Theoretical analysis of the angle-of-arrival fluctuations for optical wavext propagation through solar corona turbulence

Guanjun Xu; Zhaohui Song

doi:10.1364/OE.25.028022

1. Introduction

Nowadays, deep space optical communication has attracted numerous attention with increasing activities in deep space exploration. It not only provides a good working solution for the fast growing data rate in deep space communication, but also further affords an incentive for further exploration to extend the range to deep space [1,2]. As we all know, the Earth and the probe that revolves around the target planet, such as the Mars, Saturn, Venus, etc., move with different angular speed. Therefore, the communication link between the Earth and the probe may encounters solar scintillation during superior solar conjunction when the Sun lies between the Earth and the probe [3].

During this period, the performance of the communication link becomes severely degraded and the link will be even completely disrupted due to the coronal solar wind irregularities. The apparent angles of the optical signal wander around the centeroid of the beam, leading the defocusing of the signal source, and even result in image jittering in the focal plane of an imaging or laser systems. Due to these impacts, giving the prediction model of angle-of-arrival fluctuations for the optical wave passing through the solar corona has become quite imperative and meaningful.

Angle-of-arrival fluctuations play a significant role in diverse fields and have been extensively studied for the optical wave propagation in atmospheric turbulence theoretically and experimentally for a long time. Cheon presented close-form expression of the angle-of-arrival fluctuations for both plane and spherical waves passing through homogeneous and isotropic turbulence by Rytov approximation [4]. Pawar proposed a modified frequency spectrum of angle-of-arrival fluctuations and studies its characteristics for the parallel laser light propagates through the axially homogeneous, axisymmetric buoyancy-driven turbulence [5]. Image motion monitoring method has been used to investigate the inhomogeneity and anisotropy atmospheric air turbulence on angle-of-arrival fluctuations [6]. Nevertheless, most of these studies mainly focused on optical wave propagation in atmospheric turbulence, while the properties of optical wave propagation in coronal solar wind plasma have seldom been taken into consideration. According to the summarization in [7], the classical ways to investigate the angle-of-arrival fluctuations are geometrical optics based method and covariance of angle-of-arrival fluctuations based method. Both have been extensively applied in the calculations of angle-of-arrival fluctuations. According to [7,8], the angle-of-arrival fluctuations results in Kolmogorov turbulence are only theoretically valid in the inertial sub-range. Therefore, many researchers pay attention to the angle-of-arrival fluctuations with the non-Kolmogorov spectrum and achieve different propagation phenomenons [8,9]. Both the theories of Kolmogorov and non-Kolmogorov spectrum models have been well studied, however, these turbulence spectrum models are originally developed for atmospheric turbulence, cannot be directly applied in the circumstances when the optical wave propagates in the solar coronal turbulence.

To the best of our knowledge, the effects of non-Kolmogorov solar wind turbulence of coronal environment and the aperture averaging effect on angle-of-arrival fluctuations have not been reported. Therefore, the aim of this paper is to propose a theoretical expression of angle-of-arrival fluctuations for optical wave propagation through the coronal plasma. The solar corona turbulence spectrum with a general spectral index, ranging from 3 to 4, instead of the conventional Kolmogorov value of $\frac{11}{3}$ has been used in this paper. Using this spectrum and considering the special link model, both the angle-of-arrival fluctuations and its temporal power spectrum have been developed, and then the effects of spectral index, turbulent outer scale, wavelength, and antenna radius have been analyzed.

2. Background

2.1. Deep space optical model during superior solar conjunction

During the superior solar conjunction, optical wave transmitting between the Earth station and the spacecraft will propagate through the turbulence, such as the solar corona, ionosphere and atmosphere. In state-of-the-art optical communication, the effect of turbulence in the ionosphere and atmosphere on optical wave propagation has been extensively studied. However, little attention has been put on the impact of solar corona plasma. For these reasons, we pay attention to the angle-of-arrival fluctuation caused by solar corona plasma when the optical link gets close to the Sun.

In order to analyze the effect of solar corona irregularities on angle-of-arrival fluctuations, the optical communication geometrical link must be taken into consideration. As depicted in Fig. 1, the Sun-Earth-Probe (SEP) angle α and the Sun-Probe-Earth (SPE) angle β are both far away from 90°during the superior solar conjunction. The distance between the Earth and the probe is L, L_se denotes the distance between the Earth and the Sun, therefore, the closest distance between the Sun and the optical link (heliocentric distance) can be obtained as

L = L_{se} cos α + \frac{L_{se} sin α}{tan β}

Affected by the large irregularities which are filled by the out flow of solar wind, the received signal in place r will be moved to r+vt. Therefore, the optical wave connecting the transmitter and the receiver is wandering along the optical path.

2.2. Generalized spectrum model

During the superior solar conjunction, optical wave transmitted from the spacecraft will pass through the solar corona irregularities. Considering the mass of the ions is much greater than the electrons, we use the solar wind density to represent the density of free electrons N_e in this paper. According to [10], the relationship between the fluctuations of the relative dielectric constant and the fluctuations of the solar wind density is

δ ε = - \frac{r_{e} λ^{2}}{π} δ N_{e}

where, r_e is the classical electron radius, λ is the wavelength of the optical wave. Therefore, the variance of the relative dielectric constant fluctuations can be written as

〈 δ ε δ ε^{'} 〉 = \frac{{r_{e}}^{2} λ^{4}}{π^{2}} 〈 δ N_{e} δ N_{e}^{'} 〉

Fig. 1 A geometric diagram for deep space optical communication during superior conjunction and schematic used for angle-of-arrival fluctuations with baseline vt, solar wind speed v, and the observation axis x.

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Since the relationship between the refractive index and the dielectric constant is ε = n², their fluctuations are linearly related as δε = 2n₀δn [11], where n₀ is the ambient value and normally omitted since it is close to 1. Therefore, the variance of the refractive index fluctuations can be expressed as:

〈 δ n (r, t) δ n (r^{'}, t) 〉 = \frac{1}{4} 〈 δ ε (r, t) δ ε (r^{'}, t) 〉

Using the spectrum to represent the spatial covariance, the relationship between the refractive index spectrum Φ_n (κ, z) and the electron density irregularities spectrum Φ_N (κ, z) can be obtained by Eqs. (3) and (4), and further transformed by the Fourier transform. As a result, we have

Φ_{n} (κ, z) = \frac{{r_{e}}^{2} λ^{4}}{4 π^{2}} Φ_{N} (κ, z)

where, Φ_N (κ, z) is the spatial power spectrum of the solar wind density fluctuations in the coronal. In most investigations of the corona irregularity spectrum by many probes, a power law spectrum, Φ_N (κ) ∝ κ^−p, κ_o < κ < κ_i, has been demonstrated as the most suitable approximation for describing the irregularities. Where, κ is the magnitude of the spatial wavenumber, 3 < p < 4 denotes the spectral power law exponent and is named as standard Kolmogorov spectral index when

p = \frac{11}{3} \cdot κ_{o} = \frac{2 π}{l_{o}}, κ_{i} = \frac{2 π}{l_{i}}

, l_o and l_i represent the outer scale and inner scale, respectively. According to [12–14], the solar corona turbulent spectrum with the non-Kolmogorov spectral index takes the form as

Φ_{N} (κ, z) = \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} κ^{- p} δ N_{e}^{2} (z)

Here, Γ( ) is the gamma function, $δ N_{e}^{2} (z)$ is the variance of the electron density fluctuations. Note that, this spectrum model is the function of κ and z since the solar corona turbulence is inhomogeneous. According to the geometric diagram in Fig. 1, we have r = (z² + L_se² − 2zL_se cos α)^1/2. Therefore, $δ N_{e}^{2} (z)$ can be recast as the function of r. Since we mainly consider the angle-of-arrival fluctuations caused by the coronal solar wind irregularities, the solar wind density fluctuations in this area have a typical steep radial fall of density turbulence. According to the observation and measurement by many astronomers [15], we use the following relationship in this paper.

δ N_{e} (r) = a_{0} r^{- 2}

3. Angle-of-arrival fluctuations variance and spectrum

3.1. Variance of angle-of-arrival fluctuations

The variance of angle-of-arrival fluctuations has been derived for both plane wave and spherical wave. For the large distance deep space communication, the optical wave transmits between the Earth station and the spacecraft can be treated as a plane wave. Note that, the variance of angle-of-arrival fluctuations for plane wave can be found in [16] and was adopted directly in this paper for simplicity.

〈 θ^{2} 〉 = 2 π^{2} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, z) d κ {(\frac{2 J_{1} (κ a_{r})}{κ a_{r}})}^{2} \int_{0}^{L} {cos}^{2} (\frac{κ^{2} z}{2 κ}) d z

where,

k = \frac{2 π}{λ}

, L is the link distance as shown in Fig. 1. J₁ () denotes the first order Bessel function, a_r represents antenna radius. For the widely used imaging and laser system, the receiver’s diameter is 2a_r. Stem from averaging over a circular aperture,

\frac{2 J_{1} (κ a_{r})}{κ a_{r}}

is the Airy function and acts as a low pass filter to eliminate the frequency components that is higher than the aperture smoothing frequency. For mathematical analysis convenience, the Airy function can be modeled as the Gaussian function [9]:

exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \approx {(\frac{2 J_{1} (κ a_{r})}{κ a_{r}})}^{2}

According to [4], here, we set b=0.5216.

In this paper, we use the following triangle transformation,

{cos}^{2} (\frac{k^{2} z}{2 k}) = \frac{1}{2} [1 + cos (\frac{k^{2} z}{k})]

Inserting these approximation Eqs. (9)–(10) and the spatial spectrum Eqs. (5)–(6) into Eq. (8) yields the variance of angle-of-arrival fluctuations.

〈 θ^{2} 〉 = \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{4 Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} {r_{e}}^{2} λ^{4} δ N_{e}^{2} (r) {\begin{array}{l} \int_{0}^{\infty} κ^{3 - p} d κ exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) L + \\ \int_{0}^{\infty} κ^{3 - p} d κ exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \int_{0}^{L} cos (\frac{κ^{2} z}{k}) d z \end{array}}

As we can see, both of the two integrations are taken over to κ and determined by the aperture by wave number. For analysis purposes, the integrations in Eq. (11) are divided into two parts and the variance of angle-of-arrival fluctuations is recast as:

\begin{array}{l} 〈 θ^{2} 〉 = \frac{1}{4} \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} {r_{e}}^{2} λ^{4} δ N_{e}^{2} (r) (I_{1} L + I_{2}) \\ {\begin{array}{l} I_{1} = \int_{0}^{\infty} κ^{3 - p} d κ exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \\ I_{2} = \int_{0}^{\infty} κ^{3 - p} d κ exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \int_{0}^{L} cos (\frac{κ^{2} z}{k}) d z \end{array} \end{array}

Considering the following integral with the gamma function [17],

\int_{0}^{\infty} exp (- s t) t^{(x - 1)} d t = \frac{Γ (x)}{s^{x}}, x > 0, s > 0

And letting $s = \frac{b^{2} {a_{r}}^{2}}{4}$ , $x = 2 - \frac{p}{2}$ , we can obtain

I_{1} = \frac{1}{2} Γ (2 - \frac{p}{2}) {(\frac{b a_{r}}{2})}^{p - 4}

In order to evaluate the role of aperture smoothing in I₂ as shown in Eq. (12), we further use the following formula

\int_{0}^{\infty} t^{μ - 1} exp (- a t) sin (c t) d t = \frac{Γ (μ)}{{(a^{2} + c^{2})}^{μ / 2}} sin [μ arctan (\frac{c}{a})]

And let $μ = 1 - \frac{p}{2}$ , $a = \frac{b^{2} {a_{r}}^{2}}{4}$ , $c = \frac{L}{k}$ , then I₂ can be rewritten as:

I_{2} = \frac{1}{2} k \frac{Γ (1 - \frac{p}{2})}{{[{(\frac{b^{2} {a_{r}}^{2}}{4})}^{2} + {(\frac{L}{K})}^{2}]}^{\frac{2 - p}{4}}} sin [(1 - \frac{p}{2}) arctan (\frac{4 L}{k b^{2} {a_{r}}^{2}})]

Thus the final expression for the variance of angle-of-arrival fluctuations is of this form:

\begin{array}{l} 〈 θ^{2} 〉 = & \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{8 Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} {r_{e}}^{2} λ^{4} δ N_{e}^{2} (r) \times \\ {\frac{Γ (2 - \frac{p}{2})}{{(\frac{b a_{r}}{2})}^{4 - p}} L + k \frac{Γ (1 - \frac{p}{2})}{{[{(\frac{b a_{r}}{2})}^{4} + {(\frac{L}{k})}^{2}]}^{\frac{2 - p}{4}}} sin [(1 - \frac{p}{2}) arctan (\frac{4 L}{k b^{2} {a_{r}}^{2}})]} \end{array}

It is necessary to be noted that the above derived angle-of-arrival fluctuations variance is concise close-form. When the spectral index is set to $\frac{11}{3}$ , we have the angle-of-arrival fluctuations variance for Kolmogorov spectrum model as

\begin{array}{l} 〈 θ^{2} 〉 = & \frac{{(2 π)}^{\frac{8}{3}} {l_{o}}^{- \frac{2}{3}} Γ (\frac{11}{6})}{8 Γ (\frac{3}{2}) Γ (\frac{1}{3})} {r_{e}}^{2} λ^{4} δ N_{e}^{2} (r) \times \\ {\frac{Γ (\frac{1}{6})}{{(\frac{b a_{r}}{2})}^{1 / 3}} L + k \frac{Γ (- \frac{5}{6})}{{[{(\frac{b a_{r}}{2})}^{4} + {(\frac{L}{k})}^{2}]}^{- \frac{5}{12}}} sin [- \frac{5}{6} arctan (\frac{4 L}{k b^{2} {a_{r}}^{2}})]} \end{array}

4. Power spectrum of angle-of-arrival fluctuations

According to the Rytov approximation and the Taylor frozen turbulence hypothesis [11,18], the power spectrum of angle-of-arrival fluctuations for a plane wave has been derived by various researchers [9,23,24] and is given by

\begin{array}{l} W_{θ} (w) = & 8 π^{2} \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ, z) {(\frac{2 J_{1} (κ a_{r})}{κ a_{r}})}^{2} [J_{0} (ν κ t) - cos (2 φ) J_{2} (ν κ t)] \\ \times \int_{0}^{L} {cos}^{2} (\frac{κ^{2} z}{2 k}) d z \int_{0}^{\infty} cos (ω t) d t \end{array}

where J₀ (vκt) and J₂ (vκt) are the zero order and second order Bessel functions, respectively. vt denotes the geometrical separation between two points (s, s+vt) in the plane transverse to the direction of propagation due to the fast solar wind speed v as shown in Fig. 1. φ is the angle between the angle-of-arrival observation axis and the baseline. Substituting the spatial spectrum Eqs. (5) and (6) into Eq. (19), the temporal power spectrum of angle-of-arrival fluctuations can be written as

\begin{array}{l} W_{θ} (w) = & 2 {r_{e}}^{2} λ^{4} \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} δ N_{e}^{2} (r) \int_{0}^{L} {cos}^{2} (\frac{κ^{2} z}{2 k}) d z {(\frac{2 J_{1} (κ a_{r})}{κ a_{r}})}^{2} \\ \times \int_{0}^{\infty} d κ κ^{3 - p} \int_{0}^{\infty} [J_{0} (ν κ t) - cos (2 β) J_{2} (ν κ t)] cos (ω t) d t \end{array}

According to the integration table [19],

\int_{0}^{\infty} J_{m} (a x) cos (b x) d x = {\begin{array}{l} \frac{cos [m {sin}^{- 1} (b / a)]}{\sqrt{a^{2} - b^{2}}} & for 0 < b < a \\ \frac{- a^{m} sin (m π / 2)}{[b + \sqrt{b^{2} - a^{2}}]}, & for 0 < a < b, ν > - 1 \end{array}

Let a = vκ, b = ω, therefore, the last integration in Eq. (20) with respect to t are evaluated under the condition vκ > ω,

\begin{array}{l} \int_{0}^{\infty} [J_{0} (ν κ t) - cos (2 β) J_{2} (ν κ t)] cos (ω t) d t \\ = \frac{1}{\sqrt{{(ν κ)}^{2} - ω^{2}}} - cos (2 β) \frac{1 - 2 {(\frac{ω}{ν κ})}^{2}}{\sqrt{{(ν κ)}^{2} - ω^{2}}} \end{array}

Substituting Eqs. (9)–(10) and (22) into Eq. (20), thus the power spectrum of angle-of-arrival fluctuations can be further recast as

\begin{array}{l} W_{θ} (w) = r_{e}^{2} λ^{4} \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} δ N_{e}^{2} (r) \times \\ {\begin{array}{l} L \int_{0}^{\infty} d κ κ^{3 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) [\frac{1 - cos (2 β)}{\sqrt{{(ν κ)}^{2} - ω^{2}}} + \frac{2 cos (2 β) {(\frac{ω}{ν κ})}^{2}}{\sqrt{{(ν κ)}^{2} - ω^{2}}}] \\ + k \int_{0}^{\infty} sin (\frac{κ^{2}}{κ} L) d κ κ^{1 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) [\frac{1 - cos (2 β)}{\sqrt{{(ν κ)}^{2} - ω^{2}}} + \frac{2 cos (2 β) {(\frac{ω}{ν κ})}^{2}}{\sqrt{{(ν κ)}^{2} - ω^{2}}}] \end{array}} \end{array}

For mathematical analysis convenience, we define the following four formulas,

M_{1} = [1 - cos (2 β)] \int_{0}^{\infty} d κ κ^{3 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \frac{1}{\sqrt{{(ν κ)}^{2} - ω^{2}}}

M_{2} = 2 cos (2 β) \int_{0}^{\infty} d κ κ^{3 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \frac{{(\frac{ω}{ν κ})}^{2}}{\sqrt{{(ν κ)}^{2} - ω^{2}}}

N_{1} = (1 - cos (2 β)) \int_{0}^{\infty} sin (\frac{κ^{2}}{k} L) d κ κ^{1 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \frac{1}{\sqrt{{(ν κ)}^{2} - ω^{2}}}

N_{2} = 2 cos (2 β) \int_{0}^{\infty} sin (\frac{κ^{2}}{k} L) d κ κ^{1 - p} exp (- \frac{b^{2} {a_{r}}^{2} κ^{2}}{4}) \frac{{(\frac{ω}{ν κ})}^{2}}{\sqrt{{(ν κ)}^{2} - ω^{2}}}

Therefore, the power spectrum of angle-of-arrival can be expressed as:

W_{θ} (w) = {r_{e}}^{2} λ^{4} \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} δ N_{e}^{2} (r) \times [L (M_{1} + M_{2}) + k (N_{1} + N_{2})]

Using Euler’s formula, sin (x)= Im[exp(−ix)]. Therefore,

N_{1} = \frac{(1 - cos (2 β))}{2 ν} Im \int_{0}^{\infty} {(κ^{2})}^{2 (\frac{2 - p}{4}) - 1} {(κ^{2} - \frac{ω^{2}}{ν^{2}})}^{2 (\frac{1}{4}) - 1} exp [- (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) κ^{2}] d κ^{2}

N_{2} = cos (2 β) \frac{ω^{2}}{ν^{3}} Im \int_{0}^{\infty} {(κ^{2})}^{2 (- \frac{p}{4}) - 1} {(κ^{2} - \frac{ω^{2}}{ν^{2}})}^{2 \times \frac{1}{4} - 1} exp [- (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) κ^{2}] d κ^{2}

Considering the following integral relation [19],

\begin{array}{l} \int_{0}^{\infty} {(t + a)}^{2 μ - 1} {(t - b)}^{2 r - 1} exp (- s t) d t \\ = {\begin{cases} 0, 0 < t < b \\ Γ (2 r) {(a + b)}^{u + r - 1} s^{- μ - r} exp [\frac{s (a - b)}{2}] W_{μ - r, μ + r - \frac{1}{2}} (b s + a s), t > b \end{cases} \end{array}

We first make the following substitution, letting a = 0, $μ = 1 - \frac{p}{4}$ , $b = \frac{ω^{2}}{v^{2}}$ , $r = \frac{1}{4}$ , $s = \frac{b^{2} {a_{r}}^{2}}{4}$ , we have

M_{1} = \frac{[1 - cos (2 β)]}{2 ν} Γ (\frac{1}{2}) {(\frac{ω}{ν})}^{\frac{1 - p}{2}} {(\frac{b a_{r}}{2})}^{\frac{p - 5}{2}} exp [- \frac{b^{2} {a_{r}}^{2} ω^{2}}{8 ν^{2}}] W_{\frac{3 - p}{4}, \frac{3 - p}{4}} (\frac{b^{2} {a_{r}}^{2} ω^{2}}{4 ν^{2}})

Letting a = 0, $μ = \frac{1}{2} - \frac{p}{4}$ , $b = \frac{ω^{2}}{v^{2}}$ , $r = \frac{1}{4}$ , $s = \frac{b^{2} {a_{r}}^{2}}{4}$ , we have

M_{2} = cos (2 β) \frac{ω^{2}}{ν^{3}} Γ (\frac{1}{2}) {(\frac{ω}{ν})}^{- \frac{1 + p}{2}} {(\frac{b a_{r}}{2})}^{\frac{p - 3}{2}} exp (- \frac{b^{2} {a_{r}}^{2} ω^{2}}{8 ν^{2}}) W_{\frac{1 - p}{4}, \frac{1 - p}{4}} (\frac{b^{2} {a_{r}}^{2} ω^{2}}{4 ν^{2}})

Letting a = 0, $μ = \frac{2 - p}{4}$ , $b = \frac{ω^{2}}{v^{2}}$ , $r = \frac{1}{4}$ , $s = \frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}$ , we have

N_{1} = \frac{(1 - cos (2 β))}{2 ν} Im {\begin{cases} Γ (\frac{1}{2}) {(\frac{ω^{2}}{ν^{2}})}^{- \frac{p + 1}{4}} {(\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})}^{\frac{p - 3}{4}} exp [\frac{1}{2} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) (- \frac{ω^{2}}{ν^{2}})] \\ W_{\frac{1 - p}{4}, \frac{1 - p}{4}} [\frac{ω^{2}}{ν^{2}} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})] \end{cases}}

Letting a = 0, $μ = - \frac{p}{4}$ , $b = \frac{ω^{2}}{v^{2}}$ , $r = \frac{1}{4}$ , $s = \frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}$ , we have

N_{2} = cos (2 β) \frac{ω^{2}}{ν^{3}} Im {\begin{cases} Γ (\frac{1}{2}) {(\frac{ω^{2}}{ν^{2}})}^{- \frac{p + 3}{4}} {(\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})}^{\frac{p - 1}{4}} exp [\frac{1}{2} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) (- \frac{ω^{2}}{ν^{2}})] \\ W_{- \frac{p + 1}{4}, - \frac{p + 1}{4}} [\frac{ω^{2}}{ν^{2}} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})] \end{cases}}

Note that, W_a,b (c) is Whittaker’s confluent hypergeometric function [19]. As a result, the final solution to the power spectrum of angle-of-arrival fluctuations is obtained by inserting Eqs. (32)–(35) into Eq. (23),

\begin{array}{l} W_{θ} (w) = {r_{e}}^{2} λ^{4} \frac{{(2 π)}^{p - 1} {l_{o}}^{3 - p} Γ (\frac{p}{2}) Γ (\frac{1}{2})}{Γ (\frac{3}{2}) Γ (\frac{p - 3}{2})} δ N_{e}^{2} (r) \times \\ {\begin{cases} L \frac{[1 - cos (2 β)]}{2 ν} exp [- \frac{b^{2} {a_{r}}^{2} ω^{2}}{8 ν^{2}}] {(\frac{ω}{ν})}^{\frac{1 - p}{2}} {(\frac{b a_{r}}{2})}^{\frac{p - 5}{2}} W_{\frac{3 - p}{4}, \frac{3 - p}{4}} (\frac{b^{2} {a_{r}}^{2} ω^{2}}{4 ν^{2}}) + \\ L cos (2 β) \frac{ω^{2}}{ν^{3}} exp (- \frac{b^{2} {a_{r}}^{2} ω^{2}}{8 ν^{2}}) {(\frac{ω}{ν})}^{- \frac{1 + p}{2}} {(\frac{b a_{r}}{2})}^{\frac{p - 3}{2}} W_{\frac{1 - p}{4}, \frac{1 - p}{4}} (\frac{b^{2} {a_{r}}^{2} ω^{2}}{4 ν^{2}}) + \\ k \frac{(1 - cos (2 β))}{2 ν} Im {\begin{cases} {(\frac{ω}{ν})}^{- \frac{p + 1}{2}} {(\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})}^{\frac{p - 3}{4}} exp [\frac{1}{2} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) (- \frac{ω^{2}}{ν^{2}})] \\ W_{\frac{1 - p}{4}, \frac{1 - p}{4}} [\frac{ω^{2}}{ν^{2}} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})] \end{cases}} + \\ k cos (2 β) \frac{ω^{2}}{ν^{3}} Im {\begin{cases} {(\frac{ω}{ν})}^{- \frac{p + 3}{2}} {(\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})}^{\frac{p - 1}{4}} exp [\frac{1}{2} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k}) (- \frac{ω^{2}}{ν^{2}})] \\ W_{- \frac{p + 1}{4}, - \frac{p + 1}{4}} [\frac{ω^{2}}{ν^{2}} (\frac{b^{2} {a_{r}}^{2}}{4} + i \frac{L}{k})] \end{cases}} \end{cases}} \end{array}

Considering the optical wave propagation through solar corona turbulence and finally received by the finite aperture antenna on the Earth, the analytical expressions of both the variance of angle-of-arrival fluctuations and its corresponding power spectrum have been obtained by Eqs. (17) and (36), respectively. Intuitively, these equations contain variable parameters, such as, wavelength, receiver antenna radius, outer scale of the plasma irregularities, and spectral index. Both parameters play key roles on angle-of-arrival fluctuations.

5. Numerical results

In this section, angle-of-arrival fluctuations of optical wave propagation through the solar corona during superior solar conjunction are numerically discussed based on the theoretical derivations above. We first discuss the validity of Gaussian approximation. Then some simulations are conducted to analyze the impacts of the variable parameters (optical wavelength, antenna radius, solar wind turbulence outer scale, and spectral index) on both the variance of the angle-of-arrival fluctuations and its spatial power spectrum.

Physically, solar wind speed v and the solar wind density fluctuations coefficient a₀ in the generalized solar corona should take different values for different SEP angles and should depend on the measurement results. However, with the limitations imposed by the measurements, it is very hard to achieve these accurate data. In the following calculation, they are set to be 300km/s and 8.75×10⁷⁴ correspondingly according to the observation data reported by [15] and the assumed data by [20].

5.1. Gaussian approximation of the Airy function

In this section, we first discuss the discrepancy between the Gaussian function and the Airy function. As shown in Fig. 2, the vertical coordinate in the left side denotes the value of Airy function and the Gaussian approximation function, the discrepancy between these two functions is shown in the right side of the vertical coordinate. Obviously, both the airy function and the Gaussian approximation function have the same tendency and decrease fast from 1 to 0 with the increasing of a_r κ. Even though airy function drops to its minimum at a_r κ is 3.826 and 7.021, respectively, and increases to the maximum at a_r κ is 5.131, the largest discrepancy appears in these points is only 0.036. Apart from these points, the Gaussian function agrees well with the Airy function. Therefore, we summarize that the Gaussian approximation function also plays an antenna cutoff role and is suitable to be employed in the calculation of angle-of-arrival fluctuations.

5.2. Effect of SEP angle and SPE angle on angle-of-arrival fluctuations

Since the angle-of-arrival fluctuations also depends on the link distance between the Earth and the probe as shown in Eq. (17), the impact of SEP angle and SPE angle on angle-of-arrival fluctuations is evaluated in this subsection. Fig. 3 depicts the normalized angle-of-arrival fluctuations with different SEP angles and SPE angles. Note that the color bar on the right side of Fig. 3 denotes the normalized angle-of-arrival fluctuations. We fix λ=1064nm and l_o=2×10⁶m. Without loss of generality, the conventional Kolmogorov spectral index, $p = \frac{11}{3}$ , is chosen here.

As shown above, variable angles bring diverse effects on the angle-of-arrival fluctuations. With the increase of SEP angle and SPE angle, the coronal turbulence induces less effect on angle-of-arrival fluctuations. This is reasonable since the effect of the solar wind irregularities decrease with the SEP angle and SPE angle becoming larger, therefore, the solar wind irregularities produce less effect on the optical wave propagation. Besides, link distance that is jointly determined by the SEP angle and SPE angle will get larger when the angle becomes small. As a result, the optical waves suffer more serious effect on the angle-of-arrival fluctuations. Furthermore, the angle-of-arrival fluctuations decrease more obviously with the SEP angle ranging from 0.26 degree to 0.36 degree than the variation of SPE angle. Therefore, we set SPE angle at a constant and SEP angle is in the range of 0.26 to 1.32 degree in the following study.

Fig. 2 Function value and the discrepancy between the Gaussian function and the Airy function.

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Fig. 3 Normalized angle-of-arrival fluctuations dependence on SEP angle and SPE angle.

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5.3. Effect of outer scale and spectral index on angle-of-arrival fluctuations

We further estimate the effect of both solar wind turbulence outer scale and the spectral index on the variance of angle-of-arrival fluctuations. According to the astronomical observation [21], the spectral index fluctuated in the range of 3 to 4 under different distance from the Sun and cannot just be treated as Kolmogorov spectral index. Therefore, the spectral index in this subsection is in the range of 3 to 4. On the basis of observation, the outer scale is in the range of 1×10⁴m to 1×10⁷m. The optical wavelength is 1064nm, antenna radius is 1m. SPE angle is 0.5 degree. From Fig. 4, it is clear that with the increase in spectral index, the angle-of-arrival fluctuations decrease rapidly. This trend also can be seen with the increase of the outer scale. This result can be further interpreted from the derived variance of angle-of-arrival fluctuations in equation (17).

In addition, Fig. 5 illustrates the impacts of solar wind turbulent outer scale and spectral index on the spatial power spectrum of angle-of-arrival fluctuations, respectively. It can be seen from both Fig. 5(a) and 5(b) that, the power spectrum of the angle-of-arrival fluctuations decrease with the increase of frequency. As shown, the power spectrum of angle-of-arrival fluctuations also decrease with the rise of outer scale for the constant frequency. It is interesting that the spectrum also decrease with the increasing of the spectral index before 0.7Hz, while reversed at larger frequency.

Fig. 4 Angle-of-arrival fluctuations as a function of spectral index and outer scale.

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Fig. 5 The spatial power of angle-of-arrival fluctuations scaled by the corresponding variance in Fig. 4 as a function of the frequency for several outer scales and spectral indexes.

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5.4. Effect of optical wavelength on angle-of-arrival fluctuation

In this subsection, we evaluate the effect of optical wavelength on both the variance and the spatial power spectrum of angle-of-arrival fluctuations. Here, we take the antenna radius as a_r=1m and fix the outer scale as l_o=1×10⁷m. Unless otherwise specified, the spectral index is setting to be as p=3.8. The optical wavelength, λ, is setting as 1550nm, 1064nm, 850nm, and 632nm, respectively. Simulation results are depicted in Fig. 6. Expectedly, both the angle-of-arrival fluctuations and its correponding power spectrum are sensitive to the optical wavelength. Increasing wavelength results in larger fluctuations value, and it is obvious results in the value increase in its power spectrum. This is in agreement with previous research results for electromagnetic wave propagation in solar corona plasma [20]. Both of these results can be explained directly from the derived angle-of-arrival fluctuations in Eqs. (18) and (36) due to the approximate relationship of 〈θ〉 ∼ λ² and W_θ (κ, p) ∼ λ². Therefore, employing higher frequency is an efficient way to minimize the angle-of-arrival fluctuations due to the solar corona turbulence during the future deep space optical communication.

Fig. 6 Angle-of-arrival fluctuations (a) and its power spectrum (b) dependence on optical wavelength.

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Fig. 7 Angle-of-arrival fluctuations (a) and its power spectrum (b) dependence on antenna radius.

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5.5. Effect of antenna radius on angle-of-arrival fluctuations

To analyze antenna radius’s influence on the variance of angle-of-arrival fluctuations and its power spectrum, the solar corona turbulence and the wavelength are fixed to constant values of l_o=1×10⁷m, λ=1064nm. According to the deep space communication design, the antenna radius is setting as 1m, 4m, and 10m, respectively [22].

We can see that variable antenna radii give rise to different effects on the variance of angle-of-arrival fluctuations. With the increase of the antenna radius, the variance of angle-of-arrival fluctuations decreases obviously. The small wave front wanderings caused by the solar wind irregularities will be averaged out by the antenna smoothing process. Therefore, the larger antenna radius generates smaller angle-of-arrival fluctuations due to the aperture average effect. Apart from this, the power spectrum of angle-of-arrival fluctuations also decreases with the increasing of the antenna radius as it has been revealed in Fig. 7(b). On the other hand, the angle-of-arrival power spectrum becomes obviously small along with decrease of frequency. According to these discussions, we can conclude that another way to mitigate the effect of angle-of-arrival fluctuations caused by the solar corona irregularities during superior solar conjunction is to enlarge the received antenna radius.

From Fig. 6(a) and Fig. 7(a), the variance of angle-of-arrival fluctuations gets smaller when the SEP angle becomes larger. This result further verifies the conclusion that is summarized in Fig. 3.

6. Conclusion

In this paper, both the angle-of-arrival fluctuations and its corresponding power spectrum have been obtained for optical waves propagation in solar corona turbulence using a coronal spectrum model with the generalized spectral index. All of the developed expressions are analytical and are used to analyze the effect of non-Kolmogorov spectral index, turbulent outer scale, wavelength, and antenna radius on angle-of-arrival fluctuations.

Numerical results revealed that the angle-of-arrival fluctuations as a function of SEP angle and SPE angle decrease with the rising of these angles. Apart from this, the angle-of-arrival fluctuations decrease with the increase of the spectral index. In addition, the turbulence outer scale results in obvious impact on angle-of-arrival fluctuations and its power spectrum. These phenomenons are consistent with the results as the optical wave propagation in atmospheric turbulence. Also, calculation indicated that strong averaging effects are produced with the increasing of the antenna radius, and it will alleviate the value of angle-of-arrival fluctuations. Moreover, the influence of the wavelength was also taken into consideration in the proposed angle-of-arrival fluctuations expressions and demonstrated that the fluctuations decrease along the decrease of the wavelength. Finally, it should be noteworthy that these results will contribute to the investigations of optical wave propagation in coronal turbulence during superior solar conjunction.

Funding

Shandong Space Innovation Fund (No. 2013JJ008).

References and links

1. H. Hemmati, A. Biswas, and I. B. Djordjevic, “Deep-Space Optical Communications: Future Perspectives and Applications,” Proc. IEEE 99(11), 2020–2039 (2011). [CrossRef]

2. A. J. Hashmi, A. A. Eftekhar, A. Adibi, and F. Amoozegar, “Analysis of telescope array receivers for deep-space inter-planetary optical communication link between Earth and Mars,” Opt. Commun. 283(10), 2032–2042 (2010). [CrossRef]

3. G. Xu and Z. Song, “A New Model of Amplitude Fluctuations for Radio Propagation in Solar Corona during Superior Solar Conjunction,” Radio Sci. 51(2), 2015RS005769 (2016). [CrossRef]

4. 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]

5. S. S. Pawar and J. H. Arakeri, “Intensity and angle-of-arrival spectra of laser light propagating through axially homogeneous buoyancy-driven turbulence,” Appl. Opt. 55(22), 5945–5952 (2016). [CrossRef] [PubMed]

6. E. M. Razi and S. Rasouli, “Investigation of inhomogeneity and anisotropy in near ground layers of atmospheric air turbulence using image motion monitoring method,” Opt. Commun. 383, 255–259 (2017). [CrossRef]

7. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011). [CrossRef] [PubMed]

8. M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015). [CrossRef] [PubMed]

9. W. Du, L. Tan, J. Ma, and Y. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(6), 5763–5775 (2010). [CrossRef] [PubMed]

10. Z. Xu, J. Wu, and Z. Wu, “A survey of ionospheric effects on space-based radar,” Waves in Random Media 14(2), S189–S274 (2004). [CrossRef]

11. A. D. Wheelon, Electromagnetic scintillation. Part I. Geometrical optics (Cambridge University, 2001). [CrossRef]

12. A. I. Efimov, L. Samoznaev, M. Bird, I. Chashei, and D. Plettemeier, “Solar wind turbulence during the solar cycle deduced from Galileo coronal radio-sounding experiments,” Adv. Space Res. 42(1), 117–123 (2008). [CrossRef]

13. G. Jandieri, A. Ishimaru, K. Yasumoto, A. Khantadze, and V. Jandieri, “Angle-of-arrival of radio waves scattered by turbulent collisional magnetized plasma layer,” International J. Microw. Opt. Technol. 4, 160–169 (2009).

14. E. D. Tereshchenko, M. T. Rietveld, A. Brekke, B. Khudukon, B. Isham, and T. Hagfors, “The relationship between small-scale and large-scale ionospheric electron density irregularities generated by powerful HF electromagnetic waves at high latitudes,” Annales Geophysicae 24(11), 2901–2909 (2006). [CrossRef]

15. P. K. Manoharan, “Three-dimensional evolution of solar wind during solar cycles 22–24,” Astrophys. J. 751(2), 128 (2012). [CrossRef]

16. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009). [CrossRef]

17. L. C. Andrews, Special functions of mathematics for engineer (SPIEOptical Engineering, 1998).

18. K. C. Yeh and C.-H. Liu, “Radio wave scintillations in the ionosphere,” Proc. IEEE 70(4), 324–360 (1982). [CrossRef]

19. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

20. C. Ho, D. Morabito, and R. Woo, “Solar corona effects on angle of arrival fluctuations for microwave telecommunication links during superior solar conjunction,” Radio Sci. 43(2), 627–646 (2008). [CrossRef]

21. G. M. Calvés, S. Pogrebenko, G. Cimò, D. Duev, T. Bocanegra-Bahamón, J. Wagner, J. Kallunki, P. de Vicente, G. Kronschnabl, and R. Haas, “Observations and analysis of phase scintillation of spacecraft signal on the interplanetary plasma,” Astron. Astrophys. 564(6), 275–291 (2014).

22. P. Brandl, T. Plank, and E. Leitgeb, “Optical wireless links in future space communications with high data rate demands,” in Proceedings of International Workshop on Satellite and Space Communications, (Academic Press, 2009), pp. 305–309.

23. L. Cui, “Analysis of angle of arrival fluctuations for optical waves propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015) [CrossRef] [PubMed]

24. R. Conan, J. Borgnino, A. Ziad, and F. Martin, “Analytical solution for the covariance and for the decorrelation time of the angle of arrival of a wave front corrugated by atmospheric turbulence,” J. Opt. Soc. Am. A 17(10), 1807–1818, 2000. [CrossRef]

Theoretical analysis of the angle-of-arrival fluctuations for optical wavext propagation through solar corona turbulence

Abstract

1. Introduction

2. Background

2.1. Deep space optical model during superior solar conjunction

2.2. Generalized spectrum model

3. Angle-of-arrival fluctuations variance and spectrum

3.1. Variance of angle-of-arrival fluctuations

4. Power spectrum of angle-of-arrival fluctuations

5. Numerical results

5.1. Gaussian approximation of the Airy function

5.2. Effect of SEP angle and SPE angle on angle-of-arrival fluctuations

5.3. Effect of outer scale and spectral index on angle-of-arrival fluctuations

5.4. Effect of optical wavelength on angle-of-arrival fluctuation

5.5. Effect of antenna radius on angle-of-arrival fluctuations

6. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (36)

Optics Express