Correlation between light-flux fluctuations of two counter-propagating waves in weak atmospheric turbulence

Chunyi Chen; Huamin Yang

doi:10.1364/OE.25.012779

1. Introduction

Over the years, the scientific community has paid much attention to optical wave propagation in atmospheric turbulence. Considerable research effort has been devoted to the cases of single-direction propagation. However, the situations where two optical waves counter-propagate along a common path can also be encountered in many practical applications [1–6]; among them are bidirectional free-space optical (FSO) links, in which two waves emanating from two separated transmitters propagate in opposite directions. For a bidirectional FSO link, the correlation coefficient between turbulence-induced light-flux fluctuations of the two counter-propagating waves is actually a statistical quantity of practical interest. On the one hand, the correlation between received light-flux fluctuations at two link ends may be utilized to obtain the instantaneous channel state directly at the transmitter, hence eliminating the need for a dedicated channel-state feedback link; the instantaneous channel-state information is useful for adaptive transmission in FSO communication systems impaired by atmospheric turbulence [5,7]. On the other hand, the correlation between received light-flux fluctuations at two link ends reveals that a turbulent FSO channel has the potential for generating common randomness, which can be used to extract random secret keys shared by the two link ends [8]; shared random secret keys are valuable in cryptographic applications for secure communications. To better exploit the correlation between received light-flux fluctuations at two link ends, an understanding of its behavior is imperative.

For several decades, the point-source point-receiver (PSPR) reciprocity has proven to remain valid even when atmospheric turbulence exists over the propagation path [9–11], meaning that the received light signals at the two ends of a bidirectional FSO link with two point transceivers are perfectly correlated, irrespective of atmospheric turbulence. Nevertheless, experimental measurement has shown that the correlation coefficient between received light-flux fluctuations at two link ends may obviously decrease below 1 when the two transceivers have a finite-size aperture [12]. An important exception reported in [1,4,5] is that the registered light signals of two transceivers consisting of single-mode fiber collimators (SMFC) in a bidirectional FSO link can always preserve prefect correlations, regardless of atmospheric turbulence; the experimental observations presented in [1,13] provide evidence to support this theoretical prediction. Coupling an optical wave into a single-mode fiber in an FSO receiver may result in a significant power loss [14–18], implying that the light signal received by an SMFC transceiver is not equivalent to the light flux entering its aperture; to overcome this problem, power-in-the-bucket (PIB) receivers [1], which can measure all the light flux collected by a receiving aperture, are commonly used in FSO communication systems. Accordingly, it is interesting to explore the behavior of the statistical correlation between light-flux fluctuations received at the two ends of a bidirectional FSO link through atmospheric turbulence.

By taking into account finite-size receiving apertures, Perlot and Giggenbach [3] addressed the correlation between light-flux fluctuations of two counter-propagating spherical waves in weak atmospheric turbulence; they formulated the correlation coefficient based on the spatial covariance function of irradiance fluctuations and obtained a relevant expression whose numerator includes a five-dimensional integral (see Eq. (22) in [3] where b_F (·) and H_FR (·) are essentially expressed in the form of a one- and two-dimensional integral, respectively). To·derive simpler and more insightful mathematical models, unlike the traditional treatment of light-flux fluctuations, we will first develop the relationship between the normalized light flux and aperture-averaged first-order log-amplitude fluctuations, and then formulate the correlation coefficient between light-flux fluctuations of two counter-propagating waves in terms of the statistics of aperture-averaged first-order log-amplitude fluctuations. By doing so, the spatial covariance function of irradiance fluctuations is not necessary for our theoretical development, hence simplifying the derivation. Furthermore, although a zero inner scale of turbulence has actually been implicitly assumed in [3], the inner scale may have an impact on optical scintillations. In this work, we will incorporate the inner scale of turbulence into our theoretical derivation and elucidate its role on the correlation coefficient between light-flux fluctuations of two counter-propagating waves. Compared with the existing effort, we intend to deal with the correlation between light-flux fluctuations of two counter-propagating waves in weak turbulence from different points of view. Specifically, we attempt to understand the physical aspects of this issue in both wavenumber and path-space domains.

2. Theoretical analysis

Figure 1 illustrates the counter-propagation geometry under consideration here for two waves with the same wavelength in atmospheric turbulence. The propagation from the plane at z = 0 to the plane at z = L is referred to as “forward propagation” and that from the plane at z = L to the plane at z = 0 is called “inverse propagation”. Plane, spherical and Gaussian-beam waves are the common model waves considered in the existing literature [19–21] related to optical propagation through atmospheric turbulence. Among them, the Gaussian-beam model may be a tool most appropriate for describing the propagated waves from the perspective of many practical applications. However, for the sake of tractability, here we restrict our analysis to the first two kinds of model waves, i.e., the plane and spherical waves. Charnotskii [22] performed a comprehensive asymptotic analysis of beam-wave scintillations under both weak-and strong-turbulence conditions and presented a complete set of asymptotes. As shown in [23], under the near- and far-field conditions, i.e., the initial-beam-size Fresnel parameter is much larger and smaller, respectively, than 1, scintillation of collimated-Gaussian-beam waves in weak-turbulence cases, viz., when the coherence-radius Fresnel parameter is much greater than 1, approaches that of plane and spherical waves, respectively; additionally, the light-flux fluctuation is closely related to the scintillation. With these in mind, our later treatment of the correlation between light-flux fluctuations of two counter-propagating plane and spherical waves in weak turbulence would be expected to provide some somewhat reasonable estimates for the cases of two identical counter-propagating collimated Gaussian beams in the near- and far-field regions, respectively.

Fig. 1 Counter-propagation geometry for two waves along a common path in atmospheric turbulence.

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2.1. Two counter-propagating plane waves

By making use of the Rytov approximation, the first-order complex phase perturbation of a plane wave propagating in atmospheric turbulence from z = 0 to z = L can be expressed by [19,20]

ψ_{1, pl, F} (r, z = L) = \int_{0}^{L} dz \int \int_{- \infty}^{\infty} d v (κ, z) H_{pl, F} (κ, z) \exp (i κ \cdot r)

with

H_{pl, F} (κ, z) = i k \exp [- i \frac{κ^{2} (L - z)}{2 k}],

where r denotes a point in the plane at z = L, k is the optical wavenumber of the propagating wave, κ = |κ| is the magnitude of κ, dv (κ, z) is related to the refractive-index fluctuation n₁ (r′, z) by

n_{1} (r^{'}, z) = \int \int_{- \infty}^{\infty} \exp (i κ \cdot r^{'}) d v (κ, z) .

Performing the substitution z → (L − z) in Eq. (2) leads to the expression for the first-order complex phase perturbation of a plane wave propagating from z = L to z = 0

ψ_{1, pl, I} (ρ, z = 0) = \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) H_{pl, I} (κ, z) \exp (i κ \cdot ρ)

with

H_{pl, I} (κ, z) = i k \exp (- i \frac{κ^{2} z}{2 k}),

where ρ is a point in the plane at z = 0. Notice that, the subscripts “F” and “I” in the above expressions imply “forward propagation” and “inverse propagation”, respectively.

Note that, ψ_1,pl,_F (r, z = L) = χ_1,pl,_F (r, z = L) + iS_1,pl,_F (r, z = L), where χ_1,pl,_F (·) and S_1,pl,_F (·) are the first-order log-amplitude and phase fluctuations of the forward-propagating plane wave, respectively. By recalling the line of reasoning presented in Sec. 17–6 of [20], after some manipulations, the first-order log-amplitude fluctuation for the forward-propagating wave can be written by

χ_{1, pl, F} (r, z = L) = \frac{1}{2} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) [H_{pl, F} (κ, z) + H_{pl, F}^{*} (κ, z)] \exp (i κ \cdot r),

where the asterisk stands for the complex conjugate. In a similar manner, one can formulate the first-order log-amplitude fluctuation for the inverse-propagating wave by

χ_{1, pl, I} (ρ, z = 0) = \frac{1}{2} \int_{0}^{L} dz \int \int_{- \infty}^{\infty} d v (κ, z) [H_{pl, I} (κ, z) + H_{pl, I}^{*} (κ, z)] \exp (i κ \cdot ρ) .

In what follows, we assume that two circular apertures with the same size, centered on the z-axis and located at z = 0 and z = L, respectively, are used to collect the counter-propagated waves; the aperture planes are perpendicular to the z-axis. It is well known that a finite-size receiving aperture may cause a decrease in optical scintillations, which is called aperture averaging. Conceptually, to determine the statistics of light-flux fluctuations received by an aperture, one needs to integrate the irradiance instead of the log-amplitude fluctuation within the aperture. Thus, it would seem that the statistics of light-flux fluctuations cannot be directly derived from the log-amplitude fluctuation within the aperture. To proceed further, below we first deduce a relationship between the normalized light flux and aperture-averaged first-order log-amplitude fluctuation under weak-turbulence conditions. For an incident plane wave distorted by atmospheric turbulence, the normalized light flux received by a circular aperture can be expressed as

\hat{P} = \frac{\int \int_{D} I (r) d^{2} r}{\int \int_{D} I_{0} d^{2} r} = S_{A}^{- 1} \int \int_{D} I (r) / I_{0} d^{2} r,

where I (r) denotes the turbulence-induced irradiance at position r,

I_{0} = a_{0}^{2}

is the plane-wave irradiance at the aperture plane in the absence of atmospheric turbulence, a₀ is the amplitude of the plane wave in vacuum which is a constant within the aperture, D represents the region of the aperture, and S_A is the area of the aperture. In the Rytov approximation, it follows that

I (r) / I_{0} \approx 1 + 2 [χ_{1, pl} (r) + χ_{1, pl}^{2} (r) + χ_{2, pl} (r) + \dots]

, where χ_1,pl (r) and χ_2,pl (r) denote the first- and second-order log-amplitude fluctuations of a plane wave, respectively; 〈χ_1,pl (r)〉 = 0, 〈χ_2,pl (r)〉 is of the same order of magnitude as

〈 χ_{1, pl}^{2} (r) 〉

, and the angle brackets represent an ensemble average. The second-order Rytov perturbation is essential to evaluation of various statistical moments of optical fields; however, it can be omitted in calculation of the logamplitude variance and irradiance correlation function. In this paper, we only focus on statistics related to the log-amplitude variance and irradiance correlation function. Hence, thereinafter, we leave the terms

χ_{1, pl}^{2} (r)

, χ_2,pl (r), ⋯ out of I (r)/I₀. With these in mind, one finds

\hat{P} \approx 1 + 2 S_{A}^{- 1} \int \int_{D} χ_{1, pl} (r) d^{2} r = 1 + 2 {\hat{χ}}_{1, pl},

where

{\hat{χ}}_{1, pl}

represents the aperture-averaged value of χ_1,pl (r). Equation (9) manifests the relationship between the instantaneous normalized light flux and aperture-averaged first-order log-amplitude fluctuation. In accordance with the statistical theory, the correlation coefficient between light-flux fluctuations of two counter-propagating plane waves can be formulated by

μ_{pl} \approx \frac{B_{pl, F I}}{σ_{pl, F} σ_{pl, I}},

where

B_{pl, F I} = 〈 {\hat{χ}}_{1, pl, F} {\hat{χ}}_{1, pl, I}^{*} 〉

,

σ_{pl, F}^{2} = 〈 {\hat{χ}}_{1, pl, F} {\hat{χ}}_{1, pl, F}^{*} 〉

and

σ_{pl, I}^{2} = 〈 {\hat{χ}}_{1, pl, I} {\hat{χ}}_{1, pl, I}^{*} 〉

;

{\hat{χ}}_{1, pl, F}

and

{\hat{χ}}_{1, pl, I}

represent the aperture-averaged values of χ_1,pl,_F (r, z = L) and χ_1,pl,_I (ρ, z = 0), respectively. Notice that, in fact, the cross-covariance and variances of normalized light-flux fluctuations of two counter-propagating plane waves are 4 times B_pl,_FI,

σ_{pl, F}^{2}

and

σ_{pl, I}^{2}

, respectively; similar relation holds true for the case of two counter-propagating spherical waves that will be addressed later. It should be emphasized here that both

{\hat{χ}}_{1, pl, F}

and

{\hat{χ}}_{1, pl, I}

have a mean value of zero and are real, implying

{\hat{χ}}_{1, pl, F} \equiv {\hat{χ}}_{1, pl, F}^{*}

and

{\hat{χ}}_{1, pl, I} \equiv {\hat{χ}}_{1, pl, I}^{*}

which can be readily deduced from the definition of the first-order log-amplitude fluctuation. However, for convenience of later mathematical treatments, here we formally use the complex-conjugate operation.

{\hat{χ}}_{1, pl, F}

and

{\hat{χ}}_{1, pl, I}

can be expressed by

{\hat{χ}}_{1, pl, F} = \int \int_{- \infty}^{\infty} d^{2} r χ_{1, pl, F} (r, z = L) W_{a} (r),

{\hat{χ}}_{1, pl, I} = \int \int_{- \infty}^{\infty} d^{2} ρ χ_{1, pl, I} (ρ, z = 0) W_{a} (ρ)

with

W_{a} (r) = {\begin{matrix} 1 / (π R_{a}^{2}) & | r | < R_{a} \\ 0 & | r | \geq R_{a} \end{matrix},

where R_a denotes the radius of the aperture used to collect the wave. By introducing Eqs. (6) and (7) into Eqs. (11) and (12), respectively, and using the Fourier transform relation

{\tilde{W}}_{a, pl} (κ) = \int \int_{- \infty}^{\infty} d^{2} r W_{a} (r) \exp (i κ \cdot r) = jinc (κ R_{a}),

where the jinc function is defined by jinc (x) = 2J₁ (x) /x with J₁ (·) being a Bessel function of the first kind, one finds

{\hat{χ}}_{1, pl, F} = \frac{1}{2} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) [H_{pl, F} (κ, z) + H_{pl, F}^{*} (κ, z)] {\tilde{W}}_{a, pl} (κ),

{\hat{χ}}_{1, pl, I} = \frac{1}{2} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) [H_{pl, I} (κ, z) + H_{pl, I}^{*} (κ, z)] {\tilde{W}}_{a, pl} (κ) .

With Eqs. (15) and (16) in hand, one can develop the expressions for B_pl,_FI,

σ_{pl, F}^{2}

and

σ_{pl, I}^{2}

. After some tedious mathematical manipulations, we find

B_{pl, F I} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) {jinc}^{2} (R_{a} κ) \sin (\frac{κ^{2} z}{2 k}) \sin [\frac{κ^{2} (L - z)}{2 k}],

σ_{pl, F}^{2} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) {jinc}^{2} (R_{a} κ) \sin^{2} [\frac{κ^{2} (L - z)}{2 k}],

σ_{pl, I}^{2} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) {jinc}^{2} (R_{a} κ) \sin^{2} (\frac{κ^{2} z}{2 k}),

where Φ_n (κ, z) represents the three-dimensional turbulence spectrum. As in the mathematical treatment done by Tatarski [19] and Ishimaru [20], in arriving at Eqs. (17) – (19), we have used the following: the first is the relationship

〈 d v (κ, z) d v^{*} (κ^{'}, z^{'}) 〉 = F_{n} (| z - z^{'} |, κ) δ (κ - κ^{'}) d^{2} κ d^{2} κ^{'}

with F_n (·) representing the two-dimensional turbulence spectrum and δ () being the delta function; the second is the fact that F_n () takes a negligible value when |z − z′| is beyond the correlation distance of refractive-index fluctuations; the third is the assumption that atmospheric turbulence is statistically homogeneous and isotropic in each plane perpendicular to the z-axis. If we let R_a = 0,

σ_{pl, F}^{2}

and

σ_{pl, I}^{2}

become the variances of log-amplitude fluctuations for the forward- and inverse-propagating waves, respectively. It is straightforward to find that our result for

σ_{pl, F}^{2}

with R_a = 0 is consistent with the formula for the variance of plane-wave log-amplitude fluctuations presented in [19].

According to the existing literature [19–21], the expression for the turbulence spectrum can generally be separated into two terms, i.e., $Φ_{n} (κ, z) = C_{n}^{2} (z) {\hat{Φ}}_{n} (κ)$ , where the first term, referred to as the refractive-index structure constant, characterizes the strength of refractive-index fluctuations at the path location z, and the second term depicts the shape of the spectrum. With this observation in mind, by interchanging the order of integrations in Eqs. (17) – (19), we can elucidate the underlying physics associated with light-flux fluctuations of two counter-propagating plane waves from different perspectives.

2.1.1. Wavenumber domain perspective

When $C_{n}^{2} (z)$ is independent of the path location z, we can first evaluate the integrations on z in Eqs. (17) – (19) and hence eliminate the dependence of B_pl,_FI, $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ on z. Performing this operation permits us to express B_pl,_FI, $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ in the forms of an integral over the wavenumber domain, which are given by

B_{pl, F I} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) f_{pl} (κ \sqrt{L / k}),

σ_{pl, F}^{2} = σ_{pl, I}^{2} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) {f^{'}}_{pl} (κ \sqrt{L / k})

with

f_{pl} (\hat{κ}) = {jinc}^{2} (q_{R}^{1 / 2} \hat{κ}) h_{pl} (\hat{κ}),

{f^{'}}_{pl} (\hat{κ}) = {jinc}^{2} (q_{R}^{1 / 2} \hat{κ}) {h^{'}}_{pl} (\hat{κ}),

h_{pl} (\hat{κ}) = sinc ({\hat{κ}}^{2} / 2) - \cos ({\hat{κ}}^{2} / 2),

{h^{'}}_{pl} (\hat{κ}) = 1 - sinc ({\hat{κ}}^{2}),

where

\hat{κ} = κ {(L / k)}^{1 / 2}

is the scaled wavenumber,

q_{R} = k R_{a}^{2} / L

is called the aperture Fresnel parameter, and the sinc function is defined by sinc (x) = sin (x)/x. It is straightforward to find that f_pl(·) and f′_pl(·) actually act as two spectral weighting functions, which may emphasize or deemphasize the role of refractive-index fluctuations associated with certain spatial wavenumber κ, implying that they can filter a specific portion of

{\hat{Φ}}_{n} (κ)

appearing in Eqs. (17) – Eqs. (21) and Eqs. (17) – (22). In this sense, they can be regarded as spectral filter functions.

We note that f_pl(·) and f′_pl(·) factorize into a product of two different terms. The first term ${jinc}^{2} (q_{R}^{1 / 2} \hat{κ})$ emphasizes the role of refractive-index fluctuations associated with the scaled wavenumber $\hat{κ} ≲ 1 / (0.5216 q_{R}^{1 / 2})$ and basically neglects those associated with the scaled wavenumber $\hat{κ} ≳ 1 / (0.5216 q_{R}^{1 / 2})$ . This statement becomes apparent if we recognize the approximation jinc²(x) ≈ exp(− β²x²) with β = 0.5216 [24]. With this in mind, one can deduce that the portion of ${\hat{Φ}}_{n} (κ)$ with κ > 1/(βR_a) actually contributes little to both B_pl,_FI and $σ_{pl, F}^{2}$ . Indeed, the term ${jinc}^{2} (q_{R}^{1 / 2} \hat{κ})$ formulates the role of a receiving aperture in the treatment of the scintillation problem. It is obvious that a greater R_a leads to a smaller portion of ${\hat{Φ}}_{n} (κ)$ that can produce scintillations effectively. This is the essential reason for the reduction in scintillations induced by aperture averaging. Strictly speaking, the use of the aforementioned approximation of jinc² (·) means that the “hard aperture” described by Eqs. (13) is replaced by a Gaussian aperture, which has been previously employed by Kon [25]; doing so will facilitate our theoretical development. It is noted that f_pl() is distinguished from f′_pl(·) only by the two terms h_pl(·) and h′_pl(·). To obtain some specific idea about the difference between B_pl,_FI and $σ_{pl, F}^{2}$ , below we compare the behaviors of h_pl(·) and h′_pl(·) in terms of $\hat{κ}$ . Figure 2 shows the shapes of the functions h_pl(·) and h′_pl(). It is found from Fig. 2 that, with an increasing value of $\hat{κ}$ , both h_pl(·) and h′_pl () first grow monotonically from 0 to a peak in the vicinity of $\hat{κ} = 2$ ; after that h_pl(·) begins to oscillate around 0 with almost constant amplitude and h′_pl(), however, begins to oscillate around 1 with diminishing amplitude. Hence, refractive-index·fluctuations associated with all spatial wavenumbers make positive contribution to $σ_{pl, F}^{2}$ , whereas there may exist refractive-index fluctuations related to some spatial wavenumbers that make negative contribution to B_pl,_FI. It is apparent that $B_{pl, F I} < σ_{pl, F}^{2}$ , implying μ_pl < 1, because the curve corresponding to h′_pl(·) basically always lies above that corresponding to h_pl(·). As a result, perfect correlation between light-flux fluctuations of two counter-propagating plane waves in atmospheric turbulence is impossible in general cases. Moreover, by recalling the aforesaid approximation of jinc²(·) and the curves shown in Fig. 2, it is straightforward to infer that, if $q_{R}^{1 / 2} β ≳ 1$ , the specific behaviors of h_pl(·) and h′_pl(·) beyond $\hat{κ} = 1 / (q_{R}^{1 / 2} β)$ only play a trivial role in determining B_pl,_FI, and $σ_{pl, F}^{2}$ ; it is also noted·that $h_{pl} (\hat{κ}) ≃ {\hat{κ}}^{4} / 12$ and ${h^{'}}_{pl} (\hat{κ}) ≃ {\hat{κ}}^{4} / 6$ for $\hat{κ} ≪ 1$ . Hence, we can reason from the above analysis that $B_{pl, F I} ≃ σ_{pl, F}^{2} / 2$ and thus μ_pl ≃ 1/2 when $q_{R}^{1 / 2} ≫ 1 / β$ , irrespective of what mathematical form ${\hat{Φ}}_{n} (κ)$ takes.

Fig. 2 h_pl(·) and h′_pl(·) in terms of $\hat{κ} = κ {(L / k)}^{1 / 2}$ .

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To gain more details about μ_pl, we need to consider the specifics of the turbulence spectrum. In the literature, the von Kármán spectrum is usually used to describe the refractive-index fluctuations, and its spectral shape is characterized by ${\hat{Φ}}_{n} (κ) = 0.033 \exp (- κ^{2} / κ_{m}^{2}) {(- κ^{2} + κ_{0}^{2})}^{- 11 / 6}$ , where κ₀ = 2π/L₀ and κ_m = 5.92/l₀ with l₀ and L₀ being the inner and outer scales of turbulence, respectively. However, examination of the integrands in Eqs. (17) – (19) reveals that L₀ can be reasonably specified by ∞ in our case because both R_a and $\sqrt{L / k}$ are much smaller than L₀ for typical optical-wave propagation scenarios, implying that indeed we can use the Tatarskii spectrum, which is equivalent to the von Kármán spectrum with L₀ → ∞, in the following theoretical development. To facilitate the mathematical treatment, we divide h_pl(·) into two terms, i.e., $h_{pl} (\hat{κ}) = h_{l, pl} (\hat{κ}) - {h^{'}}_{pl} (\hat{κ} / 2^{1 / 2})$ with $h_{l, pl} (\hat{κ}) = 1 - \cos ({\hat{κ}}^{2} / 2)$ . Accordingly, B_pl,_FI is expressed as B_pl,_FI = b₁ − b₂ with b₁ given by

\begin{array}{c} b_{1} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) {jinc}^{2} (R_{a} κ) h_{1, pl} (κ \sqrt{L / k}) \\ \approx {(q_{p}^{2} + \frac{1}{4})}^{5 / 12} \cos [\frac{5}{6} \arctan (\frac{1}{2 q_{p}})] C_{1} - q_{p}^{5 / 6} C_{1}, \end{array}

where q_p = q_m + β²q_R,

q_{m} = k κ_{m}^{- 2} / L

is called the inner-scale Fresnel parameter,

C_{1} = 11 σ_{χ, pl}^{2} / [6 \cos (5 π / 12)]

, and

σ_{χ, pl}^{2} = 0.307 C_{n}^{2} k^{7 / 6} L^{11 / 6}

. We have used the aforesaid approximation of jinc² (·) in the second step of Eqs. (27), meaning that the Gaussian-aperture approximation has been employed; indeed, this is the reason for the fact that q_m and q_R can be combined together to form q_p. The term b₂ is expressed by

\begin{array}{c} b_{2} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) {jinc}^{2} (R_{a} κ) {h^{'}}_{pl} (κ \sqrt{L / k} / 2^{1 / 2}) \\ \approx \frac{12}{11} {(\frac{1}{4} + q_{p}^{2})}^{11 / 12} \sin [\frac{11}{6} \arctan (2 q_{p}) + \frac{π}{12}] C_{1} - q_{p}^{5 / 6} C_{1} . \end{array}

This result is obtained by recognizing that, with the use of the said approximation of jinc² (·), the integral in the first step of Eqs. (28) is functionally of a form akin to that dealt with by Ishimaru ([20], Eqs. (17–97) and (17–98)). In a similar way, one can work out the integral associated with

σ_{pl, F}^{2}

to yield

σ_{pl, F}^{2} \approx \frac{6}{11} {(1 + q_{p}^{2})}^{11 / 12} \sin [\frac{11}{6} \arctan (q_{p}) + \frac{π}{12}] C_{1} - q_{p}^{5 / 6} C_{1} .

With the above results in hand, we find

\begin{array}{c} μ_{pl} (q_{p}) = {\frac{6}{11} {(1 + q_{p}^{2})}^{11 - 12} \sin [\frac{11}{6} \arctan (q_{p}) + \frac{π}{12}] - q_{p}^{5 / 6}}^{- 1} {{(q_{p}^{2} + \frac{1}{4})}^{5 / 12} \\ \times \cos [\frac{5}{6} \arctan (\frac{1}{2 q_{p}})] - \frac{12}{11} {(\frac{1}{4} + q_{p}^{2})}^{11 / 12} \sin [\frac{11}{6} \arctan (2 q_{p}) + \frac{π}{12}]}, \end{array}

where, on the left-hand side, we explicitly show the dependence of μ_pl on the nondimensional parameter q_p.

Equation (30) shows our closed-form expression for the correlation coefficient between light-flux fluctuations of two counter-propagating plane waves in weak turbulence. It is worth noting that μ_pl is actually completely determined by the nondimensional parameter q_p. By examining the mathematical form of q_p, we can deduce that the inner scale l₀ plays a role similar to that the aperture radius R_a does on the correlation coefficient μ_pl. It is easy to find that μ_pl ≃ 0.468 when q_p → 0, i.e., both q_m → 0 and q_R → 0. Further, it should be pointed out that the use of the aforementioned approximation of jinc² (·) has actually no impact on the value of μ_pl with q_p → 0, thus meaning q_R → 0, because ${jinc}^{2} (q_{R}^{1 / 2} \hat{κ}) \equiv \exp (- q_{R} β^{2} {\hat{κ}}^{2}) \equiv 1$ with q_R = 0. Figure 3 shows the correlation coefficient μ_pl in terms of $q_{R}^{1 / 2}$ under the condition that the turbulence spectrum has no path dependence. It is found from Fig. 3 that μ_pl is peaked at $q_{R}^{1 / 2} \approx 0.45$ with its maximum value equal to 0.574, and μ_pl achieves its minimum value at $q_{R}^{1 / 2} = 0$ . Furthermore, μ_pl tends to 0.5 when $q_{p}^{1 / 2} ≫ 1$ , which is consistent with the previous deduction (i.e., μ_pl ≃ ½ when $q_{R}^{1 / 2} ≫ 1 / β$ ). The above analyses and calculations provide us a clear understanding of the effects of various propagation parameters on μ_pl when atmospheric turbulence is statistically homogeneous over the path.

Fig. 3 μ_pl in terms of $q_{p}^{1 / 2}$ with a turbulence spectrum independent of path locations.

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2.1.2. Path-space domain perspective

Unlike the treatment above, at this point, we first evaluate the integrations on κ in Eqs. (17) – (19) and then express B_pl,_FI, $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ in the forms of an integral over the normalized path-space domain, which are given by

B_{pl, F I} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) g_{pl} (ξ),

σ_{pl, F}^{2} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) {g^{'}}_{pl} (ξ),

σ_{pl, I}^{2} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) {g^{″}}_{pl} (ξ)

with ξ = z/L being the normalized path location,

g_{pl} (ξ) = {\tilde{h}}_{pl} (1 / 2, 1 / 2 - ξ)

,

{g^{'}}_{pl} (ξ) = {\tilde{h}}_{pl} (1 - ξ, 0)

,

{g^{″}}_{pl} (ξ) = {\tilde{h}}_{pl} (ξ, 0)

, and

{\tilde{h}}_{pl} (a, b) = \frac{3}{5} Γ (\frac{1}{6}) [{(q_{p} + i a)}^{5 / 6} + {(q_{p} - i a)}^{5 / 6} - {(q_{p} + i b)}^{5 / 6} - {(q_{p} - i b)}^{5 / 6}] .

In arriving at Eqs. (31) – (34), the said approximation of jinc² (·) has been utilized.

It is worth noting that g_pl(ξ), g′_pl(ξ) and g″_pl(ξ) play the role of a path weighting function which emphasizes or deemphasizes the contributions of turbulent eddies at different locations to B_pl,_FI, $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ , respectively. The path weighting functions appearing in Eqs. (31) – (33) in terms of ξ are plotted in Fig. 4 with various q_p. It is seen from Fig. 4 that g_pl(ξ) is symmetric about ξ = 0.5, and g′_pl(ξ) is the reflection of g″_pl(ξ) in the vertical line ξ = 0.5. Examination of Fig. 4 reveals that, on the one hand, turbulent eddies located in the vicinity of the path midpoint are most effective in increasing B_pl,_FI and those located in the vicinities of the two path endpoints nearly do not contribute to B_pl,_FI ; on the other hand, turbulent eddies near the transmitting plane are weighted most strongly for $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ , and those near the receiving plane contribute little to $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ . By recalling Eqs. (10), a statement can be made that turbulent eddies near the path midpoint and endpoints tend to make positive and negative contributions to μ_pl, respectively. Note that, the integrals in Eqs. (31) – (33) can be regarded as an inner product of the two terms contained in the integrands. It is obvious that the shape of $C_{n}^{2} (ξ L)$ in terms of ξ has an important impact on the results of the inner products, implying that μ_pl will depend on the path variations of $C_{n}^{2} (ξ L)$ . If the strength of refractive-index fluctuations over the path featured a single large sharp peak at the midpoint, μ_pl would be likely to take a value close to 1. Indeed, it is observed from Fig. 4 that, with the same q_p, the curves corresponding to g_pl(ξ), g′_pl(ξ) and g″_pl(ξ), respectively, come together at ξ = 0.5. This phenomenon is consistent with what could be expected by intuition; i.e., if only one random phase screen with a negligible thickness is positioned at the midpoint of a path in vacuum, one can find μ_pl ≡ 1. In the same context, if the random phase screen is moved away from the midpoint, we can infer from Fig. 4 that μ_pl < 1 and $σ_{pl, F}^{2} \neq σ_{pl, I}^{2}$ . By the way, there are researches on experimental simulations of laser beam propagation through atmospheric turbulence in the laboratory by using a liquid-crystal spatial light modulator acting as a random phase screen [26]; according to the above analysis, it becomes apparent that the position where the random phase screen is located plays an important role in the simulated optical scintillations, meaning that the position of the random phase screen needs to be chosen carefully in experiments in order to properly emulate the target propagation scenarios. Finally, one can see from Fig. 4 that a larger q_p leads to smaller magnitude of the path weighting functions. This means that an increase in q_p generally lowers the absolute magnitude of contributions from turbulent eddies over the path to B_pl,_FI, $σ_{pl, F}^{2}$ and $σ_{pl, I}^{2}$ .

Fig. 4 The path weighting functions g_pl(ξ), g′_pl(ξ) and g″_pl(ξ) in terms of ξ with various q_p. (a) q_p = 0; (b) q_p = 1; (c) q_p = 100.

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2.2. Two counter-propagating spherical waves

Now we turn to the situation of two counter-propagating spherical waves. Following an approach similar to that used in Sec. 2.1, the aperture-averaged values of the first-order log-amplitude fluctuations of the forward- and inverse-propagating spherical waves can be formulated by

{\hat{χ}}_{1, sp, F} = \frac{1}{2} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) [H_{sp, F} (κ, z) + H_{sp, F}^{*} (κ, z)] {\tilde{W}}_{a, sp} (\frac{z}{L} κ),

{\hat{χ}}_{1, sp, I} = \frac{1}{2} \int_{0}^{L} d z \int \int_{- \infty}^{\infty} d v (κ, z) [H_{sp, I} (κ, z) + H_{sp, I}^{*} (κ, z)] {\tilde{W}}_{a, sp} (\frac{L - z}{L} κ),

respectively, where

H_{sp, F} (κ, z) = H_{sp, I} (κ, z) = i k \exp [- i \frac{κ^{2} z (L - z)}{2 L k}],

{\tilde{W}}_{a, sp} (γ κ) = \int \int_{- \infty}^{\infty} d^{2} r W_{a} (r) \exp (i γ κ \cdot r) = jinc (γ κ R_{a}) .

Based on Eqs. (35) and (36), analogous to the plane-wave case, we can further develop the expressions for

B_{sp, F I} = 〈 {\hat{χ}}_{1, sp, F} {\hat{χ}}_{1, sp, I}^{*} 〉

,

σ_{sp, F}^{2} = 〈 {\hat{χ}}_{1, sp, F} {\hat{χ}}_{1, sp, F}^{*} 〉

and

σ_{sp, I}^{2} = 〈 {\hat{χ}}_{1, sp, I} {\hat{χ}}_{1, sp, I}^{*} 〉

, which take the forms

\begin{matrix} B_{sp, F I} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) jinc (\frac{z}{L} R_{a} κ) \\ \times jinc (\frac{L - z}{L} R_{a} κ) \sin^{2} [\frac{κ^{2} z (L - z)}{2 L k}], \end{matrix}

σ_{sp, F}^{2} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) {jinc}^{2} (\frac{z}{L} R_{a} κ) \sin^{2} [\frac{κ^{2} z (L - z)}{2 L k}],

σ_{sp, I}^{2} = 4 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} d κ κ Φ_{n} (κ, z) {jinc}^{2} (\frac{L - z}{L} R_{a} κ) \sin^{2} [\frac{κ^{2} z (L - z)}{2 L k}] .

The correlation coefficient between light-flux fluctuations of two counter-propagating spherical waves is formulated by

μ_{sp} \approx \frac{B_{sp, F I}}{σ_{sp, F} σ_{sp, I}} .

2.2.1. Wavenumber domain perspective

If the turbulence spectrum Φ_n (·) has no path dependence, we can reformulate B_sp,_FI, $σ_{sp, F}^{2}$ and $σ_{sp, I}^{2}$ as follows:

B_{sp, F I} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) f_{sp} (κ \sqrt{L / k}),

σ_{sp, F}^{2} \equiv σ_{sp, I}^{2} = 2 π^{2} k^{2} L C_{n}^{2} \int_{0}^{\infty} d κ κ {\hat{Φ}}_{n} (κ) f^{'}_{sp} (κ \sqrt{L / k})

with

f_{sp} (\hat{κ}) = \frac{1}{2} {\hat{κ}}^{4} \int_{0}^{1} d ξ ξ^{2} {(1 - ξ)}^{2} jinc [(1 - ξ) q_{R}^{1 / 2} \hat{κ}] jinc (ξ q_{R}^{1 / 2} \hat{κ}) {sinc}^{2} [\frac{{\hat{κ}}^{2}}{2} ξ (1 - ξ)],

f^{'}_{sp} (\hat{κ}) = \frac{1}{2} {\hat{κ}}^{4} \int_{0}^{1} d ξ ξ^{2} {(1 - ξ)}^{2} {jinc}^{2} (ξ q_{R}^{1 / 2} \hat{κ}) {sinc}^{2} [\frac{{\hat{κ}}^{2}}{2} ξ (1 - ξ)] .

Equations (45) and (46) show the spectral filter functions for B_sp,_FI,

σ_{sp, F}^{2}

and

σ_{sp, I}^{2}

, respectively. When R_a = 0, i.e., q_R = 0, it follows that

f_{sp} (\hat{κ}) \equiv f^{'}_{sp} (\hat{κ})

, implying that

B_{sp, F I} \equiv σ_{sp, F}^{2} \equiv σ_{sp . I}^{2}

and thus μ_sp ≡ 1, regardless of the mathematical form of

{\hat{Φ}}_{n} (κ)

. This result is indeed what we will expect from the reciprocity principle [9–11] by recognizing that R_a = 0 means a point receiver. Under general conditions, it is formidable to analytically evaluate the integral of Eq. (45) in a simple closed form. An analytical expression for Eq. (46) can be developed if the said approximation of jinc² (·) is employed; however, here we do not show it in detail. Thereinafter, this approximation will not be used. In addition, for the same reason stated in Sec. 2.1, we utilize the Tatarskii spectrum in the following theoretical analysis.

The profiles of $f_{sp} (\hat{κ})$ and $f^{'}_{sp} (\hat{κ})$ are illustrated in Fig. 5 with various q_R. It is found from Fig. 5 that the curves corresponding to $f_{sp} (\hat{κ})$ and $f^{'}_{sp} (\hat{κ})$ , respectively, basically coincide with each other when q_R = 0.04 and obviously deviate from each other within a range of the scaled wavenumber $\hat{κ}$ when q_R = 1 or 16. In fact, it is the deviation of $f_{sp} (\hat{κ})$ from $f^{'}_{sp} (\hat{κ})$ that causes μ_sp to decrease below 1. Further, a larger q_R moves the said deviation deeper into the low-wavenumber range. Notice that, ${\hat{Φ}}_{n} (κ)$ has a greater value with a smaller κ in the inertial range. With these observations in mind, it is straightforward to deduce that a larger q_R generally leads to a smaller μ_sp. On the other hand, as for the two curves associated with q_R = 1 in Fig. 5, it is seen that they begin to obviously deviate from each other only when $\hat{κ} = κ {(L / k)}^{1 / 2} ≳ 2$ . Hence, if κ_m = 5.92/l₀ ≲ 2/(L/k)^1/2, the deviation portion of these two curves only has a negligible impact on μ_sp because ${\hat{Φ}}_{n} (κ)$ rapidly drops close to 0 when κ increases beyond κ_m. This leads us to an inference that a nonzero inner scale l₀ may make the decrease in μ_sp caused by a nonzero aperture radius R_a negligible. As a result, it is necessary to consider the role of a nonzero inner scale l₀ besides the receiver aperture radius R_a in determining μ_sp under some conditions. Moreover, it is seen from Fig. 5 that $f^{'}_{sp} (\hat{κ})$ always takes a positive value, whereas $f_{sp} (\hat{κ})$ with q_R > 0 may take a negative value at some scaled wavenumbers. Thus, like the plane-wave case, refractive-index fluctuations related to some spatial wavenumbers may make negative contributions to B_sp,_FI.

Fig. 5 The spectral filter functions $f_{sp} (\hat{κ})$ and $f^{'}_{sp} (\hat{κ})$ with various q_R. (a) q_R= 0.04; (b) q_R = 1; (c) q_R = 16.

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According to Eqs. (43) – (46), one can examine the correlation coefficient in the spherical-wave case by making use of numerical calculations. By considering the Tatarskii spectrum and making the change of variable κ = t/(L/k)^1/2 in Eqs. (43) and (44), it is easy to find that μ_sp is completely determined by the two nondimensional parameters q_m and q_R. Figure 6 depicts the contours of μ as a function of $q_{m}^{1 / 2}$ and $q_{R}^{1 / 2}$ . It is seen from Fig. 6 that, with the same q_m, μ_sp decreases when q_R increases; in contrast, with the same q_R, a larger q_m results in a greater μ_sp. Hence, an increase in q_m may cancel out the reduction in μ_sp caused by enlarging the value of q_R. The underlying physical reason for this behavior of μ_sp has been elaborated in the analysis of Fig. 5. Comparison of these results with those in the plane-wave case shows that the inner scale l₀ plays a different role in determining the correlation coefficient. Notice that, Fig. 6 indeed illustrates the behavior of μ_sp in most parameter regions of practical interest to us. The contour lines in Fig. 6 are nearly straight lines except for the parts corresponding to relatively small $q_{m}^{1 / 2}$ and $q_{R}^{1 / 2}$ .

Fig. 6 Contours of μ_sp as a function of $q_{m}^{1 / 2}$ and $q_{R}^{1 / 2}$ with a turbulence spectrum independent of path locations. The subplot (b) shows a zoomed detail of the region with relatively small $q_{m}^{1 / 2}$ and $q_{R}^{1 / 2}$ .

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At this point, a question that needs to be addressed further is what value μ_sp approaches when q_R → ∞. By introducing Eqs. (45) and (46), respectively, into Eqs. (43) and (44), taking the Tatarskii spectrum into account and making the change of variable κ = t′/R_a = t′/(q_R L/k)^1/2, for a large enough q_R, it follows that exp(−q_mt′²/q_R) ≈ 1 and sinc[ξ (1 − ξ) t′2/(2q_R)] ≈ 1 within most of the integration range of practical interest. With this observation in mind, we can deduce that $B_{sp, F I} ≃ 0.571 σ_{χ, sp}^{2} q_{R}^{- 7 / 6}$ and $σ_{sp, F}^{2} ≃ 4.683 σ_{χ, sp}^{2} q_{R}^{- 7 / 6}$ when q_R ≫ 1, where $σ_{χ, sp}^{2} = 0.124 C_{n}^{2} k^{7 / 6} L^{11 / 6}$ denotes the spherical-wave log-amplitude variance. Consequently, μ_sp approaches 0.122 as q_R tends to infinity. For two counter-propagating spherical waves in weak atmospheric turbulence, Perlot and Giggenbach inferred the same conclusion from their numerical results (see page 2891 in [3]). In addition, it is easy to find from our above results that the aperture-averaging factor for spherical-wave scintillations with q_R ≫ 1 actually approaches ${\hat{A}}_{sp} = σ_{sp, F}^{2} / σ_{χ, sp}^{2} = 4.683 q_{R}^{- 7 / 6}$ ; a similar result has been presented in [19] previously.

2.2.2. Path-space domain perspective

Like the plane-wave case, B_sp,_FI, $σ_{sp, F}^{2}$ and $σ_{sp, I}^{2}$ can also be expressed in the forms of an integral over the normalized path-space domain, i.e.,

B_{sp, F I} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) g_{sp} (ξ),

σ_{sp, F}^{2} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) g^{'}_{sp} (ξ),

σ_{sp, I}^{2} = 0.033 π^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ C_{n}^{2} (ξ L) g^{″}_{sp} (ξ),

where

g_{sp} (ξ) = {\tilde{h}}_{sp} (ξ, ξ, 1 - ξ)

,

g^{'}_{sp} (ξ) = {\tilde{h}}_{sp} (ξ, ξ, ξ)

, and

g^{″}_{sp} (ξ) = {\tilde{h}}_{sp} (ξ, 1 - ξ, 1 - ξ)

with

\begin{matrix} {\tilde{h}}_{sp} (x, a, b) = x^{2} {(1 - x)}^{2} \int_{0}^{\infty} d \hat{κ} {\hat{κ}}^{4 / 3} \exp (- q_{m} {\hat{κ}}^{2}) jinc (a q_{R}^{1 / 2} \hat{κ}) \\ \times jinc (b q_{R}^{1 / 2} \hat{κ}) {sinc}^{2} [x (1 - x) {\hat{κ}}^{2} / 2] . \end{matrix}

It can be deduced from Eq. (50) that both q_m and q_R play a role in determining the shapes of g_sp(ξ), g′_sp(ξ) and g″_sp(ξ) as a function of ξ. Different from the plane-wave case, q_m and q_R in Eq. (50) cannot be combined together to define a single nondimensional parameter. It is obvious that g_sp(ξ) ≡ g′_sp(ξ) ≡ g″_sp(ξ) when q_R = 0. This indeed corresponds to the case of the PSPR reciprocity. Figure 7 demonstrates the path weighting functions g_sp(ξ), g′_sp(ξ) and g″_sp(ξ) in terms of the normalized path location ξ with various q_m and q_R. Similar to the plane-wave case, in general, g_sp(ξ) always features a symmetric shape with its peak located at ξ = 0.5 and g′_sp(ξ) is the reflection of g″_sp(ξ) in the vertical line ξ = 0.5; further, g′_sp(ξ) and g″_sp(ξ) are peaked at a path location dependent on q_m and q_R. It is seen from Fig. 7 that g_sp(ξ), g′_sp(ξ) and g″_sp(ξ) basically take on the same shape with q_R much smaller than 1, implying that μ_sp is close to 1. One can observe that, as q_R grows larger with q_m fixed, although g_sp(ξ) remains peaked at ξ = 0.5, the peaks of g′_sp(ξ) and g″_sp(ξ) gradually move from the midpoint to the left and right sides of the abscissa, respectively; this behavior means that turbulent eddies at the path midpoint are always weighted most heavily for B_sp,_FI, whereas the location of turbulent eddies that are weighted most strongly for $σ_{sp, F}^{2}$ or $σ_{sp, I}^{2}$ gradually moves toward the transmitting plane with an increasing value of q_R. In addition, it can be seen from Fig. 7 that q_m produces effects opposite to q_R, which has been found in the preceding analysis. This reveals that, from comprehensive points of view, the inner scale of turbulence should be considered in the treatment of the correlation between light-flux fluctuations of two counter-propagating waves in atmospheric turbulence. Finally, like the plane-wave case, the shape of $C_{n}^{2} (ξ L)$ in terms of ξ obviously plays an important role in determining μ_sp; if $C_{n}^{2} (ξ L)$ featured a single large sharp peak at the midpoint of a path, μ_sp would be close to 1 even with a somewhat great q_R.

Fig. 7 The path weighting functions g_sp(ξ), g′_sp(ξ) and g″_sp(ξ) in terms of ξ with various q_m and q_R. (a) q_R = 0.01; (b) q_R = 1; (c) q_R = 100. The solid curves correspond to q_m = 0; the dashed curves correspond to q_m = 0.01; the dotted curves correspond to q_m = 1.

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3. Discussions

The optical signal collected by the aperture of a PIB receiver can be viewed as the light flux entering it [19,27–29], which is equal to the integrated irradiance over the aperture. The problem as to aperture-collected optical scintillations has been commonly treated based on the spatial covariance function of irradiance fluctuations. In the previous section, we employed the relation given by Eq. (9) to relate the normalized light flux with the aperture-averaged first-order log-amplitude fluctuation, and subsequently developed the expressions for the correlation coefficient between light-flux fluctuations of two counter-propagating plane or spherical waves in weak turbulence. In our theoretical treatment, by utilizing the Fourier transform, the role of a finite receiving aperture is formulated as the spectral filtering factors presented by Eq. (14) and (38). This permits us to use a straightforward procedure to deal with the light-flux fluctuations.

In Sec. 2, we have occasionally analyzed the asymptotic behavior of the correlation coefficient between light-flux fluctuations of two counter-propagating waves. Charnotskii [28] presented a method for performing asymptotic analysis of the variance of light-flux fluctuations. Thereinafter, we follow the procedure used by this method to find a full set of asymptotes of the correlation coefficient. By considering the Tatarskii spectrum with a refractive-index structure constant independent of the path location, Eqs. (17) – (19) and (39) – (41) can be rewritten in the form

\begin{array}{l} G (α_{1}, α_{2}, α_{3}, α_{4}) = 0.033 \times 4 π^{2} k^{2} L C_{n}^{2} \int_{0}^{1} d ξ \int_{0}^{\infty} d κ κ^{- 8 / 3} \exp (- κ^{2} / κ_{m}^{2}) \\ \times jinc (α_{1} κ R_{a}) jinc (α_{2} κ R_{a}) \sin [κ^{2} L α_{3} / (2 k)] \\ \times \sin [κ^{2} L α_{4} / (2 k)] . \end{array}

Specifically, B_pl,_FI = G (1, 1, ξ, 1 − ξ),

σ_{pl, F}^{2} = G (1, 1, 1 - ξ, 1 - ξ)

,

σ_{pl, I}^{2} = G (1, 1, ξ, ξ)

, B_sp,_FI = G (ξ, 1 − ξ, ξ (1 − ξ), ξ (1 − ξ)),

σ_{sp, F}^{2} = G (ξ, ξ, ξ (1 - ξ), ξ (1 - ξ))

, and

σ_{sp, I}^{2} = G (1 - ξ, 1 - ξ, ξ (1 - ξ), ξ (1 - ξ))

. There are three spatial scales included in Eq. (51), i.e., R_a,

κ_{m}^{- 1}

and

\sqrt{L / k}

. Three asymptotic cases are related to three different situations in which one of these spatial scales determines the effective integration domain of the inside integral in Eq. (51). They are given as follows. When q_m ≪ 1 and q_R ≪ 1, one finds

\begin{matrix} G (α_{1}, α_{2}, α_{3}, α_{4}) ≃ 0.033 \times 4 π^{2} C_{n}^{2} k^{7 / 6} L^{11 / 6} \int_{0}^{1} d ξ \int_{0}^{\infty} d x x^{- 8 / 3} \\ \times \sin (x^{2} α_{3} / 2) \sin (x^{2} α_{4} / 2) = O (σ_{χ, sp}^{2}) . \end{matrix}

When q_m ≪ q_R and q_R ≫ 1, one finds

\begin{matrix} G (α_{1}, α_{2}, α_{3}, α_{4}) ≃ 0.033 π^{2} C_{n}^{2} k^{7 / 6} L^{11 / 6} q_{R}^{- 7 / 6} \int_{0}^{1} d ξ α_{3} α_{4} \int_{0}^{\infty} d x x^{4 / 3} \\ \times jinc (α_{1} x) jinc (α_{2} x) = O (σ_{χ, sp}^{2} q_{R}^{- 7 / 6}) . \end{matrix}

When q_m ≫ q_R and q_m ≫ 1, one finds

\begin{matrix} G (α_{1}, α_{2}, α_{3}, α_{4}) ≃ 0.033 π^{2} C_{n}^{2} k^{7 / 6} L^{11 / 6} q_{m}^{- 7 / 6} \int_{0}^{1} d ξ α_{3} α_{4} \int_{0}^{\infty} d x x^{4 / 3} \\ \times \exp (- x^{2}) = O (σ_{χ, sp}^{2} q_{m}^{- 7 / 6}) . \end{matrix}

A 2D map of asymptotes, previously introduced in [28], for our case is plotted in Fig. 8(a) by combining the above three asymptotes together. From Fig. 8(a), we can verify that the three asymptotes completely cover the parameter space. Asymptotes of μ_pl and μ_sp found immediately from Eqs. (52) – (54) are exhibited by Figs. 8(b) and 8(c). Indeed, Fig. 6 reflects the structure of Fig. 8(c). For many practical optical-wave propagation scenarios, q_R is larger than q_m; it can be deduced from Figs. 8(b) and 8(c) that the inner scale tends to become unimportant if q_R > q_m. Additionally, from Figs. 8(b) and 8(c), one can find that in many situations the correlation between light-flux fluctuations of two counter-propagating waves is lower than 1 obviously. However, as shown in [1,30], good correlation between two counter-propagating waves measured by SMFC transceivers can be preserved for large receiving apertures even under strong-turbulence conditions; as far as the preservation of good correlation is concerned, bidirectional FSO systems based on SMFC transceivers appear more advantageous than those based on PIB receivers.

Fig. 8 Set of Asymptotes. (a) asymptotes of Eq. (51); (b) asymptotes of the correlation coefficient μ_pl; (c) asymptotes of the correlation coefficient μ_sp.

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As an important comment, in arriving at Eq. (8), the irradiance in vacuum is indeed thought of as being independent of r. This is reasonable for the plane- and spherical-wave cases under the paraxial approximation. Because the irradiance of a collimated-Gaussian-beam wave in vacuum does depend on r, strictly speaking, the result shown by Eq. (8) is not completely applicable to the collimated-Gaussian-beam cases. However, if the beam radius at the receiving plane is much greater than the aperture radius, the variation of the irradiance in the absence of atmospheric turbulence within the aperture will be small, meaning that the use of Eq. (8) to estimate the normalized light flux of a collimated-Gaussian-beam wave will not incur a significant error. It has been stated previously that the results obtained in Sec. 2 may provide some somewhat reasonable estimates for the collimated-Gaussian-beam cases under the near-and far-field conditions; with the above discussion in mind, it is apparent that an additional condition necessary for this statement is that the aperture radius is much smaller than the beam radius at the receiving plane.

4. Conclusions

We have formulated the correlation coefficient between light-flux fluctuations of two counter-propagating plane or spherical waves in weak atmospheric turbulence. Based on the obtained formulae, by considering the Tatarskii turbulence spectrum, we have elucidated the behavior of the correlation coefficient from both the wavenumber domain and path-space domain perspectives. If the turbulence spectrum has no path dependence, the correlation coefficient is completely determined by the aperture and inner-scale Fresnel parameters. If the turbulence spectrum is dependent on path locations, the aperture and inner-scale Fresnel parameters completely determine the shape of the path weighting functions for the cross-covariance and variances of normalized light-flux fluctuations. For two counter-propagating plane waves, turbulent eddies located in the vicinities of the path midpoint and endpoints are most effective in making positive and negative contributions to the correlation coefficient, respectively. On the other hand, for two counter-propagating spherical waves, turbulent eddies located in the vicinity of the path midpoint are still weighted most strongly in producing the correlation, whereas the role of those near the path endpoints on the correlation coefficient depends on the value of the aperture Fresnel parameter.

Funding

National Natural Science Foundation of China (61007046, 61275080 and 61475025); Natural Science Foundation of Jilin Province of China (20150101016JC); Specialized Research Fund for the Doctoral Program of Higher Education of China (20132216110002).

Acknowledgments

The authors are very grateful to the reviewers for valuable comments.

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Correlation between light-flux fluctuations of two counter-propagating waves in weak atmospheric turbulence

Abstract

1. Introduction

2. Theoretical analysis

2.1. Two counter-propagating plane waves

2.1.1. Wavenumber domain perspective

2.1.2. Path-space domain perspective

2.2. Two counter-propagating spherical waves

2.2.1. Wavenumber domain perspective

2.2.2. Path-space domain perspective

3. Discussions

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (54)

Optics Express