## Abstract

The pointing jitter originated from the random mechanical vibration of the optical platform and the noise associated with the optomechanical sensor system unquestionably deteriorate the performance of the inter-satellite optical link. The impact of the jitter changes significantly with the receiver architecture. In this paper, we present a mathematical model to investigate the link performance in the presence of angular pointing jitters for different receiver architectures. Alongside the statistical pointing error model, the derived model incorporates key receiver design parameters such as detector radius, receiver aperture size, $ f $-number of the lens system, and beam compression ratio to study the impact of receiver architecture on the pointing jitter. As an example, a CubeSat optical receiver is analyzed. We show that by careful selection of beam compression ratio and $ f $-number, more than 5 orders of magnitude bit error rate improvement is achievable even at large pointing error.

© 2020 Optical Society of America

## 1. INTRODUCTION

Small satellite technology continues to evolve to tackle more advanced and challenging space missions. The realization of a small satellite constellation facilitated by high-speed optical cross-link is being investigated to ensure the success of such missions [1–3]. One of the major challenges of an inter-satellite optical link (ISOL) is the necessity of the complex pointing, acquisition, and tracking (PAT) system. A well-engineered PAT mechanism is crucial in achieving maximum connectivity in a constellation where every satellite expects to sustain multiple data communication links simultaneously as shown in Fig. 1 [4,5]. The state-of-the-art ISOL usually operates in the presence of random angular pointing error (also known as pointing jitter). This pointing error arises primarily from the noise associated with the tracking sensors, the disturbance originated from the mechanical vibration, and the base motion of the satellites [6–8]. As a result, ISOL suffers from performance degradation. Up to date, several analytical models have been presented to understand the impact of pointing jitter on the performance of the digital and analog optical links [9,10]. In addition, authors studied the impact of random pointing and tracking error on coherent and incoherent optical satellite links in [11]. Several other works also investigated the performance degradation of the optical link due to the pointing jitters [6,12,13]. The main focus of the majority of these works was given toward the modeling of the nature of pointing jitters and, therefore, to estimate the overall link performance in terms of bit error rate (BER) or bit error probability (BEP). However, the effect of the receiver architecture on the communication performance in the presence of pointing jitter is yet to be studied. Since the receiver architecture plays a critical role in the ISOL’s immunity to the random pointing error, it is desirable to have an analytical study that facilitates a quick estimation of the link performance by using realistic receiver architecture parameters.

In this paper, we present a complete analytical model that incorporates both the receiver architecture parameters and statistical distribution of the pointing errors. In particular, we derive a mathematical performance model of a direct detection optical receiver to analyze the effect of receiver design parameters on the link immunity to the pointing jitters. The presented analytical model includes key receiver design parameters, such as detector radius, receiver aperture size, $ f $-number of the focusing lens (FL) system, the beam compression ratio of the telescope, and the aberration parameter. Both instantaneous and average link performances are analyzed and presented using the derived model. As a case study, a CubeSat direct detection optical receiver is being studied. We show that by optimizing the beam compression ratio and minimizing the optical aberrations of the telescope more than 3 orders of magnitude BER improvement can be realized in a given ISOL. Moreover, our study indicates that 6 orders of magnitude BER improvement is realizable for a given ISOL by designing a receiver system with optimal selection of telescope beam compression ratio, $ f $-number of the focusing system, and aberration parameter. Hence, the presented study facilitates the selection of the optimum design parameters for the best performance in the presence of random pointing errors.

## 2. STATISTICAL MODEL OF POINTING JITTERS AND THE EFFECT OF RECEIVER ARCHITECTURE ON PERFORMANCE

#### A. Model of Pointing Jitters and Receiver Architecture

The pointing errors depends on the electrical noises in the PAT sensors and the mechanical vibrations that couple to optical subsystems. The fundamental expression of the tracking sensor noise ${\sigma _{\theta S}}$ (standard deviation) has been studied before, and it has been expressed in terms slope factor of the transfer function $\zeta$, which is a function of signal-to-noise ratio (SNR) $S/N$ of the system, ${\sigma _{\theta S}} = \frac{1}{{\sqrt {\zeta (S/N)}}}$ [14]. Mechanical vibrations cause pointing error ${\sigma _{\theta M}}$ that can be modeled as $\sigma _{\theta M}^2 = \frac{1}{{2\pi}}{\int }{S_\theta}(\omega)|1 - {H_T}(\omega {)|^2}{\rm d}\omega$, where ${S_\theta}(\omega)$ is the power spectral density of the mechanical noise and ${H_T}(\omega)$ is the closed loop transfer function of the spatial tracking loop [11]. The combined effect of the electrical noise in PAT sensors and mechanical vibrations creates elevation and azimuth pointing angle errors. The probability density function (PDF) of both the elevation pointing error ${\theta _{{\rm ET}}}$ and azimuth pointing error ${\theta _{{\rm AT}}}$ has been modeled as normal distribution as $p({\theta _{{\rm ET}}}) = \frac{1}{{\sqrt {2\pi \sigma _E^2}}}{\exp }\left(- \frac{{\theta _{{\rm ET}}^2}}{{2\sigma _E^2}}\right)$ and $p({\theta _{{\rm AT}}}) = \frac{1}{{\sqrt {2\pi \sigma _A^2}}}{\exp }\left(- \frac{{\theta _{{\rm AT}}^2}}{{2\sigma _A^2}}\right)$, respectively [9,11]. Here, $\sigma _E^2$ and $\sigma _A^2$ represent the variance of the elevation and the azimuth pointing error, respectively. Assuming the distribution of ${\theta _{{\rm ET}}}$ and ${\theta _{{\rm AT}}}$ are independent, but alike, the PDF of the radial pointing error ${\theta _T}$ ($\theta _T^2 = \theta _{{\rm ET}}^2 + \theta _{{\rm AT}}^2$) is a Rayleigh distribution that can be expressed in terms of the variance of the radial pointing error $\sigma _T^2$ [9,11], $p({\theta _T}) = \frac{{{\theta _T}}}{{\sigma _T^2}}{\exp }(- \frac{{\theta _T^2}}{{2\sigma _T^2}})$. Due to the radial symmetry, we assume ${\sigma _T} = {\sigma _A} = {\sigma _E}$ [12,15]. The pointing jitters can affect the communication performance, especially at a large angle of incidence (AOI) $\theta$ as shown in Fig. 1. The initial communication link between the transmitter and the receiver is usually established with the help of ephemerides data and advanced PAT system [16]. The transmitter and receiver usually achieve mutual line of sight tracking of each other by body pointing, coarse pointing, and fine pointing mechanisms [17–19]. However, due to the uncertainty of the satellite positions and the limited resolution of the PAT systems, $\theta$ can be significantly large between certain satellites in a constellation. For instance, in Fig. 1, a large AOI $\theta$ exists between ${S_1}$ and ${S_2}$ as the transmit beam of ${S_1}$ creates a large angle with respect to the receiver normal ${\vec n_2}$. In contrast, a smaller $\theta$ exists between ${S_1}$ and ${S_3}$ since the transmit beam of ${S_1}$ aligns well with the receiver normal of ${S_3}$ (${\vec n_3}$). For effective long-distance data communication, the AOI needs to be smaller than the receiver’s field of view (FOV) $\phi$, i.e., $\theta \lt \phi$. The instantaneous AOI at the receiver optical aperture can be written as ${\theta _i} = \theta + {\theta _T}$. For a given $\theta$, the instantaneous received optical power by the receiver aperture ${P_{{\rm rcv}}}({\theta _T})$ can be estimated from Friis transmission equation [20],

Here, ${P_T}$, ${G_T}$, ${G_R}$, ${L_P}$, $\lambda$, $R$, ${\alpha _T}$, and ${\alpha _{\!R}}$ represent average transmit power, transmit antenna gain, receiver antenna gain, pointing loss, operating wavelength, communication range, transmitter feeder loss, and receiver plumbing loss, respectively. The loss due to random pointing jitter ${L_{{\rm PJ}}}({\theta _T})$ can be given by [11], ${L_{{\rm PJ}}}({\theta _T}) = {\exp }(- {G_T} \theta _T^2)$. Assuming $\theta$ varies much slower than the pointing jitter ${\theta _T}$, the loss related to $\theta$ is lumped into the pointing loss ${L_p}$ in the above-mentioned range equation. Hence, the instantaneous signal photo current ${i_{{\rm PD}}}(\theta ,{\theta _T},{\mathbb S})$ depends on the receiver architecture that can be expressed in terms of the responsivity of the detector ${R_\lambda}$, the intrinsic gain of the photodetector $G$, and the performance degradation factor $K(\theta ,{\theta _T},{\mathbb S})$. Here, the performance degradation factor is determined by a set of receiver design parameters ${\mathbb S}$ = {aperture size, detector size, lens focal length, etc.},

The major causes of the performance degradation factor are random beam walk-off at the detector plane ${\eta _{{\rm BO}}}(\theta ,{\theta _T},{\mathbb S})$, angle dependent detector responsivity ${\eta _\lambda}(\theta ,{\theta _T})$, reflection due to anti-reflection (AR) coating ${\eta _{{\rm AR}}}(\theta ,{\theta _T})$, and free space to focusing lens (FL) coupling efficiency ${\eta _C}({\mathbb A})$ (${\mathbb A} \subset {\mathbb S}$). Hence, the performance degradation factor can be expressed as

In a high-speed free space communication system, due to the wavefront error of the received beam and the aberrations in the optical system, the focal spot size of the received beam is usually comparable to the active area of the high-speed detectors. Hence, in the presence of pointing jitter, the random beam walk-off at the detector plane becomes a dominant degradation factor. The performance degradation factor in the free space direct detection system can therefore be simplified and written as $K(\theta ,{\theta _T},{\mathbb S}) \approx {\eta _{{\rm BO}}}(\theta ,{\theta _T},{\mathbb S}){\eta _C}({\mathbb A})$. Assuming the photodetector’s radius ${r_d}$ is close to but larger than the focal spot radius ${r_{\!f}}$, i.e., ${r_d} \gt {r_{\!f}}({d_B},f,\xi)$, the radial displacement of the focal spot center from the detector center $\rho (\kappa \theta ,\kappa {\theta _T})$, angle magnification factor of the receiver telescope $\kappa$, and $\theta \lt \phi$, the $K(\theta ,{\theta _T},{\mathbb S})$ can be approximated as

The focal spot radius ${r_{\!f}}({d_B},f,\xi)$ can be approximated from the beam
diameter ${d_B}$, the beam mode quality $M$, and the focal length of the FL $f$ and is given by ${r_{\!f}}({d_B},f,\xi)
\approx 0.64{M^2}\lambda \frac{f}{d_B} + \xi
\big(\frac{{d_B^3}}{{2{f^2}}}\big)$. The aberration factor $\xi$ of the lens system depends on the
materials of optical lenses, the curvature of the lens surfaces, and
the receiver lens system design. A detail Gaussian beam propagation
simulation or experimental validation is required to approximate $\xi$ for a given optical system. For
instance, $\xi$ of a singlet plano–convex lens made
from N-BK7 glass can be estimated by detail optical simulations (in
Zemax) as $\xi = 0.1943 +
0.00202{d_B} - 0.000798f$. In this model, the numerical values
of ${d_B}$ (in mm) and $f$ (in mm) need to be used to estimate
the unit-less parameter $\xi$. Assuming $\theta$ is well controlled by the PAT system
over the considered time duration to analyze the impact of the
pointing jitter, the average SNR (${{\rm SNR}_{{\rm
avg}}}$_{)} can be estimated from the
instantaneous received power, the instantaneous noise power $\sigma _N^2(\theta
,{\theta _T})$, and ${\sigma _T}$ as

Considering a communication system that receives a pulsed light with a duty cycle $D$ and an extinction ratio ${r_{{\rm ex}}}$ in a direct detection method, the instantaneous signal current for bits 1 can be expressed as ${i_{PD - 1}}(\theta ,{\theta _T},{\mathbb S}) = \gamma .{i_{{\rm PD}}}(\theta ,{\theta _T},{\mathbb S})$. Similarly, the instantaneous signal current for bits 0 is given as ${i_{PD - 0}}(\theta ,{\theta _T},{\mathbb S}) = \gamma {r_{{\rm ex}}}{i_{{\rm PD}}}(\theta ,{\theta _T},{\mathbb S})$. Here, $\gamma = \frac{1}{{D(1 - {r_{{\rm ex}}}) + {r_{{\rm ex}}}}}$. The instantaneous noise variances $\sigma _{N - 1}^2$ (for 1) and $\sigma _{N - 0}^2$ (for 0) are a function of signal currents, receiver dark current ${I_d}$, excess noise factor ${F_A}$, and the thermal noise $\sigma _{{\rm TH}}^2$. The noise variances can be expressed as

Here, $ q $ is the elementary charge. Consequently, the instantaneous Gaussian-$ Q $ function can be written as $Q(\theta ,{\theta _T},{\mathbb S}) = \frac{{{i_{PD - 1}}(\theta ,{\theta _T},{\mathbb S}) - {i_{PD - 0}}(\theta ,{\theta _T},{\mathbb S})}}{{{\sigma _{N - 1}}(\theta ,{\theta _T},{\mathbb S}) + {\sigma _{N - 0}}(\theta ,{\theta _T},{\mathbb S})}}$ [21]. Therefore, for a given $\theta$ and ${\mathbb S}$, the instantaneous error probability can be calculated from $Q(\theta ,{\theta _T},{\mathbb S})$,

Finally, the system BER can be obtained by averaging the instantaneous BEP ${p_e}({\theta _T})$ and can be written as

#### B. Model of a Sample Direct Detection Receiver

The general formulation presented in the previous section can be modified according to a specific receiver architecture. As an illustration, here, we present the performance estimation model of a sample direct detection receiver in a small satellite platform as shown in Fig. 2.

The sample receiver has a clear aperture diameter of ${d_R}$, and the telescope beam compressor reduces the beam diameter to ${d_B}$. In a long-distance communication, we can assume a plane wave illuminates the entire receiver aperture. Therefore, the beam compression ratio ${m_T}$ can be defined as $m_T^{- 1} = \frac{d_B}{d_R}$ and $\kappa \approx {m_T}$ in Eq. (4). The detector diameter and the focal spot diameter are denoted as $2{r_d}$ and $2{r_F}$, respectively. A beam sampler (BS) is usually used to sample a fraction beam for the quadrant detector (QD) to facilitate the pointing and tracking mechanisms [2]. In Fig. 2, the on-axis received beam is shown as a solid blue line, and the off-axis received beam is shown as a dashed orange line. All angles are measured w.r.t. the direction normal to the receiver $\vec n$. Besides, ${\theta _T}$ is the instantaneous pointing error, and ${\theta _r}$ is the instantaneous total angular variation of the compressed beam. ${\theta _r}$ can be expressed as ${\theta _r} = {m_T}(\theta + {\theta _T})$. Finally, a FL of diameter ${d_{{\rm FL}}}$ focuses the beam on the detector (D). To simplify the model, we lump the impact of optical aberration in $\xi$. Hence, the coupling efficiency is approximated as ${\eta _C}({\mathbb A}) = (\frac{{{m_T}{d_{{\rm FL}}}}}{d_R}{)^2}$ if ${m_T}{d_{{\rm FL}}} \le {d_R}$; otherwise, ${\eta _C} \approx 1.0$. ${\mathbb A} = \{{m_T},{d_R},{d_{{\rm FL}}}\}$. The radial displacement of the center of the focal spot from the center of the detector can be expressed as $\rho (\kappa \theta ,\kappa {\theta _T}) \approx f{m_T}(\theta + {\theta _T})$. Under small-angle approximation, the instantaneous performance degradation factor of the sample receiver architecture ${K_s}(\theta ,{\theta _T},{\mathbb S})$ is derived from Eq. (4) and is given by

Equation (9) reveals the
effect of receiver design parameters, ${\mathbb S} =
\{{m_T},{r_d},{d_R},f,{d_{{\rm FL}}},\xi \}$, on the degradation factor. Depending
on the random pointing jitter and the receiver parameters, ${K_s}$ can vary between 0.0 and 1.0, $0 \le {K_S}(\theta
,{\theta _T},{\mathbb S}) \le 1$. Once we estimate the received power
on the detector, the instantaneous BEP and the BER of the sample
receiver can be calculated using Eqs. (7) and (8), respectively. Generally in a receiver design, ${d_{{\rm FL}}} \ge
{d_B}$_{;} therefore, we can assume ${\eta _C}({\mathbb A})
\approx 1.0$. Additionally, assuming a Gaussian
beam with ${M^2} \approx
1.0$ and the incorporated FL has an $ f $-number of $N$, the performance degradation factor
expression in Eq. (9)
can be simplified further and given as

Here, $\tau$ is the ratio of the AOI to the receiver’s FOV, i.e., $\tau = \frac{\theta}{\phi}$. The beam compression ratio ${m_T}$ of the sample receiver architecture as shown in Fig. 2 is limited by the diffraction-limited focus spot size of the aperture lens and the manufacturable optics. The diffraction-limited spot size (airy disk radius) after the aperture lens can be calculated from the focal length of the aperture lens ${f_R}$, and can be given as ${r_a} = \frac{{0.64{M^2}\lambda {f_R}}}{d_R}$. Hence, the realizable compression ratio of the sample telescope can be expressed as ${m_T} \ll \frac{{d_R^2}}{{1.28{M^2}\lambda {f_R}}}$.

## 3. PERFORMANCE ANALYSIS AND SIMULATION RESULTS

The link parameters and the receiver design parameters are summarized in Table 1. In our analyses, we consider values for each parameter that are typically used in the state-of-the-art systems based on a literature survey and manufacturer specifications [22–24]. Although the considered design parameters are pertinent to CubeSat omnidirectional optical link [4,24], all analyses apply to any satellite optical link with the appropriate design and link parameters.

#### A. Instantaneous Performance Degradation Analyses

Equation (10) reveals the dependence of the instantaneous performance degradation factor ${K_s}$ on the detector size ${r_d}$, the receiver aperture size ${d_R}$, the $N$ of the FL, the beam compression ratio ${m_T}$, the ratio of the AOI to the FOV $\tau$, and the aggregated aberration factor $\xi$ of the lens system. To demonstrate the instantaneous performance degradation, we choose an omnidirectional CubeSat platform [24] with a 15 mm receiver aperture, i.e., ${d_R} = 15.0\;{\rm mm} $ and the instantaneous pointing error, ${\theta _T} = 0.5\;\unicode {x00B5}{\rm rad}$. As ${\theta _T} \ll \tau \phi$, the effect of instantaneous ${\theta _T}$ is negligible on ${K_s}$. However, the distribution of ${\theta _T}$ significantly affects the average link performance as expected from Eqs. (5) and (8).

A sample analysis to demonstrate the dependence of ${K_s}$ on $\tau$ and ${r_d}$ (assuming $\xi = 0.15$) is presented in Fig. 3. Figures 3(a) and 3(b) show the performance degradation for $N = 2.0$ and $N = 3.0$, respectively. It can be seen from Fig. 3(a) that ${K_s}$ drops to 0.4 from 0.99 almost linearly within a range of $0 \le \tau \le 1$ for ${r_d} = 0.1\;{\rm mm} $ and ${m_T} = 3.0$. On the contrary, for ${r_d} = 0.5\;{\rm mm} $ and ${m_T} = 3.0$, ${K_s}$ remains stable (close to 1.0) for $\tau \le 0.81$. Hence, increasing ${r_d}$ improves the receiver tolerance to $\tau$ for a given ${m_T}$, as expected. Further improvement of receivers’ tolerance by increasing ${m_T}$ is evident from Fig. 3(a). Although the angular variation increases due to the beam compression, i.e., ${\theta _r} \approx {m_T}{\theta _i}$, the impact of ${m_T}$ can be counterbalanced by the FL parameters. The resultant beam walk-off at the detector plane is found to be $\rho \approx \zeta {\theta _i}N{d_R}$, where $\zeta = \frac{{{d_{{\rm FL}}}}}{d_B}$. Besides, in an aberration limited optical receiver, ${r_{\!f}} \propto d_B^3 \propto 1/m_T^3$. Therefore, for a given aperture size, a larger ${m_T}$ tends to create a smaller focal spot. As a result, the probability of beam clipping at the detector decreases due to lateral movement of focal spot that improves the achievable average SNR in the presence of pointing jitter. For example, if a beam compressor with ${m_T} = 5.0$ is incorporated, then ${K_s}$ is close to 1.0 for $\tau \le 0.43$ for ${r_d} = 0.1\;{\rm mm} $ and ${K_s}$ gradually decreases to 0.25 when $\tau$ approaches 1.0 as shown in Fig. 3(a).

A higher $N$ also helps to make the receiver system less sensitive to pointing errors. For instance, it can be seen from Fig. 3(b) that for ${m_T} = 3.0$, $N = 3.0$, and ${r_d} = 0.1\;{\rm mm} $, ${K_s}$ remains close to 1.0 for $\tau \le 0.57$ and then gradually drops to 0.45 when $\tau$ gradually approaches to 1.0. Visually, the sensitivity of ${K_s}$ on the design parameters can be compared based on the area of the quasi-flat region (relatively constant ${K_s}$ region) ${A^\prime _F}$ of the plots where ${K_s} \ge 0.98$. It can be qualitatively realized that ${A^\prime _F}(3,2) \lt {A^\prime _F}(5,2) \lt {A^\prime _F}(3,3) \lt {A^\prime _F}(5,3)$. Here, ${A^\prime _F}({m_T},N)$ represents the area of the quasi-flat region of the plots for given ${m_T}$ and $N$ as shown in Fig. 3. A larger ${A^\prime _F}$ implies higher immunity to incident angle errors. Therefore, we can say that among the considered combination of design parameters, the receiver system with ${m_T} = 3.0$ and $N = 2.0$ has the lowest immunity to pointing jitters, whereas the system with ${m_T} = 5.0$ and $N = 3.0$ demonstrates the highest immunity.

The effect of $\tau$ and $\xi$ on the instantaneous performance degradation for two different detector radiuses, ${r_d} = \{0.1 \;{\rm mm},0.25 \;{\rm mm}\}$, is presented in Fig. 4. Figure 4(a) shows the impact of $\xi$ and $\tau$ on ${K_s}$ for ${m_T} = 3.0$ and $N = 3.0$. For instance, it can be seen from Fig. 4(a) that at $\xi = 0.15$ and ${r_d} = 0.25\;{\rm mm} $, the factor ${K_s}$ is close to 1.0 for $\tau \le 0.95$. In contrast, a 0.1 mm detector shows ${K_s} \ge 0.98$ for $\tau \le 0.8$ at the same aberration, $\xi = 0.15$. It is also realizable from Fig. 4(a) that ${K_s}$ can vary from 1.0 to 0.4 over the range $0.2 \lt \tau \le 1.0$ based on the receiver parameters ${r_d}$ and $\xi$. Evidently, a receiver with larger ${r_d}$ is more immune to variation of $\tau$ and $\xi$. The instantaneous performance degradation can further be reduced by incorporating a beam compressor with a higher ${m_T}$ as can be realized from Fig. 4(b). As applied to Fig. 3, similar visual analyses of the receivers’ tolerance to $\tau$ and $\xi$ based on the relatively constant ${K_s}$ region is also applicable here. For example, it can be seen that ${A_F}(3,3) \lt {A_F}(3,5)$ for all detector sizes. Here, ${A_F}(N,{m_T})$ represents the area of the flat (constant ${K_s}$) region. Hence, for a given $N$, a system with a higher ${m_T}$ demonstrates higher immunity to pointing errors due to vibrations.

In this analysis, we use $\tau$, which is the ratio between the incident angle and the FOV, to compare the performance of different design choices. The receivers’ instantaneous FOV (half-angle) can be calculated as $\phi \approx \frac{r_d}{{N\zeta {d_R}}}$. Assuming $\zeta \approx 1.0$, the estimated instantaneous $\phi$ of the receiver designed with the parameters given in Table 1 is given in Table 2.

#### B. Performance of the Communication Link

The communication link performance for a given aperture size can be characterized by (a) link limited (LL) performance and (b) architecture limited (AL) performance. The LL performance is calculated from the link budget equation that is restrained by communication distance, transmit power, limited receiver optical aperture, etc. The LL performance assumes an optimum receiver design. We calculate the LL lowest achievable BER of the given link to be around ${10^{- 10}}$ in the absence of pointing jitters that are caused by satellite vibration. However, the performance degrades due to the unavoidable pointing jitters as given in Eqs. (5) and (8). The performance degradation depends on the architecture of the receiver; therefore, architecture-dependent link performance is achieved. The optical architecture of the receiver must be designed carefully to ensure the required BER for seamless data communication. The sample receiver performance of the satellite link in the presence of pointing jitters for different receiver design parameters is shown in Fig. 5. As shown in Figs. 5(a) and 5(b), considering random angular pointing jitters of $0.25\;\unicode {x00B5}{\rm rad} \le {\sigma _T} \le 0.75\;\unicode {x00B5}{\rm rad}$ and $\xi = 0.2$, the LL BER for the given link ${{\rm BER}_{{\rm LL}}}$ is calculated as ${10^{- 10.7}} \le {{\rm BER}_{{\rm LL}}} \le {10^{- 8}}$.

Figure 5 also shows that the achievable AL BER [calculated from Eq. (8)] varies notably depending on ${r_d}$, ${m_T}$, $N$, and $\tau$. Figure 5(a) presents the impact of pointing jitter on the achievable BER at $\tau = 0.5$. For instance, the achievable BER varies between about ${10^{- 7.5}}$ and ${10^{- 5.6}}$ when a 0.1 mm detector, a 5x beam compressor, and a FL of $N = 2.0$ are incorporated in the sample receiver architecture (as shown in Fig. 2). It can be realized that the AL performance can be improved by implementing a receiver with larger ${r_d}$, higher ${m_T}$, and larger $N$ (limited by the lens design parameters and available volume). The selection of ${r_d}$ depends on the communication bandwidth, wavelength, and noise equivalent power (NEP). Frequently, these criteria will limit the options. The bandwidth (BW) of the detector is inversely proportional to the detector size, ${\rm BW} \propto 1/{r_d}$. In addition, in many detectors such as PIN photodiodes, ${\rm NEP} \propto {r_d}$ [25,26]. Even though the NEP of most PIN photodiodes increases with the size of the detectors, many communication detectors show opposite behavior, especially avalanche photodiode (APD). For example, two commercially available InGaAs APDs with a diameter of 0.2 mm and 75 µm have NEP of $0.45\;{\rm pW}/\sqrt {{\rm Hz}}$ and $1.1\;{\rm pW}/\sqrt {{\rm Hz}}$, respectively [27]. Similarly, 1.0 mm, 0.5 mm, and 0.2 mm Si APDs show $2.5\;{\rm fW}/\sqrt {{\rm Hz}}$, $0.09\;{\rm pW}/\sqrt {{\rm Hz}}$, and $0.15\;{\rm pW}/\sqrt {{\rm Hz}}$ NEP respectively. As a result, the performance gap between 0.1 mm and 0.25 mm as shown in Figs. 5 and 6 may decrease or increase slightly based on the detector types and integrated bandwidth. We assume almost constant NEP for both detectors, and the integrated receiver’s bandwidth is fixed at 1.0 GHz by electrical filters. This allows us to compare different receiver designs effectively based on the pointing error.

The LL performance is achievable with certain sets of parameters, e.g., $\{{r_d} = 0.1,{m_T} = 5,N = 3\}$ and $\{{r_d} = 0.25,{m_T} = 5,N = 3\}$. The effect of receiver architecture is more prominent for a larger AOI as shown in Fig. 5(b). For instance, the same receiver architecture (${r_d} = 0.1\;{\rm mm}$, ${m_T} = 5$, and $N = 2$) shows significantly poor BER, ${10^{- 4.6}} \le {\rm BER} \le {10^{- 3.5}}$, at $\tau = 0.75$. However, the LL performance is still achievable with $\{{r_d} = 0.25,{m_T} = 3,N = 3\}$ and $\{{r_d} = 0.25,{m_T} = 5,N = 3\}$. Consequently, an optimal design parameters set $\{{r_d},{m_T},N\}$ must be chosen to achieve better link performance over a wide variation of $\tau$ in the presence of pointing errors.

Alongside the parameter selection, an optimum optical design that reduces the optical aberrations is also necessary to improve the performance of the satellite optical links as shown in Fig. 6. Figure 6(a) presents the achievable BER for different aberration factor $\xi$, $0.05 \le \xi \le 0.2$ at ${m_T} = 3.0$. It can be realized that an optical receiver design with low $\xi$ facilitates stable ISOL even at large $\tau$ (e.g., $\tau = 0.75$) in the presence of pointing error. For example, considering the sample receiver architecture as shown in Fig. 2 that incorporates ${m_T} = 3.0$ and $N = 2.0$, numerical analysis shows that about 6 orders of magnitude ($\approx {10^6}$) BER performance improvement is attainable by lowering $\xi$ from 0.2 to 0.05 at low jitter scenario, $\sigma = 0.3\;\unicode {x00B5}{\rm rad}$. Similarly, in the high pointing jitter scenario, $\sigma = 0.8\;\unicode {x00B5}{\rm rad}$, about ${10^4}$ times BER performance improvement is achievable by lowering $\xi$ from 0.2 to 0.05 as shown in Fig. 6(a). Indeed, a low $\xi$ mandates superior optics, complex optical design, higher cost, higher mass, and larger volume. Hence, there is an inevitable trade-off among size, weight, cost, and performance. The performance can be improved by increasing $N$ of the FL while keeping the beam compression ratio unchanged, ${m_T} = 3.0$. For example, at $\sigma = 0.8\;\unicode {x00B5}{\rm rad}$ and $\xi = 0.15$, the calculated BERs are about $1.6 \times {10^{- 3}}$ and $8 \times {10^{- 6}}$ for $N = 2.0$ and $N = 3.0$, respectively. Further performance optimization is possible by increasing ${m_T}$ as shown in Fig. 6(b). For instance, in the low jitter case ($\sigma = 0.3\;\unicode {x00B5}{\rm rad}$), the achievable BER of a receiver that incorporates $N = 3.0$ and $\xi = 0.2$ can be improved to $2.6 \times {10^{- 3}}$ ($\approx {10^{- 8.6}}$) from $1.5 \times {10^{- 6}}$ ($\approx {10^{- 5.8}}$) by increasing ${m_T}$ from 3.0 to 5.0.

Figure 7 presents the performance space of the sample receiver architecture with a given beam compression ratio and $ f $-number of the FL system while assuming a detector radius of 0.1 mm. The solid curves represent the best-case BER for given ${m_T}$ and $N$, whereas the dashed curves present the worst-case performance. Based on the optical design, lens selection, and aberration compensations, different BER curves are achievable that depend on jitter amplitude $\sigma$. However, the performance will be bounded by the solid and dashed curves as shown in Fig. 7. One can realize the receiver’s immunity to vibration jitters and AOI by analyzing the performance space. For example, the best-case BER of the system that incorporates ${m_T} = 3.0$ and $N = 2.0$ as shown in Fig. 7(a) starts rolling off around $\tau = 0.7$, and the worst-case performance fails to establish error-free communication [considering forward error correction (FEC)] when $\tau \ge 0.65$. On the contrary, it can be seen from Fig. 7(d) that if the receiver system is designed with ${m_T} = 5.0$ and $N = 3.0$, the best-case BER and the worst-case BER are immune to the variation of $\sigma$ and $\tau$ up to $\tau \approx 0.9$ and $\tau \approx 0.68$. A similar analysis of the performance roll-off can be done with Figs. 7(b) and 7(c). It is evident that ${{\rm BER}_{3 - 2}} \lt {{\rm BER}_{5 - 2}} \lt {{\rm BER}_{3 - 3}} \lt {{\rm BER}_{5 - 3}}$, where ${{\rm BER}_{{m_T} - N}}$ denotes the overall expected communication performance based on BER for given ${m_T}$ and $N$ in the presence of pointing jitters caused by satellite vibrations. We can notice that a receiver design with a larger $N$ (for a given ${m_T}$) as well as a higher ${m_T}$ (for a given $N$) is expected to perform better over a wider AOI in the presence of jitters as can be seen from Fig. 7. The above-mentioned analyses assume a 100 km communication distance with link parameters given in Table 1. An optimum optical receiver can be designed for a given satellite platform and communication requirement by analyzing major design parameters of the architecture as presented in this manuscript.

## 4. CONCLUSION

In summary, in this paper, we present an analytical model to estimate the impact of optical receiver design parameters on the ISOL in the presence of angular pointing jitters. As a case study, we present the effect of receiver design parameters on a direct detection CubeSat optical link to demonstrate the receiver design techniques with high pointing jitter rejection. Our study points out that a well-engineered optical receiver can have significant tolerance to random pointing errors that are originated from the satellite vibration and noise in the sensor systems. The analyses carried out here can be the basis for optimum optical receiver design to achieve the best ISOL performance.

## Funding

Office of Naval Research (N00014-18-1-2845).

## Disclosures

The authors declare no conflicts of interest.

## REFERENCES

**1. **J. E. Velazco, J. Griffin, D. Wernicke, J. Huleis, A. DeNucci, O. Boyraz, and I. U. Zaman, “Inter-satellite
omnidirectional optical communicator for remote
sensing,” Proc. SPIE **10769**, 157–162
(2018). [CrossRef]

**2. **K. Cahoy, P. Grenfell, A. Crews, M. Long, P. Serra, A. Nguyen, R. Fitzgerald, C. Haughwout, R. Diez, A. Aguilar, J. Conklin, C. Payne, J. Kusters, C. Sackier, M. LaRocca, and L. Yenchesky, “The CubeSat Laser Infrared
CrosslinK mission (CLICK),” Proc.
SPIE **11180**,
358–369 (2019). [CrossRef]

**3. **I. U. Zaman, A. W. Janzen, R. Torun, J. E. Velazco, and O. Boyraz, “Design tradeoffs and
challenges of omnidirectional optical antenna for high speed, long
range Inter CubeSat data communication,” in
*Small Satellite Conference, Delivering Mission Success
(SSC18-WKII-06)* (2018).

**4. **I. U. Zaman, J. E. Velazco, and O. Boyraz, “Realization of omnidirectional
CubeSat crosslink by wavelength-selective optical
transceiver,” IEEE J. Miniat. Air Space
Syst. **1**, 47–55
(2020). [CrossRef]

**5. **J. E. Velazco, J. Griffin, A. W. Janzen, J. Huleis, M. Peng, A. DeNucci, I. U. Zaman, and O. Boyraz, “High data rate inter-satellite
omnidirectional optical communicator,” in *32nd
Annual AIAA/USU Conference on Small Satellites*
(2018), paper SSC18-WKI-02.

**6. **J. Ma, X. Li, S. Yu, L. Tan, and Q. Han, “Influence of satellite
vibration on optical communication performance for intersatellite
laser links,” Opt. Rev. **19**, 25–28
(2012). [CrossRef]

**7. **M. E. Wittig, L. V. Holtz, D. E. L. Tunbridge, and H. C. Vermeulen, “In-orbit measurements of
microaccelerations of ESA’s communication satellite
Olympus,” Proc. SPIE **1218**, 205–214
(1990). [CrossRef]

**8. **F. Sansone, A. Francesconi, R. Corvaja, G. Vallone, R. Antonello, F. Branz, and P. Villoresi, “LaserCube optical
communication terminal for nano and micro satellites,”
Acta Astronaut. **173**,
310–319 (2020). [CrossRef]

**9. **S. Arnon, S. R. Rotman, and N. S. Kopeika, “Performance limitations of
free-space optical communication satellite networks due to vibrations:
direct-detection digital mode,” Proc.
SPIE **3110**,
357–368 (1997). [CrossRef]

**10. **S. Arnon and N. S. Kopeika, “The performance limitations of
free space optical communication satellite networks due to
vibrations-analog case,” in *19th Convention of
Electrical and Electronics Engineers in Israel*
(1996),
pp. 287–290.

**11. **C.-C. Chen and C. Gardner, “Impact of random pointing and
tracking errors on the design of coherent and incoherent optical
intersatellite communication links,” IEEE
Trans. Commun. **37**,
252–260 (1989). [CrossRef]

**12. **J. D. Barry and G. S. Mecherle, “Beam pointing error as a
significant design parameter for satellite-borne, free-space optical
communication systems,” Opt. Eng. **24**, 1049–1054
(1985). [CrossRef]

**13. **T. Song, Q. Wang, M.-W. Wu, T. Ohtsuki, M. Gurusamy, and P.-Y. Kam, “Impact of pointing errors on
the error performance of intersatellite laser
communications,” J. Lightwave
Technol. **35**,
3082–3091 (2017). [CrossRef]

**14. **S. G. Lambert and W. L. Casey, *Laser Communications in Space*
(Artech House,
1995).

**15. **S. Arnon, S. Rotman, and N. S. Kopeika, “Beam width and transmitter
power adaptive to tracking system performance for free-space optical
communication,” Appl. Opt. **36**, 6095–6101
(1997). [CrossRef]

**16. **M. Scheinfeild, N. S. Kopeika, and R. Melamed, “Acquisition system for
microsatellites laser communication in space,”
Proc. SPIE **3932**,
166–175 (2000). [CrossRef]

**17. **J. Chang, C. M. Schieler, K. M. Riesing, J. W. Burnside, K. Aquino, and B. S. Robinson, “Body pointing, acquisition and
tracking for small satellite laser communication,”
Proc. SPIE **10910**,
144–152 (2019). [CrossRef]

**18. **M. Guelman, A. Kogan, A. Kazarian, A. Livne, M. Orenstein, and H. Michalik, “Acquisition and pointing
control for inter-satellite laser communications,”
IEEE Trans. Aerosp. Electron. Syst. **40**, 1239–1248
(2004). [CrossRef]

**19. **H. Liu, Y. Tian, F. L. Lewis, Y. Wan, and K. P. Valavanis, “Robust formation flying
control for a team of satellites subject to nonlinearities and
uncertainties,” Aerospace Sci.
Technol. **95**, 105455
(2019). [CrossRef]

**20. **H. T. Friis, “A note on a simple
transmission formula,” Proc. IRE **34**, 254–256
(1946). [CrossRef]

**21. **G. P. Agrawal, *Fiber-Optic Communication
Systems*, 3rd ed.
(Wiley, 2004),
pp. 162–167.

**22. **R. W. Kingsbury, “Optical communications for small
satellites,” Ph.D. thesis (Massachusetts
Institute of Technology,
2015).

**23. **M. Toyoshima, “Trends in satellite
communications and the role of optical free-space communications
[Invited],” J. Opt. Netw. **4**, 300–311
(2005). [CrossRef]

**24. **I. U. Zaman, J. E. Velazco, and O. Boyraz, “Omnidirectional optical
crosslinks for CubeSats: transmitter optimization,”
IEEE Trans. Aerospace Electron. Syst.
(2020). [CrossRef]

**25. **HAMAMATSU, “*Characteristics and use of infrared
detectors*,” 2004, https://mmrc.caltech.edu/FTIR/Literature/General/Hamamatus%20IR%20dectors.pdf.

**26. **Thorlabs, “*Photodiodes*,” 2020,
https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=285.

**27. **Thorlabs, “*InGaAs Avalanche
Photodetectors/Photodiodes*,” 2020, https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=4047.