Influence of oceanic turbulence on propagation of a radial Gaussian beam array

Lu Lu; Pengfei Zhang; Chengyu Fan; Chunhong Qiao

doi:10.1364/OE.23.002827

1. Introduction

The optical propagation through the turbulent atmosphere is a very important subject in the case of remote sensing, imaging, and communication systems and has attracted considerable theoretical and practical interest in the past decades. However, light propagation through seawater is relatively unexplored compared to that in atmospheric turbulence. With the development of underwater imaging and communicating systems, influence of oceanic turbulence on propagation of lights has already been attracted much attention. Recently, it is much convenient to research homogeneous and isotropic water based on the power spectrum of oceanic turbulence [1]. Dr. Korotkova has studied the intensity and coherence properties of light, the effect of polarization characteristics and numerically researched light scintillation in oceanic turbulence, respectively [2–4]. Scintillations of optical plane and spherical waves in underwater turbulence have been published [5]. The analytic formula of wave structure function for oceanic turbulence has been obtained [6]. The attenuation coefficient of all water (pure, distilled, or natural) varies markedly with wavelength has already researched by S. Q. Duntley since 1960s [7]. However, the wavelength is absorbed much less by seawater when wavelength approximately equals to 417nm [8]. Due to its chaotic characteristics (including the absorption and scattering phenomena), the optical propagation range in underwater media is a few tens of meters [9–11]. Considering this reason, we try to use the array beams to get high power which can achieve a relatively long propagation distance. In practice, the laser beam array consisting of more than one beam is often encountered and widely used in many fields, such as the high-power system and the inertial confinement fusion. Up to now a variety of linear, radial and rectangular laser arrays have been developed to achieve high system powers [12,13]. Until now, less work has been reported that laser array beams propagate in oceanic turbulence. In this paper, taken the radial Gaussian array beam as a typical example of laser array beams, we study the properties of spreading of a radial Gaussian beam array in the ocean. It is believed that the results will benefit to study propagation properties on oceanic turbulence.

2. Rms beam width

2.1 Analytical formulae

Assuming that the a radial Gaussian beam array consists of $N$ equal Gaussian beams, which are located symmetrically on a ring with the radius $r_{0}$ , and the separation angle between two adjacent Gaussian beams is $ϕ_{0} = 2 π / N$ , as shown in Fig. 1. For the coherent combination case, the cross-spectral density function of the radial Gaussian array beams at the source plane $z = 0$ can be expressed as [14]

\begin{array}{l} W^{(0)} (x_{1}, y_{1}, x_{2}, y_{2}, z = 0) = \sum_{m = 0}^{N - 1} \sum_{n = 0}^{N - 1} \exp [- \frac{{(x_{1} - r_{0} \cos α_{m})}^{2} + {(y_{1} - r_{0} \sin α_{m})}^{2}}{w_{0}^{2}}] \\ \begin{matrix}  \end{matrix} \times \exp [- \frac{{(x_{2} - r_{0} \cos α_{n})}^{2} + {(y_{2} - r_{0} \sin α_{n})}^{2}}{w_{0}^{2}}], \end{array}

where

α_{j} = j φ_{0}, (j = 0, 1, 2, ..., N - 1, N \geq 2)

, and

w_{0}

is the waist width of Gaussian beams.

Fig. 1 Schematic diagram of the radial beam array.

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Based on the extended Huygens-Fresnel principle, the average intensity of the coherent combined Gaussian array beams represented by Eq. (1) propagating through oceanic turbulence at the receiver plane z reads as [14]

\begin{array}{l} 〈 I (x, z) 〉 = {(\frac{k}{2 π z})}^{2} \iint \iint d x_{1} d y_{1} d x_{2} d y_{2} W^{(0)} (x_{1}, y_{1}, x_{2}, y_{2}, z = 0) \\ \begin{matrix}  \end{matrix} \times \exp {\frac{i k}{2 z} [{(x - x_{1})}^{2} + {(y - y_{1})}^{2} - {(x - x_{2})}^{2} - {(y - y_{2})}^{2}]} \\ \begin{matrix}  \end{matrix} \times {〈 \exp [ψ^{*} (x, y, x_{1}, y_{1}, z) + ψ (x, y, x_{2}, y_{2}, z)] 〉}_{m}, \end{array}

where k is the wave number related to the wavelength

λ

by

k = 2 π / λ

.

< • >_{m}

denotes average over the ensemble of the turbulent medium, and [14]

\begin{array}{l} {〈 \exp [ψ^{*} (x, y, x_{1}, y_{1}, z) + ψ (x, y, x_{2}, y_{2}, z)] 〉}_{m} \\ \begin{matrix}  \end{matrix} = \exp {- 4 π^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) [1 - J_{0} (κ ξ \sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}}) d κ d ξ]}, \end{array}

where

J_{0} (•)

is the Bessel function of the first kind and order zero.

Φ_{n} (κ)

is the power spectrum of oceanic turbulence,

κ

is the magnitude of spatial wave number.

According to [2], the power spectrum for homogeneous and isotropic oceanic water when the eddy thermal diffusivity and the diffusion of salt are assumed to be equal is given by the expression

Φ_{n} (κ) = 0.388 \times 10^{- 8} ε^{- 1 / 3} κ^{- 11 / 3} [1 + 2.35 {(κ η)}^{2 / 3}] \frac{χ_{T}}{w^{2}} (w^{2} e^{- A_{T} δ} + e^{- A_{S} δ} - 2 w e^{- A_{T S} δ}),

where

ε

is the rate of dissipation of kinetic energy per unit mass of fluid ranging from

10^{- 1} m^{2} / s^{3}

to

10^{- 10} m^{2} / s^{3}

,

χ_{T}

is the rate of dissipation of mean-squared temperature and has the range

10^{- 4} K^{2} / s

to

10^{- 10} K^{2} / s

,

w

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 [2]. Additionally,

η

is the Kolmogorov micro scale (inner scale), and

A_{T} = 1.863 \times 10^{- 2}

,

A_{S} = 1.9 \times 10^{- 4}

,

A_{T S} = 9.41 \times 10^{- 3}

,

δ = 8.284 {(κ η)}^{4 / 3} + 12.978 {(κ η)}^{2}

[2].

Substituting from Eq. (1), (3)-(4) into Eq. (2), the Rms beam width of Gaussian array beams propagating in oceanic turbulence, i.e., [14]

W_{t u r b} = \sqrt{A + (B / k^{2}) z^{2} + F z^{3}},

where

A = \frac{1}{G} \sum_{m = 0}^{N - 1} \sum_{n = 0}^{N - 1} S {w_{0}^{2} + r_{0}^{2} [1 + \cos (α_{m} - α_{n})]},

B = \frac{1}{G} \sum_{m = 0}^{N - 1} \sum_{n = 0}^{N - 1} \frac{2 S}{w_{0}^{4}} {w_{0}^{2} - r_{0}^{2} [1 - \cos (α_{m} - α_{n})]},

S = \exp {r_{0}^{2} [1 - \cos (α_{m} - α_{n})] / w_{0}^{2}},

G = \sum_{m = 0}^{N - 1} \sum_{n = 0}^{N - 1} S,

F = 0.388 \times 10^{- 8} (4 π^{2} / 3 w^{2}) ε^{- 1 / 3} χ_{T} (67.832 w^{2} - 176.699 w + 475.692) .

in this paper, Eq. (10) which denotes the effect of the turbulence on the beam can be deduced in Appendix.

2.2 Numerical calculation results and analysis

In our paper, wavelength $λ = 0.417 μm$ , waist width $w_{0} = 0.01 m$ , radius $r_{0} = 0.02 m$ , inner scale $η = 10^{- 3} m$ , propagation distance $z = 50 m$ and $N = 10$ are chosen. In order to examine the accuracy of the calculation method on oceanic turbulence, we give a comparison of results of $F$ calculated by the analytical expression obtained in Eq. (10) and by numerical calculation of the formula of Eq. (11) in [14] (i. e., $F = \frac{4 π^{2}}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ$ ), it can be seen that the two results are in agreement with each other exactly with three oceanic parameters in Fig. 2.

Fig. 2 Changing of F versus (a) $\log χ_{T}$ , (b) $\log ε$ and (c) w.

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Further, Rms beam width for oceanic turbulence as a function of oceanic parameter for radial beam array with coherent combination are analyzed in Fig. 3. Figure 3(a) is shown that Rms beam width increases as $\log χ_{T}$ increases, and the curve rises rapidly when strong turbulence (i.e., $χ_{T} > 10^{- 6} K^{2} / s$ ) is satisfied. It is worth to be mentioned that the higher values of $χ_{T}$ generally determine the more energetic turbulence in [15,16]. Rms beam width is inversely proportional to $\log ε$ in Fig. 3(b). It is known that $ε$ has a closed connection with turbulence scale. The smaller $ε$ is, the larger turbulence scale is. Meanwhile, the spreading of beam will be wider. Based on experimental datum [16], the surface of ocean has large $ε$ (or the small turbulence scale) which brings to weak turbulence. The salinity-induced influence is much stronger than temperature-induced. From Fig. 3(c), it is shown that the spreading of laser beams when w is very close to 0 (i.e., dominating salinity-induced optical turbulence) is much larger than the case of w equals to −5 (i.e., dominating temperature-induced optical turbulence). Besides, from Fig. 3(a)-(c), at the point where $χ_{T} = 10^{- 7} K^{2} / s$ , $ε = 10^{- 5} m^{2} / s^{3}$ and $w = - 3$ , the Rms beam width approximates to 0.02878m. It is practical that the beam spreading is less than 0.03m when light propagates through seawater approximately 50m.

Fig. 3 Rms beam width for oceanic turbulence as a function of (a) $\log χ_{T}$ , (b) $\log ε$ and (c) w for radial beam array with coherent combination.

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3. Effective radius of curvature

3.1 Formulae

The transformation of the mean squared beam width of an arbitrary field by the optical ABCD system can be expressed as [17]

〈 r_{2}^{2} 〉 = 〈 {(A r_{1} + B θ_{1})}^{2} 〉,

where A, B, C and D are elements of the transfer matrix

(\begin{matrix} A & B \\ C & D \end{matrix})

,

θ \equiv (θ_{x}, θ_{y})

,

k θ_{x}

and

k θ_{y}

are the wave vector components along the x-axis and y-axis respectively. The parameters with the subscripts “1” and “2” denote those before and after the optical ABCD system respectively.

Comparing the propagation equation of the mean squared beam width of an arbitrary field (i.e., Eq. (11)) with that of an ideal Gaussian beam, the effective radius of curvature is defined by [17]

R = 〈 r^{2} 〉 / 〈 r \cdot θ 〉,

the general formulae of the second moments

〈 r^{2} 〉

and

〈 r \cdot θ 〉

of partially coherent beams propagating through turbulence can be expressed as [18]

〈 r^{2} 〉 = {〈 r^{2} 〉}_{0} + 2 〈 r \cdot θ 〉 z + {〈 θ^{2} 〉}_{0} z^{2} + F z^{3},

〈 r \cdot θ 〉 = {〈 r \cdot θ 〉}_{0} + {〈 θ^{2} 〉}_{0} z + (3 / 2) F z^{2},

the effective radius of curvature of a radial Gaussian beam array propagating through atmospheric turbulence [19], in this paper, which for oceanic turbulence can be expressed as

R = \frac{A + (B / k^{2}) z^{2} + F z^{3}}{(B / k^{2}) z + (3 / 2) F z^{2}} .

3.2. Numerical analysis

It has been shown that effective radius of curvature R decreases due to the strongly strength of turbulence for coherent combination. The power of turbulence determines the value of effective radius of curvature. As we know, the weak turbulence has the following characteristics (i.e., small $χ_{T}$ , larger $ε$ and salinity-induced predominant) in this part. In Fig. 4(a), the effective radius of curvature decreases as $\log χ_{T}$ increases. Especially, when strong turbulence can be satisfied (i.e., $χ_{T} > 10^{- 6} K^{2} / s$ ), the effective radius of curvature decreases slightly. In this case, the smaller effective radius of curvature is equivalent to the wider beam spreading. When propagating through weak to strong turbulence, the beam width will be spread gradually. Meanwhile, the quality of the beam will be affected much seriously in stronger turbulence. In Fig. 4(b) the effective radius of curvature increases with $\log ε$ increased. In Fig. 4(c), the effective radius of curvature has a smaller value when salinity-induced optical turbulence dominates.

Fig. 4 Effective radius of curvature for oceanic turbulence as a function of (a) $\log χ_{T}$ , (b) $\log ε$ and (c) w for radial beam array.

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5. Conclusions

In this paper, the analytical expression of Rms beam width of a radial Gaussian beam array has been derived. For effect of the turbulence on the beam, much work has just been researched by numerical calculation on $F = \frac{4 π^{2}}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ$ . The analytical expression of F has been obtained in our paper. Giving a comparison of results of F calculated by the analytical expression and the previously integrating one, it can be seen that the two results are in agreement with each other exactly. Besides, the influences of $χ_{T}$ , $ε$ and w are investigated. Rms beam width increases as $\log χ_{T}$ increased, $\log ε$ decreased and salinity-induced dominant. The spreading of laser beams will strengthen when larger $χ_{T}$ , smaller $ε$ and larger w satisfied. Further, the changings of effective radius of curvature with three parameters are studied. Effective radius of curvature decreases when $χ_{T}$ and w increase, and $ε$ decreases. The power of turbulence determines the value of effective radius of curvature. The stronger turbulence leads to the much smaller effective radius of curvature and wider beam spreading. It is believed that the findings of this study may be useful in applications for underwater propagating, imaging system.

Appendix

Based on the expression of F [14], substituting Eq. (4) into it, we can obtain the expression as

\begin{array}{l} F = 0.388 \times 10^{- 8} π^{2} ε^{- 1 / 3} \frac{χ_{T}}{w^{2}} [\int_{0}^{\infty} κ^{- 2 / 3} (w^{2} e^{- a κ^{\frac{4}{3}} - b κ^{2}} + e^{- c κ^{\frac{4}{3}} - d κ^{2}} - 2 w e^{- e κ^{\frac{4}{3}} - f κ^{2}}) d κ \\ \begin{matrix}  \end{matrix} + \int_{0}^{\infty} 2.35 η^{2 / 3} (w^{2} e^{- a κ^{\frac{4}{3}} - b κ^{2}} + e^{- c κ^{\frac{4}{3}} - d κ^{2}} - 2 w e^{- e κ^{\frac{4}{3}} - f κ^{2}}) d κ], \end{array}

where

a = 8.284 A_{T} η^{4 / 3}

,

b = 12.978 A_{T} η^{2}

,

c = 8.284 A_{S} η^{4 / 3}

,

d = 12.978 A_{S} η^{2}

,

e = 8.284 A_{T S} η^{4 / 3}

,

f = 12.978 A_{T S} η^{2}

, and

g = 2.35 η^{2 / 3}

. above integrals can be expressed as [6]

\begin{array}{l} \int_{0}^{\infty} κ^{- \frac{2}{3}} e^{- Q κ^{\frac{4}{3}} - R κ^{2}} d κ = \frac{1}{4} R^{- \frac{3}{2}} {2 R^{\frac{4}{3}} Γ (\frac{1}{6}) {}_{2}F_{2} (\frac{1}{12}, \frac{7}{12}; \frac{1}{3}, \frac{2}{3}; - \frac{4 Q^{3}}{27 R^{2}}) \\ \begin{matrix} \begin{array}{l} - 2 Q R^{\frac{2}{3}} Γ (\frac{5}{6}) {}_{2}F_{2} (\frac{5}{12}, \frac{11}{12}; \frac{2}{3}, \frac{4}{3}; - \frac{4 Q^{3}}{27 R^{2}}) \\ + Q^{2} Γ (\frac{3}{2}) {}_{2}F_{2} (\frac{3}{4}, \frac{5}{4}; \frac{4}{3}, \frac{5}{3}; - \frac{4 Q^{3}}{27 R^{2}})}, \end{array} \end{matrix} \end{array}

\begin{array}{l} \int_{0}^{\infty} e^{- Q κ^{\frac{4}{3}} - R κ^{2}} d κ = \frac{1}{4} R^{- \frac{11}{6}} {2 R^{\frac{4}{3}} Γ (\frac{1}{2}) {}_{2}F_{2} (\frac{1}{4}, \frac{3}{4}; \frac{1}{3}, \frac{2}{3}; - \frac{4 Q^{3}}{27 R^{2}}) \\ \begin{matrix} \begin{array}{l} - 2 Q R^{\frac{2}{3}} Γ (\frac{7}{6}) {}_{2}F_{2} (\frac{7}{12}, \frac{13}{12}; \frac{2}{3}, \frac{4}{3}; - \frac{4 Q^{3}}{27 R^{2}}) \\ + Q^{2} Γ (\frac{11}{6}) {}_{2}F_{2} (\frac{11}{12}, \frac{17}{12}; \frac{4}{3}, \frac{5}{3}; - \frac{4 Q^{3}}{27 R^{2}})}, \end{array} \end{matrix} \end{array}

where

Γ (•)

is the Gamma function and

{}_{p}F_{q} (a_{1}, ..., a_{p}; c_{1}, ... c_{q}; x)

is the generalized hypergeometric function, where p and q are positive integers. For our case (i.e., power spectrum in Eq. (4)), we have proved that

\frac{4 Q^{3}}{27 R^{2}} ≪ 1

is always satisfied. Thus, for our case we can simplify the integration result from Eq. (10) by using the following formula (see Eq. (8) in [20]) for the

| x | ≪ 1

case, i.e.

{}_{2}F_{2} (α, β; γ, ς; - x) \approx 1 - \frac{α β x}{γ ς}, \begin{matrix} \begin{matrix} | x | ≪ 1 \end{matrix} \end{matrix}

Subtracting Eq. (17)-(19) into Eq. (16), the analytical expression is

F = 0.388 \times 10^{- 8} (4 π^{2} / 3 w^{2}) ε^{- 1 / 3} χ_{T} (67.832 w^{2} - 176.699 w + 475.692) .

It is clear that Eq. (20) is the same as Eq. (10) in the text.

Acknowledgment

The authors are very thankful to the reviewers for valuable comments. Chengyu Fan, Lu Lu, Pengfei Zhang and Chunhong Qiao acknowledge the support by the National Natural Science Foundation of China (NSFC) under grant 61405205 and 61475105.

References and links

1. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000). [CrossRef]

2. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285(6), 872–875 (2012). [CrossRef]

3. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011). [CrossRef]

4. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012). [CrossRef]

5. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014). [CrossRef] [PubMed]

6. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence,” Opt. Express 22(22), 27112–27122 (2014). [CrossRef] [PubMed]

7. S. Q. Duntley, “Light in the sea,” J. Opt. Soc. Am. A 53(2), 214–233 (1963). [CrossRef]

8. R. M. Pope and E. S. Fry, “Absorption spectrum (380-700 nm) of pure water. II. Integrating cavity measurements,” Appl. Opt. 36(33), 8710–8723 (1997). [CrossRef] [PubMed]

9. F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49(16), 3224–3230 (2010). [CrossRef] [PubMed]

10. F. Hanson and S. Radic, “High bandwidth underwater optical communication,” Appl. Opt. 47(2), 277–283 (2008). [CrossRef] [PubMed]

11. M. C. Haller, “Review of selected oceanic EM/EO scattering problems,” Proc. SPIE 7588(75880C), 75880C (2010). [CrossRef]

12. 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 wave guide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996). [CrossRef]

13. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24 channel radial array slab RF-excited carbon dioxide laser, gas and chemical lasers and applications,” Proc. SPIE 2987, 13–21 (1997). [CrossRef]

14. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011). [CrossRef] [PubMed]

15. D. Bogucki, A. Domaradzki, J. R. V. Zaneveld, and T. D. Dickey, “Light Scattering Induced by Turbulent Flow,” Proc. SPIE 2258, 247–255 (1994). [CrossRef]

16. S. A. Thorpe, An Introduction to Oceanic Turbulence (Cambridge University, 2007).

17. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027–S1049 (1992). [CrossRef]

18. Y. Dan and B. Zhang, “Second moments of partially coherent beams in atmospheric turbulence,” Opt. Lett. 34(5), 563–565 (2009). [CrossRef] [PubMed]

19. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010). [PubMed]

20. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Influence of oceanic turbulence on propagation of a radial Gaussian beam array

Abstract

1. Introduction

2. Rms beam width

2.1 Analytical formulae

2.2 Numerical calculation results and analysis

3. Effective radius of curvature

3.1 Formulae

3.2. Numerical analysis

5. Conclusions

Appendix

Acknowledgment

References and links

Cited By

Figures (4)

Equations (20)

Optics Express