjugation, and the indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. (x, y, p, q) represents the random part of the complex phase of a spherical wave that propagates from point (x, y, 0) at the source plane to the point (p, q, z) at the receiver plane and can be expressed as [18]
ˇj ˇl + (4/r04 )
H(q, z) =
H(p, z)H(q, z) exp(i(
+
(2z)2
a K + p K 2 n l 3
(4)
where N is the expansion order, Bj and Cj are the complex expansion coefficients which can be obtained by optimization computation directly. By using the extended Huygens–Fresnel principle, the average intensity distribution at the receiver I(p,q,z) can be expressed as I(p, q, z) =
k2 Bj Bl
(1 + Cj )b2m ω02
−
(1 + Cl )b2n ω02
ˇl
× exp
−
(2bm ˇl Kj r02 + 2q ˇl K3 r02 − 2bn Kl − 2q K3 )
2
4(ˇj + 4/ˇl r04 )ˇl2 r04 (11)
where Kj = 1/ω02 + Cj /ω02 , Kl = 1/ω02 + Cl /ω02 , p = p + (am − an )/K3 , q = q + (bm − bn )/K3 , K3 = ik/2z, ˇj = (1/ω02 ) + (Cj /ω02 ) − (ik/2z) + (1/r02 ), ˇl = (1/ω02 ) + (Cl /ω02 ) − (ik/2z) + (1/r02 ).
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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The received beam with phase compensated by AO2 is used as a new source in the optical system, power and beam quality are key factors. Power efficiency of the new source can be expressed as T=
PL P0
(12)
where PL = s I(p, q, z) dpdq denotes power of the received beam, S denotes the window area of the receiver, P0 denotes power of the laser array source. Although phase of the received beam can be ideally compensated by AO2, intensity distribution of the received beam has great impact on the new source power centrality in far field [19]. Due to the influences of diffraction and the turbulence, the beam intensity distribution on the way up becomes disorderly, this degrades the power centrality in far field of the new source. We introduce Strehl ratio of the new source induced by intensity disorderly distribution to show the beam quality, which is defined as the ratio of the peak intensity in far field to the peak intensity in far field with an ideal flattened source [19]. SR =
IP IP-ideal
IP = [Ut (, )Ut (, )∗ ]|→0,→0 ∗
IP-ideal = [Ut-ideal (, )Ut-ideal (, ) ]|→0,→0
(13)
After obtaining the optimal piston phase distributions of the laser array { im }, intensity distribution of the received beam can be obtained by using equations from Eq. (7) to Eq. (11), and the performance evaluation factor of the new source can be obtained by using equations from Eq. (12) to Eq. (17).
(14)
3. Numerical calculations
(15)
where IP denotes the peak optical intensity at the focal plane, IP-ideal denotes the peak optical intensity at the focal plane with an ideal flattened source, Ut (, ) denotes the optical field at the focal plane and Ut-ideal (, ) denotes the optical field at the focal plane with an ideal flattened source. After derivation, we can obtain
SR =
s
s0 ×
2
I(p, q, z)dpdq
s
I(p, q, z)dpdq
(16)
where s0 = (D2 − d2 )/4 is the area of the receiver, d and D are the inner diameter and the outer diameter, respectively. In the optical system, the piston phase of each beam at the source plane is optimized by AO1 and aimed to improve performance of the received beam. In the present paper, we take AO1 an optimization algorithm based adaptive optics controlled by the stochastic parallel gradient descent algorithm, and define the performance evaluation factor used in the phase optimization process as the product of power efficiency and beam quality of the new source, shown as = T × SR
Fig. 2. Intensity distribution of the laser array.
(17)
The whole control loop in AO1 can be described as the following steps [20] (1) Define the evaluation function of the optimization process . (2) Define cycle index of the optimization process m and samples number of phase distribution M. Set i = 0, i = 1, 2, . . ., M. (3) Generate statistically independent random perturbations ı i , i = 1, 2, . . ., M, all |ı i | are small values that are typically chosen as statistically independent variables having zero mean and equal variances, ı k =0, ı k ı l = 2 ıkl where ıkl is the Kronecker symbol. (4) Compute the difference between two evaluations of the system performance ı = + − − , where + = ( 1 + ı 1 , 2 + ı 2 , . . . M + ı M ) and − = ( 1 − ı 1 , 2 − ı 2 , . . . M − ı M ) (5) Update the value of i according to the following equation n+1 = in + ı in ı, where n denotes the steps of iteration, i
is the update gain, > 0 accords to the procedure of maximization. (6) Repeat steps (3)–(5) until algorithms converge and the optimal optical phase distribution { im }, i = 1, 2, . . ., M is obtained.
3.1. Parameters Parameters of the optical system are set as: the laser array is composed of 6 coherent Gaussian beams with 3.8 m wavelength and 0.15 m beam width, intensity distribution of the source is shown in Fig. 2, radius of the sub-transmitter = 0.15 m, the receiver is a telescope with 0.15 m-inner radius and 0.60 m-outer radius, the zenith angle of the propagation path = 0, the propagation distance is 30 km, samples number of phase distribution in AO1 M = 6, the update gain = 0.15, cycle index of the optimization process m = 1000. Distribution of the atmospheric structure constant is described as the H-V 5/7 turbulent model. Cn2 (h) = 8.2 × 10−56 V (h) h10 exp 2
+ 2.7 × 10−16 exp
V (h) = 5 + 30 exp
−h 1000
−h 1500
+ C0 exp
2 h − 9400 −
4800
−h 100
(18)
(19)
where h is the altitude from the ground, V(h) is the wind speed along the vertical path, C0 is the nominal value of at ground level (the typical value is 4.0 × 10−14 m−2/3 ). 3.2. Calculations and results analysis In order to show performance of the new source in the optical system, we perform some calculations. Fig. 3 illustrates the intensity distributions of the new source before and after phase optimization, the circular white broken lines represent the hardedge of the receiver. Fig. 4 illustrates the curve of performance evaluation factor on the steps of the phase optimization algorithm iteration, and Table 1 shows performances of the new source before and after phase optimization. Shown in Fig. 3 and Table 1, before phase optimization, power converges to the center due to the influences of diffraction and the turbulence, serious power losses are caused by receiver inner obstruction and power efficiency of the new source is 82%. After phase optimization, optical intensity in the central area is degraded and power efficiency of the new source is improved from 82% to 89%, meanwhile uniformity of the intensity is improved from 0.90
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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Table 2 Performance of the new source with different propagation distances. Propagation distance (km) 5 10 15 20 25 30 40 50
Power efficiency 98% 97% 95% 93% 91% 89% 82% 75%
Beam quality
Performance evaluation factor
0.69 0.78 0.84 0.86 0.90 0.91 0.92 0.92
0.68 0.76 0.79 0.80 0.82 0.81 0.75 0.70
Table 3 Values of the performance evaluation factor with certain laser wavelength and propagation distance. Propagation distance (km)/wavelength (m)
1.064
1.315
3.8
5 10 15 20 25 30 40 50
0.73 0.79 0.81 0.81 0.80 0.79 0.78 0.76
0.71 0.78 0.79 0.80 0.81 0.80 0.78 0.76
0.68 0.76 0.79 0.80 0.82 0.81 0.75 0.70
Fig. 3. Intensity distribution of the received beam.
to 0.91. Fig. 4 shows that only after 65 steps of iteration, the performance evaluation factor approaches to an absolute maximum value, which illustrate the astringency of phase optimization in AO1. To show the evolution of the new source performance with the increase of propagation distance, we take different values of beam propagation distance and calculate performance of the optical system. The coming results are listed in Table 2. Form Table 2, we can get that power efficiency of the new source is high and beam quality is low when propagation distance is short. With the increase of propagation distance, power
efficiency of the new source decreases, and beam quality of the new source increases. Performance evaluation factor of the new source approaches to the maximum value at a certain propagation distance, for the determinate optical system in the present paper, the propagation distance is 25 km, and the maximum performance evaluation factor is 0.82. To show the influences of beam wavelength on the new source performance, we take beam wavelength as 1.064 m, 1.315 m, 3.8 m, and calculate performances of the optical system. Values of the performance evaluation factor with certain laser wavelength and propagation distance are listed in Table 3. Results show that optical system with 1.064 m source approaches the optimal performance factor 0.81 at 15–20 km beam propagation distance; optical system with 1.315 m source approaches the optimal performance factor 0.81 at 25 km beam propagation distance; the optical system with 3.8 m source approaches the optimal performance factor 0.82 at 25 km beam propagation distance.
4. Conclusions
Fig. 4. Performance evaluation factor on the steps of the phase optimization algorithm iteration.
Table 1 Performance of the new source. Power efficiency Before phase optimization After phase optimization
Beam quality
Performance evaluation factor
82%
0.90
0.74
89%
0.91
0.81
In this paper, we have studied performance of an optical system with coherent laser array source and phase optimization in turbulent atmosphere. Model of an optical system with coherent laser array source and the piston phase distributions optimized by the stochastic parallel gradient descent algorithm was established. Theory of beam propagation through the optical system in turbulent atmosphere was analyzed, and the analytical formulas of the beam average intensity along the propagation path were derived. Power efficiency and Strehl ratio induced by intensity disorderly distribution were introduced to evaluate performance of the new source. Under the H-V 5/7 atmospheric turbulent model, performances of an optical system with determinate parameters were calculated, and the influences of the propagation distance and the laser wavelength on the system performance were numerically analyzed, respectively. Results in the present paper may provide references for designing and analyzing of optical systems with coherent laser array source and phase optimization.
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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Acknowledgement The authors are indebted to the National High Technology Research and Development Program of China (2012AA022007). References [1] X. Chu, Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere, Opt. Express 24 (2007) 17613–17618. [2] E. Ronen, A.A. Ishaaya, Phase locking a fiber laser array via diffractive coupling, Opt. Express 19 (2011) 1510–1515. [3] S.J. Augst, T.Y. Fan, A. Sanchez, Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers, Opt. Lett. 29 (2004) 474–476. [4] C.J. Corcoran, F. Durville, Experimental demonstration of a phase-locked laser array using a self-Fourier cavity, Appl. Phys. Lett. 86 (2005) 201118. [5] A. Ishaaya, N. Davidson, A. Friesem, Passive laser beam combining with intracavity interferometric combiners, IEEE J. Sel. Top. Quantum Electron. 15 (2009) 301–311. [6] X. Chu, Z. Liu, Y. Wu, Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path, J. Opt. Soc. Am. A 25 (2008) 74–79. [7] X. Wang, Study on Optimization Algorithm Based Adaptive Optics in Laser Phased Array, National University of Defense Technology, Changsha, China, 2011, pp. 5–24. [8] P. Zhou, Z. Liu, X. Xu, F. Xi, X. Chu, MaF H., Propagation performance of adaptive phase-locked fiber laser array in turbulent atmosphere, Chin. J. Lasers 36 (2009) 1442–1446.
5
[9] D.O. Caplan, Laser communication transmitter and receiver design, J. Opt. Fiber Commun. Res. 4 (2007) 225–226. [10] C. Paterson, Atmospheric turbulence and orbital angular momentum of single photons for optical communication, Phys. Rev. Lett. 94 (2005) 153901. [11] W. Holmes, NASA’s tracking and data relay satellite system, Commun. Soc. Mag. 16 (1978) 13–20. [12] M. Hartman, S. Restaino, J. Baker, D. Payne, J. Bukley, EAGLE/relay mirror technology development, Proc. SPIE 4724 (2002) 110–117. [13] G.-g. Ren, J.-j. Huang, Major progress of U.S. 2005’ high-energy laser technology, Laser Optoelectron. Prog. 43 (2006) 5–6. [14] J. Mansell, Beam shaping for relay mirrors, Proc. SPIE 6290 (2006), 62900K62900K-10. [15] E.A. Duff, D.C. Washburn, The magic of relay mirrors, Proc. SPIE 5413 (2004) 139–143. [16] S.G. Leonard, Laser Options for National Missile Defense, Air Command and Staff College Air University, Alabama, 1998, pp. 75–84. [17] H. Wu, H. Zhao, X. Xu, W. Wu, J. Chen, Y. Zhao, Beam shaping for uplink transmission of a relay mirror system, Appl. Opt. 49 (2010) 3245–3249. [18] X. Chu, G. Zhou, Power coupling of a two-Cassegrain telescope system in turbulent atmosphere in a slant path, Opt. Express 15 (2007) 7697–7707. [19] H. Wu, H. Zhao, W. Wu, J. Chen, X. Xu, C. Xiong, J. Chen, Analysis of the evaluation function used in optimization of a relay mirror system, Proc. SPIE 8192 (2011) 819209. [20] M.A. Vorontsov, V.P. Sivokon, Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction, J. Opt. Soc. Am. A 15 (1998) 2745–2758.
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
ARTICLE IN PRESS Optik xxx (2013) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere Huiyun Wu a,∗ , Xin Li a , Shen Sheng a , Zhisong Huang a , Shuhai Huang a , Siqing Zhao a , Hua Wang a , Zhenhai Sun a,∗∗ , Xiegu Xu a , Rui Xiao b,∗∗ a b
Department of Science and Technology, Academy of Military Medical Sciences, Beijing 100071, PR China Beijing Institute of Radiation Medicine, Academy of Military Medical Sciences, Beijing 100850, PR China
a r t i c l e
i n f o
Article history: Received 8 December 2012 Accepted 1 May 2013 Available online xxx Keywords: Optical system Coherent laser array Phase optimization Turbulent atmosphere
a b s t r a c t Model of an optical system with coherent laser array source and the piston phase optimized by the stochastic parallel gradient descent algorithm is established. With this model, theory of beam propagation through the optical system in turbulent atmosphere is analyzed, and the analytical formulas of the beam average intensity along the propagation path are derived. Strehl ratio of the received beam induced by intensity disorderly distribution and power efficiency of the received beam are introduced to evaluate performance of the optical system. Under the H-V 5/7 atmospheric turbulent model, performance of an optical system with determinate parameters was calculated, and the influences of the propagation distance and the laser wavelength were numerically analyzed, respectively. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Considerable interest has been exhibited in coherent laser array due to their importance in basic science and some attractive applications [1–5]. The propagation properties of coherent laser array in turbulent atmosphere have been widely investigated, Chu et al. have studied the propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path and analyzed the average intensity along the propagation path [6], Wang has studied performance of the active phase locking of coherently combined beam array by using the optimization algorithm based adaptive optics and suggested that the optimization algorithm based adaptive optics can be an effective way in phase locking of coherently combined beam array [7], Zhou et al. have numerically studied the propagation performance of adaptive phase-locked fiber laser array in turbulent atmosphere and suggested that the adaptive phaselocked fiber laser array can improve the power centrality in far field [8]. In recent years, the optical systems in which beam propagates long-distance through the turbulent atmosphere and is then received as a new source for further propagation, such as the laser relay mirror system and the free-space optical communication system, have attracted much attention [9–14]. And the potential
∗ Corresponding author. Tel.: +86 13366069770. ∗∗ Corresponding authors. E-mail addresses: [email protected] (H. Wu), [email protected] (Z. Sun), [email protected] (R. Xiao).
application of phase-locked coherent laser array in the aforementioned optical systems has become an attractive area of scientific research. As a new area of scientific research, study on performance of the optical system with phase-locked coherent laser array source is a key work. In the present paper, we theoretically analyze the optical system with coherent laser array source and the piston phases optimized by the stochastic parallel gradient descent algorithm, and calculate the optical system performance under the H-V 5/7 turbulent model. 2. Models and theoretical analysis 2.1. Models Schematic diagram of beam propagation in an optical system with coherent laser array source and phase optimization is shown in Fig. 1. The optical system is composed of the laser array source, the transmitter, the receiver, two adaptive optics installations (AO1 at the source and AO2 at the receiver) [15,16]. AO1 optimizes the piston phase of each beam before entering into the transmitter, which aims to improve performance of the received beam. The laser array propagates from the transmitter to the receiver through the turbulence on the way up and is then received by the Cassegrain receiver. The received beam is cleaned up by AO2 and used as a new source for further propagation. Shown in Refs. [16,17], when using a high-precision adaptive optics system, phase of the received beam can be effectively corrected. Thus, in the present paper, we assume that phase of the received beam can be ideally compensated by AO2.
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.018
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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ARTICLE IN PRESS
IJLEO-53359; No. of Pages 5
H. Wu et al. / Optik xxx (2013) xxx–xxx
2
Fig. 1. Schematic diagram of beam propagation in an optical system with coherent laser array source.
The optical field at the source plane U0 (x, y, 0) can be expressed as U0 (x, y, 0) =
M
Um (x, y, 0) × tm (x, y, 0)
2
Um (x, y, 0) = exp −
tm (x, y, 0) =
(x − am ) + (y − bm ) ω02
1 0
= exp
(1)
m 2
× exp(
(2)
m)
x 2 + y2 ≤
(3)
−
k
−D (x − w, y − v) w 2
[(x − w)2 + (y − v)2 ]
(6)
constant of the refractive index. By inserting Eqs. (1)–(4) and Eq. (6) into Eq. (5), we can express the average intensity at the receiver as I(p, q, z) =
mnjl (p, q, z) =
1 r02
(w, v, p, q)] = exp
where Dw (x − w, y − v) is the wave structure function, r0 = −3/5 is the coherence length of spherical wave propa(0.545C¯ n2 k2 L) gating in the turbulent medium with C¯ n2 being averaged structure
else
where Um (x, y, 0) denotes the mth Gaussian beam with beam width ω0 and central coordinate (am , bm , 0), m denotes the piston phase
∗
exp[ (x, y, p, q) +
M N N M
(7)
mjnl (p, q, z)
m=1 j=1 n=1 l=1
where
2
(2z)
Bj Bl
2
∞
∞
∞
∞
× −∞
× exp
−∞
−∞
2
2
−
exp
ω02
−∞
ik 2 2 2 2 [(p − x) + (q − y) − (p − w) − (q − v) ] 2z
exp
−
1 r02
2
2
2
2
[(x − am ) + (y − bm ) + (w − an ) + (v − bn )]
−
2
2
Cj [(x − am ) + (y − bm ) ] + Cl [(w − an ) + (v − bn ) ]
2
(8)
2
[(x − w) + (y − v) ]
exp[i(
m
−
n )]dxdydwd
v
After calculation, we can obtain of the mth beam, tm (x, y, 0) is the hard-aperture truncation function of the sub-transmitter, M denotes the number of beams in the array, denotes the sub-aperture radius. By using the Gaussian functions, the truncation function of the sub-transmitter can be expressed as
tm (x, y, 0) =
N
Cj
−
Bj exp
2
j=1
H(p, z) =
2
mnjl (p, q, z) =
2
[(x − am ) + (y − bm ) ]
k2 (2z) exp
2
ik 2z
∞
−∞
∞
−∞
∞
−∞
∞
U(x, y, 0)U ∗ (w, v, 0)
−∞
[(p − x)2 + (q − y)2 − (p − w)2 − (q − v)2 ]
× exp[ (x, y, p, q) +
∗
(w, v, p, q)]dxdydwdv
−
exp
m
(1 + Cj )a2m ω02
−
−
n ))
(9)
(1 + Cl )a2n ω02
ˇl
× exp
(2am ˇl Kj r02 + 2p ˇl K3 r02 − 2an Kl − 2p K3 )
2
4(ˇj + 4/ˇl r04 )ˇl2 r04 (10)
ˇj ˇl + (4/r04 )
exp
b K + p K 2 n l 3
+ (5)
where k is the wave number, the asterisk denotes the complex conjugation, and the indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. (x, y, p, q) represents the random part of the complex phase of a spherical wave that propagates from point (x, y, 0) at the source plane to the point (p, q, z) at the receiver plane and can be expressed as [18]
ˇj ˇl + (4/r04 )
H(q, z) =
H(p, z)H(q, z) exp(i(
+
(2z)2
a K + p K 2 n l 3
(4)
where N is the expansion order, Bj and Cj are the complex expansion coefficients which can be obtained by optimization computation directly. By using the extended Huygens–Fresnel principle, the average intensity distribution at the receiver I(p,q,z) can be expressed as I(p, q, z) =
k2 Bj Bl
(1 + Cj )b2m ω02
−
(1 + Cl )b2n ω02
ˇl
× exp
−
(2bm ˇl Kj r02 + 2q ˇl K3 r02 − 2bn Kl − 2q K3 )
2
4(ˇj + 4/ˇl r04 )ˇl2 r04 (11)
where Kj = 1/ω02 + Cj /ω02 , Kl = 1/ω02 + Cl /ω02 , p = p + (am − an )/K3 , q = q + (bm − bn )/K3 , K3 = ik/2z, ˇj = (1/ω02 ) + (Cj /ω02 ) − (ik/2z) + (1/r02 ), ˇl = (1/ω02 ) + (Cl /ω02 ) − (ik/2z) + (1/r02 ).
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The received beam with phase compensated by AO2 is used as a new source in the optical system, power and beam quality are key factors. Power efficiency of the new source can be expressed as T=
PL P0
(12)
where PL = s I(p, q, z) dpdq denotes power of the received beam, S denotes the window area of the receiver, P0 denotes power of the laser array source. Although phase of the received beam can be ideally compensated by AO2, intensity distribution of the received beam has great impact on the new source power centrality in far field [19]. Due to the influences of diffraction and the turbulence, the beam intensity distribution on the way up becomes disorderly, this degrades the power centrality in far field of the new source. We introduce Strehl ratio of the new source induced by intensity disorderly distribution to show the beam quality, which is defined as the ratio of the peak intensity in far field to the peak intensity in far field with an ideal flattened source [19]. SR =
IP IP-ideal
IP = [Ut (, )Ut (, )∗ ]|→0,→0 ∗
IP-ideal = [Ut-ideal (, )Ut-ideal (, ) ]|→0,→0
(13)
After obtaining the optimal piston phase distributions of the laser array { im }, intensity distribution of the received beam can be obtained by using equations from Eq. (7) to Eq. (11), and the performance evaluation factor of the new source can be obtained by using equations from Eq. (12) to Eq. (17).
(14)
3. Numerical calculations
(15)
where IP denotes the peak optical intensity at the focal plane, IP-ideal denotes the peak optical intensity at the focal plane with an ideal flattened source, Ut (, ) denotes the optical field at the focal plane and Ut-ideal (, ) denotes the optical field at the focal plane with an ideal flattened source. After derivation, we can obtain
SR =
s
s0 ×
2
I(p, q, z)dpdq
s
I(p, q, z)dpdq
(16)
where s0 = (D2 − d2 )/4 is the area of the receiver, d and D are the inner diameter and the outer diameter, respectively. In the optical system, the piston phase of each beam at the source plane is optimized by AO1 and aimed to improve performance of the received beam. In the present paper, we take AO1 an optimization algorithm based adaptive optics controlled by the stochastic parallel gradient descent algorithm, and define the performance evaluation factor used in the phase optimization process as the product of power efficiency and beam quality of the new source, shown as = T × SR
Fig. 2. Intensity distribution of the laser array.
(17)
The whole control loop in AO1 can be described as the following steps [20] (1) Define the evaluation function of the optimization process . (2) Define cycle index of the optimization process m and samples number of phase distribution M. Set i = 0, i = 1, 2, . . ., M. (3) Generate statistically independent random perturbations ı i , i = 1, 2, . . ., M, all |ı i | are small values that are typically chosen as statistically independent variables having zero mean and equal variances, ı k =0, ı k ı l = 2 ıkl where ıkl is the Kronecker symbol. (4) Compute the difference between two evaluations of the system performance ı = + − − , where + = ( 1 + ı 1 , 2 + ı 2 , . . . M + ı M ) and − = ( 1 − ı 1 , 2 − ı 2 , . . . M − ı M ) (5) Update the value of i according to the following equation n+1 = in + ı in ı, where n denotes the steps of iteration, i
is the update gain, > 0 accords to the procedure of maximization. (6) Repeat steps (3)–(5) until algorithms converge and the optimal optical phase distribution { im }, i = 1, 2, . . ., M is obtained.
3.1. Parameters Parameters of the optical system are set as: the laser array is composed of 6 coherent Gaussian beams with 3.8 m wavelength and 0.15 m beam width, intensity distribution of the source is shown in Fig. 2, radius of the sub-transmitter = 0.15 m, the receiver is a telescope with 0.15 m-inner radius and 0.60 m-outer radius, the zenith angle of the propagation path = 0, the propagation distance is 30 km, samples number of phase distribution in AO1 M = 6, the update gain = 0.15, cycle index of the optimization process m = 1000. Distribution of the atmospheric structure constant is described as the H-V 5/7 turbulent model. Cn2 (h) = 8.2 × 10−56 V (h) h10 exp 2
+ 2.7 × 10−16 exp
V (h) = 5 + 30 exp
−h 1000
−h 1500
+ C0 exp
2 h − 9400 −
4800
−h 100
(18)
(19)
where h is the altitude from the ground, V(h) is the wind speed along the vertical path, C0 is the nominal value of at ground level (the typical value is 4.0 × 10−14 m−2/3 ). 3.2. Calculations and results analysis In order to show performance of the new source in the optical system, we perform some calculations. Fig. 3 illustrates the intensity distributions of the new source before and after phase optimization, the circular white broken lines represent the hardedge of the receiver. Fig. 4 illustrates the curve of performance evaluation factor on the steps of the phase optimization algorithm iteration, and Table 1 shows performances of the new source before and after phase optimization. Shown in Fig. 3 and Table 1, before phase optimization, power converges to the center due to the influences of diffraction and the turbulence, serious power losses are caused by receiver inner obstruction and power efficiency of the new source is 82%. After phase optimization, optical intensity in the central area is degraded and power efficiency of the new source is improved from 82% to 89%, meanwhile uniformity of the intensity is improved from 0.90
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Table 2 Performance of the new source with different propagation distances. Propagation distance (km) 5 10 15 20 25 30 40 50
Power efficiency 98% 97% 95% 93% 91% 89% 82% 75%
Beam quality
Performance evaluation factor
0.69 0.78 0.84 0.86 0.90 0.91 0.92 0.92
0.68 0.76 0.79 0.80 0.82 0.81 0.75 0.70
Table 3 Values of the performance evaluation factor with certain laser wavelength and propagation distance. Propagation distance (km)/wavelength (m)
1.064
1.315
3.8
5 10 15 20 25 30 40 50
0.73 0.79 0.81 0.81 0.80 0.79 0.78 0.76
0.71 0.78 0.79 0.80 0.81 0.80 0.78 0.76
0.68 0.76 0.79 0.80 0.82 0.81 0.75 0.70
Fig. 3. Intensity distribution of the received beam.
to 0.91. Fig. 4 shows that only after 65 steps of iteration, the performance evaluation factor approaches to an absolute maximum value, which illustrate the astringency of phase optimization in AO1. To show the evolution of the new source performance with the increase of propagation distance, we take different values of beam propagation distance and calculate performance of the optical system. The coming results are listed in Table 2. Form Table 2, we can get that power efficiency of the new source is high and beam quality is low when propagation distance is short. With the increase of propagation distance, power
efficiency of the new source decreases, and beam quality of the new source increases. Performance evaluation factor of the new source approaches to the maximum value at a certain propagation distance, for the determinate optical system in the present paper, the propagation distance is 25 km, and the maximum performance evaluation factor is 0.82. To show the influences of beam wavelength on the new source performance, we take beam wavelength as 1.064 m, 1.315 m, 3.8 m, and calculate performances of the optical system. Values of the performance evaluation factor with certain laser wavelength and propagation distance are listed in Table 3. Results show that optical system with 1.064 m source approaches the optimal performance factor 0.81 at 15–20 km beam propagation distance; optical system with 1.315 m source approaches the optimal performance factor 0.81 at 25 km beam propagation distance; the optical system with 3.8 m source approaches the optimal performance factor 0.82 at 25 km beam propagation distance.
4. Conclusions
Fig. 4. Performance evaluation factor on the steps of the phase optimization algorithm iteration.
Table 1 Performance of the new source. Power efficiency Before phase optimization After phase optimization
Beam quality
Performance evaluation factor
82%
0.90
0.74
89%
0.91
0.81
In this paper, we have studied performance of an optical system with coherent laser array source and phase optimization in turbulent atmosphere. Model of an optical system with coherent laser array source and the piston phase distributions optimized by the stochastic parallel gradient descent algorithm was established. Theory of beam propagation through the optical system in turbulent atmosphere was analyzed, and the analytical formulas of the beam average intensity along the propagation path were derived. Power efficiency and Strehl ratio induced by intensity disorderly distribution were introduced to evaluate performance of the new source. Under the H-V 5/7 atmospheric turbulent model, performances of an optical system with determinate parameters were calculated, and the influences of the propagation distance and the laser wavelength on the system performance were numerically analyzed, respectively. Results in the present paper may provide references for designing and analyzing of optical systems with coherent laser array source and phase optimization.
Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018
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Please cite this article in press as: H. Wu, et al., Study on an optical system with coherent laser array source and phase optimization in turbulent atmosphere, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.018