Performance Analysis of Space-to-Ground Downlink for Polarization Shift Keying Optical Communications with a Gaussian-Schell Model Beam

Abstract

Free-space optical communication has emerged as a pivotal technology for space-to-ground downlinks; however, signal degradation caused by atmospheric turbulence continues to pose a significant challenge. In this study, a model for the polarization transmission characteristics of a Gaussian-Schell model (GSM) beam in downlink was established, and conditions sufficient for maintaining the polarization transmission characteristics were derived. The impact of the source spatial coherence on the performance of optical communication systems using circular polarization shift keying (CPolSK) modulation was investigated. Additionally, models for the probability density distribution and scintillation index of the optical intensity under atmospheric turbulence were developed along with a bit error rate model for the optical communication system. The effects of the laser spatial coherence on these models were also analyzed. The results indicate that the optimal performance in the turbulent downlink is achieved with fully coherent light, where the GSM-beam-based CPolSK-modulated system demonstrates a reduction of 1.51 dB in the required power compared to that of an on–off keying system. The implications of this study suggest that optimizing spatial coherence could significantly enhance the reliability of space-to-ground communication systems under atmospheric disturbances.

1. Introduction

Free-space optical (FSO) communication, which uses lasers as signal carriers, has gained significant attention as a complementary solution to radio frequency communication for numerous applications and services [1]. The key benefits of FSO communication include high data rates, broad bandwidth, strict security, and compact components [2].

However, the development of FSO systems is constrained by atmospheric conditions. The transmitted signal is highly sensitive to atmospheric turbulence. Specifically, turbulence-induced intensity fluctuations, known as optical scintillation, are regarded as the primary limitation of the system performance [3,4]. To counteract the effects of atmospheric turbulence, various strategies have been suggested, such as partially coherent light, adaptive optics, and spatial diversity [5,6,7].

The application of spatial diversity and adaptive optics technologies significantly increases the system complexity and production costs. In comparison, the application of partially coherent light as a source in spatial optical communication systems offers a simpler and more cost-effective solution [8]. Partially coherent light, as exemplified by Gaussian Schell-model (GSM) laser beams, is less susceptible to atmospheric turbulence than fully coherent light. Consequently, numerous researchers have concentrated on utilizing partially coherent light to minimize the bit error rate (BER) of systems [9,10].

The emerging polarization shift keying (PolSK) technique has been proposed for partially counteracting the effects of turbulent atmosphere on space optical communication systems [11]. This method encodes information onto a carrier using the polarization state of a beam, reducing the required optical power by 3 dB compared to that of the on–off keying (OOK) modulation scheme [12].

Most previous studies on PolSK primarily focused on the polarization characteristics of horizontal-link communications [13,14,15]. To fully utilize the benefits of a PolSK system using GSM laser beams, it is essential to identify the conditions necessary for preserving the polarization characteristics of the downlink [16]. Nevertheless, existing PolSK-based space laser communication systems require further enhancement, particularly in terms of communication performance and data transmission rate.

This study innovatively integrates the advantages of partially coherent light and PolSK technology to enhance the performance of FSO-based space communication systems. A GSM-beam-based polarization transmission characteristic model for downlink was developed, and sufficient conditions for maintaining the polarization transmission characteristics were derived. To elucidate the impact of source spatial coherence on the performance of an optical communication system using a GSM-beam-based circular PolSK (CPolSK) modulation scheme, probability density distribution and scintillation index models of optical intensity were derived for an atmospheric turbulence environment. Additionally, a BER model for the CPolSK-modulated optical communication system was established. The simulation results demonstrated that, compared to an OOK system, the GSM-beam-based CPolSK-modulated system achieved a power reduction of 1.51 dB when using fully coherent light under identical conditions. These findings highlight the potential of PolSK-based optical communication for improving system reliability in turbulent environments. This work focuses on circular PolSK; an alternative of multiplexing linear OOK polarization channels is left for future study.

2. Theoretical Model of Polarization Characteristics

To determine the performance of a CPolSK laser communication system, it is crucial to understand the polarization characteristics of the propagated signal in a turbulent atmosphere. This section discusses the polarization transmission characteristic model of a GSM beam for downlink and the sufficient conditions required to maintain the polarization transmission characteristics.

In isotropic turbulence, polarization states are preserved on average, so a circularly polarized beam remains circular during propagation. Circular polarization is chosen here for convenience of modulation and detection; in the isotropic turbulence considered, orthogonal linear polarization (horizontal/vertical) would yield a similar performance. The cross-spectral density matrix components at the transmitter can be obtained as follows [17]:

$$W_{ij}\left({\rho }_1,{\rho }_2,0;\omega \right)=A_i{A}_j{B}_{ij}\mathit{exp}\left[-{\displaystyle \frac{{\rho }_1^2+{\rho }_2^2}{4w_0^2}}\right]\mathit{exp}\left[-{\displaystyle \frac{({\rho }_2-{\rho }_1{)}^2}{2{\delta }_{ij}^2}}\right]$$

(1)

Here, ρ₁ and ρ₂ denote the position vector on the source plane. i and j index the Cartesian polarization components; δ_ij and w₀ represent the coherence lengths and half-width, respectively. The terms A_i and A_j denote two orthogonal electric field amplitude components, and B_ij corresponds to the correlation coefficient between them. Notably, the coefficients A_i, B_ij, w₀, and δ_ij are positive, frequency-dependent, and position-independent.

In particular, B_ij and δ_ij must satisfy the following expressions [18]:

$$\begin{array}{c}{B}_{ij}=1,wheni=j,\\ \left|B_{ij}\right|\le 1,wheni\ne j\\ B_{ij}=B_{ji}^{*},{\delta }_{ij}={\delta }_{ji}.\end{array}$$

(2)

The degree of polarization (DOP) is defined as the ratio of the intensity of the polarized portion to the total intensity [19], which is expressed as

$$P\left(\mathit{r},z;\omega \right)=\sqrt{1-{\displaystyle \frac{4\mathit{det}\left[\stackrel{⃡}{W}\left(\mathit{r},z;\omega \right)\right]}{{\left[Tr\stackrel{⃡}{W}\left(\mathit{r},z;\omega \right)\right]}^2}}},$$

(3)

where det represents the determinant of the matrix and Tr represents its trace. The state of polarization (SOP) of the polarized portion of the beam is characterized by the polarization ellipse, which, in turn, is characterized by the degree of ellipticity and the orientation angle [20].

$$\alpha \left(\mathit{r},z;\omega \right)={\displaystyle \frac{1}{2}}\mathit{arctan}\left({\displaystyle \frac{2\mathit{Re}\left[W_{xy}\left(\mathit{r},z;\omega \right)\right]}{W_{xx}\left(\mathit{r},z;\omega \right)-W_{yy}\left(\mathit{r},z;\omega \right)}}\right),-{\displaystyle \frac{\pi }{2}}\le \alpha \le {\displaystyle \frac{\pi }{2}}$$

(4)

$$\epsilon ={\displaystyle \frac{A_{minor}}{A_{major}}}=\sqrt{{\displaystyle \frac{{\left[{\left(W_{xx}-W_{yy}\right)}^2+4{\left|W_{xy}\right|}^2\right]}^{\frac{1}{2}}-{\left[{\left(W_{xx}-W_{yy}\right)}^2+4{\left[\mathit{Re}{W}_{xy}\right]}^2\right]}^{\frac{1}{2}}}{{\left[{\left(W_{xx}-W_{yy}\right)}^2+4{\left|W_{xy}\right|}^2\right]}^{\frac{1}{2}}+{\left[{\left(W_{xx}-W_{yy}\right)}^2+4{\left[\mathit{Re}{W}_{xy}\right]}^2\right]}^{\frac{1}{2}}}}}$$

(5)

The terms A_major and A_minor correspond to the major and minor semiaxes, respectively. For a circularly polarized GSM laser beam, the term α(r, z; ω) varies within the range of −π/2–π/2, whereas the degree of ellipticity ε is unity [21]. Consequently, the DOP and SOP at the source plane can be expressed as follows.

$$P\left(\mathit{\rho },z=0;\omega \right)=\sqrt{{\displaystyle \frac{{\left(A_x^2-A_y^2\right)}^2+4A_x^2{A}_y^2{\left|B_{xy}\right|}^2}{{\left(A_x^2+A_y^2\right)}^2}}}$$

(6)

$$\epsilon ={\displaystyle \frac{A_{minor}}{A_{major}}}=\sqrt{{\displaystyle \frac{{\left[{\left(A_x^2-A_y^2\right)}^2+4A_x^2{A}_y^2{\left|B_{xy}\right|}^2\right]}^{\frac{1}{2}}-{\left[{\left(A_x^2-A_y^2\right)}^2+4A_x^2{A}_y^2{\mathit{Re}\left|B_{xy}\right|}^2\right]}^{\frac{1}{2}}}{{\left[{\left(A_x^2-A_y^2\right)}^2+4A_x^2{A}_y^2{\left|B_{xy}\right|}^2\right]}^{\frac{1}{2}}+{\left[{\left(A_x^2-A_y^2\right)}^2+4A_x^2{A}_y^2{\mathit{Re}\left|B_{xy}\right|}^2\right]}^{\frac{1}{2}}}}}=1$$

(7)

The parameters of GSM beam satisfy the following conditions.

$$A_x=A_y, \mathit{Re}\left|B_{xy}\right|=0$$

(8)

The elements of the matrix in the downlink are expressed as [22] as follows.

$$\begin{array}{c}{W}_{ij}\left({\mathit{r}}_1,{\mathit{r}}_2,z;\omega \right)={\left({\displaystyle \frac{k}{2\pi z}}\right)}^2\iint d^2{\mathit{\rho }}_1\iint d^2{\mathit{\rho }}_2{W}_{ij}\left({\mathit{\rho }}_1,{\mathit{\rho }}_2,0;\omega \right)\\ \times \mathit{exp}\left[-ik{\displaystyle \frac{{\left({\mathit{r}}_1-{\mathit{\rho }}_1\right)}^2-{\left({\mathit{r}}_2-{\mathit{\rho }}_2\right)}^2}{2z}}\right]\\ \times {⟨\mathit{exp}\left[{\psi }^{*}\left({\mathit{r}}_1,{\mathit{\rho }}_1,z\right)+\psi \left({\mathit{r}}_2,{\mathit{\rho }}_2,z\right)\right]⟩}_m\end{array}$$

(9)

Here, ρ₁ and ρ₂ represent the transverse coordinates at the transmitter plane, whereas r₁ and r₂ denote those at the receiver plane. The wave number is given by k = 2π/λ. The final term, 〈…〉_m, represents the ensemble average over statistical realizations in the turbulent downlink and can be expressed as [18] follows.

$$\begin{array}{c}{⟨\mathit{exp}\left[{\psi }^{*}\left({\mathit{r}}_1,{\mathit{\rho }}_1,z\right)+\psi \left({\mathit{r}}_2,{\mathit{\rho }}_2,z\right)\right]⟩}_m\\ \approx \mathit{exp}\left\{-{\displaystyle \frac{1}{2}}{B}_0\left[B_1{\left({\mathit{\rho }}_1-{\mathit{\rho }}_2\right)}^2+B_2\left({\mathit{\rho }}_1-{\mathit{\rho }}_2\right)\left({\mathit{r}}_1-{\mathit{r}}_2\right)+B_3{\left({\mathit{r}}_1-{\mathit{r}}_2\right)}^2\right]\right\}\end{array}$$

(10)

with

$$\left\{\begin{array}{c}{B}_0=2{\pi }^2{k}^2\mathit{sec}\theta {\int }_0^{\infty }{\kappa }^3{\Phi }_n\left(\kappa \right)d\kappa \hfill \\ B_1={\int }_{h_0}^H{C}_n^2\left(h\right){\left(1-\xi \right)}^{2}dh\hfill \\ B_2=2{\int }_{h_0}^H{C}_n^2\left(h\right)\left(1-\xi \right)\xi dh\hfill \\ B_3={\int }_{h_0}^H{C}_n^2\left(h\right){\xi }^{2}dh\hfill \end{array}\right.$$

(11)

In the downlink channel, H and h₀ represent the altitudes of the transmitter and receiver relative to the horizontal plane, respectively. The zenith angle is denoted as θ (as illustrated in Figure 1). The power spectrum of the refractive index fluctuations follows the Kolmogorov’s power law [23].

$${\Phi }_n\left(\kappa \right)=0.033C_n^2{\kappa }^{-11/3},$$

(12)

where 1/l₀ ≤ κ ≤ 1∕L₀; L₀ and l₀ represent the outer and inner turbulence scales, respectively. C_n²(h) is the refractive-index structure parameter at altitude h and quantifies the turbulence intensity downlink. A widely adopted model for C_n²(h) is expressed as follows [24].

$$\begin{array}{c}{C}_n^2\left(h\right)=0.00594{\left({\displaystyle \frac{\upsilon }{27}}\right)}^2{\left(10^{-5}h\right)}^{10}\mathit{exp}\left(-{\displaystyle \frac{h}{1000}}\right)\\ +2.7\times 10^{-16}\mathit{exp}\left({\displaystyle \frac{-h}{1500}}\right)+C_0\mathit{exp}\left(-{\displaystyle \frac{h}{100}}\right)\end{array}$$

(13)

Here, C₀ denotes the refractive index structure parameter and v denotes the wind velocity [25]. Substituting Equations (10)–(13) into Equation (9) and performing the necessary calculations, the following equations are obtained.

$$\begin{array}{c}{W}_{ij}\left(r_1,r_2,z\right)={\displaystyle \frac{A_i{A}_j{B}_{ij}}{{\Delta }_{ij}^2\left(z\right)}}\mathit{exp}\left[-{\displaystyle \frac{{\left(r_1+r_2\right)}^2}{8w_0^2{\Delta }_{ij}^2\left(z\right)}}\right]\mathit{exp}\left[{\displaystyle \frac{ik\left(r_2^2-r_1^2\right)}{2R_{ij}\left(z\right)}}\right]\\ \times \mathit{exp}\left\{-\left[{\displaystyle \frac{1}{2{\Omega }_{ij}^2{\Delta }_{ij}^2\left(z\right)}}+\left(M_1+{\displaystyle \frac{M_2+M_3}{{\Delta }_{ij}^2\left(z\right)}}\right)-{\displaystyle \frac{M_2^2{z}^2}{2k^2{w}_0^2{\Delta }_{ij}^2\left(z\right)}}\right]{\left(r_1-r_2\right)}^2\right\}\end{array}$$

(14)

with

$$\left\{\begin{array}{c}{\Delta }_{ij}^2\left(z\right)=1+{\left({\displaystyle \frac{z}{k{w}_0{\Omega }_{ij}}}\right)}^2+{\displaystyle \frac{2M_3{z}^2}{k^2{w}_0^2}}{M}_1={\displaystyle \frac{1}{2}}{B}_0{B}_1\hfill \\ {\displaystyle \frac{1}{{\Omega }_{ij}^2}}={\displaystyle \frac{1}{4w_0^2}}+{\displaystyle \frac{1}{{\delta }_{ij}^2}}{M}_2={\displaystyle \frac{1}{2}}{B}_0{B}_2\hfill \\ R_{ij}(z)={\displaystyle \frac{z{k}^2{w}_0^2{\Delta }_{ij}^2\left(z\right)}{k^2{w}_0^2{\Delta }_{ij}^2\left(z\right)+M_2{z}^2-k^2{w}_0^2}}{M}_3={\displaystyle \frac{1}{2}}{B}_0{B}_3\hfill \end{array}\right.,$$

(15)

where Δ_ij²(z) refers to the spatial expansion coefficient, and R_ij(z) represents the radius of curvature of the wavefront [26]. When considering a single point Q(r, z) at the receiver plane and assuming r₁ = r₂ = r, Equation (14) can be simplified as follows:

$$W_{ij}\left(\mathit{r},z;\omega \right)={\displaystyle \frac{A_i{A}_j{B}_{ij}}{{\Delta }_{ij}^2\left(z\right)}}\mathit{exp}\left[-{\displaystyle \frac{{\mathit{r}}^2}{2w_0^2{\Delta }_{ij}^2\left(z\right)}}\right]$$

(16)

The polarization characteristics at the receiver plane can be expressed as follows.

$$P\left(\mathit{r},z;\omega \right)=\sqrt{{\displaystyle \frac{{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)-\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2+4\frac{A_x^2{A}_y^2{\left|B_{xy}\right|}^2}{{\Delta }_{xy}^4}\mathit{exp}\left(-\frac{r^2}{{\sigma }^2{\Delta }_{xy}^2}\right)}{{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)+\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2}}}$$

(17)

$$\epsilon \left(\mathit{r},z;\omega \right)=\sqrt{{\displaystyle \frac{\begin{array}{c}{\left[{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)-\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2+4\frac{A_x^2{A}_y^2{\left|B_{xy}\right|}^2}{{\Delta }_{xy}^4}\mathit{exp}\left(-\frac{r^2}{{\sigma }^2{\Delta }_{xy}^2}\right)\right]}^{\frac{1}{2}}-\\ {\left[{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)-\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2+4\frac{A_x^2{A}_y^2\mathit{Re}\left[B_{xy}\right]}{{\Delta }_{xy}^4}\mathit{exp}\left(-\frac{r^2}{{\sigma }^2{\Delta }_{xy}^2}\right)\right]}^{\frac{1}{2}}\end{array}}{\begin{array}{c}{\left[{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)-\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2+4\frac{A_x^2{A}_y^2{\left|B_{xy}\right|}^2}{{\Delta }_{xy}^4}\mathit{exp}\left(-\frac{r^2}{{\sigma }^2{\Delta }_{xy}^2}\right)\right]}^{\frac{1}{2}}+\\ {\left[{\left[\frac{A_x^2}{{\Delta }_{xx}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{xx}^2}\right)-\frac{A_y^2}{{\Delta }_{yy}^2}\mathit{exp}\left(-\frac{r^2}{2{\sigma }^2{\Delta }_{yy}^2}\right)\right]}^2+4\frac{A_x^2{A}_y^2\mathit{Re}\left[B_{xy}\right]}{{\Delta }_{xy}^4}\mathit{exp}\left(-\frac{r^2}{{\sigma }^2{\Delta }_{xy}^2}\right)\right]}^{\frac{1}{2}}\end{array}}}}$$

(18)

The SOP and DOP remain invariant during downlink propagation if the following conditions hold.

$$P\left(\mathit{r},z=0;\omega \right)=P\left(\mathit{r},z;\omega \right),\epsilon \left(\mathit{r},z=0;\omega \right)=\epsilon \left(\mathit{r},z;\omega \right)=1$$

(19)

By solving the equations above, the following results are obtained.

$${\delta }_{xx}={\delta }_{xy}={\delta }_{yy},\mathit{Re}\left|B_{xy}\right|=0,A_x=A_y$$

(20)

Therefore, conditions sufficient to maintain the polarization characteristics of a circularly polarized GSM laser beam in the downlink were derived. The conditions depend on the source parameters and are unrelated to the turbulence parameters, modeled by the extended Huygens–Fresnel integral. This finding is crucial for establishing a design foundation for CPolSK optical communication systems.

3. Performance Analysis of Direct Detection CPolSK System

3.1. Direct Detection CPolSK System Model

Figure 2 shows the block diagram of a direct-detection CPolSK system using a GSM beam. The beam propagation is simulated by the extended Huygens–Fresnel integral; the scintillation index is computed using the standard Rytov variance formula for a GSM beam. First, a phase diffuser converts the incident light into a GSM beam, which then passes through a polarization controller (PC). Thus, a linearly polarized GSM beam oriented +45° relative to the optical axis is modulated. By controlling the terminal voltages of a birefringence crystal-based external polarization modulator (PM), +45° or −45° linearly polarized lights are obtained. A quarter-wave plate converts a left- or right-circular polarization into two orthogonal linear polarizations, which are then separated by the PBS for detection. Finally, after passing through a quarter-wave plate (QWP), the emitted light is transmitted via an antenna as either left- or right-circularly polarized light. In this work, beam propagation is modeled by the extended Huygens–Fresnel integral, and the scintillation index is calculated via theoretical expressions.

At the receiver, a QWP converts the incoming signal into linearly polarized light without requiring polarization coordinate alignment between the transmitter and receiver. After passing through the polarization beam splitter, the ±45° linearly polarized light is received by two photodetectors (PDs). Finally, the signals are demodulated using a differential circuit. The output current of the receiver can be expressed as follows.

$$i=i_{D2}-i_{D1}=\left(i_R+i_{n2}\right)-\left(i_L+i_{n1}\right)=\left\{\begin{array}{c}{i}_s+i_{n}symbol“1”\\ -i_s+i_{n}symbol“0”’\end{array}\right.$$

(21)

where i_L and i_R are the detected currents of the left and right circularly polarized optical signals, respectively; i_n1 and i_n2 are the thermal noise currents. Taking equal normal distributions N(0, σ_n²) for i_n1 and i_n2, we obtained i_n = i_n2−i_n1, N(0, 2σ_n²).

3.2. Bit Error Rate

Atmospheric turbulence induces fluctuations in the received intensity of a GSM laser beam, leading to variations in the BER [27]. The initial circular polarization condition derived above is applied here, ensuring the input beam satisfies that condition. To evaluate the performance of a CPolSK optical communication system in a downlink channel, the intensity scintillation probability distribution is assumed to follow the gamma–gamma distribution model, expressed as

$$f\left(I\right)={\displaystyle \frac{2{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)I}}{\left({\displaystyle \frac{I}{⟨I⟩}}\right)}^{{\displaystyle \frac{\alpha +\beta }{2}}}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta {\displaystyle \frac{I}{⟨I⟩}}}\right),\mathrm{I}\text{>}0.$$

(22)

Here, K_α-β(•) denotes the modified Bessel function of the second kind of order m, and Γ(•) represents the gamma function. I is the received intensity, and <I> denotes the mean received intensity, which is normalized to unity. The parameters α and β correspond to the effective numbers of small- and large-scale scatters, respectively. They are expressed as follows.

$$\alpha ={\displaystyle \frac{1}{\mathit{exp}{\sigma }_{\mathit{ln}x}^2-1}}$$

(23)

$$\beta ={\displaystyle \frac{1}{\mathit{exp}{\sigma }_{\mathit{ln}y}^2-1}}$$

(24)

Here, ${\sigma }_{\mathit{ln}x}^2$ and ${\sigma }_{\mathit{ln}y}^2$ denote the variances associated with the large- and small-scale fluctuations, respectively. For a GSM laser beam source, the flux variance ${\sigma }_I^2(z,d)$ of the intensity fluctuations at the receiver plane can be formulated as

$${\sigma }_I^2=\mathit{exp}[{\sigma }_{\mathit{ln}x}^2+{\sigma }_{\mathit{ln}y}^2]-1$$

(25)

For a downlink channel, the scintillation ${\sigma }_{\mathit{ln}x}^2$ and ${\sigma }_{\mathit{ln}y}^2$ can be expressed as follows.

$${\sigma }_{\mathit{ln}x}^2={\displaystyle \frac{0.49{\sigma }_B^2}{{\left[1+{\left[0.98/{\sigma }_B^{4/5}\gamma \right]}^{6/7}{\sigma }_B^{12/5}\right]}^{7/6}}}$$

(26)

$${\sigma }_{\mathit{ln}y}^2={\displaystyle \frac{0.51{\sigma }_B^2}{{\left(1+0.69{\sigma }_B^{12/5}\right)}^{5/6}}}$$

(27)

${\sigma }_B^2$ is the scintillation index of the beam propagating through the downlink channel under weak turbulence, which is given by

$${\sigma }_{1,\mathit{weak}}^2={\sigma }_B^2=8.702{\mu }_1{k}^{7/6}{\left(H-h_0\right)}^{5/6}{sec}^{11/6}\left(\theta \right),$$

(28)

where ${\mu }_1$ can be expressed as

$${\mu }_1=\mathit{Re}{\int }_{h_0}^H{C}_n^2\left(h\right)\left\{{\xi }^{5/6}{\left[\Lambda \xi +i\left(1-\overline{\Theta }\xi \right)\right]}^{5/6}-{\Lambda }^{5/6}{\xi }^{5/3}\right\}\mathrm{d}h.$$

(29)

$\gamma ={\sigma }_{1,strong}^2-1$, where ${\sigma }_{1,strong}^2$ is the scintillation index under strong fluctuation conditions, Re denotes the real-part operator. It is expressed as

$${\sigma }_{1,strong}^2=1+{\displaystyle \frac{1.23L^{-1/3}{\mathit{sec}}^{-2/5}\left(\theta \right)}{{\left[k^{1/3}{\int }_{h_0}^H{C}_n^2\left(h\right)dh\right]}^{7/5}}}{\int }_{h_0}^H{C}_n^2\left(h\right){\displaystyle \frac{{\left(1-\overline{\Theta }\xi \right)}^2}{{\xi }^{1/3}{\left[{\left(1-\overline{\Theta }\xi \right)}^{5/3}-{\left(1-\overline{\Theta }\right)}^{8/3}\right]}^{7/5}}}\mathrm{d}h.$$

(30)

where L is the propagation distance. The other parameters are given by

$$\mathsf{\Theta }={\displaystyle \frac{R\left(z\right)+z}{R\left(z\right)}},\overline{\mathsf{\Theta }}=1-\mathsf{\Theta },\mathsf{\Lambda }={\displaystyle \frac{2z}{k{w}^2\left(z\right)}},$$

(31)

where R(z) and w(z) are the wavefront curvature radius of the GSM laser beam and beam width at the receiver plane in free space, respectively.

For a direct-detection CPolSK modulation scheme, the BER can be expressed by

$$P_{se,CPolSK}={\displaystyle \frac{1}{2}}p\left(e|0\right)+{\displaystyle \frac{1}{2}}p\left(e|1\right)={\displaystyle \frac{1}{2}}erfc\left({\displaystyle \frac{i_s}{2{\sigma }_n}}\right)={\displaystyle \frac{1}{2}}erfc\left({\displaystyle \frac{I\overline{\gamma }}{2}}\right),$$

(32)

where $\overline{\gamma }={\displaystyle \frac{⟨i_s⟩}{{\sigma }_n}}$ represents the equivalent signal-to-noise ratio (SNR), p denotes the optical power, erfc denotes the complementary error function, and $I={\displaystyle \frac{i_s}{⟨i_s⟩}}$ denotes the normalized optical intensity. Consequently, the average BER of the transmitted optical signal can be formulated as follows:

$$⟨P_{se,CPolSK}⟩={\int }_0^{\infty }f\left(I\right)P_{se,CPolSK}\left(I\right)dI.$$

(33)

4. Numerical Results and Analysis

4.1. GSM Beam Scintillation Index in Space-to-Ground Link

Table 1. Simulation parameters.

Title 1	Title 2	Title 3
radius of curvature	F0	∞
source coherence parameter	ζs	1, 2, 5, ∞
laser wavelength	λ	1550 nm
receiving aperture	D	1–10 cm
transmission distance	z	10 km
beam waist width	W0	5 cm
zenith angle	θ	0–88°
wind velocity	v	21 m/s
refractive index parameter	C0	1.7 × 10−14 m−2/3
transmission altitude	H0	20 km

presents the simulation parameters of GSM-beam propagation through a downlink channel. Here, F₀ = πW₀²/(λL) is the Fresnel number of the link, and ζ_s is the source’s normalized spatial coherence width. The Fresnel number F₀ and coherence parameter ζ_s (as defined above) are used to compute the turbulence-induced phase variance in the simulations.

Figure 3 illustrates the variation in the scintillation index for a GSM beam in downlink as a function of transmission distance under different ζ_s and θ values. The simulation parameters are listed in Table 1. The solid red line corresponds to fully coherent light with a source coherence parameter of ζ_s = 1. By contrast, the blue and orange dashed lines indicate partially coherent light with ζ_s = 2 and ζ_s = 5, respectively, and the dashed green line represents fully incoherent light with an infinite source coherence parameter.

As evident from Figure 3A–E, at a zenith angle of θ = 0°, the scintillation index for all four beams increases with transmission distance and decreases as the source coherence parameter increases.

The scintillation index does not peak until a transmission distances of 10 km. In addition, partially coherent GSM beams exhibit lower scintillation indices than fully coherent beams. Both partially coherent and incoherent GSM beams exhibit lower scintillation indices than fully coherent beams. However, although incoherent beams yield the lowest scintillation index, their received optical intensity is extremely low, resulting in poor system performance. Therefore, incoherent cases are not emphasized in our main analysis.

4.2. GSM Beam Probability Density Distribution of Receiving Optical Intensity

Figure 4 shows the probability density distribution of the normalized central receiving optical intensity in the downlink at a transmission distance of 10 km under various source coherence parameters. At a fixed zenith angle of θ = 60° and a transmission distance at z = 10 km, along with other simulation parameters from Table 1, the curves are normalized using the average receiving optical intensity of fully coherent light.

As shown in Figure 4, increasing the source coherence parameter causes the probability density distribution of the central receiving optical intensity of the GSM beam to shift to the left, the peak value to increase, and the curve to become sharper. This observation suggests that the partially coherent GSM beam demonstrates an improved resistance to atmospheric turbulence in the downlink. These results indicate that the partially coherent beam is more resistant to turbulence: Its larger coherence radius causes turbulence-induced phase distortions to be averaged out, reducing scintillation. However, if coherence is too low, increased beam spread can reduce SNR. Therefore, an intermediate coherence yields optimal performance.

4.3. Performance of CPolSK-Modulated System

Figure 5 shows the BER of the GSM-beam-based CPolSK-modulated optical communication system as a function of the mean-receiving SNR in the downlink for different source coherence parameters. For reference, the BER curve of a fully coherent light source in an OOK-modulated optical communication system is also shown (solid black line).

As shown in Figure 5, an increase in the average receiving SNR leads to a decrease in the BER for both the CPolSK and OOK systems. In downlink, when the source coherence parameter is one (i.e., the transmitting beam is fully coherent), the CPolSK system achieves optimal performance. When the source coherence parameter was set to two, the BER of the CPolSK system slightly outperformed that of the OOK-modulated system with fully coherent light. This is because, in downlink, the influence of atmospheric turbulence is relatively weak, making the reduction in the received optical intensity due to the GSM beam more significant than the scintillation suppression effect.

At a system BER of 1 × 10⁻⁹, the OOK modulated system requires an average receiving SNR of 14 dB, whereas the CPolSK system using fully coherent light achieves the same BER at 12.49 dB, demonstrating a 1.51 dB reduction in the required SNR. The coherent beam achieves a lower BER at the same transmit power; this is because the partially coherent beam has a reduced mean intensity at the receiver, offsetting its lower scintillation. This indicates that using fully coherent light enhances the performance of the CPolSK-modulated system in the downlink under weak turbulence conditions.

5. Conclusions

This study explores a polarization invariance model for GSM laser beam propagation for optical communication in a downlink channel based on the unified theory of polarization coherence and the Huygens–Fresnel principle. It developed models for the probability density distribution and scintillation index of optical intensity under atmospheric turbulence, along with a BER model for a CPolSK-modulated system using a GSM beam. The influence of laser spatial coherence on these parameters was thoroughly analyzed. Numerical simulations revealed that partially coherent GSM beams exhibited enhanced robustness to atmospheric turbulence in the downlink. Using fully coherent light, the GSM-beam-based CPolSK modulated system achieved optimal performance with reduction of 1.51 dB in the required SNR compared to that of an OOK system using fully coherent light. Using an incoherent source with PolSK can approach the BER of coherent OOK at similar power levels, demonstrating that alternative transmitter designs may offer different tradeoffs. The proposed partially coherent scheme requires higher transmitted power than the fully coherent case to achieve the same BER. Note that this study focuses on theoretical channel performance; hardware implementation and system design are beyond its scope. It is worth noting that the multi-phase screen transmission method is an alternative simulation approach that can complement our analysis by providing real-time and statistical insights. Its implementation will be considered in future work.