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The analytical formulae for the wave structure functions (WSF) and the spatial coherence radius of plane and spherical waves propagating through oceanic turbulence are derived. It is found that the Kolmogorov five-thirds power law of WSF is also valid for oceanic turbulence in the inertial range. The changes of the WSF and the spatial coherence radius versus different parameters of oceanic turbulence are examined.
© 2014 Optical Society of America
1. Introduction
Knowledge of beam spreading is important in a free space optics (FSO) communications link because it determines the loss of power at the receiver, while the spatial coherence radius defines the effective receiver aperture size in a heterodyne detection system [1]. Until now, numerous works have been carried out concerning both the beam spreading and the spatial coherence radius in atmospheric turbulence [1–6]. Because of the complexity of oceanic turbulence, light propagation through oceanic turbulence is relatively less explored compared to that in atmospheric turbulence. Since recently the interest in active optical underwater communications, imaging and sensing appeared [7, 8], it has become important to deeply understand how the oceanic turbulence affects light propagation [9–17]. Recently the structure functions in underwater media are evaluated numerically [18]. However, until now the spatial coherence radius in oceanic turbulence, through the use of the wave structure function (WSF), has not been examined.
Atmospheric turbulence is primarily caused by fluctuating temperature, while oceanic turbulence (without impurities, bubbles and the suspended particles) is induced by the temperature and salinity fluctuations. The temperature and salinity spectra have similar “bumped” profiles, with bumps occurring at different wave numbers. In 2000, Nikishov developed an analytical model for such a power spectrum of oceanic turbulence [9]. In this paper, based on this power spectrum of oceanic turbulence, we derived the analytical formulae for the spatial coherence radius of plane and spherical waves propagating through oceanic turbulence by first obtaining the WSF, and study the changes of the spatial coherence radius versus the parameters determining oceanic turbulence.
2. Analytical formulae
The power spectrum introduced in [9] for homogeneous and isotropic oceanic water is given by the expression
where is the rate of dissipation of kinetic energy per unit mass of fluid ranging from to, is the rate of dissipation of mean-squared temperature and has the range to, is the Kolmogorov micro scale (inner scale), and , , , [9]. Additionally, defines the ratio of temperature and salinity contributions to the refractive index spectrum, which in the ocean waters can vary in the interval [-5; 0], with −5 and 0 corresponding to dominating temperature-induced and salinity-induced optical turbulence, respectively [9].It is mentioned that Eq. (1) in this paper is equivalent to Eq. (41) in [9], which is shown in Appendix A. Equation (1) is invalid for the case (i.e., dominating salinity-induced optical turbulence). For thecase, the derivation of the power spectrum from Eq. (1) is given in Appendix B.
Under Rytov approximation, the WSF of a plane wave propagating through isotropic and homogeneous turbulence is defined by [1]
whereis the separation distance between two points on the phase front transverse to the axis of propagation, is the optical wave-number related to the wavelength by , is the path length and is the zero-order Bessel function.By expanding the zero-order Bessel function in power series, we can write the WSF in the form
where we have used the power spectrum given by Eq. (1) and interchanged the order of summation and integration. In addition, , , , , , , and are taken in Eq. (3).In order to perform the integration in Eq. (3), we apply the following the integral formulae derived by using the software of Mathematica 8.0, i.e.,
where is the Gamma function and is the generalized hypergeometric function, where p and q are positive integers. For our case (i.e., power spectrum in Eq. (1)), we have proved that is always satisfied. Thus, for our case we can simplify the integration result from Eq. (3) by using the following formula (see Eq. (8) in [19]) for the case, i.e.Then, applying the definition of Pochhammer symbol (see Appendix I in [1], i.e., ), the relation of Pochhammer symbol (see in Chapter 9 Eq. (b) in [20], i.e., , and the definition of the generalized hypergeometric function (see Eq. (11.1) in [20], i.e., ), it can be shown that the last expression for WSF of a plane wave in oceanic turbulence reduces to
Equation (7) is the general analytical formula of WSF. In many cases of interest, it suffices to know the form of the WSF only in certain asymptotic regimes. We apply the relation (see Eq. (6) in [19]) to eliminate the minus in the first parameter of in Eq. (7). Then, recalling the asymptotic relations in Eq. (6) and the confluent hypergeometric function (CH4) of Appendix I in [1], i.e.,
after very tedious calculations, we obtain the WSF of a plane wave in certain asymptotic regimes asIt is noted that the WSF for the case in Eq. (9) is derived by using the first formula in Eqs. (6) and (8) for case, while the WSF for the case in Eq. (9) is derived by using the second formula in Eqs. (6) and (8) for the case.
The separation distance at which the modulus of the complex degree of coherence (DOC) falls to defines the spatial coherence radius , i.e., . Based on the expressions given in Eq. (9), we can obtain the plane-wave spatial coherence radius as
Under Rytov approximation, the WSF of a spherical wave is defined by [1]
Similarly, we can derive the WSF of a spherical wave to be
and the spherical-wave spatial coherence radius asBased on the second formula of Eqs. (9), (10), (12) and (13), the WSF of both a plane wave and a spherical wave can be rewritten asEquation (14) indicates that the spatial coherence radius is the only parameter characterizing the WSF, and under Rytov approximation the Kolmogorov five-thirds power law of WSF is also valid for oceanic turbulence in the inertial range if the power spectrum of oceanic turbulence proposed by Nikishov is adopted.
It is noted that Eqs. (9), (10), (12) and (13) are invalid for the case (i.e., dominating salinity-induced optical turbulence). For the case, the analytical formulae of the WSF and the spatial coherence radius derived from Eqs. (9), (10), (12) and (13) are shown in Appendix C.
According to Ref [1], under Rytov approximation, the definitions of WSF of a plane wave and a spherical wave are given by Eq. (2) and Eq. (11), respectively. Rytov approximation is limited to weak fluctuations. It is known that the expression for WSF depends on the mutual coherence function (MCF). However, for the special cases of a plane wave and a spherical wave, it has been shown MCF predicted by strong fluctuation theories is the same as that predicted by Rytov approximation [1]. Only a plane wave and a spherical wave cases are considered in this paper. Thus, the results of the WSF and the spatial coherence radius obtained in this paper are valid both in weak and strong fluctuations.
3. Numerical calculation results and analysis
In order to examine the correctness of the analytical results obtained in this paper, we give a comparison of results of WSF calculated by the analytical formulae obtained in this paper and by the definitions of Eq. (2) and Eq. (11). Curves of the WSF versus , and are shown in Figure 1, Fig. 2, and Fig. 3, respectively, where , and are taken and is satisfied. It can be seen that the two results are in agreement with each other exactly.
For both plane waves and spherical waves, changes of the spatial coherence radius versus , and are shown in Fig. 4, Fig. 5, and Fig. 6, respectively, where , and are chosen and is satisfied. It should be stated that we have shown that the laws of versus , and are similar for different values of wavelength. For the case, the absorption coefficient of pure water is 0.00442m−1 measured by using the integrating cavity absorption meter (ICAM) [21], and the drop in intensity due to absorption at 30m leaves 87.55% [22].
In Fig. 4 the smallest value of w is . When , we have which is very small value. In physics, the salinity-induced turbulence is much stronger than the temperature-induced turbulence, which results in being very small when approaches to zero (e. g., ). From Figs. 4-6 it can be seen that both for plane waves and spherical waves, decreases as and increase and as decreases. In addition, the spatial coherence radius of a spherical wave is larger than that of a plane wave, and the difference of between a spherical wave and a plane wave decreases as and increase and as decreases.
4. Concluding remarks
In summary, the analytical formulae for the WSF and the spatial coherence radius of a plane wave and a spherical wave propagating through oceanic turbulence have been derived in this paper, which are valid both in weak and strong fluctuations. It has been shown that under Rytov approximation, the Kolmogorov five-thirds power law of WSF is also valid for the oceanic turbulence in the inertial range if the power spectrum of oceanic turbulence proposed by Nikishov is adopted, and the salinity-induced turbulence is stronger than temperature-induced turbulence.
It is mentioned that WSF is actually a sum of the structure functions (i.e., the log-amplitude function and the phase structure function). Not only the spatial coherence radius can be derived from the WSF, but also the root-mean-square (rms) angle-of-arrival and rms image jitter are both derived from the phase structure function. Our results are of considerable theoretical and practical interest for operations in communication, imaging and sensing systems involving turbulent underwater channels.
Appendix A: Derivation of Eq. (1) from Eq. (41) in [9]
According to Eq. (41) in [9], the spectrum of fluctuation of the refraction index distribution is expressed as
where , , , , , is the eddy diffusion coefficients of heat, and is diffusion of the salt. It is noted that only symbols are changed in this paper, i.e., in Eq. (41) of [9] is replaced by , and is replaced by .Substituting from into Eq. (A1), we obtain
For homogeneous and isotropic oceanic water, we have , which results in . Under this condition, Eq. (16) reduces to
According to the relation between the spatial power spectrum of the refractive index and its scalar spectrum of fluctuation of the refractive index distribution, we have [10]
where ,, and are the same as those in the text. It is clear that Eq. (18) is the same as Eq. (1) in the text.Appendix B: Derivation of the power spectrum from Eq. (1) for the w=0 case
According to [9], the definitions of and can be expressed as
where , and are the differences in temperature and salinity between top and bottom boundaries of domain under study [9], is the vertical coordinate.Substituting from Eqs. (19) and (20) into Eq. (1), we obtain
If the water is isothermal (i.e., ), we have . Thus, Eq. (B3) reduces to
where is the rate of dissipation of mean-squared salinity [10]. For homogeneous and isotropic oceanic water, is used to derive Eq. (21).Appendix C: Analytical formulae of the WSF and the spatial coherence radius for the w=0 case
Substituting from Eqs. (19) and (20) into Eqs. (9), (10), (12) and (13) and simplifying them, we obtain the analytical formulae of the WSF and the spatial coherence radius for the case, i.e.,
Plane wave
Spherical wave
Acknowledgments
Xiaoling Ji and Lu Lu acknowledge the support by the National Natural Science Foundation of China (NSFC) under grants 61475105 and 61178070, and by the financial support from Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province under grant 12TD008. 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.” The authors are very thankful to the reviewers for their very valuable comments.
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