New model of the fiber coupling efficiency of a partially coherent Gaussian beam in an ocean to fiber link

Beibei Hu; Yixin Zhang; Yun Zhu

doi:10.1364/OE.26.025111

1. Introduction

Knowledge of the efficiency with which random light can be coupled to optical fibers is important for the receiver design of oceanic and atmospheric link [1]. Since Winze and Leeb discussed the fiber coupling efficiency of random light and its application in lidar [1], numerous studies have attempted to explore the fiber coupling efficiency for the link of turbulent atmosphere to fiber [2–7]. And with the development of free space optical communication technology, the propagation of laser beams through oceanic turbulence has received increasingly attention in recent years [8–15]. For the link of turbulent ocean to fiber, Hanson et al. [9] measured the coupling efficiency of a focused beam into a single-mode fiber in laboratory experiment. Their experiment of simple tip-tilt control system shown to maintain good coupling efficiency with the beam radius comparable to the transverse coherence length. Hu et al. [8] discussed the impact of turbulence on the fiber-coupling efficiency of Gaussian-Schell model beams propagating in the link of ocean to fiber. This work shown that the salinity fluctuation has a greater impact on the fiber-coupling efficiency than temperature fluctuation does, and the fiber-coupling efficiency can be achieved the maximum by choosing design parameter according specific oceanic turbulence condition. However, this models ignore the effect of turbulence on average light intensity. It makes these models to be only applicable in weak turbulent link of ocean/atmosphere to fiber. In order to be able to calculate the coupling efficiency of ocean/atmosphere to fiber in moderate to strong turbulent region, we must find a new model.

The purpose of this work is to develop a theoretical model for the fiber coupling efficiency to include the effect of turbulence on average light intensity, and obtain the knowledge of the efficiency for a partially coherent Gaussian (PCG) beam propagating in the link of anisotropic oceanic turbulence to fiber. In the analysis, we only consider the influence of anisotropic oceanic turbulence due to temperature and salinity fluctuations on the coupling efficiency. This paper is organized as follows: Section 2 gives an expression of fiber coupling efficiency of a PCG beam in the link of seawater to fiber. The analysis of the effect of link parameters on the coupling efficiency of a PCG beam is given in Section 3. The conclusions are drawn in Section 4.

2. Fiber coupling efficiency of a PCG beam in the link of anisotropic oceanic turbulence to fiber

The fiber coupling efficiency in the link of free space to fiber is defined as the ratio of the average optical power coupled into the fiber 〈P_c〉, to the average available optical power in the receiver aperture plane 〈P_a〉 [1]. For PCG beam, wave propagation through oceanic turbulence is characterized in terms of the cross-spectral density function [16], so we have

η = \frac{〈 P_{c} 〉}{〈 P_{a} 〉} = \frac{〈 {| \iint_{A} E_{A} (r) F_{A}^{*} (r) |}^{2} d r 〉}{〈 {| \iint_{A} E_{A} (r) |}^{2} d r 〉} = \frac{\iint_{A} F_{A}^{*} (r_{1}) F_{A} (r_{2}) W (r_{1}, r_{2}, z) d r_{1} d r_{2}}{〈 {| \iint_{A} E_{A} (r) |}^{2} d r 〉},

where E_A(r) characterizes the incident optical field in the receiver aperture plane A, * indicates the complex conjugate, W(r₁, r₂, z) is the cross-spectral density function of the GSM beams propagating through oceanic turbulence at the propagation distance z, and F_A(r) is the backpropagated fiber-mode profile and is given by [4,8]

F_{A} (r) = \frac{\sqrt{2 π} ω_{0}}{λ f} exp [- {(\frac{π ω_{0} r}{λ f})}^{2}],

where λ is the laser wavelength, f is the focal length of the coupling lens, and ω₀ is the fiber-mode field radius.

At z = 0 the free-space electric field of a unit-amplitude, lowest-order paraxial Gaussian beam propagating predominantly along the z axis can be represented in the form [17]

u (ρ, 0) = exp [- (\frac{1}{w_{0}^{2}} + i \frac{k}{2 R_{0}}) ρ^{2}],

where w₀ is the transmitter beam radius (beam size), R₀ is the radius of curvature of the phase front, k = 2π/λ is the optical wave number, and ρ = (x² + y²)^1/2 is the radial distance from the beam center line; for simplicity we introduce the notation ρ² = |ρ|².

The complex field of the distance z from the transmitter can be represented, using the generalized Huygens–Fresnel principle, as [16]

u (r, z) = - \frac{i k}{2 π z} \iint d^{2} ρ u (ρ, 0) exp [- i k z + \frac{i k {| r - ρ |}^{2}}{2 z} + ψ (ρ, r)],

where ψ(ρ, r) represents the random part of the complex phase of a spherical wave due to propagation in a turbulent medium and r has the same definition in the plane z as ρ in the z = 0 plane.

Now, the cross-spectral density at the receiver in Eq. (1) can be represented as

\begin{array}{l} W (r_{1}, r_{2}, z) = & {(\frac{k}{2 π z})}^{2} \iint d^{2} ρ_{1} \iint d^{2} ρ_{2} W (ρ_{1}, ρ_{2}, 0) {〈 exp [ψ^{*} (ρ_{1}, r_{1}, z) + ψ (ρ_{2}, r_{2}, z)] 〉}_{o} \\ \times exp {[i k \frac{{(r_{1} - ρ_{1})}^{2} - {(r_{2} - ρ_{2})}^{2}}{2 z}]}^{2}, \end{array}

where symbol 〈...〉_o represents the ensemble average of the oceanic turbulence medium.

In Eq. (5) the quantity W(ρ₁, ρ₂, 0) is the cross-spectral density at the transmitter, which can be represented as [16]

W (ρ_{1}, ρ_{2}, 0) = u (ρ_{1}, 0) u^{*} (ρ_{2}, 0) exp [- \frac{{(ρ_{1} - ρ_{2})}^{2}}{2 σ_{0}^{2}}] .

In Eq. (6) the quantity σ₀ denotes the spatial correlation length, which describes the partial coherence properties of the transmitter source. Note that when $σ_{0}^{2}$ is infinite, the cross-spectral density is completely described by the deterministic field.

Using the sum and difference vector notation [16]

r_{s} = \frac{1}{2} (r_{1} + r_{2}), r_{d} = r_{1} - r_{2}, ρ_{s} = \frac{1}{2} (ρ_{1} + ρ_{2}), ρ_{d} = ρ_{1} - ρ_{2},

we can express the cross-spectral density at the transmitter as

W (ρ_{d}, ρ_{s}, 0) = exp {- \frac{1}{w_{0}^{2}} [\frac{1}{2} (ρ_{d}^{2} + 4 ρ_{s}^{2})]} - i k \frac{ρ_{d} \cdot ρ_{s}}{R_{0}} - \frac{ρ_{d}^{2}}{2 σ_{0}^{2}},

and we can also write

exp {\frac{i k}{2 z} [{(r_{1} - ρ_{1})}^{2} - {(r_{2} - ρ_{2})}^{2}]} = exp [\frac{i k}{z} (ρ_{s} - r_{s}) - (ρ_{d} - r_{d})] .

In Eq. (5), the random part of the complex phase of a spherical wave propagating in turbulence can be approximated by [16]

{〈 exp [ψ^{*} (ρ_{1}, r_{1}, z) + ψ (ρ_{2}, r_{2}, z)] 〉}_{o} ≅ exp (- \frac{r_{d}^{2} + ρ_{d}^{2} + r_{d} \cdot ρ_{d}}{ρ_{0}^{2}}),

where ρ₀ is the coherence length of a spherical wave propagating in anisotropic oceanic turbulence and is given by

ρ_{0} = {[\frac{π^{2} k^{2} z}{3} \iint_{0}^{\infty} κ^{3} Φ_{n} (κ_{ζ}, ζ) d κ]}^{- 1 / 2},

and Φ_n (κ_ζ, ζ) is the spatial wavenumber power spectrum of the refractive index fluctuations of the anisotropy turbulent ocean. As mentioned earlier, we consider the influence of anisotropic oceanic turbulence due to temperature and salinity fluctuations on the beam, using the spatial wavenumber power spectrum of anisotropic turbulence model in [15,18]

Φ_{n} (κ_{ζ}, ζ) = 0.388 \times 10^{- 8} ε^{- 1 / 3} χ_{t} ζ^{2} κ_{ζ}^{- 11 / 3} [1 + 2.35 {(κ_{ζ} l_{0})}^{2 / 3}] f (κ_{ζ}, ϖ),

where

κ_{ζ} = {(κ_{z}^{2} + ζ^{2} κ^{2})}^{1 / 2}

,

κ = {(κ_{x}^{2} + κ_{y}^{2})}^{1 / 2}

, κ is the spatial frequency of turbulent fluctuations, ε is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 10⁻¹⁰ m²/s³ to 10⁻¹ m²/s³, χ_t is the rate of dissipation of mean-square temperature variance and has the range 10⁻¹⁰ K²/s to 10⁻⁴ K²/s, ϖ is the ratio of temperature and salinity contributions to the refractive index spectrum, which in the seawater can vary from −5 to 0, with −5 and 0 corresponding to dominating temperature-induced and salinity-induced optical turbulence respectively, l₀ and ζ are the inner scale of oceanic turbulence and anisotropic anisotropic factor of oceanic turbulence respectively;

f (κ_{ζ}, ϖ) = exp (- A_{T} δ) + ϖ^{- 2} exp (- A_{S} δ) - 2 ϖ^{- 1} exp (- A_{T S} δ),

where δ = 8.284(κ_ζl₀)^4/3 + 12.978(κ_ζl₀)², A_T = 1.863 × 10⁻², A_S = 1.9 × 10⁻⁴, and A_TS = 9.41 × 10⁻³.

By substituting Eqs. (11) and (12) into Eq. (10), performing several steps of calculations, we can obtain [13,19]

ρ_{0}^{- 2} = 18.02 C_{m}^{2} k^{2} ζ^{- 2} z l_{0}^{- 1 / 3} (0.483 - 0.835 ϖ^{- 1} + 3.38 ϖ^{- 2}),

where

C_{m}^{2} = 10^{- 8} ε^{- 1 / 3}

χ_t is define as the “equivalent” temperature structure constant.

From the above discussion, we can now express the cross-spectral density at the field point:

\begin{array}{l} W (r_{d}, r_{s}, z) = & {(\frac{k}{2 π z})}^{2} \iint d^{2} ρ_{s} \iint d^{2} ρ_{d} exp [- \frac{2 ρ_{s}^{2}}{w_{0}^{2}} - i k \frac{ρ_{s} \cdot ρ_{d}}{R_{0}} + i k \frac{(ρ_{d} - r_{d})}{z}] \\ \times exp [- \frac{ρ_{d}^{2}}{2 w_{0}^{2}} - \frac{ρ_{d}^{2}}{2 σ_{0}^{2}} - \frac{ρ_{d}^{2} + ρ_{d} \cdot r_{d} + r_{d}^{2}}{ρ_{0}^{2}} - i k \frac{r_{s} \cdot (ρ_{d} - r_{d})}{z}] . \end{array}

Evaluating this integral, we obtain the expression for the cross-spectral density at the receiver:

W (r_{d}, r_{s}, z) = \frac{w_{0}^{2}}{w^{2} (z)} exp [- r_{d}^{2} L - \frac{2 r_{s}^{2}}{w^{2} (z)} + i M (r_{d} \cdot r_{s})],

where

w (z) = w_{0} {[{(1 - \frac{z}{R_{0}})}^{2} + \frac{4 z^{2}}{k^{2} w_{0}^{4}} (1 + \frac{w_{0}^{2}}{σ_{0}^{2}} + \frac{2 w_{0}^{2}}{ρ_{0}^{2}})]}^{1 / 2}

,

ϕ = \frac{k w_{0}^{2}}{2 z} (1 - \frac{z}{R_{0}}) - \frac{2 z}{k ρ_{0}^{2}}

,

L = \frac{1}{ρ_{0}^{2}} + \frac{k^{2} w_{0}^{2}}{8 z^{2}} - \frac{ϕ^{2}}{2 w^{2} (z)}

, and

M = \frac{k}{z} - \frac{2 ϕ}{w^{2} (z)}

.

Form Eq. (15), we can obtain the average light intensity

I (r, z) = W (r_{d} = 0, r_{s} = r, z) = \frac{w_{0}^{2}}{w^{2} (z)} exp (- \frac{2 r^{2}}{w^{2} (z)}) .

By Eq. (16), we obtain the analytical expression of the denominator in the Eq. (1):

〈 \iint_{A} {| E_{A} (r) |}^{2} d r 〉 = \frac{π w_{0}^{2}}{2} [1 - exp (- \frac{D^{2}}{2 w^{2} (z)})],

where D is the aperture diameter of the receiver.

Substituting Eqs. (17), (15) and (2) into Eq. (1), we express the fiber coupling efficiency of the PCG beam through anisotropic oceanic turbulence

\begin{array}{l} η_{1} = \frac{4 ω_{0}^{2}}{λ^{2} f^{2} w^{2} (z) [1 - exp (- \frac{D^{2}}{2 w^{2} (z)})]} \int_{0}^{D / 2} \int_{0}^{D / 2} \int_{0}^{2 π} \int_{0}^{2 π} exp [- (L + \frac{π^{2} ω_{0}^{2}}{2 λ^{2} f^{2}}) r_{d}^{2} \\ - (\frac{2}{w^{2} (z)} + \frac{2 π^{2} ω_{0}^{2}}{λ^{2} f^{2}}) r_{s}^{2}] exp [i M r_{d} r_{s} cos (φ_{d} - φ_{s})] r_{d} r_{s} d r_{d} d r_{s} d φ_{d} d φ_{s} . \end{array}

The results in the double integral over the angle variables φ_d and φ_s is given by

\int_{0}^{2 π} \int_{0}^{2 π} exp [i M r_{d} r_{s} cos (φ_{d} - φ_{s})] d φ_{d} d φ_{s} = 4 π^{2} J_{0} (M r_{d} r_{s}),

where J₀(...) denotes the Bessel function of first kind and zero order. Substituting Eq. (19) into Eq. (18), we arrive at

\begin{array}{l} η_{1} = \frac{16 π^{2} ω_{0}^{2}}{λ^{2} f^{2} w^{2} (z) [1 - exp (- \frac{D^{2}}{2 w^{2} (z)})]} \int_{0}^{D / 2} \int_{0}^{D / 2} exp [- (L + \frac{π^{2} ω_{0}^{2}}{2 λ^{2} f^{2}}) r_{d}^{2} \\ - (\frac{2}{w^{2} (z)} + \frac{2 π^{2} ω_{0}^{2}}{λ^{2} f^{2}}) r_{s}^{2}] J_{0} [M r_{d} r_{s}] r_{d} r_{s} d r_{d} d r_{s} . \end{array}

By normalizing the radial integration variables to the receiver lens radius, we define x₁ = 2r_d/D, and x₂ = 2r_s/D, η₁ is then rewritten as

\begin{array}{l} η_{1} = & \frac{D^{4} π^{2} ω_{0}^{2}}{λ^{2} f^{2} w^{2} (z) [1 - exp (- \frac{D^{2}}{2 w^{2} (z)})]} \int_{0}^{1} \int_{0}^{1} exp [- (\frac{D^{2} L}{4} + \frac{D^{2} π^{2} ω_{0}^{2}}{8 λ^{2} f^{2}}) x_{1}^{2} \\ - (\frac{D^{2}}{2 w^{2} (z)} + \frac{D^{2} π^{2} ω_{0}^{2}}{2 λ^{2} f^{2}}) x_{2}^{2}] J_{0} (M D^{2} \frac{x_{1} x_{2}}{4}) x_{1} x_{2} d x_{1} d x_{2} . \end{array}

In the old model, the average input optical intensity is considered independent of r₁ or r₂, so the the denominator in Eq. (1) equal to πD²I/4, by this approximation, we have

\begin{array}{l} η_{2} = & \frac{2 D^{2} π^{2} ω_{0}^{2}}{λ^{2} f^{2}} \int_{0}^{1} \int_{0}^{1} exp [- (\frac{D^{2} L}{4} - \frac{D^{2}}{8 w^{2} (z)} + \frac{D^{2} π^{2} ω_{0}^{2}}{8 λ^{2} f^{2}}) x_{1}^{2} - \frac{D^{2} π^{2} ω_{0}^{2}}{2 λ^{2} f^{2}} x_{2}^{2}] \\ \times J_{0} (M D^{2} \frac{x_{1} x_{2}}{4}) x_{1} x_{2} d x_{1} d x_{2} . \end{array}

3. Analysis of fiber coupling efficiency of a PCG beam in the link of seawater to fiber with a weak to strong anisotropic oceanic turbulence

In this section, we present the numerical discussions for the fiber coupling efficiency of PCG beams through anisotropic oceanic turbulence as the functions of source’s and turbulent parameters. On account of the mismatching between the incident beam and axial positioning of fiber is the main factor to reduce coupling efficiency [2], and the mismatching between the collimated beams and fiber axial positioning is minimum, so in the following calculations, we choose R₀ = +∞. Furthermore, in the discussion below, unless otherwise specified all plots have w₀ = 0.014 m, D = 0.05 m, λ = 532 nm, ϖ = −3, ζ = 1, σ₀ = +∞, $C_{m}^{2} = 10^{- 16} m^{- 2 / 3} K^{2}$ , l₀ = 1 mm, ω₀ = 5 μm, f = 0.4 m and z = 50 m. In addition, we use the ⋄, ▸ and ○ marked curve to represent the curves calculated by the model in this paper, and the unmarked curve to represent the curves calculated by the model in [1,4].

The impact of focal length of the coupling lens f and wavelength λ on two models of fiber coupling efficiency η is shown in Fig. 1. We set f changes from 0.1 to 0.8 m for the conditions λ = 417, 488, and 532 nm. Note that these wavelengths are typical oceanic transmission window wavelengths due to their small absorption loss [20]. It is clear from Fig. 1 that as the focal length increases, the fiber coupling efficiency of two models increase too and η also increases as the wavelength λ increases. There is an optimum focal length corresponding to the maximum coupling efficiency for each given λ. Therefore, we can choose an optimum focal length f and a longer wavelength λ in transmission window to improve the fiber coupling efficiency.

Fig. 1 The fiber coupling efficiency η of two models as a function of focal length f with different wavelength λ.

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Figure 2 demonstrates the influence of propagation distance z and coherent length σ₀ on the fiber coupling efficiency η. When σ₀ = +∞, the laser beam is fully coherent beam while the partially coherent beam is given in other cases. As expected, the fiber coupling efficiency of the new model is still greater than the previous one, and η decreases as the increase of z but increases as σ₀ increasing. This is consistent with the work of [6,8], which showed that the lower degree of coherent and longer propagation distance will degrade the coupling efficiency of PCG beam in atmospheric and oceanic turbulence. It manifests that the coherent laser beam may suppress the effect of oceanic turbulence to some extent, so we can come to the conclusion that fully coherent beam is a good choice in terms of coupling efficiency.

Fig. 2 The fiber coupling efficiency η of two models as a function of the propagation distance z with different coherent length σ₀.

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Next we consider the effect of beam size and anisotropic factor on the coupling efficiency. For this purpose, we change w₀ from 0.01 to 0.08 m with ζ = 1, 2, and 3. Figure 3 shows how the coupling efficiency varies with w₀ for three different values of ζ. As one can see in Fig. 3, the fiber coupling efficiency increases with the increment of anisotropic factor ζ. This conclusion is derived from the fact that the wavefront distortion of a light wave in anisotropic turbulent media is less than that in isotropic turbulence [14,19,21]. It indicates that the anisotropy can restrain the effects of turbulence to some extent in the oceanic channel. Furthermore, the coupling efficiency increases with the increase of the beam size and decreases reversely when the maximum is reached. This law is the opposite to the variation of the scintillation of on-axis collimated beam with the beam radius w₀ [17]. The reason is that the scintillation is corresponding to the short-term pointing jitter, and the short-term directional jitter is caused by the short-term wavefront distortion; when scintillation increases, the short-term pointing jitter increases, which will increase the mismatching between the incident beam and axial positioning of fiber. Therefore, the coupling efficiency will decrease. In term of the coupling efficiency improvement by reducing the mismatching between the incident beam and axial positioning of fiber, choosing a suitable beam size w₀ and adopting adaptive tracking technology is beneficial.

Fig. 3 The fiber coupling efficiency η of two models as a function of beam size w₀ with different anisotropic factor ζ.

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In Fig. 4, we curve the fiber coupling efficiency η of PCG beam as a function of receiver lens diameter D and the equivalent temperature structure constant $C_{m}^{2}$ , where D varies from 0 to 0.2 m with $C_{m}^{2} = 10^{- 14}$ , 10⁻¹⁵, and 10⁻¹⁶ m^−2/3K². Figure 4 shows that as the value of D increases, both two models increase to a peak value first, then the old model drop rapidly, while the new model becomes stable after a little reduction. This results due to the misalignment between the incident beam and axial positioning of fiber, and that is why the maximum coupling efficiency occurs at a smaller value of D as the $C_{m}^{2}$ increases; accordingly, alignment between the incident beam and axial positioning of fiber will exist only for a smaller area in this case. One can also note that the new model significantly higher than the previous model when the value of D is approximately greater than 0.05 m. This is because the previous model assumes that the average input optical intensity is independent of r, but that is not the case when D is bigger, the value of the denominator in Eq. (1) is greater than the actual value. This results manifest that when the receiving threshold allows, choosing a smaller receiver lens diameter D is beneficial to improve the signal-to-fiber coupling in the oceanic turbulence link. As we know that $C_{m}^{2}$ denotes the strength of the oceanic turbulence for given salinity, hence we can obtain an relatively higher fiber coupling efficiency when weaker oceanic turbulence strength is satisfied.

Fig. 4 The fiber coupling efficiency η of two models as a function of the receiver lens diameter D with different temperature structure constant $C_{m}^{2}$ .

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To illustrate the effect of relative strength of temperature and salinity fluctuations and inner scale on the fiber coupling efficiency, in Fig. 5, we plot the numeration curves of the fiber coupling efficiency η as a function of the relative strength of temperature and salinity fluctuations ϖ and the inner scale l₀ of laser beams. In Fig. 5, the ranges of the value of ϖ changes from −4.5 to −0.5, the inner scale l₀ is 1, 5, and 10 mm. One can see from the Fig. 5 that the coupling efficiency η will increase if l₀ increases. The inner scale l₀ determines the lower limit of turbulence’s inertial rang, and a larger l₀ means fewer effective turbulence eddies in the inertial range and thus weaker turbulence strength. When l₀ increases, the scattering effect decreases, which will reduce the effect of beam spreading, that is, a lager l₀ leads to a higher fiber coupling efficiency. Meanwhile, due to the stronger effect of refraction introduced by the higher salt, the fiber coupling efficiency decreases rapidly when the value of ϖ is close to −0.5. The results reveal that the fiber coupling efficiency has better immunity to the effect of ocean temperature variation than salinity variation.

Fig. 5 The fiber coupling efficiency η of two models as a function of relative strength of temperature and salinity fluctuations ϖ with different inner scale l₀.

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4. Conclusion

In this paper we propose a new model of the fiber coupling efficiency for a PCG beam propagating through an anisotropic oceanic turbulence link to analyze the effects of turbulence and beam parameters. Our results show that the new model of fiber coupling efficiency always higher than the previous model, in other words, the old coupling efficiency model ignores the effect of turbulence on average light intensity and underestimates the coupling efficiency of the light from wireless turbulence channels to optical fiber. The relationships between fiber coupling efficiency and light source and turbulence are as follows: the fiber coupling efficiency reduces with the increment of equivalent temperature structure constant $C_{m}^{2}$ , propagate distance z, receiver lens diameter D, and temperature-salinity contribution ratio ϖ; however, when a large inner scale l₀ and anisotropic factor ζ is satisfied, the higher coupling efficiency will be achieved. And the relationships show that one of the key factors affecting fiber coupling efficiency is the mismatching between the incident beam and axial positioning of fiber, therefore, one ought to select a longer wavelength and a higher coherent degree of laser beam with an optimum beam size w₀, adaptive tracking technology and optimum focal length of the coupling lens f to improve the coupling efficiency between the wireless turbulence channel to optical fiber. Our results are beneficial for the design of fiber-coupled-based oceanic communication link with PCG beam.

Funding

National Natural Science Foundation of China (NSFC) (61701196); Postgraduate Research & Practice Innovation Program of Jiangsu Provence (SJCX17_0495); Fundamental Research Funds for the Central Universities (Grant No. JUSRP51716A).

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New model of the fiber coupling efficiency of a partially coherent Gaussian beam in an ocean to fiber link

Abstract

1. Introduction

2. Fiber coupling efficiency of a PCG beam in the link of anisotropic oceanic turbulence to fiber

3. Analysis of fiber coupling efficiency of a PCG beam in the link of seawater to fiber with a weak to strong anisotropic oceanic turbulence

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (23)

Optics Express