Experimental verification of fiber coupling characteristics for FSO downlinks from the International Space Station

Phuc V. Trinh; Dimitar R. Kolev; Koichi Shiratama; Alberto Carrasco-Casado; Yasushi Munemasa; Yasushi Munemasa; Hiroaki Yamazoe; Hiromitsu Komatsu; Toshiaki Kamata; Takashi Nakao; Shinji Ohta; Kyohei Iwamoto; Mikio Fujiwara; Hiroyuki Tsuji; Morio Toyoshima

doi:10.1364/OE.484512

1. Introduction

In the development towards the sixth-generation (6G) era, high-capacity connections with ubiquitous coverage services are required to enable augmented reality (AR) and virtual reality (VR), autonomous driving, and telemedicine, to name a few [1]. Unfortunately, terrestrial infrastructures are not sufficient to catch up with these growing demands due to the limited resources in difficult terrains. Benefited from the worldwide coverage and high visibility, the satellite constellation networks are instrumental to make 6G services available ubiquitously and accessible to citizens all over the world [2]. In fact, many mega-constellation programs have been planned recently to deliver global communication and sensing services by Non-GeoStationary Orbit (NGSO) satellite systems [3], i.e. satellites operating in low-Earth orbit (LEO) and medium-Earth orbit (MEO). Simultaneously, radio frequency (RF) bands are becoming exhausted and suffering from severe interference issues. This is where free-space optical (FSO) systems come into play, offering large capacity as well as the immunity to interference while requiring no international frequency coordination. Using both satellite-based FSO and terrestrial optical fiber links, all-optical seamless connections at high data rates are achievable, forming an integrated space-ground backbone network. Furthermore, space FSO links are also attractive for reliable and wide-band communications from spacecraft and space stations to the Earth, such as the Laser-Enhanced Mission and Navigation Operational Services (LEMNOS) for use on Orion spacecraft [4], and Small Optical Link for International Space Station (SOLISS) for establishing a bidirectional laser communication link between the International Space Station (ISS) and the optical ground station (OGS) of NICT [5–7].

At the OGS, it is important to compensate for the severe channel loss over the atmosphere and geometrical loss when receiving a widely broadened beam through a limited-aperture telescope by using a low-noise optical amplifier (LNA), e.g. erbium-doped fiber amplifier (EDFA). This apparently requires coupling the free-space optical beam into a single-mode fiber (SMF). Nevertheless, this could possibly result in a high coupling loss (10$\sim$19 dB) for a LEO-to-ground FSO downlink [8]. This is because the SMF has a very small core diameter (typically 9 $\mu$m) that is susceptible to angle-of-arrival (AoA) fluctuations of the laser beam observed at the focal plane of the optical receiving system. In addition, atmospheric turbulence and misalignment errors excite several high-order spatial modes that cannot be coupled into the SMF as it only supports the fundamental Gaussian mode. Fortunately, recent advancements in mode multiplexing technologies, such as photonic lanterns [9,10] and multi-plane light conversion (MPLC) with coherent optical combination [11,12], have enabled the use of a multi-mode fiber (MMF) for free-space-to-fiber coupling before combining all coupled modes into a single SMF for further signal processing. The MMF can considerably increase the achievable coupling efficiency (CE) since it has a much larger core diameter compared to the SMF, which makes it more resilient to AoA fluctuations while concurrently supporting various high-order spatial modes.

To evaluate the performance of communication systems, the average CE is often used to evaluate the signal-to-noise ratio (SNR) and bit-error rate (BER) metrics [13]. However, it was shown by experimental data that the CE randomly fluctuates within a wide range around its mean [14] and the SNR and BER are nonlinear to the CE [15]. As a result, it is inaccurate to estimate link performance metrics by the average CE, since the collected signal may still experience strong fluctuations even with a high average CE. Therefore, it is of critical importance to determine the probability density function (PDF) of the CE for the estimation of the average SNR and BER performance [16]. In the literature, only a few studies developed the analytical PDF of the SMF CE and verified with experimental data in the laboratory [16–18]. It was found that the Gamma PDF is the best fit for the SMF CE considering the modified von Karman turbulence spectrum [17], while the Rician PDF governs the randomness of the SMF CE under non-Kolmogorov turbulence [16,18]. Although the average CE into an MMF has been experimentally reported for an 8-km horizontal FSO link [19], the CE PDF was not studied. To the best of authors’ knowledge, such PDF verification has never been performed for any actual FSO links coupled into an MMF.

Motivated by the above-mentioned fact and the potential use of MMF in mode multiplexing technologies, in this paper, we report the statistical verification study of the MMF CE using experimental data from an FSO link connecting the ISS in LEO and NICT’s OGS in Tokyo, Japan. The onboard optical communication terminal, namely SOLISS, was jointly developed by Sony Computer Science Laboratories (Sony CSL) and Japan Aerospace Exploration Agency (JAXA), and installed on the IVA-replaceable small exposed experiment platform (i-SEEP) attached to the exposed facility of "KIBO" Japanese experiment module of the ISS [5–7]. To assist the fiber coupling process at the receiving end, both SOLISS and the OGS are supported by a closed-loop fine-tracking system consisting of a quadrant detector (QD) and a fast-steering mirror (FSM) to correct the AoA fluctuations due to pointing and tracking errors, i.e. misalignment errors. Our contributions are therefore threefold. Firstly, the world’s first successful FSO downlink from the ISS to an OGS coupled into a 200-$\mu$m MMF is reported. Secondly, using the AoA and received power data at the OGS, statistical characteristics such as channel coherence time, power spectral density, spectrogram, and the PDFs of AoA, beam misalignments, and atmospheric turbulence-induced fluctuations are verified and compared with the state-of-the-art theoretical background. Finally, we, for the first time, empirically show that the random fluctuations of the MMF CE could be accurately characterized by a Gamma PDF. These results are considered as useful references for the practical design of communication modems and the performance evaluation of future high-speed FSO links from space.

2. Theoretical background

2.1 Composite PDF for channel modeling

2.1.1 Beckmann PDF characterizing AoA fluctuations

The random AoA displacements of the optical link in both horizontal $x$ and vertical $y$ axes are commonly modeled as independent Gaussian random variables (RVs) with means $\{\mu _x,\mu _y\}$ and variances $\{\sigma _x,\sigma _y\}$, respectively [20]. In the most general case, AoA displacements in both axes are non-zero-mean Gaussian RVs with different angle jitters, i.e. $(\mu _x\neq \mu _y,\sigma _x\neq \sigma _y)$, thus a total radial AoA, denoted as $\epsilon$, follows a four-parameter Beckmann distribution, written as [21]

(1)$$f_{\epsilon}\left ( \epsilon \right )=\frac{\epsilon}{2\pi\sigma _{x}\sigma _{y}}\int_{0}^{2\pi}\exp\left ( -\frac{\left (\epsilon\cos\left ( \theta \right )-\mu _{x} \right )^{2}}{2\sigma _{x}^{2}}-\frac{\left (\epsilon\sin\left ( \theta \right )-\mu _{y} \right )^{2}}{2\sigma _{y}^{2}}\right )\textrm{d}\theta.$$

It is noted that the AoA displacements in the optical link can be caused by the atmospheric turbulence, platform vibrations, and telescope pointing inaccuracy in both transmitter (i.e. SOLISS) and receiver (i.e. NICT’s OGS) [20]. The AoA misalignments can result in interruptions of the coupled power into the fiber tip with a limited field of view (FoV). The corresponding loss due to the AoA misaligments, denoted as $h_\text {AoA}$, can be expressed as [22]

(2)$$h_\text{AoA}=\left\{\begin{matrix} 1\,\;\;\;\text{if}\,\;\;\theta _\text{AoA}\leq\theta _\text{FoV} \\ 0\,\;\;\;\text{if}\,\;\;\theta _\text{AoA}>\theta _\text{FoV} \end{matrix}\right., $$

where $\theta _\text {AoA}$ and $\theta _\text {FoV}$ represent the instantaneous AoA and the half-angle FoV of the fiber tip, respectively. Eq. (2) holds if the small power contributions from the side lobes in the Airy pattern of the focused beam spot are ignored. Since our OGS is equipped with a closed-loop fine-tracking system, the received optical beam is always stabilized within the FoV of the optical fiber as long as $\theta _\text {AoA}$ is smaller than $\theta _\text {FoV}$ of the fine-tracking sensor, i.e. the QD. As detailed in Section 4, the measured AoA values during the experiment are always within the QD’s $\theta _\text {FoV}$, leading to $h_\text {AoA}=1$ in our analysis.

2.1.2 Approximated Beckmann PDF characterizing beam misaligments

Similar to the AoA displacements, the optical beam received at the OGS is also randomly displaced from the aligned position with the center of the receiving telescope. This causes random fractions of power collected by the telescope over time. The radial beam displacement, denoted as $r$, can also be modeled by a four-parameter Beckmann distribution for the most general case when the displacements in both horizontal $x$ and vertical $y$ axes are modeled as independent Gaussian RVs with non-zero means $\{\mu _{x,r},\mu _{y,r}\}$ and different beam jitters $\{\sigma _{x,r},\sigma _{y,r}\}$. The PDF of $r$ follows the same mathematical form as in Eq. (1), rewritten as [21]

(3)$$f_{r}\left (r \right )=\frac{r}{2\pi\sigma _{x,r}\sigma _{y,r}}\int_{0}^{2\pi}\exp\left ( -\frac{\left (r\cos\left ( \theta \right )-\mu _{x,r} \right )^{2}}{2\sigma _{x,r}^{2}}-\frac{\left (r\sin\left ( \theta \right )-\mu _{y,r} \right )^{2}}{2\sigma _{y,r}^{2}}\right )\textrm{d}\theta.$$

Since a closed-form solution for the integral in Eq. (3) is unknown, the Beckmann distribution could be approximated by a modified Rayleigh distribution, which is more tractable [23]. Now, Eq. (3) can be rewritten as [23]

(4)$$f_{r}\left ( r \right )\approx\frac{r}{\sigma _{\text{mod}}^{2}}\exp\left ( -\frac{r^{2}}{2\sigma _{\text{mod}}^{2}} \right ),$$

where $\sigma _\text {mod}=\left ( \frac {3\mu _{x,r}^{2}\sigma _{x,r}^{4}+3\mu _{y,r}^{2}\sigma _{y,r}^{4}+\sigma _{x,r}^{6}+\sigma _{y,r}^{6}}{2} \right )^{1/3}$ is the modified beam jitter variance approximation. The PDF characterizing the random fraction of power captured by the telescope under beam misalignments, denoted as $h_{\text {p}}$, can be expressed as [23]

(5)$$f_{h_{\text{p}} }\left ( h_{\text{p}} \right )=\frac{\varphi _{\text{mod}}^{2}}{A_{\text{mod}}^{\varphi _{\text{mod}}^{2}}}h _{\text{p}}^{\varphi _{\text{mod}}^{2}-1},$$

where $\varphi _\text {mod}=\frac {w_{L\text {eq}}}{2\sigma _\text {mod}}$ is the ratio between the equivalent beam-width $w_{L\text {eq}}$ and the modified radial beam-jitter variance $\sigma _\text {mod}$. $w_{L\text {eq}}$ is given as

(6)$$w_{L\text{eq}}^2=w_L^2\frac{\sqrt{\pi}\text{erf}\left(\vartheta \right)}{2\vartheta \exp\left( -\vartheta ^2\right)},$$

where $\textrm {erf}\!\left (x\right ) \!=\! \frac {2}{\sqrt {\pi }}\int _{0}^{x}\exp \!\left (- t^{2} \right )\textrm {d}t$ is the Gauss error function, $\vartheta =\frac {\sqrt {\pi }D}{2\sqrt {2}w_L}$ with $w_L$ the beam waist at a distance $L$, $D$ the telescope aperture diameter. $w_L=W\sqrt {1+T}$, where $W$ is the beam waist due to pure diffraction for a collimated beam and $T$ is the beam broadening coefficient due to turbulence. For downlink optical beams at high elevation angles, we have $T\approx 1$ and $w_L\approx W$, which means that the turbulence-induced beam broadening effect is negligible [24]. $A_\text {mod}=A_{0}G$ where $G=\exp \left ( \frac {1}{\varphi _\text {mod}^{2}}-\frac {1}{2\varphi _{x}^{2}}-\frac {1}{2\varphi _{y}^{2}}-\frac {\mu _{x,r}^2}{2\sigma _{x,r}^{2}\varphi _{x}^{2}} -\frac {\mu _{y,r}^2}{2\sigma _{y,r}^{2}\varphi _{y}^{2}} \right )$, and $A_0=\left [ \text {erf} \left ( \vartheta \right )\right ]^2$, which is the maximum fraction of collected power over the receiving aperture when there are no misalignments. $\varphi _{x}=\frac {w_{L\text {eq}}}{2\sigma _{x,r}}$ and $\varphi _{y}=\frac {w_{L\text {eq}}}{2\sigma _{y,r}}$ are the ratios between the equivalent beam-width $w_{L\text {eq}}$ and the beam-jitter variances for the horizontal $x$ and vertical $y$ axes, respectively. It is noted that $\varphi _\text {mod}$ in Eq. (5) is also an indicator of the strength of beam misalignment-induced fluctuations, with $1\leftarrow \varphi _\text {mod}$ and $\varphi _\text {mod}\to \infty$ indicating strong and negligible misalignments, respectively.

2.1.3 LN PDF characterizing atmospheric turbulence

For a downlink beam at high elevation angles, the turbulence-induced signal fluctuations are typically weak and characterized by the LN distribution [24]. Thus, the PDF of a turbulence channel state $h_{\text {t}}$ that follows the LN distribution can be expressed as [25]

(7)$$f_{ h_{\text{t}}}\left ( h_{\text{t}} \right )=\frac{1}{h_{\text{t}}\sqrt{2\pi\sigma _{\text{R}}^{2}}}\exp\left ( -\frac{\left ( \ln\left (h_{\text{t}} \right )+\frac{\sigma _{\text{R}}^{2}}{2} \right )^2}{2\sigma _{\text{R}}^{2}} \right ),$$

where $\sigma _{\text {R}}^{2}$ denotes the Rytov variance for the downlink path over the atmosphere, readily given as [24]

(8)$$\sigma _{\text{R}}^{2}=2.25\!\left (\! \frac{2\pi}{\lambda } \!\right )^{\!7/6}\!\sec^{\!11/6}\!\!\left ( \xi \right )\!\!\int_{H_{\text{OGS}}}^{H_{\text{atm}}}\!C_{n}^{2}\!\left ( h \right )\!\left ( h- H_{\text{OGS}}\right )^{5/6}\!\text{d}h,$$

where $\xi$ denotes the zenith angle, $\lambda$ is the wavelength of the optical beam, $H_{\text {OGS}}$ is the altitude of the OGS above sea level, $H_{\text {atm}}$ is the maximum altitude where the atmosphere exists, $C_{\text {n}}^{2}\!\left ( h \right )$ is the altitude-dependent refractive index structure parameter profile. It is important to emphasize that when the size of the OGS telescope is larger than the transverse correlation width, the aperture-averaging effect can occur, where the scintillation index (SI) quantifying the turbulence strength is reducing with increasing aperture sizes [24]. The SI is defined as $\sigma _\text {SI}^2 \overset {\Delta }= (\mathbb {E}\big [ h_{\text {t}} ^2\big ]/\mathbb {E}\big [ h_{\text {t}} \big ]^2)-1$, where $\mathbb {E}[\cdot ]$ represents the statistical expectation. As described later in Section 3, our considered downlink propagation path is established at 50-degree elevation angle, thus the transverse correlation width is on the order of 7$\sim$10 cm [24]. Our 40-cm sub-aperture telescope at the OGS infrastructure is hence significantly larger than the transverse correlation width, indicating the occurrence of aperture-averaging effect. Conveniently, the SI for a downlink with aperture-averaging effect, which is now equivalent to the Rytov variance $\sigma _{\text {R}}^{2}$, is readily given as [24]

(9)$$\begin{aligned} \sigma _{SI}^{2}\left ( D\right )\approx\sigma _\text{R}^{2}=&8.7 k^{7/6}\left ( \!H_{\text{atm}}-H_{\text{OGS}} \right )^{5/6}\sec^{11/6}\left ( \xi \right )\\ &\times\text{Re}\int_{H_{\text{OGS}}}^{H_{\text{atm}}}C_{\text{n}}^{2}\left ( h \right )\left [ \left ( \frac{kD^2}{16L}+i\frac{h-H_{\text{OGS}}}{H_{\text{atm}}-H_{\text{OGS}}} \right )^{5/6}-\left (\frac{kD^2}{16L} \right )^{5/6} \right ]\text{d}h, \end{aligned}$$

where $C_{\text {n}}^{2}\left ( h \right )$ is defined in Eq. (8) and calculated from the modified Hufnagel-Valley model for our OGS site [26]. For weak turbulence conditions, the SI value is much less than unity, i.e. $\sigma _{SI}^{2}\left ( D\right )\ll 1$.

2.1.4 Composite channel PDF: LN and approximated Beckmann model

The composite channel coefficient, denoted as $h_c$, can be written as $h_c=h_\text {l}h_\text {AoA}h_\text {p}h_\text {t}$, where $h_\text {l}$ denotes the deterministic channel loss. Since $h_\text {AoA}=1$ in our analysis, we then have $h_c=h_\text {l}h_\text {p}h_\text {t}$, given that $h_\text {p}$ and $h_\text {t}$ are statistically independent. With the help of Eqs. (5) and (7) and performing a normalization to the statistical mean of the composite channel fluctuations for fitting with the histogram of the normalized received power data, the LN & approximated Beckmann composite PDF for the downlink can be finally written as [27]

(10)$$f\left ( h_\text{c} \right )=\frac{\varphi_\text{mod}^{2}}{2\left ( \Psi\right )^{\varphi_\text{mod}^{2}}}h_\text{c}^{\varphi_\text{mod} ^{2}-1}\text{erfc}\left ( \frac{\text{ln}\left (\frac{h_\text{c}}{\Psi} \right )+\varpi}{\sqrt{2\sigma _{\text{SI}}^{2}\left ( D\right )}} \right ) \exp\left (\frac{\sigma _{\text{SI}}^{2}\left ( D\right )}{2}\varphi_\text{mod}^{2}\left ( 1+\varphi_\text{mod} ^{2} \right )\right ),$$

where $\varpi =\frac {\sigma _{\text {SI}}^{2}\left ( D\right )}{2}\left ( 1+2\varphi _\text {mod} ^{2} \right )$. In Eq. (10), the fitting parameters with histogram data include $\varphi _\text {mod}$, $\sigma _{\text {SI}}^{2}\left ( D\right )$, and $\Psi$. It is noted that $\Psi$ is fitted to the histogram data as an estimation of the ratio $A_\text {mod}/\mathbb {E}\left [ h_{\text {p}} \right ]$, where $A_\text {mod}$ is defined in Eq. (5) and $\mathbb {E}\left [ h_{\text {p}} \right ]$ that defines the statistical mean of $h_\text {p}$ is given as [21]

(11)$$\mathbb{E}\left [ h_{\text{p}} \right ]=\frac{A_0\varphi_{x}\varphi_{y}}{\sqrt{\left ( 1+\varphi_{x}^{2} \right )\left ( 1+\varphi_{y}^{2} \right )}}\exp\left ( - \frac{2}{w_{L\text{eq}}^2}\left [ \frac{\mu _{x,r}^{2}}{1+\frac{1}{\varphi_{x}^{2}}}+ \frac{\mu _{y,r}^{2}}{1+\frac{1}{\varphi_{y}^{2}}}\right ]\right ).$$

2.2 Time-frequency channel characteristics

2.2.1 Channel coherence time

For a high-capacity FSO communication link, an outage event caused by atmospheric turbulence, which often lasts for several milliseconds (ms), i.e. equivalent to channel coherence time, will cause a burst of packet erasures, also known as burst errors. Therefore, when designing a communication modem, the use of an interleaver is necessary to change the order of transmitted data so that the consecutive data bytes can be re-distributed across various codewords, thereby decreasing the chances of burst errors occurring within the data. As a matter of fact, knowledge on the channel coherence time is very critical for the optimization design of the forward error-correction code (FEC) codeword length and interleaving depth. The channel coherence time, denoted as $T_\text {c}$, can be directly calculated from the normalized auto-covariance function (ACF) of the received power data, denoted as $\hat {\varrho }$, and determined at $1/e$ of the peak of the ACF at zero lag, written as $\hat {\varrho }\left ( T_\text {c} \right )=\frac {1}{e}$ [28], where $\hat {\varrho }\left ( m \right )\!=\!\frac {\varrho \left ( m \right )}{\varrho \left ( 0 \right )}$ with

(12)$$\begin{aligned} \varrho\!\left ( m \right )\!=\!\!\left\{\begin{matrix} \sum_{n=1}^{M-m}\hat{P}_{\text{Rx},n+m}\hat{P}_{\text{Rx},n}, & \!\!\!\!\!\!\,M\!-\!1\geq m\geq 0\\ \varrho\left ({-}m \right ), & \!\!\!\!\!\!\,-\!M\!+\!1\leq m< 0 \end{matrix}\right., \end{aligned}$$

being the ACF. In Eq. (12), $\hat {P}_{\text {Rx},n}=P_{\text {Rx},n}-\frac {1}{M}\sum _{i=1}^{M}P_{\text {Rx},i}$ where $P_{\text {Rx},i}$ is the $i$th-sample of the total $M$ samples of the received power data.

2.2.2 Power spectral density

To investigate the temporal statistics of the received power fluctuations, the power spectral density (PSD) is an important figure of merit. Based on Taylor’s fronzen turbulence hypothesis that allows to convert spatial statistics into temporal statistics [24] and assuming point receivers, the PSD of an optical signal is defined by the Fourier transform of the temporal covariance function, expressed as

(13)$$S\!\left (\omega \right )\!\!=\!2\!\int_{\!-\infty }^{\infty }\!\!B\!\left ( \tau,\!L \right )\!\exp\!\left (\! -i\omega \tau \right )\!\text{d}\tau \!\!=\!4\!\int_{\!0}^{\infty }\!\!\!B\!\left ( \tau,\!L \right )\!\cos\!\left ( \omega \tau \right )\!\text{d}\tau,$$

where $\omega$ denotes the angular frequency and $B\left ( \tau,L \right )$ represents the temporal covariance function [24]. There also exists a cut-off frequency, denoted as $f_c$, that divides the spectrum into two parts, where the part below $f_c$ has a relatively flat PSD and the part above $f_c$ decays following a power-law exponent $\kappa =8/3$. $f_c$ can be related to the channel coherence time $T_\text {c}$ as $f_c\!\!=\!\!\sqrt {2\pi }/T_\text {c}$. It should be noted that typical receivers for space-to-ground FSO links have large aperture sizes, i.e. non-point receivers, thus there should be a slight deviation in the power-law exponent of the experimental spectrum compared to the theoretical one. From the received power data, the PSD can be conveniently estimated by its periodogram, i.e. the Fourier transform of the biased estimate of the autocorrelation sequence, defined as [29]

(14)$$\hat{P}\!\left ( f \right )\!=\!\frac{\Delta t}{M}\!\left | \sum_{n=0}^{M-1}P_{\text{Rx},n}\!\exp\!\left({-}j2\pi f\Delta t n\right) \right |^{2},$$

where $-1/2\Delta t<f\leq 1/2\Delta t$ and $\Delta t$ is the sampling interval.

2.2.3 Spectrogram

For a general view of how the spectrum of frequencies of a signal varies with time, the spectrogram is often used, which is defined as an intensity plot of the short-time Fourier transform (STFT) magnitude. The STFT is a sequence of fast Fourier transforms of windowed data segments, where the windows are overlapped in time [29]. Given that the received power data $P_{\text {Rx}}$ is a discrete time signal with $M$ samples and a window size $m$, let $\textbf {S}\in \mathbb {R}^{m\times \left ( M-m+1 \right )}$ be the matrix with the consecutive segments as consecutive columns, i.e. $\left [P_{\text {Rx}}\left [ 0 \right ] ,P_{\text {Rx}}\left [ 1 \right ],\ldots,P_{\text {Rx}}\left [ m-1 \right ] \right ]^{T}$ is the first column, and $\left [P_{\text {Rx}}\left [ 1 \right ] ,P_{\text {Rx}}\left [ 2 \right ],\ldots,P_{\text {Rx}}\left [ m \right ] \right ]^{T}$ is the second column, and so forth, with the rows and columns indexed by time. The spectrogram of $P_{\text {Rx}}$ is then defined as the matrix $\mathbf {\hat {S}}$ whose columns are the discrete Fourier transform of the columns of $\textbf {S}$, given as $\mathbf {\hat {S}}\!=\!\mathbf {\bar {F}}\mathbf {S}$, where $\mathbf {\bar {F}}$ is the complex conjugate of the Fourier matrix $\mathbf {F}$, with $\mathbf {F}$ given as [29]

(15)$$\begin{aligned} \mathbf{F}=&\begin{bmatrix} 1 & \!\!\!\! 1 & \!\!\!\! 1 & \!\!\!\! \cdots & \!\!\!\!1 \\ 1 & \!\!\!\! e^{\left ( j\frac{2\pi }{M} \right )} & \!\!\!\!e^{\left ( j\frac{4\pi }{M} \right )} & \!\!\!\! \cdots & \!\!\!\!e^{\left ( j\frac{2\pi\left ( M-1 \right ) }{M} \right )} \\ 1 & \!\!\!\! e^{\left ( j\frac{4\pi }{M} \right )} & \!\!\!\! e^{\left ( j\frac{8\pi }{M} \right )} & \!\!\!\! \cdots & \!\!\!\!e^{\left ( j\frac{4\pi \left ( M-1 \right )}{M} \right )} \\ \vdots & \!\!\!\! \vdots & \!\!\!\! \vdots & \!\!\!\! \ddots & \!\!\!\! \vdots \\ 1 & \!\!e^{\left ( j\frac{2\pi\left ( M-1 \right ) }{M} \right )} & \!\!\!\! e^{\left ( j\frac{4\pi \left ( M-1 \right )}{M} \right )} & \!\!\!\! \cdots & \!\!\!\! e^{\left ( j\frac{2\pi \left ( M-1 \right )^{2}}{M} \right )} \end{bmatrix}. \end{aligned}$$

2.3 Theoretical PDFs under consideration for MMF CE

A list of well-known PDFs under consideration for characterizing the random distribution of the MMF CE is summarized in Supplement 1 for the sake of conciseness. The best fitted PDF model can be considered as an empirical PDF that provides a good approximation to the statistical behavior of the MMF CE. Further investigations based on solid theoretical developments to derive an analytical PDF model for the MMF CE, taking into account all channel effects, should be addressed in the future.

2.4 Goodness-of-fit (GoF) evaluation metric

The GoF evaluation metric, denoted as $R^2$, is a statistical metric popularly used to examine a probability distribution’s fitness, mathematically defined as

(16)$$R^2 = 1 - \frac{\sum_{i = 1}^N\left(f_{\text{m}, i} - f_{p, i}\right)^2}{\sum_{i =1}^N\left(f_{\text{m}, i} - \bar{f}\right)^2},$$

where $N$ denotes the number of bins of the histogram data, $f_{m, i}$ and $f_{p, i}$ are respectively the measured and predicted probability values corresponding to the $i$th-bin, and $\bar {f} = \frac {\sum _{i = 1}^Nf_{\text {m}, i}}{N}$ is the mean of the measured data. If $R^2\rightarrow 1$, it indicates that the probability distribution better fits the measured data and vice versa. We will use this GoF metric to evaluate the validity of the examined PDFs throughout this paper.

3. Experiment descriptions

3.1 SOLISS configurations

Over the past few years, NICT has been developing miniaturized terminals that could be installed in very small satellites, e.g. CubeSats, while also compatible with a variety of other different platforms such as drones and high-altitude platform stations (HAPS) for applications in the future non-terrestrial networks [30]. Similarly, Sony CSL and JAXA have jointly developed SOLISS for micro-satellites and tested the terminal in orbit using the i-SEEP attached to the Japanese experiment module on the ISS as shown in Fig. 1(a). The flight model of SOLISS is depicted in Fig. 1(b), consisting of an optical communication unit (OCU) installed on a bi-axial gimbal, control units (CU) 1 and 2, and a monitor camera. All components are placed on a temperature-controlled base plate of i-SEEP. The CU1 includes a system-bus control unit, a power supplying unit, and a communication interface to i-SEEP. CU2 is the control unit of the monitor camera, which is a 360-degree camera to monitor the bi-axial gimbal holding the 1.2-kg OCU. This gimbal has a wide range of motion with an azimuth angle range of 110 degrees and an elevation angle range of 120 degrees at a maximum speed of 2 degrees/second. The OCU is further detailed in Fig. 1(c), housing a 40-mm aperture for transmitting (Tx), a 26-mm aperture for receiving (Rx), and a 26-mm aperture for a wide-angle image sensor. The Tx and Rx signals could be separated by using different wavelengths (1550-nm and 1565-nm) and polarizations (RHCP and LHCP), respectively. An EDFA is installed inside the CU1 and connected to OCU through a polarization-maintaining optical fiber cable with FC/APC connector. The EDFA could provide a maximum transmitted power of 500 mW to close the link with the OGS.

Fig. 1. Overview of SOLISS configurations. (a) Installed position of SOLISS on i-SEEP attached to the Japanese experimental module on the ISS; (b) Flight model of SOLISS; (c) SOLISS’s optical communication unit (OCU); (d) a high-definition (HD) photo transmitted from SOLISS and received at NICT’s OGS through 100-Mbps Ethernet.

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The tracking and pointing mechanism of SOLISS comprises 3 states: acquisition, coarse tracking, and fine pointing. In the acquisition state, CU1 calculates the ISS trajectory based on position and attitude information and derives a table of angles for the bi-axial gimbal to point toward the OGS. In the coarse-tracking state, the wide-angle image sensor has a full FoV of 125.2 mrad to acquire the 1565-nm beacon light transmitted from the OGS site, which then works with the bi-axial gimbal in a closed-loop operation governed by a proportional-integral-derivative (PID) controller to further correct the pointing angles of the gimbal. The designed accuracy of the coarse-tracking system is 500 $\mu$rad. Finally, in the fine-pointing state, an FSM based on Sony’s optical disc technology and a QD perform the PID closed-loop fine-pointing control to accurately correct the misaligned angles of the transmitted beam due to platform vibrations. The QD has a 3-mrad full FoV and the fine-pointing accuracy is 30 $\mu$rad. The divergence half-angle of the transmitted beam through the 40-mm Tx aperture is 135.9 $\mu$rad, measured at $\exp (-2)$ intensity. SOLISS can transmit data at 100 Mbps using intensity modulation with non-return-to-zero (NRZ) format over Ethernet. Fig. 1(d) shows a high-definition (HD) image of SOLISS taken in-orbit by the 360-degree monitor camera, which was transmitted from SOLISS and successfully received at NICT’s OGS on 11 March 2020. This confirmed the effectiveness of miniaturization technologies for optical tracking and pointing between LEO and ground, and demonstrated for the first time the communication feasibility between a space station and ground networks via FSO links.

3.2 OGS configurations

Figure 2(a) shows the aerial view of NICT’s 1-m OGS infrastructure in Tokyo. However, the full 1-m telescope aperture was not used. Instead, a 40-cm sub-aperture telescope is installed inside the 1-m telescope as the receiving aperture for downlinks, as shown in Fig. 2(b). For uplinks, the Tx beam size covering 30 cm of the aperture. This configuration was selected after a careful consideration of the trade-off between the aperture size and system performance/complexity, primarily for NICT’s future HICALI (High Speed Communication with Advanced Laser Instrument) experiments to demonstrate a 10-Gbps optical feeder link system between geosynchronous equatorial orbit (GEO) and ground [31]. To facilitate tracking and pointing with satellites, this OGS is also equipped with a high-power beacon setup, mounted underneath the 1-m telescope. The optical bench is mounted on the telescope Nasmyth as shown in Fig. 2(c). The optics design that was used in the MMF-coupling experiment is then shown in Fig. 3. More specifically, after receving the downlink beam through the 40-cm telescope, the optical beam is reduced to about 1.62 cm as illustrated in Fig. 3(a). This is optimized for the utilization of optical elements in the receiving optics system, which is further detailed in Fig. 3(b). The key components of this receiving system are the FSM and QD, working in a closed-loop operation for finely correcting the AoA fluctuations of the receiving beam. The QD has an active area diameter of 1.5 mm and was designed to have a full FoV of about 608 $\mu$rad. As long as the incoming beam falls within the QD’s FoV, the beam spot incident onto the QD gives feedback about changes in the beam position and thus the angular displacements compared to the aligned position can be compensated by the FSM via a closed-loop PID operation. The FSM has a 2.39-cm clear aperture with angular resolution of less than 0.6 $\mu$rad and is placed at 45 degrees relative to the telescope’s optical axis. It is also equipped with a built-in optical sensor to monitor the AoA fluctuations of the incoming beam at the telescope entrance. The bandwidth of the closed-loop system is 300 Hz, which is defined as the frequency at which the closed-loop magnitude response is equal to -3 dB. This means the inputs with frequencies below 300 Hz can be tracked reasonably well by the closed-loop system. At the fiber tip position, the focused beam spot diameter is 10.5 $\mu$m, simulated by OpticStudio Zemax software. The fine-tracking system was designed to have sufficient accuracy for coupling into a SMF at about -3.7 dB loss in the laboratory environment. In our experiment with SOLISS, we specifically choose a 200-$\mu$m MMF for the coupling test to verify the MMF coupling efficiency for the potential use of mode multiplexing technologies.

Fig. 2. Overview of NICT’s OGS configurations. (a) 1-m OGS at NICT headquarters in Tokyo; (b) OGS in operation with a 40-cm sub-aperture telescope inside and a beacon setup underneath; (c) optical receiver equipped with a fine-tracking system.

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Fig. 3. Overview of OGS’s optics design using OpticStudio Zemax software. (a) 3D layout of the telescope and receiving optics; (b) 3D layout of the receiving optics with detailed components. VIS CAM: Visible camera, IR CAM: infrared camera, BS: beam splitter, OPM: optical power meter, DM: deformable mirror, WFS: wave-front sensor. The OPM, DM, and WFS were not used during the experiment with SOLISS.

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3.3 Experiment setup

Figure 4 shows an illustration of the experiment setup with SOLISS for the ISS pass over Tokyo on 21 September 2021, where the closest distance with the OGS site was about 536 km at 50-degree elevation angle. With SOLISS’s divergence half-angle of 135.9 $\mu$rad, the beam footprint when reaching the ground is about 146 m. We prepared a 1-m OGS with a 40-cm sub-aperture for the MMF-coupling experiment supported by the fine-tracking system as described in Section 3.2. For the link acquisition with SOLISS as described in Section 3.1, a beacon setup emits a 1565-nm beam with LHCP polarization at an average transmitted power of 5 W towards the predicted ISS orbit. However, it is important to note that the beacon setup underneath the 1-m OGS was not used in this experiment, and the 1565-nm beacon was launched from a nearby telescope 20-m away from the 1-m OGS infrastructure, due to our experiment conditions. Therefore, the beam received by the 1-m OGS with a 40-cm sub-aperture was relatively misaligned as SOLISS was trying to direct the beam center towards the beacon telescope location. Nevertheless, using the receiving optics with the fine-tracking system shown in Fig. 2(c), we were able to correct the misaligned AoA and successfully couple the free-space beam into a 200-$\mu$m MMF. This demonstrated the first fiber-coupling experiment of an FSO link from a space station to ground, serving as a valuable reference for the future integrated space-to-ground FSO-based networks.

Fig. 4. Illustration of the experiment setup with SOLISS on 21 September 2021.

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4. Results and analyses

The data obtained from the experiment setup described in Section 3.3 when the distance between SOLISS and our OGS site was closest (approximately 536 km) at 50-degree elevation angle are chosen for statistical analyses. Each analyzed dataset contains the data accumulated over 3 seconds at sampling rates of 6 kHz and 10 kHz for AoA and MMF-coupled power data, respectively. This is because two different data-acquisition systems were used for the fine-tracking system to collect AoA data and for the power sensor to collect MMF-coupled power data, respectively.

The AoA data of the incoming beam received at the 40-cm sub-aperture telescope in the 1-m OGS infrastructure are shown in Fig. 5(a). After the OGS’s fine-tracking corrections, the corresponding AoA values are shown in Fig. 5(b). It is clearly seen in Fig. 5(a) that the received beam from SOLISS was relatively misaligned with the mean values in both axes being $\mu _X=257.4571$ $\mu$rad and $\mu _Y=293.6384$ $\mu$rad. This is expectable since SOLISS acquisition, tracking and pointing systems were trying to align with the nearby beacon telescope using the 1565-nm beacon uplink. The 1-m OGS telescope pointing direction errors due to orbit prediction inaccuracy also contribute to the displaced AoA. As a result, the received beam at the 1-m OGS infrastructure should have a fixed radial displacement angle, i.e. $\mu =\sqrt {\mu _X^2+\mu _Y^2}=390.5223$ $\mu$rad. Nevertheless, the AoA standard deviation values are much smaller, i.e. $\sigma _X=4.7505$ $\mu$rad and $\sigma _Y=3.9258$ $\mu$rad, thanks to SOLISS’s fine-pointing system that keeps the beam stable against platform vibrations and beam deflections over the atmosphere. The fine-tracking system at the OGS was able to operate in closed-loop to correct the AoA displacements since the AoA of the incoming beam was within the QD’s full FoV of about 608 $\mu$rad. It is noted that the AoA data after fine-tracking corrections are well within the full FoV of the 200-$\mu$m MMF, which is about 112 $\mu$rad. This validates the condition $h_\text {AoA}=1$ in our analysis as stated in Section 2.1.1. As seen in Fig. 5(b), the AoA after corrections has the mean values in both axes very close to the aligned position at 0, i.e. $\mu _X=-0.3659$ $\mu$rad and $\mu _Y=0.5765$ $\mu$rad. However, the tracking noises still cause some deviations in the corrected AoA, i.e. $\sigma _X=0.3935$ $\mu$rad and $\sigma _Y=0.4124$ $\mu$rad. The radial AoA values before and after corrections are finally depicted in Fig. 5(c) and used for further statistical analyses.

Fig. 5. AoA data before and after fine-tracking corrections during 3 seconds. (a) AoA at the 40-cm sub-aperture telescope before corrections; (b) Corresponding AoA after corrections; (c) Radial AoA data before and after corrections.

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Figure 6 shows the statistical verification of the AoA data in Fig. 5(c) with the Beckmann PDF that characterizes the generalized AoA fluctuations as described in Section 2.1.1. This PDF model is chosen since the data in Fig. 5 reveal that the misaligned AoA values in both axes are two RVs with different means and variances. The values of GoF metric and fitting parameters are summarized in Table 1. As seen in Figs. 6(a) and 6(c), the Beckmann PDFs fit very well with the histogram data with GoF $R^2$ values higher than 0.999, indicating that the theoretical Beckmann PDF is a valid model for characterizing the random AoA fluctuations in LEO-to-ground FSO links and the corresponding random AoA fluctuations after OGS’s fine-tracking corrections. The fitted parameters shown in Table 1 are also very close with the experimental values in Fig. 5. To further examine the small-scale mismatch at the tails of all the fitted PDFs, we re-plot the results in log-log scale as shown in Figs. 6(b) and 6(d). It is generally observed that the mismatches at both tails are very small. We only notice a slightly higher mismatch at the lower tail in Fig. 6(d) when the corrected radial AoA is very close to zero. This is because at some instances the corrected AoA value was zero, while the PDF only characterizes non-zero values, leading to an overestimation of the PDF values at the lower tail near zero.

Fig. 6. Statistical verifications of the AoA data. Number of histogram bins $N=50$. (a) Radial AoA histogram data versus the fitted Beckmann PDF; (b) Radial AoA histogram data versus the fitted Beckmann PDF in log-log scale; (c) Corrected radial AoA histogram data versus the fitted Beckmann PDF; (d) Corrected radial AoA histogram data versus the fitted Beckmann PDF in log-log scale.

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Table 1. GoF metric and fitting parameters of the AoA fluctuations.

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In Fig. 7, we select the composite LN & approximated Beckmann model described in Section 2.1.4 for verifying the statistical distribution of the received power coupled into the MMF. Fig. 7(a) then shows the raw data of the MMF-coupled power over 3 seconds at the sampling rate of 10 kHz, and Figs. 7(b) and 7(c) depict the fitting results of the LN and approximated Beckmann PDF with the histogram data in linear and log-log scales, respectively. GoF metric and fitting parameters are summarized in Table 2. It is noted that the received-power data in Fig. 7(a) were normalized to the mean in Figs. 7(b) and 7(c) for the characterization of composite fading. It is clearly observed that the composite LN and approximated Beckmann PDF fits very well with the histogram data in both linear and log-log scales, highlighting that the theoretical model is very accurate in describing the random fluctuations of the received power. The fitting parameters in Table 2 also reveal separated effects from atmospheric turbulence and beam misalignment, indicating a weak turbulence condition (i.e. $\sigma _\text {SI}^2(D)=0.003$) and weak misalignment errors (i.e. $\varphi _\text {mod}=6.0532$). The fitted SI with aperture averaging effect $\sigma _\text {SI}^2(D)$ is also compared with the SI directly calculated from the data, denoted as $\sigma _\text {SI,exp}^2(D)$, which is defined as

(17)$$\sigma _{\text{SI,exp}}^{2}(D) \overset{\Delta}= \frac{\frac{1}{M}\sum_{i=1}^{M}P_{\text{Rx},i}^2}{\big[ \frac{1}{M}\sum_{i=1}^{M}P_{\text{Rx},i} \big]^2}-1.$$

Fig. 7. MMF-coupled power statistical analyses. (a) MMF-coupled power data accumulated over 3 seconds at 10-kHz sampling frequency; (b) Normalized MMF-coupled power histogram data versus the fitted LN and approximated Beckmann PDF, number of bins N=20; (c) Normalized MMF-coupled power histogram data versus the fitted LN and approximated Beckmann PDF in log-log scale, number of bins N=20; (d) Normalized ACF of the MMF-coupled power; (e) PSD of the MMF-coupled power; (f) Spectrogram of the MMF-coupled power, window size $m=128$.

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Table 2. GoF metric and fitting parameters of the MMF-coupled power.

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It is seen from Table 2 that the values of $\sigma _\text {SI}^2(D)$ and $\sigma _\text {SI,exp}^2(D)$ are approximately the same, certifying the accuracy of the fitted distribution.

To look closer into the physical characteristics of the power fluctuations, Figs. 7(d), 7(e), and 7(f) investigate the channel coherence time, the power spectral density, and the spectrogram of the MMF-coupled power, respectively. The channel coherence time can be deduced from the normalized ACF as defined in Section 2.2.1. The channel coherence time can be found in Fig. 7(d) as $T_\text {c}\approx 15.5$ ms, when the coherence time is determined at $1/e$ roll-off of the ACF. This again confirms the weak turbulence condition and slow-fading characteristic of the channel. Another important figure of merit for an FSO link is the PSD of the random power fluctuations, which is investigated in Fig. 7(e). As the power data were sampled at 10 kHz, we are able to see the PSD with frequency spectrum up to 5 kHz. It is evident that considerable magnitudes mostly appear in frequencies below the cut-off frequency $f_\text {c}=161.718$ Hz, which has a relatively flat spectrum. The part above $f_\text {c}$ quickly decays with a power-law exponent $\kappa =2.9159$, estimated by a linear regression method. This power-law exponent is slightly deviated from the theoretical $\kappa =8/3$ in Section 2.2.1 as expected, due to the non-point 40-cm receiving aperture. Finally, for a general view in both frequency and time, we present the spectrogram in Fig. 7(f). Only the spectrum up to 2 kHz is shown to highlight the significant frequency components. It is also confirmed that the main frequency components are below the cut-off frequency, while higher frequency components with considerable magnitudes intermittently reach 1 kHz, indicating weak turbulence effects.

We finally examine the MMF-coupling efficiency of the optical receiving system, which is defined as the ratio of the power coupled into the MMF to the power at the optical lens plane before the fiber tip. It is noted that all power data were acquired when the fine-tracking system was in closed-loop operations. The power at the optical lens plane before the fiber tip can be inferred from the received power data at the QD, since we know the deterministic optical loss from the QD to the lens position in Fig. 3. The MMF coupling loss and coupling efficiency data are then plotted in Fig. 8(a). It is noteworthy that the coupling data collected over 3 seconds are normalized to the sampling frequency at 2 kHz. This is because the QD power data and MMF-coupled power data were acquired at different frequencies, i.e. 6 kHz and 10 kHz, respectively, thus they are both normalized to 2 kHz for the sake of comparison. It is observed that the average coupling loss over 3 seconds is about 5.45 dB, corresponding to 28.478% CE. It should be reminded that this average coupling performance was attained when the 1-m OGS infrastructure was not in the optimal position to receive the downlink beam as described in Section 3.3.

Fig. 8. MMF coupling efficiency statistical verifications. (a) MMF-coupling data over 3 seconds at 2-kHz sampling rate; (b) MMF-coupling efficiency histogram data versus fitted theoretical PDFs, number of bins $N=50$; (c) MMF-coupling efficiency histogram data versus fitted theoretical PDFs in log-log scale, number of bins $N=50$.

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In Figs. 8(b) and 8(c), we empirically examine several well-known PDFs to characterize the random fluctuations in the MMF CE. The PDF of the MMF CE under physical channel effects including misalignment and atmospheric turbulence is not known in the literature to the best of authors’ knowledge. Since deriving an analytical PDF expression of the CE into an MMF is very challenging under the influences of both atmospheric turbulence and misalignment errors, we aim to empirically investigate among well-known PDFs to find the best-fitted one, serving as a practical reference for future theoretical developments. Figs. 8(b) and 8(c) respectively show the fitted PDFs versus histogram data in linear and log-log scales. Details of the GoF metric and fitting parameters are given in Table 3. Detailed descriptions of the fitting parameters can be found in Supplement 1. It is obvious from the GoF metric $R^2$ that the Gamma PDF provides the best fit to the histogram data in both linear and log-log scales, displaying a good accuracy at both lower and upper tails of the PDF. This certifies that the Gamma PDF is valid for characterizing the random fluctuations of the MMF CE, and could be used for further estimations of SNR and BER metrics. It is also observed that the Nakagami-$m$, LN, and log-logistic PDFs achieve a high $R^2$ value of more than 0.99, hinting an acceptable accuracy when matching with the histogram data. The Weibull and Rician PDFs attain a slightly lower $R^2$ value of more than 0.98, showing considerable deviations in the lower tails of the PDFs compared with histogram data. Interestingly, the Gamma-Gamma PDF, which is well-known for characterizing moderate-to-strong turbulence-induced fading, exhibits a very low accuracy with $R^2=0.05124$. This could be possibly due to the difference in the nature of fluctuations in the CE where it is derived from the overlapping of the random optical mode fields at the receiving optics, rather than a product of small-scale and large-scale fading components forming the Gamma-Gamma PDF. Finally, it should be noted that although the Gamma PDF shows the best fit, it might only serve as an acceptable approximation for the analytical PDF of the MMF CE, which indeed deserves further investigation and theoretical development. The fitting parameters in Table 3 could then be utilized as practical estimation values for future theoretical analyses.

Table 3. GoF metric and fitting parameters of the PDFs of MMF CE.

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5. Conclusion

In this paper, we reported the MMF-coupling experiment between the SOLISS optical terminal onboard the ISS and a side 1-m OGS, where the link alignment is not optimal compared to the aligned position with the nearby beacon telescope. Using a 40-cm sub-aperture telescope at the 1-m OGS and a fine-tracking system, we were able to couple the free-space beam into a 200-$\mu$m MMF with an average loss of 5.45 dB. From the AoA and received power data, we provided valuable knowledge on the statistical characteristics of the AoA fluctuations, beam misalignments, atmospheric turbulence, and time-frequency fading channel. In addition, the Gamma PDF was found to be a suitable choice for characterizing the random fluctuations of the MMF CE in a LEO-to-ground FSO link, which has never been verified in the literature. Our statistical results are essential for the design of channel error-correction codes and interleaver and for SNR and BER evaluations of space-to-ground FSO communication links.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request, given the permission of the Japanese National Institute of Information and Communications Technology (NICT). The raw data are not publicly available and subject to export control.

Supplemental document

See Supplement 1 for supporting content.

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	$R^{2}$	$μ_{X}$	$μ_{Y}$	$σ_{X}$	$σ_{Y}$
Radial AoA	0.99988	256.5856 $μ$ rad	294.46 $μ$ rad	5.5276 $μ$ rad	4.777 $μ$ rad
Corrected radial AoA	0.99931	-0.36258 $μ$ rad	0.53946 $μ$ rad	0.28927 $μ$ rad	0.56064 $μ$ rad

PDFs	GoF metric	Fitting parameters
Gamma	$R^{2} = 0.9998$	$α = 6.29$ , $β = 21.2509$
Nakagami- $m$	$R^{2} = 0.99597$	$m = 1.7955$ , $Ω = 0.093888$
LN	$R^{2} = 0.99205$	$μ = - 1.2585$ , $σ = 0.41605$
Log-logistic	$R^{2} = 0.9919$	$α = 0.28353$ , $β = 3.9298$
Weibull	$R^{2} = 0.98614$	$k = 2.8107$ , $λ = 0.31418$
Rician	$R^{2} = 0.98168$	$ν = 0.24778$ , $σ = 0.11674$
Gamma-Gamma	$R^{2} = 0.05124$	$α = 140.263$ , $β = 1.0895$

	$R^{2}$	$μ_{X}$	$μ_{Y}$	$σ_{X}$	$σ_{Y}$
Radial AoA	0.99988	256.5856 $μ$ rad	294.46 $μ$ rad	5.5276 $μ$ rad	4.777 $μ$ rad
Corrected radial AoA	0.99931	-0.36258 $μ$ rad	0.53946 $μ$ rad	0.28927 $μ$ rad	0.56064 $μ$ rad

PDFs	GoF metric	Fitting parameters
Gamma	$R^{2} = 0.9998$	$α = 6.29$ , $β = 21.2509$
Nakagami- $m$	$R^{2} = 0.99597$	$m = 1.7955$ , $Ω = 0.093888$
LN	$R^{2} = 0.99205$	$μ = - 1.2585$ , $σ = 0.41605$
Log-logistic	$R^{2} = 0.9919$	$α = 0.28353$ , $β = 3.9298$
Weibull	$R^{2} = 0.98614$	$k = 2.8107$ , $λ = 0.31418$
Rician	$R^{2} = 0.98168$	$ν = 0.24778$ , $σ = 0.11674$
Gamma-Gamma	$R^{2} = 0.05124$	$α = 140.263$ , $β = 1.0895$

Experimental verification of fiber coupling characteristics for FSO downlinks from the International Space Station

Abstract

1. Introduction

2. Theoretical background

2.1 Composite PDF for channel modeling

2.1.1 Beckmann PDF characterizing AoA fluctuations

2.1.2 Approximated Beckmann PDF characterizing beam misaligments

2.1.3 LN PDF characterizing atmospheric turbulence

2.1.4 Composite channel PDF: LN and approximated Beckmann model

2.2 Time-frequency channel characteristics

2.2.1 Channel coherence time

2.2.2 Power spectral density

2.2.3 Spectrogram

2.3 Theoretical PDFs under consideration for MMF CE

2.4 Goodness-of-fit (GoF) evaluation metric

3. Experiment descriptions

3.1 SOLISS configurations

3.2 OGS configurations

3.3 Experiment setup

4. Results and analyses

5. Conclusion

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (8)

Tables (3)

Equations (17)

Optics Express

Phuc V. Trinh	https://orcid.org/0000-0001-8087-6739
Dimitar R. Kolev	https://orcid.org/0000-0002-5773-0317
Alberto Carrasco-Casado	https://orcid.org/0000-0002-3412-7945