Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence Yongping Huang,1,2 Bin Zhang,1,* Zenghui Gao,2 Guangpu Zhao,2 and Zhichun Duan2 2
1 College of Electronics Information, Sichuan University, Chengdu, Sichuan 610064, China Computational Physics Key Laboratory of Sichuan Province, Physics Institute, Yibin University, Yibin, Sichuan 644007, China * [email protected]
Abstract: The analytical expressions for the cross-spectral density and average intensity of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence are obtained by using the extended Huygens–Fresnel principle and the spatial power spectrum of the refractive index of ocean water. The evolution behavior of GSM vortex beams through oceanic turbulence is studied in detail by numerical simulation. It is shown that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence including the rate of dissipation of turbulent kinetic energy per unit mass of fluid, rate of dissipation of mean-square temperature, relative strength of temperature salinity fluctuations, and beam parameters including the spatial correlation length and topological charge of the beams, as well as the propagation distance. © 2014 Optical Society of America OCIS codes: (010.4455) Oceanic propagation; (260.6042) Singular optics; (010.3310) Laser beam transmission; (030.0030) Coherence and statistical optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008). T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008). J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011). J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009). X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011). Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014). X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013). R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978). V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007). O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010). O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011). E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011). O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012). N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
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16. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006). 17. J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014). 18. Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012). 19. M. Tang and D. Zhao, “Spectral changes in stochastic anisotropic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014). 20. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). 21. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007). 22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 23. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003). 24. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
1. Introduction Recently, much attention has been paid to the laser beams with phase singularities propagating through ideal atmospheric turbulence and non-Kolmogorov turbulence [1–7]. It was shown that the spreading of partially coherent vortex beam is less affected by atmospheric turbulence than partially coherent non-vortex, and the topological charge can be used as the information carrier in optical communication [1, 2]. The intensity distribution, coherent vortex evolution and orbital angular momentum entangled states of vortex beam in non-Kolmogorov turbulence were investigated in [5–7]. On the other hand, the oceanic turbulence is another important random medium, but the turbulent properties of the underwater turbulence are determined by temperature and salinity fluctuation of the oceanic water [8–10]. It was shown that the degree of polarization, coherence, spectral changes, scintillation index and intensity distributions of electromagnetic beams through oceanic turbulence are affected by oceanic turbulence [11–19]. This paper is devoted to study the evolution behavior of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence. The analytical expressions for the oceanic turbulence quantity T (η , ε , χT , ω ) and the cross-spectral density as well as average intensity of GSM vortex beams through oceanic turbulence are derived in Sec.2 and Sec.3. Then, Sec.3 and Sec.4 illustrate the evolution behavior of GSM vortex beams through oceanic turbulence numerically. Finally, the main conclusion is drawn in Sec.5. 2. The cross-spectral density of Gaussian Schell-model vortex beams through oceanic turbulence In Cartesian coordinates the field distribution of a vortex beam at the z = 0 source plane is expressed as m
U ( ρ, z = 0) = u ( ρ)[ ρx + isgn(m) ρ y ] ,
(1)
where u(ρ) represents the profile of the background beam envelope, ρ≡(ρx, ρy) is two-dimensional coordinate vectors at the plane z = 0, sgn(.) is the sign function, m is the topological charge. The cross-spectral density of GSM vortex beams at the plane z = 0 is written as W ( 0) ( ρ1 , ρ2 , 0) = [( ρ1x ρ 2 x + ρ1 y ρ 2 y ) + i sgn ( m ) ( ρ1x ρ 2 y − ρ 2 x ρ1 y )]m 2
ρ −ρ ρ2 + ρ2 × exp − 1 2 2 exp(− 1 2 2 ). (2) w 2σ 0 0 In Eq. (2) ρ1 = (ρ1x, ρ1y) and ρ2 = (ρ2x, ρ2y) are positions of two points at the plane z = 0, respectively. In this paper the m is assumed to be ± 1, in the case for m = 0, Eq. (2) degenerates
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17724
into the cross spectral density of GSM non-vortex beam, w0 is the waist width for the Gaussian part, σ0 is spatial correlation length. According to the extended Huygens–Fresnel principle, the cross-spectral density function at the z plane (z>0) of GSM vortex beams through oceanic turbulence is written as W ( ρ1′, ρ2′ , z ) = (
k
) 2 d 2 ρ1 d 2 ρ2W ( 0) ( ρ1 , ρ2 , 0)
2π z 2 2 ik × exp − ( ρ1′ − ρ1 ) − ( ρ2′ − ρ2 ) 2z ∗ × exp[ψ ( ρ1′, ρ1 ) + ψ ( ρ2′ , ρ2 )] ,
(3)
where k denotes the wave number related to the wavelength by k = 2π/λ. ρ1′ and ρ2′ are the positions of two points at the z plane, respectively. The asterisk * specifies the complex conjugate, ⋅ represents the ensemble average. ψ(ρ'i, ρi) (i = 1, 2) represents the random part of the complex phase of a spherical wave due to the existence of the turbulence, whose ensemble average of the turbulent ocean water can be expressed as [20] exp[ψ ∗ ( ρ1′, ρ1 ) + ψ ( ρ2′ , ρ2 )]
= exp {− 4π 2 k 2 z
1
0
∞
0
)}
(
d κ d ξκΦn (κ ) 1 − J 0 κ (1 − ξ )( ρ1′ − ρ2′ ) + ξ ( ρ1 − ρ2 )
π 2k 2 z ∞ 3 2 2 = exp − κ Φn (κ )d κ ( ρ1′ − ρ2′ ) + ( ρ1′ − ρ2′ )( ρ1 − ρ2 ) + ( ρ1 − ρ2 ) 0 3
{
}
2 2 = exp −k 2 zT (η , ε , χT , ω ) ( ρ1′ − ρ2′ ) + ( ρ1′ − ρ2′ )( ρ1 − ρ2 ) + ( ρ1 − ρ2 )
, (4)
where J0 (.) is the Bessel function of the first kind and zero order, and T (η , ε , χT , ω ) =
π2 3
∞
0
κ 3 Φn (κ ) d κ ,
(5)
denoting the strength of the oceanic turbulence. Φn(κ) in Eq. (5) is the spatial power spectrum of the refractive-index fluctuations of the turbulent ocean water. For the sake of simplicity, we consider the influence of oceanic turbulence due to temperature and salinity fluctuations on the beam, using the spatial power spectrum of oceanic turbulence model in [9, 11]
χT [1 + 2.35(κη ) 2/3 ] ω2 − 2ω e − A δ ) ,
Φ n (κ ,η , ε , χT , ω ) = 0.388 × 10 −8 ε −1/3κ −11/3 × (ω 2 e − AT δ + e − AS δ
TS
(6)
where
δ = 8.284(κη ) 4/3 + 12.978(κη ) 2 ,
(7)
η is the Kolmogorov micro scale, ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in range from 10−4m2/s3 to10−10m2/s3, χT being the rate of dissipation of mean-square temperature, taking values in the range from 10−4K2/s to 10−10K2/s [10]. ω is the relative strength of temperature and salinity fluctuation, which in the ocean water the value can range from 0 to −5, the minus sign of the parameter ω denotes that there is a reduction in temperature and an increase in salinity with depth. 0 corresponding to the case
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17725
when temperature-driven turbulence dominates, −5 corresponding to the situation when salinity-driven turbulence prevails [10]. Substituting Eqs. (6) and (7) and AT = 1.863 × 10−2 , AS = 1.9 × 10−4 as well as
ATS = 9.41× 10−3 [14] into Eq. (5), Eq. (5) can be simplified to T (η , ε , χT , ω ) = 1.2765 × 10−8 ω −2ε −1/3η −1/3 χT (47.5708 − 17.6701ω + 6.78335ω 2 ). (8)
Obviously, the T (η , ε , χT , ω ) increases with the increasing χT and decreasingω, ε and η. Introducing two variables of integration s = (ρ1 + ρ2)/2, t = ρ1-ρ2 and with Eqs. (4)–(8) substituting into Eq. (3), we can obtain ik ( ρ1′2 − ρ2′2 )]exp[−T (η , ε , χT , ω ) k 2 z ( ρ1′ − ρ2′ ) 2 ] 2π z 2z t2 × d 2 s d 2 t s 2 − i ( sx t y − s y t x ) 4
W ( ρ1′, ρ2′ , z ) = (
k
) 2 exp[−
2 ik ik × exp −at 2 − st exp − 2 s 2 exp ( ρ1′ − ρ2′ ) s z w z 0 ik × exp t ( ρ1′ + ρ2′ ) − T (η , ε , χT , ω ) k 2 z ( ρ1′ − ρ2′ ) z 2
, (9)
where a=
1 1 1 ( 2 + 2 ) + k 2 zT (η , ε , χT , ω ) , 2 w0 σ 0
(10)
sx = (ρ1x + ρ2x)/2, tx = ρ1x-ρ2x, sy = (ρ1y + ρ2y)/2, ty = ρ1y-ρ2y, the symbol ‘ ’ corresponds to m = ± 1 at the source plane z = 0. Using the integral formulae [21] q2 π ) , p p
(11)
q2 π q ) ( ), p p p
(12)
2 exp(− px + 2qx)dx = exp(
x exp(− px
2
+ 2qx)dx = exp(
2 2 x exp(− px + 2qx)dx =
q2 π 1 2q 2 exp( ) (1 + ), p p p 2p
(13)
after tedious but straightforward integral calculations, we deliver k 2 ik ) exp[− ( ρ1′2 − ρ2′2 )] 2π z 2z 2 × exp − k 2 zT (η , ε , χT , ω ) ρ1′ − ρ2′ × ( N1 − N 2 ) ( N 3 − N 4 ) , (14)
W ( ρ1′, ρ2′ , z ) = (
where N1 =
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π2 aC
Ex E y exp(
Fx2 + Fy2 C
)(
Fx2 + Fy2 C
2
+
1 ), C
(15)
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17726
N2 =
N3 =
N4 =
π 2 w02 8D
exp(−
iπ 2 w0 2aCD iπ 2 w0 2aCD
Ex
Ey
Gx2 + Gy2 Gx2 + G y2 1 k 2 w02 2 ′ ′ − × + ), ) exp( )( ρ ρ 1 2 8z 2 D D2 D
(16)
F 2 Gy2 2 k 2 w2 exp − 20 ( ρ1′y − ρ 2′ y ) exp x + , C CD D 8z
(17)
Fy2 Gx2 k 2 w02 2 ′ ′ − + ρ ρ exp − exp , ( ) 1x 2x 2 CD D 8z C
(18)
Fx Gy
Fy Gx
ik , z
(19)
2 b2 − , w02 4a
(20)
b 2 w02 , 8
(21)
b=
C=
D = a−
2 1 ik Ex = exp ( ρ1′x + ρ 2′ x ) − T (η , ε , χT , ω ) k 2 z ( ρ1′x − ρ 2′ x ) , 4a 2 z
(22)
1 ik ikb bk 2 Fx = [ ( ρ1′x − ρ2′ x ) − ( ρ1′x + ρ2′ x ) + T (η , ε , χT , ω ) ( ρ1′x − ρ2′ x )], 2 z 4az 2a
(23)
b 2 w02 1 ik ( ρ1′x − ρ2′ x )]. (24) Gx = [ ( ρ1′x + ρ2′ x ) − T (η , ε , χT , ω ) k 2 z ( ρ1′x − ρ2′ x ) − 2 2z 4 Using of the symmetry, Ey, Fy and Gy are obtained from Ex, Fx and Gx by replacing ρ′1x and ρ′2x with ρ′1y and ρ′2y, respectively. Equations (14)–(24) provide the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 propagating through oceanic turbulence. It follows from Eq. (14) that the cross–spectral density of GSM vortex beams depends on the oceanic turbulence and beam parameters such as the Kolmogorov micro scaleη, rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, rate of dissipation of mean-square temperatureχT, relative strength of temperature salinity fluctuation ω, spatial correlation length σ0, waist width w0, sign and values of topological charge m, as well as the propagation distance z, and the positions of two points at the z plane. In addition, for the case of T(η, ε, χT, ω) = 0 in Eqs. (14)–(24), which can simplify to the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 in free space.
3. The evolution behavior of average intensity of Gaussian Schell-model vortex beams through oceanic turbulence
Substituting ρ1′ = ρ2′ = ρ′ into Eq. (14), the average intensity at the z plane is expressed as
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17727
I ( ρ ′ , z ) = W ( ρ ′ , ρ′ , z ) =(
t2 ) 2 d 2 s d 2 t[ s 2 − i ( sx t y − s y t x )] 2π z 4 k
2 ik ik × exp −at 2 − st exp − 2 s 2 exp( ρ′t ), z z w0
(25)
after straightforward integral calculations deliver the expression k 2 ρ ′2 1 + ) 2 2 C 2π z aC 4az C π 2 w02 ik ρ ′2 1 k2 ik ρ ′2 1 2 ′ ( 1)] ( + ) exp( )}. ρ (26) × exp[ − − C 4az 2 8D 2 zD 2 D 2 zD Eq. (26) is the analytical expression of a GSM vortex in oceanic turbulence average intensity, Eq. (26) together with Eqs. (8), (10), (19)–(21) imply that the average intensity depends on correlation length σ0, waist width w0, topological charge |m|, and parameter T(η, ε, χT, ω) (including the Kolmogorov micro scaleη, the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, the rate of dissipation of mean-square temperatureχT and the relative strength of temperature and salinity fluctuation ω), but is independent of the sign of m. It’s worth noting that Eq. (26) reduces to the analytical expression of a GSM vortex in free space for the case of T(η, ε, χT, ω) = 0 in Eq. (10), namely, I ( ρ′ , z ) = (
k
)2 {
π2
(
a′ =
1 1 1 ( + ), 2 w02 σ 02
(27)
2 b2 − , w02 4a ′
(28)
b 2 w02 , 8
(29)
C′ =
D′ = a ′ −
k 2 ρ ′2 1 + ) 2π z a ′C ′ 4a ′z 2 C ′2 C ′ π 2 w02 ik ρ ′2 1 k2 ik ρ ′2 1 × exp[ ( + ) exp( )}, ρ ′2 ( − 1)] − 2 2 C′ 4a ′z 8D ′ 2 zD′ D′ 2 zD′
I ′( ρ′ , z ) = (
k
)2 {
π2
(
(30)
which is consistent with the Eq. (9) in [4] for the case of Cn2 = 0. Illustration numerical examples are given in Figs. 1(a)–1(e), which plot normalized intensity distributions I/Imax (Imax—maximum intensity) of a GSM vortex beam versus the slanted axis ρ′r ( ρ r′ = ρ x′2 + ρ y′2 ) for different values of (a) propagation distance z, (b) relative strength of temperature and salinity fluctuation ω, (c) rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (d) rate of dissipation of mean-square temperature χT, (e) spatial correlation length σ0, where the calculation parameters w0 = 3cm, λ = 1060nm and η = 10−3m are fixed in every figure, while the rest parameters including σ0 = 2.5cm, ω = −2.5, ε =
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17728
Fig. 1. Evolution of normalized intensity profiles of GSM vortex beams propagating in oceanic turbulence.
10−7 m2/s3, χT = 10−9 K2/s and z = 300m are allowed to vary in different figures. As can be seen from Fig. 1(a), the average intensity of GSM vortex beams propagating through oceanic turbulence undergoes several stages of evolution like as GSM vortex beams propagating through ideal Kolmogorov atmospheric turbulence [4]. At the z = 0 plane there exists a zero intensity at the center ρ′r = 0, where the phase becomes singular. With increasing propagation distance to zflat = 560m, a flat-topped intensity profile occurs, and finally evolves into a Gaussian one at zGau = 700m. In Figs. 1(b)–1(e), we can see that at the given propagation distance (e.g., z = 300m), the evolution behavior of average intensity of GSM vortex beams in
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oceanic turbulence is significantly affected by the oceanic turbulence parameters includingω, ε and χT, and beam parameter σ0. From Figs. 1(a)–1(e), it is interesting to find that the beam profile will first take on a hollow shape, and then approach a Gaussian distribution, for example, when ω = −0.5 in Fig. 1(b), ε = 10−10 m2/s3in Fig. 1(c), χT = 10−7 K2/s in Fig. 1(d) and σ0 = 1cm, the I/Imax of a GSM vortex beam in oceanic turbulence extend to a Gaussian profile. The larger χT, the smallerω, ε and spatial correlation length and the longer propagation distance z are, the faster the change of the beam profile is. We can see from Eq. (8) and Figs. 2(a) and 2(b) (the calculation parameters are the same as those of Fig. 1) that the larger χT and the smallerω and ε are, the stronger the oceanic turbulence is. Therefore, Figs. 1(a)–1(e) indicate that the evolution behavior of intensity depends on the oceanic turbulence strength, the beam coherent degree and the propagation distance. At the same time, it is shown that the larger χT and z or the smaller ω and ε are, the larger the spreading of the beam width is, but the effect on the beam width by the spatial correlation length is not significant for the GSM vortex beams through oceanic turbulence.
Fig. 2. T(η,ω,ε,χT) versus the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (a) for different values of the rate of dissipation of mean-square temperatureχT and (b) for different values of the relative strength of temperature and salinity fluctuation ω.
4. The evolution behavior of coherent vortices of Gaussian Schell-model vortex beams through oceanic turbulence
The spectral degree of coherence is defined as [22]
μ ( ρ1′, ρ2′ , z ) =
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W ( ρ1′, ρ2′ , z ) , [ I ( ρ1′, z ) I ( ρ2′ , z )]1/ 2
(31)
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17730
and the position of coherence vortices is determined by [23] Re[ μ ( ρ1′, ρ2′ , z )] = 0,
(32)
Im[ μ ( ρ1′, ρ2′ , z )] = 0.
(33)
where Re and Im stand for the real and imaginary parts of μ ( ρ1′, ρ2′ , z ) , respectively. The topological charge and its sign of coherence vortices are determined by the sign principle [24]. Figure 3 gives curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 through oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m, where ρ′1 = (6cm, 9cm), m = + 1, σ0 = 2.5cm are kept fixed, and the other calculation parameters are the same as in Fig. 1. Figure 4 plots contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m, where the calculation parameters are the same as in Fig. 3. Figures 3(a), 3(b), 4(a), and 4(b) indicate that there exists a coherent vortex A with m = + 1 at z = 50m whose position is (0.3783cm,-0.1178cm) and two coherent vortices A and B with m = + 1, −1 appear at z = 220m. The singularity A moves to the position (3.27cm, 0.2321cm), while a new singularity B with m = −1 appears at the position (−7.351cm, 11.88cm). Therefore, the position and number of coherent vortices change with increasing propagation distance through oceanic turbulence.
Fig. 3. Curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 in oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m.
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17731
Fig. 4. Contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m.
Figure 5 plots that the position and number of coherent vortices of a GSM vortex beam with m = + 1 through oceanic turbulence for different values ofω,ε, χT, where z = 300m is kept fixed, the other calculation parameters are same as those of Fig. 1, the white and black dots represent m = −1 and m = + 1, respectively. It can be seen that there appears only a coherent vortex with m = + 1 and the topological charge is conserved within a certain propagation distance. But as the propagation distance increases, two coherent vortices with m = + 1 and m = −1 appear, whose positions change and gradually move closer. From Figs. 5(a)–5(c), we can see that the larger ω is, the larger the distance zc for the conservation of the topological charge. It can be seen from Figs. 5(b), 5(d) and 5(e) that the larger ε is, the larger zc is, for the case of ε = 10-10, 10−7 and 10−5, zc corresponding to 70m, 150m and 250m, respectively. The position and number of coherent vortices of GSM vortex beams with m = + 1 propagation through oceanic turbulence for different values of χT are plotted in Figs. 5(b), 5(f) and 5(g). We can see that the position and number of coherent vortices change with propagation distance, and two coherent vortices with m = + 1 and m = −1 appear at z = 150m in Fig. 5(b), 70m in Fig. 5(f) and 40m in Fig. 5(g) respectively. Obviously, the smaller the χT is, the larger the distance zc for the conservation of the topological charge, which indicates that the distance for the conservation of the topological charge increases with increasingω, ε and decreasingχT. The physical reason can be seen from Fig. 2 that the largerω, ε and smaller χT are, the smaller the value of T(η, ε, χT, ω) is, which indicates that the weaker the oceanic turbulence is, the less effect on the conservation of the topological charge by the weaker oceanic turbulence. And then the positions of two singularities constantly change and the separation distance between them gradually becomes smaller, and this condition is more significant for the larger the rate of dissipation of mean-square temperature χT in Fig. 5(g).
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17732
Fig. 5. Position and number of coherent vortices of a GSM with m = + 1 propagation through oceanic turbulence for different values of ω, ε, χT.
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17733
5. Concluding remarks
In this paper, we have studied the evolution behavior of coherent vortices and average intensity of GSM vortex beams propagating through oceanic turbulence in detail. It has been shown analytically and numerically that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence and beam parameters, including the rate of dissipation of turbulent kinetic energy per unit mass of fluidε, rate of dissipation of mean-square temperatureχT, relative strength of temperature salinity fluctuationsω, spatial correlation length and topological charge of the beams, as well as the propagation distance. The larger χT, the smaller ω, ε and the shorter spatial correlation length and the longer propagation distance z are, the faster the evolution of the beam intensity profile is. The smaller χT and the larger ω and ε are, the longer the distance appearing two coherent vortices with m = + 1 and m = −1, namely, the larger the distance zc for the conservation of the topological charge. The results obtained in this paper would be useful for potential applications of vortex beams in oceanic optical communications. Acknowledgments
This work is supported by the National Natural Science Foundation of China under grant No.61275203, National Natural Science Foundation of China and Civil Aviation Administration of China under grant No. 61079023. A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department under grant No.14ZA0268, the Program of Yibin Municipal Science and Technology Bureau under grant No.2013SF020, and the Scientific Research Project of Yibin University under grant No. 2013YY04.
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17734
1 College of Electronics Information, Sichuan University, Chengdu, Sichuan 610064, China Computational Physics Key Laboratory of Sichuan Province, Physics Institute, Yibin University, Yibin, Sichuan 644007, China * [email protected]
Abstract: The analytical expressions for the cross-spectral density and average intensity of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence are obtained by using the extended Huygens–Fresnel principle and the spatial power spectrum of the refractive index of ocean water. The evolution behavior of GSM vortex beams through oceanic turbulence is studied in detail by numerical simulation. It is shown that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence including the rate of dissipation of turbulent kinetic energy per unit mass of fluid, rate of dissipation of mean-square temperature, relative strength of temperature salinity fluctuations, and beam parameters including the spatial correlation length and topological charge of the beams, as well as the propagation distance. © 2014 Optical Society of America OCIS codes: (010.4455) Oceanic propagation; (260.6042) Singular optics; (010.3310) Laser beam transmission; (030.0030) Coherence and statistical optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008). T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008). J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011). J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009). X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011). Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014). X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik (Stuttg.) 124(17), 2635–2638 (2013). R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. A 68(8), 1067–1072 (1978). V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuation of the sea-water refractive index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2007). O. Korotkova and N. Farwell, “Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence,” Proc. SPIE 7588, 75880S (2010). O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011). E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105(2), 415–420 (2011). O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Wave Random Complex 22(2), 260–266 (2012). N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012).
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16. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006). 17. J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014). 18. Y. Zhou, K. Huang, and D. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109(2), 289–294 (2012). 19. M. Tang and D. Zhao, “Spectral changes in stochastic anisotropic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014). 20. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). 21. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007). 22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 23. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003). 24. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
1. Introduction Recently, much attention has been paid to the laser beams with phase singularities propagating through ideal atmospheric turbulence and non-Kolmogorov turbulence [1–7]. It was shown that the spreading of partially coherent vortex beam is less affected by atmospheric turbulence than partially coherent non-vortex, and the topological charge can be used as the information carrier in optical communication [1, 2]. The intensity distribution, coherent vortex evolution and orbital angular momentum entangled states of vortex beam in non-Kolmogorov turbulence were investigated in [5–7]. On the other hand, the oceanic turbulence is another important random medium, but the turbulent properties of the underwater turbulence are determined by temperature and salinity fluctuation of the oceanic water [8–10]. It was shown that the degree of polarization, coherence, spectral changes, scintillation index and intensity distributions of electromagnetic beams through oceanic turbulence are affected by oceanic turbulence [11–19]. This paper is devoted to study the evolution behavior of Gaussian Schell-model (GSM) vortex beams propagating through oceanic turbulence. The analytical expressions for the oceanic turbulence quantity T (η , ε , χT , ω ) and the cross-spectral density as well as average intensity of GSM vortex beams through oceanic turbulence are derived in Sec.2 and Sec.3. Then, Sec.3 and Sec.4 illustrate the evolution behavior of GSM vortex beams through oceanic turbulence numerically. Finally, the main conclusion is drawn in Sec.5. 2. The cross-spectral density of Gaussian Schell-model vortex beams through oceanic turbulence In Cartesian coordinates the field distribution of a vortex beam at the z = 0 source plane is expressed as m
U ( ρ, z = 0) = u ( ρ)[ ρx + isgn(m) ρ y ] ,
(1)
where u(ρ) represents the profile of the background beam envelope, ρ≡(ρx, ρy) is two-dimensional coordinate vectors at the plane z = 0, sgn(.) is the sign function, m is the topological charge. The cross-spectral density of GSM vortex beams at the plane z = 0 is written as W ( 0) ( ρ1 , ρ2 , 0) = [( ρ1x ρ 2 x + ρ1 y ρ 2 y ) + i sgn ( m ) ( ρ1x ρ 2 y − ρ 2 x ρ1 y )]m 2
ρ −ρ ρ2 + ρ2 × exp − 1 2 2 exp(− 1 2 2 ). (2) w 2σ 0 0 In Eq. (2) ρ1 = (ρ1x, ρ1y) and ρ2 = (ρ2x, ρ2y) are positions of two points at the plane z = 0, respectively. In this paper the m is assumed to be ± 1, in the case for m = 0, Eq. (2) degenerates
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17724
into the cross spectral density of GSM non-vortex beam, w0 is the waist width for the Gaussian part, σ0 is spatial correlation length. According to the extended Huygens–Fresnel principle, the cross-spectral density function at the z plane (z>0) of GSM vortex beams through oceanic turbulence is written as W ( ρ1′, ρ2′ , z ) = (
k
) 2 d 2 ρ1 d 2 ρ2W ( 0) ( ρ1 , ρ2 , 0)
2π z 2 2 ik × exp − ( ρ1′ − ρ1 ) − ( ρ2′ − ρ2 ) 2z ∗ × exp[ψ ( ρ1′, ρ1 ) + ψ ( ρ2′ , ρ2 )] ,
(3)
where k denotes the wave number related to the wavelength by k = 2π/λ. ρ1′ and ρ2′ are the positions of two points at the z plane, respectively. The asterisk * specifies the complex conjugate, ⋅ represents the ensemble average. ψ(ρ'i, ρi) (i = 1, 2) represents the random part of the complex phase of a spherical wave due to the existence of the turbulence, whose ensemble average of the turbulent ocean water can be expressed as [20] exp[ψ ∗ ( ρ1′, ρ1 ) + ψ ( ρ2′ , ρ2 )]
= exp {− 4π 2 k 2 z
1
0
∞
0
)}
(
d κ d ξκΦn (κ ) 1 − J 0 κ (1 − ξ )( ρ1′ − ρ2′ ) + ξ ( ρ1 − ρ2 )
π 2k 2 z ∞ 3 2 2 = exp − κ Φn (κ )d κ ( ρ1′ − ρ2′ ) + ( ρ1′ − ρ2′ )( ρ1 − ρ2 ) + ( ρ1 − ρ2 ) 0 3
{
}
2 2 = exp −k 2 zT (η , ε , χT , ω ) ( ρ1′ − ρ2′ ) + ( ρ1′ − ρ2′ )( ρ1 − ρ2 ) + ( ρ1 − ρ2 )
, (4)
where J0 (.) is the Bessel function of the first kind and zero order, and T (η , ε , χT , ω ) =
π2 3
∞
0
κ 3 Φn (κ ) d κ ,
(5)
denoting the strength of the oceanic turbulence. Φn(κ) in Eq. (5) is the spatial power spectrum of the refractive-index fluctuations of the turbulent ocean water. For the sake of simplicity, we consider the influence of oceanic turbulence due to temperature and salinity fluctuations on the beam, using the spatial power spectrum of oceanic turbulence model in [9, 11]
χT [1 + 2.35(κη ) 2/3 ] ω2 − 2ω e − A δ ) ,
Φ n (κ ,η , ε , χT , ω ) = 0.388 × 10 −8 ε −1/3κ −11/3 × (ω 2 e − AT δ + e − AS δ
TS
(6)
where
δ = 8.284(κη ) 4/3 + 12.978(κη ) 2 ,
(7)
η is the Kolmogorov micro scale, ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in range from 10−4m2/s3 to10−10m2/s3, χT being the rate of dissipation of mean-square temperature, taking values in the range from 10−4K2/s to 10−10K2/s [10]. ω is the relative strength of temperature and salinity fluctuation, which in the ocean water the value can range from 0 to −5, the minus sign of the parameter ω denotes that there is a reduction in temperature and an increase in salinity with depth. 0 corresponding to the case
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17725
when temperature-driven turbulence dominates, −5 corresponding to the situation when salinity-driven turbulence prevails [10]. Substituting Eqs. (6) and (7) and AT = 1.863 × 10−2 , AS = 1.9 × 10−4 as well as
ATS = 9.41× 10−3 [14] into Eq. (5), Eq. (5) can be simplified to T (η , ε , χT , ω ) = 1.2765 × 10−8 ω −2ε −1/3η −1/3 χT (47.5708 − 17.6701ω + 6.78335ω 2 ). (8)
Obviously, the T (η , ε , χT , ω ) increases with the increasing χT and decreasingω, ε and η. Introducing two variables of integration s = (ρ1 + ρ2)/2, t = ρ1-ρ2 and with Eqs. (4)–(8) substituting into Eq. (3), we can obtain ik ( ρ1′2 − ρ2′2 )]exp[−T (η , ε , χT , ω ) k 2 z ( ρ1′ − ρ2′ ) 2 ] 2π z 2z t2 × d 2 s d 2 t s 2 − i ( sx t y − s y t x ) 4
W ( ρ1′, ρ2′ , z ) = (
k
) 2 exp[−
2 ik ik × exp −at 2 − st exp − 2 s 2 exp ( ρ1′ − ρ2′ ) s z w z 0 ik × exp t ( ρ1′ + ρ2′ ) − T (η , ε , χT , ω ) k 2 z ( ρ1′ − ρ2′ ) z 2
, (9)
where a=
1 1 1 ( 2 + 2 ) + k 2 zT (η , ε , χT , ω ) , 2 w0 σ 0
(10)
sx = (ρ1x + ρ2x)/2, tx = ρ1x-ρ2x, sy = (ρ1y + ρ2y)/2, ty = ρ1y-ρ2y, the symbol ‘ ’ corresponds to m = ± 1 at the source plane z = 0. Using the integral formulae [21] q2 π ) , p p
(11)
q2 π q ) ( ), p p p
(12)
2 exp(− px + 2qx)dx = exp(
x exp(− px
2
+ 2qx)dx = exp(
2 2 x exp(− px + 2qx)dx =
q2 π 1 2q 2 exp( ) (1 + ), p p p 2p
(13)
after tedious but straightforward integral calculations, we deliver k 2 ik ) exp[− ( ρ1′2 − ρ2′2 )] 2π z 2z 2 × exp − k 2 zT (η , ε , χT , ω ) ρ1′ − ρ2′ × ( N1 − N 2 ) ( N 3 − N 4 ) , (14)
W ( ρ1′, ρ2′ , z ) = (
where N1 =
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π2 aC
Ex E y exp(
Fx2 + Fy2 C
)(
Fx2 + Fy2 C
2
+
1 ), C
(15)
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17726
N2 =
N3 =
N4 =
π 2 w02 8D
exp(−
iπ 2 w0 2aCD iπ 2 w0 2aCD
Ex
Ey
Gx2 + Gy2 Gx2 + G y2 1 k 2 w02 2 ′ ′ − × + ), ) exp( )( ρ ρ 1 2 8z 2 D D2 D
(16)
F 2 Gy2 2 k 2 w2 exp − 20 ( ρ1′y − ρ 2′ y ) exp x + , C CD D 8z
(17)
Fy2 Gx2 k 2 w02 2 ′ ′ − + ρ ρ exp − exp , ( ) 1x 2x 2 CD D 8z C
(18)
Fx Gy
Fy Gx
ik , z
(19)
2 b2 − , w02 4a
(20)
b 2 w02 , 8
(21)
b=
C=
D = a−
2 1 ik Ex = exp ( ρ1′x + ρ 2′ x ) − T (η , ε , χT , ω ) k 2 z ( ρ1′x − ρ 2′ x ) , 4a 2 z
(22)
1 ik ikb bk 2 Fx = [ ( ρ1′x − ρ2′ x ) − ( ρ1′x + ρ2′ x ) + T (η , ε , χT , ω ) ( ρ1′x − ρ2′ x )], 2 z 4az 2a
(23)
b 2 w02 1 ik ( ρ1′x − ρ2′ x )]. (24) Gx = [ ( ρ1′x + ρ2′ x ) − T (η , ε , χT , ω ) k 2 z ( ρ1′x − ρ2′ x ) − 2 2z 4 Using of the symmetry, Ey, Fy and Gy are obtained from Ex, Fx and Gx by replacing ρ′1x and ρ′2x with ρ′1y and ρ′2y, respectively. Equations (14)–(24) provide the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 propagating through oceanic turbulence. It follows from Eq. (14) that the cross–spectral density of GSM vortex beams depends on the oceanic turbulence and beam parameters such as the Kolmogorov micro scaleη, rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, rate of dissipation of mean-square temperatureχT, relative strength of temperature salinity fluctuation ω, spatial correlation length σ0, waist width w0, sign and values of topological charge m, as well as the propagation distance z, and the positions of two points at the z plane. In addition, for the case of T(η, ε, χT, ω) = 0 in Eqs. (14)–(24), which can simplify to the analytical expression for the cross–spectral density of GSM vortex beams with m = ± 1 in free space.
3. The evolution behavior of average intensity of Gaussian Schell-model vortex beams through oceanic turbulence
Substituting ρ1′ = ρ2′ = ρ′ into Eq. (14), the average intensity at the z plane is expressed as
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17727
I ( ρ ′ , z ) = W ( ρ ′ , ρ′ , z ) =(
t2 ) 2 d 2 s d 2 t[ s 2 − i ( sx t y − s y t x )] 2π z 4 k
2 ik ik × exp −at 2 − st exp − 2 s 2 exp( ρ′t ), z z w0
(25)
after straightforward integral calculations deliver the expression k 2 ρ ′2 1 + ) 2 2 C 2π z aC 4az C π 2 w02 ik ρ ′2 1 k2 ik ρ ′2 1 2 ′ ( 1)] ( + ) exp( )}. ρ (26) × exp[ − − C 4az 2 8D 2 zD 2 D 2 zD Eq. (26) is the analytical expression of a GSM vortex in oceanic turbulence average intensity, Eq. (26) together with Eqs. (8), (10), (19)–(21) imply that the average intensity depends on correlation length σ0, waist width w0, topological charge |m|, and parameter T(η, ε, χT, ω) (including the Kolmogorov micro scaleη, the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, the rate of dissipation of mean-square temperatureχT and the relative strength of temperature and salinity fluctuation ω), but is independent of the sign of m. It’s worth noting that Eq. (26) reduces to the analytical expression of a GSM vortex in free space for the case of T(η, ε, χT, ω) = 0 in Eq. (10), namely, I ( ρ′ , z ) = (
k
)2 {
π2
(
a′ =
1 1 1 ( + ), 2 w02 σ 02
(27)
2 b2 − , w02 4a ′
(28)
b 2 w02 , 8
(29)
C′ =
D′ = a ′ −
k 2 ρ ′2 1 + ) 2π z a ′C ′ 4a ′z 2 C ′2 C ′ π 2 w02 ik ρ ′2 1 k2 ik ρ ′2 1 × exp[ ( + ) exp( )}, ρ ′2 ( − 1)] − 2 2 C′ 4a ′z 8D ′ 2 zD′ D′ 2 zD′
I ′( ρ′ , z ) = (
k
)2 {
π2
(
(30)
which is consistent with the Eq. (9) in [4] for the case of Cn2 = 0. Illustration numerical examples are given in Figs. 1(a)–1(e), which plot normalized intensity distributions I/Imax (Imax—maximum intensity) of a GSM vortex beam versus the slanted axis ρ′r ( ρ r′ = ρ x′2 + ρ y′2 ) for different values of (a) propagation distance z, (b) relative strength of temperature and salinity fluctuation ω, (c) rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (d) rate of dissipation of mean-square temperature χT, (e) spatial correlation length σ0, where the calculation parameters w0 = 3cm, λ = 1060nm and η = 10−3m are fixed in every figure, while the rest parameters including σ0 = 2.5cm, ω = −2.5, ε =
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17728
Fig. 1. Evolution of normalized intensity profiles of GSM vortex beams propagating in oceanic turbulence.
10−7 m2/s3, χT = 10−9 K2/s and z = 300m are allowed to vary in different figures. As can be seen from Fig. 1(a), the average intensity of GSM vortex beams propagating through oceanic turbulence undergoes several stages of evolution like as GSM vortex beams propagating through ideal Kolmogorov atmospheric turbulence [4]. At the z = 0 plane there exists a zero intensity at the center ρ′r = 0, where the phase becomes singular. With increasing propagation distance to zflat = 560m, a flat-topped intensity profile occurs, and finally evolves into a Gaussian one at zGau = 700m. In Figs. 1(b)–1(e), we can see that at the given propagation distance (e.g., z = 300m), the evolution behavior of average intensity of GSM vortex beams in
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oceanic turbulence is significantly affected by the oceanic turbulence parameters includingω, ε and χT, and beam parameter σ0. From Figs. 1(a)–1(e), it is interesting to find that the beam profile will first take on a hollow shape, and then approach a Gaussian distribution, for example, when ω = −0.5 in Fig. 1(b), ε = 10−10 m2/s3in Fig. 1(c), χT = 10−7 K2/s in Fig. 1(d) and σ0 = 1cm, the I/Imax of a GSM vortex beam in oceanic turbulence extend to a Gaussian profile. The larger χT, the smallerω, ε and spatial correlation length and the longer propagation distance z are, the faster the change of the beam profile is. We can see from Eq. (8) and Figs. 2(a) and 2(b) (the calculation parameters are the same as those of Fig. 1) that the larger χT and the smallerω and ε are, the stronger the oceanic turbulence is. Therefore, Figs. 1(a)–1(e) indicate that the evolution behavior of intensity depends on the oceanic turbulence strength, the beam coherent degree and the propagation distance. At the same time, it is shown that the larger χT and z or the smaller ω and ε are, the larger the spreading of the beam width is, but the effect on the beam width by the spatial correlation length is not significant for the GSM vortex beams through oceanic turbulence.
Fig. 2. T(η,ω,ε,χT) versus the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε, (a) for different values of the rate of dissipation of mean-square temperatureχT and (b) for different values of the relative strength of temperature and salinity fluctuation ω.
4. The evolution behavior of coherent vortices of Gaussian Schell-model vortex beams through oceanic turbulence
The spectral degree of coherence is defined as [22]
μ ( ρ1′, ρ2′ , z ) =
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W ( ρ1′, ρ2′ , z ) , [ I ( ρ1′, z ) I ( ρ2′ , z )]1/ 2
(31)
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17730
and the position of coherence vortices is determined by [23] Re[ μ ( ρ1′, ρ2′ , z )] = 0,
(32)
Im[ μ ( ρ1′, ρ2′ , z )] = 0.
(33)
where Re and Im stand for the real and imaginary parts of μ ( ρ1′, ρ2′ , z ) , respectively. The topological charge and its sign of coherence vortices are determined by the sign principle [24]. Figure 3 gives curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 through oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m, where ρ′1 = (6cm, 9cm), m = + 1, σ0 = 2.5cm are kept fixed, and the other calculation parameters are the same as in Fig. 1. Figure 4 plots contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m, where the calculation parameters are the same as in Fig. 3. Figures 3(a), 3(b), 4(a), and 4(b) indicate that there exists a coherent vortex A with m = + 1 at z = 50m whose position is (0.3783cm,-0.1178cm) and two coherent vortices A and B with m = + 1, −1 appear at z = 220m. The singularity A moves to the position (3.27cm, 0.2321cm), while a new singularity B with m = −1 appears at the position (−7.351cm, 11.88cm). Therefore, the position and number of coherent vortices change with increasing propagation distance through oceanic turbulence.
Fig. 3. Curves of Reμ = 0 and Imμ = 0 of a GSM vortex beam with m = + 1 in oceanic turbulence at the propagation distance (a) z = 50m and (b) z = 220m.
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17731
Fig. 4. Contour lines of phase of a GSM vortex beam with m = + 1 in oceanic turbulence at the plane (a) z = 50m and (b) z = 220m.
Figure 5 plots that the position and number of coherent vortices of a GSM vortex beam with m = + 1 through oceanic turbulence for different values ofω,ε, χT, where z = 300m is kept fixed, the other calculation parameters are same as those of Fig. 1, the white and black dots represent m = −1 and m = + 1, respectively. It can be seen that there appears only a coherent vortex with m = + 1 and the topological charge is conserved within a certain propagation distance. But as the propagation distance increases, two coherent vortices with m = + 1 and m = −1 appear, whose positions change and gradually move closer. From Figs. 5(a)–5(c), we can see that the larger ω is, the larger the distance zc for the conservation of the topological charge. It can be seen from Figs. 5(b), 5(d) and 5(e) that the larger ε is, the larger zc is, for the case of ε = 10-10, 10−7 and 10−5, zc corresponding to 70m, 150m and 250m, respectively. The position and number of coherent vortices of GSM vortex beams with m = + 1 propagation through oceanic turbulence for different values of χT are plotted in Figs. 5(b), 5(f) and 5(g). We can see that the position and number of coherent vortices change with propagation distance, and two coherent vortices with m = + 1 and m = −1 appear at z = 150m in Fig. 5(b), 70m in Fig. 5(f) and 40m in Fig. 5(g) respectively. Obviously, the smaller the χT is, the larger the distance zc for the conservation of the topological charge, which indicates that the distance for the conservation of the topological charge increases with increasingω, ε and decreasingχT. The physical reason can be seen from Fig. 2 that the largerω, ε and smaller χT are, the smaller the value of T(η, ε, χT, ω) is, which indicates that the weaker the oceanic turbulence is, the less effect on the conservation of the topological charge by the weaker oceanic turbulence. And then the positions of two singularities constantly change and the separation distance between them gradually becomes smaller, and this condition is more significant for the larger the rate of dissipation of mean-square temperature χT in Fig. 5(g).
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Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17732
Fig. 5. Position and number of coherent vortices of a GSM with m = + 1 propagation through oceanic turbulence for different values of ω, ε, χT.
#211431 - $15.00 USD (C) 2014 OSA
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17733
5. Concluding remarks
In this paper, we have studied the evolution behavior of coherent vortices and average intensity of GSM vortex beams propagating through oceanic turbulence in detail. It has been shown analytically and numerically that the evolution behavior of coherent vortices and average intensity depends on the oceanic turbulence and beam parameters, including the rate of dissipation of turbulent kinetic energy per unit mass of fluidε, rate of dissipation of mean-square temperatureχT, relative strength of temperature salinity fluctuationsω, spatial correlation length and topological charge of the beams, as well as the propagation distance. The larger χT, the smaller ω, ε and the shorter spatial correlation length and the longer propagation distance z are, the faster the evolution of the beam intensity profile is. The smaller χT and the larger ω and ε are, the longer the distance appearing two coherent vortices with m = + 1 and m = −1, namely, the larger the distance zc for the conservation of the topological charge. The results obtained in this paper would be useful for potential applications of vortex beams in oceanic optical communications. Acknowledgments
This work is supported by the National Natural Science Foundation of China under grant No.61275203, National Natural Science Foundation of China and Civil Aviation Administration of China under grant No. 61079023. A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department under grant No.14ZA0268, the Program of Yibin Municipal Science and Technology Bureau under grant No.2013SF020, and the Scientific Research Project of Yibin University under grant No. 2013YY04.
#211431 - $15.00 USD (C) 2014 OSA
Received 6 May 2014; revised 7 Jun 2014; accepted 22 Jun 2014; published 14 Jul 2014 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017723 | OPTICS EXPRESS 17734
