1552
J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
Y. Ata and Y. Baykal
Scintillations of optical plane and spherical waves in underwater turbulence Yalçın Ata1 and Yahya Baykal2,* 1
Defense Industries Research and Development Institute (TÜBI_TAK SAGE), P.K. 16 06261 Mamak, Ankara, Turkey 2 Department of Electronic and Communication Engineering, Çankaya University, Eskişehir Yolu, 29. km. 06810 Yenimahalle Ankara, Turkey *Corresponding author: [email protected] Received November 25, 2013; revised May 25, 2014; accepted May 25, 2014; posted June 2, 2014 (Doc. ID 201969); published June 23, 2014 The scintillation indices of optical plane and spherical waves propagating in underwater turbulent media are evaluated by using the Rytov method, and the variations in the scintillation indices are investigated when the rate of dissipation of mean squared temperature, the temperature and salinity fluctuations, the propagation distance, the wavelength, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy are varied. Results show that as in the atmosphere, also in underwater media the plane wave is more affected by turbulence as compared to the spherical wave. The underwater turbulence effect becomes significant at 5–10 m for a plane wave and at 20–25 m for a spherical wave. The turbulence effect is relatively small in deep water and is large at the surface of the water. Salinity-induced turbulence strongly dominates the scintillations compared to temperature-induced turbulence. © 2014 Optical Society of America OCIS codes: (010.0010) Atmospheric and oceanic optics; (010.4455) Oceanic propagation; (290.5930) Scintillation; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/JOSAA.31.001552
1. INTRODUCTION Underwater media are of interest and have been analyzed for many years. Turbulence in underwater media is also an important concern for researchers [1–3]. Mainly the temperature and the salinity fluctuations generate turbulence in the underwater medium, whereas temperature, wind, and humidity are the main drivers of turbulence in the atmosphere. Random optical intensity fluctuations called scintillation are among the important factors that degrade the optical system performance. Optical propagation [4–7] and scintillation [8–11] in atmospheric turbulence are examined in numerous studies. In the underwater turbulent medium, there are relatively few studies. Due to its chaotic characteristics, the optical propagation range in underwater media is a few tens of meters [12,13]. Hill [4] calculated the structure function and the coherence length of a plane wave using the fluctuations of temperature and salinity. In their study, Nikishov and Nikishov introduced [14] a unitless variable w that defines the contributions of the temperature and the salinity fluctuations to the refractive index spectrum. In underwater, w changes in the interval [−5, 0], w −5 gives the temperature-induced turbulence, and w 0 gives the salinity-induced turbulence. Lu et al. [5] investigated the underwater turbulence effect on the beamwidth and spreading. Farwell and Korotkova [7] described the intensity and the coherence characteristics of optical propagation and showed that underwater turbulence has a significant effect on the beam as compared to atmospheric turbulence. Recently, Korotkova et al. studied the light scintillation in oceanic turbulence by representing the scintillation index by the longitudinal and radial components [15]. In this paper, by employing the Rytov method, we formulate and evaluate the on-axis scintillation index of optical plane 1084-7529/14/071552-05$15.00/0
and spherical waves propagating horizontally in an underwater turbulent medium. As also shown in [15], we reproduce and represent the underwater scintillations versus the propagation distance for various rates of dissipation of mean squared temperature and parameters that determine the relative strength of temperature and salinity. Additionally, we report the very important variations of scintillations versus the wavelength, the rate of dissipation of mean squared temperature, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy for various parameters that determine the relative strength of temperature and salinity. Also, another crucial dependence of the underwater scintillations versus the rate of dissipation of mean squared temperature and the propagation distance is revealed for different wavelengths.
2. UNDERWATER TURBULENCE The power spectrum of the atmosphere is given by the wellknown Kolmogorov −11∕3 power law [16]. This model is widely used to characterize optical propagation in the atmosphere, but underwater, because of the existence of different constituents, turbulence attains different properties and power spectra. The power spectrum of the oceanic water [7,14] in isotropic and homogeneous turbulence at the stable stratification where thermal diffusivity K T and the diffusion of the salt K S are equal to each other is given as ϕn κ 0.388 × 10−8 ε−1∕3 κ−11∕3 1 2.35κη2∕3 ×
X T 2 −AT δ w e e−AS δ − 2we−AT S δ ; w2
© 2014 Optical Society of America
(1)
Y. Ata and Y. Baykal
Vol. 31, No. 7 / July 2014 / J. Opt. Soc. Am. A
where κ is the magnitude of the spatial frequency, ε is the rate of dissipation of the turbulent kinetic energy varying from 10−1 m2 ∕s3 in surface water to 10−10 m2 ∕s3 in deep water [1], X T is the rate of dissipation of the mean squared temperature altering from 10−2 K2 ∕s in surface water to 10−10 K2 ∕s in deep water [1], η is the Kolmogorov microscale length expressed as η v3 ∕ε0.25 [12], v is the kinematic viscosity in m2 ∕s3 , 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 0 when the salinity-driven turbulence dominates, and w −5 when the temperature-driven turbulence dominates.
3. SCINTILLATION INDEX FOR PLANE WAVE The log-amplitude correlations of a general beam are given in [17]. The scintillation index at the origin of the propagation distance L is associated with the log-amplitude correlation function, which is found to be [8] m2 L 4Bχ L Z Z L 4π Re dz 0
∞ 0
Z κdκ
2π 0
dθM 1 L; z; κ; θ
M 2 L; z; κ; θϕn κ ;
(2)
where Bχ L is the log-amplitude correlation function, z is the distance variable along the propagation axis, θ is the angular component, M 1 L; z; κ; θ NL; η; κ; θNL; z; −κ; θ;
(3)
M 2 L; z; κ; θ NL; z; κ; θN L; z; κ; θ;
(4)
NL; z; κ; θ ik exp
iz − L 2 κ ; 2k
Bχ L 2π2
Z 0
L
Z dz
∞ 0
κdκjH r j2 ϕn κ;
1553
(7)
where jH r j2 k2 sin2 0.5γL − zκ2 ∕k and γ z∕L. As is known, the scintillation index is related to the logamplitude correlation function by the relation m2 L 4Bχ L. Substituting the power spectrum for oceanic water given in Eq. (1), we obtain the scintillation index for the spherical wave in the underwater medium as X m2 L 6.208 × 10−8 ε−1∕3 k2 π 2 T2 w ZL Z∞ × dz dκκ −8∕3 1 2.35κη2∕3 0 0 z L − z 2 × sin2 κ w2 e−AT δ e−AS δ − 2we−ATS δ : L 2k (8)
5. RESULTS AND DISCUSSION The results presented in this section are obtained by numerically evaluating Eqs. (6) and (8). We restrict the scintillation index value smaller than one (m2 < 1); thus our results are valid in the weak turbulence regime. Figure 1 shows the scintillation index versus the propagation distance for plane and spherical waves for various rates of dissipation of mean squared temperature. As expected, for any rate of dissipation of mean squared temperature, large propagation distances yield large scintillations for both plane and spherical waves. Under the same turbulence parameters, the scintillations for the plane wave underwater are larger than the scintillations for the spherical wave. As the rate of dissipation of the mean squared temperature X T increases,
(5)
i −10.5 , k 2π∕λ, and λ is the wavelength. Substituting Eqs. (1) and (3)–(5) into Eq. (2), the on-axis scintillation index for the optical plane wave in underwater turbulence is obtained as Z L X m2 L 1.552π × 10−8 ε−1∕3 T2 k2 Re dz w 0 Z 2π Z∞ κ −8∕3 1 2.35κη2∕3 dκ dθ × 0 0 iz − L 2 × exp κ k ×w2 e−AT δ e−AS δ − 2we−ATS δ :
(6)
4. SCINTILLATION INDEX FOR SPHERICAL WAVE The log-amplitude correlation function of a spherical wave based on the Rytov solution is given in [18] as
Fig. 1. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus propagation distance for various rates of dissipation of mean squared temperature.
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J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
the scintillation index increases. Since the rate of dissipation of the mean squared temperature is higher in the regions closer to the surface of the water, the results in Fig. 1 mean that in deep water, similarly to in [5], as X T attains a large value, the beam has a wider spreading, which indicates a stronger turbulence effect. In Fig. 2 we can see that the salinity-driven turbulence increases the scintillation much more than the temperaturedriven turbulence. Scintillation increases drastically for both w −0.1 and w −0.5; the effect of turbulence becomes much stronger in a few meters. Similar to the results presented in Fig. 1, the plane wave tends to be affected more as compared to the spherical wave. The longest weak turbulence distance is about 10 m for a plane wave and 25 m for a spherical wave for temperature-driven turbulence. Figure 3 provides the scintillations versus the rate of dissipation of the mean squared temperature. We note that the upper plot for the plane wave is at the propagation distance of 10 m, whereas the lower plot for the spherical wave is at the propagation distance of 30 m. At the fixed depth of water (indicated by the corresponding rate of dissipation of the mean squared temperature) and at the same w, it is seen that the same scintillation index level is attained by the plane wave at a much smaller propagation distance. Due to absorption and scattering phenomena, the use of the optical signals in underwater media is limited to the blue– green spectrum [5]. Absorption and scattering values are small enough to communicate optically about 100 m. When turbulence is involved, this range decreases to several tens of meters as implied in Figs. 2 and 3. Smaller wavelength optical waves confront a larger turbulence effect. The variations of the scintillations are given in Fig. 4, from which it is observed as expected that as the wavelength increases, the scintillations decrease at any w. In Fig. 4 we again observe
Fig. 2. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus propagation distance for various temperature and salinity contribution parameters.
Y. Ata and Y. Baykal
Fig. 3. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus rate of dissipation of mean squared temperature for various temperature and salinity contribution parameters.
that the effect of salinity fluctuations is much larger than the effect of temperature fluctuations at any wavelength. In order to make an easier comparison between the plane and spherical wave scintillation indices for different wavelengths, in Figs. 5 and 6 we provide both plane and spherical wave results in the same plot. It is clearly seen from Figs. 5
Fig. 4. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus wavelength for various temperature and salinity contribution parameters.
Y. Ata and Y. Baykal
Vol. 31, No. 7 / July 2014 / J. Opt. Soc. Am. A
1555
Fig. 5. Scintillation index of plane and spherical waves versus rate of dissipation of mean squared temperature for various wavelengths.
Fig. 8. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus the rate of dissipation of the turbulent kinetic energy for various temperature and salinity contribution parameters.
Fig. 6. Scintillation index of plane and spherical waves versus propagation distance for various wavelengths.
and 6 that in an underwater medium, at any wavelength and propagation distance, the plane wave scintillation index is always larger than the scintillation index of the spherical wave no matter what the underwater turbulence parameters are. Figure 7 is plotted by inserting ε v3 ∕η4 in Eqs. (6) and (8) for the plane (upper plot) and spherical (lower plot) waves, respectively, where the viscosity, v, is taken as 10−6 m2 ∕s. Figure 7 indicates that being valid for all w, an increase in the Kolmogorov microscale length causes the scintillations to increase, and at a fixed Kolmogorov microscale length, the scintillations are higher when w attains lower values, i.e., when underwater turbulence is dominated by the salinity fluctuations. Figure 8 is plotted by inserting η v3 ∕ε0.25 in Eqs. (6) and (8) for the plane (upper plot) and spherical (lower plot) waves, respectively, where the viscosity, v, is again taken as 10−6 m2 ∕s. Figure 8 shows that for all w, when the rate of dissipation of the turbulent kinetic energy increases, the scintillations decrease. At a fixed rate of dissipation of the turbulent kinetic energy, the scintillations are higher for lower w values, i.e., again when underwater turbulence is dominated by the salinity fluctuations.
6. CONCLUSION
Fig. 7. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus Kolmogorov microscale length for various temperature and salinity contribution parameters.
The scintillation index for plane and spherical waves is evaluated in underwater turbulence. The effects of the wavelength, propagation distance, and underwater turbulence parameters such as the rate of dissipation of the mean squared temperature, the relative strength of temperature and salinity, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy on the scintillations are examined. The general conclusion is that the underwater turbulence affects the optical wave propagation significantly. The propagation distance for weak underwater turbulence
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J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
is only a few tens of meters for both plane and spherical waves, whereas a comparable propagation distance to reach the same scintillation value is several kilometers in atmospheric turbulence [8]. Salinity fluctuations are much more effective as compared to temperature fluctuations in the determination of the scintillations. The turbulence effect becomes stronger as the horizontal propagation comes close to the surface of the water, whereas the turbulence effect is quite small in the deep water boundary. Shorter wavelengths are more affected from underwater turbulence, but due to the absorption and scattering effects, blue and green wavelengths are preferred in underwater communications. As in atmospheric turbulence, in order to obtain smaller scintillations, a spherical wave is preferred as compared to a plane wave at any underwater turbulence parameter, propagation distance, and wavelength.
ACKNOWLEDGMENTS Yahya Baykal gratefully acknowledges the support provided by Çankaya University, Tübitak, for project no. 113E589 and the ICT COST Action IC1101 entitled “Optical Wireless Communications—An Emerging Technology.”
REFERENCES 1. 2. 3. 4.
S. A. Thorpe, The Turbulent Ocean (Cambridge University, 2005). Y. I. Kopilevich, N. V. Aleksejev, B. V. Kurasov, and V. A. Yakovlev, “Diagnostics of seawater refractive turbulence,” Proc. SPIE 2208, 35–43 (1997). C. H. Gibson, “Turbulence in the ocean, atmosphere, galaxy, and universe,” Appl. Mech. Rev. 49, 299–315 (1996). R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68, 1067–1072 (1978).
Y. Ata and Y. Baykal 5. 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 8, 1052–1058 (2006). 6. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulent ocean,” Appl. Phys. B 105, 415–420 (2011). 7. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012). 8. H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. 46, 1099–1106 (2007). 9. S. A. Arpali, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of higher-order cos-Gaussian, cosh-Gaussian and annular beams,” J. Mod. Opt. 55, 227–239 (2008). 10. N. Perlot, “Evaluation of the scintillation loss for optical communication systems with direct detection,” Opt. Eng. 46, 025003 (2007). 11. K. S. Gochelashvily and V. I. Shishov, “Laser beam scintillation beyond a turbulent layer,” Opt. Acta 18, 313–320 (1971). 12. F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49, 3224–3230 (2010). 13. F. Hanson and S. Radic, “High bandwidth underwater optical communication,” Appl. Opt. 47, 277–283 (2008). 14. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). 15. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012). 16. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). 17. Y. Baykal, “Formulation of correlations for general-type beams in atmospheric turbulence,” J. Opt. Soc. Am. A 23, 889–893 (2006). 18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley, 1999).
J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
Y. Ata and Y. Baykal
Scintillations of optical plane and spherical waves in underwater turbulence Yalçın Ata1 and Yahya Baykal2,* 1
Defense Industries Research and Development Institute (TÜBI_TAK SAGE), P.K. 16 06261 Mamak, Ankara, Turkey 2 Department of Electronic and Communication Engineering, Çankaya University, Eskişehir Yolu, 29. km. 06810 Yenimahalle Ankara, Turkey *Corresponding author: [email protected] Received November 25, 2013; revised May 25, 2014; accepted May 25, 2014; posted June 2, 2014 (Doc. ID 201969); published June 23, 2014 The scintillation indices of optical plane and spherical waves propagating in underwater turbulent media are evaluated by using the Rytov method, and the variations in the scintillation indices are investigated when the rate of dissipation of mean squared temperature, the temperature and salinity fluctuations, the propagation distance, the wavelength, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy are varied. Results show that as in the atmosphere, also in underwater media the plane wave is more affected by turbulence as compared to the spherical wave. The underwater turbulence effect becomes significant at 5–10 m for a plane wave and at 20–25 m for a spherical wave. The turbulence effect is relatively small in deep water and is large at the surface of the water. Salinity-induced turbulence strongly dominates the scintillations compared to temperature-induced turbulence. © 2014 Optical Society of America OCIS codes: (010.0010) Atmospheric and oceanic optics; (010.4455) Oceanic propagation; (290.5930) Scintillation; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/JOSAA.31.001552
1. INTRODUCTION Underwater media are of interest and have been analyzed for many years. Turbulence in underwater media is also an important concern for researchers [1–3]. Mainly the temperature and the salinity fluctuations generate turbulence in the underwater medium, whereas temperature, wind, and humidity are the main drivers of turbulence in the atmosphere. Random optical intensity fluctuations called scintillation are among the important factors that degrade the optical system performance. Optical propagation [4–7] and scintillation [8–11] in atmospheric turbulence are examined in numerous studies. In the underwater turbulent medium, there are relatively few studies. Due to its chaotic characteristics, the optical propagation range in underwater media is a few tens of meters [12,13]. Hill [4] calculated the structure function and the coherence length of a plane wave using the fluctuations of temperature and salinity. In their study, Nikishov and Nikishov introduced [14] a unitless variable w that defines the contributions of the temperature and the salinity fluctuations to the refractive index spectrum. In underwater, w changes in the interval [−5, 0], w −5 gives the temperature-induced turbulence, and w 0 gives the salinity-induced turbulence. Lu et al. [5] investigated the underwater turbulence effect on the beamwidth and spreading. Farwell and Korotkova [7] described the intensity and the coherence characteristics of optical propagation and showed that underwater turbulence has a significant effect on the beam as compared to atmospheric turbulence. Recently, Korotkova et al. studied the light scintillation in oceanic turbulence by representing the scintillation index by the longitudinal and radial components [15]. In this paper, by employing the Rytov method, we formulate and evaluate the on-axis scintillation index of optical plane 1084-7529/14/071552-05$15.00/0
and spherical waves propagating horizontally in an underwater turbulent medium. As also shown in [15], we reproduce and represent the underwater scintillations versus the propagation distance for various rates of dissipation of mean squared temperature and parameters that determine the relative strength of temperature and salinity. Additionally, we report the very important variations of scintillations versus the wavelength, the rate of dissipation of mean squared temperature, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy for various parameters that determine the relative strength of temperature and salinity. Also, another crucial dependence of the underwater scintillations versus the rate of dissipation of mean squared temperature and the propagation distance is revealed for different wavelengths.
2. UNDERWATER TURBULENCE The power spectrum of the atmosphere is given by the wellknown Kolmogorov −11∕3 power law [16]. This model is widely used to characterize optical propagation in the atmosphere, but underwater, because of the existence of different constituents, turbulence attains different properties and power spectra. The power spectrum of the oceanic water [7,14] in isotropic and homogeneous turbulence at the stable stratification where thermal diffusivity K T and the diffusion of the salt K S are equal to each other is given as ϕn κ 0.388 × 10−8 ε−1∕3 κ−11∕3 1 2.35κη2∕3 ×
X T 2 −AT δ w e e−AS δ − 2we−AT S δ ; w2
© 2014 Optical Society of America
(1)
Y. Ata and Y. Baykal
Vol. 31, No. 7 / July 2014 / J. Opt. Soc. Am. A
where κ is the magnitude of the spatial frequency, ε is the rate of dissipation of the turbulent kinetic energy varying from 10−1 m2 ∕s3 in surface water to 10−10 m2 ∕s3 in deep water [1], X T is the rate of dissipation of the mean squared temperature altering from 10−2 K2 ∕s in surface water to 10−10 K2 ∕s in deep water [1], η is the Kolmogorov microscale length expressed as η v3 ∕ε0.25 [12], v is the kinematic viscosity in m2 ∕s3 , 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 0 when the salinity-driven turbulence dominates, and w −5 when the temperature-driven turbulence dominates.
3. SCINTILLATION INDEX FOR PLANE WAVE The log-amplitude correlations of a general beam are given in [17]. The scintillation index at the origin of the propagation distance L is associated with the log-amplitude correlation function, which is found to be [8] m2 L 4Bχ L Z Z L 4π Re dz 0
∞ 0
Z κdκ
2π 0
dθM 1 L; z; κ; θ
M 2 L; z; κ; θϕn κ ;
(2)
where Bχ L is the log-amplitude correlation function, z is the distance variable along the propagation axis, θ is the angular component, M 1 L; z; κ; θ NL; η; κ; θNL; z; −κ; θ;
(3)
M 2 L; z; κ; θ NL; z; κ; θN L; z; κ; θ;
(4)
NL; z; κ; θ ik exp
iz − L 2 κ ; 2k
Bχ L 2π2
Z 0
L
Z dz
∞ 0
κdκjH r j2 ϕn κ;
1553
(7)
where jH r j2 k2 sin2 0.5γL − zκ2 ∕k and γ z∕L. As is known, the scintillation index is related to the logamplitude correlation function by the relation m2 L 4Bχ L. Substituting the power spectrum for oceanic water given in Eq. (1), we obtain the scintillation index for the spherical wave in the underwater medium as X m2 L 6.208 × 10−8 ε−1∕3 k2 π 2 T2 w ZL Z∞ × dz dκκ −8∕3 1 2.35κη2∕3 0 0 z L − z 2 × sin2 κ w2 e−AT δ e−AS δ − 2we−ATS δ : L 2k (8)
5. RESULTS AND DISCUSSION The results presented in this section are obtained by numerically evaluating Eqs. (6) and (8). We restrict the scintillation index value smaller than one (m2 < 1); thus our results are valid in the weak turbulence regime. Figure 1 shows the scintillation index versus the propagation distance for plane and spherical waves for various rates of dissipation of mean squared temperature. As expected, for any rate of dissipation of mean squared temperature, large propagation distances yield large scintillations for both plane and spherical waves. Under the same turbulence parameters, the scintillations for the plane wave underwater are larger than the scintillations for the spherical wave. As the rate of dissipation of the mean squared temperature X T increases,
(5)
i −10.5 , k 2π∕λ, and λ is the wavelength. Substituting Eqs. (1) and (3)–(5) into Eq. (2), the on-axis scintillation index for the optical plane wave in underwater turbulence is obtained as Z L X m2 L 1.552π × 10−8 ε−1∕3 T2 k2 Re dz w 0 Z 2π Z∞ κ −8∕3 1 2.35κη2∕3 dκ dθ × 0 0 iz − L 2 × exp κ k ×w2 e−AT δ e−AS δ − 2we−ATS δ :
(6)
4. SCINTILLATION INDEX FOR SPHERICAL WAVE The log-amplitude correlation function of a spherical wave based on the Rytov solution is given in [18] as
Fig. 1. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus propagation distance for various rates of dissipation of mean squared temperature.
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the scintillation index increases. Since the rate of dissipation of the mean squared temperature is higher in the regions closer to the surface of the water, the results in Fig. 1 mean that in deep water, similarly to in [5], as X T attains a large value, the beam has a wider spreading, which indicates a stronger turbulence effect. In Fig. 2 we can see that the salinity-driven turbulence increases the scintillation much more than the temperaturedriven turbulence. Scintillation increases drastically for both w −0.1 and w −0.5; the effect of turbulence becomes much stronger in a few meters. Similar to the results presented in Fig. 1, the plane wave tends to be affected more as compared to the spherical wave. The longest weak turbulence distance is about 10 m for a plane wave and 25 m for a spherical wave for temperature-driven turbulence. Figure 3 provides the scintillations versus the rate of dissipation of the mean squared temperature. We note that the upper plot for the plane wave is at the propagation distance of 10 m, whereas the lower plot for the spherical wave is at the propagation distance of 30 m. At the fixed depth of water (indicated by the corresponding rate of dissipation of the mean squared temperature) and at the same w, it is seen that the same scintillation index level is attained by the plane wave at a much smaller propagation distance. Due to absorption and scattering phenomena, the use of the optical signals in underwater media is limited to the blue– green spectrum [5]. Absorption and scattering values are small enough to communicate optically about 100 m. When turbulence is involved, this range decreases to several tens of meters as implied in Figs. 2 and 3. Smaller wavelength optical waves confront a larger turbulence effect. The variations of the scintillations are given in Fig. 4, from which it is observed as expected that as the wavelength increases, the scintillations decrease at any w. In Fig. 4 we again observe
Fig. 2. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus propagation distance for various temperature and salinity contribution parameters.
Y. Ata and Y. Baykal
Fig. 3. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus rate of dissipation of mean squared temperature for various temperature and salinity contribution parameters.
that the effect of salinity fluctuations is much larger than the effect of temperature fluctuations at any wavelength. In order to make an easier comparison between the plane and spherical wave scintillation indices for different wavelengths, in Figs. 5 and 6 we provide both plane and spherical wave results in the same plot. It is clearly seen from Figs. 5
Fig. 4. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus wavelength for various temperature and salinity contribution parameters.
Y. Ata and Y. Baykal
Vol. 31, No. 7 / July 2014 / J. Opt. Soc. Am. A
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Fig. 5. Scintillation index of plane and spherical waves versus rate of dissipation of mean squared temperature for various wavelengths.
Fig. 8. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus the rate of dissipation of the turbulent kinetic energy for various temperature and salinity contribution parameters.
Fig. 6. Scintillation index of plane and spherical waves versus propagation distance for various wavelengths.
and 6 that in an underwater medium, at any wavelength and propagation distance, the plane wave scintillation index is always larger than the scintillation index of the spherical wave no matter what the underwater turbulence parameters are. Figure 7 is plotted by inserting ε v3 ∕η4 in Eqs. (6) and (8) for the plane (upper plot) and spherical (lower plot) waves, respectively, where the viscosity, v, is taken as 10−6 m2 ∕s. Figure 7 indicates that being valid for all w, an increase in the Kolmogorov microscale length causes the scintillations to increase, and at a fixed Kolmogorov microscale length, the scintillations are higher when w attains lower values, i.e., when underwater turbulence is dominated by the salinity fluctuations. Figure 8 is plotted by inserting η v3 ∕ε0.25 in Eqs. (6) and (8) for the plane (upper plot) and spherical (lower plot) waves, respectively, where the viscosity, v, is again taken as 10−6 m2 ∕s. Figure 8 shows that for all w, when the rate of dissipation of the turbulent kinetic energy increases, the scintillations decrease. At a fixed rate of dissipation of the turbulent kinetic energy, the scintillations are higher for lower w values, i.e., again when underwater turbulence is dominated by the salinity fluctuations.
6. CONCLUSION
Fig. 7. Scintillation index of plane (upper plot) and spherical (lower plot) waves versus Kolmogorov microscale length for various temperature and salinity contribution parameters.
The scintillation index for plane and spherical waves is evaluated in underwater turbulence. The effects of the wavelength, propagation distance, and underwater turbulence parameters such as the rate of dissipation of the mean squared temperature, the relative strength of temperature and salinity, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy on the scintillations are examined. The general conclusion is that the underwater turbulence affects the optical wave propagation significantly. The propagation distance for weak underwater turbulence
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J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
is only a few tens of meters for both plane and spherical waves, whereas a comparable propagation distance to reach the same scintillation value is several kilometers in atmospheric turbulence [8]. Salinity fluctuations are much more effective as compared to temperature fluctuations in the determination of the scintillations. The turbulence effect becomes stronger as the horizontal propagation comes close to the surface of the water, whereas the turbulence effect is quite small in the deep water boundary. Shorter wavelengths are more affected from underwater turbulence, but due to the absorption and scattering effects, blue and green wavelengths are preferred in underwater communications. As in atmospheric turbulence, in order to obtain smaller scintillations, a spherical wave is preferred as compared to a plane wave at any underwater turbulence parameter, propagation distance, and wavelength.
ACKNOWLEDGMENTS Yahya Baykal gratefully acknowledges the support provided by Çankaya University, Tübitak, for project no. 113E589 and the ICT COST Action IC1101 entitled “Optical Wireless Communications—An Emerging Technology.”
REFERENCES 1. 2. 3. 4.
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