Research Article
Vol. 54, No. 11 / April 10 2015 / Applied Optics
3407
Regions of spreading of Gaussian array beams propagating through oceanic turbulence MIAOMIAO TANG
AND
DAOMU ZHAO*
Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received 12 January 2015; revised 14 March 2015; accepted 14 March 2015; posted 16 March 2015 (Doc. ID 232422); published 9 April 2015
The spreading of Gaussian array beams for two types of beam combination propagating through turbulent ocean is investigated. Analytical formulas for the Rayleigh range zR and turbulence distances zT and zTT are derived. Numerical results show that the spreading regions are influenced by the oceanic turbulence parameters and the source parameters. Moreover, the Rayleigh range of the coherent combination case tends to be more sensitive to parameters of the source and oceanic turbulence than that for the incoherent combination one, while in the case of turbulence distances, it is the reverse. © 2015 Optical Society of America OCIS codes: (140.3295) Laser beam characterization; (010.4455) Oceanic propagation; (010.4450) Oceanic optics; (030.7060) Turbulence. http://dx.doi.org/10.1364/AO.54.003407
1. INTRODUCTION Array beams have attracted high interest in recent years due to their wide applications in high-power systems, inertial confinement fusion, high-energy weapons, etc. [1–3]. A variety of linear, rectangular, and radial laser arrays has been developed and investigated [3–15]. In particular, as an important limiting factor in applications, spreading of arrays induced by turbulence has been examined in a number of papers [7–15]. Ji et al. explored the effective Rayleigh range of Gaussian array in ideal atmospheric turbulence [7], and investigated the influence of turbulence on the effective radius of curvature of radial Gaussian array beams [8]. Ai and Dan studied the turbulence-negligible propagation of Gaussian-Schell-model array beams [10]. In addition, Huang et al. reported both the turbulence distance and the effective radius of curvature for partly coherent Hermite–Gaussian linear array (PCHGLA) beams through non-Kolmogorov turbulence [14,15]. Turbulent ocean, as one kind of random medium, is of interest and has been investigated for several decades. While optical turbulence in ocean turbulence is driven by temperature and salinity fluctuations, the two power spectra have been separately measured a fairly long time ago; an analytical model for combination has been developed only recently [16,17]. With this model some work has been carried out on the optical propagation and scintillation in turbulent ocean [17–22]. However, to the best of our knowledge, there is no present literature dealing with the spreading of array beams through oceanic turbulence. In this paper, based on the mean-squared beam width, we study the Rayleigh range z R and the turbulence distances 1559-128X/15/113407-05$15/0$15.00 © 2015 Optical Society of America
(e.g., z T and z T T ) of array beams in the clear-water turbulent ocean. To illustrate the theory, a Gaussian array beam discussed as a typical example of laser array beams, and the dependences of the spreading regions on oceanic turbulence parameters and array beam parameters are studied in detail.
2. GAUSSIAN ARRAY BEAMS PROPAGATING THROUGH OCEANIC TURBULENCE As shown in Fig. 1, a one-dimensional linear array consists of N individual off-axis Gaussian beams arranged along the x-axis with the equal separation x d , propagating into the positive half-space z ≥ 0, filled with oceanic turbulence. First we consider the case of coherent combination, thus the cross-spectral density function of the source plane is given by [8] W 0 x 10 ; x 20 ; 0 N −1
2 X
x 0 − j x 2 x 20 − j2 x d 2 exp − 1 1 d ; w20 N −1
N −1
2 X
j1 −N2−1 j2 −
(1)
2
where w0 is the waist width of individual off-axis Gaussian beam. By using the extended Huygens–Fresnel diffraction integral, the intensity of coherently combined Gaussian array beams propagating through oceanic turbulence reads as [23] ZZ k x − x 10 2 − x − x 20 2 0 0 0 I x;z W x 1 ; x 2 ; 0 exp −ik 2z 2πz × hexpψ x; x 10 ; z ψx; x 20 ; zim dx 10 dx 20 ;
(2)
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Vol. 54, No. 11 / April 10 2015 / Applied Optics
N −1
β Fig. 1. Schematic diagram of the one-dimensional Gaussian array beam.
where k 2π∕λ is the wavenumber of light with λ being wavelength, and h…im implies that the average is taken over the ensemble of statistical realizations of the turbulent medium, the term in the sharp brackets with the subscript m in Eq. (2) can be written as [24] hexpψ x; x 10 ; z ψx; x 20 ; zim 2 2 0 Z π k zx 1 − x 20 2 ∞ 3 exp − κ Φκdκ ; 3 0
(3)
where Φκ is the spatial power spectrum of the turbulent refractive-index fluctuations. In our analysis we will employ the model developed in [16] for clear-water oceanic turbulence, which combines effects of temperature and salinity fluctuations in the water column. A particular case is considered here, when the eddy thermal diffusivity and the diffusion of the salt are equal, then Φn κ 0.388 × 10−8 ε−1∕3 κ −11∕3 1 2.35κη2∕3 f κ; w; χ T ;
(4)
where ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid, which may vary in range from 10−1 m2 ∕s3 to 10−10 m2 ∕s3 , η 10−3 m being the Kolmogorov microscale (inner scale), and χ f κ; w; χ T T2 w2 e −AT δ e −−AS δ 2we −AT S δ ; w
(5)
with χ T being the rate of dissipation of mean-square temperature, taking values in the range from 10−2 K 2 ∕s in surface water to 10−10 K 2 ∕s in deep water, AT 1.863 × 10−2 , AS 1.9 × 10−4 , ATS 9.41 × 10−3 , and δ 8.284κη4∕3 12.978κη2 . w −5; 0 is the parameter that determines the relative strength of temperature and salinity in driving the index fluctuations, where w −5 when the temperature-driven turbulence dominates, and w 0 when the salinity-driven turbulence dominates [17]. The mean-squared beam width is defined as [25] R 4 x 2 hI x; zidx hx 2 i R : (6) hI x; zidx
2 X
N −1 N −1 X 2 2 X 4S 1 2 2 1 − ∕ j − j x S; 2 d 2 2 2 1 w0 k w0 N −1 N −1 N −1
N −1
2 X
j1 −N2−1 j2 −
j1 −
2
α
βz 2
γz 3 ;
8 γ π2 3
α
Z
∞
2
κ 3 Φκdκ;
(10)
0
1 S exp − 2 j1 − j2 2 x 2d : 2w0
(11)
The first two terms in Eq. (7) represent the spreading of Gaussian array beams due to free-space diffraction, while spreading of Gaussian array beams as a result of oceanic turbulence deterioration is represented by the third term in the formula [26]. For the situation of the Gaussian array beams involving incoherent combination, the cross-spectral density function of the individual off-axis Gaussian beam centered in source plane reads as [8] x 10 − jx d 2 x 20 − jx d 2 0 0 0 ; W j x 1 ; x 2 ; 0 exp − w20 N −1 N −1 ; j∈ − : (12) 2 2 The intensity of the individual off-axis Gaussian beam propagating in oceanic turbulence reads as ZZ k x − x 10 2 − x − x 20 2 0 0 0 W j x 1 ; x 2 ;0 exp −ik I j x; z 2z 2πz × hexpψ x; x 10 ; z ψx; x 20 ; zim dx 10 dx 20 :
(13)
From Eq. (13), we can obtain the intensity of the incoherently combined Gaussian array beam in oceanic turbulence, which is given by N −1
2 X
I x; z
I j x; z:
(14)
j−N2−1
Upon substituting from Eq. (14) into Eq. (6), and following the treatment for coherent combination case, we obtain the mean-squared beam width of incoherently combined Gaussian array beams in turbulent ocean: hx 2 i α 0 β 0 z 2 γz 3 ;
(15)
N 2 − 1x 2d ; 3
(16)
α 0 w20
β0
(7)
where
j2 −
(9)
Upon substituting from Eq. (2) into Eq. (6), the expression for the mean-squared beam width of coherently combined Gaussian array beams can be derived (see [8], Appendix A): hx 2 i
2
4 w20 k 2
:
(17)
3. REGIONS OF SPREADING N −1 2
X
N −1 2
X
j1 −N2−1 j2 −N2−1
Sw20 j1 j2 2 x 2d ∕
N −1 2
X
N −1 2
X
j1 −N2−1 j2 −N2−1
S; (8)
A. Rayleigh Range
It is known that the Rayleigh range is defined as the propagation distance at which the cross-sectional area of a beam
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Vol. 54, No. 11 / April 10 2015 / Applied Optics
doubles [27]. From Eqs. (7) and (15), the Rayleigh range of Gaussian array beams with coherent combination and incoherent combination propagating in oceanic turbulence can be obtained, respectively,
G
27 η 1 − η2 2 αγ 2 η32 β3 2 2
3 1 − η2 γ81η22 1 − η22 α2 γ 2 12η42 αβ3 1∕2 2
1∕3
: (27)
1 β T − β; T 3γ
(18)
1 β 02 T 0 0 − β 0 ; T 3γ
(19)
z T in
R 0 η21 β 02 ∕R 0 η1 β 0 ; 31 − η1 γ
(28)
(20)
z T T in
G 0 η22 β 02 ∕G 0 η2 β 0 ; 31 − η2 γ
(29)
z Rco
z Rin
3409
2
Similarly, for the case of incoherent combination, the turbulence distance z T and z T T of Gaussian array beams propagating through oceanic turbulence can be obtained as
where
27 2 3 αγ − β3 γ81α2 γ 2 − 12αβ3 1∕2 T 2 2 T0
1∕3
;
with
27 0 2 3 α γ − β 03 γ81α 02 γ 2 − 12α 0 β 03 1∕2 2 2
1∕3
R0 : (21)
27 η 1 − η1 2 α 0 γ 2 η31 β 03 2 1 3 1 − η1 γ81η21 1 − η21 α 02 γ 2 12η41 α 0 β 03 1∕2 2
1∕3
(30)
B. Turbulence Distances
We apply turbulence distances z T and z T T to indicate the effects of oceanic turbulence on the spreading of Gaussian array beams. The turbulence distances z T and z T T are defined as [26] hx 2 z
T i − 2
hx 2free z T i
hx z T i
η1 ;
hx 2 z T T i − hx 2free z T T i η2 : hx 2 z T T i
;
0 < η1 ≤ 0.1;
0.9 ≤ η2 < 1;
(22)
27 η 1 − η2 2 α 0 γ 2 η32 β 03 G 2 2 0
3 1 − η2 γ81η22 1 − η22 α 02 γ 2 12η42 α 0 β 03 1∕2 2
1∕3
:
(31)
(23)
where hx 2free z T i and hx 2free z T T i are the mean-squared beam width in free space at z z T and z z T T planes, respectively. The influence of the turbulence on laser beams is small enough and can be neglected when z < z T , while the beam spreading is dominated by turbulence when z > z T T [26]. On substituting from Eq. (7) into Eqs. (22) and (23), respectively, one finds the turbulence distances z T and z T T of coherently combined Gaussian array beams, that z T co
R η21 β2 ∕R η1 β ; 31 − η1 γ
(24)
z T T co
G η22 β2 ∕G η2 β ; 31 − η2 γ
(25)
4. NUMERICAL CALCULATION RESULTS AND ANALYSIS In this section, numerical calculations are carried out for the Rayleigh range and turbulence distances of Gaussian array beams passing through turbulent ocean. Some typical results are compiled in Figs. 2–5. The following values of parameters are used in the calculations unless other values are specified in the captions: λ 1.06 μm, w0 x d 5 mm, N 9, η 10−3 m, ε 10−7 m2 ∕s3 , w −2.5, χ T 10−8 K 2 ∕s, η1 0.1, η2 0.9. The influence of the rate of dissipation of the mean squared temperature χ T on z R and z T for coherent and incoherent combination cases are given in Figs. 2(a) and 2(b), respectively.
with R
27 η 1 − η1 2 αγ 2 η31 β3 2 1
3 1 − η1 γ81η21 1 − η21 α2 γ 2 12η41 αβ3 1∕2 2
1∕3
; (26)
Fig. 2. (a) z R for coherent combination and incoherent combination cases versus χ T ; (b) z T for coherent combination and incoherent combination cases versus χ T .
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Research Article
Fig. 3. (a) z R and z T for coherent combination and incoherent combination cases versus χ T ; (b) z R and z T for coherent combination and incoherent combination cases versus w; (c) z R and z T for coherent combination and incoherent combination cases versus ε.
Fig. 4. (a) z R and z T for coherent combination and incoherent combination cases versus w0 ; (b) z R and z T for coherent combination and incoherent combination cases versus x d ; (c) z R and z T for coherent combination and incoherent combination cases versus N .
Fig. 5. (a) z T T for coherent combination and incoherent combination cases versus χ T ; (b) z T T for coherent combination and incoherent combination cases versus w; (c) z T T for coherent combination and incoherent combination cases versus ε.
As expected, z R and z T decrease monotonically with the increasing χ T for both types of beam combinations, which means in the regions closer to the surface of the water, the array has a wider spatial spreading. It is shown that z R of coherent combined Gaussian array beam drops more rapidly as compared to the incoherent combination case, while in the case of z T , it is the reverse; the incoherent combination source tends to be more sensitive to the index χ T . Figure 3 illustrates the behaviors of z R and z T versus another two major parameters of oceanic turbulence: (a) temperaturesalinity balance parameter w and (b) energy dissipation rate ε. It is seen from Fig. 3(a) that for the two types of beam combination, z R and z T show similar variation behavior. As the
temperature-salinity balance parameter increases z R and z T get lower, which implies that the turbulence effect of salinity fluctuations is much stronger than that of temperature fluctuations. In Fig. 3(b), it is clearly displayed the influence of ε on the regions of spreading, as the parameter ε increases, z R and z T increase for any combination case. Figure 4(a) reveals that the waist width w0 may markedly affect z R and z T for both coherent and incoherent combination cases. As the value of w0 increases, z R increases while z T gets lower, but when the index w0 is large enough, it has almost no impact on the spreading regions. From Figs. 4(b) and 4(c) one finds that for both coherent and incoherent combination cases, z R and z T increase with increasing x d and N . Similar to the
Research Article results presented in Figs. 2 and 3, we again observe that the Rayleigh range of the coherent combination case tends to be affected more as compared to the incoherent combination one, as for z T , the incoherent combination source tends to be more sensitive to the beam parameters. From Fig. 5 one finds that the influences of the turbulence parameters on z T T are similar to those on z T , and as a result of the accumulation effects of turbulence, z T T is much larger than z R and z T at the same underwater turbulence parameter. 5. CONCLUSION In this paper, the regions of spreading of Gaussian array beams propagating through turbulent ocean have been examined by using the Rayleigh range and turbulence distances (e.g., z T and z T T ). The effects of the array beam parameters and oceanic turbulence parameters, such as the rate of dissipation of the mean squared temperature, the temperature-salinity balance parameter and the energy dissipation rate on spreading regions, are stressed, and both coherent and incoherent combination cases are taken into consideration. The general conclusion is that for the two types of beam combination, z R , z T , and z T T increase with increasing χ T , w and decreasing ε. Besides, the waist width w0 has significant impact on the spreading regions, as well as the beam number N and the relative beam separation distance x d . It is also shown that z R for the coherent combination case tends to be more sensitive to parameters of the source and oceanic turbulence than that for the incoherent combination one, while in the cases of z T and z T T , it is the reverse. This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253). REFERENCES 1. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996). 2. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997). 3. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000). 4. X. Y. Du and D. M. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008). ˘ Y. Baykal, and Y. J. Cai, “Scintillations of laser array 5. H. T. Eyyuboglu, beams,” Appl. Phys. B 91, 265–271 (2008). 6. Y. B. Zhu, D. M. Zhao, and X. Y. Du, “Propagation of stochastic Gaussian–Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008). ˘ and Y. Baykal, “Influence of turbulence on 7. X. L. Ji, H. T. Eyyuboglu, the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
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8. X. L. Ji and Z. C. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283, 3884–3890 (2010). 9. X. L. Ji and Z. C. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmosphere turbulence,” Appl. Phys. B 93, 915–923 (2008). 10. Y. L. Ai and Y. Q. Dan, “Range of turbulence-negligible propagation of Gaussian Schell-model array beams,” Opt. Commun. 284, 3216– 3220 (2011). 11. X. L. Ji and X. Q. Li, “Effective radius of curvature of partially coherent Hermite-Gaussian array beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010). 12. G. H. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian–Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35, 715–717 (2010). 13. Y. P. Huang, Z. H. Gao, F. H. Wang, and Z. C. Duan, “Effective radius of curvature of partially coherent Hermite-Gaussian beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 59, 1674–1679 (2012). 14. Y. P. Huang and B. Zhang, “Turbulence distance for laser beams propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 30, 2339–2346 (2013). 15. Y. P. Huang, Z. H. Gao, F. H. Wang, G. P. Zhao, Z. C. Duan, X. Xiao, and Q. Liu, “The effective radius of curvature of partially coherent Hermite-Gaussian linear array beams passing through nonKolmogorov turbulence,” Opt. Commun. 315, 130–137 (2014). 16. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). 17. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284, 1740–1746 (2011). 18. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011). 19. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012). 20. Y. Zhou, K. K. Huang, and D. M. Zhao, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence,” Appl. Phys. B 109, 289–294 (2012). 21. M. M. Tang and D. M. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014). 22. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31, 1552– 1556 (2014). 23. X. Y. Du, D. M. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). 24. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). 25. X. L. Ji, Z. C. Pu, G. M. Ji, and X. H. Jia, “Regions of spreading of partially coherent beams propagating through non-Kolmogorov turbulence,” Opt. Commun. 322, 109–113 (2014). 26. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1597 (2002). 27. A. E. Siegman, Lasers (University Science Books, 1986).
Vol. 54, No. 11 / April 10 2015 / Applied Optics
3407
Regions of spreading of Gaussian array beams propagating through oceanic turbulence MIAOMIAO TANG
AND
DAOMU ZHAO*
Department of Physics, Zhejiang University, Hangzhou 310027, China *Corresponding author: [email protected] Received 12 January 2015; revised 14 March 2015; accepted 14 March 2015; posted 16 March 2015 (Doc. ID 232422); published 9 April 2015
The spreading of Gaussian array beams for two types of beam combination propagating through turbulent ocean is investigated. Analytical formulas for the Rayleigh range zR and turbulence distances zT and zTT are derived. Numerical results show that the spreading regions are influenced by the oceanic turbulence parameters and the source parameters. Moreover, the Rayleigh range of the coherent combination case tends to be more sensitive to parameters of the source and oceanic turbulence than that for the incoherent combination one, while in the case of turbulence distances, it is the reverse. © 2015 Optical Society of America OCIS codes: (140.3295) Laser beam characterization; (010.4455) Oceanic propagation; (010.4450) Oceanic optics; (030.7060) Turbulence. http://dx.doi.org/10.1364/AO.54.003407
1. INTRODUCTION Array beams have attracted high interest in recent years due to their wide applications in high-power systems, inertial confinement fusion, high-energy weapons, etc. [1–3]. A variety of linear, rectangular, and radial laser arrays has been developed and investigated [3–15]. In particular, as an important limiting factor in applications, spreading of arrays induced by turbulence has been examined in a number of papers [7–15]. Ji et al. explored the effective Rayleigh range of Gaussian array in ideal atmospheric turbulence [7], and investigated the influence of turbulence on the effective radius of curvature of radial Gaussian array beams [8]. Ai and Dan studied the turbulence-negligible propagation of Gaussian-Schell-model array beams [10]. In addition, Huang et al. reported both the turbulence distance and the effective radius of curvature for partly coherent Hermite–Gaussian linear array (PCHGLA) beams through non-Kolmogorov turbulence [14,15]. Turbulent ocean, as one kind of random medium, is of interest and has been investigated for several decades. While optical turbulence in ocean turbulence is driven by temperature and salinity fluctuations, the two power spectra have been separately measured a fairly long time ago; an analytical model for combination has been developed only recently [16,17]. With this model some work has been carried out on the optical propagation and scintillation in turbulent ocean [17–22]. However, to the best of our knowledge, there is no present literature dealing with the spreading of array beams through oceanic turbulence. In this paper, based on the mean-squared beam width, we study the Rayleigh range z R and the turbulence distances 1559-128X/15/113407-05$15/0$15.00 © 2015 Optical Society of America
(e.g., z T and z T T ) of array beams in the clear-water turbulent ocean. To illustrate the theory, a Gaussian array beam discussed as a typical example of laser array beams, and the dependences of the spreading regions on oceanic turbulence parameters and array beam parameters are studied in detail.
2. GAUSSIAN ARRAY BEAMS PROPAGATING THROUGH OCEANIC TURBULENCE As shown in Fig. 1, a one-dimensional linear array consists of N individual off-axis Gaussian beams arranged along the x-axis with the equal separation x d , propagating into the positive half-space z ≥ 0, filled with oceanic turbulence. First we consider the case of coherent combination, thus the cross-spectral density function of the source plane is given by [8] W 0 x 10 ; x 20 ; 0 N −1
2 X
x 0 − j x 2 x 20 − j2 x d 2 exp − 1 1 d ; w20 N −1
N −1
2 X
j1 −N2−1 j2 −
(1)
2
where w0 is the waist width of individual off-axis Gaussian beam. By using the extended Huygens–Fresnel diffraction integral, the intensity of coherently combined Gaussian array beams propagating through oceanic turbulence reads as [23] ZZ k x − x 10 2 − x − x 20 2 0 0 0 I x;z W x 1 ; x 2 ; 0 exp −ik 2z 2πz × hexpψ x; x 10 ; z ψx; x 20 ; zim dx 10 dx 20 ;
(2)
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Vol. 54, No. 11 / April 10 2015 / Applied Optics
N −1
β Fig. 1. Schematic diagram of the one-dimensional Gaussian array beam.
where k 2π∕λ is the wavenumber of light with λ being wavelength, and h…im implies that the average is taken over the ensemble of statistical realizations of the turbulent medium, the term in the sharp brackets with the subscript m in Eq. (2) can be written as [24] hexpψ x; x 10 ; z ψx; x 20 ; zim 2 2 0 Z π k zx 1 − x 20 2 ∞ 3 exp − κ Φκdκ ; 3 0
(3)
where Φκ is the spatial power spectrum of the turbulent refractive-index fluctuations. In our analysis we will employ the model developed in [16] for clear-water oceanic turbulence, which combines effects of temperature and salinity fluctuations in the water column. A particular case is considered here, when the eddy thermal diffusivity and the diffusion of the salt are equal, then Φn κ 0.388 × 10−8 ε−1∕3 κ −11∕3 1 2.35κη2∕3 f κ; w; χ T ;
(4)
where ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid, which may vary in range from 10−1 m2 ∕s3 to 10−10 m2 ∕s3 , η 10−3 m being the Kolmogorov microscale (inner scale), and χ f κ; w; χ T T2 w2 e −AT δ e −−AS δ 2we −AT S δ ; w
(5)
with χ T being the rate of dissipation of mean-square temperature, taking values in the range from 10−2 K 2 ∕s in surface water to 10−10 K 2 ∕s in deep water, AT 1.863 × 10−2 , AS 1.9 × 10−4 , ATS 9.41 × 10−3 , and δ 8.284κη4∕3 12.978κη2 . w −5; 0 is the parameter that determines the relative strength of temperature and salinity in driving the index fluctuations, where w −5 when the temperature-driven turbulence dominates, and w 0 when the salinity-driven turbulence dominates [17]. The mean-squared beam width is defined as [25] R 4 x 2 hI x; zidx hx 2 i R : (6) hI x; zidx
2 X
N −1 N −1 X 2 2 X 4S 1 2 2 1 − ∕ j − j x S; 2 d 2 2 2 1 w0 k w0 N −1 N −1 N −1
N −1
2 X
j1 −N2−1 j2 −
j1 −
2
α
βz 2
γz 3 ;
8 γ π2 3
α
Z
∞
2
κ 3 Φκdκ;
(10)
0
1 S exp − 2 j1 − j2 2 x 2d : 2w0
(11)
The first two terms in Eq. (7) represent the spreading of Gaussian array beams due to free-space diffraction, while spreading of Gaussian array beams as a result of oceanic turbulence deterioration is represented by the third term in the formula [26]. For the situation of the Gaussian array beams involving incoherent combination, the cross-spectral density function of the individual off-axis Gaussian beam centered in source plane reads as [8] x 10 − jx d 2 x 20 − jx d 2 0 0 0 ; W j x 1 ; x 2 ; 0 exp − w20 N −1 N −1 ; j∈ − : (12) 2 2 The intensity of the individual off-axis Gaussian beam propagating in oceanic turbulence reads as ZZ k x − x 10 2 − x − x 20 2 0 0 0 W j x 1 ; x 2 ;0 exp −ik I j x; z 2z 2πz × hexpψ x; x 10 ; z ψx; x 20 ; zim dx 10 dx 20 :
(13)
From Eq. (13), we can obtain the intensity of the incoherently combined Gaussian array beam in oceanic turbulence, which is given by N −1
2 X
I x; z
I j x; z:
(14)
j−N2−1
Upon substituting from Eq. (14) into Eq. (6), and following the treatment for coherent combination case, we obtain the mean-squared beam width of incoherently combined Gaussian array beams in turbulent ocean: hx 2 i α 0 β 0 z 2 γz 3 ;
(15)
N 2 − 1x 2d ; 3
(16)
α 0 w20
β0
(7)
where
j2 −
(9)
Upon substituting from Eq. (2) into Eq. (6), the expression for the mean-squared beam width of coherently combined Gaussian array beams can be derived (see [8], Appendix A): hx 2 i
2
4 w20 k 2
:
(17)
3. REGIONS OF SPREADING N −1 2
X
N −1 2
X
j1 −N2−1 j2 −N2−1
Sw20 j1 j2 2 x 2d ∕
N −1 2
X
N −1 2
X
j1 −N2−1 j2 −N2−1
S; (8)
A. Rayleigh Range
It is known that the Rayleigh range is defined as the propagation distance at which the cross-sectional area of a beam
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Vol. 54, No. 11 / April 10 2015 / Applied Optics
doubles [27]. From Eqs. (7) and (15), the Rayleigh range of Gaussian array beams with coherent combination and incoherent combination propagating in oceanic turbulence can be obtained, respectively,
G
27 η 1 − η2 2 αγ 2 η32 β3 2 2
3 1 − η2 γ81η22 1 − η22 α2 γ 2 12η42 αβ3 1∕2 2
1∕3
: (27)
1 β T − β; T 3γ
(18)
1 β 02 T 0 0 − β 0 ; T 3γ
(19)
z T in
R 0 η21 β 02 ∕R 0 η1 β 0 ; 31 − η1 γ
(28)
(20)
z T T in
G 0 η22 β 02 ∕G 0 η2 β 0 ; 31 − η2 γ
(29)
z Rco
z Rin
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2
Similarly, for the case of incoherent combination, the turbulence distance z T and z T T of Gaussian array beams propagating through oceanic turbulence can be obtained as
where
27 2 3 αγ − β3 γ81α2 γ 2 − 12αβ3 1∕2 T 2 2 T0
1∕3
;
with
27 0 2 3 α γ − β 03 γ81α 02 γ 2 − 12α 0 β 03 1∕2 2 2
1∕3
R0 : (21)
27 η 1 − η1 2 α 0 γ 2 η31 β 03 2 1 3 1 − η1 γ81η21 1 − η21 α 02 γ 2 12η41 α 0 β 03 1∕2 2
1∕3
(30)
B. Turbulence Distances
We apply turbulence distances z T and z T T to indicate the effects of oceanic turbulence on the spreading of Gaussian array beams. The turbulence distances z T and z T T are defined as [26] hx 2 z
T i − 2
hx 2free z T i
hx z T i
η1 ;
hx 2 z T T i − hx 2free z T T i η2 : hx 2 z T T i
;
0 < η1 ≤ 0.1;
0.9 ≤ η2 < 1;
(22)
27 η 1 − η2 2 α 0 γ 2 η32 β 03 G 2 2 0
3 1 − η2 γ81η22 1 − η22 α 02 γ 2 12η42 α 0 β 03 1∕2 2
1∕3
:
(31)
(23)
where hx 2free z T i and hx 2free z T T i are the mean-squared beam width in free space at z z T and z z T T planes, respectively. The influence of the turbulence on laser beams is small enough and can be neglected when z < z T , while the beam spreading is dominated by turbulence when z > z T T [26]. On substituting from Eq. (7) into Eqs. (22) and (23), respectively, one finds the turbulence distances z T and z T T of coherently combined Gaussian array beams, that z T co
R η21 β2 ∕R η1 β ; 31 − η1 γ
(24)
z T T co
G η22 β2 ∕G η2 β ; 31 − η2 γ
(25)
4. NUMERICAL CALCULATION RESULTS AND ANALYSIS In this section, numerical calculations are carried out for the Rayleigh range and turbulence distances of Gaussian array beams passing through turbulent ocean. Some typical results are compiled in Figs. 2–5. The following values of parameters are used in the calculations unless other values are specified in the captions: λ 1.06 μm, w0 x d 5 mm, N 9, η 10−3 m, ε 10−7 m2 ∕s3 , w −2.5, χ T 10−8 K 2 ∕s, η1 0.1, η2 0.9. The influence of the rate of dissipation of the mean squared temperature χ T on z R and z T for coherent and incoherent combination cases are given in Figs. 2(a) and 2(b), respectively.
with R
27 η 1 − η1 2 αγ 2 η31 β3 2 1
3 1 − η1 γ81η21 1 − η21 α2 γ 2 12η41 αβ3 1∕2 2
1∕3
; (26)
Fig. 2. (a) z R for coherent combination and incoherent combination cases versus χ T ; (b) z T for coherent combination and incoherent combination cases versus χ T .
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Research Article
Fig. 3. (a) z R and z T for coherent combination and incoherent combination cases versus χ T ; (b) z R and z T for coherent combination and incoherent combination cases versus w; (c) z R and z T for coherent combination and incoherent combination cases versus ε.
Fig. 4. (a) z R and z T for coherent combination and incoherent combination cases versus w0 ; (b) z R and z T for coherent combination and incoherent combination cases versus x d ; (c) z R and z T for coherent combination and incoherent combination cases versus N .
Fig. 5. (a) z T T for coherent combination and incoherent combination cases versus χ T ; (b) z T T for coherent combination and incoherent combination cases versus w; (c) z T T for coherent combination and incoherent combination cases versus ε.
As expected, z R and z T decrease monotonically with the increasing χ T for both types of beam combinations, which means in the regions closer to the surface of the water, the array has a wider spatial spreading. It is shown that z R of coherent combined Gaussian array beam drops more rapidly as compared to the incoherent combination case, while in the case of z T , it is the reverse; the incoherent combination source tends to be more sensitive to the index χ T . Figure 3 illustrates the behaviors of z R and z T versus another two major parameters of oceanic turbulence: (a) temperaturesalinity balance parameter w and (b) energy dissipation rate ε. It is seen from Fig. 3(a) that for the two types of beam combination, z R and z T show similar variation behavior. As the
temperature-salinity balance parameter increases z R and z T get lower, which implies that the turbulence effect of salinity fluctuations is much stronger than that of temperature fluctuations. In Fig. 3(b), it is clearly displayed the influence of ε on the regions of spreading, as the parameter ε increases, z R and z T increase for any combination case. Figure 4(a) reveals that the waist width w0 may markedly affect z R and z T for both coherent and incoherent combination cases. As the value of w0 increases, z R increases while z T gets lower, but when the index w0 is large enough, it has almost no impact on the spreading regions. From Figs. 4(b) and 4(c) one finds that for both coherent and incoherent combination cases, z R and z T increase with increasing x d and N . Similar to the
Research Article results presented in Figs. 2 and 3, we again observe that the Rayleigh range of the coherent combination case tends to be affected more as compared to the incoherent combination one, as for z T , the incoherent combination source tends to be more sensitive to the beam parameters. From Fig. 5 one finds that the influences of the turbulence parameters on z T T are similar to those on z T , and as a result of the accumulation effects of turbulence, z T T is much larger than z R and z T at the same underwater turbulence parameter. 5. CONCLUSION In this paper, the regions of spreading of Gaussian array beams propagating through turbulent ocean have been examined by using the Rayleigh range and turbulence distances (e.g., z T and z T T ). The effects of the array beam parameters and oceanic turbulence parameters, such as the rate of dissipation of the mean squared temperature, the temperature-salinity balance parameter and the energy dissipation rate on spreading regions, are stressed, and both coherent and incoherent combination cases are taken into consideration. The general conclusion is that for the two types of beam combination, z R , z T , and z T T increase with increasing χ T , w and decreasing ε. Besides, the waist width w0 has significant impact on the spreading regions, as well as the beam number N and the relative beam separation distance x d . It is also shown that z R for the coherent combination case tends to be more sensitive to parameters of the source and oceanic turbulence than that for the incoherent combination one, while in the cases of z T and z T T , it is the reverse. This work was supported by the National Natural Science Foundation of China (NSFC) (11274273 and 11474253). REFERENCES 1. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagating characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400–407 (1996). 2. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997). 3. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000). 4. X. Y. Du and D. M. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008). ˘ Y. Baykal, and Y. J. Cai, “Scintillations of laser array 5. H. T. Eyyuboglu, beams,” Appl. Phys. B 91, 265–271 (2008). 6. Y. B. Zhu, D. M. Zhao, and X. Y. Du, “Propagation of stochastic Gaussian–Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008). ˘ and Y. Baykal, “Influence of turbulence on 7. X. L. Ji, H. T. Eyyuboglu, the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18, 6922–6928 (2010).
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