Fiber-coupling efficiency of Gaussian Schell model for optical communication through atmospheric turbulence Liying Tan, Mengnan Li,* Qingbo Yang, and Jing Ma National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin 150001, China *Corresponding author: [email protected] Received 24 November 2014; revised 11 February 2015; accepted 15 February 2015; posted 17 February 2015 (Doc. ID 228420); published 16 March 2015

In practice, due to the laser device and the inevitable error of the processing technique, the laser source emitted from the communication terminal is partially coherent, and is represented as a Gaussian Schell model (GSM). The cross-spectral density function based on the Gaussian model in previous research is replaced by the GSM. Thus the fiber-coupling efficiency equation of the GSM laser source through atmospheric turbulence is deduced. The GSM equation presents the effect of the source coherent parameter ζ on the fiber-coupling efficiency, which was not included previously. The effects of the source coherent parameter ζ on the spatial coherent radius and the fiber-coupling efficiency through atmospheric turbulence are numerically simulated and analyzed. The result manifests that the fiber-coupling efficiency invariably degrades with increasing ζ. The work in this paper is aimed to improve the redundancy design of fiber-coupling receiver systems by analyzing the fiber-coupling efficiency with the source coherent parameters. © 2015 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (060.2430) Fibers, single-mode; (010.3310) Laser beam transmission; (060.4510) Optical communications. http://dx.doi.org/10.1364/AO.54.002318

1. Introduction

With the development of the information technique, the demand for higher link capacities is indubitable [1–3]. Laser communication technology has provided many advantages over microwave communication, such as wider bandwidth, larger capacity, more compact terminal, lower power consumption, greater security against eavesdropping, immunity from interference, and no regulatory restrictions for using frequencies and bandwidths. Due to the potential advantages for both commercial and military applications, a host of research has been performed in connection with the study of laser communication technology [2–6].

1559-128X/15/092318-08$15.00/0 © 2015 Optical Society of America 2318

APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015

In order to improve the performance of the laser communication system, a slice of fiber-optic components has been introduced, such as the fiber laser transmitter, erbium-doped fiber amplifier (EDFA), and fiber modem [7,8]. The single-mode fiber (SMF) with an optical preamplifier [9] is a promising solution for laser communication. Adopting auto heterodyne detection technology based on differential phase shift keying (DPSK) has a potential advantage because it poses relatively lax requirements on laser linewidth and promises high spectral efficiency in dense wavelength division multiplexed (DWDM) systems [10,11]. A host of research has focused on the fiber-coupling efficiency of the SMF [12–21]. The fiber-coupling efficiency of an ideally coherent laser source through atmospheric turbulence has been discussed previously, and the results have been applied to lidar and free-space optical communication [12,13]. However,

in practice, the laser emitted is partially coherent, and is represented as Gaussian Schell model (GSM) [22,23]. The effect of the GSM laser source on the fiber-coupling efficiency through atmospheric turbulence has not been studied yet. For the laser communication system (shown in Fig. 1), based on the fiber-coupling technique, the optical signal is eventually coupled into the SMF at the receiver aperture. Besides the effect of the GSM laser source, propagation through the atmosphere turbulence also degrades the matching degree between the focused laser field and the SMF [12,13]. The reasons mentioned above limit the fiber-coupling efficiency and degrade the communication performance. This paper investigates the fiber-coupling efficiency with a GSM laser source. In previous research, the fiber-coupling efficiency equation based on the Gaussian model has been deduced [13]. Replacing the Gaussian model adopted in the cross-spectral density function of the equation in Ref. [13] by GSM, we obtain the new cross-spectral density function. Thus applying the new cross-spectral density function, the fiber-coupling efficiency equation of the GSM laser source through atmospheric turbulence is deduced in this paper. With the variation of the source coherent parameter ζ, the refractive index structure constant C2n , the aperture diameter D, and the propagation distance z, the fiber-coupling efficiency is numerically simulated and the effects of the four parameters on the fiber-coupling efficiency are presented. The structure of this paper is as follows. In Section 2, after reviewing the fiber-coupling efficiency equation in previous research, as a significant improvement, the equation of the GSM laser source is deduced containing the source coherent parameter ζ. In Section 3, the effect of the four parameters mentioned above on the spatial coherent radius is obtained, and the reason behind the variation is explained. In Section 4, the numerical simulation of the fiber-coupling efficiency is performed and the effect on it is discussed in detail. The conclusions are summarized in Section 5. 2. Fiber-Coupling Efficiency of GSM Laser Source A. Review of Fiber-Coupling Efficiency through Atmospheric Turbulence

Propagation through atmosphere turbulence degrades the spatial coherent degree of laser beam and limits the fiber-coupling efficiency [12,13]. As shown in Fig. 1, the optical signal with phase distortion caused by atmosphere turbulence is coupled into a SMF by a receiver telescope, which is equivalent to a thin diffraction-limited lens with focal length f and diameter D located at pupil plane A. The SMF end is placed at plane B, which is the focal plane of telescope A. The fiber-coupling efficiency is defined as the ratio of the average optical power after coupling into the fiber hPc i to the average optical power hPa i at the

EA,FA Atmospheric path

A

EB,FB

B

D/2

2ωa

Single-Mode Fiber GSM laser source wavefront

Distorted x=0 wavefront

x=f

Fig. 1. Illustration of coupling optical signal into single-mode fiber through atmospheric turbulence.

receiver aperture plane. It can be calculated at either incidence pupil plane A or focal plane B shown in Fig. 1 [12,13]. The fiber-coupling efficiency in this paper is calculated at plane A, which is given by [12] 2    RR     A EA rF A rdr  hPc i   ;  ηH  RR hPa i 2 A jEA rj dr

(1)

where EA r denotes the incident optical field at the receiver aperture plane A, F A r is the normalized fiber-mode profile, and F A r is the conjugate of F A r. Factorizing the squared integration in Eq. (1), interchanging the ensemble average and integration, and assuming that the fiber mode is deterministic and the average incidence optical intensity is independent of r, we arrive at [12] ηH 

1 AR

ZZ A

F A r1 F A r2 μA r1 ; r2 dr1 dr2 ;

(2)

where AR  πD2 ∕4 denotes the receiver aperture area. Assuming that the fiber is positioned at the focal plane of the receiver lens and centered on the optical axis to maximize the coupling efficiency, the fiber-mode profile is then given by [21] F A r 

p    2π ωa rπωa 2 exp − ; λf λf

(3)

where λ is the laser wavelength, ωa is the field radius of the fiber mode at the fiber end surface, and f is the focal length of the coupling lens. In Eq. (2), μA r1 ; r2  denotes the complex degree of coherence of the incident optical field at plane A. In Ref. [12], μA r1 ; r2  is deduced based on the Gaussian model, which will be replaced by GSM in the next section. B. Cross-Spectral Density Function of GSM Laser Source

In order to obtain μA r1 ; r2  based on GSM, knowledge of the GSM laser source is required. The crossspectral density function of the GSM laser source at the transmitter is given by [24] 20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS

2319

 2   ρ  ρ2 ρ − ρ 2 Wρ1 ; ρ2 ; 0  I 0 exp − 1 2 2 exp − 1 2 2 ; (4) w0 2δ0 where I 0 is the average optical intensity, w0 is the beam width, and δ20 is the variance of the GSM laser source describing the ensemble average of spatially dependent random phases at the transmitter. Applying the generalized Huygens–Fresnel principle, the cross-spectral density function of the GSM laser source propagating through atmospheric turbulence at propagation distance z is given by [25]  Wr1 ; r2 ; z 

k 2πz

2 ZZ

d2 ρ1

ZZ

d2 ρ2 Wρ1 ; ρ2 ; 0

 r1 − ρ1 2 − r2 − ρ2 2 × exp −ik 2z × hexpψ  r1 ; ρ1 ; z  ψr2 ; ρ2 ; zim ;

(5)

where ρ1 and ρ2 are the horizontal coordinate vectors at plane z  0 and r1 and r2 are the horizontal coordinate vectors at plane z > 0. Wave number k  2π∕λ and symbol h…im denote the ensemble average and integration. For brevity, the mathematical description of Eq. (5), explained in detail in Ref. [25], is not presented here. In Eq. (5), h…im is approximated as [26]

≈ expf−Mρ1 − ρ2 2  ρ1 − ρ2 r1 − r2   r1 − r2 2 g; (6) where M is an expression with the strength of atmospheric turbulence considered. ϕn k is the power spectrum of the refractive index fluctuations. Substituting Eqs. (4) and (6) into Eq. (5), performing several steps of calculations, we arrive at

Wr1 ; r2 ; z 

k 2πz

2 ZZ

d2 ρ1

ZZ

where the parameters presented in Eq. (8) are 8 Lz  1  z2n ζp ; > > > < ζ  1  w2 ∕δ2  2Mw2 p 0 0R 0 > M  0.33π 2 k2 z 0∞ k3 ϕn kdk; > > : 2 θ  w20 δ20 ∕w20  δ20 : Here zn  2z∕kw20 and zn is the normalized distance. Customarily, when the variables r1 and r2 are independent of each other, the third term expikr22 − r21 ∕Jz is not considered in the paper. The GSM laser source is considered in Eq. (8), and ζ is defined as the source coherent parameter, the value of which is equal to 1  w20 ∕δ20 [25]. When ζ  1 the laser source turns to the ideally coherent laser source while the partially coherent laser source is given in other cases. Of note, the source coherent degree becomes worse with increasing ζ. C.

Fiber-Coupling Efficiency of GSM Laser Source

For the quasi-monochromatic field, the complex degree of coherence of the laser field is defined as [26]

d2 ρ2 I 0

× expf−Mρ1 − ρ2 2  ρ1 − ρ2 r1 − r2   r1 − r2 2 g:

(7)

After some algebraic manipulations and integral operation, Eq. (7) is rewritten as APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015

(9)

Substituting Eq. (8) into Eq. (9), the equation is rewritten as  

 1 2 2M 2 z2 μr1 ; r2 ;z  exp − M 1 − 2 2 2 Lz k w0 Lz 2θ Lz  ikr22 − r21  1 : r1 − r2 2 × exp − 2 Jz 2w0 Lz (10) For convenient analysis, we take some algebraic manipulations and define a parameter H presented as follows:   1 2 2M 2 z2 1 M 1 − : − 2 2 H 2 Lz k w0 Lz 2w20 Lz 2θ Lz

 2   ρ  ρ2 ρ − ρ 2 × exp − 1 2 2 exp − 1 2 2 w0 2δ0  2 r1 − ρ1  − r2 − ρ2 2 × exp −ik 2z

2320

   I0 1 2 exp −  M 1  Lz Lz 2θ2 Lz 2 2 2M z r1 − r2 2 − 2 2 k w0 Lz   ikr22 − r21  r  r 2 ; (8) exp × exp − 1 2 2 Jz 2w0 Lz

Wr1 ; r2 ; z p : μr1 ; r2 ; z  p Wr1 ; r1 ; z Wr2 ; r2 ; z

hexpψ  r1 ; ρ1 ; z  ψr2 ; ρ2 ; zim



Wr1 ; r2 ; z 

(11) Thus the complex degree of coherence at receiver aperture A with the source coherent parameter ζ considered is given by μA r1 ; r2   exp−Hr1 − r2 2 :

(12)

Substituting Eqs. (3) and (12) into Eq. (2), the fibercoupling efficiency of the GSM laser source through atmospheric turbulence is expressed as

2 ηH  2 πwa AR



ZZ

 r2  r2 exp−Hr1 − r2   exp − 1 2 2 dr1 dr2 : wa 2

A

(13) We use the law of cosines given by r1 − r2 2  r21  r22 − 2r1 r2 cosφ1 − φ2 ;

(14)

where r1 and r2 are the horizontal coordinate vectors at the propagation plane z > 0. φ1 and φ2 are the angles between r1 and r2 and the centered axis at the propagation plane. The fiber-coupling efficiency ηH is then rewritten as ηH 

2 πw2a AR

Z

D∕2

Z

0

D∕2

Z

0



Z

0

2π 0

exp−Hr21  r22 

 2Hr1 r2 cosφ1 − φ2   2  r1  r22 × exp − r1 r2 dφ1 dφ2 dr1 dr2 : w2a



Z

0



0

exp2Hr1 r2 cosφ1 − φ2 dφ1 dφ2

 4π 2 I 0 2Hr1 r2 ;

(16)

where I 0 … denotes the modified Bessel function of first kind and zero order. Substituting Eq. (16) into Eq. (15), we arrive at ηH 

8π w2a AR

Z

D∕2 0

Z

D∕2 0

   1 exp − H  2 r21  r22  wa

× I 0 2Hr1 r2 r1 r2 dr1 dr2 :

(17)

Normalizing the radial integration variables to the radius of the receiver lens and introducing x1  2r1 ∕D, x2  2r2 ∕D, ηH is expressed as    2D HD2 D2 2 2  x1  x2  exp − ηH  2 4 wa 0 0 4w2a   HD2 x x x x dx dx ; (18) × I0 2 1 2 1 2 1 2 2

Z 1Z

1

(19)

The parameter β representing the coupling geometry is defined as the ratio of the receiver lens radius to the radius of the backpropagated fiber mode, which is given by [13] β

D πDωa :  2λf 2ωa

(20)

Adopting the defining method of Ref. [13], parameter AR  πD2 ∕4 denotes the receiver aperture area and AH denotes the speckle area in a circle. Therefore AR ∕AH is defined as the number of speckles at the receiver aperture A, which is equal to the ratio of the receiver aperture area to the speckle area. Substituting parameter H of Eq. (11) into Eq. (18), we arrive at   1 M 2 M 2 z2 1  −  D2 : Lz 8δ20 Lz 4 2k2 w20 Lz

 (15)

The second term exp2Hr1 r2 cosφ1 − φ2  in the double integral over the angle variables φ1 and φ2 is given by [13] Z

   A exp − R  β2 x21  x22  AH 0 0   A × I 0 2 R x1 x2 x1 x2 dx1 dx2 : AH Z 1Z

ηH  8β2

1

where D is the diameter of the receiver aperture. Finally introducing parameters AR , AH , and β into Eq. (18), the fiber-coupling efficiency of the GSM laser source is given by

AR ∕AH 

(21) We define the value of the speckle area AH as AH  πρ2H . ρH denotes the radius of speckle area AH in a circle, which is called the spatial coherent radius. ρH is given by s    1 2 2M 2 z2 M 1 ; (22) − 2 2 ρH  1∕ Lz k w0 Lz 2δ20 Lz where the parameters presented in Eqs. (21) and (22) are 8 2 < Lz  1  zn ζp ; ζ  1  w20 ∕δ20  2Mw20 R : p M  0.33π 2 k2 z 0∞ k3 ϕn kdk: When the coupling geometry parameter β and the aperture diameter D are fixed in the communication system, in order to obtain the fiber-coupling efficiency of the GSM laser source, AR ∕AH calculated by Eq. (21) needs to be substituted into Eq. (19). 3. Analysis of Spatial Coherent Radius

In order to analyze the fiber-coupling efficiency of the GSM laser source through atmosphere turbulence, the spatial coherent radius ρH obtained by Eq. (22) is discussed in this section. Assume that the Tatarski spectrum is the power spectrum of the refractive index fluctuations, which is given by [27] ϕn k 

0.033C2n k−11∕3

 2 k exp − 2 ; km

20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS

(23)

2321

Symbol

Value

Laser wavelength Propagation distance Wave number Beam width Inner scale of turbulence Outer scale of turbulence Refractive index structure constant

λ z k w0 l0 L0 C2n

Source coherent parameter Receiver aperture diameter

ζ D

1550 nm ≤10 km 2π∕λ 4 cm 5 mm 5m 5 × 10−16 ∼ 1 ×10−14 m−2∕3 1–20 10–50 cm

where k ≥ 1∕L0 and km  5.92∕l0 . C2n is the refractive index structure parameter. In Eq. (22), ρH is affected by the source coherent parameter ζ, the refractive index structure constant C2n , and the propagation distance z. Numerical results are based on the parameters of Table 1, which summarizes all the parameters presented in the numerical simulation. The effect of beam width w0 is not discussed in this paper, which is assumed to be 4 cm. In order to verify the feasibility of Eq. (22), the comparison of the spatial coherent radius ρH obtained by Eq. (22) and ρC obtained by the previous equation [12,13] is plotted in Fig. 2. In the figure, ρH turns out to be the spatial coherent radius of the ideally coherent laser source when ζ  1. ρC denotes the spatial coherent radius obtained by the equation of the ideally coherent laser source in Ref. [13], the expression of which is ρC  1.4572C2n k2 z−3∕5. As shown in Fig. 2, the tendency of the two curves is consistent, which always falls with increasing z. When the propagation z is fixed, the difference value of ρH and ρC is so tiny—almost one percent of the value of ρC —that it can be ignored. Therefore, when ζ  1, the laser source is referred to as ideally coherent and Eq. (22) is assumed to be equivalent to the previous equation of ρC [12,13]. The result is invariably suitable when other parameters are selected in the simulation. When both β and D are fixed, in the case of ζ  1, Eq. (19) is also assumed

200

ζ=1 ζ=2 ζ=5 ζ=17

150 100

Zp

150

ζ=1 ζ=2 ζ=5 ζ=17

125

ρH (mm)

Parameter Description

to be equivalent to the previous equation of ηC in Refs. [12] and [13] because of the same spatial coherent radius obtained, which is not presented here. Compared with the fiber-coupling efficiency equation in Refs. [12] and [13], Eq. (19) presents the effect of the source coherent parameter ζ on the fibercoupling efficiency through the atmosphere. The fiber-coupling efficiency equation obtained in Refs. [12] and [13] is also assumed to be a special case of the fiber-coupling efficiency equation when ζ  1 presented in this paper. In Fig. 3, with the variation of parameters ζ, z and C2n , the spatial coherent radius ρH is presented. The black solid curve shown in Fig. 3 represents the spatial coherent radius ρH when ζ  1, which is referred to as the ideally coherent laser source emitted from the communication terminal. The other dotted curves represent the spatial coherent radius ρH when ζ > 1, which is referred to as the GSM laser source. As shown in Fig. 3(a), when C2n is fixed, the four curves with various values of ζ dispersed at z  0 tend to converge at a point Zp with increasing z. The rule is also suitable to the curves in the other three figures. Meanwhile, in Figs. 3(c) and 3(d), when C2n becomes bigger, the four curves shows different propagation rules before converging. As an example, when ζ  1, the value of ρH invariably decreases with increasing z, while the other three curves show different propagation rules before converging. When C2n  5 × 10−16 m−2∕3 and ζ is set to 2, 5, and 17, respectively, the curves rise before converging and then decline after converging. When C2n  5 × 10−15 m−2∕3 and ζ is set to 5 and 17, respectively, the curves rise before converging and then decline after converging,

H

Simulation Parameters

ρ (mm)

Table 1.

100 75

Zp

50

50 25 0 0

2

4

6

8

0

10

0

2

3

H

100

C

80

24

ζ=1 ζ=2 ζ=5 ζ=17

60 Zp

40

16 0 0

2

4

6

8

3

z (10 m)

0

0

2

4

6

8

10

3

z (10 m) Fig. 2. Spatial coherent radius as a function of z, where ζ  1 and C2n  1 × 10−15 m−2∕3 . 2322

APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015

8

10

ζ=1 ζ=2 ζ=5 ζ=17

40 30

Zp

20

20

8

50

ρH (mm)

ρ

ρH (mm)

Coherent radius (cm)

ρ

32

6

(b) Cn2=1×10-15m-2/3

(a) Cn2=5×10-16m-2/3

40

4

z (103m)

z (10 m)

(c) Cn2=5×10-15m-2/3

10

10

0

2

4

6

8

10

z (103m)

(d) Cn2=1×10-14m-2/3

Fig. 3. ρH as a function of propagation distance z for various values of C2n and ζ. (a) C2n  5 × 10−16 m−2∕3 , (b) C2n  1 × 10−15 m−2∕3 , (c) C2n  5 × 10−15 m−2∕3 , and (d) C2n  1 × 10−14 m−2∕3 .

180

20

A /A

120

H

150

60

10

ζ=17 ζ=5 ζ=2 ζ=1

5

30 0 -16 10

15

R

90

H

ρ (mm)

25

ζ=1 ζ=2 ζ=5 ζ=17

10

-15

10 C2 (m-2/3)

-14

10

n

Fig. 4. ρH as a function of parameter ζ, where z  5 km.

C2n

0 0.1

-13

for various values of coherent

but when ζ is set to 1 and 2, the curves invariably decline. With the increasing C2n shown from Figs. 3(a)–3(d), the converging point Zp decreases from 8000 m [shown in Fig. 3(a)] to 2500 m [shown in Fig. 3(d)]. The variation of ρH is large before the curves with various values of ζ converge at Zp . The explanation behind the variation is discussed in detail later. When z  5 km, with the variation of the refractive index structure constant C2n and the coherent parameter ζ, the spatial coherent radius ρH is presented in Fig. 4. As is shown, the four curves with various values of ζ dispersed at C2n  1 × 10−16 m−2∕3 tend to converge and invariably decline with increasing C2n . The variation of ρH with various values of ζ is large before converging when C2n is small (such as C2n  1 × 10−16 m−2∕3 to C2n  1 × 10−15 m−2∕3 ). The variation of ρH is so tiny that it can be ignored when C2n is large enough, such as when C2n  1 × 10−13 m−2∕3 . The results above indicate that the spatial coherent radius ρH is closely related to the strength of the atmospheric turbulence, the propagation distance, and the coherent parameter. In the figures above, the variation of ρH depends on both the coherent parameter ζ and the refractive index structure constant C2n . In previous research [28], the effect of the coherent parameter ζ on the spatial coherent radius ρH is mainly reflected on the beam broadening, and ρH will invariably rise with the increasing beam broadening. In the absence of turbulence, the beam width is expressed as wz  w20 1  z2n ζ1∕2 [13], where w0 is the beam width at the transmitter and zn  2z∕kw20 is the normalized distance. As is shown, the beam width wz will invariably rise with the increase of the source coherent parameter ζ and the propagation distance z. Meanwhile, the spatial coherent radius ρH also decreases with the increase of the strength of the atmospheric turbulence [28]. When propagating through atmospheric turbulence, both the effects of the beam broadening and atmospheric turbulence work. As is shown from Figs. 2–4, when the effect of the beam broadening is dominant, the spatial coherent

0.2

0.3

0.4

0.5

D (m) Fig. 5. AR ∕AH as a function of D for various values of ζ, where z  5 km and C2n  1 × 10−15 m−2∕3 .

radius ρH increases with the increase of the propagation distance z. But when the effect of atmospheric turbulence is dominant, ρH decreases with the increase of the propagation distance z. 4. Analysis of Fiber-Coupling Efficiency

In Section 2, the fiber-coupling efficiency equation of the GSM laser source through the atmospheric turbulence is given by    AR 2 2 2 exp −  β x1  x2  ηH  8β AH 0 0   A × I 0 2 R x1 x2 x1 x2 dx1 dx2 : AH 2

Z 1Z

1

(24)

In Eq. (24), the fiber-coupling efficiency ηH is mainly affected by the number of speckles AR ∕AH and the coupling geometry β. When β is fixed, AR ∕AH is obtained by Eq. (21). In order to calculate the fiber-coupling efficiency of the GSM laser source, AR ∕AH needs to be substituted into Eq. (24). With the variation of the aperture diameter D and the coherent parameter ζ, the number of speckles AR ∕AH is presented in Fig. 5. As is shown, when z  5 km, C2n  1 × 10−15 m−2∕3 , the four curves with various values of ζ converged at D  10 cm tend to disperse and keep rising with increasing D. The results indicate that the number of speckles AR ∕AH is closely related to the aperture diameter D and the coherent parameter ζ when the propagation distance and the strength of atmospheric turbulence are fixed. In Ref. [12] the optimum value of β is 1.12 and the effect of different β on the fiber-coupling efficiency is not obvious. The maximum coupling efficiency obtained by optimizing β is 0.8145 [12]. Therefore in the section all the parameters in the numerical simulation are the same as illustrated in Table 1. Besides, D  10 cm and β  1.12. With the variation of the aperture diameter D and the coherent parameter ζ, the fiber-coupling efficiency ηH is presented in Fig. 6. As is shown, the fiber-coupling efficiency curves invariably decline with increasing D. When the aperture diameter D 20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS

2323

50

ζ=17 ζ=5 ζ=2 ζ=1

30

H

η (%)

40

20 10 0 0.1

0.2

0.3

0.4

0.5

D (m) Fig. 6. ηH as a function of D for various values of ζ, where z  5 km and C2n  1 × 10−15 m−2∕3 .

is fixed, ηH degrades with increasing ζ and the four curves tend to converge with increasing D. With the variation of parameters ζ, z and C2n , the fiber-coupling efficiency ηH is presented in Fig. 7. As is shown, when z and C2n are fixed, the value of ηH invariably decreases with increasing ζ. With the increasing C2n shown from Figs. 7(a) to 7(d), the value of ηH decreases slower. In practice, the best fiber-coupling efficiency is in need. As shown in Figs. 7(a)–7(d), when z  2.5 km, the value of ηH is not invariably the best with the variation of ζ. As an example, in Fig. 7(a) compared with the other three curves, the value of ηH at z  2.5 km is bigger than the others when ζ is less than 2, but smaller when ζ is greater than 2. In Fig. 7(b), the value of ηH at z  2.5 km is bigger than that at z  10 km when ζ is less than 6, but smaller when ζ is greater than 6. In the remaining figures, when the effect of atmospheric turbulence is dominant, the z=2.5km z=5.0km z=7.5km z=10km

65

ηH (%)

60 55 50

55

z=2.5km z=5.0km z=7.5km z=10km

50

ηH (%)

70

45

45 40

40 35

1

5 10 15 Coherent parameter ζ

35

20

1

(a) Cn2=5×10-16m-2/3 z=2.5km z=5.0km z=7.5km z=10km

ηH (%)

30 25 20 15 10

20

(b) Cn2=1×10-15m-2/3 30

z=2.5km z=5.0km z=7.5km z=10km

25

ηH (%)

35

5 10 15 Coherent parameter ζ

20 15 10

1

5 10 15 Coherent parameter ζ

(c) Cn2=5×10-15m-2/3

20

5

1

5 10 15 Coherent parameter ζ

20

(d) Cn2=1×10-14m-2/3

Fig. 7. ηH as a function of ζ for various values of C2n and z, where D  10 cm. (a) C2n  5 × 10−16 m−2∕3 , (b) C2n  1 × 10−15 m−2∕3 , (c) C2n  5 × 10−15 m−2∕3 , and (d) C2n  1 × 10−14 m−2∕3 . 2324

APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015

value of ηH at z  2.5 km invariably shows better performance than the others [shown in Figs. 7(c) and 7(d)]. The results are obtained when the receiver aperture is fixed for D  10 cm, which is not considered for other cases. The best fiber-coupling efficiency can be obtained on the specific engineering issue in detail. The result indicates that the fiber-coupling efficiency ηH is closely related to the strength of the atmosphere turbulence, the propagation distance, and the coherent parameter. From previous research we performed, increasing the coherent parameter ζ, increasing of the strength of atmospheric turbulence, or applying a larger receiver aperture will seriously degrade the fiber-coupling efficiency and the communication performance. 5. Conclusion

In this paper, the cross-spectral density function is given based on the GSM instead of the Gaussian model. The fiber-coupling efficiency of the GSM laser source through atmospheric turbulence is deduced by applying the cross-spectral density function based on GSM. Compared with the previous equation, the equation based on GSM presents the effect of the source coherent parameter on the fiber-coupling efficiency. The spatial coherent radius and the fibercoupling efficiency with various values of the parameters proposed in Table 1 (Section 3) are analyzed. The results show that the spatial coherent radius is mainly related to the coherent parameter, the refractive index structure constant, and the propagation distance. When the coupling geometry parameter is fixed, the fiber-coupling efficiency is also affected by the parameters mentioned above and the aperture diameter. When the effect of the beam broadening is dominant, the spatial coherent radius and the fiber-coupling efficiency invariably rise with the increase of the propagation distance. But when the effect of atmospheric turbulence is dominant, the two parameters mentioned above invariably decline with the increase of the propagation distance. The best fiber-coupling efficiency can be obtained on the specific engineering issue in detail. In the presence of degradation of the source coherent degree, a higher incidence optical power is needed to obtain the desired optical power after coupling into the fiber. The resulting degradation of the fibercoupling efficiency is a significant part of the freespace laser communication link’s power budget and needs to be taken into account for links that use a fiber-coupling receiver. We hope to use this work to analyze the effect of the fiber-coupling efficiency and improve the redundancy design of the fibercoupling receiver system. Experimental research concerning the fibercoupling efficiency of the GSM laser source will be performed in the near future. A spatial light modulator (SLM) will be applied to generate a GSM laser source. With a SMF at the receiver, measurement of the fiber-coupling efficiency of the GSM laser source

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20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS

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Fiber-coupling efficiency of Gaussian Schell model for optical communication through atmospheric turbulence.

In practice, due to the laser device and the inevitable error of the processing technique, the laser source emitted from the communication terminal is...
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