Fiber coupling efficiency for a Gaussian-beam wave propagating through non-Kolmogorov turbulence

Chao Zhai; Liying Tan; Siyuan Yu; Jing Ma

doi:10.1364/OE.23.015242

1. Introduction

It is well-known that atmospheric turbulence severely degrades the performance of imaging and laser systems [1–3]. For a long time, the Kolmogorov model for atmospheric turbulence has been widely applied to estimate the performance of imaging and laser systems operating in the atmosphere, which has been confirmed by numerous experimental evidence.

Despite the success of the Kolmogorov model, recently both the experimental data [4–8] and the theoretical investigations [9–12] have shown that it is not the only possible turbulent one in the atmosphere. This has prompted the scientist community to research optical propagation in non-Kolmogorov atmospheric turbulence. Beland developed the expressions of log-amplitude variance and the coherence length for optical wave propagating through weak isotropic non-Kolmogorov turbulence [13]. Stribling et al. presented an analysis of the wave structure function and the Strehl ratio as the refractive-index fluctuations deviated from Kolmogorov statistics [14]. Boreman and Dainty studied the expressions of non-Kolmogorov turbulence in the light of Zernike polynomials [15]. Gurvich and Belen’kii introduced a model for the power spectrum of stratospheric non-Kolmogorov turbulence and investigated the stratospheric turbulence on the scintillation and the coherence of starlight and on the degradation of star image [16]. Toselli et al. introduced a non-Kolmogorov theoretical power spectrum model and analyzed long term beam spread, scintillation index, probability of fade, mean SNR, and mean BER as variations of the spectrum exponent for horizontal link [17]. And then they analyzed the angle-of-arrival fluctuations for free space laser beam again [18]. Baykal et al. found the equivalence of the structure constants in non-Kolmogorov and Kolmogorov spectra in a turbulent atmosphere [19]. Chen et al. developed the expressions of temporal averaged pulse intensity for optical pulses propagating through non-Kolmogorov turbulence under the strong fluctuation conditions and the narrow-band assumption [20]. So far all of theoretical investigations for optical propagation in non-Kolmogorov atmospheric turbulence have focused on the unbounded plane or spherical wave models. However, in many applications the plane and spherical wave approximations do not suffice to describe the propagation properties of optical wave. Therefore, it is very necessary to extend the investigation of optical propagation in non-Kolmogorov atmospheric turbulence to Gaussian-beam wave model.

In this paper, a non-Kolmogorov theoretical power spectrum is considered [18], which has a generalized power law that takes all the values ranging from 3 to 4. As the power law α is set to the standard Kolmogorov value 11/3, the spectrum reduces to the conventional von Kármán spectrum for Kolmogorov turbulence. Based on this spectrum and the method of effective beam parameters [21], the fiber coupling efficiency for a Gaussian-beam wave has been derived in both weak and strong fluctuation regimes for a horizontal path and the influence of spectral power law variations on the fiber coupling efficiency has been analyzed.

2. Non-Kolmogorov spectrum

For the purpose of this paper, a theoretical power spectrum model that describes non-Kolmogorov optical turbulence is considered [18], which obeys a more general power law and in which the power law exponent can take all the values ranging from 3 to 4,

Φ_{n} (κ, α) = A (α) {\tilde{C}}_{n}^{2} \frac{\exp (- κ^{2} / κ_{m}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, \begin{matrix} 0 \leq κ < \infty \end{matrix}, \begin{matrix} 3 < α < 4, \end{matrix}

where

κ

is the magnitude of three dimensional wave number vector (in units of rad/m), α is the spectrum power law exponent,

{\tilde{C}}_{n}^{2}

is a generalized refractive-index structure constant (in units of m^3-^α) that describes the strength of the turbulence along the path,

A (α) = Γ (α - 1) \cos (α π / 2) / 4 π^{2}

, Γ(x) is the gamma function,

κ_{0} = 2 π / L_{0}

,

L_{0}

is the outer scale parameter,

κ_{m} = c (α) / l_{0}

,

l_{0}

is the inner scale parameter and

c (α) = {[(2 π / 3) Γ (2.5 - 0.5 α) A (α)]}^{1 / (α - 5)}

. In the case of α = 11/3,

{\tilde{C}}_{n}^{2}

becomes the known structure constant

C_{n}^{2}

of the Kolmogorov spectrum and Eq. (1) reduces to the conventional von Kármán spectrum for Kolmogorov turbulence. Nowadays, an equivalence between the generalized structure constant in non-Kolmogorov turbulence and the structure constant in Kolmogorov turbulence has been devised, which is given by Eq. (7) of [19] (note that a negative sign in D(α) should be added). Utilizing this equivalence expression and rearranging, Eq. (1) is rewritten as

Φ_{n} (κ, α) = h (α) \frac{\exp (- κ^{2} / κ_{m}^{2})}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, \begin{matrix} 0 \leq κ < \infty \end{matrix}, \begin{matrix} 3 < α < 4, \end{matrix}

where

h (α) = - \frac{Γ (α) {(k / L)}^{α / 2 - 11 / 6} C_{n}^{2}}{8 π^{2} Γ (1 - 0.5 α) {[Γ (0.5 α)]}^{2} \sin (0.25 π α)} .

Here, L is the propagation distance, k = 2π/λ, λ is the wavelength.

3. Fiber coupling efficiency for a Gaussian-beam wave

The fiber coupling efficiency for an optical wave is defined as the ratio of the average power coupled into the fiber, $〈 P_{c} 〉$ , to the average power in the receiver aperture plane, $〈 P_{a} 〉$ , and is given by [22, 23]

η = \frac{〈 P_{c} 〉}{〈 P_{a} 〉} = \frac{〈 {| \int_{A} U_{i} (r) U_{m}^{*} (r) d r |}^{2} 〉}{〈 \int_{A} {| U_{i} (r) |}^{2} d r 〉},

where

U_{i} (r)

is the incident optical field in the receiver aperture plane and

U_{m} (r)

is the normalized fiber mode profile. The overlap integral in the numerator of Eq. (4) is evaluated in the receiver aperture plane A, because it is more convenient to do so. The numerator of Eq. (4) can be rearranged by expanding the squared integration to write the coupling efficiency as

η = \frac{1}{〈 P_{a} 〉} \iint_{A} Γ_{i} (r_{1}, r_{2}) U_{m}^{*} (r_{1}) U_{m} (r_{2}) d r_{1} d r_{2},

where

Γ_{i} (r_{1}, r_{2}) = 〈 U_{i} (r_{1}) U_{i}^{*} (r_{2}) 〉

is the mutual coherence function of the incident field.

Assuming that the fiber end face is positioned in the focal plane of the receiver lens and centered on the optical axis to maximize the coupling efficiency, the fiber mode profile propagated to the front surface of the lens can be given by [23]

U_{m} (r) = \frac{k W_{m}}{\sqrt{2 π} f} \exp [- {(\frac{k W_{m}}{2 f})}^{2} r^{2}],

where W_m is the fiber mode field radius at the fiber end face, and f is the focal length of the receiver lens. The amplitude factor in Eq. (6) is included to normalize the power carried by the mode to unity. The fiber mode profile was approximated by a Gaussian function in the derivation of Eq. (6). This approximation is commonly used in calculations of fiber coupling efficiency and does not lead to an appreciable loss of accuracy [24].

For a Gaussian-beam wave, the mutual coherence function in the weak fluctuation regime is given by [25]

\begin{array}{l} Γ_{i} (r_{1}, r_{2}) = Γ_{i}^{0} (r_{1}, r_{2}) \exp 〚 - 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ) \\ \begin{matrix} \begin{matrix} \times {1 - \exp (- \frac{Λ L κ^{2} ξ^{2}}{k}) J_{0} [| (1 - \bar{Θ} ξ) p - 2 i Λ ξ r | κ]} d κ d ξ 〛 \end{matrix} \end{matrix}, \end{array}

where

Γ_{i}^{0} (r_{1}, r_{2}) = \frac{W_{0}^{2}}{W^{2}} \exp (- \frac{2 r^{2}}{W^{2}} - \frac{ρ^{2}}{2 W^{2}} - i \frac{k}{F} p \cdot r),

W is the beam spot size radius at the receiver, F is the phase front radius of curvature at the receiver,

Φ_{n} (κ)

is the power spectrum for refractive-index fluctuations, J₀(x) is a Bessel function of the first kind, the complementary parameter

\bar{Θ} = 1 - Θ

,

r = (r_{1} + r_{2}) / 2

,

p = r_{1} - r_{2}

,

r = | r |

,

ρ = | p |

. Θ and Λ are the output plane (or receiver) beam parameters that are related to the input plane (or transmitter) beam parameters Θ₀ and Λ₀ by

Θ = 1 + \frac{L}{F} = \frac{Θ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}}, \begin{matrix} Λ = \frac{2 L}{k W^{2}} = \frac{Λ_{0}}{Θ_{0}^{2} + Λ_{0}^{2}} \end{matrix},

where Θ₀ and Λ₀ are defined by

Θ_{0} = 1 - L / F_{0}

and

Λ_{0} = 2 L / k W_{0}^{2}

, respectively. Here, W₀ and F₀ denote the beam spot size radius and phase front radius of curvature at the transmitter. The parameter Θ₀ is also called the curvature parameter and Λ₀ is called the Fresnel ratio at the input plane, while the quantity Θ is called the curvature parameter and Λ is called the Fresnel ratio at the output plane. Either the transmitter beam parameters or the receiver beam parameters can describe the diffractive characteristics of Gaussian-beam wave.

For non-Kolmogorov turbulence, Eq. (7) can be written as

\begin{array}{l} Γ_{i} (r_{1}, r_{2}, α) = Γ_{i}^{0} (r_{1}, r_{2}) \exp 〚 - 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} & \times {1 - \exp (- \frac{Λ L κ^{2} ξ^{2}}{k}) J_{0} [| (1 - \bar{Θ} ξ) p - 2 i Λ ξ r | κ]} d κ d ξ 〛 \end{matrix} \end{matrix} . \end{array}

According to [21], we can set $r_{1} = - r_{2}$ to simplify the mutual coherence function for a Gaussian-beam wave, which does not lead to an appreciable loss of accuracy. Then the mutual coherence function becomes a function of only the scalar distance ρ,

\begin{array}{l} Γ_{i} (ρ, α) = \frac{W_{0}^{2}}{W^{2}} \exp {- \frac{1}{4} Λ (\frac{k ρ^{2}}{L}) - 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) \\ \begin{matrix} \begin{matrix} \times [1 - \exp (- \frac{Λ L κ^{2} ξ^{2}}{k}) J_{0} (| 1 - \bar{Θ} ξ | ρ κ)] d κ d ξ} \end{matrix} \end{matrix} . \end{array}

Using the method of effective beam parameters [21], the mutual coherence function for a Gaussian-beam wave meeting both weak and strong fluctuation conditions can be obtained,

\begin{array}{l} Γ_{i} (ρ, α) = \frac{W_{0}^{2}}{W_{L T}^{2}} \exp [- \frac{1}{4} Λ_{e} (\frac{k ρ^{2}}{L}) - 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) d κ d ξ \\ \begin{matrix} \begin{matrix}  \end{matrix} & + 4 π^{2} k^{2} L \int_{0}^{1} \int_{0}^{\infty} κ Φ_{n} (κ, α) \exp (- \frac{Λ {}_{e}L κ^{2} ξ^{2}}{k}) J_{0} (| 1 - {\bar{Θ}}_{e} ξ | ρ κ) d κ d ξ], \end{matrix} \end{array}

where

Θ_{e} = 1 + \frac{L}{F_{L T}} = \frac{Θ - 2 q Λ / 3}{1 + 4 q Λ / 3}, \begin{matrix} Λ_{e} \end{matrix} = \frac{2 L}{k W_{L T}^{2}} = \frac{Λ}{1 + 4 q Λ / 3},

are the effective receiver beam parameters, W_LT is the effective beam spot size radius at the receiver, F_LT is the effective phase front radius of curvature at the receiver,

q = L / k ρ_{p}^{2}

, and

ρ_{p}

is the coherence length of a plane wave. For non-Kolmogorov turbulence, the coherence length of a plane wave is given by [26]

ρ_{p} (α) = {[- 2^{3 - α} h (α) π^{2} k^{2} L \frac{Γ (1 - α / 2)}{Γ (α / 2)}]}^{- 1 / (α - 2)} .

Substituting Eq. (2) into Eq. (12), and using the integral relation,

U (a; c; z) = \frac{1}{Γ (a)} \int_{0}^{\infty} e^{- z t} t^{a - 1} {(1 + t)}^{c - a - 1} d t, \begin{matrix} a > 0 \end{matrix}, \begin{matrix} Re (z) > 0 \end{matrix},

and the series representation of the Bessel function of the first kind [25],

J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} {(x / 2)}^{2 n + p}}{n! Γ (n + p + 1)}, \begin{matrix} | x | < \infty \end{matrix},

where

U (a; c; z)

is the confluent hypergeometric function of the second kind and p denotes the order of the Bessel function of the first kind, Eq. (12) can be expressed as

\begin{array}{l} Γ_{i} (ρ, α) = \frac{W_{0}^{2}}{W_{L T}^{2}} \exp [- \frac{1}{4} Λ_{e} (\frac{k ρ^{2}}{L}) - 2 π^{2} k^{2} h (α) L κ_{0}^{2 - α} U (1; 2 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}}) \\ \begin{matrix} + 2 π^{2} k^{2} h (α) L κ_{0}^{2 - α} \int_{0}^{1} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} {(| 1 - {\bar{Θ}}_{e} ξ | ρ κ_{0} / 2)}^{2 n}}{n!} U (n + 1; n + 2 - \frac{α}{2}; \frac{κ_{0}^{2}}{κ_{m}^{2}} + \frac{Λ {}_{e}L ξ^{2} κ_{0}^{2}}{k}) d ξ] . \end{matrix} \end{array}

Based on Eq. (9) and Eq. (13), it can be obtained that the condition $κ_{0}^{2} / κ_{m}^{2} + Λ {}_{e}L ξ^{2} κ_{0}^{2} / k ≪ 1$ , roughly the same as

\frac{l_{0}^{2}}{L_{0}^{2}} + \frac{1}{1 + 4 q Λ / 3} \cdot \frac{2 π^{2} W_{0}^{2} ξ^{2}}{(1 + \frac{Θ_{0}^{2}}{Λ_{0}^{2}}) L_{0}^{2}} ≪ 1,

is satisfied for most cases of interest, with the possible exception of a large aperture focused beam. Then using the asymptotic formula [25],

U (a; c; z) \sim \frac{Γ (1 - c)}{Γ (1 + a - c)} + \frac{Γ (c - 1)}{Γ (a)} z^{1 - c}, \begin{matrix} | z | ≪ 1 \end{matrix},

the series representation of the modified Bessel function of the first kind,

I_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(x / 2)}^{2 n + p}}{n! Γ (n + p + 1)}, \begin{matrix} | x | < \infty \end{matrix},

and the series representation of the confluent hypergeometric function of the first kind,

{}_{1}F_{1} (a; c; z) = \sum_{n = 0}^{\infty} \frac{{(a)}_{n}}{{(c)}_{n}} \frac{z^{n}}{n!}, \begin{matrix} | z | < \infty \end{matrix},

where p denotes the order of the modified Bessel function of the first kind, the Eq. (17) can be expressed as

\begin{array}{l} Γ_{i} (ρ, α) = \frac{W_{0}^{2}}{W_{L T}^{2}} exp {- \frac{1}{4} Λ_{e} (\frac{k ρ^{2}}{L}) - 2 π^{2} k^{2} h (α) L [\frac{Γ (\frac{α}{2} - 1)}{Γ (\frac{α}{2})} κ_{0}^{2 - α} + \frac{Γ (1 - \frac{α}{2})}{Γ (1)} κ_{m}^{2 - α}] \\ \begin{matrix} + 2 π^{2} k^{2} h (α) L {\int_{0}^{1} (\frac{Λ {}_{e}L ξ^{2}}{k} + \frac{1}{κ_{m}^{2}})}^{\frac{α}{2} - 1} {}_{1}F_{1} [1 - \frac{α}{2}; 1; - \frac{{(1 - {\bar{Θ}}_{e} ξ)}^{2} ρ^{2} κ_{m}^{2}}{4 (1 + Λ {}_{e}L ξ^{2} κ_{m}^{2} / k)}] Γ (1 - \frac{α}{2}) d ξ \end{matrix} \\ \begin{matrix} - 2 π^{2} k^{2} h (α) L κ_{0}^{2 - α} \int_{0}^{1} I_{1 - \frac{α}{2}} (| 1 - {\bar{Θ}}_{e} ξ | ρ κ_{0}) Γ (1 - \frac{α}{2}) {(\frac{| 1 - {\bar{Θ}}_{e} ξ | ρ κ_{0}}{2})}^{\frac{α}{2} - 1} d ξ} . \end{matrix} \end{array}

Finally, using the asymptotic formulas [25],

{}_{1}F_{1} (a; c; - z) \sim \frac{Γ (c)}{Γ (c - a)} z^{- a}, \begin{matrix} Re (z) ≫ 1 \end{matrix},

and

I_{p} (x) \sim \frac{{(x / 2)}^{p}}{Γ (1 + p)}, \begin{matrix} p \neq - 1, - 2, - 3, \dots \end{matrix}, \begin{matrix} z \to 0^{+} \end{matrix},

the mutual coherence function for a Gaussian-beam wave propagating through non-Kolmogorov atmospheric turbulence in both weak and strong fluctuation regimes is given by

Γ_{i} (ρ, α) = \frac{W_{0}^{2}}{W_{L T}^{2}} \exp [- \frac{1}{4} Λ_{e} (\frac{k ρ^{2}}{L}) + M ρ^{α - 2} - B], \begin{matrix} l_{0} ≪ ρ ≪ L_{0}, \end{matrix}

where

B = 2 π^{2} k^{2} h (α) L [(\frac{Γ (\frac{α}{2} - 1)}{Γ (\frac{α}{2})} + \frac{Γ (1 - \frac{α}{2})}{Γ (2 - \frac{α}{2})}) κ_{0}^{2 - α} + \frac{Γ (1 - \frac{α}{2})}{Γ (1)} κ_{m}^{2 - α}],

M = 2^{3 - α} π^{2} k^{2} h (α) L \frac{Γ (1 - \frac{α}{2}) a_{e}}{Γ (\frac{α}{2}) (α - 1)},

a_{e} = {\begin{matrix} \frac{1 - Θ_{e}^{α - 1}}{1 - Θ_{e}}, \begin{matrix} Θ_{e} \geq 0, \end{matrix} \\ \frac{1 + {| Θ_{e} |}^{α - 1}}{1 - Θ_{e}}, \begin{matrix} Θ_{e} < 0. \end{matrix} \end{matrix}

Here it is noted that the analytic approximation Eq. (25) is sufficiently accurate for most cases of interest, with the possible exception of a large aperture focused beam. And the limiting condition $l_{0} ≪ ρ ≪ L_{0}$ is commonly used in derivations of mutual coherence function and does not lead to an appreciable loss of accuracy for calculations of fiber coupling efficiency [23, 25].

For a Gaussian-beam wave, the average power in the receiver aperture plane meeting both weak and strong fluctuation conditions is given by [25]

〈 P_{a} 〉 = \int_{0}^{\frac{D}{2}} \int_{0}^{2 π} \frac{W_{0}^{2}}{W_{L T}^{2}} \exp (- \frac{2 r^{2}}{W_{L T}^{2}}) r d θ d r = \frac{π W_{0}^{2}}{2} [1 - \exp (- \frac{D^{2}}{2 W_{L T}^{2}})],

where D is the receiver lens diameter. Inserting Eq. (25) and Eq. (29) into Eq. (5), the fiber coupling efficiency for a Gaussian-beam wave is obtained,

\begin{array}{l} η = \frac{4 W_{m}^{2}}{λ^{2} f^{2} W_{L T}^{2} [1 - \exp (- \frac{D^{2}}{2 W_{L T}^{2}})]} \\ \begin{matrix} \times \end{matrix} \int_{0}^{\frac{D}{2}} \int_{0}^{\frac{D}{2}} \int_{0}^{2 π} \int_{0}^{2 π} \exp [- {(\frac{π W_{m}}{λ f})}^{2} (r_{1}^{2} + r_{2}^{2}) - \frac{1}{4} Λ_{e} (\frac{k ρ^{2}}{L}) + M ρ^{α - 2} - B] r_{1} r_{2} d θ_{1} d θ_{2} d r_{1} d r_{2} . \end{array}

Then the law of cosines given by

ρ^{2} = {| r_{1} - r_{2} |}^{2} = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2}),

is used to expand

ρ^{2}

and

ρ^{α - 2}

. This results in a double integral over the angle variables θ₁ and θ₂ that needs to be evaluated first and is given by

I = \int_{0}^{2 π} \int_{0}^{2 π} \exp {- \frac{k Λ_{e}}{4 L} [r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})] + M {[r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{1} - θ_{2})]}^{\frac{α}{2} - 1}} d θ_{1} d θ_{2} .

Making a change of variables to $θ_{d} = θ_{1} - θ_{2}$ and $θ = θ_{2}$ and evaluating the integral over θ yields

I = 4 π \int_{0}^{π} \exp {- \frac{k Λ_{e}}{4 L} (r_{1}^{2} + r_{2}^{2}) [1 - \frac{2 r_{1} r_{2}}{r_{1}^{2} + r_{2}^{2}} \cos (θ_{d})] + M {(r_{1}^{2} + r_{2}^{2})}^{\frac{α}{2} - 1} {[1 - \frac{2 r_{1} r_{2}}{r_{1}^{2} + r_{2}^{2}} \cos (θ_{d})]}^{\frac{α}{2} - 1}} d θ_{d} .

Normalizing the radial integration variables to the receiver lens radius and defining $x_{1} = 2 r_{1} / D$ and $x_{2} = 2 r_{2} / D$ , the integral I over the angle variables is reduced to

I = 4 π \int_{0}^{π} \exp {- \frac{k Λ_{e}}{4 L} v [1 - u \cos (θ_{d})] + M v^{\frac{α}{2} - 1} {[1 - u \cos (θ_{d})]}^{\frac{α}{2} - 1}} d θ_{d},

where

v = \frac{D^{2}}{4} (x_{1}^{2} + x_{2}^{2})

,

u = \frac{2 x_{1} x_{2}}{x_{1}^{2} + x_{2}^{2}}

. Substituting the result for the integral I given by Eq. (34) into Eq. (30) and using the normalized integration variables x₁ and x₂, the fiber coupling efficiency for a Gaussian-beam wave propagating through non-Kolmogorov atmospheric turbulence in both weak and strong fluctuation regimes is given by

η = \frac{4 β^{2} D^{2} \exp (- B)}{π W_{L T}^{2} [1 - \exp (- \frac{D^{2}}{2 W_{L T}^{2}})]} \int_{0}^{1} \int_{0}^{1} \exp [- β^{2} (x_{1}^{2} + x_{2}^{2})] F (\frac{D^{2}}{4} (x_{1}^{2} + x_{2}^{2}), \frac{2 x_{1} x_{2}}{x_{1}^{2} + x_{2}^{2}}, α, M) x_{1} x_{2} d x_{1} d x_{2},

where β is the design parameter and

β = \frac{D}{2} \frac{π W_{m}}{λ f}

, F is a function of four variables given by the integral

F (v, u, α, M) = \int_{0}^{π} \exp {- \frac{k Λ_{e}}{4 L} v [1 - u \cos (θ)] + M v^{\frac{α}{2} - 1} {[1 - u \cos (θ)]}^{\frac{α}{2} - 1}} d θ .

4. Results and discussions

In this section, numerical calculations are conducted to analyze the influences of Θ₀, Λ₀ and α on the fiber coupling efficiency and the optimum design parameter for a Gaussian-beam wave propagating through non-Kolmogorov turbulence. The values of turbulence and optical wave parameters used in the following calculations are as follows: C_n² = 1 × 10⁻¹⁴m^-2/3, D = 0.1m, L = 5km, λ = 1.55μm, l₀ = 1mm, L₀ = 1m, unless other values are specified in the figures. Since the path length L and optical wavelength λ are fixed (L = 5km and λ = 1.55μm. In fact, other values can also be chosen), all changes in the Fresnel ratio at the transmitter Λ₀ = 2L/kW₀² correspond to variations in the transmitter beam spot size radius W₀.

In Fig. 1, the coupling efficiency for a collimated beam is plotted against the Fresnel ratio at the transmitter Λ₀ for various values of power law α. It can be seen from Fig. 1 that for an optimum design parameter and some fixed power law α for lower Λ₀ values than 10 the coupling efficiency for a collimated beam increases up to a maximum value that occurs near Λ₀ = 1. At the maximum point the curve changes its slope and the coupling efficiency begins to decrease slightly. In addition, the coupling efficiency for a collimated beam decreases with the increment of power law α. This comment is similar to that on the coupling efficiency for a plane or spherical wave in [27]. In order to further analyze the influences of Θ₀, Λ₀ and α on the optimum design parameter, the optimum design parameter for a collimated beam for various values of power law α is plotted in Fig. 2 as a function of Λ₀ for the same case as Fig. 1. As it is shown in Fig. 2, for some fixed power law α for lower Λ₀ values than 10 the optimum design parameter for a collimated beam decreases down to a minimum value that occurs near Λ₀ = 1. At the minimum point the curve changes its slope and the optimum design parameter begins to increase slightly. It can also be deduced from Fig. 2 that the optimum design parameter for a collimated beam increases with the increment of power law α. This is consistent with the comment about the optimum design parameter for a plane or spherical wave in [27].

Fig. 1 Maximum coupling efficiency for a collimated beam (Θ₀ = 1) as a function of Λ₀ for various values of α.

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Fig. 2 Optimum design parameter for a collimated beam (Θ₀ = 1) as a function of Λ₀ for various values of α.

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The coupling efficiency and the optimum design parameter for a convergent beam are plotted in Figs. 3 and 4, respectively, in a similar manner to that for the collimated beam in Figs. 1 and 2 for the curvature parameter at the transmitter Θ₀ = 0.1 for a special case. Figure 3 is much like Fig. 1, thus the same comments as for the collimated beam can also be deduced from Fig. 3, except that the value of Λ₀ corresponding to the maximum point of coupling efficiency decays with the decreasing of the curvature parameter at the transmitter Θ₀. Figure 4 shows that for some fixed power law α the optimum design parameter for a convergent beam decreases down to a minimum value near Λ₀ = 0.1, then remains the minimum value until Λ₀ approaches 10, finally the curve begins to increase slightly with the increment of Λ₀. In addition, the value of Λ₀ at which the optimum design parameter approaches the minimum value decays with the decrement of the curvature parameter at the transmitter Θ₀. And, for a convergent beam, the optimum design parameter also increases with the increment of power law α.

Fig. 3 Maximum coupling efficiency for a convergent beam (Θ₀ = 0.1) as a function of Λ₀ for various values of α.

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Fig. 4 Optimum design parameter for a convergent beam (Θ₀ = 0.1) as a function of Λ₀ for various values of α.

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The coupling efficiency for a divergent beam is plotted in Fig. 5 as a function of the Fresnel ratio at the transmitter Λ₀ for various values of power law α for a special case. As shown in Fig. 5, we can get the same comments as for Figs. 1 and 3, except that the value of Λ₀ corresponding to the maximum point of coupling efficiency will increase with the increment of the curvature parameter at the transmitter Θ₀. Figure 6 shows the optimum design parameter for a divergent beam as a function of the Fresnel ratio at the transmitter Λ₀ for various values of power law α. As shown in Fig. 6, for some fixed power law α, the optimum design parameter for a divergent beam has a minimum value near Λ₀ = 10 and increases slightly as Λ₀ approaches 100. In addition, the value of Λ₀ corresponding to the minimum point of the optimum design parameter will increase with the rising of the curvature parameter at the transmitter Θ₀. And, for a divergent beam, the optimum design parameter still increases with the increment of power law α.

Fig. 5 Maximum coupling efficiency for a divergent beam (Θ₀ = 10) as a function of Λ₀ for various values of α.

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Fig. 6 Optimum design parameter for a divergent beam (Θ₀ = 10) as a function of Λ₀ for various values of α.

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It can be obtained easily from Figs. 2, 4 and 6 that the optimum design parameter has different trends with the changing of the curvature parameter at the transmitter Θ₀. In order to further analyze the reason, the optimum design parameter and the coherence length are plotted in Figs. 7 and 8 as a function of the Fresnel ratio at the transmitter Λ₀ for various values of the curvature parameter at the transmitter Θ₀ for α = 11/3, respectively, for a special case, the coherence length is calculated by the Eq. (59) of chapter 7 in [25]. Comparing Fig. 7 with Fig. 8, we can observe that for some fixed curvature parameter at the transmitter Θ₀ the optimum design parameter and the coherence length have the opposite trend exactly. This comment is similar to that on the optimum design parameter for a plane or spherical wave in [27]. Hence, we can conclude that for a plane wave, spherical wave or Gaussian beam an increase in the coherence length will cause the optimum design parameter to decrease. Considering that the optimum design parameter will increase as the receiver lens diameter increases (on account of the limited space, the figure is not listed in this paper), the conclusions obtained in this paper are consistent with the comments in [22] which showed that the optimum design parameter would increase with the increment of $\sqrt{A_{R} / A_{c}}$ (A_R/A_c represents the number of speckles over the receiver aperture area). In addition, we can also observe from Figs. 1, 3, 5 and 8 that for some fixed curvature parameter at the transmitter Θ₀ the coupling efficiency and the coherence length have different trends, therefore the coupling efficiency for a Gaussian beam is not similar to that for a plane wave in [23] which decreases with the increment of $\sqrt{A_{R} / A_{c}}$ exactly. This phenomenon can be explained that the equation of coupling efficiency for a Gaussian beam in this paper is more complex than that for a plane wave in [23], so the coupling efficiency for a Gaussian beam will not only depend on the value of $\sqrt{A_{R} / A_{c}}$ .

Fig. 7 Optimum design parameter as a function of Λ₀ for various values of Θ₀, where α = 11/3.

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Fig. 8 Coherence length as a function of Λ₀ for various values of Θ₀, where α = 11/3.

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5. Conclusion

In this paper, the fiber coupling efficiency for a Gaussian-beam wave propagating through non-Kolmogorov turbulence is derived in both weak and strong fluctuation regimes for a horizontal path using the method of effective beam parameters [21] and a non-Kolmogorov theoretical power spectrum which owns a generalized power law and in which the power law exponent varies from 3 to 4. The derived expressions are used to analyze the effect of spectral power law variations on the fiber coupling efficiency.

The results show that for an optimum design parameter and some fixed power law α, the fiber coupling efficiency increases up to a peak value, and at the maximum point the curve changes its slope and begins to decrease slightly with the increase of the Fresnel ratio at the transmitter Λ₀. The value of Λ₀ corresponding to the maximum point of coupling efficiency increases with the increment of the curvature parameter at the transmitter Θ₀. And the coupling efficiency decreases with the increment of power law α. The optimum design parameter has the opposite trend of the coupling efficiency with the increase of the Fresnel ratio at the transmitter Λ₀ for some fixed power law α. The value of Λ₀ at which the optimum design parameter approaches the minimum value increases with the rising of the curvature parameter at the transmitter Θ₀. In addition, the optimum design parameter increases with the increment of power law α. The results will contribute to the investigations of atmospheric turbulence and optical wave propagation in the atmospheric turbulence.

Acknowledgments

This work is supported by the program of excellent team in Harbin Institute of Technology.

References and links

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Fiber coupling efficiency for a Gaussian-beam wave propagating through non-Kolmogorov turbulence

Abstract

1. Introduction

2. Non-Kolmogorov spectrum

3. Fiber coupling efficiency for a Gaussian-beam wave

4. Results and discussions

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (36)

Optics Express