Optical downlink propagation from space-to-earth: aperture-averaged power fluctuations, temporal covariance and power spectrum

H. T. Yura

doi:10.1364/OE.26.026787

1. Introduction

Over the next decade, it is anticipated that there will be a proliferation of laser communication links between GEO/LEO space-based platforms to ground/ airborne terminals in order to expand data channel capacities and reduce the probabilities of interception or detection compared to traditional RF links. As is well known atmospheric turbulence induces received power fluctuations that degrade optical links. Here, we quantitatively consider the effect of the finite sized downlink circular receiver apertures regarding the reduction of the collected power fluctuations, and both the corresponding scintillation power spectra density (PSD) and the increase in the temporal correlation time of the signal. It is tacitly assumed that all optical sources considered here are in space, well above the atmosphere.

In contrast to slant path propagation, aperture averaging for constant turbulence conditions (e.g., horizontal propagation) has been well documented in the literature for both plane and spherical waves [1–5]. Although formal expressions for both the aperture averaged PSD and corresponding temporal covariance function are given in Refs. [1] and [5], numerical results only are presented for constant turbulence conditions. Additionally, for constant turbulence conditions aperture-averaged scintillation, temporal covariance, and PSD is available in the literature [6]. In contrast to the work presented here, the analysis given in [6] assumes a Gaussian shaped aperture rather than a hard circular aperture treated here, nor were there any results regarding scaling laws of the temporal correlation time or the corresponding characteristic fall-off frequency of the PSD (i.e., scintillation bandwidth). On the other hand, the corresponding aperture averaging effects for slant path propagation is sparsely documented in the literature. Numerical results for the temporal characteristics and the aperture-averaging factor of slant path propagation have recently been documented [7]. An approximate engineering expression for the aperture-averaging factor that introduced an effective propagation range for slant paths has been derived [8], while experimental observations of the temporal effects of aperture size on the statistical properties of collected power is also available in the literature [9], [10]. In particular, there does not appear to be a published comprehensive quantitative analysis of aperture averaging effects for slant path propagation through the atmosphere on the temporal properties of the collected power. It is the aim of this paper to present just such an analysis. In particular, we present general expressions for the downlink aperture-averaging factor, the corresponding temporal covariance of collected power and PSD, and the aperture averaged quasi-frequency, as well as accurate “user friendly” engineering approximations for these quantities. Furthermore, simplified analytic approximations are presented for slant propagation both for the aperture averaged characteristic roll-off frequency of the PSD (i.e., a measure of the scintillation bandwidth) and the temporal correlation time of collected power. Engineers designing laser communication systems can readily use these expressions to develop bandwidth and level crossing considerations, and simulations, which would be used to derive system requirements. It should be stressed that the presentation presented here is necessarily incomplete in that the results are primarily theoretical, the validity of which requires more experimental evidence.

Because aperture-averaging effects are not an issue for uplink scenarios we focus our attention on downlink propagation, where for all cases of practical concern the optical wave front impinging on the turbulent atmosphere can be considered as a plane wave. The methodology for obtaining engineering expressions for aperture averaged downlink propagation is as follows. We first consider aperture-averaged plane wave propagation for constant turbulence conditions and propagation distance $L$ . The corresponding result for slant path propagation is then obtained by substituting the effective propagation range, $L_{e f f}$ , derived in [8] for $L$ . Here, we tacitly assume that both the Fresnel spatial scale (i.e., $\sqrt{L_{e f f} / k}$ , where $k$ is the optical wave number) and aperture radius are much less than the outer scale of turbulence. Thus, all spatial scales of interest here are in the so-called inertial sub-range, where the Kolmogorov spectrum applies. The validity of these engineering approximations are obtained by comparing them to the corresponding exact numerical integrated along the slant path results, where in all cases very good agreement is obtained.

Thus, this paper should be of interest to both the Free Space Optical laser communication and remote sensing communities and is organized as follows. Based on the Kolmogorov spectrum and assuming weak-to-moderate scintillations conditions where lognormal statistics apply, we present in Sec. 2 a general expression based on the Rytov approximation for the reduction of the variance of collected power due to spatial (aperture) averaging from which we obtain both exact and approximate results for the aperture- averaging factor. In Sec. 3 and 4, we derive both exact and highly accurate engineering approximations for the aperture-averaged signal temporal correlation function and corresponding PSD. Further, the present analysis is in good agreement with available experimental observations. In Sec. 5 we apply the present analysis to the calculation of the so-called quasi-frequency that is central to the determination of the level crossing rates and mean duration of fades and surges in a propagation channel. In particular, we derive a closed form expression for the aperture-averaged quasi-frequency for constant turbulence conditions and for the downlink an integral expression over the propagation path for a given turbulence height profile. Finally, in Sec. 6 we present some concluding remarks.

2. Aperture averaging of scintillation

A central parameter in the characterization of the statistical distribution of received power is the scintillation index, SI, defined as the variance of received power, normalized to the square of its mean value. We have

S I = \frac{〈 P^{2} 〉 - {〈 P 〉}^{2}}{{〈 P 〉}^{2}},

where angular brackets denote the ensemble average. Assuming lognormal scintillation statistics, the scintillation index is given by

\begin{array}{l} S I = \exp [σ_{\ln I}^{2}] - 1 \\ ≅ σ_{\ln I}^{2}, ​ for σ_{\ln I}^{2} < < 1, \end{array}

where

σ_{\ln I}^{2}

is the log-intensity variance [1], [5]. For downlinks and weak-to-moderate scintillation conditions it can be shown that the aperture-averaged scintillation index in the Rytov perturbation approximation can be expressed as [5]

S I (a) ≅ σ_{\ln I}^{2} (a) = 16 π^{2} k^{2} \int_{0}^{L} d s \int_{0}^{\infty} d K K \sin^{2} [\frac{K^{2} s}{2 k}] Φ_{n} (K, s) {(\frac{2 J_{1} (K a)}{K a})}^{2},

where

a

is the (hard) circular aperture radius,

k

is the optical wave number ( =

2 π / λ

, where

λ

is the wavelength),

K

is the spatial wave number,

s

denotes path length,

L

is the propagation range,

J_{1} (\cdot)

is the Bessel function of the first kind of order unity,

C_{n}^{2} (h)

is the index structure constant profile as function of height

h

above ground, and

Φ_{n}

is the spatial refractive index spectrum. For a spherical earth, the height above ground as a function of path length is given by

h (s) = \sqrt{{(R_{E} + H)}^{2} + 2 s (R_{E} + H) \sin θ + s^{2}} - R_{E},

where the mean radius of the earth

R_{E}

is about 6371 km,

θ

is the elevation angle, and

H

is the height of the receiving aperture above ground. For the scintillation effects considered here, it is tacitly assumed that both the spatial Fresnel length and the aperture size lies well within the inertial sub-range: both inner and outer scale effects are negligible and can be ignored. Thus, in the following we use the Kolmogorov spectrum (see [5]), where

Φ_{n} (K, s) = \frac{0.033 C_{n}^{2} [h (s)])}{K^{11 / 3}},

where

C_{n}^{2}

is the index structure constant as a function of height above ground. Substituting Eq. (2.3) and Eq. (2.4) into Eq. (2.2) and performing the integration over spatial frequency yields

\begin{array}{l} S I (a) = 2.25 k^{7 / 6} \int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a) \\ = A A (a) S I (0), \end{array}

where the dimensionless quantity

G (s, a)

is given by

G (s, a) = \frac{0.4442 k^{5 / 6}}{s^{5 / 6}} (\begin{array}{l} 2.251 (^{s / k) 5 / 6}_{4} F_{5} (- \frac{5}{12}, \frac{1}{12}, \frac{3}{4}, \frac{5}{4}; \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; - \frac{a^{4} k^{2}}{4 s^{2}}) \\ + 1.750 (^{k / s) 1 / 6} a^{2}_{4} F_{5} (- \frac{1}{12}, \frac{7}{12}, \frac{5}{4}, \frac{7}{4}; 2, 2, \frac{5}{2}, 2; - \frac{a^{4} k^{2}}{4 s^{2}}) - 2.774 a^{5 / 3} \end{array})

_{$_{p} F_{q}$} are generalized hypergeometric functions [11],

S I (0) = 2.25 k^{7 / 6} \int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6}

is the scintillation index for a “point receiver” and

A A (a) = \frac{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a)}{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6}},

is the plane wave aperture-averaging factor (

\leq 1

) that reduces the scintillation index of a receiving aperture of radius

a

from that of a “point receiver”. Although Mathematica can readily evaluate generalized hypergeometric functions to an arbitrary numerical precision, other analysis computer programs cannot (e.g., MATLAB, Excel). For this reason, and to the extent possible, we endeavor to derive “user friendly” accurate engineering approximations for the various statistical quantities of interest here and in what follows. Accordingly, an accurate global approximation for

G (s, a)

based on Eq. (2.6) is given by

G (s, a) ≅ \frac{1}{1 + 0.754 {(a^{2} k / s)}^{7 / 6}}

Below various exact analytic expressions are presented containing multiple argument hypergeometric functions that are used here globally to compare numerically with the corresponding engineering approximations. In general, these exact expressions are not recommended to be used by potential users in practice and are additionally documented here for reference purposes. Rather, the primary intent and purpose of this paper is to recommend and provide the reader with accurate user friendly engineering approximations for the various statistical quantities presented below.

2.1 Constant turbulence conditions

Consider the aperture-averaging factor for constant turbulence conditions (i.e. $C_{n}^{2}$ = Constant). Integrating Eq. (2.8) over path length and simplifying yields

A A (η) = \frac{4.24}{η^{2}} (\begin{array}{l} 1 / (192 \sqrt{2}) η^{2} Γ (- 11 / 6) (24 \sqrt{3} - 1)_{4} F_{5} (- \frac{11}{12}, - \frac{5}{12}, \frac{3}{4}, \frac{5}{4}; \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; - \frac{η^{4}}{4}) \\ + 11 (1 + \sqrt{3}) η^{2}_{4} F_{5} (- \frac{5}{12}, \frac{1}{12}, \frac{5}{4}, \frac{7}{4}; \frac{3}{2}, \frac{3}{2}, 2, 2, \frac{5}{2}; - \frac{η^{4}}{4}) + \frac{η^{11 / 3} Γ (- 5 / 6) Γ (7 / 3)}{\sqrt{π} Γ (17 / 6) Γ (23 / 6)} \end{array})

where

Γ (\cdot)

is the gamma function and

η = a \sqrt{\frac{k}{L}}

Hence, for constant turbulence conditions the exact aperture-averaging factor be expressed in analytic form, and is plotted as a function of

η

by the blue curve in Fig. 1. An accurate global approximation based on Eq. (2.10), which is adequate for engineering applications is given by

A A (η) ≅ \frac{1}{{(1 + 1.07 η^{2})}^{7 / 6}},

and is plotted in Fig. 1 by the red curve. Examination of Fig. 1 reveals that Eq. (2.12) is an excellent approximation to the exact aperture-averaging factor. When the aperture radius is much greater than the Fresnel length it can be shown that the

S I (a) \propto a^{- 7 / 3}

instead of

S I (a) \propto a^{- 2}

that would be expected of an average of statistically independent correlation cells collected by the aperture [1,3,5]. This related to fact that the correlation function of irradiance goes negative at some separation over the aperture and energy conservation, and is discussed in detail by Churnside in [3].

Fig. 1 Comparison of the exact and approximate aperture-averaging factor for plane waves and constant turbulence conditions.

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2.2 Downlink slant propagation paths

For all downlink cases considered here the laser optical source in space is well above the atmosphere and the upper limit, $L$ , on integral over propagation path can be set equal to infinity. For downlink propagation and weak to moderate scintillation conditions the predominant turbulent eddy size involved in producing scintillation is of the order Fresnel size, $\sqrt{L_{e f f} / k}$ , where $L_{e f f}$ is the independent of wavelength effective scintillation propagation range. Generalizing the results in [8], $L_{e f f}$ for a spherical earth is given by

L_{e f f} = {(\frac{18 \int_{0}^{L} d s C_{n}^{2} [h (s)] s^{2}}{11 \int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6}})}^{6 / 7}

where in the flat earth approximation

L_{e f f} (θ) ≅ L_{e f f} (π / 2) \csc θ

, and for constant

C_{n}^{2}

(e.g., horizontal propagation over level terrain)

L_{e f f} = L

. For example, for propagation at nadir

L_{e f f} (π / 2)

is about 11.9 and 14.6 km for the HV-5/7 and Maui nighttime turbulence profile, respectively [12].

Next to obtain an analytic approximation for the downlink aperture-averaging factor we just replace $L$ in Eq. (2.11) $L_{e f f}$ For illustrative purposes we compare in Figs. 2 the exact results of Eq. (2.9) to that obtained from Eq. (2.12) with $L$ replaced by $L_{e f f}$ for various turbulence profiles and propagation conditions. Examination of Figs. 2(a)-2(d) reveals very good agreement between the exact and approximate analytic results over a wide range of aperture sizes and propagation conditions. In particular, the relative error in Figs. 2(a), 2(b) and 2(d) is better than 10%, while the worst agreement is in Fig. 2(c) near an aperture diameter of about 5 cm with an error of about 16%.

Fig. 2 (a) Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 30°, and an Maui Day turbulence profile (https://www.amostech.com/TechnicalPapers/2010/Posters/Bradford.pdf) (b) Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 45°, and an Maui Night turbulence profile. (c). Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1064 nm., an elevation angle of 60°, and the Hufnagle-Valley 5/7 turbulence profile. (d). Comparison of the exact and approximate aperture-averaging factor as a function of aperture diameter for downlink propagation at 1550 nm, an elevation angle of 15°, and the Clear1 turbulence profile [12]. In this example the effective wind speed is assumed due to a high-altitude aircraft flying at a speed of 200 m/s normal to the optical line of sight.

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3. Aperture-averaged temporal correlation

Here we use the usual assumption that the wind bears the turbulent eddies across the line of sight at a uniform speed V(s). This is the Taylor’s frozen-random medium hypothesis and is found to be valid in many situations [13]. The plane wave aperture averaged temporal auto correlation of power, $μ (τ; a)$ , for weak to moderate turbulence conditions can be expressed from a generalization of the results given in Sec. 5 of Ref [5]. as

μ (τ, a) = \frac{\int_{0}^{L} d s C_{n}^{2} [h (s)] \int_{0}^{\infty} d K \sin^{2} (K^{2} s / 2 k) Φ n (K) {(\frac{2 J_{1} (K a)}{K a})}^{2} J_{0} (K V (s) τ)}{\int_{0}^{L} d s C_{n}^{2} [h (s)] \int_{0}^{\infty} d K \sin^{2} (K^{2} s / 2 k) Φ n (K) {(\frac{2 J_{1} (K a)}{K a})}^{2}},

where

J_{0} (\cdot)

is the Bessel Function of the first kind of order zero, and

V (s)

is the wind speed normal to the line of sight at path length

s

.

3.1 Constant turbulence conditions

Assuming both constant $C_{n}^{2}$ and $V$ , the integral over path length in Eq. (3.1) can be obtained in closed form and $μ (τ; a)$ is given by

μ (τ; η) = \frac{1}{N (η)} \int_{0}^{\infty} d x (1 - \sin c x^{2}) {(\frac{2 J_{1} (x η)}{x η})}^{2} J_{0} (x τ_{n}) / x^{8 / 3},

where

sinc x \equiv \sin x / x

,

τ_{n} = τ V / \sqrt{k / L}

is normalized time delay. Except for

η = 0 and η \to \infty

Eq. (3.2) cannot be obtained in closed for and numerical integration is required. For a “point receiver” (i.e.,

η = 0

) the temporal auto correlation, which was first obtained by Tatarskii (see [1], Chap. 3), is obtained from Eq. (3.2) and given by

\begin{array}{l} μ (τ; 0) = \frac{11}{24} (2 + \sqrt{3}) τ_{n}^{2}_{2} F_{3} (- \frac{5}{12}, \frac{1}{12}; 1, \frac{3}{2}, \frac{3}{2}; - \frac{τ_{n}^{4}}{64}) +_{2} F_{3} (- \frac{11}{12}, - \frac{5}{12}; \frac{1}{2}, \frac{1}{2}, 1; - \frac{τ_{n}^{4}}{64}) \\ - \frac{121 (2 + \sqrt{3}) Γ (- 11 / 6) τ_{n}^{5 / 3}}{144 π 2^{1 / 6}} \end{array}

On the other hand, for very large values of

η

, the receiver filter function controls the integrand falls off rapidly for

x > 1 / η

and we can approximate

1 - sinc x^{2}

by

x^{4}

. The resulting integral can then by obtained in closed form given by

μ (τ; a) = \frac{2^{4 / 3} Γ {(7 / 6)}_{3} F_{2} (\frac{7}{6}, \frac{7}{6}, \frac{3}{2}; 2, 3; \frac{4}{τ_{n}^{2}})}{Γ (- 1 / 6) τ_{n}^{7 / 3}}

where for

η \to \infty

,

τ_{n} \to τ / (a / V)

. That is, for sufficiently large aperture sizes the dominant spatial scale becomes the aperture radius, which is in contrast to

η ~

1 where the dominant spatial scale is the Fresnel length

\sqrt{L / k}

. This feature is illustrated in Figs. 3 and 4, where the temporal auto correlation function is plotted for

η ~ 1

and

η > > 1

, respectively.

Fig. 3 The temporal auto correlation function as function of normalized time delay plotted for $η ~ unity$ .

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Fig. 4 The temporal auto correlation function as function of normalized time delay plotted for $η > > 1$ .

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A quantity of interest that is needed in system simulations is the characteristic aperture-averaged correlation time $τ_{C}$ , which to the best of my knowledge has not been treated in the literature. For a “point receiver” and, $η > > 1$ $τ_{c} = \sqrt{L / k} / V and a / V$ , respectively. From these asymptotic values we assume that an approximate expression for the aperture-averaged correlation time, $τ_{C} (η)$ , is given by

\begin{array}{l} τ_{C} (η) = τ_{C} (0) \sqrt{1 + η^{2}} \\ \to \sqrt{L / k} / V, for η < < 1 \\ \to a / V, for η > > 1 \end{array}

where,

τ_{c} (0) = \sqrt{L / k} / V

. Physically, it is expected that the temporal correlation time increases for increasing aperture size. This is because spatial scales less than the order of the dimensions of the aperture are averaged out, and only spatial scales larger than the aperture contribute, which takes a longer time to pass over the aperture. Now, we have from Eq. (3.3) that at

τ = τ_{C}

(i.e.,

τ_{n} = 1

)

μ (τ; 0) = 0.315

, which we take here as the value of the aperture-averaged temporal correlation time. Then in order to access the validity of the approximation given by Eq. (3.5) we compare in Fig. 5 the numerically exact temporal correlation time, obtained from Eq. (3.2) to that analytic approximation of Eq. (3.5). Examination of this figure reveals that Eq. (3.5) is in excellent agreement to the exact results.

Fig. 5 Temporal correlation time: a comparison of the numerically exact obtained from Eq. (3.2) to the analytic approximation Eq. (3.5).

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3.2 Downlink slant propagation paths

In general, for downlink propagation the temporal coherence is given by Eq. (3.1), which is a double integral over path length and spatial wave number. In order to obtain results, numerical methods must be employed, which are rather time consuming, and not particularly adapted to obtain estimates of the aperture-averaged correlation time. Therefore, in view of the success of using Eq. (3.5) to obtain an estimate of the correlation time for constant turbulence and cross wind conditions we use as a candidate for an engineering approximation for the downlink aperture-averaged correlation time Eq. (3.5) $L$ and $V$ replaced by $L_{e f f}$ and the rms value of $V (s)$ given by

V_{r m s} = \sqrt{\frac{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a) V^{2} [h (s)]}{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a)}} .

Note in general that the rms value of

V (s)

includes both the effects of the prevailing atmospheric wind and the pseudo-wind due to slewing (e.g., a receiving system tracking the download signal from a LEO satellite). For most cases of practical concern

V_{r m s}

is dominated by atmospheric winds and slewing for GEO and LEO targets, respectively.

Specifically, this engineering approximation for the aperture-averaged down link temporal correlation is given by

τ_{C} (a) = \frac{\sqrt{L_{e f f} / k}}{V_{r m e}} \sqrt{1 + η_{e f f}^{2}}

where

η_{e f f} = a \sqrt{k / L_{e f f}}

In Figs. 6, we show a comparison of the exact numerical temporal correlation function given by Eq. (3.1) to the engineering approximation given by Eq. (3.3) for various propagation and crosswind conditions. For a variable crosswind condition, we use here for illustrative purposes the Bufton wind model with a ground speed of 5 m/s given by

V (h) = 5 + 30 \exp [- {(\frac{h - 9400}{4800})}^{2}],

where

h

is the height above ground in meters [14]. Assuming that the characteristic temporal correlation time is given for a delay time where the temporal correlation falls to 0.315, examination of these figures reveals very good agreement (with a maximum error of 2.3% indicated in Fig. 6c) for using Eq. (3.5) for estimating “realistic)” down link aperture-averaged correlation times [15].

Fig. 6 (a) Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions. (b). Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions (c). Comparison of the exact numerical and engineering approximation to the temporal autocorrelation, obtained from Eq. (3.1) and Eq. (3.2), respectively for the conditions

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4. Aperture-averaged power spectral density

The temporal power spectrum is given by the Fourier transform of the temporal auto correlation function (see, for example [1], and [5]). Following the derivation given by Tatarskii in [1], the general expression for the aperture-averaged temporal power spectrum, $W (ω; a)$ , can be expressed as

W (ω; a) = S I (a) W_{n} (ω; a),

where

ω

denotes angular frequency in rad/s and the normalized PSD

W_{n} (ω; a)

is given by (see the Appendix)

W_{n} (ω; a) = \frac{\int_{0}^{L} d s C_{n}^{2} [h (s)] V^{- 1} (s) \int_{0}^{\infty} d K \frac{\sin^{2} [\frac{s}{2 k} (ω^{2} / V^{2} (s) + K^{2})]}{{(ω^{2} / V^{2} (s) + K^{2})}^{11 / 6}} {(\frac{2 J_{1} (\sqrt{(ω^{2} / V^{2} (s) + K^{2}) a})}{(ω^{2} / V^{2} (s) + K^{2})})}^{2}}{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a)}

With

W_{n} (ω; a)

given by Eq. (4.2) we have, as indicated in the Appendix, that

\int_{0}^{\infty} d ω W (ω; a) = S I (a)

.

4.1 Constant turbulence conditions

For both a constant $C_{n}^{2}$ and cross wind speed, the integral the propagation path can be obtained in closed form and it can be shown that the normalized PSD becomes

W_{n} (ω; a) = \frac{1}{N (η)} \int_{0}^{\infty} d q \frac{1 - \sin c [q^{2} + {(ω / ω_{F})}^{2}]}{ω_{F} {[q^{2} + {(ω / ω_{F})}^{2}]}^{11 / 6}} {(\frac{2 J_{1} [\sqrt{q^{2} + {(ω / ω_{F})}^{2}} η]}{\sqrt{q^{2} + {(ω / ω_{F})}^{2}} η})}^{2}

where the normalization constant

N (η)

is given in Appendix by Eq. (A-7). This expression introduces the so-called “Fresnel frequency” that is defined by the Fresnel length and the crosswind speed:

ω_{F} = V \sqrt{\frac{k}{L}}

For sufficiently small values of aperture size, where aperture averaging can be ignored, this provides a natural metric for scaling the scintillation frequencies. In particular, it is where the PSD to rolls off at high angular frequencies, which is a measure of the scintillation bandwidth (the corresponding frequency in Hz is given by

ω_{F} / 2 π

). Because the PSD and the temporal correlation covariance function are Fourier transform pairs, it follows directly that the corresponding measure of the aperture averaged scintillation bandwidth (in Hz), given by

f_{F} (η) = \frac{f_{F} (0)}{\sqrt{1 + η^{2}}},

where

f_{F} (0) = V \sqrt{k / L} / 2 π = V / \sqrt{2 π λ L}

.

For a “point receiver” the PSD, which was first obtained by Tatarskii (see [1], Chap. 4), is obtained from Eq. (4.2) and given by

W_{n} (ω, 0) = \frac{2.577}{ω_{n}^{8 / 3}} (1 - \frac{16 π Γ (\frac{17}{6}) Im (e i ω_{n}^{2} (\frac{_{1} F_{1} (\frac{1}{2}; \frac{4}{3}; - i ω_{n}^{2})}{Γ (- \frac{4}{3}) Γ (\frac{17}{6})} - \frac{{(- i ω_{n}^{2})}^{7 / 3}_{1} F_{1} (\frac{17}{6}; \frac{10}{3}; - i ω_{n}^{2})}{\sqrt{π} Γ (\frac{10}{3})}))}{11 \sqrt{3} Γ (\frac{7}{3}) ω_{n}^{2}}),

where “Im” denotes the imaginary part,

ω_{n} = ω / ω_{F}

, and is plotted in Fig. 7(a) as a function of the scaled scintillation angular frequency

ω_{n}

. As discussed in [1] and [5], the PSD is relatively constant for

ω_{n} < ω_{F}

and falls off asymptotically as

ω^{- 8 / 3}

for higher frequencies.

Fig. 7 (a) Numerical evaluation of $W (ω)$ as a function of $ω / ω_{F}$ for various values of $η = a \sqrt{k / L}$ (b) Numerical evaluation of $ω W (ω)$ as a function of $ω / ω_{F}$ for various values of $η = a \sqrt{k / L}$ .

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For arbitrary values of $η$ , Eq. (4.2) cannot be obtained in closed form, and numerical methods must be employed. For illustrative purposes, both the PSD and $ω W_{n} (ω; η)$ , for various values of $η$ , are plotted in Fig. 7(a) and 7(b) as a function of $ω_{n}$ . Examination of Fig. 7(a) reveals that the asymptotic high frequency roll off of the PSD for $η > 0$ decreases much faster than that for a “point receiver”. Qualitatively, for $η$ greater than about ½-1 the PSD has a power law fall-off dependence as $ω^{- q}$ , where q is near 17/3, which is in accord with experimental observations (see Fig. 2 of [7]). Additionally, examination of Fig. 7(b) reveals that, as expected physically, the characteristic roll-off frequency decreases for increasing values of $η$ . In Fig. 8, the characteristic aperture-averaged Fresnel frequency, $f_{F} (η)$ , is compared to that predicted by the analytic approximation of Eq. (4.4). Examination of this figure reveals very good agreement between the numerically exact and the analytic approximation.

Fig. 8 The normalized aperture-averaged PSD plotted as a function of normalized frequency for $η > > 1$ .

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As $η \to \infty$ , it can be shown that at high frequencies the PSD falls off as $ω^{- 17 / 3}$ . This asymptotic dependence has also been predicted in [9]. An analytic form for the PSD can also be obtained. For $η > > 1$ , it can be shown that we can ignore the “sinc” function in Eq. (4.2). Then the integral can be solved yielding

\begin{array}{l} W_{n} (ω, a) = \frac{1}{2 \sqrt{π}} (\frac{π Γ {(n)}_{2} F_{3} (- \frac{5}{6}, \frac{3}{2}; - \frac{1}{3}, 2, 3; - ω_{n}^{2})}{Γ (\frac{11}{6}) ω_{n}^{8 / 3}} + \\ \frac{4 Γ (- \frac{4}{3}) Γ {(- \frac{17}{6})}_{2} F_{3} (- \frac{1}{2}, \frac{17}{6}; \frac{7}{3}, \frac{10}{3}, \frac{13}{3}; - ω_{n}^{2})}{Γ (\frac{10}{3}) Γ (\frac{13}{3})}), \end{array}

where for

η > > 1

,

ω_{n} = V / a

, and is plotted in Fig. 8, which clearly shows the asymptotic

ω^{- 17 / 3}

fall-off at high frequencies. In contrast, it can be shown that the corresponding asymptotic frequency dependence for the soft Gaussian aperture of diameter

D_{G}

treated in [6] decays as

ω^{- 8 / 3} \exp [- D_{G}^{2} ω^{2} / 4 V^{2}]

, where such a decay is not in accord with measured data [9].

4.2 Downlink slant propagation paths

In general, for downlink propagation the PSD is given by Eq. (4.2), which is a double integral over path length and spatial wave number. Similar to the procedure used in Sec. 3B for obtaining temporal scaling laws, we therefore use as a candidate for an engineering approximation for the downlink aperture-averaged characteristic Fresnel frequency Eq. (4.4) where $L$ and $V$ are replaced by $L_{e f f}$ and $V_{r m s}$ , respectively.

Specifically, this engineering approximation for the aperture-averaged down link characteristic scintillation angular frequency bandwidth is given by

\begin{array}{l} ω_{F} (η_{e f f}) = \frac{ω_{F} (0)}{\sqrt{1 + η_{e f f}^{2}}} \\ ​ ​ ​ ​ ​ \to ω_{F} (0), for η_{e f f} < < 1 \\ \to V / a, for η_{e f f} > > 1, \end{array}

where

ω_{F} (0) = V_{r m s} \sqrt{k / L_{e f f}}

, and

η_{e f f} = a \sqrt{k / L_{e f f}}

. Correspondingly, as an engineering approximation for the aperture averaged downlink PSD we use Eq. (4.2), where

ω_{F}

and

V

are replaced by

ω_{F} (η)

and

V_{r m s}

, respectively. Astronomical observations were conducted at the La Palma Observatory at an altitude of 2360 m in the Canary Islands. A sequence of evening to midnight 550 nm power spectra measurements was made of Vega at an elevation angle of

45^{\circ}

for a variety of telescope openings. In particular, in Fig. 3 of [9] experimental results are presented for

f W (f)

, and a measure of the scintillation bandwidth is obtained from the frequency where

f W (f)

peaks. In order to compare these results to that predicted by Eq. (4.8) we use the Maui night time turbulence profile, and because no wind speed information is available, we determine a wind speed to use in Eq. (4.8) that matches the position of the peak frequency observed for the smallest aperture of 2.5 cm, and these results are compared to the corresponding experimental values in Table 1. Examination of this table reveals that the theoretical and experimental results are in good agreement, which lends credence to the general use of Eq. (4.8) for predicting the downlink aperture-averaged scintillation bandwidth.

Table 1. A comparison of the measured scintillation bandwidth reported in [8] to that predicted by Eq. (4.8). For an elevation angle of 45_°and the Maui night time profile $L_{e f f}$ = 20.6 km, and an rms wind speed of 16.6 m/s were used to obtain a frequency match for a telescope diameter of 2.5 cm. For the aperture diameters shown above the scintillation index is less than about 0.15, which is within the weak scintillation regime.

View Table

Figures 9, present comparisons of the downlink aperture-averaged exact numerical and the engineering approximation for various propagation conditions and turbulence profiles. Examination of these figures show very good agreement of the engineering approximation to the corresponding exact numerical results, we conclude that this approximation will provide systems engineers accurate results for realistic operational conditions.

Fig. 9 (a) Comparison of the exact numerical and engineering approximation to the PSD, obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated. (b). Comparison of the exact numerical and engineering approximation to the PSD obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated. (c). Comparison of the exact numerical and engineering approximation to the PSD obtained from Eq. (4.2) and Eq. (4.3), respectively for the conditions indicated.

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5. Aperture-averaged downlink quasi frequency for level crossing statistics

We apply the results presented above to the calculation of the downlink aperture-averaged angular quasi frequency, $ω_{Q}$ , which is central to the calculation of level crossing rates and the mean duration of fades and surges of collected power. In general, this quantity if given by [16]

ω_{Q} = \sqrt{\frac{\int_{0}^{\infty} d ω ω^{2} W (ω)}{\int_{0}^{\infty} d ω W (ω)}},

where

W (ω)

is given by Eq. (4.2). Alternatively, it is shown in [16] that an equivalent expression for

ω_{Q}

is given by

ω_{Q} = \sqrt{\frac{- \ddot{B} (0, a)}{B (0, a)}},

where the corresponding numerator and denominator are the second derivative of the temporal correlation function evaluated at

τ = 0

and variance of collected power, respectively. Here, however, Eq. (5.1a) is used to obtain quantitative results.

As shown in the Appendix, closed form expressions for both the integration over $ω$ and the inherent integration over spatial wave number in Eq. (4.2) can be obtained for both the numerator and denominator in Eq. (5.1). The final result for $ω_{Q}$ is then given by the square root of the ratio of two integrations over the propagation path, weighted by the index structure constant profile. We have,

ω_{Q} (a) = \sqrt{\frac{\int_{0}^{L} d s V^{2} (s) C_{n}^{2} [h (s)] s^{5 / 6} Q (s, a)}{\int_{0}^{L} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a)}},

where the aperture-averaged path dependent weighting functions

Q

and

G

are given by Eq. (A-5) and Eq. (2.6), respectively. For any value of

η > 0

the integrals in Eq. (5.2) are finite. Note, however, for a “point receiver” and the Kolmogorov spectrum the numerator of Eq. (5.2) is divergent, and a refractive index spectrum that employs an outer scale of turbulence must be used to obtain finite results. However, “point receivers” are of no practical concern and only finite sized apertures are considered here.

In a similar manner to procedure indicated in Sections. 2-4 an approximate analytic expression for the quasi frequency can be obtained from the case of plane wave propagating through constant turbulence conditions. The results are given by

ω_{Q}^{a p} (a) = \frac{V_{r m s}}{\sqrt{L_{e f f} / k}} \sqrt{\frac{Θ (η_{e f f})}{Δ (η_{e f f})}},

where

\begin{array}{l} Θ (η) = - \frac{π}{η^{2}} (\frac{1}{384 \sqrt{2}} η^{2} Γ (- \frac{5}{6}) (5 (\sqrt{3} - 1) η^{2}_{4} F_{5} (\frac{1}{12}, \frac{7}{12}, \frac{5}{4}, \frac{7}{4}; \\ \frac{3}{2}, \frac{3}{2}, 2, 2, \frac{5}{2}; \\ - \frac{η^{2}}{4}) - 24 {(1 + \sqrt{2})}_{4} F_{5} (- \frac{5}{12}, \frac{1}{12}, \frac{3}{4}, \frac{5}{4}; \\ \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; \\ - \frac{η^{2}}{4})) - \frac{η^{5 / 3} Γ (\frac{1}{3}) Γ (\frac{7}{6})}{\sqrt{π} Γ (\frac{11}{6}) Γ (\frac{17}{6})} \end{array}

And

\begin{array}{l} Δ (η) = - \frac{2 π}{η^{2}} (- \frac{1}{384 \sqrt{2}} η^{2} Γ (- \frac{11}{6}) (24 {(\sqrt{3} - 1)}_{4} F_{5} (\frac{11}{12}, - \frac{7}{12}, \frac{3}{4}, \frac{5}{4}; \\ \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; \\ - \frac{η^{4}}{4}) + 11 (1 + \sqrt{3}) η^{2}_{4} F_{5} (- \frac{5}{12}, \frac{1}{12}, \frac{5}{4}, \frac{7}{4}; \\ \frac{3}{2}, \frac{3}{2}, 2, 2 \frac{5}{2}; \\ - \frac{η^{2}}{4})) - \frac{η^{11 / 3} Γ (- \frac{5}{6}) Γ (\frac{7}{3})}{2 \sqrt{π} Γ (\frac{17}{6}) Γ (\frac{23}{6})} \end{array}

As shown in Fig. 10, an excellent “user friendly” approximation for the reduced quasi frequency, $\sqrt{Θ (η_{e f f}) / Δ (η_{e f f})}$ , is given by

\sqrt{\frac{Θ (η_{e f f})}{Δ (η_{e f f})}} ≅ 1.30 η_{e f f}^{- 0.228} \exp [- 0.879 η_{e f f}] + 0.772 η_{e f f}^{- 0.358} (1 - \exp [- 0.879 η_{e f f}])

As an illustration of the utility of calculating the aperture-averaged quasi frequency we consider a 10 cm diameter receiving located in a aircraft at an altitude 40 kft traveling at a speed of 200 m/s receiving downlink signals from space, and at a wavelength of 1550 nm. In Fig. 11 the exact and analytic approximate quasi frequency is plotted as a function of elevation angle, by the blue and red curve, respectively. Examination of this figure reveals good agreement, where relative errors over the range of elevation angles indicated is less than about 5%. For a worst-case condition, we assume an elevation angle of 15 degrees and model the turbulence conditions by 4 times the nominal Clear1 profile, where we obtain an aperture-averaged SI = 0.165 and a quasi frequency from Eq. (5.2) of about 434 Hz. Level crossing statistics for lognormal statistics has been previously treated in the literature [17]. Figs. 12(a) and 12(b) are plots of the level crossing rate and mean duration of a fade as a function of the fade level in dB. For example, a 6 dB fade occurs at a rate of about 6

s^{- 1}

and persist for about 20 nsec.

Fig. 10 A comparison of the exact and approximate reduced quasi frequency as a function of the normalized aperture radius.

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Fig. 11 The exact and analytic approximation quasi frequency obtained from Eq. (5.2) and Eq. (5.3), respectively plotted as a function of elevation angle for the conditions indicated.

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Fig. 12 (a) The level crossing rate as a function of the fade level for and elevation angle of 15° and the conditions indicated in Fig. 10. (b) The mean duration of a fade as a function of the fade level for an elevation angle of 15 degrees and the conditions indicated in Fig. 10.

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6. Conclusion

For downlink space-based optical lasers and weak to moderate scintillation conditions expressions have been presented and quantitatively evaluated for the aperture-averaged collected power fluctuations, and also for both the temporal covariance of the corresponding signal and the power spectral density. Additionally, closed form expressions for the aperture-averaged quasi-frequency that is central to the determination of level crossing rates and duration of fades and surges in a propagation channel are derived. Specifically, for downlink paths an integral expression over the propagation path for a given turbulence height profile is derived. To the best of our knowledge, this is the first comprehensive analysis of aperture-averaged quantities that has been derived that also addresses downlink slant propagation paths where the turbulence varies over the path. The analysis developed here is based on physical considerations that have been historically used previously [1], [5]. To the extent possible we have obtained accurate “user friendly” engineering expressions that quantitatively yields aperture-averaged statistical quantities that are useful to system design engineers in predicting system performance under operational conditions. The validity of these engineering approximations are obtained by comparing them to the corresponding exact numerical integrated along the slant path results, where in all cases very good agreement is obtained. Additionally, elementary analytic scaling relations have been obtained for both the aperture-averaged signal correlation time, and corresponding scintillation bandwidth given by Eq. (3.6) and Eq. (4.8), respectively. A comparison is made to available measured data, which lends credence to the general use of these scaling relations. Finally, in the weak to moderate scintillation regime “user friendly” accurate closed form expressions have been obtained for the quasi frequency, which are valid for arbitrary downlink propagation paths and turbulence profiles.

Appendix evaluation of integrals

First, we calculate the integration over angular frequency and spatial wave number of the numerator in Eq. (5.1). Because we are dealing with a ratio of terms both we omit below all common factors. Consider the numerator given in Eq. (4.2). We note that in general the PSD is an even function of both $ω$ and $K$ , and therefore we can extend both of the integrations to $\pm \infty$ , and include a multiplicative factor of ¼. We then transform to polar coordinates where $ω / V (s) = r \cos ϕ$ , $K = r \sin ϕ$ , $ϕ$ is the polar angle and $r = \sqrt{ω^{2} / V {(s)}^{2} + K^{2}}$ . Then we have

\int_{0}^{\infty} d ω W (ω) \propto \int_{0}^{\infty} d r r \int_{0}^{2 π} d ϕ r^{- 11 / 3} \sin^{2} (r^{2} s / 2 k) {(2 J_{1} (r a) / r a)}^{2}

Both integrations of the numerator can now be solved in closed form, and the final result is given by

\int_{0}^{\infty} d ω W (ω) = \int_{0}^{\infty} d s C_{n}^{2} [h (s)] s^{5 / 6} G (s, a),

where

G (s, a)

is given be Eq. (2.6), and is both identical to the denominator in Eq. (4.2) and is the appropriate term to appear in the denominator of Eq. (5.2). That is,

\int_{0}^{\infty} d ω W_{n} (ω) = 1

, from which it follows directly that

\int_{0}^{\infty} d ω W (ω) = S I (a)

.

\Next we calculate the integration over angular frequency and spatial wave number of the numerator in eq. (5.1). We then again transform to polar coordinates where $K = r \sin ϕ$ , $ϕ$ is the polar angle and $r = \sqrt{ω^{2} / V {(s)}^{2} + K^{2}}$ . We have

\int_{0}^{\infty} d ω ω^{2} W (ω) \to \frac{1}{4} \int_{0}^{\infty} d r r \int_{0}^{2 π} d ϕ \frac{V^{2} (s) r^{2} \cos^{2} ϕ}{r^{11 / 3}} \sin^{2} (r^{2} s / 2 k) {(2 J_{1} (r a) / r a)}^{2} .

Both integrations can be solved in closed form, and the final result is given by

\int_{0}^{\infty} d ω ω^{2} W (ω; a) = \int_{0}^{L} d s C_{n}^{2} [h (s)] V^{2} (s) s^{5 / 6} Q (s; a),

where the dimensions of

Q (s, a)

is 1/(length)² and is given by

\begin{array}{l} Q (s, a) = \frac{1}{16896 a^{17 / 6} s^{5 / 6}} π Γ (\frac{1}{6}) \\ (\frac{11 \sqrt{2} (\sqrt{3} - 1) a^{4}_{4} F_{5} (\frac{7}{12}, \frac{13}{12}, \frac{5}{4}, \frac{7}{4}, \frac{3}{2}, \frac{3}{2}, 2, 2, \frac{5}{2}; - \frac{a^{4} k^{2}}{4 s^{2}})}{{(\frac{s^{2}}{k^{2}})}^{7 / 12}} + \\ \frac{264 \sqrt{2} (1 + \sqrt{3}) a^{4}_{4} F_{5} (\frac{1}{12}, \frac{7}{12}, \frac{3}{4}, \frac{5}{4}; \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; - \frac{a^{4} k^{2}}{4 s^{2}})}{12 \sqrt{\frac{s^{2}}{k^{2}}}} + \\ \frac{2304 a^{5 / 3} Γ (\frac{4}{3})}{\sqrt{π} Γ {(\frac{11}{6})}^{2}}) \end{array}

Next, we evaluate the normalization factor,

N (η)

, appearing in the denominator of Eq. (4.2). This quantity is given by the integral over angular frequency of the corresponding integral factor on the right-hand side of Eq. (4.2). We then transform to polar coordinates where

ω / ω_{F} = r \cos ϕ

,

q = r \sin ϕ

,

ϕ

is the polar angle, and

r = \sqrt{q^{2} + {(ω / ω_{F})}^{2}}

. This integral becomes

N (η) = \frac{π}{2} \int_{0}^{\infty} d r r \frac{1 - \sin c(r^{2})}{r^{11 / 3}} {[2 J_{1} (r η) / r η]}^{2},

which can be evaluated in closed form and is given by

\begin{array}{l} N (η) = \\ - \frac{1}{Γ (\frac{23}{6})} \\ (\frac{7}{576} π^{2} (12 \sqrt{2} {(\sqrt{3} - 1)}_{4} F_{5} (- \frac{11}{12}, - \frac{5}{12}, \frac{3}{4}, \frac{1}{4}; \frac{1}{2}, 1, \frac{3}{2}, \frac{3}{2}, 2; - \frac{η^{4}}{4}) + \\ 11 \sqrt{2 \sqrt{3}} η^{5}_{4} F_{5} (- \frac{5}{12}, \frac{1}{12}, \frac{5}{4}, \frac{7}{4}; \frac{3}{2}, \frac{3}{2}, 2, 2; \frac{5}{2}; - \frac{η^{4}}{4}) + \\ \frac{\sqrt{π} η^{5 / 3} Γ (\frac{- 5}{6}) Γ (\frac{7}{3})}{Γ (\frac{17}{6})}) \end{array}

References and links

1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, U. S. Department of Commerce, Springfield, Virginia, 1971, Sec 53.

2. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57(2), 169–178 (1967). [CrossRef]

3. J. H. Churnside, “Aperture Averaging of Optical Scintillations in the Turbulent Atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991). [CrossRef] [PubMed]

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

5. A. D. Wheelon, Electromagnetic Scintillation, II. Weak Scattering, University Press, Cambridge, Secs. 2.2.3 and 4.1.4, 2003.

6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and temporal spectrum,” Waves Random Media 10(1), 53–70 (2000). [CrossRef]

7. H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014). [CrossRef]

8. H. T. Yura and W. G. McKinley, “Aperture averaging of scintillation for space-to-ground optical communication applications,” Appl. Opt. 22(11), 1608–1609 (1983). [CrossRef] [PubMed]

9. D. Dravins, L. Lindgren, E. Mezey, and A. T. Young, “Atmospheric Intensity Scintillation of Stars, III. Effects for Different Telescope Apertures,” Publ. Astron. Soc. Pac. 110(747), 610–633 (1998). [CrossRef]

10. E. Vilar, J. Haddon, P. Lo, and T. J. Moulsley, “Measurements and Modeling of Amplitude and Phase Scintillations in an Earth-Space path,” Journal of the Institution of Electronic and Radio Engineers 55(3), 87–96 (1985). [CrossRef]

11. S. Wolfram, Mathematica, (Cambridge University Press, 2012), Version 9.

12. The Infrared & Electro-Optical Handbook Vol, 3 Atmospheric Propagation of Radiation, F. G. Smith, Editor, Chapter. 2, SPIE Optical Engineering Press, Bellingham WA, 1993. Note, the HV-5/7 model is also known as the HV-21 model.

13. A. D. Wheelon and I. Electromagnetic Scintillation, Geometrical Optics (University Press, Cambridge, 2001), Sec. 6.1.2.

14. J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12(8), 1785–1793 (1973). [CrossRef] [PubMed]

15. Realistic to the extent of knowing the correct index-structure constant profile.

16. P. Beckmann, “Probability in Communication Engineering”, Harcourt Brace & World, Inc., 1967, Sec. 6.7. ”

17. H. T. Yura and W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22(21), 3353–3358 (1983). [CrossRef] [PubMed]

Telescope Aperture Diameter (cm)	[9]: Scintillation Bandwidth (Hz)	Eq. (4.8): Scintillation Bandwidth (Hz)
2.5	58.3	58.3
5	52.3	55.4
10	32.8	30.2
20	18.9	21.4
60	~10	12.4

Optical downlink propagation from space-to-earth: aperture-averaged power fluctuations, temporal covariance and power spectrum

Abstract

1. Introduction

2. Aperture averaging of scintillation

2.1 Constant turbulence conditions

2.2 Downlink slant propagation paths

3. Aperture-averaged temporal correlation

3.1 Constant turbulence conditions

3.2 Downlink slant propagation paths

4. Aperture-averaged power spectral density

4.1 Constant turbulence conditions

4.2 Downlink slant propagation paths

5. Aperture-averaged downlink quasi frequency for level crossing statistics

6. Conclusion

Appendix evaluation of integrals

References and links

Cited By

Figures (12)

Tables (1)

Equations (45)

Optics Express