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Fiber-coupling efficiency of Gaussian–Schell model beams through an ocean to fiber optical communication link
Optics Communications 417 (2018) 14–18
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Fiber-coupling efficiency of Gaussian–Schell model beams through an ocean to fiber optical communication link Beibei Hu a , Haifeng Shi a , Yixin Zhang a,b, * a b
School of Science. Jiangnan University, Wuxi 214122, China Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, Wuxi 214122, China
a r t i c l e
i n f o
Keywords: Fiber-coupling efficiency Oceanic turbulence Gaussian–Schell model beams Cross-spectral density function Free-space optical communication
a b s t r a c t We theoretically study the fiber-coupling efficiency of Gaussian–Schell model beams propagating through oceanic turbulence. The expression of the fiber-coupling efficiency is derived based on the spatial power spectrum of oceanic turbulence and the cross-spectral density function. Our work shows that the salinity fluctuation has a greater impact on the fiber-coupling efficiency than temperature fluctuation does. We can select longer 𝜆 in the ‘‘ocean window’’ and higher spatial coherence of light source to improve the fiber-coupling efficiency of the communication link. We also can achieve the maximum fiber-coupling efficiency by choosing design parameter according specific oceanic turbulence condition. Our results are able to help the design of optical communication link for oceanic turbulence to fiber sensor.
1. Introduction Knowledge of the efficiency with which random light can be coupled to optical fiber is important for the design of link receivers of optical communication [1–4]. Since Winze and Leeb discussed the fibercoupling efficiency of random light and its application in lidar [1], a large number of research papers on the fiber-coupling efficiency of the single-mode fiber (SMF) for the link of free-space to fiber in atmospheric turbulence have been carried out [5–8]. In [5], Toyoshima gave a model of the maximum fiber-coupling efficiency under some random jitter environment, which is a function of only random angular jitter and the Airy beam size of the optical beam. By this model one can calculated the fiber-coupling efficiency simply without any complicated numerical calculation. The paper also investigated the average bit error ratio and fade level of fiber-coupled signals at desired fade probability. Tan et al. [6] studied the effect of spatial coherence of light source on the fiber-coupling efficiency and derived the crossspectral density function of the Gaussian–Schell model (GSM) and the fiber-coupling efficiency of GSM beams through atmospheric turbulence was deduced. Zhai et al. [7] researched the influence of non-Kolmogorov turbulence and turbulence inhomogeneity on fiber-coupling efficiency based on the non-Kolmogorov satellite downlinks and uplinks. They found the fiber-coupling efficiency decreases as the non-Kolmogorov power law exponent increases. Ma et al. [8] derived a statistical model for coupling efficiency of spatial light coupling into a SMF which can *
be used to estimate the signal-to-noise ratio and bit error rate of a freespace optical (FSO) communication link. As the FSO communication in turbulent ocean has received increasingly attention in recent years, the propagation of optical beams through oceanic turbulence has been become a subject of considerable importance [9–17]. Such as, Hanson et al. [18] measured the coupling efficiency of a focused beam into a SMF in laboratory experiment. Their experiment of simple tip-tilt control system shown to maintain good coupling efficiency, the beam radius need equal to the transverse coherence length. However, the theoretical model for laser beam from oceanic turbulence link coupled into SMF is a relatively unexplored topic, but for communication link design, this is very necessary. In this paper, we derived a theoretical model of the fiber-coupling efficiency to obtain the knowledge of the efficiency with which GSM beams propagating in oceanic turbulence and coupled to optical fiber. In the analysis, we assume that ocean was pure seawater which is free of suspended organic and inorganic impurities. That is to say, we only consider the influence of oceanic turbulence due to temperature and salinity fluctuations on the coupling efficiency. The rest of this paper is organized as follows. Section 2 gives a brief overview of the fiber-coupling efficiency equation in previous work, and then estimated the cross-spectral density function of the GSM beams in oceanic turbulence and a new fiber-coupling efficiency equation. The numerical simulations and analysis are given in Section 3. Our conclusions are drawn in Section 4.
Corresponding author at: School of Science. Jiangnan University, Wuxi 214122, China. E-mail address: [email protected] (Y. Zhang).
https://doi.org/10.1016/j.optcom.2018.02.031 Received 16 November 2017; Received in revised form 10 February 2018; Accepted 12 February 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
B. Hu et al.
Optics Communications 417 (2018) 14–18
of mean-square temperature variance and has the range 10−10 K 2 ∕s to 10−4 K 2 ∕s, 𝜅 is the spatial frequency of turbulent fluctuations, 𝜂 is the inner scale of oceanic turbulence, 𝜛 is the ratio of temperature and salinity contributions to the refractive index spectrum, which in the seawater can vary from −5 to 0, with −5 and 0 corresponding to dominating temperature-induced and salinity-induced optical turbulence respectively, and 𝜙(𝜅, 𝜛) is given as
2. Fiber-coupling efficiency of GSM beams through oceanic turbulence The fiber-coupling efficiency in FSO is defined as the ratio of the average optical power coupled into the fiber, ⟨𝑷𝑎 ⟩, to the average available optical power in the system’s aperture plane, ⟨𝑷𝑎 ⟩, and is given by [1] 𝜂𝑐 =
∬ 𝑨 𝑭𝑨∗ (𝒓1 )𝑭𝑨 (𝒓2 )𝑾 (𝒓1 , 𝒓2 , 𝑧)d𝒓1 d𝒓2 ⟨𝑷𝑐 ⟩ = , ⟨ ⟩ ⟨𝑷𝑎 ⟩ ∬ |𝑬𝑨 (𝒓)|2 d𝒓
( ) ( ) ( ) 𝜙(𝜅, 𝜛) = exp −𝐴𝑇 𝛿 + 𝜛 −2 exp −𝐴𝑆 𝛿 − 2𝜛 −1 exp −𝐴𝑇 𝑆 𝛿 ,
(1)
8.284(𝜅𝜂)4∕3
where 𝑬𝑨 (𝒓) characterizes the incident optical field in the receiver aperture plane 𝐀, ∗ indicates the complex conjugate, 𝑾 (𝒓1 , 𝒓2 , 𝑧) is the crossspectral density function of the GSM beams propagating through oceanic turbulence at propagation distance 𝑧; 𝑭𝑨 (𝒓) is the backpropagated fibermode profile and is given by [4] ( ) √ 2 𝑟2 exp − , (2) 𝑭𝑨 (𝒓) = 𝜋𝜔2𝑎 𝜔2𝑎
𝑁 = 3.603 × 10−7 𝑘2 𝑧
(9)
(10)
where ⎧ 𝐿(𝑧) = 1 + 4𝑧2 𝜉∕𝑘2 𝑤40 ⎪ 2 ⎨ 𝜉 = 𝜁𝑠 + 2𝑁𝑤0 ⎪ 2 2 ⎩ 𝜁𝑠 = 1 + 𝑤0 ∕𝛿0 .
where 𝝆1 and 𝝆2 are the transverse coordinate vectors at plane 𝑧 = 0, 𝒓1 and 𝒓2 are the transverse coordinate vectors at the 𝑧 (𝑧 > 0) plane, 𝑘 = 2𝜋∕𝜆 is the wave number, and symbol ⟨⋯ ⟩𝑜 represents the ensemble average in oceanic turbulence; 𝑾 (𝝆1 , 𝝆2 , 0) is the cross-spectral density function of the GSM beams at the transmitter plane, which is given by [20] ( ) [ ] 𝝆 2 + 𝝆2 2 (𝝆 − 𝝆2 )2 𝑾 (𝝆1 , 𝝆2 , 0) = 𝐼0 exp − 1 exp − 1 , (4) 𝑤20 2𝛿02
(12)
As a rule, [when the variables] 𝒓1 and 𝒓2 are independent of each other, the term exp 𝑖𝑘(𝒓2 2 − 𝒓1 2 )∕𝐽 (𝑧) is not considered in the paper, and 𝜁𝑠 is defined as the source coherent parameter [21]. 2.2. Fiber-coupling efficiency We substituting Eqs. (2) and (11) into Eq. (1), the fiber-coupling efficiency of the GSM beams propagating through oceanic turbulence is expressed as { [ ( ) 2 2 1 +𝑁 1+ exp − 𝜂𝑐 = 𝐿(𝑧) 𝜋𝜔2𝑎 𝑨𝑅 ∬ 𝑨 2𝛿02 𝐿(𝑧) } ] )2 2𝑁 2 𝑧2 ( (13) − 𝒓1 − 𝒓2 2 𝑘2 𝑤0 𝐿(𝑧)
where 𝐼0 is the average optical intensity, 𝑤0 is the waist width for the source, and 𝛿0 denotes the transverse coherence width. In Eq. (3), ⟨⋯ ⟩𝑜 can be written as [14] ⟨ [ ]⟩ exp 𝜓 ∗ (𝒓1 , 𝝆1 , 𝑧) + 𝜓(𝒓2 , 𝝆2 , 𝑧) 𝑜 { } [( )2 ( )( ) ( )2 ] = exp −𝑁 𝝆1 − 𝝆2 + 𝝆1 − 𝝆2 𝒓1 − 𝒓2 + 𝒓1 − 𝒓2 , (5)
( × exp
where 𝑁 is an expression with the strength of oceanic turbulence considered, and it can be written as
−
𝑟21 + 𝑟22 𝜔2𝑎
) d𝒓1 d𝒓2 ,
where parameter 𝑨𝑅 = 𝜋𝐷2 ∕4 denotes the receiver aperture area, and 𝐷 is the diameter of receiver aperture. To simplify expression, we define a parameter 𝑀 as follows: ( ) 1 2 2𝑁 2 𝑧2 +𝑁 1+ . (14) 𝑀= − 2 𝐿(𝑧) 2𝛿0 𝐿(𝑧) 𝑘2 𝑤20 𝐿(𝑧)
∞
(6)
where 𝛷𝑛 is the spatial power spectrum of the refractive index fluctuations of the turbulent ocean water. As mentioned earlier, we consider the influence of oceanic turbulence generated by temperature and salinity fluctuations on the beam. The spatial power spectrum of oceanic turbulence model can be given by [15] [ ] 𝛷𝑛 (𝜅) = 0.388 × 10−8 𝜀−1∕3 𝜒𝑡 𝜅 −11∕3 1 + 2.35(𝜅𝜂)2∕3 𝜙(𝜅, 𝜛),
(0.483𝜛 2 − 0.835𝜛 + 3.38).
Substituting Eqs. (4), (5) and (10) into Eq. (3), Eq. (3) can be simplified to { [ ] ( ) 𝑤20 + 𝛿02 𝐼 2 𝑾 (𝒓1 , 𝒓2 , 𝑧) = 0 exp − +𝑁 1+ 𝐿(𝑧) 𝐿(𝑧) 2𝐿(𝑧)𝑤20 𝛿02 } )2 2𝑁 2 𝑧2 ( 𝒓1 − 𝒓2 − (11) 2 2 𝑘 𝑤0 𝐿(𝑧) )2 ) )] ( ( [ ( 2 𝒓1 + 𝒓2 𝑖𝑘 𝒓2 − 𝒓1 2 × exp − exp , 𝐽 (𝑧) 2𝑤20 𝐿(𝑧)
Based on the generalized Huygens–Fresnel principle, the crossspectral density function of the GSM beams propagating through oceanic turbulence at the 𝑧 plane is defined by [19] ( ) 𝑘 2 𝑾 (𝒓1 , 𝒓2 , 𝑧) = d2 𝝆1 d2 𝝆2 𝑾 (𝝆1 , 𝝆2 , 0) ∬ 2𝜋𝑧 ∬ [ ] (𝒓 − 𝝆1 )2 − (𝒓2 − 𝝆2 )2 (3) × exp −𝑖𝑘 1 2𝑧 ⟨ [ ]⟩ × exp 𝜓 ∗ (𝒓1 , 𝝆1 , 𝑧) + 𝜓(𝒓2 , 𝝆2 , 𝑧) 𝑜 ,
𝜅 𝛷𝑛 (𝜅)d𝜅,
2𝜛 2
𝑁 = 18.02𝐶𝑚2 𝑘2 𝑧 𝜂 −1∕3 (0.483 − 0.835𝜛 −1 + 3.38𝜛 −2 ).
2.1. The cross-spectral density function of the GSM beams in oceanic turbulence
3
(𝜀𝜂)−1∕3 𝜒𝑡
1.863 × 10−2 , 𝐴𝑆
To simplify the writing, we define 𝐶𝑚2 = 10−8 𝜀−1∕3 𝜒𝑡 as the ‘‘equivalent’’ temperature structure constant. Now Eq. (9) can be written as
where 𝜔𝑎 is the mode-field radius of the backpropagated fiber mode given by 𝜔𝑎 = 𝜆𝑓 ∕𝜋𝜔0 , 𝜆 is the laser wavelength, 𝜔0 is the fiber-mode field radius and 𝑓 is the focal length of the coupling lens.
𝜋 2 𝑘2 𝑧 𝑁= 3 ∫0
+ 12.978(𝜅𝜂)2 , 𝐴𝑇
(8)
where 𝛿 = = = 1.9 × 10−4 , −3 and 𝐴𝑇 𝑆 = 9.41 × 10 . Substituting Eqs. (7) and (8) into Eq. (6), we can obtain [10]
𝑨
Thus the fiber-coupling efficiency is written as [ ] ( 2 ) 𝑟 + 𝑟22 ( )2 2 𝜂𝑐 = exp −𝑀 𝒓1 − 𝒓2 exp − 1 d𝒓1 d𝒓2 . 𝜋𝜔2𝑎 𝑨𝑅 ∬ 𝑨 𝜔2𝑎
(7)
(15)
By the law of cosines:
where 𝜀 is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 10−10 m2 ∕s3 to 10−1 m2 ∕s3 , 𝜒𝑡 is the rate of dissipation
( )2 𝒓1 − 𝒓2 = 𝑟21 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ), 15
(16)
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Optics Communications 417 (2018) 14–18
Fig. 1. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the design parameter 𝛽 for different values of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 (a) and the coherent parameter 𝜁𝑠 (b).
we can write the fiber-coupling efficiency as 𝜂𝑐 =
2𝜋 2𝜋 ) [ ( exp −𝑀 𝑟21 + 𝑟22 ∫0 ∫0 ] + 2𝑀𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ) ( 2 ) 𝑟 + 𝑟22 × exp − 1 𝑟1 𝑟2 d𝜑1 d𝜑2 d𝑟1 d𝑟2 . 𝜔2𝑎 𝐷∕2
2
𝜋𝜔2𝑎 𝑨𝑅
∫0
we finally obtain the fiber-coupling efficiency 𝜂𝑐 as ( ) ∞ 𝑨𝑅 2 𝑥22 ( ) 4𝛽 2 exp −𝐻𝑥22 1 𝐹1 1; 1; 𝜂𝑐 = 𝑥2 d𝑥2 , 𝐻 ∫0 𝑨𝐻 2 𝐻
𝐷∕2
∫0
where 1 𝐹1 (⋅) is a confluent hypergeometric function and 𝐻 = 𝑀𝐷2 ∕4 + 𝐷2 ∕4𝜔2𝑎 = 𝑨𝑅 ∕𝑨𝐻 + 𝛽 2 . Adopting the defining method of Ref. [1], 𝑨𝑅 ∕𝑨𝐻 is defined as the number of speckles at the receiver aperture 𝑨, 𝑨𝐻 = 𝜋𝜌2𝐻 stands for the coherence area of 𝑬𝑨 , also known as speckle size, where 𝜌𝐻 denotes the spatial coherent radius, which is given by √ [ ] √ ) ( √ 2𝑁 2 𝑧2 1 2 √ 𝜌𝐻 = 1∕ − +𝑁 1+ . (24) 𝐿(𝑧) 2𝛿02 𝐿(𝑧) 𝑘2 𝑤20 𝐿(𝑧)
(17)
In Eqs. (16) and (17), 𝜑1 and 𝜑2 are the angles between 𝒓1 and 𝒓2 and the centered axis at the propagation plane, respectively. The double integral over the angle variables 𝜑1 and 𝜑2 can be given by [4] 2𝜋
∫0
2𝜋
∫0
[ ] ( ) exp 2𝑀𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ) d𝜑1 d𝜑2 = 4𝜋 2 𝐼0 2𝑀𝑟1 𝑟2 ,
The design parameter 𝛽 is defined as the ratio of the aperture radius to the radius of the backpropagated fiber mode, which is given by [4]
(18)
where 𝐼0 (⋯ ) denotes the modified Bessel function of first kind and zero order. Substituting Eq. (18) into Eq. (17), we arrive at [ ( ) ] 𝐷∕2 𝐷∕2 ) 8𝜋 1 ( 2 2 𝜂𝑐 = exp − 𝑀 + 𝑟1 + 𝑟2 ∫0 𝜔2𝑎 𝑨𝑅 ∫0 𝜔2𝑎 ( ) 𝐼0 2𝑀𝑟1 𝑟2 𝑟1 𝑟2 d𝑟1 d𝑟2 . (19)
𝛽=
(25)
In this section, we present the numerical discussions for the fibercoupling efficiency of the GSM beams propagating through oceanic turbulence as the functions of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 , the ratio of temperature and salinity contributions to the refractive index spectrum 𝜛, the inner scale 𝜂 of oceanic turbulence, the dissipation rate of temperature variance 𝜒𝑡 , the rate of dissipation of kinetic energy per unit mass of fluid 𝜀, the coherent parameter 𝜁𝑠 , the source radius 𝑤0 , the propagation distance 𝑧, the wavelength 𝜆, and the design parameter 𝛽. In order to study the effect of above parameters on the fiber-coupling efficiency exactly, we plot Figs. 1–5 based on the Eq. (20) in the following calculations, and we set the parameters 𝑤0 , 𝐷 as 𝑤0 = 0.04 m, and 𝐷 = 0.1 m. To illustrate the influence of the design parameter 𝛽 on the system performance, we plot the fiber-coupling efficiency of GSM beams as a function of 𝛽 ranging from 0 to 4, for the case of 𝐶𝑚2 = 0, 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−15 m−2∕3 in Fig. 1(a) and 𝜁𝑠 = 1, 𝜁𝑠 = 2, 𝜁𝑠 = 5 in Fig. 1(b). The other parameters of link are set as: 𝑧 = 50 m, 𝜆 = 417 nm, 𝜛 = −3, 𝜁𝑠 = 2 in Fig. 1(a), and 𝐶𝑚2 = 10−17 m−2∕3 in Fig. 1(b). We notice that, in Fig. 1, when the value of 𝛽 increases, the fiber-coupling efficiency increases at first and then decreases for small 𝐶𝑚2 and 𝜁𝑠 , whereas it changes slowly for larger 𝐶𝑚2 and 𝜁𝑠 . It implies that the design parameter 𝛽 has an influence on the fiber-coupling efficiency to some extent, and there is an optimum value of 𝛽 exists under different strength of 𝐶𝑚2 and 𝜁𝑠 . Meanwhile, the maximum 𝜂𝑐 decreases with 𝐶𝑚2 and 𝜁𝑠 increasing, and the optimum value of design parameter 𝛽 is 1.21,1.22, 1.57 for 𝐶𝑚2 = 0, 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−15 m−2∕3 respectively in Fig. 1(a); 1.14, 1.22, 1.36 for 𝜁𝑠 = 1, 𝜁𝑠 = 2, 𝜁𝑠 = 5 respectively in
By defining new variables 𝑥1 = 2𝑟1 ∕𝐷, 𝑥2 = 2𝑟2 ∕𝐷, the 𝜂𝑐 is expressed
(20)
In order to simplify the calculation, we extend the range of the integration from 0 to 1 in Eq. (20) to that from 0 to infinity in the condition of strong oceanic turbulence, and its error less than or equal to 10%. Hence we obtain an approximate expression in the form [ ( ) ] ∞ ∞ ) 𝑀𝐷2 𝐷2 ( 2 2𝐷2 𝜂𝑐 ≈ exp − + 𝑥1 + 𝑥22 4 𝜔2𝑎 ∫0 ∫0 4𝜔2𝑎 ( ) 𝑀𝐷2 𝐼0 𝑥1 𝑥2 𝑥1 𝑥2 d𝑥1 d𝑥2 . (21) 2 Recalling the integral formula [22]: ( ) 𝑥𝛼−1 exp −𝑝𝑥2 𝐼𝜈 (𝑐𝑥) d𝑥 = 𝐴𝛼𝜈 ∫0 [ ] Re 𝑝, Re (𝛼 + 𝜈) > 0; |arg 𝑐| < 𝜋 [ ] ( ) (𝛼 + 𝜈) ∕2 𝛼+𝜈 𝑐2 𝐴𝛼𝜈 = 2−𝜈−1 𝑐 𝜈 𝑝−(𝛼+𝜈)∕2 𝛤 𝐹 ; 𝜈 + 1; , 𝜈+1 1 1 2 4𝑝
𝐷𝜋𝜔0 𝐷 = . 2𝜔𝑎 2𝜆𝑓
3. Analysis of fiber-coupling efficiency
as
[ ( ) ] 1 1 ) 2𝐷2 𝑀𝐷2 𝐷2 ( 2 2 exp − + 𝜂𝑐 = 𝑥1 + 𝑥2 4 𝜔2𝑎 ∫0 ∫0 4𝜔2𝑎 ) ( 𝑀𝐷2 𝑥1 𝑥2 𝑥1 𝑥2 d𝑥1 d𝑥2 . 𝐼0 2
(23)
∞
(22)
16
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Optics Communications 417 (2018) 14–18
Fig. 2. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the ratio of temperature and salinity contributions to the refractive index spectrum 𝜛 for different values of wavelength 𝜆.
Fig. 3. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the rate of dissipation of mean-square temperature variance 𝜒𝑡 for different values of the rate of dissipation of kinetic energy per unit mass of fluid 𝜀.
Fig. 1(b). We can infer that the maximum fiber-coupling efficiency can be achieved by choosing the appropriate design parameter. Fig. 2 indicates that the fiber-coupling efficiency of GSM beams as a function of temperature and salinity contributions to the refractive index spectrum 𝜛 and the wavelength 𝜆. In Fig. 2, we set the range of 𝜛 changes from −4.5 to −0.5, and the 𝜆 are 417 nm, 488 nm and 532 nm, which the value usually used in other papers [12–15]. The other system parameters of link are set as: 𝛽 = 1.23, 𝑧 = 50 m, 𝜁𝑠 = 2, and 𝐶𝑚2 = 10−17 m−2∕3 . As expected, 𝜂𝑐 decreases as the increasing of 𝜛 but increases as 𝜆 increasing. We also find that the fiber-coupling efficiency has a clear downtrend when the value of 𝜛 is closer to zero. It means that the salinity fluctuation has a greater impact on the fibercoupling efficiency than temperature fluctuation does. On one hand, the increase in salinity will contribute a stronger oceanic turbulence [17] and induce the increase of scintillation. On the other hand, the beam with small wavelength would have stronger scintillation effect [10]. Both of them will reduce the fiber-coupling efficiency. This result is similar to the effect of atmospheric turbulence on the fiber-coupling efficiency [6]. In addition, we know that there is a spectral window in the ocean similar to that present in the atmosphere, which has attenuation of the blue–green light in the 400 nm–580 nm band less than the attenuation of the other bands [23], hence we select the longer 𝜆 in the ‘‘ocean window’’ can improve the fiber-coupling efficiency and the communication performance for other parameters being given. In Fig. 3, we investigate the influence of 𝜀 and 𝜒𝑡 on the fibercoupling efficiency of GSM beams for the case 𝜀 = 10−1 m2 ∕s3 , 𝜀 = 10−3 m2 ∕s3 , and 𝜀 = 10−6 m2 ∕s3 , where 𝜒𝑡 changes from 10−9 K 2 ∕s to 10−8 K 2 ∕s. The other system parameters of link are set as: 𝛽 = 1.55, 𝑧 = 50 m, 𝜁𝑠 = 2, 𝜆 = 417 nm, and 𝜛 = −3. From Fig. 3, we see that the fibercoupling efficiency of GSM beams decreases with the increasing of 𝜒𝑡 but increases as the increasing of 𝜀. We know that 𝜒𝑡 express the strength of the small scale temperature gradient and 𝜀 is inversely proportional to the size of the smallest flow structure, it means that the higher values of 𝜒𝑡 and small value of 𝜀 both will leads to strong oceanic turbulence [9–11]. Consequently, the fiber-coupling efficiency will be degrades. In Fig. 4, the fiber-coupling efficiency of GSM beams is plotted as a function of 𝑧 for 𝜁𝑠 =1, 2, and 5. The other system parameters of link are set as: 𝛽 = 1.23, 𝜂 = 1 mm, 𝜛 = −3, 𝜆 = 417 nm, and 𝐶𝑚2 = 10−17 m−2∕3 . The solid line means that the laser source is fully coherent, and the other two lines represent the partially coherence source which is the GSM beams as referred above. We find that the value of the fiber-coupling efficiency of GSM beams is largest when 𝜁𝑠 = 1, and it decreases with the increasing of the propagating distance 𝑧 and the coherence parameter 𝜁𝑠 . This is for the reason that degrade of the beam coherence make the
Fig. 4. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of propagating distance 𝑧 for different values of the coherent parameter 𝜁𝑠 .
fiber-coupling efficiency decrease, and it is similar to the effect of a beam in atmospheric turbulence [6]. It manifest that the higher spatial coherence of the incident light can improve the fiber-coupling efficiency when the propagation distance is fixed. Fig. 5 presents the curves of the fiber-coupling efficiency of GSM beams as a function of the inner scale 𝜂 ranging from 1 mm to 5 mm for the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−16 m−2∕3 , and 𝐶𝑚2 = 10−15 m−2∕3 . The other system parameters of link are set as: 𝛽 = 1.40, 𝑧 = 50 m, 𝜁𝑠 = 2, 𝜛 = −3, and 𝜆 = 417 nm. As we can see that the fiber-coupling efficiency degrades sharply with the increasing of the 𝐶𝑚2 . Therefore, the stronger oceanic turbulence is, the greater effect it has on the fiber-coupling efficiency. In addition, when 𝐶𝑚2 = 10−17 m−2∕3 , the value of 𝜂𝑐 is almost constant, it indicated that we can ignore the effect of the inner scale 𝜂 of oceanic turbulence on the fiber-coupling efficiency in the weak oceanic turbulence. However, with the enhancement of turbulence strength, the influence of internal scale on fiber-coupling efficiency cannot be neglected. 4. Conclusion In this paper, we have molded the fiber-coupling efficiency of the GSM beams propagating through oceanic turbulence. The results showed that the fiber-coupling efficiency of the GSM beams increases as 17
B. Hu et al.
Optics Communications 417 (2018) 14–18 [2] Q. Liu, C. Qiao, G. Mitchell, S. Stanton, Optical wireless communication networks for first- and last-mile broadband access, J. Opt. Netw. 4 (2005) 807–828 (invited). [3] Z. Ghassemlooy, W. Popoola, S. Rajbhandari, Optical Wireless Communications System and Channel Modelling with Matlab, CRC Press, 2013. [4] Y. Dikmelik, F.M. Davidson, Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence, Appl. Opt. 44 (2005) 4946–4952. [5] M. Toyoshima, Maximum fiber coupling efficiency and optimum beam size in the presence of random angular jitter for free-space laser systems and their applications, J. Opt. Soc. Am. A. 23 (2006) 2246–2250. [6] L. Tan, M. Li, Q. Tang, J. Ma, Fiber-coupling efficiency of Gaussian Schell model for optical communication through atmospheric turbulence, Appl. Opt. 54 (2015) 2318–2325. [7] C. Zhai, L. Tan, S. Yu, Y. Cao, J. Ma, Fiber coupling efficiency in non-Kolmogorov satellite links, Opt. Commun. 336 (2015) 93–97. [8] J. Ma, L. Ma, Q. Yang, Q. Ran, Statistical model of the efficiency for spatial light coupling into a single-mode fiber in the presence of atmospheric turbulence, Appl. Opt. 54 (2015) 9287–9293. [9] N. Farwell, O. Korotkova, Intensity and coherence properties of light in oceanic turbulence, Opt. Commun. 285 (2012) 872–875. [10] Y. Wu, Y. Zhang, Y. Li, Z. Hu, Beam wander of Gaussian–Schell model beams propagating through oceanic turbulence, Opt. Commun. 371 (2016) 59–66. [11] Y. Baykal, Expressing oceanic turbulence parameters by atmospheric turbulence structure constant, Appl. Opt. 55 (2016) 1228–1231. [12] Y. Li, L. Yu, Y. Zhang, Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite–Gaussian vortex beam in the ocean, Opt. Express 25 (2017) 12203–12215. [13] X. Chen, D. Zhao, Propagation properties of electromagnetic rectangular multiGaussian Schell-model beams in oceanic turbulence, Opt. Commun. 372 (2016) 137– 143. [14] Y. Wu, Y. Zhang, Y. Zhu, Average intensity and directionality of partially coherent model beams propagating in turbulent ocean, J. Opt. Soc. Am. A. 33 (2016) 1451– 1458. [15] L. Yu, Y. Zhang, Analysis of modal crosstalk for communication in turbulent ocean using Lommel–Gaussian beam, Opt. Express 25 (2017) 22565–22574. [16] W. Zhu, M. Tang, D. Zhao, Propagation of multi-Gaussian Schell model beams in oceanic turbulence, Optik 127 (2016) 3775–3778. [17] Z. Zhou, M. Guo, D. Zhao, Propagation of specular and anti-specular Gaussian Schellmodel beams in oceanic turbulence, Opt. Commun. 383 (2017) 287–293. [18] F. Hanson, M. Lasher, Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber, Appl. Opt. 49 (2010) 3224–3230. [19] L.C. Andrews, R.L. Phillips, Laser Beam Propagation Through Random Media, seconed ed., SPIE Press, Washington, 2005. [20] B. Lü, Laser Optics: Beam Characterization, Propagation and Transformation, Resonator Technology and Physics, Higher Education Press, Beijing, 2002. [21] J.C. Ricklin, F.M. Davidson, Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication, J. Opt. Soc. Am. A. 19 (2002) 1794–1802. [22] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marechev, Integrals and Series, Gordon and Breach Science Publishers, 1986. [23] R.C. Smith, K.S. Baker, Optical properties of the clearest natural waters (200– 800 nm), Appl. Opt. 20 (1981) 177–184.
Fig. 5. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the inner scale 𝜂 for different values of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 .
the increasing of the 𝜆 in the ‘‘ocean window’’, 𝜂, and 𝜀, but decreases as the increasing of the 𝜁𝑠 , 𝑧, 𝐶𝑚2 , 𝜛, and 𝜒𝑡 . There is an optimum design parameter 𝛽 exists under different system source coherent parameter and turbulence conditions, it manifests that we can choose an optimum design parameter and higher source coherent parameter to maximize the fiber-coupling efficiency. The fiber-coupling efficiency of the GSM beams is weak dependence on the inner scale 𝜂 in weak turbulence. The influence of salinity fluctuation on the fiber-coupling efficiency is stronger than the influence of temperature fluctuation for given other parameters. Our results can be beneficial for the design of fiber-coupledbased FSO communication system in oceanic environment. Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51716A) and Postgraduate Research & Practice Innovation Program of Jiangsu Provence (Grant No. SJCX17_0495). The authors would like to thanks reviewers for valuable comments. References [1] P.J. Winze, W.R. Leeb, Fiber coupling efficiency for random light and its applications to lidar, Opt. Lett. 23 (1998) 986–988.
18
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Fiber-coupling efficiency of Gaussian–Schell model beams through an ocean to fiber optical communication link Beibei Hu a , Haifeng Shi a , Yixin Zhang a,b, * a b
School of Science. Jiangnan University, Wuxi 214122, China Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, Wuxi 214122, China
a r t i c l e
i n f o
Keywords: Fiber-coupling efficiency Oceanic turbulence Gaussian–Schell model beams Cross-spectral density function Free-space optical communication
a b s t r a c t We theoretically study the fiber-coupling efficiency of Gaussian–Schell model beams propagating through oceanic turbulence. The expression of the fiber-coupling efficiency is derived based on the spatial power spectrum of oceanic turbulence and the cross-spectral density function. Our work shows that the salinity fluctuation has a greater impact on the fiber-coupling efficiency than temperature fluctuation does. We can select longer 𝜆 in the ‘‘ocean window’’ and higher spatial coherence of light source to improve the fiber-coupling efficiency of the communication link. We also can achieve the maximum fiber-coupling efficiency by choosing design parameter according specific oceanic turbulence condition. Our results are able to help the design of optical communication link for oceanic turbulence to fiber sensor.
1. Introduction Knowledge of the efficiency with which random light can be coupled to optical fiber is important for the design of link receivers of optical communication [1–4]. Since Winze and Leeb discussed the fibercoupling efficiency of random light and its application in lidar [1], a large number of research papers on the fiber-coupling efficiency of the single-mode fiber (SMF) for the link of free-space to fiber in atmospheric turbulence have been carried out [5–8]. In [5], Toyoshima gave a model of the maximum fiber-coupling efficiency under some random jitter environment, which is a function of only random angular jitter and the Airy beam size of the optical beam. By this model one can calculated the fiber-coupling efficiency simply without any complicated numerical calculation. The paper also investigated the average bit error ratio and fade level of fiber-coupled signals at desired fade probability. Tan et al. [6] studied the effect of spatial coherence of light source on the fiber-coupling efficiency and derived the crossspectral density function of the Gaussian–Schell model (GSM) and the fiber-coupling efficiency of GSM beams through atmospheric turbulence was deduced. Zhai et al. [7] researched the influence of non-Kolmogorov turbulence and turbulence inhomogeneity on fiber-coupling efficiency based on the non-Kolmogorov satellite downlinks and uplinks. They found the fiber-coupling efficiency decreases as the non-Kolmogorov power law exponent increases. Ma et al. [8] derived a statistical model for coupling efficiency of spatial light coupling into a SMF which can *
be used to estimate the signal-to-noise ratio and bit error rate of a freespace optical (FSO) communication link. As the FSO communication in turbulent ocean has received increasingly attention in recent years, the propagation of optical beams through oceanic turbulence has been become a subject of considerable importance [9–17]. Such as, Hanson et al. [18] measured the coupling efficiency of a focused beam into a SMF in laboratory experiment. Their experiment of simple tip-tilt control system shown to maintain good coupling efficiency, the beam radius need equal to the transverse coherence length. However, the theoretical model for laser beam from oceanic turbulence link coupled into SMF is a relatively unexplored topic, but for communication link design, this is very necessary. In this paper, we derived a theoretical model of the fiber-coupling efficiency to obtain the knowledge of the efficiency with which GSM beams propagating in oceanic turbulence and coupled to optical fiber. In the analysis, we assume that ocean was pure seawater which is free of suspended organic and inorganic impurities. That is to say, we only consider the influence of oceanic turbulence due to temperature and salinity fluctuations on the coupling efficiency. The rest of this paper is organized as follows. Section 2 gives a brief overview of the fiber-coupling efficiency equation in previous work, and then estimated the cross-spectral density function of the GSM beams in oceanic turbulence and a new fiber-coupling efficiency equation. The numerical simulations and analysis are given in Section 3. Our conclusions are drawn in Section 4.
Corresponding author at: School of Science. Jiangnan University, Wuxi 214122, China. E-mail address: [email protected] (Y. Zhang).
https://doi.org/10.1016/j.optcom.2018.02.031 Received 16 November 2017; Received in revised form 10 February 2018; Accepted 12 February 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.
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Optics Communications 417 (2018) 14–18
of mean-square temperature variance and has the range 10−10 K 2 ∕s to 10−4 K 2 ∕s, 𝜅 is the spatial frequency of turbulent fluctuations, 𝜂 is the inner scale of oceanic turbulence, 𝜛 is the ratio of temperature and salinity contributions to the refractive index spectrum, which in the seawater can vary from −5 to 0, with −5 and 0 corresponding to dominating temperature-induced and salinity-induced optical turbulence respectively, and 𝜙(𝜅, 𝜛) is given as
2. Fiber-coupling efficiency of GSM beams through oceanic turbulence The fiber-coupling efficiency in FSO is defined as the ratio of the average optical power coupled into the fiber, ⟨𝑷𝑎 ⟩, to the average available optical power in the system’s aperture plane, ⟨𝑷𝑎 ⟩, and is given by [1] 𝜂𝑐 =
∬ 𝑨 𝑭𝑨∗ (𝒓1 )𝑭𝑨 (𝒓2 )𝑾 (𝒓1 , 𝒓2 , 𝑧)d𝒓1 d𝒓2 ⟨𝑷𝑐 ⟩ = , ⟨ ⟩ ⟨𝑷𝑎 ⟩ ∬ |𝑬𝑨 (𝒓)|2 d𝒓
( ) ( ) ( ) 𝜙(𝜅, 𝜛) = exp −𝐴𝑇 𝛿 + 𝜛 −2 exp −𝐴𝑆 𝛿 − 2𝜛 −1 exp −𝐴𝑇 𝑆 𝛿 ,
(1)
8.284(𝜅𝜂)4∕3
where 𝑬𝑨 (𝒓) characterizes the incident optical field in the receiver aperture plane 𝐀, ∗ indicates the complex conjugate, 𝑾 (𝒓1 , 𝒓2 , 𝑧) is the crossspectral density function of the GSM beams propagating through oceanic turbulence at propagation distance 𝑧; 𝑭𝑨 (𝒓) is the backpropagated fibermode profile and is given by [4] ( ) √ 2 𝑟2 exp − , (2) 𝑭𝑨 (𝒓) = 𝜋𝜔2𝑎 𝜔2𝑎
𝑁 = 3.603 × 10−7 𝑘2 𝑧
(9)
(10)
where ⎧ 𝐿(𝑧) = 1 + 4𝑧2 𝜉∕𝑘2 𝑤40 ⎪ 2 ⎨ 𝜉 = 𝜁𝑠 + 2𝑁𝑤0 ⎪ 2 2 ⎩ 𝜁𝑠 = 1 + 𝑤0 ∕𝛿0 .
where 𝝆1 and 𝝆2 are the transverse coordinate vectors at plane 𝑧 = 0, 𝒓1 and 𝒓2 are the transverse coordinate vectors at the 𝑧 (𝑧 > 0) plane, 𝑘 = 2𝜋∕𝜆 is the wave number, and symbol ⟨⋯ ⟩𝑜 represents the ensemble average in oceanic turbulence; 𝑾 (𝝆1 , 𝝆2 , 0) is the cross-spectral density function of the GSM beams at the transmitter plane, which is given by [20] ( ) [ ] 𝝆 2 + 𝝆2 2 (𝝆 − 𝝆2 )2 𝑾 (𝝆1 , 𝝆2 , 0) = 𝐼0 exp − 1 exp − 1 , (4) 𝑤20 2𝛿02
(12)
As a rule, [when the variables] 𝒓1 and 𝒓2 are independent of each other, the term exp 𝑖𝑘(𝒓2 2 − 𝒓1 2 )∕𝐽 (𝑧) is not considered in the paper, and 𝜁𝑠 is defined as the source coherent parameter [21]. 2.2. Fiber-coupling efficiency We substituting Eqs. (2) and (11) into Eq. (1), the fiber-coupling efficiency of the GSM beams propagating through oceanic turbulence is expressed as { [ ( ) 2 2 1 +𝑁 1+ exp − 𝜂𝑐 = 𝐿(𝑧) 𝜋𝜔2𝑎 𝑨𝑅 ∬ 𝑨 2𝛿02 𝐿(𝑧) } ] )2 2𝑁 2 𝑧2 ( (13) − 𝒓1 − 𝒓2 2 𝑘2 𝑤0 𝐿(𝑧)
where 𝐼0 is the average optical intensity, 𝑤0 is the waist width for the source, and 𝛿0 denotes the transverse coherence width. In Eq. (3), ⟨⋯ ⟩𝑜 can be written as [14] ⟨ [ ]⟩ exp 𝜓 ∗ (𝒓1 , 𝝆1 , 𝑧) + 𝜓(𝒓2 , 𝝆2 , 𝑧) 𝑜 { } [( )2 ( )( ) ( )2 ] = exp −𝑁 𝝆1 − 𝝆2 + 𝝆1 − 𝝆2 𝒓1 − 𝒓2 + 𝒓1 − 𝒓2 , (5)
( × exp
where 𝑁 is an expression with the strength of oceanic turbulence considered, and it can be written as
−
𝑟21 + 𝑟22 𝜔2𝑎
) d𝒓1 d𝒓2 ,
where parameter 𝑨𝑅 = 𝜋𝐷2 ∕4 denotes the receiver aperture area, and 𝐷 is the diameter of receiver aperture. To simplify expression, we define a parameter 𝑀 as follows: ( ) 1 2 2𝑁 2 𝑧2 +𝑁 1+ . (14) 𝑀= − 2 𝐿(𝑧) 2𝛿0 𝐿(𝑧) 𝑘2 𝑤20 𝐿(𝑧)
∞
(6)
where 𝛷𝑛 is the spatial power spectrum of the refractive index fluctuations of the turbulent ocean water. As mentioned earlier, we consider the influence of oceanic turbulence generated by temperature and salinity fluctuations on the beam. The spatial power spectrum of oceanic turbulence model can be given by [15] [ ] 𝛷𝑛 (𝜅) = 0.388 × 10−8 𝜀−1∕3 𝜒𝑡 𝜅 −11∕3 1 + 2.35(𝜅𝜂)2∕3 𝜙(𝜅, 𝜛),
(0.483𝜛 2 − 0.835𝜛 + 3.38).
Substituting Eqs. (4), (5) and (10) into Eq. (3), Eq. (3) can be simplified to { [ ] ( ) 𝑤20 + 𝛿02 𝐼 2 𝑾 (𝒓1 , 𝒓2 , 𝑧) = 0 exp − +𝑁 1+ 𝐿(𝑧) 𝐿(𝑧) 2𝐿(𝑧)𝑤20 𝛿02 } )2 2𝑁 2 𝑧2 ( 𝒓1 − 𝒓2 − (11) 2 2 𝑘 𝑤0 𝐿(𝑧) )2 ) )] ( ( [ ( 2 𝒓1 + 𝒓2 𝑖𝑘 𝒓2 − 𝒓1 2 × exp − exp , 𝐽 (𝑧) 2𝑤20 𝐿(𝑧)
Based on the generalized Huygens–Fresnel principle, the crossspectral density function of the GSM beams propagating through oceanic turbulence at the 𝑧 plane is defined by [19] ( ) 𝑘 2 𝑾 (𝒓1 , 𝒓2 , 𝑧) = d2 𝝆1 d2 𝝆2 𝑾 (𝝆1 , 𝝆2 , 0) ∬ 2𝜋𝑧 ∬ [ ] (𝒓 − 𝝆1 )2 − (𝒓2 − 𝝆2 )2 (3) × exp −𝑖𝑘 1 2𝑧 ⟨ [ ]⟩ × exp 𝜓 ∗ (𝒓1 , 𝝆1 , 𝑧) + 𝜓(𝒓2 , 𝝆2 , 𝑧) 𝑜 ,
𝜅 𝛷𝑛 (𝜅)d𝜅,
2𝜛 2
𝑁 = 18.02𝐶𝑚2 𝑘2 𝑧 𝜂 −1∕3 (0.483 − 0.835𝜛 −1 + 3.38𝜛 −2 ).
2.1. The cross-spectral density function of the GSM beams in oceanic turbulence
3
(𝜀𝜂)−1∕3 𝜒𝑡
1.863 × 10−2 , 𝐴𝑆
To simplify the writing, we define 𝐶𝑚2 = 10−8 𝜀−1∕3 𝜒𝑡 as the ‘‘equivalent’’ temperature structure constant. Now Eq. (9) can be written as
where 𝜔𝑎 is the mode-field radius of the backpropagated fiber mode given by 𝜔𝑎 = 𝜆𝑓 ∕𝜋𝜔0 , 𝜆 is the laser wavelength, 𝜔0 is the fiber-mode field radius and 𝑓 is the focal length of the coupling lens.
𝜋 2 𝑘2 𝑧 𝑁= 3 ∫0
+ 12.978(𝜅𝜂)2 , 𝐴𝑇
(8)
where 𝛿 = = = 1.9 × 10−4 , −3 and 𝐴𝑇 𝑆 = 9.41 × 10 . Substituting Eqs. (7) and (8) into Eq. (6), we can obtain [10]
𝑨
Thus the fiber-coupling efficiency is written as [ ] ( 2 ) 𝑟 + 𝑟22 ( )2 2 𝜂𝑐 = exp −𝑀 𝒓1 − 𝒓2 exp − 1 d𝒓1 d𝒓2 . 𝜋𝜔2𝑎 𝑨𝑅 ∬ 𝑨 𝜔2𝑎
(7)
(15)
By the law of cosines:
where 𝜀 is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 10−10 m2 ∕s3 to 10−1 m2 ∕s3 , 𝜒𝑡 is the rate of dissipation
( )2 𝒓1 − 𝒓2 = 𝑟21 + 𝑟22 − 2𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ), 15
(16)
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Optics Communications 417 (2018) 14–18
Fig. 1. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the design parameter 𝛽 for different values of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 (a) and the coherent parameter 𝜁𝑠 (b).
we can write the fiber-coupling efficiency as 𝜂𝑐 =
2𝜋 2𝜋 ) [ ( exp −𝑀 𝑟21 + 𝑟22 ∫0 ∫0 ] + 2𝑀𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ) ( 2 ) 𝑟 + 𝑟22 × exp − 1 𝑟1 𝑟2 d𝜑1 d𝜑2 d𝑟1 d𝑟2 . 𝜔2𝑎 𝐷∕2
2
𝜋𝜔2𝑎 𝑨𝑅
∫0
we finally obtain the fiber-coupling efficiency 𝜂𝑐 as ( ) ∞ 𝑨𝑅 2 𝑥22 ( ) 4𝛽 2 exp −𝐻𝑥22 1 𝐹1 1; 1; 𝜂𝑐 = 𝑥2 d𝑥2 , 𝐻 ∫0 𝑨𝐻 2 𝐻
𝐷∕2
∫0
where 1 𝐹1 (⋅) is a confluent hypergeometric function and 𝐻 = 𝑀𝐷2 ∕4 + 𝐷2 ∕4𝜔2𝑎 = 𝑨𝑅 ∕𝑨𝐻 + 𝛽 2 . Adopting the defining method of Ref. [1], 𝑨𝑅 ∕𝑨𝐻 is defined as the number of speckles at the receiver aperture 𝑨, 𝑨𝐻 = 𝜋𝜌2𝐻 stands for the coherence area of 𝑬𝑨 , also known as speckle size, where 𝜌𝐻 denotes the spatial coherent radius, which is given by √ [ ] √ ) ( √ 2𝑁 2 𝑧2 1 2 √ 𝜌𝐻 = 1∕ − +𝑁 1+ . (24) 𝐿(𝑧) 2𝛿02 𝐿(𝑧) 𝑘2 𝑤20 𝐿(𝑧)
(17)
In Eqs. (16) and (17), 𝜑1 and 𝜑2 are the angles between 𝒓1 and 𝒓2 and the centered axis at the propagation plane, respectively. The double integral over the angle variables 𝜑1 and 𝜑2 can be given by [4] 2𝜋
∫0
2𝜋
∫0
[ ] ( ) exp 2𝑀𝑟1 𝑟2 cos(𝜑1 − 𝜑2 ) d𝜑1 d𝜑2 = 4𝜋 2 𝐼0 2𝑀𝑟1 𝑟2 ,
The design parameter 𝛽 is defined as the ratio of the aperture radius to the radius of the backpropagated fiber mode, which is given by [4]
(18)
where 𝐼0 (⋯ ) denotes the modified Bessel function of first kind and zero order. Substituting Eq. (18) into Eq. (17), we arrive at [ ( ) ] 𝐷∕2 𝐷∕2 ) 8𝜋 1 ( 2 2 𝜂𝑐 = exp − 𝑀 + 𝑟1 + 𝑟2 ∫0 𝜔2𝑎 𝑨𝑅 ∫0 𝜔2𝑎 ( ) 𝐼0 2𝑀𝑟1 𝑟2 𝑟1 𝑟2 d𝑟1 d𝑟2 . (19)
𝛽=
(25)
In this section, we present the numerical discussions for the fibercoupling efficiency of the GSM beams propagating through oceanic turbulence as the functions of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 , the ratio of temperature and salinity contributions to the refractive index spectrum 𝜛, the inner scale 𝜂 of oceanic turbulence, the dissipation rate of temperature variance 𝜒𝑡 , the rate of dissipation of kinetic energy per unit mass of fluid 𝜀, the coherent parameter 𝜁𝑠 , the source radius 𝑤0 , the propagation distance 𝑧, the wavelength 𝜆, and the design parameter 𝛽. In order to study the effect of above parameters on the fiber-coupling efficiency exactly, we plot Figs. 1–5 based on the Eq. (20) in the following calculations, and we set the parameters 𝑤0 , 𝐷 as 𝑤0 = 0.04 m, and 𝐷 = 0.1 m. To illustrate the influence of the design parameter 𝛽 on the system performance, we plot the fiber-coupling efficiency of GSM beams as a function of 𝛽 ranging from 0 to 4, for the case of 𝐶𝑚2 = 0, 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−15 m−2∕3 in Fig. 1(a) and 𝜁𝑠 = 1, 𝜁𝑠 = 2, 𝜁𝑠 = 5 in Fig. 1(b). The other parameters of link are set as: 𝑧 = 50 m, 𝜆 = 417 nm, 𝜛 = −3, 𝜁𝑠 = 2 in Fig. 1(a), and 𝐶𝑚2 = 10−17 m−2∕3 in Fig. 1(b). We notice that, in Fig. 1, when the value of 𝛽 increases, the fiber-coupling efficiency increases at first and then decreases for small 𝐶𝑚2 and 𝜁𝑠 , whereas it changes slowly for larger 𝐶𝑚2 and 𝜁𝑠 . It implies that the design parameter 𝛽 has an influence on the fiber-coupling efficiency to some extent, and there is an optimum value of 𝛽 exists under different strength of 𝐶𝑚2 and 𝜁𝑠 . Meanwhile, the maximum 𝜂𝑐 decreases with 𝐶𝑚2 and 𝜁𝑠 increasing, and the optimum value of design parameter 𝛽 is 1.21,1.22, 1.57 for 𝐶𝑚2 = 0, 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−15 m−2∕3 respectively in Fig. 1(a); 1.14, 1.22, 1.36 for 𝜁𝑠 = 1, 𝜁𝑠 = 2, 𝜁𝑠 = 5 respectively in
By defining new variables 𝑥1 = 2𝑟1 ∕𝐷, 𝑥2 = 2𝑟2 ∕𝐷, the 𝜂𝑐 is expressed
(20)
In order to simplify the calculation, we extend the range of the integration from 0 to 1 in Eq. (20) to that from 0 to infinity in the condition of strong oceanic turbulence, and its error less than or equal to 10%. Hence we obtain an approximate expression in the form [ ( ) ] ∞ ∞ ) 𝑀𝐷2 𝐷2 ( 2 2𝐷2 𝜂𝑐 ≈ exp − + 𝑥1 + 𝑥22 4 𝜔2𝑎 ∫0 ∫0 4𝜔2𝑎 ( ) 𝑀𝐷2 𝐼0 𝑥1 𝑥2 𝑥1 𝑥2 d𝑥1 d𝑥2 . (21) 2 Recalling the integral formula [22]: ( ) 𝑥𝛼−1 exp −𝑝𝑥2 𝐼𝜈 (𝑐𝑥) d𝑥 = 𝐴𝛼𝜈 ∫0 [ ] Re 𝑝, Re (𝛼 + 𝜈) > 0; |arg 𝑐| < 𝜋 [ ] ( ) (𝛼 + 𝜈) ∕2 𝛼+𝜈 𝑐2 𝐴𝛼𝜈 = 2−𝜈−1 𝑐 𝜈 𝑝−(𝛼+𝜈)∕2 𝛤 𝐹 ; 𝜈 + 1; , 𝜈+1 1 1 2 4𝑝
𝐷𝜋𝜔0 𝐷 = . 2𝜔𝑎 2𝜆𝑓
3. Analysis of fiber-coupling efficiency
as
[ ( ) ] 1 1 ) 2𝐷2 𝑀𝐷2 𝐷2 ( 2 2 exp − + 𝜂𝑐 = 𝑥1 + 𝑥2 4 𝜔2𝑎 ∫0 ∫0 4𝜔2𝑎 ) ( 𝑀𝐷2 𝑥1 𝑥2 𝑥1 𝑥2 d𝑥1 d𝑥2 . 𝐼0 2
(23)
∞
(22)
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Optics Communications 417 (2018) 14–18
Fig. 2. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the ratio of temperature and salinity contributions to the refractive index spectrum 𝜛 for different values of wavelength 𝜆.
Fig. 3. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the rate of dissipation of mean-square temperature variance 𝜒𝑡 for different values of the rate of dissipation of kinetic energy per unit mass of fluid 𝜀.
Fig. 1(b). We can infer that the maximum fiber-coupling efficiency can be achieved by choosing the appropriate design parameter. Fig. 2 indicates that the fiber-coupling efficiency of GSM beams as a function of temperature and salinity contributions to the refractive index spectrum 𝜛 and the wavelength 𝜆. In Fig. 2, we set the range of 𝜛 changes from −4.5 to −0.5, and the 𝜆 are 417 nm, 488 nm and 532 nm, which the value usually used in other papers [12–15]. The other system parameters of link are set as: 𝛽 = 1.23, 𝑧 = 50 m, 𝜁𝑠 = 2, and 𝐶𝑚2 = 10−17 m−2∕3 . As expected, 𝜂𝑐 decreases as the increasing of 𝜛 but increases as 𝜆 increasing. We also find that the fiber-coupling efficiency has a clear downtrend when the value of 𝜛 is closer to zero. It means that the salinity fluctuation has a greater impact on the fibercoupling efficiency than temperature fluctuation does. On one hand, the increase in salinity will contribute a stronger oceanic turbulence [17] and induce the increase of scintillation. On the other hand, the beam with small wavelength would have stronger scintillation effect [10]. Both of them will reduce the fiber-coupling efficiency. This result is similar to the effect of atmospheric turbulence on the fiber-coupling efficiency [6]. In addition, we know that there is a spectral window in the ocean similar to that present in the atmosphere, which has attenuation of the blue–green light in the 400 nm–580 nm band less than the attenuation of the other bands [23], hence we select the longer 𝜆 in the ‘‘ocean window’’ can improve the fiber-coupling efficiency and the communication performance for other parameters being given. In Fig. 3, we investigate the influence of 𝜀 and 𝜒𝑡 on the fibercoupling efficiency of GSM beams for the case 𝜀 = 10−1 m2 ∕s3 , 𝜀 = 10−3 m2 ∕s3 , and 𝜀 = 10−6 m2 ∕s3 , where 𝜒𝑡 changes from 10−9 K 2 ∕s to 10−8 K 2 ∕s. The other system parameters of link are set as: 𝛽 = 1.55, 𝑧 = 50 m, 𝜁𝑠 = 2, 𝜆 = 417 nm, and 𝜛 = −3. From Fig. 3, we see that the fibercoupling efficiency of GSM beams decreases with the increasing of 𝜒𝑡 but increases as the increasing of 𝜀. We know that 𝜒𝑡 express the strength of the small scale temperature gradient and 𝜀 is inversely proportional to the size of the smallest flow structure, it means that the higher values of 𝜒𝑡 and small value of 𝜀 both will leads to strong oceanic turbulence [9–11]. Consequently, the fiber-coupling efficiency will be degrades. In Fig. 4, the fiber-coupling efficiency of GSM beams is plotted as a function of 𝑧 for 𝜁𝑠 =1, 2, and 5. The other system parameters of link are set as: 𝛽 = 1.23, 𝜂 = 1 mm, 𝜛 = −3, 𝜆 = 417 nm, and 𝐶𝑚2 = 10−17 m−2∕3 . The solid line means that the laser source is fully coherent, and the other two lines represent the partially coherence source which is the GSM beams as referred above. We find that the value of the fiber-coupling efficiency of GSM beams is largest when 𝜁𝑠 = 1, and it decreases with the increasing of the propagating distance 𝑧 and the coherence parameter 𝜁𝑠 . This is for the reason that degrade of the beam coherence make the
Fig. 4. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of propagating distance 𝑧 for different values of the coherent parameter 𝜁𝑠 .
fiber-coupling efficiency decrease, and it is similar to the effect of a beam in atmospheric turbulence [6]. It manifest that the higher spatial coherence of the incident light can improve the fiber-coupling efficiency when the propagation distance is fixed. Fig. 5 presents the curves of the fiber-coupling efficiency of GSM beams as a function of the inner scale 𝜂 ranging from 1 mm to 5 mm for the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 = 10−17 m−2∕3 , 𝐶𝑚2 = 10−16 m−2∕3 , and 𝐶𝑚2 = 10−15 m−2∕3 . The other system parameters of link are set as: 𝛽 = 1.40, 𝑧 = 50 m, 𝜁𝑠 = 2, 𝜛 = −3, and 𝜆 = 417 nm. As we can see that the fiber-coupling efficiency degrades sharply with the increasing of the 𝐶𝑚2 . Therefore, the stronger oceanic turbulence is, the greater effect it has on the fiber-coupling efficiency. In addition, when 𝐶𝑚2 = 10−17 m−2∕3 , the value of 𝜂𝑐 is almost constant, it indicated that we can ignore the effect of the inner scale 𝜂 of oceanic turbulence on the fiber-coupling efficiency in the weak oceanic turbulence. However, with the enhancement of turbulence strength, the influence of internal scale on fiber-coupling efficiency cannot be neglected. 4. Conclusion In this paper, we have molded the fiber-coupling efficiency of the GSM beams propagating through oceanic turbulence. The results showed that the fiber-coupling efficiency of the GSM beams increases as 17
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Fig. 5. The fiber-coupling efficiency 𝜂𝑐 of GSM beams as a function of the inner scale 𝜂 for different values of the ‘‘equivalent’’ temperature structure constant 𝐶𝑚2 .
the increasing of the 𝜆 in the ‘‘ocean window’’, 𝜂, and 𝜀, but decreases as the increasing of the 𝜁𝑠 , 𝑧, 𝐶𝑚2 , 𝜛, and 𝜒𝑡 . There is an optimum design parameter 𝛽 exists under different system source coherent parameter and turbulence conditions, it manifests that we can choose an optimum design parameter and higher source coherent parameter to maximize the fiber-coupling efficiency. The fiber-coupling efficiency of the GSM beams is weak dependence on the inner scale 𝜂 in weak turbulence. The influence of salinity fluctuation on the fiber-coupling efficiency is stronger than the influence of temperature fluctuation for given other parameters. Our results can be beneficial for the design of fiber-coupledbased FSO communication system in oceanic environment. Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51716A) and Postgraduate Research & Practice Innovation Program of Jiangsu Provence (Grant No. SJCX17_0495). The authors would like to thanks reviewers for valuable comments. References [1] P.J. Winze, W.R. Leeb, Fiber coupling efficiency for random light and its applications to lidar, Opt. Lett. 23 (1998) 986–988.
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