Abstract

Expressions for the correlation coefficient between light-flux fluctuations of two waves counter-propagating along a common path in weak turbulence are developed. Only the aperture and inner-scale Fresnel parameters are needed for evaluation of the correlation coefficient if the turbulence spectrum has no path dependence, and of the path weighting functions for the cross-covariance and variances of normalized light-flux fluctuations if the turbulence spectrum is dependent on path locations. Under the condition that atmospheric turbulence is statistically homogeneous over a path, although good correlation between light-flux fluctuations of two counter-propagating spherical waves may be achieved for a relatively small aperture Fresnel parameter or relatively large inner-scale Fresnel parameter, the correlation coefficient between light-flux fluctuations of two counter-propagating plane waves is always lower than 1 obviously. When the aperture Fresnel parameter becomes larger than the inner-scale Fresnel parameter, the inner scale of turbulence tends to play an unimportant role in determining the correlation coefficient.

© 2017 Optical Society of America

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 bF (·) and HFR (·) 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]

ψ1,pl,F(r,z=L)=0Ldzdv(κ,z)Hpl,F(κ,z)exp(iκr)
with
Hpl,F(κ,z)=ikexp[iκ2(Lz)2k],
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
n1(r,z)=exp(iκr)dv(κ,z).
Performing the substitution z → (Lz) 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
ψ1,pl,I(ρ,z=0)=0Ldzdv(κ,z)Hpl,I(κ,z)exp(iκρ)
with
Hpl,I(κ,z)=ikexp(iκ2z2k),
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, ψ1,pl,F (r, z = L) = χ1,pl,F (r, z = L) + iS1,pl,F (r, z = L), where χ1,pl,F (·) and S1,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

χ1,pl,F(r,z=L)=120Ldzdv(κ,z)[Hpl,F(κ,z)+Hpl,F*(κ,z)]exp(iκr),
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
χ1,pl,I(ρ,z=0)=120Ldzdv(κ,z)[Hpl,I(κ,z)+Hpl,I*(κ,z)]exp(iκρ).

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

P^=DI(r)d2rDI0d2r=SA1DI(r)/I0d2r,
where I (r) denotes the turbulence-induced irradiance at position r, I0=a02 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 SA is the area of the aperture. In the Rytov approximation, it follows that I(r)/I01+2[χ1,pl(r)+χ1,pl2(r)+χ2,pl(r)+], where χ1,pl (r) and χ2,pl (r) denote the first- and second-order log-amplitude fluctuations of a plane wave, respectively; 〈χ1,pl (r)〉 = 0, 〈χ2,pl (r)〉 is of the same order of magnitude as χ1,pl2(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 χ1,pl2(r), χ2,pl (r), ⋯ out of I (r)/I0. With these in mind, one finds
P^1+2SA1Dχ1,pl(r)d2r=1+2χ^1,pl,
where χ^1,pl represents the aperture-averaged value of χ1,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
μplBpl,FIσpl,Fσpl,I,
where Bpl,FI=χ^1,pl,Fχ^1,pl,I*, σpl,F2=χ^1,pl,Fχ^1,pl,F* and σpl,I2=χ^1,pl,Iχ^1,pl,I*; χ^1,pl,F and χ^1,pl,I represent the aperture-averaged values of χ1,pl,F (r, z = L) and χ1,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 Bpl,FI, σpl,F2 and σpl,I2, 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 χ^1,pl,F and χ^1,pl,I have a mean value of zero and are real, implying χ^1,pl,Fχ^1,pl,F* and χ^1,pl,Iχ^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. χ^1,pl,F and χ^1,pl,I can be expressed by
χ^1,pl,F=d2rχ1,pl,F(r,z=L)Wa(r),
χ^1,pl,I=d2ρχ1,pl,I(ρ,z=0)Wa(ρ)
with
Wa(r)={1/(πRa2)|r|<Ra0|r|Ra,
where Ra 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
W˜a,pl(κ)=d2rWa(r)exp(iκr)=jinc(κRa),
where the jinc function is defined by jinc (x) = 2J1 (x) /x with J1 (·) being a Bessel function of the first kind, one finds
χ^1,pl,F=120Ldzdv(κ,z)[Hpl,F(κ,z)+Hpl,F*(κ,z)]W˜a,pl(κ),
χ^1,pl,I=120Ldzdv(κ,z)[Hpl,I(κ,z)+Hpl,I*(κ,z)]W˜a,pl(κ).
With Eqs. (15) and (16) in hand, one can develop the expressions for Bpl,FI, σpl,F2 and σpl,I2. After some tedious mathematical manipulations, we find
Bpl,FI=4π2k20Ldz0dκκΦn(κ,z)jinc2(Raκ)sin(κ2z2k)sin[κ2(Lz)2k],
σpl,F2=4π2k20Ldz0dκκΦn(κ,z)jinc2(Raκ)sin2[κ2(Lz)2k],
σpl,I2=4π2k20Ldz0dκκΦn(κ,z)jinc2(Raκ)sin2(κ2z2k),
where Φn (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
dv(κ,z)dv*(κ,z)=Fn(|zz|,κ)δ(κκ)d2κd2κ
with Fn (·) representing the two-dimensional turbulence spectrum and δ () being the delta function; the second is the fact that Fn () takes a negligible value when |zz′| 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 Ra = 0, σpl,F2 and σpl,I2 become the variances of log-amplitude fluctuations for the forward- and inverse-propagating waves, respectively. It is straightforward to find that our result for σpl,F2 with Ra = 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., Φn(κ,z)=Cn2(z)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 Cn2(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 Bpl,FI, σpl,F2 and σpl,I2 on z. Performing this operation permits us to express Bpl,FI, σpl,F2 and σpl,I2 in the forms of an integral over the wavenumber domain, which are given by

Bpl,FI=2π2k2LCn20dκκΦ^n(κ)fpl(κL/k),
σpl,F2=σpl,I2=2π2k2LCn20dκκΦ^n(κ)fpl(κL/k)
with
fpl(κ^)=jinc2(qR1/2κ^)hpl(κ^),
fpl(κ^)=jinc2(qR1/2κ^)hpl(κ^),
hpl(κ^)=sinc(κ^2/2)cos(κ^2/2),
hpl(κ^)=1sinc(κ^2),
where κ^=κ(L/k)1/2 is the scaled wavenumber, qR=kRa2/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 fpl(·) and fpl(·) 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 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 fpl(·) and fpl(·) factorize into a product of two different terms. The first term jinc2(qR1/2κ^) emphasizes the role of refractive-index fluctuations associated with the scaled wavenumber κ^1/(0.5216qR1/2) and basically neglects those associated with the scaled wavenumber κ^1/(0.5216qR1/2). 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 Φ^n(κ) with κ > 1/(βRa) actually contributes little to both Bpl,FI and σpl,F2. Indeed, the term jinc2(qR1/2κ^) formulates the role of a receiving aperture in the treatment of the scintillation problem. It is obvious that a greater Ra leads to a smaller portion of 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 fpl() is distinguished from fpl(·) only by the two terms hpl(·) and hpl(·). To obtain some specific idea about the difference between Bpl,FI and σpl,F2, below we compare the behaviors of hpl(·) and hpl(·) in terms of κ^. Figure 2 shows the shapes of the functions hpl(·) and hpl(). It is found from Fig. 2 that, with an increasing value of κ^, both hpl(·) and hpl () first grow monotonically from 0 to a peak in the vicinity of κ^=2; after that hpl(·) begins to oscillate around 0 with almost constant amplitude and hpl(), however, begins to oscillate around 1 with diminishing amplitude. Hence, refractive-index·fluctuations associated with all spatial wavenumbers make positive contribution to σpl,F2, whereas there may exist refractive-index fluctuations related to some spatial wavenumbers that make negative contribution to Bpl,FI. It is apparent that Bpl,FI<σpl,F2, implying μpl < 1, because the curve corresponding to hpl(·) basically always lies above that corresponding to hpl(·). 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 qR1/2β1, the specific behaviors of hpl(·) and hpl(·) beyond κ^=1/(qR1/2β) only play a trivial role in determining Bpl,FI, and σpl,F2; it is also noted·that hpl(κ^)κ^4/12 and hpl(κ^)κ^4/6 for κ^1. Hence, we can reason from the above analysis that Bpl,FIσpl,F2/2 and thus μpl ≃ 1/2 when qR1/21/β, irrespective of what mathematical form Φ^n(κ) takes.

 

Fig. 2 hpl(·) and hpl(·) in terms of κ^=κ(L/k)1/2.

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To gain more details about μpl, 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 Φ^n(κ)=0.033exp(κ2/κm2)(κ2+κ02)11/6, where κ0 = 2π/L0 and κm = 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 Ra and 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 hpl(·) into two terms, i.e., hpl(κ^)=hl,pl(κ^)hpl(κ^/21/2) with hl,pl(κ^)=1cos(κ^2/2). Accordingly, Bpl,FI is expressed as Bpl,FI = b1b2 with b1 given by

b1=2π2k2LCn20dκκΦ^n(κ)jinc2(Raκ)h1,pl(κL/k)(qp2+14)5/12cos[56arctan(12qp)]C1qp5/6C1,
where qp = qm + β2qR, qm=kκm2/L is called the inner-scale Fresnel parameter, C1=11σχ,pl2/[6cos(5π/12)], and σχ,pl2=0.307Cn2k7/6L11/6. 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 qm and qR can be combined together to form qp. The term b2 is expressed by
b2=2π2k2LCn20dκκΦ^n(κ)jinc2(Raκ)hpl(κL/k/21/2)1211(14+qp2)11/12sin[116arctan(2qp)+π12]C1qp5/6C1.
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 σpl,F2 to yield
σpl,F2611(1+qp2)11/12sin[116arctan(qp)+π12]C1qp5/6C1.
With the above results in hand, we find
μpl(qp)={611(1+qp2)1112sin[116arctan(qp)+π12]qp5/6}1{(qp2+14)5/12×cos[56arctan(12qp)]1211(14+qp2)11/12sin[116arctan(2qp)+π12]},
where, on the left-hand side, we explicitly show the dependence of μpl on the nondimensional parameter qp.

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 μpl is actually completely determined by the nondimensional parameter qp. By examining the mathematical form of qp, we can deduce that the inner scale l0 plays a role similar to that the aperture radius Ra does on the correlation coefficient μpl. It is easy to find that μpl ≃ 0.468 when qp → 0, i.e., both qm → 0 and qR → 0. Further, it should be pointed out that the use of the aforementioned approximation of jinc2 (·) has actually no impact on the value of μpl with qp → 0, thus meaning qR → 0, because jinc2(qR1/2κ^)exp(qRβ2κ^2)1 with qR = 0. Figure 3 shows the correlation coefficient μpl in terms of qR1/2 under the condition that the turbulence spectrum has no path dependence. It is found from Fig. 3 that μpl is peaked at qR1/20.45 with its maximum value equal to 0.574, and μpl achieves its minimum value at qR1/2=0. Furthermore, μpl tends to 0.5 when qp1/21, which is consistent with the previous deduction (i.e., μpl ≃ ½ when qR1/21/β). The above analyses and calculations provide us a clear understanding of the effects of various propagation parameters on μpl when atmospheric turbulence is statistically homogeneous over the path.

 

Fig. 3 μpl in terms of qp1/2 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 Bpl,FI, σpl,F2 and σpl,I2 in the forms of an integral over the normalized path-space domain, which are given by

Bpl,FI=0.033π2k7/6L11/601dξCn2(ξL)gpl(ξ),
σpl,F2=0.033π2k7/6L11/601dξCn2(ξL)gpl(ξ),
σpl,I2=0.033π2k7/6L11/601dξCn2(ξL)gpl(ξ)
with ξ = z/L being the normalized path location, gpl(ξ)=h˜pl(1/2,1/2ξ), gpl(ξ)=h˜pl(1ξ,0), gpl(ξ)=h˜pl(ξ,0), and
h˜pl(a,b)=35Γ(16)[(qp+ia)5/6+(qpia)5/6(qp+ib)5/6(qpib)5/6].
In arriving at Eqs. (31)(34), the said approximation of jinc2 (·) has been utilized.

It is worth noting that gpl(ξ), gpl(ξ) and gpl(ξ) play the role of a path weighting function which emphasizes or deemphasizes the contributions of turbulent eddies at different locations to Bpl,FI, σpl,F2 and σpl,I2, respectively. The path weighting functions appearing in Eqs. (31)(33) in terms of ξ are plotted in Fig. 4 with various qp. It is seen from Fig. 4 that gpl(ξ) is symmetric about ξ = 0.5, and gpl(ξ) is the reflection of gpl(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 Bpl,FI and those located in the vicinities of the two path endpoints nearly do not contribute to Bpl,FI ; on the other hand, turbulent eddies near the transmitting plane are weighted most strongly for σpl,F2 and σpl,I2, and those near the receiving plane contribute little to σpl,F2 and σpl,I2. 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 μpl, 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 Cn2(ξL) in terms of ξ has an important impact on the results of the inner products, implying that μpl will depend on the path variations of Cn2(ξL). If the strength of refractive-index fluctuations over the path featured a single large sharp peak at the midpoint, μpl would be likely to take a value close to 1. Indeed, it is observed from Fig. 4 that, with the same qp, the curves corresponding to gpl(ξ), gpl(ξ) and gpl(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 μpl ≡ 1. In the same context, if the random phase screen is moved away from the midpoint, we can infer from Fig. 4 that μpl < 1 and σpl,F2σpl,I2. 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 qp leads to smaller magnitude of the path weighting functions. This means that an increase in qp generally lowers the absolute magnitude of contributions from turbulent eddies over the path to Bpl,FI, σpl,F2 and σpl,I2.

 

Fig. 4 The path weighting functions gpl(ξ), gpl(ξ) and gpl(ξ) in terms of ξ with various qp. (a) qp = 0; (b) qp = 1; (c) qp = 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

χ^1,sp,F=120Ldzdv(κ,z)[Hsp,F(κ,z)+Hsp,F(κ,z)]W˜a,sp(zLκ),
χ^1,sp,I=120Ldzdv(κ,z)[Hsp,I(κ,z)+Hsp,I(κ,z)]W˜a,sp(LzLκ),
respectively, where
Hsp,F(κ,z)=Hsp,I(κ,z)=ikexp[iκ2z(Lz)2Lk],
W˜a,sp(γκ)=d2rWa(r)exp(iγκr)=jinc(γκRa).
Based on Eqs. (35) and (36), analogous to the plane-wave case, we can further develop the expressions for Bsp,FI=χ^1,sp,Fχ^1,sp,I, σsp,F2=χ^1,sp,Fχ^1,sp,F and σsp,I2=χ^1,sp,Iχ^1,sp,I, which take the forms
Bsp,FI=4π2k20Ldz0dκκΦn(κ,z)jinc(zLRaκ)×jinc(LzLRaκ)sin2[κ2z(Lz)2Lk],
σsp,F2=4π2k20Ldz0dκκΦn(κ,z)jinc2(zLRaκ)sin2[κ2z(Lz)2Lk],
σsp,I2=4π2k20Ldz0dκκΦn(κ,z)jinc2(LzLRaκ)sin2[κ2z(Lz)2Lk].
The correlation coefficient between light-flux fluctuations of two counter-propagating spherical waves is formulated by
μspBsp,FIσsp,Fσsp,I.

2.2.1. Wavenumber domain perspective

If the turbulence spectrum Φn (·) has no path dependence, we can reformulate Bsp,FI, σsp,F2 and σsp,I2 as follows:

Bsp,FI=2π2k2LCn20dκκΦ^n(κ)fsp(κL/k),
σsp,F2σsp,I2=2π2k2LCn20dκκΦ^n(κ)f sp(κL/k)
with
fsp(κ^)=12κ^401dξξ2(1ξ)2jinc[(1ξ)qR1/2κ^]jinc(ξqR1/2κ^)sinc2[κ^22ξ(1ξ)],
f sp(κ^)=12κ^401dξξ2(1ξ)2jinc2(ξqR1/2κ^)sinc2[κ^22ξ(1ξ)].
Equations (45) and (46) show the spectral filter functions for Bsp,FI, σsp,F2 and σsp,I2, respectively. When Ra = 0, i.e., qR = 0, it follows that fsp(κ^)f sp(κ^), implying that Bsp,FIσsp,F2σsp.I2 and thus μsp ≡ 1, regardless of the mathematical form of Φ^n(κ). This result is indeed what we will expect from the reciprocity principle [9–11] by recognizing that Ra = 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 fsp(κ^) and f sp(κ^) are illustrated in Fig. 5 with various qR. It is found from Fig. 5 that the curves corresponding to fsp(κ^) and f sp(κ^), respectively, basically coincide with each other when qR = 0.04 and obviously deviate from each other within a range of the scaled wavenumber κ^ when qR = 1 or 16. In fact, it is the deviation of fsp(κ^) from f sp(κ^) that causes μsp to decrease below 1. Further, a larger qR moves the said deviation deeper into the low-wavenumber range. Notice that, 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 qR generally leads to a smaller μsp. On the other hand, as for the two curves associated with qR = 1 in Fig. 5, it is seen that they begin to obviously deviate from each other only when κ^=κ(L/k)1/22. Hence, if κm = 5.92/l0 ≲ 2/(L/k)1/2, the deviation portion of these two curves only has a negligible impact on μsp because Φ^n(κ) rapidly drops close to 0 when κ increases beyond κm. This leads us to an inference that a nonzero inner scale l0 may make the decrease in μsp caused by a nonzero aperture radius Ra negligible. As a result, it is necessary to consider the role of a nonzero inner scale l0 besides the receiver aperture radius Ra in determining μsp under some conditions. Moreover, it is seen from Fig. 5 that f sp(κ^) always takes a positive value, whereas fsp(κ^) with qR > 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 Bsp,FI.

 

Fig. 5 The spectral filter functions fsp(κ^) and f sp(κ^) with various qR. (a) qR= 0.04; (b) qR = 1; (c) qR = 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 μsp is completely determined by the two nondimensional parameters qm and qR. Figure 6 depicts the contours of μ as a function of qm1/2 and qR1/2. It is seen from Fig. 6 that, with the same qm, μsp decreases when qR increases; in contrast, with the same qR, a larger qm results in a greater μsp. Hence, an increase in qm may cancel out the reduction in μsp caused by enlarging the value of qR. The underlying physical reason for this behavior of μsp 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 μsp 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 qm1/2 and qR1/2.

 

Fig. 6 Contours of μsp as a function of qm1/2 and qR1/2 with a turbulence spectrum independent of path locations. The subplot (b) shows a zoomed detail of the region with relatively small qm1/2 and qR1/2.

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At this point, a question that needs to be addressed further is what value μsp approaches when qR → ∞. 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′/Ra = t′/(qR L/k)1/2, for a large enough qR, it follows that exp(−qmt2/qR) ≈ 1 and sinc[ξ (1 − ξ) t′2/(2qR)] ≈ 1 within most of the integration range of practical interest. With this observation in mind, we can deduce that Bsp,FI0.571σχ,sp2qR7/6 and σsp,F24.683σχ,sp2qR7/6 when qR ≫ 1, where σχ,sp2=0.124Cn2k7/6L11/6 denotes the spherical-wave log-amplitude variance. Consequently, μsp approaches 0.122 as qR 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 qR ≫ 1 actually approaches A^sp=σsp,F2/σχ,sp2=4.683qR7/6; a similar result has been presented in [19] previously.

2.2.2. Path-space domain perspective

Like the plane-wave case, Bsp,FI, σsp,F2 and σsp,I2 can also be expressed in the forms of an integral over the normalized path-space domain, i.e.,

Bsp,FI=0.033π2k7/6L11/601dξCn2(ξL)gsp(ξ),
σsp,F2=0.033π2k7/6L11/601dξCn2(ξL)g sp(ξ),
σsp,I2=0.033π2k7/6L11/601dξCn2(ξL)g sp(ξ),
where gsp(ξ)=h˜sp(ξ,ξ,1ξ), g sp(ξ)=h˜sp(ξ,ξ,ξ), and g sp(ξ)=h˜sp(ξ,1ξ,1ξ) with
h˜sp(x,a,b)=x2(1x)20dκ^κ^4/3exp(qmκ^2)jinc(aqR1/2κ^)×jinc(bqR1/2κ^)sinc2[x(1x)κ^2/2].

It can be deduced from Eq. (50) that both qm and qR play a role in determining the shapes of gsp(ξ), gsp(ξ) and gsp(ξ) as a function of ξ. Different from the plane-wave case, qm and qR in Eq. (50) cannot be combined together to define a single nondimensional parameter. It is obvious that gsp(ξ) ≡ gsp(ξ) ≡ gsp(ξ) when qR = 0. This indeed corresponds to the case of the PSPR reciprocity. Figure 7 demonstrates the path weighting functions gsp(ξ), gsp(ξ) and gsp(ξ) in terms of the normalized path location ξ with various qm and qR. Similar to the plane-wave case, in general, gsp(ξ) always features a symmetric shape with its peak located at ξ = 0.5 and gsp(ξ) is the reflection of gsp(ξ) in the vertical line ξ = 0.5; further, gsp(ξ) and gsp(ξ) are peaked at a path location dependent on qm and qR. It is seen from Fig. 7 that gsp(ξ), gsp(ξ) and gsp(ξ) basically take on the same shape with qR much smaller than 1, implying that μsp is close to 1. One can observe that, as qR grows larger with qm fixed, although gsp(ξ) remains peaked at ξ = 0.5, the peaks of gsp(ξ) and gsp(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 Bsp,FI, whereas the location of turbulent eddies that are weighted most strongly for σsp,F2 or σsp,I2 gradually moves toward the transmitting plane with an increasing value of qR. In addition, it can be seen from Fig. 7 that qm produces effects opposite to qR, 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 Cn2(ξL) in terms of ξ obviously plays an important role in determining μsp; if Cn2(ξL) featured a single large sharp peak at the midpoint of a path, μsp would be close to 1 even with a somewhat great qR.

 

Fig. 7 The path weighting functions gsp(ξ), gsp(ξ) and gsp(ξ) in terms of ξ with various qm and qR. (a) qR = 0.01; (b) qR = 1; (c) qR = 100. The solid curves correspond to qm = 0; the dashed curves correspond to qm = 0.01; the dotted curves correspond to qm = 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

G(α1,α2,α3,α4)=0.033×4π2k2LCn201dξ0dκκ8/3exp(κ2/κm2)×jinc(α1κRa)jinc(α2κRa)sin[κ2Lα3/(2k)]×sin[κ2Lα4/(2k)].
Specifically, Bpl,FI = G (1, 1, ξ, 1 − ξ), σpl,F2=G(1,1,1ξ,1ξ), σpl,I2=G(1,1,ξ,ξ), Bsp,FI = G (ξ, 1 − ξ, ξ (1 − ξ), ξ (1 − ξ)), σsp,F2=G(ξ,ξ,ξ(1ξ),ξ(1ξ)), and σsp,I2=G(1ξ,1ξ,ξ(1ξ),ξ(1ξ)). There are three spatial scales included in Eq. (51), i.e., Ra, κm1 and 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 qm ≪ 1 and qR ≪ 1, one finds
G(α1,α2,α3,α4)0.033×4π2Cn2k7/6L11/601dξ0dxx8/3×sin(x2α3/2)sin(x2α4/2)=O(σχ,sp2).
When qmqR and qR ≫ 1, one finds
G(α1,α2,α3,α4)0.033π2Cn2k7/6L11/6qR7/601dξα3α40dxx4/3×jinc(α1x)jinc(α2x)=O(σχ,sp2qR7/6).
When qmqR and qm ≫ 1, one finds
G(α1,α2,α3,α4)0.033π2Cn2k7/6L11/6qm7/601dξα3α40dxx4/3×exp(x2)=O(σχ,sp2qm7/6).

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 μpl and μsp 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, qR is larger than qm; it can be deduced from Figs. 8(b) and 8(c) that the inner scale tends to become unimportant if qR > qm. 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 μpl; (c) asymptotes of the correlation coefficient μsp.

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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.

References and links

1. J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013). [CrossRef]  

2. M. A. Vorontsov, “Conservation laws for counter-propagating optical waves in atmospheric turbulence with application to directed energy and laser communications,” in Proceedings of Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2013), paper PW3F.1.

3. N. Perlot and D. Giggenbach, “Scintillation correlation between forward and return spherical waves,” Appl. Opt. 51(15), 2888–2893 (2012). [CrossRef]   [PubMed]  

4. J. H. Shapiro and A. L. Puryear, “Reciprocity-enhanced optical communication through atmospheric turbulence -part I: reciprocity proofs and far-field power transfer optimization,” J. Opt. Commun. Netw. 4(12), 947–954 (2012). [CrossRef]  

5. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity-enhanced optical communication through atmospheric turbulence - part II: communication architectures and performance,” J. Opt. Commun. Netw. 5(8), 888–900 (2013). [CrossRef]  

6. C. Chen, H. Yang, S. Tong, and Y. Lou, “Mean-square angle-of-arrival difference between two counter-propagating spherical waves in the presence of atmospheric turbulence,” Opt. Express 23(19), 24657–24668 (2015). [CrossRef]   [PubMed]  

7. A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012). [CrossRef]  

8. N. Wang, X. Song, J. Cheng, and V. C. M. Leung, “Enhancing the security of free-space optical communications with secret sharing and key agreement,” J. Opt. Commun. Netw. 6(12), 1072–1081 (2014). [CrossRef]  

9. J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am. 61(4), 492–495 (1971). [CrossRef]  

10. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10(7), 1652–1658 (1971). [CrossRef]   [PubMed]  

11. V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982). [CrossRef]  

12. D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. 51(16), 3145–3152 (2012). [CrossRef]   [PubMed]  

13. R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express 20(19), 21635–21644 (2012). [CrossRef]   [PubMed]  

14. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23(13), 986–988 (1998). [CrossRef]  

15. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005). [CrossRef]   [PubMed]  

16. M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010). [CrossRef]  

17. C. Chen, H. Yang, H. Wang, S. Tong, and Y. Lou, “Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence,” Appl. Opt. 50(3), 307–312 (2011). [CrossRef]   [PubMed]  

18. C. Chen and H. Yang, “Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence,” Opt. Express 24(17), 19713–19727 (2016). [CrossRef]   [PubMed]  

19. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

20. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).

21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005). [CrossRef]  

22. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994). [CrossRef]  

23. M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011). [CrossRef]  

24. Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007). [CrossRef]  

25. A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970). [CrossRef]  

26. J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006). [CrossRef]  

27. L. C. Andrews, “Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9(4), 597–600(1992). [CrossRef]  

28. M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991). [CrossRef]  

29. C. Chen, H. Yang, Z. Zhou, W. Zhang, M. Kavehrad, S. Tong, and T. Wang, “Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence,” Opt. Express 21(24), 29731–29743 (2013). [CrossRef]  

30. M. A. Vorontsov, S. L. Lachinova, and A. K. Majumdar, “Target-in-the-loop remote sensing of laser beam and atmospheric turbulence characteristics,” Appl. Opt. 55(19), 5172–5179 (2016). [CrossRef]   [PubMed]  

References

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  1. J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
    [Crossref]
  2. M. A. Vorontsov, “Conservation laws for counter-propagating optical waves in atmospheric turbulence with application to directed energy and laser communications,” in Proceedings of Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2013), paper PW3F.1.
  3. N. Perlot and D. Giggenbach, “Scintillation correlation between forward and return spherical waves,” Appl. Opt. 51(15), 2888–2893 (2012).
    [Crossref] [PubMed]
  4. J. H. Shapiro and A. L. Puryear, “Reciprocity-enhanced optical communication through atmospheric turbulence -part I: reciprocity proofs and far-field power transfer optimization,” J. Opt. Commun. Netw. 4(12), 947–954 (2012).
    [Crossref]
  5. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity-enhanced optical communication through atmospheric turbulence - part II: communication architectures and performance,” J. Opt. Commun. Netw. 5(8), 888–900 (2013).
    [Crossref]
  6. C. Chen, H. Yang, S. Tong, and Y. Lou, “Mean-square angle-of-arrival difference between two counter-propagating spherical waves in the presence of atmospheric turbulence,” Opt. Express 23(19), 24657–24668 (2015).
    [Crossref] [PubMed]
  7. A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
    [Crossref]
  8. N. Wang, X. Song, J. Cheng, and V. C. M. Leung, “Enhancing the security of free-space optical communications with secret sharing and key agreement,” J. Opt. Commun. Netw. 6(12), 1072–1081 (2014).
    [Crossref]
  9. J. H. Shapiro, “Reciprocity of the turbulent atmosphere,” J. Opt. Soc. Am. 61(4), 492–495 (1971).
    [Crossref]
  10. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10(7), 1652–1658 (1971).
    [Crossref] [PubMed]
  11. V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
    [Crossref]
  12. D. Giggenbach, W. Cowley, K. Grant, and N. Perlot, “Experimental verification of the limits of optical channel intensity reciprocity,” Appl. Opt. 51(16), 3145–3152 (2012).
    [Crossref] [PubMed]
  13. R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express 20(19), 21635–21644 (2012).
    [Crossref] [PubMed]
  14. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23(13), 986–988 (1998).
    [Crossref]
  15. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005).
    [Crossref] [PubMed]
  16. M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
    [Crossref]
  17. C. Chen, H. Yang, H. Wang, S. Tong, and Y. Lou, “Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence,” Appl. Opt. 50(3), 307–312 (2011).
    [Crossref] [PubMed]
  18. C. Chen and H. Yang, “Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence,” Opt. Express 24(17), 19713–19727 (2016).
    [Crossref] [PubMed]
  19. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  20. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).
  21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
    [Crossref]
  22. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
    [Crossref]
  23. M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
    [Crossref]
  24. Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
    [Crossref]
  25. A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
    [Crossref]
  26. J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
    [Crossref]
  27. L. C. Andrews, “Aperture-averaging factor for optical scintillations of plane and spherical waves in the atmosphere,” J. Opt. Soc. Am. A 9(4), 597–600(1992).
    [Crossref]
  28. M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
    [Crossref]
  29. C. Chen, H. Yang, Z. Zhou, W. Zhang, M. Kavehrad, S. Tong, and T. Wang, “Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence,” Opt. Express 21(24), 29731–29743 (2013).
    [Crossref]
  30. M. A. Vorontsov, S. L. Lachinova, and A. K. Majumdar, “Target-in-the-loop remote sensing of laser beam and atmospheric turbulence characteristics,” Appl. Opt. 55(19), 5172–5179 (2016).
    [Crossref] [PubMed]

2016 (2)

2015 (1)

2014 (1)

2013 (3)

2012 (5)

2011 (2)

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

C. Chen, H. Yang, H. Wang, S. Tong, and Y. Lou, “Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence,” Appl. Opt. 50(3), 307–312 (2011).
[Crossref] [PubMed]

2010 (1)

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

2007 (1)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

2006 (1)

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

2005 (1)

1998 (1)

1994 (1)

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

1992 (1)

1991 (1)

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[Crossref]

1982 (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

1971 (2)

1970 (1)

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[Crossref]

Andrews, L. C.

Baker, G. J.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

Charnotskii, M.

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

Charnotskii, M. I.

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[Crossref]

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Chen, C.

Cheng, J.

Cheon, Y.

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Cowley, W.

Cowley, W. G.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Davidson, F. M.

Dikmelik, Y.

Dolfi, D.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Duncan, B. D.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Giggenbach, D.

Goda, M. E.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Grant, K.

Greco, J. A.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).

Kavehrad, M.

Khatoon, A.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Kon, A. I.

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[Crossref]

Lachinova, S. L.

Leeb, W. R.

Letzepis, N.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

Leung, V. C. M.

Lou, Y.

Lukin, V. P.

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Lutomirski, R. F.

Majumdar, A. K.

Minet, J.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Muschinski, A.

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Parenti, R. R.

Perlot, N.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[Crossref]

Polnau, E.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

Puryear, A. L.

Puryear, L.

Roth, J. M.

Schmidt, J. D.

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

Shapiro, J. H.

Song, X.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tong, S.

Vorontsov, M. A.

M. A. Vorontsov, S. L. Lachinova, and A. K. Majumdar, “Target-in-the-loop remote sensing of laser beam and atmospheric turbulence characteristics,” Appl. Opt. 55(19), 5172–5179 (2016).
[Crossref] [PubMed]

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

M. A. Vorontsov, “Conservation laws for counter-propagating optical waves in atmospheric turbulence with application to directed energy and laser communications,” in Proceedings of Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2013), paper PW3F.1.

Walther, F. G.

Wang, H.

Wang, N.

Wang, T.

Winzer, P. J.

Yang, H.

Yura, H. T.

Zhang, W.

Zhou, Z.

Appl. Opt. (6)

J. Opt. (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15(2), 022401 (2013).
[Crossref]

J. Opt. Commun. Netw. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A. 24(2), 415–422 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (4)

A. Khatoon, W. G. Cowley, and N. Letzepis, “Capacity of adaptive free-space optical channel using bi-directional links,” Proc. SPIE 8517, 85170X (2012).
[Crossref]

J. D. Schmidt, M. E. Goda, and B. D. Duncan, “Emulating bulk turbulence with a liquid-crystal spatial light modulator,” Proc. SPIE 6306, 63060O (2006).
[Crossref]

M. Charnotskii, “Coupling turbulence-distorted wave front to fiber: wave propagation theory perspective,” Proc. SPIE 7814, 78140I (2010).
[Crossref]

M. I. Charnotskii and G. J. Baker, “Practical calculation of the beam scintillation index based on the rigorous asymptotic propagation theory,” Proc. SPIE 8038, 803804 (2011).
[Crossref]

Radiophys. Quantum. Electron. (1)

A. I. Kon, “Focusing of light in a turbulent medium,” Radiophys. Quantum. Electron. 13(1), 43–50 (1970).
[Crossref]

Sov. J. Quantum Electron. (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12(5), 602–605 (1982).
[Crossref]

Waves Random Media (2)

M. I. Charnotskii, “Asymptotic analysis of the flux fluctuations averaging and finite-size source scintillations in random media,” Waves Random Media 1(4), 223–243 (1991).
[Crossref]

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillations in a turbulent medium,” Waves Random Media 4(3), 243–273 (1994).
[Crossref]

Other (4)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, 1978).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[Crossref]

M. A. Vorontsov, “Conservation laws for counter-propagating optical waves in atmospheric turbulence with application to directed energy and laser communications,” in Proceedings of Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2013), paper PW3F.1.

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Figures (8)

Fig. 1
Fig. 1 Counter-propagation geometry for two waves along a common path in atmospheric turbulence.
Fig. 2
Fig. 2 hpl(·) and hpl(·) in terms of κ ^ = κ ( L / k ) 1 / 2 .
Fig. 3
Fig. 3 μpl in terms of q p 1 / 2 with a turbulence spectrum independent of path locations.
Fig. 4
Fig. 4 The path weighting functions gpl(ξ), gpl(ξ) and gpl(ξ) in terms of ξ with various qp. (a) qp = 0; (b) qp = 1; (c) qp = 100.
Fig. 5
Fig. 5 The spectral filter functions f sp ( κ ^ ) and f   sp ( κ ^ ) with various qR. (a) qR= 0.04; (b) qR = 1; (c) qR = 16.
Fig. 6
Fig. 6 Contours of μsp 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 .
Fig. 7
Fig. 7 The path weighting functions gsp(ξ), gsp(ξ) and gsp(ξ) in terms of ξ with various qm and qR. (a) qR = 0.01; (b) qR = 1; (c) qR = 100. The solid curves correspond to qm = 0; the dashed curves correspond to qm = 0.01; the dotted curves correspond to qm = 1.
Fig. 8
Fig. 8 Set of Asymptotes. (a) asymptotes of Eq. (51); (b) asymptotes of the correlation coefficient μpl; (c) asymptotes of the correlation coefficient μsp.

Equations (54)

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ψ 1 , pl , F ( r , z = L ) = 0 L dz d v ( κ , z ) H pl , F ( κ , z ) exp ( i κ r )
H pl , F ( κ , z ) = i k exp [ i κ 2 ( L z ) 2 k ] ,
n 1 ( r , z ) = exp ( i κ r ) d v ( κ , z ) .
ψ 1 , pl , I ( ρ , z = 0 ) = 0 L d z d v ( κ , z ) H pl , I ( κ , z ) exp ( i κ ρ )
H pl , I ( κ , z ) = i k exp ( i κ 2 z 2 k ) ,
χ 1 , pl , F ( r , z = L ) = 1 2 0 L d z d v ( κ , z ) [ H pl , F ( κ , z ) + H pl , F * ( κ , z ) ] exp ( i κ r ) ,
χ 1 , pl , I ( ρ , z = 0 ) = 1 2 0 L dz d v ( κ , z ) [ H pl , I ( κ , z ) + H pl , I * ( κ , z ) ] exp ( i κ ρ ) .
P ^ = D I ( r ) d 2 r D I 0 d 2 r = S A 1 D I ( r ) / I 0 d 2 r ,
P ^ 1 + 2 S A 1 D χ 1 , pl ( r ) d 2 r = 1 + 2 χ ^ 1 , pl ,
μ pl B pl , F I σ pl , F σ pl , I ,
χ ^ 1 , pl , F = d 2 r χ 1 , pl , F ( r , z = L ) W a ( r ) ,
χ ^ 1 , pl , I = d 2 ρ χ 1 , pl , I ( ρ , z = 0 ) W a ( ρ )
W a ( r ) = { 1 / ( π R a 2 ) | r | < R a 0 | r | R a ,
W ˜ a , pl ( κ ) = d 2 r W a ( r ) exp ( i κ r ) = jinc ( κ R a ) ,
χ ^ 1 , pl , F = 1 2 0 L d z d v ( κ , z ) [ H pl , F ( κ , z ) + H pl , F * ( κ , z ) ] W ˜ a , pl ( κ ) ,
χ ^ 1 , pl , I = 1 2 0 L d z d v ( κ , z ) [ H pl , I ( κ , z ) + H pl , I * ( κ , z ) ] W ˜ a , pl ( κ ) .
B pl , F I = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin ( κ 2 z 2 k ) sin [ κ 2 ( L z ) 2 k ] ,
σ pl , F 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin 2 [ κ 2 ( L z ) 2 k ] ,
σ pl , I 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( R a κ ) sin 2 ( κ 2 z 2 k ) ,
d v ( κ , z ) d v * ( κ , z ) = F n ( | z z | , κ ) δ ( κ κ ) d 2 κ d 2 κ
B pl , F I = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f pl ( κ L / k ) ,
σ pl , F 2 = σ pl , I 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f pl ( κ L / k )
f pl ( κ ^ ) = jinc 2 ( q R 1 / 2 κ ^ ) h pl ( κ ^ ) ,
f pl ( κ ^ ) = jinc 2 ( q R 1 / 2 κ ^ ) h pl ( κ ^ ) ,
h pl ( κ ^ ) = sinc ( κ ^ 2 / 2 ) cos ( κ ^ 2 / 2 ) ,
h pl ( κ ^ ) = 1 sinc ( κ ^ 2 ) ,
b 1 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) jinc 2 ( R a κ ) h 1 , pl ( κ L / k ) ( q p 2 + 1 4 ) 5 / 12 cos [ 5 6 arctan ( 1 2 q p ) ] C 1 q p 5 / 6 C 1 ,
b 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) jinc 2 ( R a κ ) h pl ( κ L / k / 2 1 / 2 ) 12 11 ( 1 4 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( 2 q p ) + π 12 ] C 1 q p 5 / 6 C 1 .
σ pl , F 2 6 11 ( 1 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( q p ) + π 12 ] C 1 q p 5 / 6 C 1 .
μ pl ( q p ) = { 6 11 ( 1 + q p 2 ) 11 12 sin [ 11 6 arctan ( q p ) + π 12 ] q p 5 / 6 } 1 { ( q p 2 + 1 4 ) 5 / 12 × cos [ 5 6 arctan ( 1 2 q p ) ] 12 11 ( 1 4 + q p 2 ) 11 / 12 sin [ 11 6 arctan ( 2 q p ) + π 12 ] } ,
B pl , F I = 0.033 π 2 k 7 / 6 L 11 / 6 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 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 0 1 d ξ C n 2 ( ξ L ) g pl ( ξ )
h ˜ pl ( a , b ) = 3 5 Γ ( 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 ] .
χ ^ 1 , sp , F = 1 2 0 L d z d v ( κ , z ) [ H sp , F ( κ , z ) + H sp , F ( κ , z ) ] W ˜ a , sp ( z L κ ) ,
χ ^ 1 , sp , I = 1 2 0 L d z d v ( κ , z ) [ H sp , I ( κ , z ) + H sp , I ( κ , z ) ] W ˜ a , sp ( L z L κ ) ,
H sp , F ( κ , z ) = H sp , I ( κ , z ) = i k exp [ i κ 2 z ( L z ) 2 L k ] ,
W ˜ a , sp ( γ κ ) = d 2 r W a ( r ) exp ( i γ κ r ) = jinc ( γ κ R a ) .
B sp , F I = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc ( z L R a κ ) × jinc ( L z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] ,
σ sp , F 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] ,
σ sp , I 2 = 4 π 2 k 2 0 L d z 0 d κ κ Φ n ( κ , z ) jinc 2 ( L z L R a κ ) sin 2 [ κ 2 z ( L z ) 2 L k ] .
μ sp B sp , F I σ sp , F σ sp , I .
B sp , F I = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f sp ( κ L / k ) ,
σ sp , F 2 σ sp , I 2 = 2 π 2 k 2 L C n 2 0 d κ κ Φ ^ n ( κ ) f   sp ( κ L / k )
f sp ( κ ^ ) = 1 2 κ ^ 4 0 1 d ξ ξ 2 ( 1 ξ ) 2 jinc [ ( 1 ξ ) q R 1 / 2 κ ^ ] jinc ( ξ q R 1 / 2 κ ^ ) sinc 2 [ κ ^ 2 2 ξ ( 1 ξ ) ] ,
f   sp ( κ ^ ) = 1 2 κ ^ 4 0 1 d ξ ξ 2 ( 1 ξ ) 2 jinc 2 ( ξ q R 1 / 2 κ ^ ) sinc 2 [ κ ^ 2 2 ξ ( 1 ξ ) ] .
B sp , F I = 0.033 π 2 k 7 / 6 L 11 / 6 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 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 0 1 d ξ C n 2 ( ξ L ) g   sp ( ξ ) ,
h ˜ sp ( x , a , b ) = x 2 ( 1 x ) 2 0 d κ ^ κ ^ 4 / 3 exp ( q m κ ^ 2 ) jinc ( a q R 1 / 2 κ ^ ) × jinc ( b q R 1 / 2 κ ^ ) sinc 2 [ x ( 1 x ) κ ^ 2 / 2 ] .
G ( α 1 , α 2 , α 3 , α 4 ) = 0.033 × 4 π 2 k 2 L C n 2 0 1 d ξ 0 d κ κ 8 / 3 exp ( κ 2 / κ m 2 ) × jinc ( α 1 κ R a ) jinc ( α 2 κ R a ) sin [ κ 2 L α 3 / ( 2 k ) ] × sin [ κ 2 L α 4 / ( 2 k ) ] .
G ( α 1 , α 2 , α 3 , α 4 ) 0.033 × 4 π 2 C n 2 k 7 / 6 L 11 / 6 0 1 d ξ 0 d x x 8 / 3 × sin ( x 2 α 3 / 2 ) sin ( x 2 α 4 / 2 ) = O ( σ χ , sp 2 ) .
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 0 1 d ξ α 3 α 4 0 d x x 4 / 3 × jinc ( α 1 x ) jinc ( α 2 x ) = O ( σ χ , sp 2 q R 7 / 6 ) .
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 0 1 d ξ α 3 α 4 0 d x x 4 / 3 × exp ( x 2 ) = O ( σ χ , sp 2 q m 7 / 6 ) .

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