Mean-square angle-of-arrival difference between two counter-propagating spherical waves in the presence of atmospheric turbulence

Chunyi Chen; Huamin Yang; Shoufeng Tong; Yan Lou

doi:10.1364/OE.23.024657

1. Introduction

Over the past years, many researchers have focused attention on counter-propagation of two optical waves along a common path in atmospheric turbulence because of the need for knowledge of the relation between the two counter-propagating waves in relevant engineering applications [1–11 ]. An important practical example of such engineering applications is a bidirectional optical link through the earth’s atmosphere; recently, several efforts have been made to clarify the statistical correlation between the received instantaneous signal power at one link end and that at the other end [6–11 ]. In fact, it has been found for several decades that even though there exits atmospheric turbulence, the field at an observation point, due to a point source located at a given distance from the observation point, remains unchanged if we interchange the positions of the point source and observation point [1,2 ]. This phenomenon has been referred to as the point-source point-receiver (PSPR) reciprocity [8]. Fluctuations in the angle of arrival (AOA) of an optical wave have an important impact on a heterodyne detection system [12]. For a bidirectional optical link using heterodyne detection, an analysis of the relation between the instantaneous AOA fluctuations of two counter-propagating optical waves may be useful in understanding the statistical correlation between the output signals of the detection systems at both link ends.

In accordance with the literature [13–15 ], it is found that the AOA of an optical wave at an observation point is actually determined by the corresponding transverse gradient of the wave front at that observation point. Atmospheric turbulence causes wave-front distortions in propagating optical waves, which further lead to fluctuations in the AOA of the waves at the observation plane. The variance of AOA fluctuations depends heavily on the inner scale of turbulence. Consortini et al. [16] presented a method for measuring the inner scale based on simultaneous measurements of both the AOA and irradiance fluctuations of an optical wave propagating in the atmospheric turbulence. By considering a target-in-the-loop propagation geometry, Churnside and Lataitis [17] investigated the AOA fluctuations of a reflected laser beam from a reflector at a distance from the transmitter in the presence of atmospheric turbulence; many studies of wave-front tilt measurement with a laser guide star (LGS) also considered a similar propagation geometry; i.e., a beam is launched from the ground to create a LGS at a certain altitude in the sky and the wave-front tilt of the returned wave from the LGS is measured at the ground [18–21 ]. Nevertheless, the propagation geometry under consideration in this paper is essentially a point-to-point case, meaning that the two counter-propagating spherical waves are generated by two independent point sources at each end of the path. Recently, the variance of AOA fluctuations for both plane and spherical waves propagating through non-Kolmogorov turbulence has also been theoretically formulated based on the geometrical optics method [22,23 ]. Examination of the published formulations concerning the AOA fluctuations reveals that the variances of AOA fluctuations of two counter-propagating waves at both ends of a path will be identical if the turbulence spectrum has no path dependence; however, we cannot infer directly from this fact that the instantaneous angles of arrival of the two counter-propagating waves are always the same. As is well known, the turbulence-induced AOA fluctuations have a close connection with the phase disturbances in propagating optical fields. In the aforementioned PSPR case, interchanging the point source and observation point does not lead to any changes in the phase of the wave. The above consideration stimulates us to deliberate on other new questions as to whether the mean-square difference between the instantaneous angles of arrival of two counter-propagating spherical waves can reach zero in a specific situation, e.g., the one in which the turbulence spectrum has no path dependence, and how the propagation and turbulence parameters affect it.

The purpose of this paper is to gain an insight into the relation between the instantaneous AOA fluctuations of two counter-propagating spherical waves in atmospheric turbulence, and hence to understand optical wave counter-propagation in atmospheric turbulence from a different point of view. The propagation geometry considered by us is essentially the same as that in the PSPR situation. In the later sections, we begin by detailing the development of general expressions for the mean-square difference and correlation coefficient between the instantaneous angles of arrival of two counter-propagating spherical waves in atmospheric turbulence, then present the numerical calculations and discussions, and finally give the concluding remarks.

2. Theoretical formulations

Figure 1 illustrates a diagram of the counter-propagation geometry under consideration in this work. We will refer to the propagation from a point source positioned at the point (x = 0, y = 0, z = 0) to the observation point (x = 0, y = 0, z = L) as the ‘forward propagation’ and that from a point source positioned at the point (x = 0, y = 0, z = L) to the observation point (x = 0, y = 0, z = 0) as the ‘inverse propagation’. Without loss of generality, here the z-axis has been directed from the forward-propagation point source to the inverse-propagation point source.

Fig. 1 Counter-propagation of two spherical waves along a common path in atmospheric turbulence.

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Let us first consider the forward-propagation case. By employing the Rytov approximation method [13–15,24 ], the phase fluctuations of the wave at the observation point can be given by ([13], Sec. 2.3)

{\hat{S}}_{F} (r, z = L) = k \int_{0}^{L} d z \int d v (κ, z) \cos [\frac{κ^{2} z (L - z)}{2 k L}] \exp (\frac{i z}{L} κ \cdot r),

where

k = 2 π / λ

is the optical wave number with

λ

denoting the wavelength, r = (x, y) is a two-dimensional point in the z = L plane,

d v (κ, z)

represents the random amplitude of refractive-index fluctuations,

κ

is a two-dimensional wave vector, and the subscript ‘F’ on the left-hand side means ‘forward propagation’. It is noted that

d v (κ, z)

is obtained by expressing the refractive-index fluctuations in a spectral representation, and

d v (κ, z)

=

d v^{*} (- κ, z)

. We define the AOA of a wave at an observation point (r, z ) as the angle that the local propagation direction of the wave at that observation point, i.e., the local ray direction in a geometric sense, makes with the z-axis. In accordance with the literature [13–15 ], the AOA of the forward-propagating spherical wave, shown by Fig. 1, at the point (r = 0, z = L) can be expressed in the form

β_{F} = {\nabla_{t} φ_{F} (r) |}_{r = 0},

where

\nabla_{t}

represents the transverse gradient operator, and

φ_{F} (r)

= [

S_{F}^{(0)} (r, z = L)

+

{\hat{S}}_{F} (r, z = L)

]/k denotes the wave front with

S_{F}^{(0)} (r, z = L)

being the phase of the wave at the z = L plane in the absence of atmospheric turbulence. With

\nabla_{t} S_{F}^{(0)} (r, z = L) |_{r = 0}

≡ 0 in mind, combination of Eqs. (1) and (2) leads to the expression for the AOA of the forward-propagating spherical wave at the point (r = 0, z = L) given by

β_{F} = i \int_{0}^{L} d z \int d v (κ, z) κ \frac{z}{L} \cos [\frac{κ^{2} z (L - z)}{2 k L}] .

In a similar manner, for the inverse-propagation case, the expression for the AOA of the wave at the point (r = 0, z = 0) can be developed to give

β_{I} = i \int_{0}^{L} d z \int d v (κ, z) κ \frac{L - z}{L} \cos [\frac{κ^{2} z (L - z)}{2 k L}] .

where the subscript ‘I’ on the left-hand side implies ‘inverse propagation’.

In what follows, we begin to formulate the mean-square AOA difference between two counter-propagating spherical waves in the presence of atmospheric turbulence. As in [9–11 ], for simplicity, the wavelengths associated with the two counter-propagating waves are assumed to be identical. It is noted that both $β_{F}$ and $β_{I}$ are random quantities due to the existence of atmospheric turbulence. Based on Eqs. (3) and (4) , by utilizing the Markov approximation [13–15,24 ] and following the same procedure used to obtain Eq. (2.110) in [13], the mean-square difference between $β_{F}$ and $β_{I}$ can be formulated to give (see Appendix A)

δ_{F I}^{2} = 2 π \int_{0}^{L} {(2 \frac{z}{L} - 1)}^{2} d z \int \int_{- \infty}^{\infty} d^{2} κ Φ_{n} (κ, z) κ^{2} \cos^{2} [\frac{κ^{2} z (L - z)}{2 k L}],

where

Φ_{n} (\cdot)

is the turbulence spectrum and κ = |κ|. In arriving at Eq. (5), we have indeed assumed that the atmospheric turbulence is isotropic, which can further lead us to

δ_{F I}^{2} = 4 π^{2} L \int_{0}^{1} d ξ {(1 - 2 ξ)}^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, ξ L) \cos^{2} [\frac{κ^{2} L ξ (1 - ξ)}{2 k}] .

In fact, the AOA can be conceptually separated into two orthogonal components, i.e., the x- and y-components. Notice that,

δ_{F I}^{2}

appearing in Eq. (5) denotes the total mean-square AOA difference between two counter-propagating spherical waves, which is equal to the sum of the mean-square AOA differences in the x and y directions. The magnitude of

Φ_{n} (\cdot)

for atmospheric turbulence with a finite inner scale l ₀ falls off rapidly to zero when κ increases beyond a value κ_m (~1/l ₀). In the limit of the geometrical optics approximation [24], i.e., κ ² L/k ≈0 for all κ smaller than κ_m, the squared cosine term in Eq. (6) can be replaced by 1. Below, we rewrite Eq. (6) as follows:

δ_{F I}^{2} = δ_{F I, r}^{2} - δ_{F I, d}^{2}

with

δ_{F I, r}^{2} = 4 π^{2} L \int_{0}^{1} d ξ {(1 - 2 ξ)}^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, ξ L),

δ_{F I, d}^{2} = 4 π^{2} L \int_{0}^{1} d ξ {(1 - 2 ξ)}^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, ξ L) \sin^{2} (\frac{κ^{2} L ξ (1 - ξ)}{2 k}) .

It is easy to find that

δ_{F I, r}^{2}

and

δ_{F I, d}^{2}

arise from the refraction and diffraction effects of turbulent cells, respectively. Based on Eqs. (7)–(9) , we find that the refraction and diffraction effects of turbulent cells make positive and negative contributions to

δ_{F I}^{2}

, respectively. In the two limits (L/k)^1/2 → 0 and (L/k)^1/2 → ∞, one finds

δ_{F I, d}^{2} = {\begin{matrix} 0 & \sqrt{L / k} \to 0 \\ δ_{F I, r}^{2} / 2 & \sqrt{L / k} \to \infty \end{matrix} .

Combination of Eqs. (7) and (10) leads to

δ_{F I}^{2} = {\begin{matrix} δ_{F I, r}^{2} & \sqrt{L / k} \to 0 \\ \frac{1}{2} δ_{F I, r}^{2} & \sqrt{L / k} \to \infty \end{matrix} .

Equation (11) shows that the two asymptotic values of

δ_{F I}^{2}

differ by a factor of 1/2 for any turbulence spectrum. As is well known, the diffraction phenomenon of a wave associated with a larger wavelength λ is easier to observe. In fact, for a given finite L, (L/k)^1/2 → 0 implies λ → 0, which corresponds to the case that the diffraction effects can be completely ignored; on the other hand, for a given finite L, (L/k)^1/2 → ∞ means λ → ∞, which corresponds to the situation that the diffraction effects become most prominent. As a result, we can infer that

δ_{F I, d}^{2} / δ_{F I, r}^{2} \leq 0.5

for any turbulence spectrum.

A turbulence spectrum model taking the form [22]

Φ_{n} (κ, z) = A (α) {\tilde{C}}_{n}^{2} (z) {(κ^{2} + κ_{0}^{2})}^{- α / 2} \exp (- κ^{2} / κ_{m}^{2})

has been widely employed in the literature concerning optical wave propagation through atmospheric turbulence, where

{\tilde{C}}_{n}^{2} (z)

is the refractive-index structure constant in units of m³⁻ ^α,

κ_{0} = 2 π / L_{0}

with L ₀ being the outer scale of turbulence,

κ_{m} = c (α) / l_{0}

, c(α) = {2πΓ[(5−α)/2] A(α)/3}^1/( ^α ⁻⁵⁾,

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

, Γ(∙) denotes the gamma function, and

α

is referred to as the spectral index with 3 <

α

< 4. As was well known, the inner scale of turbulence is critical for AOA fluctuations. However, the outer scale of turbulence is not critical for AOA fluctuations except when (L/k)^1/2 ~L ₀, which generally corresponds to an extremely large propagation distance and holds no interest for us. To simplify the mathematical presentation, in the following development we will drop the outer scale from the turbulence spectrum model, i.e., let L ₀ → ∞. This does not affect the final conclusions. By introducing Eq. (12) into Eqs. (8) and (9) and meanwhile letting κ ₀ = 0, one finds

δ_{F I, r}^{2} = 2 π^{2} A (α) Γ (2 - \frac{α}{2}) κ_{m}^{4 - α} L \int_{0}^{1} d ξ {\tilde{C}}_{n}^{2} (ξ L) W_{r} (ξ),

δ_{F I, d}^{2} = 2 π^{2} A (α) Γ (2 - \frac{α}{2}) κ_{m}^{4 - α} L \int_{0}^{1} d ξ {\tilde{C}}_{n}^{2} (ξ L) W_{d} (ξ)

with

W_{r} (ξ) = {(1 - 2 ξ)}^{2},

W_{d} (ξ) = \frac{1}{2} W_{r} (ξ) [1 - Q (θ)],

Q (θ) = \cos [(2 - \frac{α}{2}) arc \tan (θ)] {(1 + θ^{2})}^{α / 4 - 1},

where θ = qξ(1−ξ) and q =

κ_{m}^{2}

L/k. In what follows, q will be called the Fresnel number; W_r(ξ) and W_d(ξ) will be referred to as the refraction and diffraction path weighting functions, respectively. Physically speaking, W_r(ξ) and W_d(ξ) characterize the weight of the refraction- and diffraction-induced contributions, respectively, to the mean-square AOA difference from turbulent cells located at various positions on the propagation path. As could have been expected, W_r(ξ) is independent of the wavelength, and W_d(ξ) is a function of the wavelength. At the middle point of the path, it is found that W_r(ξ = 0.5) = W_d(ξ = 0.5) = 0, meaning that the turbulent cells located at the middle point do not contribute to the mean-square AOA difference. Moreover, examination of Eqs. (15) and (16) reveals that the curve corresponding to either W_r(ξ) or W_d(ξ) in terms of ξ has a symmetric shape with respect to ξ = 0.5. Note that Q(θ) = 0 and 1 when θ → ∞ and 0, respectively. It is easy to verify that 0 ≤ Q(θ) ≤ 1. Thus, it can be deduced that W_d(ξ) / W_r(ξ) ≤ 0.5 except for ξ = 0.5, implying that

δ_{F I, d}^{2} / δ_{F I, r}^{2} \leq 0.5

; this verifies once again our previous inference.

The unit of ${\tilde{C}}_{n}^{2} (z)$ depends on the spectral index α. This fact may cause some difficulties when one analyzes various optical-wave statistics based on numerical calculations [25,26 ]. To avoid the unit choice inconsistency [25] in the representation of calculation results concerning $δ_{F I}^{2}$ in the form of dependence on α, below we normalize the mean-square AOA difference between two counter-propagating spherical waves by the variances of AOA fluctuations. Following the procedure used to obtain Eq. (6), the variance of AOA fluctuations for the forward-propagating spherical wave at the observation point (r = 0, z = L) can be expressed as

〈 β_{F}^{2} 〉 = 4 π^{2} L \int_{0}^{1} d ξ ξ^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, ξ L) \cos^{2} [\frac{κ^{2} L ξ (1 - ξ)}{2 k}] .

Similarly, the variance of AOA fluctuations for the inverse-propagating spherical wave at the observation point (r = 0, z = 0) can be expressed by

〈 β_{I}^{2} 〉 = 4 π^{2} L \int_{0}^{1} d ξ {(1 - ξ)}^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, ξ L) \cos^{2} [\frac{κ^{2} L ξ (1 - ξ)}{2 k}] .

After evaluating the integrations over κ in Eqs. (18) and (19) , we can obtain the normalized mean-square AOA difference given by

\begin{matrix} {\hat{δ}}_{F I}^{2} = \frac{δ_{F I}^{2}}{\sqrt{〈 β_{F}^{2} 〉 〈 β_{I}^{2} 〉}} \\ = \frac{\int_{0}^{1} d ξ {\tilde{C}}_{n}^{2} (ξ L) {(1 - 2 ξ)}^{2} [1 + Q (θ)]}{\sqrt{\int_{0}^{1} d ξ^{'} {\tilde{C}}_{n}^{2} (ξ^{'} L) {ξ^{'}}^{2} [1 + Q (θ^{'})] \int_{0}^{1} d ξ {\tilde{C}}_{n}^{2} (ξ L) {(1 - ξ)}^{2} [1 + Q (θ)]}}, \end{matrix}

where

θ^{'}

= qξ′(1−ξ′). If Φ_n(∙) has no path dependence, i.e.,

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

(∙) does not vary over the propagation path and

〈 β_{F}^{2} 〉

≡

〈 β_{I}^{2} 〉

, Eq. (20) reduces to

{\hat{δ}}_{F I}^{2} = \frac{\int_{0}^{1} d ξ {(1 - 2 ξ)}^{2} [1 + Q (θ)]}{\int_{0}^{1} d ξ {(1 - ξ)}^{2} [1 + Q (θ)]} .

According to the normalized mean-square AOA difference between two counter-propagating spherical waves, the correlation coefficient between the angles of arrival in the x or y direction can be readily expressed as follows (see Appendix B):

γ_{F I, v} = \frac{〈 β_{F, v} β_{I, v} 〉}{\sqrt{〈 β_{F, v}^{2} 〉 〈 β_{I, v}^{2} 〉}} = \frac{1}{2} (\sqrt{\frac{〈 β_{F}^{2} 〉}{〈 β_{I}^{2} 〉}} + \sqrt{\frac{〈 β_{I}^{2} 〉}{〈 β_{F}^{2} 〉}}) - \frac{1}{2} {\hat{δ}}_{F I}^{2},

where β_F _, _v and β_I _, _v are the instantaneous angles of arrival in the v direction associated with the forward and inverse propagation, respectively (v = x or y). Here, we emphasize that Eq. (22) has been obtained by assuming that atmospheric turbulence is isotropic. Moreover, if Φ_n(∙) has no path dependence, one finds

γ_{F I, v} = 1 - {\hat{δ}}_{F I}^{2} / 2.

It should be pointed out that the definitions shown by Eqs. (20) and (22) are not meaningful for α = 3 because Φ_n(∙) ≡ 0 with α = 3, resulting in

〈 β_{F}^{2} 〉

=

〈 β_{I}^{2} 〉

≡ 0.

3. Numerical calculations and discussions

In this section, we discuss the results obtained in the previous section with the help of numerical calculations. Figure 2 illustrates the path weighting functions shown by Eqs. (15) and (16) in terms of ξ with various q. As has been pointed out, we can see from Fig. 2 that at the middle point of the propagation path, viz., ξ = 0.5, both the refraction and diffraction weighting functions are equal to zero. It is apparent that W_r(ξ) achieves its maximum value at ξ = 0 and 1; however, W_d(ξ) becomes zero at ξ = 0 and 1, causing its curve to take on a double-peak appearance. Furthermore, although W_r(ξ) does not depend on the Fresnel number q, it can be found from Fig. 2 that W_d(ξ) can have an appreciable value only when q ≳ 1. The physical interpretation about this phenomenon is as follows: q ≳ 1 means (L/k)^1/2 ≳ l ₀; it has been known that turbulent cells with scale sizes larger than the first Fresnel zone (L/k)^1/2 do not diffract an optical wave through a large enough angle to arrive at the observation point [24]; that is, only the diffraction effects produced by those turbulent cells with scale sizes smaller than the first Fresnel zone (L/k)^1/2 are important; meanwhile, theoretically speaking, the inner scale l ₀ describes the smallest scale size of turbulent cells; consequently, if (L/k)^1/2 < l ₀ (i.e., all the scale sizes of turbulent cells are larger than the first Fresnel zone (L/k)^1/2), none of the turbulent cells can produce important diffraction effects, resulting in that W_d(ξ) has an inappreciable value. In addition, when the Fresnel number q increases steadily beyond 1, there are increasing turbulent cells that can produce important diffraction effects, causing a rise in the value of W_d(ξ), which can be clearly seen in Figs. 2(b)–2(d). It is easy to find by numerical calculations that with a given θ > 0, a larger α leads to a greater value of Q(θ). This is the reason why Figs. 2(b)–2(d) show that W_d(ξ) diminishes as α increases with the same q and ξ, except for ξ = 0, 0.5 and 1. Indeed, when α → 4, the structure function of the turbulence tends to take a quadratic form, resulting in that the wave-front distortions mainly manifest themselves as tilts and the turbulence-induced higher-order wave-front distortions become really tiny [25,27–29 ]; the tilts can be regarded as the turbulence-induced deflections of propagating waves, which should be related to the refraction effects; accordingly, the diffraction effects become less significant when α grows larger with fixed q.

Fig. 2 The path weighting functions in terms of ξ with various q. (a) the refraction path weighting function W_r(∙); (b) the diffraction path weighting function W_d(∙) with α = 3.1; (c) the diffraction path weighting function W_d(∙) with α = 11/3; (d) the diffraction path weighting function W_d(∙) with α = 3.9.

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When ${\tilde{C}}_{n}^{2} (\cdot)$ has path dependence, Eq. (20) can be used to determine the normalized mean-square AOA difference between two counter-propagating spherical waves in atmospheric turbulence. In the literature concerning optical wave propagation, almost all the cases, in which one needs to consider the variation of ${\tilde{C}}_{n}^{2} (\cdot)$ over the propagation path, involve a vertical or slant path in the earth’s atmosphere. Strictly speaking, for the earth’s atmospheric turbulence, both the inner scale l ₀ and spectral index α are dependent on the altitude ([15,30 ], Eq. (5)), implying that these two parameters indeed change over a vertical or slant path. However, we note that the turbulence spectrum shown by Eq. (12) assumes that both the inner scale and spectral index have no path dependence. Hence, although Eq. (20) allows us formally to consider the path dependence of ${\tilde{C}}_{n}^{2} (\cdot)$ , it is not a very suitable mathematical model which can be used to carry out a rigorous analysis of the normalized mean-square AOA difference between two counter-propagating spherical waves if a vertical or slant path in the earth’s atmosphere needs to be considered. On the other hand, the variance of AOA fluctuations of a wave propagating from the earth’s ground to a position in the space is apparently different from that of a wave propagating from the position in the space to the earth’s ground; thus it is straightforward to deduce that the mean-square AOA difference between two counter-propagating waves in the vertical- or slant-propagation case cannot be zero. However, for a horizontal path, it is common to assume that the turbulence spectrum has no path dependence, and in this case, the variances of AOA fluctuations of two counter-propagating waves at both ends of the path will be the same. Below we focus our attention on the horizontal propagation case, in which the parameters ${\tilde{C}}_{n}^{2} (\cdot)$ , l ₀ and α are assumed to remain constant over the path. Figure 3 shows the normalized mean-square AOA difference between two counter-propagating spherical waves in atmospheric turbulence as a function of the Fresnel number q and spectral index α. It is seen from Fig. 3 that all the curves corresponding to ${\hat{δ}}_{F I}^{2}$ in terms of q take on a single-peak appearance; a smaller α leads to a higher peak. It is found that ${\hat{δ}}_{F I}^{2}$ quickly approaches 1 when q decreases below 1 for all α. It is also observed that ${\hat{δ}}_{F I}^{2}$ gradually becomes close to 1 when q steadily increases to a large enough value. Note that, if the first Fresnel zone (L/k)^1/2 is fixed, an increase in q means a decrease in l ₀. Furthermore, we find by numerical calculations that the maximum value of ${\hat{δ}}_{F I}^{2}$ is smaller than 1.08, implying that 1 ≤ ${\hat{δ}}_{F I}^{2}$ < 1.08. Accordingly, a statement can be made that the mean-square AOA difference between two counter-propagating spherical waves is indeed approximately equal to the variance of AOA fluctuations. Figure 4 shows the correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves as a function of q and α. As could been expected, all the curves in Fig. 4 take on a single-valley appearance. Based on the analysis of Fig. 3, it can be readily deduced that 0.46 < $γ_{F I, v}$ ≤ 0.5. These findings reveal that the angles of arrival of two counter-propagating spherical waves in atmospheric turbulence cannot be perfectly correlated under any conditions of practical interest and the correlation coefficient only shows a slight dependence on q and α.

Fig. 3 Normalized mean-square AOA difference between two counter-propagating spherical waves as a function of q and α. 3 < α < 4. The turbulence spectrum is assumed to have no path dependence.

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Fig. 4 Correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves as a function of q and α. The subscript v can be specified as x or y. 3 < α < 4. The turbulence spectrum is assumed to have no path dependence.

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At this point, we present a brief discussion about the validity domain of our theoretical formulations. Because the preceding formulations have been developed based on the Rytov approximation, it may be expected that they are applicable only in the weak fluctuation regime. On the other hand, because the theoretical technique we used to formulate the mean-square AOA difference is essentially the same as that used to develop the expressions for the variance of AOA fluctuations based on the Rytov approximation, we believe that our theoretical formulations should have the same validity domain as that of the expressions for the variance of AOA fluctuations obtained by making use of the Rytov approximation. The experimental study concerning the AOA fluctuations carried out by Gurvich and Kallistratova [31] has proved that the formulation of AOA fluctuations developed by employing the Rytov approximation remains valid within a range wider than the weak fluctuation regime. For this reason, it is believed that our formulations may also be applicable beyond the weak fluctuation regime. This result does not surprise us. Indeed, it can be found from [24,25,32 ] that for a spherical wave propagating in atmospheric turbulence, the expression for the mutual coherence function obtained by making use of the Rytov approximation is identical to that derived based on the Markov approximation, which is a method applicable to the strong fluctuation regime; in other words, the expression for the mutual coherence function of spherical waves obtained by employing the Rytov approximation is also applicable in the strong fluctuation regime. Furthermore, it is interesting to note that Eq. (21) is even independent of the refractive-index structure constant.

4. Conclusions

In this paper, general formulations for the mean-square AOA difference between two counter-propagating spherical waves in atmospheric turbulence have been developed by making use of the Rytov approximation method. Based on these formulations, expressions for the correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves have been derived. Both the refraction and diffraction effects of turbulent cells can play a role in determining the mean-square AOA difference; the former and the latter make positive and negative contributions to the mean-square AOA difference, respectively. The turbulent cells located at the midpoint of the propagation path do not contribute to the mean-square AOA difference. If the mean-square AOA difference is separated into the refraction and diffraction parts, the ratio of the diffraction part to the refraction one is never larger than 0.5 for any turbulence spectrum. By consider the cases that the turbulence spectrum has no path dependence, numerical calculations have been carried out. The findings are the following. The normalized mean-square AOA difference only changes slightly within a very narrow range when the Fresnel number or the spectral index varies, and its value can be regarded as being always approximately equal to 1, implying that the mean-square AOA difference always approximates to the variance of AOA fluctuations. The Fresnel number and spectral index only have a slight impact on the value of the correlation coefficient between the angles of arrival in the x or y direction of two counter-propagating spherical waves, which ranges from 0.46 to 0.5 for all values of the Fresnel number and spectral index. The obtained analytical expressions may still remain valid in relatively strong fluctuation regimes. As a final comment, we note that the theoretical models developed here apply to both Kolmogorov and non-Kolmogorov atmospheric turbulence.

Appendix A

According to Eqs. (3) and (4) , it follows that

\begin{matrix} δ_{F I}^{2} = 〈 {(β_{F} - β_{I})}^{2} 〉 \\ = \int_{0}^{L} d z \int_{0}^{L} d z^{'} \iint 〈 d v (κ, z) d v^{*} (κ^{'}, z^{'}) 〉 κ \cdot κ^{'} \\ \times {(2 \frac{z}{L} - 1) \cos [\frac{κ^{2} z (L - z)}{2 k L}]} {(2 \frac{z^{'}}{L} - 1) \cos [\frac{{κ^{'}}^{2} z^{'} (L - z^{'})}{2 k L}]} \end{matrix}

with

〈 d v (κ, z) d v^{*} (κ^{'}, z^{'}) 〉 = {\tilde{C}}_{n}^{2} (z / 2 + z^{'} / 2) E_{n} (κ, | z - z^{'} |) δ (κ - κ^{'}) d κ d κ^{'},

where the angle brackets stand for an ensemble average, κ′ = |κ′|,

{\tilde{C}}_{n}^{2} (\cdot)

represents the refractive-index structure constant, E_n(∙) is the two-dimensional spatial power spectrum of refractive-index fluctuations scaled by

{\tilde{C}}_{n}^{2} (\cdot)

, and

δ (\cdot)

denotes the Dirac delta function. By introducing Eq. (25) into Eq. (24), making the change of variables

z_{s} = (z + z^{'}) / 2

and

z_{d} = z - z^{'}

, employing the relation [13,24,33 ]

{\tilde{C}}_{n}^{2} (z_{s}) \int_{0}^{\infty} E_{n} (κ, z_{d}) d z_{d} = π Φ_{n} (κ, z_{s}),

and recalling the arguments on page 51 of [13], one can obtain the result shown by Eq. (5).

Appendix B

Under the condition that atmospheric turbulence is isotropic, it is easy to find that

\begin{matrix} δ_{F I}^{2} = δ_{F I, x}^{2} + δ_{F I, y}^{2} \\ = 〈 β_{F, x}^{2} 〉 + 〈 β_{I, x}^{2} 〉 - 2 〈 β_{F, x} β_{I, x} 〉 + 〈 β_{F, y}^{2} 〉 + 〈 β_{I, y}^{2} 〉 - 2 〈 β_{F, y} β_{I, y} 〉 \\ = 〈 β_{F}^{2} 〉 + 〈 β_{I}^{2} 〉 - 2 〈 β_{F, x} β_{I, x} 〉 - 2 〈 β_{F, y} β_{I, y} 〉 \\ = 〈 β_{F}^{2} 〉 + 〈 β_{I}^{2} 〉 - 4 〈 β_{F, v} β_{I, v} 〉, \end{matrix}

where β_F _, _v and β_I _, _v denote the instantaneous angles of arrival in the v direction for the forward and inverse propagation, respectively (v = x or y). Note that,

〈 β_{F, x}^{2} 〉 = 〈 β_{F, y}^{2} 〉

,

〈 β_{I, x}^{2} 〉 = 〈 β_{I, y}^{2} 〉

, and

〈 β_{F, x} β_{I, x} 〉 = 〈 β_{F, y} β_{I, y} 〉

due to the assumption of isotropic atmospheric turbulence. Moreover, it is apparent that the correlation coefficient can be written by

γ_{F I, v} = \frac{〈 β_{F, v} β_{I, v} 〉}{\sqrt{〈 β_{F, v}^{2} 〉 〈 β_{I, v}^{2} 〉}} = \frac{2 〈 β_{F, v} β_{I, v} 〉}{\sqrt{〈 β_{F}^{2} 〉 〈 β_{I}^{2} 〉}} .

Based on Eqs. (20), (27) and (28) , one can yield the result shown by Eq. (22).

Acknowledgments

The authors are very grateful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China (61007046, 61275080, 61475025), the Natural Science Foundation of Jilin Province of China (20150101016JC), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20132216110002).

References and links

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

2. 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]

3. D. L. Fried and H. T. Yura, “Telescope-performance reciprocity for propagation in a turbulent medium,” J. Opt. Soc. Am. 62(4), 600–602 (1972). [CrossRef]

4. 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]

5. R. Mahon, C. I. Moore, M. Ferraro, W. S. Rabinovich, and M. R. Suite, “Atmospheric turbulence effects measured along horizontal-path optical retro-reflector links,” Appl. Opt. 51(25), 6147–6158 (2012). [CrossRef] [PubMed]

6. 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]

7. 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]

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

9. 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]

10. 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.

11. 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]

12. K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. 25(11), 1817–1825 (1986). [CrossRef] [PubMed]

13. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007).

14. A. D. Wheelon, Electromagnetic Scintillation II: Weak Scattering (Cambridge University, 2003).

15. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Jerusalem, 1971).

16. A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, “Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation,” Opt. Commun. 216(1–3), 19–23 (2003). [CrossRef]

17. J. H. Churnside and R. J. Lataitis, “Angle-of-arrival fluctuations of a reflected beam in atmospheric turbulence,” J. Opt. Soc. Am. A 4(7), 1264–1272 (1987). [CrossRef]

18. M. S. Belen’kii, S. J. Karis, J. M. Brown 2nd, and R. Q. Fugate, “Experimental validation of a technique to measure tilt from a laser guide star,” Opt. Lett. 24(10), 637–639 (1999). [CrossRef] [PubMed]

19. V. P. Lukin, “Comparison the efficiencies of different schemes for laser guide stars forming,” Proc. SPIE 3126, 460–466 (1997). [CrossRef]

20. V. P. Lukin, “Some problems in the use of laser guide stars,” Proc. SPIE 3983, 90–100 (1999). [CrossRef]

21. R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, and R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11(1), 310–324 (1994). [CrossRef]

22. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]

23. L. Cui, B. Xue, X. Cao, and F. Zhou, “Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 31(4), 829–835 (2014). [CrossRef] [PubMed]

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

25. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29(5), 711–721 (2012). [CrossRef] [PubMed]

26. Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36(23), 4554–4556 (2011). [CrossRef] [PubMed]

27. Y. Baykal and H. T. Eyyuboğlu, “Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free-space optics links,” Opt. Eng. 45(5), 056001 (2006). [CrossRef]

28. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70(6), 745–747 (1980). [CrossRef]

29. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]

30. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]

31. A. S. Gurvich and M. A. Kallistratova, “Experimental study of the fluctuations in angle of incidence of a light beam under conditions of strong intensity fluctuations,” Radiophys. Quantum Electron. 11(1), 37–40 (1968). [CrossRef]

32. M. Charnotskii, “Extended Huygens-Fresnel principle and optical waves propagation in turbulence: discussion,” J. Opt. Soc. Am. A 32(7), 1357–1365 (2015). [CrossRef]

33. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11(6), 1399–1406 (1972). [CrossRef] [PubMed]

Mean-square angle-of-arrival difference between two counter-propagating spherical waves in the presence of atmospheric turbulence

Abstract

1. Introduction

2. Theoretical formulations

3. Numerical calculations and discussions

4. Conclusions

Appendix A

Appendix B

Acknowledgments

References and links

Cited By

Figures (4)

Equations (28)

Optics Express