Channel Modeling and Rate Analysis of Optical Inter-Satellite Link (OISL)

Non-terrestrial network (NTN), particularly those using low-Earth orbit (LEO) satellites, provide widespread wireless connectivity through radio frequency (RF) signal transmissions. It’s important to note that these satellites are expected to be interconnected to transfer data to designated ground stations for internet access. Typically, ground stations are located in fixed and constrained areas. Therefore, establishing inter-satellite links (ISLs) among satellites is crucial for various applications within NTNs.

Optical ISLs (OISLs) are developed using laser communications to achieve very high-throughput data transfer [1]. Unlike the congested wireless RF spectrum, which includes bands such as the S-band, Ka-band, and Ku-band, the infrared portion of the electromagnetic spectrum used in OISL—ranging from approximately 300 gigahertz (GHz) to around 430 terahertz (THz)—provides an exceptional bandwidth. This characteristic enhances the potential for encoding more data into the waveform. In comparison to RF links, OISL can gather more energy, allowing for a reduction in the size, weight, and power requirements of the laser transmitter and detector [2].

In contrast to laser communication between a satellite and a ground station, the propagation of the OISL signal is primarily influenced by pointing errors. These errors can significantly reduce the power of the received signal at the detector, underscoring the importance of addressing this issue. Pointing errors in OISL arise from the satellite’s jitter and tracking noise [3]. These sources of misalignment collectively contribute to the vibration of the pointing direction.

Prior Art: Several studies in the literature have explored the channel model for satellite optical communications. In [4] and [5], beam pointing errors were introduced and modeled using Gaussian distributions for both elevation and horizontal directions. The authors of [6] examined the effects of vibrations on the bit error rate in satellite optical communications. However, these works did not consider how the beam waist changes with propagation distance, reducing the detector’s received energy. Additionally, they assumed that photons radiate omnidirectionally in the channel model, overlooking the directional characteristics of lasers. In [7], the combined effects of air turbulence and jitter on outage probabilities in terrestrial free-space optical (FSO) links were investigated. Another study in [8] analyzed the impact of Hoyt-distributed pointing errors on the error performance of on-off keying optical signals. Nonetheless, these studies focused on terrestrial FSO links that accounted for atmospheric turbulence and pointing errors. The channel statistics can be revisited in more manageable forms in the context of OISL. Moreover, the existing literature should address the average achievable data rate for OISL using laser beams across various applications. Therefore, developing an analytical model for OISL is essential to fully characterize its overall communication performance in NTN.

Contributions: The main contributions of this paper are summarized as follows.

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  OISL Channel Model: We establish a channel model for OISL based on a Gaussian beam. The statistical characteristics of the OISL channel are derived by considering the effects of pointing errors and the detector’s sensitivity threshold. Additionally, we derive the simplified forms of the probability density function (PDF), the cumulative distribution function (CDF), and the average channel state for the OISL channel, using Rayleigh-distributed radial deviation. Furthermore, we determine the maximum radial deviation distance associated with a Gaussian beam.

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  Cooperative OISLs Design: We designed a cooperative OISL communication system where satellites forward data between source and destination satellites by using OISLs. With the developed OISL channel model, we can accurately derive the average achievable data rate of an OISL communication, facilitating performance analysis of a cooperative OISL communication system. Moreover, we introduce optimizing the number of OISL relaying satellites and laser frequency to guarantee a latency constraint and a targeted data transmission size.

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  OISLs Design Insights: Regarding the average achievable data rate, adjusting the laser frequency or beam waist introduces a trade-off between concentrating beam energy and balancing misalignment. A reduced beam waist could potentially increase the detector’s received power intensity. However, reducing the beam waist causes the detector to deviate from the beam center and decrease the received power intensity. Furthermore, changing the number of satellites in a cooperative OISL communication system results in a trade-off between the increased data rate of each OISL link and the sum of latency over all cooperative links.

The remainder of this paper is organized as follows. Section II provides a system model. Section III introduces the channel statistics of the OISL. Section IV demonstrates the achievable data rate and the total latency. Simulation results are given in Section V, and the paper is concluded in Section VI.

In OISL, the received signal power $y_{s}$ at the detector suffers from a fluctuation in signal power, which is modeled by

where $h$ denotes the channel state due to the expanded beam, path loss, and pointing errors, $\eta$ is the detector responsivity, ${x_{s}}$ is the power of the transmitted signal, and $n_{0}$ is signal-independent additive white Gaussian noise with variance ${\sigma_{n}}^{2}$. Specifically, $h$ depends on two factors [7] shown as follows

where $h_{PL}$ encompasses the deterministic path loss which approaches one in space, since the laser beam does not propagate omnidirectionally and space is in a vacuum state [9], and $h_{PE}$ indicates the channel random attenuation caused by pointing errors. This model is suitable for any satellite or spacecraft with OISL devices. We assume that the laser beam can be tracked and pointed with misalignment [9], and the Doppler shift due to satellite movement can be well compensated.

In Fig.1, the beam waist $w_{z}$ at distance $z$ is given by

where $\theta=\frac{\lambda}{{\pi{w_{0}}}}$ is the beam divergence angle [10], $\lambda$ denotes the wavelength, and $w_{0}$ is the beam waist at $z=0$ m. Considering the long propagation distance and diffused beam, we approximate $w_{z}$ as follows

In Fig. 1, ${\bf{r}}$ denotes the radial vector from the beam center, i.e., pointing error, and we assume a large field of view at the receiver with no angular fluctuation [11]. The power intensity distribution $I\left({{\bf{r}},z}\right)$ of a Gaussian beam [7] at distance $z$ is

Since the shapes of the transverse plane of the beam and detector are symmetrical, $h_{PE}$ depends on the radial deviation distance of the pointing error, i.e., $r=\left\|{\bf{r}}\right\|$, which is given by

where ${{\cal A}_{d}}$ is the area of the detector, $w_{d}$ is the radius of the detector, and we have ${{\cal A}_{d}}=\pi{w_{d}}^{2}$.

The main notations are summarized in Table 1.

In this section, we derive the OISL channel statistical characteristics.

Without loss of generality, we reasonably assume that the horizontal and vertical misalignment, stemming from the satellite’s jitter and tracking noise, follow independent and identical normal distributions [4, 5, 6, 7]. Since the sum of two independent normal variables is also normally distributed, we use a single normal distribution to represent the vertical or horizontal misalignment for notation simplicity.

In Theorem 2, we observe that the PDF of $h_{PE}$, i.e., ${f_{{h_{PE}}}}\left(y\right)$, is a power function of $y$. It’s worth noting that ${f_{{h_{PE}}}}\left(y\right)$ changes with the propagation distance $z$.

Denote $p_{th}$ as the sensitivity power threshold for a detector to identify an optical signal and $h_{th}$ as the corresponding average sensitivity threshold of $h_{PE}$, which is given by

Thus, the maximum radial deviation distance, i.e., $r_{max}$, is calculated by ${h_{PE}}\left({r,z}\right)={h_{th}}$, as follows

In some cases, the average channel state is essential to evaluating the average channel condition of OISL, as given in the following Theorem.

It is worth noting that in Theorem 3, ${\overline{h}_{PE}}$ can be approximated by ${{\bar{h}}_{PE}}\approx\frac{{2{w_{d}}^{2}}}{{{w_{z}}^{2}+4{\sigma_{s}}^{2}}}$ when ${{h_{th}}}$ is very small.

In this section, we design a cooperative OISL communication system where relaying satellites are strategically positioned to balance the trade-off between the increased data rate of each link and the overall transmission latency. As illustrated in Fig. 2, the channel state of the single hop between the source and destination satellites can be unfavorable due to the broader beam width and significant pointing errors that arise from the long propagation distance. Introducing additional relaying satellites enhances the data rate of each hop; however, the increased number of transmissions through these relays may lead to an increase in total transmission latency. To address this, we aim to minimize the number of cooperating satellites, denoted as $N$, while ensuring that the total latency remains below a specified threshold, $T_{th}$. This is done with the given parameters of a data size $D$ and a communication distance $L$ between the source and destination satellites.

where ${\cal R}_{n}$ is the average data rate of the n-th OISL.

In practice, the variance of pointing errors ${\sigma_{s}}^{2}$ increases with the propagation distance due to the increased difficulty of tracking. Therefore, ${\sigma_{s}}$ is a function of propagation distance. For example, we model ${\sigma_{s}}={\sigma_{s,0}}{e^{{k_{0}}\frac{\delta}{{{d_{0}}}}}}$ and ${\sigma_{s,0}}=2$, $k_{0}=0.1$, and $d_{0}=100$ km as the reference distance.

The optimal $N$ is obtained by solving the equation $ND={T_{th}}{{\cal R}_{n}}$, where we consider equally spaced relay satellites.

In this section, we conduct simulations to evaluate the OISLs’ communication performance. Unless specified otherwise, the default parameters are shown in Table I.

In Fig. 3, we compare the average channel state caused by pointing errors, i.e., ${\overline{h}_{PE}}$, as derived in Theorem 3, versus OISL laser frequency, i.e., $f$, under various values of $\sigma_{s}^{2}$ and different propagation distances $z$. It is observed that ${\overline{h}_{PE}}$ gradually increases as the laser frequency $f$ increases. This is because a higher frequency allows the laser beam to concentrate more energy on the detector. However, when $\sigma_{s}^{2}$ is large, the increase in $f$ does not significantly enhance ${\overline{h}_{PE}}$. This limitation is due to the misalignment caused by pointing errors, which leads the detector to deviate from the concentrated beam. Additionally, we observe that ${\overline{h}_{PE}}$ decreases with the increment of $\sigma_{s}^{2}$ and $z$, which aligns with our expectations. Moreover, the results from Monte Carlo simulations validate the analytical findings.

In Fig. 4, we examine the average achievable data rate, i.e., ${{\cal R}_{n}}$, derived in Theorem 5 versus OISL laser frequency, i.e., $f$, under various $\sigma_{s}^{2}$ and propagation distance $z$. We observe that there is an optimal laser frequency that maximizes ${{\cal R}_{n}}$. On the one hand, when $f$ is too small, the laser’s energy is diffused, and the power intensity received at the detector decreases. On the other hand, when $f$ is too large, the detector may deviate much from the beam center due to pointing errors. In turn, the amount of energy received at the detector also decreases. Therefore, there is a trade-off between concentrating the beam energy and balancing the misalignment.

In Fig. 5, the total latency of the cooperative OISLs system is presented concerning the number of cooperative satellites with/without frequency optimization. The optimal laser frequency solution is calculated by exhaustive searching. Optimizing the number of cooperative satellites and $f$ can significantly reduce the total latency. The reduction in total latency results from a trade-off involving balancing the increased data rate of each cooperative link with the sum of latency over all cooperative links.

In this paper, we developed an OISL channel model and derived its channel statistics. Based on the channel model, we introduced a cooperative OISL system in which the number of cooperative satellites involved and laser frequency can be optimized to minimize the total latency. Simulation results reveal the trade-offs in the design of the OISL system. Note that the proposed analytical model applies to both current and more complicated OISL systems. Future research can utilize the proposed model to assess the data rate and reliability of OISLs in more complex satellite mega-constellations, considering a pre-defined SINR threshold. This paper examines OISL channel modeling while accounting for pointing errors. Future investigations could explore additional factors impacting OISL, such as satellite orbit perturbations, beam tracking, and angular fluctuations.