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Abstract
Based on the extended Huygens-Fresnel principle, we have derived the analytical expression of the average intensity of optical coherence lattices (OCLs) in oceanic turbulence with anisotropy, and then the beam quality parameters including the Strehl ratio (SR) and the power-in-the-bucket (PIB) are obtained. One can find that the OCLs will eventually evolve into Gaussian shape with the periodicity reciprocity gradually breaking down when propagating through the anisotropic ocean water, and that the trend of evolving into Gaussian can be accelerated for increasing the ratio of temperature and salinity contributions to the refractive index spectrum ω, the lattice constant a and the rate of dissipation of mean square temperature χT or decreasing the anisotropic factor ξ and the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε. Further, the SR and PIB in the target plane under the effects of oceanic parameters are discussed in detail, and the SR and PIB can be increased for the larger ξ and ε or the smaller χT and ω, namely, the beam quality becomes better. Our results can find potential application in the future optical communication system in an oceanic environment.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the optical wireless communication underwater media being an innovative and good alternative communication method [1, 2], the propagation of optical beams through oceanic turbulence is of great importance. As beams propagate in the oceanic turbulence, the properties are influenced by the turbulence parameters, and the shape of optical fields and the beam quality get changed during propagation. As a result, investigating the average intensity and the beam quality in the oceanic turbulence is essential to design of an FSO communication link operating in the underwater environment.
In recent years, much attention has been paid to the beam propagation properties in ocean link. By using the power spectrum of oceanic turbulence which is determined by the temperature and salinity fluctuation [3], the research on the homogeneous and isotropic water has become more convenient and hotter. The intensity and coherence [4], scintillations [5], polarization properties [6–8] as well as the wave structure function of plane and spherical waves [9] have been reported, and the effects of the oceanic parameters are discussed. To reduce the turbulence effects, the laser beam array is often utilized [10–13], and the beam quality can be improved comparing with the single beam, moreover, the Rayleigh range of the coherent combination case tends to be more sensitive to the oceanic turbulence comparing with the incoherent combination one. The evolution of vortex beams in oceanic turbulence is detailly investigated, the average intensity, the topological conservation distance and the mode detection probability can be aimed to the optical communication [14]. The oceanic turbulence plays an important role in the evolution of the Shell-model beams, the oscillations in the degree of coherence profiles of cosine-Gaussian-correlated Schell-model (CGSM) beams can be weakened [15]. The comparison between different kinds of Schell-model beams in oceanic turbulence are presented [16], and GSM gives a smaller spreading comparing with others, moreover, the normalized propagation factor for four kinds of Schell-model beams demonstrates that the CGSM is more robust against the oceanic turbulence [17]. More recently, the optical coherence lattices (OCLs), a class of partially coherent beams with periodic coherence properties, was lately discovered [18], and experimental generation of OCLs carrying information was reported [19]. Later, the propagation properties of OCLs in free space have been explored [20], and a novel phenomenon of periodicity reciprocity between their intensity and coherence properties is presented; in addition, the interference between the OCLs is investigated [21], and the intensity pattern can be changed by varying the weight distribution parameter. The OCLs propagation in the turbulent atmosphere have been examined, and the periodicity reciprocity in intensity profile can be preserved over a certain distance [22]. However, to the best of our knowledge, the propagation of the OCLs through the anisotropic ocean link have not been investigated.
In the paper, the properties of OCLs in anisotropic ocean without scattering and absorbing of water are studied based on the extended Huygens-Fresnel principle, the object of the paper is to investigate the anisotropic ocean turbulence on the OCLs propagation and the conditions for periodicity reciprocity preservation distance as well as the beam quality, thus the analytical expression of the average intensity and the beam quality described by intensity Strehl ratio (SR) and power-in-the-bucket (PIB) are derived. The effects of the oceanic parameters on the average intensity and beam quality evolution are detailly discussed, and the beam quality is improved comparing with the fundamental Gaussian beams. Our results can be helpful in the design of an optical communication system in an oceanic environment. The paper is organized as follows. Firstly in Sec. 2, the cross-spectral density of OCLs in oceanic turbulence with anisotropy is derived based on the extended Huygens-Fresnel principle. Then in Sec. 3, the analytical expression of the average intensity is obtained with its evolution being discussed. Additionally in Sec. 4, we go a step further to explore the beam quality of OCLs, including the intensity Strehl ratio (SR) and power-in-the-bucket (PIB). Finally in Sec. 5, the paper is concluded.
2. The cross-spectral density of OCLs in anisotropic turbulent ocean
The cross-spectral density of OCLs at the source plane can be expressed as [18–20]:
where σ0 is the size of the beam waist, representing a transverse scale parameter, ns is a non-negative integers, vns is the power distribution of the pseudo-modes composing the lattice, and a is a lattice constant. N indicates the number of lattice lobes, from the equation, the source changes into the single fundamental Gaussian beams for N = 0. Here, we investigate the propagation properties of OCLs in oceanic turbulence, and the paraxial evolution of OCLs in the oceanic turbulence can be expressed as the extended Huygens-Fresnel integral [13, 23–25]: where k = 2π/λ is the wave number with λ being the wavelength of the source beam. ρ1 = (x1, y1) and ρ2 = (x2, y2) are two arbitrary transverse position vectors at the receiver plane, . 〈exp [ψ* (ρ1,r1) + ψ (ρ1,r1)]〉 denotes the ensemble average of the turbulent ocean water and can be represented as [2, 14, 26]: In Eq. (3), ρocξ is the lateral coherence length of the spherical wave in the anisotropic turbulent ocean. From [6], the lateral coherence length ρocξ is defined as: with and ξ representing the turbulence strength and anisotropic factor, respectively. It should be noted from Eq. (4) that the oceanic turbulence strength is inversely proportional to the coherence length ρocξ. That is to say, the smaller ρocξ means the stronger turbulent strength. For simplicity, we only consider the influence of oceanic turbulence due to temperature and salinity fluctuations on beams propagation with the scattering and absorbing of ocean water being ignored, and in Markov approximation, the two-dimensional refractive index turbulence spatial spectrum model for homogeneous and anisotropic oceanic water is: where AT = 1.863×10−2, AS = 1.9×10−4, ATS = 9.41×10−3, δ = 8.284 (κ′η)4/3 + 12.978 (κ′η)2 and . Substituting Eq. (5) into Eq. (4), the analytical expression of ρocξ can be obtained by a long but straightforward calculation: η is the Kolmogorov micro scale; χT is the rate of dissipation of mean square temperature and ranges from 10−4K2/s to 10−10K2/s; ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary in range from 10−4m2/s3 to 10−10m2/s3; ω represents their ratio of the temperature and salinity contributions to the refractive index spectrum, which may vary from −5 to 0, with −5 or 0 dominating temperature-induced or salinity-induced turbulence, respectively. Substituting Eq. (3) into Eq. (2), we have Introducing two variables of integration R = (r1 + r2)/2, T = r1 − r2 into Eq. (7), the expression of the cross-spectral density of OCLs can be expressed as: Using the formulas mentioned in [27] After tedious calculations, the analytical expression for cross-spectral density of OCLs propagating in the turbulent ocean is: where , .3. The evolution behavior of average intensity of OCLs in anisotropic turbulent ocean
Substituting ρ1 = ρ2 = ρ into Eq. (11), the average intensity at z plane can be expressed as:
Eq. (12) imply that the average intensity of OCLs depends on the initial beam waist σ0, the lattice constant a, the propagation distance z, the number of lattice lobes N and the oceanic turbulence parameters. In order to conveniently calculate the results, the coordinates are considered to be normalized: S1 = s1/σ0, , then the average intensity is: where . From Eq. (13), the expression for the intensity of OCLs in free space reduces to the form: where , and the expression of Eq. (14) is identical to that mentioned in [18].Based on Eqs. (13) and (14), the average intensity evolution during propagation can be obtained numerically, and the wavelength λ = 532nm and the inner scale η = 1mm as well as vns = 1 are fixed in the numerical simulations. In addition, σ0 is set to be 0.01m, and Z = 1 means that the actual propagation distance is: . Figure 1 presents the evolution of OCLs versus propagation distance in free space and anisotropic ocean. In Fig. 1(a), the OCLs firstly evolve from a Gaussian shape to a rectangular shape, and then breaks up into multiple fundamental Gaussian beams, and this phenomenon is known as the periodicity reciprocity; Also, the number of fundamental Gaussian beams mainly depends on the number of lattice lobes N. However, the beam shape can be largely influenced by the oceanic turbulence as shown in Fig. 1(c) where the periodicity reciprocity totally breaks down during propagation; The OCLs can maintain the rectangular lattice shape over a certain propagation distance, while this shape will eventually change into the Gaussian in Figs. 1(b) and 1(c), typical of any beam eventually succumbing to the oceanic turbulence. In addition, the periodicity reciprocity of OCLs in oceanic turbulence can be reproduced with the decrease of the lattice constant a comparing Fig. 1(b) with Fig. 1(c), and disappears after propagating through a certain distance. Therefore, adjusting the lattice constant a is a flexible approach to control the lattice structure of OCLs in oceanic turbulence. The next, the effects of the oceanic parameters on the average intensity evolution of OCLs at Z = 2 are discussed in detail, as shown in Fig. 2. Figure 2(a) gives the average intensity of OCLs under different anisotropic factors, and the larger anisotropic factor leads to the emergence of the periodicity reciprocity, thus the distance for OCLs holding periodicity reciprocity increases, also the similar conclusion can be obtained from Fig. 2(c) where the increase of ε makes OCLs begin to show the periodicity reciprocity, however, the opposite results are presented for a larger ocean parameter in Figs. 2(b) and 2(d) comparing with Fig. 2(a), and the increase of ω and χT accelerates the velocity of evolving into Gaussian beams. From Eq. (6), the increase of ξ and ε or the decrease of ω and χT leads to the larger coherence length, and then the turbulence strength becomes weak. Therefore, one result in a sharp contrast with the OCL evolution scenario in free space can be concluded that the periodicity reciprocity preservation distance for OCLs decreases for the stronger oceanic turbulence strength. For application, the periodicity reciprocity properties and rectangular shape of the OCL intensity profile can be utilized to transport information over some distances in the turbulent ocean, paving the way for optical communications underwater.
Fig. 1 The average intensity of OCLs versus propagation distance in free space for (a) a = 1, N = 10 and σ0 = 0.01m, and in anisotropic ocean for (b) a = 0.5, N = 10, σ0 = 0.01m, ξ = 1, ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s and (c) a = 1, N = 10, σ0 = 0.01m, ξ = 1, ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s.
Fig. 2 The average intensity of OCLs in anisotropic ocean at Z = 2 under different turbulence parameters with a = 1, N = 10, σ0 = 0.01m for (a) ω = −2.5, ε = 10−5m2/s3 and χT = 10−9K2/s, (b) ξ = 1, ε = 10−5m2/s3 and χT = 10−9K2/s, (c) ξ = 1, ω = −2.5 and χT = 10−9K2/s and (d) ξ = 1, ω = −2.5 and ε = 10−5m2/s3.
4. The beam quality evaluation factor of OCLs in anisotropic turbulent ocean
4.1. The intensity Strehl ratio SR
Firstly, the intensity Strehl ratio SR is chosen as the characteristic parameter of beam quality, and is defined as [28]:
where Imax and I0max are the peak intensity with and without oceanic turbulence, respectively. The higher value the Strehl ratio SR is, the less the peak intensity decreases, and then the better beam quality is confirmed.Substituting Eqs. (13) and (14) into Eq. (15), the analytical expression of Strehl ratio SR can be expressed as:
Figure 3 gives the evolution of SR versus propagation distance under the source parameters, and the decreasing trend of SR is presented during propagation, which indicates that the beam quality becomes worse with the propagation distance. In addition, adopting the array beams is an effective way to resist the turbulence comparing with the single beam, and the scintillation is weak [10, 12], thus one can find that the SR increases for the larger number of lattice lobes N. However, further increasing N significantly influences the SR, and the increment trend of SR becomes slow in Fig. 3(a) comparing the case of N = 1 with that of N = 5. From the analysis in Figs. 1(b) and 1(c), the decrease of the lattice constant a leads to the recurrence of the lattice structure that can reduce the turbulence disturbance, similar to the results in [22], the decrease of the lattice constant a helps to improve the beam quality of OCLs. The effects of the oceanic turbulence with anisotropy on the SR are shown in Fig. 4 where the beam propagation distance is set to be Z = 2. Obviously, the SR gradually decreases for increasing ω, and for ω close to the maximum value, the decreasing trend becomes sharp. That’s to say, the salinity-induced turbulence fluctuation makes greater contribution to the decrease of beam quality comparing with the temperature-induced turbulence fluctuation. On the other hand, the SR increases for increasing ξ and ε or decreasing χT, as Figs. 4(a)–4(c) show. In addition, it is noted that the change of the SR is extremely subtle with χT = 10−5K2/s in Fig. 4(c). Comparing the red curve and blue curve in Fig. 4(b) with that in Fig. 4(c), one can conclude that the SR is more sensitive for altering χT. For example, in Fig. 4(c), the SR shows a decreasing trend within a small region under χT = 10−5K2/s and largely increases when χT decreases to χT = 10−9K2/s, however, the variation of SR is smaller than that in Fig. 4(c) as ε increases from 10−9m2/s3 to 10−5m2/s3.
Fig. 3 The Strehl ratio versus propagation distance in ocean turbulence under the the effects of the number of lattice lobes N with a = 1, ξ = 1, σ0 = 0.01m, ω = −2.5, ε = 10−5m2/s3.
Fig. 4 The Strehl ratio under different oceanic turbulence parameters with Z = 2, a = 1, N = 5 and σ0 = 0.01m for (a) ε = 10−5m2/s3, χT = 10−9K2/s, (b) ξ = 1, χT = 10−9K2/s and (c) ξ = 1, ε = 10−5m2/s3.
4.2. The power-in-the-bucket (PIB)
In addition, we are also interested in the beam power focusability in the target plane when laser beams propagate through the ocean link. The power-in-the-bucket (PIB), a factor that indicates how much the fraction of the total power is within a bucket size in the target plane, is a useful method to measure the beam power focusability. In order to simplify the calculations, we choose a square integral region, then PIB is defined as [11, 28]:
where w is the effective size of the detector. Substituting Eq. (14) into Eq. (18), the PIB can be expressed as: where the analytical expression of A1 is identical to that in Eq. (13). In the discussions, the number of lattice lobes N is set to be N = 5, and the analytical result of Eq. (18) can be obtained: where erf (·) is the error function andFigure 5 presents the PIB of OCLs versus different parameters in anisotropic ocean link. It can be clearly indicated that a larger w leads to the larger detection region, as a result, more energy of OCLs can be detected, and then the PIB increases as shown in Fig. 5(a). Figures 5(b)–5(e) present the PIB under different ocean parameters with Z = 3, obviously, the PIB gradually decreases with the increase of ω, moreover, the decreasing trend becomes sharp for ω increasing to its maximum value, which means that less beam power can be captured in the target plane due to the salinity-induced turbulence fluctuation, however, the larger w, ξ and ε or the smaller χT leads to the increased PIB, resulting in the enhanced beam power focusability in the target plane, thus the beam quality can be improved. In addition, in Fig. 5(c), the PIB almost remains the identical value under ω < −2.5 when ξ increases from 3 to 6, therefore, the PIB is not sensitive to the anisotropic factor ξ when ξ goes beyond a certain value and the temperature-induced fluctuation dominates the oceanic turbulence; For ω close to its maximum value, the PIB always tends to the identical value no matter how the parameters changed; Similar to the results in Fig. 4(c), the variation of PIB is almost negeligible with χT = 10−5K2/s and presents the larger sensitivity for altering χT comparing with Fig. 5(d). It is known that the turbulence strength is inversely proportional to the coherence length, and the influences of the parameters on the beam quality can be easily explained from the coherence length in Eq. (6). The larger ξ, ε or the smaller χT and ω leads to the larger coherence length ρocξ, which indicates the weak oceanic turbulence strength, thus the turbulence-induced phase is smaller, and the wavefront distortion gets weakened, then the degradation of beam quality becomes improved, as shown in Fig. 4 and Fig. 5. In [22], the propagation of OCLs in atmosphere turbulence is investigated, the conditions for the lattice structure evolution and the scintillation under the source parameters are discussed. However, in this paper, the object is to investigate the effects of anisotropic ocean turbulence on the evolution of OCLs as well as the beam quality, the influence of the lattice lobes on the beam quality is consistent with [10, 22], besides, the investigation of the change of the periodicity reciprocity preservation distance is potential in underwater communication because this kind of lattice structure can be used to transfer information, and the two factors SR and PIB are used to measure the beam quality, thus the impacts of the oceanic turbulence on the peak intensity and energy detection are indicated.
Fig. 5 The power-in-the-bucket (PIB) of OCLs in anisotropic ocean versus different parameters of w, ξ, ε, ω and χT with σ0 = 0.01m, (a) ω = −2.5, ε = 10−5m2/s3, a = 1, ξ = 1 and χT = 10−9K2/s, (b) Z = 3, ε = 10−5m2/s3, a = 1, ξ = 1 and χT = 10−9K2/s, (c) w = 5, Z = 3, ε = 10−5m2/s3, a = 1 and χT = 10−9K2/s, (d) w = 5, Z = 3, ξ = 1, a = 1 and χT = 10−9K2/s and (e) w = 5, Z = 3, ξ = 1, a = 1 and ε = 10−5m2/s3.
5. Conclusion
In this paper, the analytical expression of average intensity and beam quality (SR and PIB) of optical coherence lattices (OCLs) propagating through the anisotropic ocean turbulence are derived based on the extended Huygens-Fresnel principle. It is shown that the average intensity profile evolves from the lattice structure to the Gaussian shape under the effect of the turbulence, and the periodicity reciprocity preservation distance decreases with the stronger oceanic turbulence strength, while decreasing the lattice constant a can increase this preservation distance. In addition, the beam quality of OCLs depends on the number of lattice lobes and the oceanic turbulence parameters. The SR and PIB are largely degraded with the effect of the salinity-induced turbulence comparing with the temperature-induced turbulence, and can be increased with increasing the number of lattice lobes N, the anisotropic factor ξ and the rate of dissipation of turbulent kinetic energy per unit mass of fluid ε or decreasing the rate of dissipation of mean square temperature χT. Namely, the bigger value the SR or the PIB is, the better the beam quality of OCLs in anisotropic oceanic turbulence is. It is believed that the conclusions in this paper may be potential in applications for underwater communication.
Funding
National Natural Science Foundation of China (61571183); Hunan Provincial Natural Science Foundation of China (2017JJ1014).
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