Avalanche Photodiode-Based Deep Space Optical Uplink Communication in the Presence of Channel Impairments

Abstract

Optical communication is a critical technology for future deep space exploration, offering substantial advantages in transmission capacity and spectrum utilization. This paper establishes a comprehensive theoretical framework for avalanche photodiode (APD)-based deep space optical uplink communication under combined channel impairments, including atmospheric and coronal turbulence induced beam scintillation, pointing errors, angle-of-arrival (AOA) fluctuations, link attenuation, and background noise. A closed-form analytical channel model unifying these effects is derived and validated through Monte Carlo simulations. Webb and Gaussian approximations are employed to characterize APD output statistics, with theoretical symbol error rate (SER) expressions for pulse position modulation (PPM) derived under diverse impairment scenarios. Numerical results demonstrate that the Webb model achieves higher accuracy by capturing APD gain dynamics, while the Gaussian approximation remains viable when APD gain exceeds a channel fading-dependent gain threshold. Key system parameters such as APD gain and field-of-view (FOV) angle are analyzed. The optimal APD gain significantly influences the achievement of optimal SER performance, and angle of FOV design balances AOA fluctuations tolerance against noise suppression. These findings enable hardware optimization under size, weight, power, and cost (SWaP-C) constraints without compromising performance. Our work provides critical guidelines for designing robust APD-based deep space optical uplink communication systems.

1. Introduction

In deep space wireless communication, optical communication is a well-appreciated method due to numerous advantages, such as large transmission capacity, license-free frequency spectrum, etc., [1]. Deep space optical communication has been utilized in the National Aeronautics and Space Administration (NASA) Lunar Laser Communication Demonstration (LLCD) project [2]. Current research in deep space optical communication has concentrated predominantly on achieving high-rate downlink transmission capabilities [3], while uplink systems remain inadequately addressed. Nevertheless, studying optical uplink communication is essential for several key reasons, summarized as follows. First, channel models for uplink and downlink are different, so receiver-focused approaches optimized for downlink prove inadequate to address cascading impairments in uplink scenarios [4]. Uplink transmissions suffer from atmospheric disturbances in the initial propagation stage, which leads to cumulative beam wandering and intensity fading [5]. Data from European Space Agency (ESA) experiments reveal that transmit-end atmospheric turbulence introduces much higher wavefront aberration variance compared to downlink [6]. Second, deep space optical uplink systems face critical limitations in both hardware miniaturization and signal processing capabilities [7]. Achieving both high photon collection and compact device design requires careful trade-offs due to size, weight, power, and cost (SWaP-C) constraints. For example, NASA’s launch standards limit optical receivers to apertures below 0.5 m and total masses below 50 kg [8]. Third, uplink communication proves indispensable in deep space missions by ensuring mission-critical command transmission (e.g., orbital maneuvers) where even 0.1% data loss risks total mission failure. Therefore, this paper focuses on addressing these critical challenges for deep space optical uplink communication.

For deep space optical channel modeling, Ivanov et al. in [3] investigated the Poisson channel in a controlled laboratory environment. Works in [9,10,11] took into account the coronal turbulence-induced beam scintillation effect. However, the uplink of deep space optical communication suffers from more impairments. As shown in Figure 1, a schematic of a deep space optical communication system is demonstrated. First, atmosphere and solar wind will result in beam scintillation [10,12]. For modeling turbulence-induced beam scintillation, various distributions were proposed, such as Log-normal [13], Gamma-Gamma [14], K-distribution [15], Exponentiated Weibull [16], Fisher–Snedecor [17], and Málaga distribution [18]. Gamma-Gamma has higher accuracy than both Log-normal and K-distribution [14]. In addition, the two-parameter structure of the Gamma-Gamma model enables higher computational efficiency than multi-parameter alternatives like Exponentiated Weibull, Fisher–Snedecor, and Málaga distribution [17,18]. This efficiency benefits real-time optical systems requiring rapid turbulence compensation with minimal latency. Second, beam misalignment is one of the inescapable impairments [19]. Due to the movement of the satellite, it is difficult for an optical beam to aim perfectly at the receiving aperture [20]. Beam misalignments include two aspects, pointing errors and angle of arrival (AOA) fluctuations. In [21], pointing errors were assumed to be identical in both vertical and horizontal axes, and are modeled by Rayleigh distribution. For AOA fluctuations, it was modeled by two-valued (1 and 0) distribution in [22,23]. Factor 1 shows that no signal power is lost and 0 indicates that complete optical signal dropout occurs. Third, link attenuation is also a serious impairment in the optical uplink [24]. The aerosol particles and molecules in the atmosphere cause the scattering and absorption of the optical beam [25,26]. Geometric diffusion-induced attenuation is related to link length, beam divergence angle, and optical receiving aperture size [24]. Last, background noise is a non-negligible damaging factor of deep space optical communication, which depends on parameters of noise sources, FOV, optical filter bandwidth, and receiving aperture area [27]. In conclusion, in the uplink of deep space optical communication, atmospheric and coronal turbulence, pointing errors, AOA fluctuations, link attenuation, and background noise are important impairments and need to be considered in the channel model.

To address SWaP-C constraints while maintaining high communication performance in miniaturized hardware platforms, pulse position modulation (PPM) and on-off keying (OOK) are widely used [2,3,5,9,28]. LLCD utilized the PPM [2], and works in [3,5,7,28] also considered the PPM in deep space optical communication. Xu et al. in [9] highlighted the use of OOK for deep space exploration. For the photodetection of PPM and OOK signals, several optical detectors are commonly deployed, including P-i-N photodiodes (PINs), avalanche photodiodes (APDs), single-photon avalanche diodes (SPADs), and superconducting nanowire single-photon detectors (SNSPDs) [29,30,31,32]. APDs exhibit a compact footprint and lower power consumption compared to PINs [30], and have higher detection efficiency compared to SPADs. In addition, SPADs operate within a temperature range of 0–25 °C [31]. SNSPDs require cryogenic cooling below −268 °C, resulting in prohibitively high power demands, prolonged startup times, and bulky system designs [32]. Thus, APDs are a preferred choice for deep space optical uplink communication. Webb [33] and Gaussian [34] can be used to approximate APD output. Webb distribution offers superior accuracy but is more complex [35]. In previous studies, the Gaussian distribution was widely used due to its simplicity [36,37,38,39,40,41]. However, when the system performance evaluated by the Gaussian model is better than the actual communication performance, the actual link budget will be lower than the analytical link budget analyzed during system design, which significantly increases the probability of link outage.

Motivated by the above analysis, this paper studies an APD-based deep space optical uplink communication in the presence of various channel impairments. The block diagram of our studied system is shown in Figure 2. Specifically, our contributions are as follows:

• We study the combined effects of critical channel impairments in deep space optical uplink systems, including atmospheric and coronal turbulence, pointing errors, AOA fluctuations, link attenuation, and background noise. A closed-form analytical uplink channel model integrating these factors is derived, and its accuracy is rigorously validated through Monte Carlo simulations.
• We use Webb and Gaussian distributions to approximate APD output. For both models, theoretical expressions of PPM communication SER performance are derived. A comparative analysis is performed for the Webb and Gaussian models’ evaluated SER performance.
• We discuss various numerical results, including characteristics of channel destructive factors, Webb and Gaussian modeled APD output and SER performance. While the Gaussian model exhibits inherent limitations in capturing the dynamics of avalanche noise, its operational constraints are analyzed. The results reveal non-monotonic dependencies of SER on the APD gain and FOV angle. Hence, these parameters are strategically optimized to improve the performance of our considered system.

The rest of this article is organized as follows. In Section 2, we analyze various channel impairments and a closed-form expression of the uplink model is derived. Section 3 first introduces the APD output and noise model. In Section 4, theoretical expressions of PPM SER performance based on Webb and Gaussian models are derived. The simulation results are shown and discussed in Section 5. Finally, Section 6 concludes this article.

2. Optical Uplink Channel Model

Considering PPM modulation, after beam propagating from ground station to deep space satellite, received optical power of a PPM slot is expressed as

$$\begin{array}{c}\hfill P_r=h{P}_t+P_b,\end{array}$$

(1)

where h, $P_t$, and $P_b$ are the channel fading coefficient, transmitted optical power, and power of background noise, respectively. In a high speed optical communication scenario [42], h is presumed to be constant over a PPM symbol duration. Various crucial factors impair optical power. We denote the power fading coefficient h as

$$\begin{array}{c}\hfill h=h_t{h}_s{h}_p{h}_a{h}_c,\end{array}$$

(2)

where $h_t$ and $h_s$ are caused by atmospheric turbulence and coronal turbulence-induced beam scintillation, respectively. Moreover, $h_p$, $h_a$, and $h_c$ denote the fading coefficients that result from pointing errors, AOA fluctuations, and link attenuation, respectively.

In the following, the statistical properties of the above channel fading factors are first introduced. Then, a closed-form expression of the probability density function (PDF) of h is derived.

2.1. Statistical Properties of Channel Fading Factors

2.1.1. Atmospheric and Coronal Turbulence-Induced Beam Scintillation

Turbulence will cause optical intensity fluctuations, i.e., beam scintillation. Numerous statistical models for beam scintillation have been proposed, such as Log-normal, Gamma-Gamma, K-distribution, Exponentiated Weibull, Fisher–Snedecor, and Málaga distribution. We adopt the commonly used Gamma-Gamma model, whose PDF is defined as [14]

$$\begin{array}{c}\hfill f_g\left(\left.x\right|\alpha ,\beta \right)={\displaystyle \frac{2{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{x}^{{\displaystyle \frac{\alpha +\beta }{2}}-1}{\mathrm{K}}_{\alpha -\beta }\left(2\sqrt{\alpha \beta x}\right),\end{array}$$

(3)

where $\alpha$ and $\beta$ represent the small-scale and large-scale turbulent eddies, respectively. $\Gamma \left(·\right)$, and ${\mathrm{K}}_{\alpha -\beta }\left(·\right)$ denote the standard gamma function and the modified Bessel function of the second kind of order $(\alpha -\beta )$, respectively. In addition, both $\alpha$ and $\beta$ are closely related to the Rytov variance ${\sigma }_g^2$ of turbulence.

For atmospheric turbulence-caused optical power fading coefficient, it is denoted as $h_t$. The distribution of $h_t$ is as (3), which is rewritten as $f_g\left(\left.h_t\right|{\alpha }_t,{\beta }_t\right)$. Rytov variance ${\sigma }_g^2$ for atmospheric turbulence is rewritten as ${\sigma }_{ht}^2$, which is given by [43]

$$\begin{array}{c}\hfill {\sigma }_{ht}^2\simeq 2.25k_n^{7/6}{sec}^{11/6}\left({\zeta }_g\right){\int }_{H_g}^{H_s}{C_n^2\left(H\right)\left(H_s-H\right)}^{{\displaystyle \frac{5}{6}}}dH,\end{array}$$

(4)

where ${\zeta }_g$ and $k_n=2\pi /\lambda$ are the zenith angle and the optical wave number with $\lambda$ being the wavelength, respectively. In addition, $H,H_s$, and $H_g$ denote the height parameter, satellite height, and ground station altitude, respectively. $C_n^2\left(H\right)$ is the structure parameter for refractive index fluctuation.

Coronal turbulence-caused fading coefficient is expressed as $h_s$. Similarly, the PDF of $h_s$ is as (3) and rewritten as $f_g\left(\left.h_s\right|{\alpha }_s,{\beta }_s\right)$. Rytov variance ${\sigma }_g^2$ for coronal turbulence is rewritten as ${\sigma }_{hs}^2$, which takes the form [9]

$$\begin{array}{cc}\hfill {\sigma }_{hs}^2& \simeq {\displaystyle \frac{1}{2}}{r}_e^2{\left(2\pi \right)}^{{\displaystyle \frac{p_l-3}{2}}}\pi \left(3-p_l\right)\Gamma \left({\displaystyle \frac{p_l}{2}}\right)\Gamma \left({\displaystyle \frac{p_l-1}{2}}\right)\Gamma \left(1+{\displaystyle \frac{p_l}{2}}\right){\zeta }_s^2{N}_e^2{L}_o^{3-p_l}{\left(L_l+\lambda \right)}^{1+{\displaystyle \frac{p_l}{2}}}sec\left({\displaystyle \frac{\pi p_l}{4}}\right),\hfill \end{array}$$

(5)

where $r_e$, $p_l$, and ${\zeta }_s$ represent the classical electron radius, spectral index, and relative solar wind density fluctuations ratio, respectively. In addition, $N_e$, $L_o$, and $L_l$ represent the solar wind density, outer scale of coronal turbulence, and communication distance, respectively.

The parameters of ${\sigma }_{ht}^2$ and ${\sigma }_{hs}^2$ are significant for the intensity of atmospheric and coronal turbulence, respectively. The typical values of ${\sigma }_{ht}^2$ and ${\sigma }_{hs}^2$ are analyzed in Appendix A.1.

2.1.2. Pointing Errors and AOA Fluctuations

The continuous motion of satellites introduces significant challenges in maintaining the precise alignment of the optical beam with the receiving aperture. Beam wander effects further exacerbate pointing inaccuracies and AOA fluctuations. As established in prior research [37], the instantaneous pointing errors can be statistically modeled using Rayleigh distribution. In particular, the optical power fading coefficient $h_p$, induced by these pointing errors, demonstrates a parametric dependence on system characteristics including beam divergence angle and receiver aperture diameter. This relationship underscores the necessity for comprehensive system parameter optimization to mitigate power degradation in space optical communications. Considering a Gaussian beam, PDF of $h_p$ is [44]

$$\begin{array}{c}\hfill f_{h_p}\left(h_p\right)={\displaystyle \frac{w_b^{2}cos{\theta }_a}{4{\sigma }_p^2}}{\left(\mathrm{erf}\left({\displaystyle \frac{\sqrt{\pi }{d}_r}{2\sqrt{2}{w}_d}}\right)\right)}^{-{\displaystyle \frac{w_b^2}{4{\sigma }_p^2}}}{h}_p^{{\displaystyle \frac{w_b^2}{4{\sigma }_p^2}}-1},\end{array}$$

(6)

where $w_d$, ${\sigma }_p^2$, and ${\theta }_a$ are the beam width, normalized variance of pointing errors, and instantaneous AOA, respectively. In addition, $\mathrm{erf}$ is the Gaussian error function, and the equivalent beam width $w_b$ is related to $w_d$.

For AOA fluctuations induced fading coefficient $h_a$, a two-valued distribution can be used, which is [23]

$$\begin{array}{c}\hfill h_a\approx \left\{\begin{array}{c}1,{\theta }_a<{\Omega }_F,\hfill \\ 0,{\theta }_a\ge {\Omega }_F,\hfill \end{array}\right.\end{array}$$

(7)

where ${\Omega }_F$ denotes the angle of FOV. The rationality of this two-valued distribution is discussed here. The intensity of light spot on the APD photon sensing aperture is expressed by the Airy pattern. Let $d_{Ai}$ be the distance between the center and any point in the airy light spot. When $d_{Ai}\ge 25\lambda$, the power of circle spot with radius $d_{Ai}$ accounts for 99% of the total optical power [45]. Taking $\lambda =1550$ nm as an example, $25\lambda =3.87$$\mathsf{\mu }$m is much smaller than the APD photon sensing aperture. Therefore, when the light beam converges at the APD photon sensing aperture, the total received power can be detected; that is, $h_a=1$. When ${\theta }_a>{\Omega }_F$, almost no optical power is detected, i.e., $h_a=0$. Figure 3 shows an example for both cases. Instantaneous AOA ${\theta }_a$ in (7) follows the Rayleigh distribution [44], which is

$$\begin{array}{c}\hfill f_{{\theta }_a}\left({\theta }_a\right)={\displaystyle \frac{{\theta }_a}{{\sigma }_{\theta }^2}}exp\left(-{\displaystyle \frac{{\theta }_a^2}{2{\sigma }_{\theta }^2}}\right),\end{array}$$

(8)

where ${\sigma }_{\theta }^2$ is the variance of AOA.

2.1.3. Link Attenuation

In deep space optical uplink, two essential factors cause the optical power attenuation, geometric diffusion and atmosphere attenuation. Due to the optical receiving aperture size being limited and the optical divergence angle being nonzero, the light spot at the receiver is much larger than the receiving aperture. Consequently, most of the optical power is lost. This is the geometric diffusion attenuation, which is denoted as ${\varsigma }_g={\left(\pi L_l{tan}^2{\theta }_d\right)}^{-1}$, where ${\theta }_d$ is the optical divergence angle. Moreover, the atmosphere contains numerous molecules and aerosol particles, which cause absorption and scattering of optical power. Specific attenuation effects vary with weather conditions, such as clear weather, foggy, hazy, rainy, and snowy. Attenuation in the atmosphere is represented by the Beers–Lambert law, which is given by ${\varsigma }_a=exp\left(-L_a\xi \right)$ [46]. Here, $L_a$ stands for the length of the atmospheric link and $\xi$ is the atmospheric attenuation coefficient related to weather conditions. In addition, interplanetary dust is not a critical impairment for the scenarios analyzed in this work. Consequently, the link attenuation coefficient $h_c$ is derived as

$$\begin{array}{c}\hfill h_c={\varsigma }_g{\varsigma }_a={\displaystyle \frac{1}{\pi L_l{tan}^2{\theta }_{d}exp\left(L_l\xi \right)}}.\end{array}$$

(9)

2.1.4. Background Noise

In deep space optical communication, usually, the optical power received is very small because of the ultra-long link distance. Therefore, background noise can easily impair communication performance. The power of received background noise $P_b$ is [27]

$$\begin{array}{c}\hfill P_b=0.25\pi d_r^2{B}_o{\Omega }_F{N}_b\left(\lambda \right),\end{array}$$

(10)

where $B_o$ is the bandwidth of optical filter, and $N_b\left(\lambda \right)$ is the background radiation spectral radiance. Moreover, $N_b\left(\lambda \right)$ is expressed as [27]

$$\begin{array}{c}\hfill N_b\left(\lambda \right)={\displaystyle \frac{2\hslash c{R}_f}{{\lambda }^{3}exp\left(\hslash c/\phantom{\hslash c{\kappa }_c{T}_b\lambda }{\kappa }_c{T}_b\lambda \right)-1}},\end{array}$$

(11)

where ℏ, c, and ${\kappa }_c$ are the Planck’s constant, light speed, and Boltzmann’s constant, respectively. The parameter $R_f$ is the spectral albedo of a celestial body, defined as the ratio of radiation reflected by the planet (including its atmosphere) to total incident light. In the solar system, the Sun is the luminous body; therefore, the Sun’s spectral albedo value is set to 1. The average spectral albedo of the Earth is $R_f=0.25$. The values of spectral albedo for other important celestial bodies can be found in [27]. It is critical to recognize that other celestial bodies inherently exhibit blackbody radiation even in the absence of solar reflection. Consequently, when such bodies act as independent radiation sources, $R_f$ is equivalently set to unity. In addition, the parameter $T_b$ denotes the average blackbody temperature of a celestial body; for example, the Sun’s $T_b=5776$ K and Earth’s $T_b=288$ K. When the specifics of a deep space mission are known (e.g., communication range, satellite position), the variables in (11) are nearly fixed, thereby determining the value of background radiation spectral radiance.

2.2. Closed-Form Expression for the Channel Model

The comprehensive channel impairments in deep space optical uplinks encompass turbulence-induced beam scintillation (atmospheric $h_t$, coronal $h_s$), pointing errors $h_p$, AOA fluctuations $h_a$, link attenuation $h_c$, and background noise $P_b$. While these parameters are critical for channel modeling, their synergistic impacts remain scarcely addressed in the existing literature. To bridge this gap, we present a novel closed-form analytical model (Equation (2)) that integrates all degradation mechanisms. The methodological framework for deriving the PDF of the composite channel coefficient h is as follows.

First, let $h_2=h_t{h}_s$ represent the beam scintillation caused by atmospheric turbulence and coronal turbulence, the cumulative distribution function (CDF) of $h_2$ is

$$\begin{array}{c}\hfill F_{h_2}\left(h_2\right)={\int }_0^{\infty }{\int }_0^{\infty }{f}_g\left(h_t\right)f_g\left(h_s\right)d{h}_{t}d{h}_s.\end{array}$$

(12)

The PDF of $h_2$ is written as $f_{h_2}\left(h_2\right)$, which is the derivative of the CDF of $h_2$. Hence, with $h_t=h_2/\phantom{h_2{h}_s}{h}_s$, $f_{h_2}\left(h_2\right)$ is derived as

$$\begin{array}{cc}\hfill f_{h_2}\left(h_2\right)& ={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{h}_2}}{\int }_0^{\infty }{\int }_0^{\infty }{f}_g\left({\displaystyle \frac{h_2}{h_s}}\right)f_g\left(h_s\right)d{\displaystyle \frac{h_2}{h_s}}d{h}_s={\int }_0^{\infty }{\displaystyle \frac{1}{h_s}}{f}_g\left({\displaystyle \frac{h_2}{h_s}}\right)f_g\left(h_s\right)d{h}_s.\hfill \end{array}$$

(13)

According to (3), (13) is rewritten as

$$\begin{array}{cc}\hfill & f_{h_2}\left(h_2\right)={\displaystyle \frac{4{\left({\alpha }_t{\beta }_t\right)}^{\frac{{\alpha }_t+{\beta }_t}{2}}{\left({\alpha }_s{\beta }_s\right)}^{\frac{{\alpha }_s+{\beta }_s}{2}}{h}_2^{\frac{{\alpha }_t+{\beta }_t}{2}-1}}{\Gamma \left({\alpha }_t\right)\Gamma \left({\beta }_t\right)\Gamma \left({\alpha }_s\right)\Gamma \left({\beta }_s\right)}}{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{K}}_{{\alpha }_t-{\beta }_t}\left(2\sqrt{{\alpha }_t{\beta }_t{h}_2/\phantom{{\alpha }_t{\beta }_t{h}_2{h}_s}{h}_s}\right){\mathrm{K}}_{{\alpha }_s-{\beta }_s}\left(2\sqrt{{\alpha }_s{\beta }_s{h}_s}\right)}{h_s^{1-\frac{{\alpha }_s+{\beta }_s-{\alpha }_t-{\beta }_t}{2}}}}d{h}_s,\hfill \end{array}$$

(14)

where $K_v\left(x\right)={\displaystyle \frac{1}{2}}{\widehat{\mathrm{G}}}_{0,2}^{2,0}\left[{\left.{\displaystyle \frac{x^2}{4}}\right|}_{v/2,-v/2}\right]$, and ${\widehat{\mathrm{G}}}_{p,q}^{m,n}\left[·\right]$ is the Meijer’s G function. Employing (07.34.21.0011.01) in [47], the result of (14) is

$$\begin{array}{cc}\hfill f_{h_2}\left(h_2\right)& ={\displaystyle \frac{{\alpha }_t{\beta }_t{\alpha }_s{\beta }_s}{h_2\Gamma \left({\alpha }_t\right)\Gamma \left({\beta }_t\right)\Gamma \left({\alpha }_s\right)\Gamma \left({\beta }_s\right)}}{\widehat{\mathrm{G}}}_{0,4}^{4,0}\left[{\left.{\alpha }_t{\beta }_t{\alpha }_s{\beta }_s{h}_2\right|}_{{\alpha }_t-1,{\beta }_t-1,{\alpha }_s-1,{\beta }_s-1}\right].\hfill \end{array}$$

(15)

The specific derivation can be found in Appendix A.2.

Next, let $h_3=h_2{h}_p$ represent the joint effect of turbulence-induced scintillation and pointing errors. Combining the PDF expressions in (6) and (15), the PDF of $h_3$ is derived as

$$\begin{array}{cc}\hfill & f_{h_3}\left(h_3\right)={\displaystyle \frac{C_{\alpha \beta }{C}_{p1}{C}_{p2}^2}{\Gamma \left({\alpha }_t\right)\Gamma \left({\beta }_t\right)\Gamma \left({\alpha }_s\right)\Gamma \left({\beta }_s\right)}}{\widehat{\mathrm{G}}}_{1,5}^{5,0}\left[C_{\alpha \beta }{C}_{p1}{h}_3\left|\begin{array}{c}{C}_{p2}^2\\ C_{p2}^2-1,{\alpha }_t-1,{\beta }_t-1,{\alpha }_s-1,{\beta }_s-1\end{array}\right.\right],\hfill \end{array}$$

(16)

where $C_{\alpha \beta }={\alpha }_t{\beta }_t{\alpha }_s{\beta }_s$, $C_{p1}=w_b^2/\phantom{w_b^{2}2d_r^2}2d_r^2$, and $C_{p2}=w_b/\phantom{w_{b}2{\sigma }_p}2{\sigma }_p$.

Finally, we take into account AOA fluctuations and link attenuation, i.e., $h=h_3{h}_a{h}_c$, where $h_a$ is modeled as (7), and $h_c$ is a constant when link length, divergence angle, and weather condition are given. Therefore, the PDF of h is derived as

$$\begin{array}{cc}\hfill f_h\left(h\right)& ={\displaystyle \frac{C_{\alpha \beta }{C}_{p1}{C}_{p2}^2}{h_c\Gamma \left({\alpha }_t\right)\Gamma \left({\beta }_t\right)\Gamma \left({\alpha }_c\right)\Gamma \left({\beta }_c\right)}}\left(1-exp\left(-{\displaystyle \frac{{\Omega }_F^2}{2{\sigma }_{\theta }^2}}\right)\right){\widehat{\mathrm{G}}}_{1,5}^{5,0}\left[{\displaystyle \frac{C_{\alpha \beta }{C}_{p1}}{h_c}}h\left|\begin{array}{c}{C}_{p2}^2\\ C_{p2}^2-1,{\alpha }_t-1,{\beta }_t-1,{\alpha }_c-1,{\beta }_c-1\end{array}\right.\right]\hfill \\ \hfill & +\delta \left(h\right)·exp\left(-{\displaystyle \frac{{\Omega }_F^2}{2{\sigma }_{\theta }^2}}\right),\hfill \end{array}$$

(17)

where $\delta \left(·\right)$ denotes the Dirac delta function.

In addition, background noise constituting an additive noise component in the received optical power, it is fundamentally governed by the spectral radiance density $N_b\left(\lambda \right)$ through the radiometric relationship shown in (10) and (11). This spectral radiance is intrinsically related to environmental radiation sources, with three dominant scenarios considered in this work: direct solar radiation, Earth-reflected sunlight, and thermal emission from the Earth’s night side, corresponding to maximum, moderate, and minimum noise conditions, respectively [27]. At an operational wavelength of $\lambda =1550$ nm, under blackbody temperature for the Sun (5776 K) and Earth (288 K) with $R_f=1$, $N_b\left(\lambda \right)$ exhibits extreme variability, ranging from $6.4\times {10}^{-20}$ W/Hz in the solar-dominated worst-case scenario to $2.5\times {10}^{-33}$ W/Hz under optimal Earth nighttime conditions. This thirteen-order-of-magnitude disparity highlights the critical role of orbital positioning and receiver pointing accuracy in mitigating background noise, particularly when operating near solar illumination zones, where adaptive filtering strategies become essential for maintaining communication fidelity.

3. APD Output Models

APD efficiently converts optical signals into electrical currents through photon-triggered carrier multiplication, making them compact and energy-efficient detectors for deep space optical uplinks. The McIntyre model [48] fundamentally describes the stochastic detection process of APD through exact conditional probabilities. Subsequent studies developed simplified Webb [33] and Gaussian [35] approximations to facilitate the analysis of communication systems, trading precision for computational tractability. Comparative studies [49] reveal that the Webb approximation better preserves detection statistics despite increased mathematical complexity. This section presents a comparative analysis of both approaches, structured as follows: First, we establish the foundational APD detection principles and noise characteristics. Second, we derive the Webb-approximated probability distribution for the APD output. Third, we formulate the Gaussian-approximated APD output model. Finally, we compare the two models and conclude that the Webb model better characterizes the gain dynamics of APD with higher precision, while the Gaussian approximation is primarily applicable in noise-dominated scenarios.

3.1. APD Detection Process and Noise Model

The photon absorption process in an APD during a PPM time slot follows Poisson statistics, where the absorbed photon count n has a mean value $\overline{n}=P_r{\eta }_a{T}_s/\phantom{P_r{\eta }_a{T}_s\hslash v}\hslash v$. Here, $P_r$ denotes the optical power derived in (1). The parameters ${\eta }_a$, $T_s$, and v are the APD quantum efficiency, PPM time slot width, and optical frequency, respectively. The resultant electron count m, governed by the McIntyre distribution, arises from stochastic avalanche multiplication processes. Therefore, the PDF of electron generation is obtained by convolution of the Poisson and McIntyre PDFs, as established in [48].

The statistical fluctuation in m, arising from both the photon absorption randomness and the avalanche multiplication uncertainty, constitutes the fundamental mechanism of the shot noise in APD-based receivers. The shot noise magnitude depends intrinsically on the quantum efficiency and multiplication statistics governed by the McIntyre model. Beyond fundamental shot noise, the output current of APD exhibits two critical noise components: thermal noise and dark current noise [30]. The thermal noise, originating from electron thermal agitation in the front-end load resistor, follows additive white Gaussian statistics with power spectral density ${\sigma }_{th}^2=4{\kappa }_c{T}_e{B}_e{T}_s^2/\phantom{4{\kappa }_c{T}_e{B}_e{T}_s^2{R}_L}{R}_L$, where $T_e$, $B_e$, and $R_L$ are the APD temperature, bandwidth, and load resistance, respectively. Dark current noise consists of surface leakage current and bulk current, obeying Gaussian statistics with mean $I_D{T}_s$ and variance ${\sigma }_D^2=2q{I}_D{B}_e{T}_s^2$. The composite noise current $i_{td}$ comprising both dark current and thermal noise components follows a composite Gaussian distribution expressed as

$$\begin{array}{c}\hfill \varphi \left(\left.i_{td}\right|{\mu }_{td},{\sigma }_{td}^2\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{td}}}exp\left(-{\displaystyle \frac{{\left(i_{td}-{\mu }_{td}\right)}^2}{2{\sigma }_{td}^2}}\right),\end{array}$$

(18)

where ${\mu }_{td}=I_D{T}_s$ and ${\sigma }_{td}^2={\sigma }_{th}^2+{\sigma }_D^2$.

By integrating the McIntyre–Poisson convolution framework for photon-generated electron statistics [48] with the Gaussian noise component (18), the comprehensive APD output current model is derived as

$$\begin{array}{c}\hfill p_m\left(i\right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{n=1}^m}\left({\displaystyle \frac{{\left(\overline{n}\right)}^n\Gamma \left(c_{ka}m/\phantom{c_{ka}m{k}_a}{k}_a+1\right){\left(c_G-c_G{k}_a\right)}^{m-n}}{\sqrt{2\pi }{\sigma }_{m}m\left(n-1\right)!\left(m-n\right)!}}{\displaystyle \frac{{\left(1/\phantom{1G}G+k_a{c}_G\right)}^{n+c_{ka}m}}{\Gamma \left(c_{ka}m+n+1\right)}}exp\left(-\overline{n}-{\displaystyle \frac{{\left(i-{\mu }_m\right)}^2}{2{\sigma }_m^2}}\right)\right),\end{array}$$

(19)

where G and $k_a$ are the APD gain and the APD ionization ratio, respectively. In addition, $c_{ka}=k_a/\phantom{k_a\left(1-k_a\right)}\left(1-k_a\right)$, $c_g=\left(G-1\right)/\phantom{\left(G-1\right)G}G$, ${\sigma }_m^2={\sigma }_{td}^2$, and ${\mu }_m=m{q}_e+{\mu }_{td}$ with $q_e$ being the electron charge.

3.2. Webb and Gaussian-Approximated APD Output Model

To balance computational efficiency with model fidelity in APD characterization, Webb model [33] provides an optimal framework for approximating the stochastic response of APD output. Given the mean received photons $\overline{n}$, the conditional PDF of electron multiplication $f_W\left(\left.m\right|\overline{n}\right)$ approximated by the Webb distribution is

$$\begin{array}{cc}\hfill f_W\left(\left.m\right|\overline{n}\right)& ={\displaystyle \frac{{\left(1+\left(m-G\overline{n}\right)/\phantom{\left(m-G\overline{n}\right)c_f\overline{n}}{c}_f\overline{n}\right)}^{2/3}}{\sqrt{2\pi \overline{n}{G}^2{F}_a}}}exp\left(-{\displaystyle \frac{{\left(m-G\overline{n}\right)}^2}{2\overline{n}{G}^2{F}_a\left(1+\left(m-G\overline{n}\right)/\phantom{\left(m-G\overline{n}\right)c_f\overline{n}}{c}_f\overline{n}\right)}}\right),\hfill \end{array}$$

(20)

where $F_a$ is the APD noise factor, and $c_F={\displaystyle \frac{G{F}_a}{F_a-1}}$. Through statistical convolution of the Webb-distributed photon response (20) with Gaussian noise component (18), the composite APD output model preserving avalanche multiplication characteristics is formulated as

$$\begin{array}{c}\hfill p_w\left(i\right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\left(1+\left(m-G\overline{n}\right)/\phantom{\left(m-G\overline{n}\right)c_f\overline{n}}{c}_f\overline{n}\right)}^{2/3}}{2\pi {\sigma }_m\sqrt{\overline{n}{G}^2{F}_a}}}exp\left(-{\displaystyle \frac{{\left(m-G\overline{n}\right)}^2}{2\overline{n}{G}^2{F}_a\left(1+\frac{m-G\overline{n}}{c_f\overline{n}}\right)}}-{\displaystyle \frac{{\left(i-{\mu }_m\right)}^2}{2{\sigma }_m^2}}\right).\end{array}$$

(21)

To further simplify the Webb modeled APD output, Gaussian approximation is strategically implemented. That is, the conditional PDF of electron multiplication $f_W\left(\left.m\right|\overline{n}\right)$ under mean photon flux $\overline{n}$ transitions to Gaussian statistics. Hence, (20) is rewritten as

$$\begin{array}{c}\hfill f_G\left(\left.m\right|\overline{n}\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{sh}^2}}exp\left(-{\displaystyle \frac{{\left(m-{\mu }_{sh}\right)}^2}{2{\sigma }_{sh}^2}}\right),\end{array}$$

(22)

where ${\sigma }_{sh}^2=2G^2{q}_e^2{F}_a\overline{n}{B}_e{T}_s$ and ${\mu }_{sh}=G{q}_e\overline{n}$ are the variance and the mean of this Gaussian distribution, respectively. Incorporating APD noise characteristics, the APD output approximated by Gaussian model is

$$\begin{array}{c}\hfill p_g\left(i\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_g^2}}exp\left(-{\displaystyle \frac{{\left(i-{\mu }_g\right)}^2}{2{\sigma }_g^2}}\right),\end{array}$$

(23)

where ${\mu }_g={\mu }_{sh}+{\mu }_m$ and ${\sigma }_g^2={\sigma }_{sh}^2+{\sigma }_m^2$.

The Webb model [33] establishes a rigorous framework for APD avalanche gain dynamics through the statistical convolution of photon absorption statistics and nonlinear multiplication processes. This approach effectively captures multiplicative shot noise mechanisms (i.e., avalanche fluctuations) that dominate in gain-saturated operational regimes, as quantified in Section 5. In contrast, the Gaussian approximation model employs an additive Gaussian noise assumption to simplify APD output analysis, sacrificing fidelity in modeling nonlinear gain dynamics. While this abstraction disregards the inherent nonlinear gain dynamics captured by the Webb model, it offers significant computational advantages for noise-limited scenarios. Specifically, under high background radiation conditions as discussed in Section 5, where system performance becomes dominated by external noise sources rather than intrinsic APD gain characteristics, the Gaussian model achieves acceptable accuracy.

4. PPM SER Performance Analysis

In deep space optical uplink communication systems constrained by power budgets, PPM is strategically implemented due to its superior photon efficiency under average transmission power constraints. To quantify communication reliability, we adopt the SER as the critical performance metric, defined as the probability of erroneous identification of the PPM symbol. This section presents a comprehensive SER analysis based on both Webb and Gaussian-approximated APD output models, specifically incorporating channel impairments: atmospheric and coronal turbulence-induced fading, pointing error fluctuations, AOA fluctuations, link attenuation, and background noise. Comprehensive analysis provides critical insights for designing robust deep space optical uplink communication systems.

The conditional SER for PPM systems under channel fading coefficient h is analytically formulated as

$$\begin{array}{c}\hfill E_{\left.p\right|h}=1-{\int }_{-\infty }^{\infty }{p}_e\left(\left.i_x\right|1\right)d{i}_x{\left({\int }_{-\infty }^{i_x}{p}_e\left(\left.i_y\right|0\right)d{i}_y\right)}^{Q-1},\end{array}$$

(24)

where Q denotes the PPM symbol size. Conditional probability densities $p_e\left(\left.i_x\right|1\right)$ and $p_e\left(\left.i_y\right|0\right)$ characterize the APD output under pulse transmission (bit 1) and silence states (bit 0), respectively. These distributions are formulated through both Webb (21) and Gaussian (23)-approximated stochastic APD output. The statistical characterization of the deep space optical uplink channel is analytically formulated as $f_h\left(h\right)$ in (17). Under the influence of various channel impairments, the unconditioned SER of the PPM system is obtained by

$$\begin{array}{c}\hfill E_p={\int }_0^{\infty }{f}_h\left(h\right)E_{\left.p\right|h}dh.\end{array}$$

(25)

Consequently, through Webb-modeled APD statistical output, the PPM SER performance is derived as

$$\begin{array}{c}\hfill E_W={\int }_0^{\infty }{f}_h\left(h\right)\left(1-\mathrm{InS}\left(i_x,{\displaystyle \frac{\frac{1}{c_2\sqrt{x}}{\left(1+\frac{c_x}{c_{f}x}\right)}^{2/3}}{exp\left(\frac{c_x^2}{c_1\overline{n}+\frac{c_x{c}_{22}\overline{n}}{c_{f}x}}+\frac{{\left(i_x-{\mu }_m\right)}^2}{2{\sigma }_m^2}\right)}}\right){\mathrm{InS}}^{Q-1}\left(i_y,{\displaystyle \frac{\frac{1}{c_2\sqrt{{\overline{n}}_b}}{\left(1+\frac{c_y}{c_f{\overline{n}}_b}\right)}^{2/3}}{exp\left(\frac{c_y^2}{c_1{\overline{n}}_b+\frac{c_y{c}_{22}{\overline{n}}_b}{c_f{\overline{n}}_b}}+\frac{{\left(i_y-{\mu }_m\right)}^2}{2{\sigma }_m^2}\right)}}\right)\right)dh.\end{array}$$

(26)

where $\mathrm{InS}\left(i,·\right)$ represents the operation of ${\int }_{-\infty }^{\infty }{\displaystyle \sum _{m=0}^{\infty }}\left(·\right)di$. Parameters $x=h{\overline{n}}_s+{\overline{n}}_b$, $c_x=m-Gx$, $c_y=m-G{\overline{n}}_b$, $c_1=2G^2{F}_a$, $c_2=\pi {\sigma }_m\sqrt{2c_1}$. The complete mathematical derivation of this SER formulation is provided in Appendix A.2. The Webb-modeled SER formulation (26) encounters analytical intractability in resolving this multidimensional integration complexity. We can use trapezoidal rules and simulation tools to obtain significant numerical computation results.

For the Gaussian-approximated APD output model operating under composite channel conditions, the SER of PPM framework is analytically formulated as

$$\begin{array}{cc}\hfill E_G=& {\displaystyle \frac{Q-1}{2}}exp\left(-{\displaystyle \frac{{\Omega }_F^2}{2{\sigma }_{\theta }^2}}\right)+{\displaystyle \frac{(Q-1)C_{\alpha \beta }{C}_{p1}{C}_{p2}^2}{2h_c\Gamma \left({\alpha }_t\right)\Gamma \left({\beta }_t\right)\Gamma \left({\alpha }_s\right)\Gamma \left({\beta }_s\right)}}\left(1-exp\left(-{\displaystyle \frac{{\Omega }_F^2}{2{\sigma }_{\theta }^2}}\right)\right)\hfill \\ \hfill & \times {\int }_0^{\infty }{\widehat{\mathrm{G}}}_{1,5}^{5,0}\left[{\displaystyle \frac{C_{\alpha \beta }{C}_{p1}}{h_c}}h\left|\begin{array}{c}{C}_{p2}^2\\ C_{p2}^2-1,{\alpha }_t-1,{\beta }_t-1,{\alpha }_s-1,{\beta }_s-1\end{array}\right.\right]erfc\left({\displaystyle \frac{G{q}_e{\overline{n}}_{s}h}{\sqrt{c_{g1}\left(2{\overline{n}}_b+h{\overline{n}}_s\right)+c_{g2}}}}\right)dh.\hfill \end{array}$$

(27)

where $c_{g1}=2q_e^2{G}^2{F}_a$, $c_{g2}=4q_e{I}_D{T}_s+{\displaystyle \frac{8{\kappa }_c{T}_e{T}_s}{R_L}}$, and $\mathrm{erfc}\left(·\right)$ is the complementary error function. Correspondingly, the mathematical derivation process underlying these results is rigorously presented in Appendix A.2.

5. Numerical Results and Discussion

This section presents various numerical results. First, the effects of various fading factors are discussed. Then, we compare Webb and Gaussian-approximated APD output and SER performance. Finally, key system parameters are properly designed to optimize communication performance.

For a single transmission, where the link length, divergence angle, and atmosphere loss factor are fixed, the channel attenuation is constant. Therefore, for simplicity, we normalize the channel attenuation coefficient, i.e., $h_c=1$. In addition,

Table 1. Simulation parameters.

Parameter	Symbol	Value
Optical wavelength	$\lambda$	1550 nm
PPM symbol size	Q	16
Slot width	$T_s$	$2\times {10}^{-8}$ s
Optical filter bandwidth	$B_o$	$12.5\times {10}^9$ Hz
APD quantum efficiency	${\eta }_a$	$0.9$
APD Bandwidth	$B_e$	$2.5\times {10}^7$ Hz
Dark current	$I_D$	$1\times {10}^{-9}$ A
Load resistance	$R_L$	$146,650$$\Omega$
Ionization ratio	$k_a$	$0.007$
Field of view	${\Omega }_F$	150 $\mathsf{\mu }$rad
Spectral albedo of the Earth	$R_f$	0.25
Optical receiving aperture diameter	$d_r$	$0.1$ m
Background radiation spectral radiance	$N_b\left(\lambda \right)$	$6.4\times {10}^{-20}$ W/Hz

summarizes other constant parameters used in simulations.

5.1. Results of the Channel Model

For atmospheric and coronal turbulence, values of their Rytov variances are shown in Appendix A.1. We use ${\sigma }_{ht}^2=0.4$ and ${\sigma }_{hc}^2=0.6$ for weak turbulence, and ${\sigma }_{ht}^2=2$ and ${\sigma }_{hc}^2=3$ for strong turbulence. Consider that the radius of light spot have been normalized at the receiver, we use ${\sigma }_p=0.1$ and ${\sigma }_p=0.3$ to represent small and large pointing errors, respectively. For simulation of AOA fluctuations, we use ${\sigma }_{\theta }=20$$\mu$rad for small fluctuations and use ${\sigma }_{\theta }=100$$\mu$rad for large fluctuations.

Our derived closed-form expression for the uplink channel model is shown as (17). The analytical results and the Monte Carlo simulation results under different ${\sigma }_{ht}^2$, ${\sigma }_{hs}^2$, ${\sigma }_p$, and ${\sigma }_{\theta }$ are shown in Figure 4. It can be seen that the analytical results agree well with Monte Carlo simulation results, which confirms the accuracy of our derived closed-form expression of the uplink model. Moreover, it is clear that the average channel fading coefficient, i.e., mean of h, decreases with the increase in ${\sigma }_{ht}^2$, ${\sigma }_{hc}^2$, ${\sigma }_p$, and ${\sigma }_{\theta }$.

In addition, as shown in Figure 4a,b, respectively, the distribution shapes of the PDF curves under various values of ${\sigma }_{ht}^2$, ${\sigma }_{hs}^2$, and ${\sigma }_p$ are different. However, in Figure 4c, the distribution shapes of the PDF curves are similar for different AOA fluctuations. Specifically, curvilinear trends are the same, and only absolute PDF values are different when ${\sigma }_{\theta }$ is 20 $\mathsf{\mu }$rad and 100 $\mathsf{\mu }$rad. The reasons for these cases are that various atmospheric turbulence, coronal turbulence, and pointing errors will cause different fading in the channel. However, in the model of AOA fluctuations, $h_a$ is reasonably assumed to be two values, 1 and 0, which correspond to the optical beam being captured successfully and unsuccessfully, respectively. That is, $h_a$ does not influence the channel distribution when $h_a=1$, and the probability of $h_a=0$ increases with increasing ${\sigma }_{\theta }$ when FOV is given. Hence, in addition to the results shown in Figure 4c, it is reasonable to infer that the probability of $h_a=0$ of the blue curve is bigger than that of the red curve.

5.2. Results of the Webb and Gaussian Modeled SER

For performance analysis of APD-based optical communication systems, many papers have shown that the Webb model is more precise than the Gaussian model [33,35,49]. Of course, the Gaussian model is also widely used for performance analysis due to the simplicity of expressions [36,37,38,39,40,41]. Here, we first investigate whether the Gaussian model is precise enough to characterize the considered deep space optical uplink system. In particular, we compare the SER estimated by Webb model and Gaussian model, with the Webb-estimated SER as the performance benchmark. SER performance is studied under various channel conditions. The applicable conditions where the Gaussian model can give a close fit to the Webb model, i.e., having a low approximation error, are discussed.

Figure 5a shows comparisons of SER using Webb and Gaussian models under different average background noise and over the lossless link, where APD gain is 40. Parameters ${\overline{n}}_b$ and ${\overline{n}}_s$ denote the average number of background noise photons and received optical photons in a PPM symbol, respectively. Under both Webb and Gaussian models, SER decreases monotonically with increasing received photons, i.e., signal-to-noise ratio (SNR), aligning with expectations. In addition, SER exhibits an inverse dependence on the background noise level. In particular, under ${\overline{n}}_b=1$, the Gaussian approximation achieves remarkable congruence with the Webb model, validating the efficacy of the Gaussian model in noise-dominated regimes as analyzed in Section 3.2. At ${\overline{n}}_b=0.01$, the Gaussian model systematically underestimates SER relative to the Webb framework, highlighting its inherent limitations in high-sensitivity regimes. However, in this scenario, the Gaussian model can still be utilized to evaluate the system performance. This is because the communication performance estimated by the Gaussian model is worse than the actual system performance. Consequently, during system design, to achieve specific communication performance targets, the designed system redundancy (referring to the performance margin, such as additional power or error correction capability) becomes higher than the actual system redundancy. For instance, the actual received signal power might be greater than pre-research design assumptions. Consequently, over the lossless link, Webb with high accuracy and Gaussian with low complexity are both applicable in optical communications.

In the fading channel, comparisons of SER using Webb and Gaussian models with APD gain being 40 are shown in Figure 5b. It can be seen that, when average received photons are given, the SER becomes worse as pointing errors, AOA fluctuations, and atmospheric and coronal turbulence intensity increase. Moreover, in Figure 5b, SER decreases with the increase in the average received photons under various ${\sigma }_{ht}^2$, ${\sigma }_{hs}^2$, and ${\sigma }_p$, but in Figure 5c, the SER first decreases, and then remains constant, which means average received photons reaching saturation point. This is because ${\sigma }_{\theta }$ can cause an optical receiving failure, which further results in SER error floors. In other words, the error floor level is decided primarily by ${\sigma }_{\theta }$. In particular, the error floor increases with increasing ${\sigma }_{\theta }$.

Moreover, when the error floor is not present in Figure 5b,c, Gaussian model estimated SER is smaller than Webb model estimated SER under a given average received photons, which differs from Figure 5a. That is, communication redundancy estimated by the Gaussian model is higher than that by the Webb model. The actual redundancy of APD-based communication system can be evaluated by the Webb model, so the Gaussian model overestimates communication performance and no longer applies under the circumstances of Figure 5b,c.

As illustrated in Figure 6a, comparative analysis of SER dynamics under varying APD gain G reveals fundamentally distinct behaviors between the Webb and Gaussian models. For fixed turbulence parameters ${\sigma }_{ht}^2$ and ${\sigma }_{hs}^2$, the Gaussian-modeled SER exhibits a non-monotonic trend: initially decreasing sharply with G, reaching a minimum, then gradually increasing due to noise amplification. The Webb modeled SER rapidly decreases before stabilizing asymptotically. This divergence is due to the models’ contrasting treatments of signal-to-shot noise ratio (SSNR) shown in Figure 6b. At low-to-moderate gains ($G<G_T$), both models predict SER reduction dominated by signal amplification, where photon-generated electrons scale quadratically with G while thermal and dark current noise remain constant. The critical threshold $G_T$, defined as the intersection point of the Webb and Gaussian SER curves, decreases logarithmically with increasing ${\sigma }_{ht}^2$ and ${\sigma }_{hs}^2$. Beyond $G_T$, Gaussian-modeled SER degrades due to its large noise amplification, whereas Webb-modeled SER stabilizes by incorporating McIntyre-distributed gain statistics that inherently bound excess noise growth. These dynamics underscore the necessity of the Webb model in gain-dominated regimes ($G<G_T$). In addition, the threshold gain $G_T$ exhibits a positive correlation with channel fading, decreasing from 760 to 300 when ${\sigma }_{ht}^2$, ${\sigma }_{hs}^2$ decrease from (2, 3) to (0.4, 0.6), corresponding to a 60.5% mitigation. Similar trends appear under pointing errors and AOA fluctuations, as detailed in Appendix A.3. These significant results reveal that adaptive model switching based on real-time gain monitoring and channel state estimation optimizes receiver efficiency for deep space optical uplink communication under various channel impairments.

In general, deep space optical uplink systems confront multi-factor channel impairments, including atmospheric and coronal turbulence, pointing errors, AOA fluctuations, and background noise. For modeling the APD output and estimating system SER performance, Webb model achieves high accuracy by greatly resolving the APD gain dynamics. Gaussian approximation becomes operationally viable when the APD gain exceeds the critical threshold $G_T$, where noise is dominated.

5.3. Parameters Design

The performance of deep space optical uplink communication can be optimized by designing key parameters of the system. First, the FOV angle ${\Omega }_F$ critically governs the performance through its dual influence on AOA fluctuations and background noise. As illustrated in (7) and (10), small ${\Omega }_F$ configurations mitigate background noise by spatially filtering extraneous photons but exacerbate signal loss from AOA fluctuations. In contrast, a large ${\Omega }_F$ improves beam capture robustness at the cost of amplified background noise. Figure 7 illustrates the relationship between the received background photon count and the field-of-view angle ${\Omega }_F$. The results demonstrate that, when implementing 16-PPM modulation with a 10 MHz slot rate, the average background photon number ${\overline{n}}_b$ exceeds 10 photons/slot once ${\Omega }_F$ exceeds 0.5 mrad. Therefore, it is necessary to consider the trade-off between AOA fluctuations and background noise when designing the angle of FOV.

From Figure 6, it is concluded that the APD gain is a significant parameter affecting system performance. As the APD gain G increases, the SER predicted by the Webb model initially decreases before stabilizing at a relatively constant level, while the Gaussian model exhibits a distinct behavior in which the SER first reduces and then subsequently increases. Given that the Webb model provides more accurate characterization of practical APD operation compared to the Gaussian approximation, the optimal APD gain determination should primarily rely on the SER trend observed through Webb modeling. Specifically, this optimal gain corresponds to the transition point where the Webb model’s SER curve approaches its stable plateau region. For example, as shown in Figure 6c, the Webb model identifies approximately 150 as the optimal value of G. However, when employing the computationally simpler Gaussian model while requiring enhanced estimation accuracy, the optimal gain selection must satisfy dual criteria: achieving minimal SER according to the Webb model reference while simultaneously meeting the Gaussian model’s operational precondition $G\ge G_T$ established in prior analysis. This explains why, under identical conditions in Figure 6c, despite the Gaussian model suggesting $G=150$ as its minimum SER point, the practically viable value of G should be elevated to 500 to comply with both performance optimization requirements and model validity constraints.

6. Conclusions

This study established a comprehensive analytical framework for APD-based deep space optical uplink systems under combined channel impairments, including atmospheric and coronal turbulence-induced scintillation, pointing errors, AOA fluctuations, link attenuation, and background noise. A closed-form analytical channel model integrating these effects was derived and validated through Monte Carlo simulations, demonstrating strong alignment with the empirical results. The Webb and Gaussian models were used to characterize the APD output statistics. The Webb model exhibited higher accuracy in capturing nonlinear gain dynamics and SER trends. Gaussian approximation introduced significant deviations in the SER estimation unless the APD gain exceeded a critical threshold, which decreased under intensified channel fading. This highlighted the need for adaptive model selection based on real-time gain monitoring and channel conditions. Key system parameters, such as APD gain, angle of FOV, and o ptimal APD gain significantly influenced the achievement of optimal SER performance, balancing signal amplification and noise saturation.FOV design required a trade-off: small FOV angles reduced background noise but increased AOA-induced signal loss, while large FOV angles improved alignment robustness at the cost of noise intrusion. These findings enabled hardware optimization under SWaP-C constraints. Our work provided critical guidelines for designing robust APD-based deep space optical uplink communication systems.