## 2 System and Channel Model
### Generalized Pointing Errors with Nonzero Boresight  
Apart from beam width and photodetector(s) size, which have been considered through the well-known zero-boresight pointing error model, first introduced in [27] for free-air communications and in [24] for TOW links, we also take into account the NZB along with nonidentical jitters for the elevation and the horizontal displacement. Consequently, since the Gaussian light beam traverses each backward transdermal path of \(\delta_{m}\) in length, the corresponding fraction of the collected power at the \(m\)-th receiver circular aperture with radius \(r_{m}\), can be approximated as [12; 27]:  
\[h_{p,m}(R_{m},\delta_{m})\approx A_{0,m}\exp\left(-\frac{2R_{m}^{2}}{w_{eq,m} ^{2}}\right),\ R_{m}\geq 0, \tag{5}\]  
where \(w_{eq,m}\) represents the equivalent beam width in the \(m\)-th receiver aperture, which is given as \(w_{eq,m}=\left[\frac{\sqrt{\pi\text{erf}}(v_{m})w_{eq,m}^{2}}{2v_{m}\exp{(-v _{m}^{2})}}\right]^{1/2}\), with \(v_{m}=\frac{\sqrt{\pi}r_{m}}{\sqrt{2}}w_{\delta,m}\) and \(\text{erf}(. )\) denoting the error function ([31], Equation (8.250.1)). Additionally, in TOW links we get [6; 11; 18]:  
\[w_{\delta,m}=\delta_{m}\tan(\theta_{m}/2) \tag{6}\]  
where \(w_{\delta,m}\) represents the \(m\)-th beam waist on the receiver(s) plane at distance \(\delta_{m}\) from the implanted retroreflective modulator, with \(\theta_{m}\) being the corresponding divergence angle [4; 25]. Moreover, \(A_{0,m}=\text{erf}^{2}(v_{m})\) is the fraction of the collected power at \(r_{m}\)=0, [5; 17]. Furthermore, \(R_{m}\) is the radial displacement at the \(m\)-th receiver, which is expressed as \(R_{m}=\left|\overset{\rightarrow}{R}_{m}\right|=\sqrt{R_{x,m}^{2}+R_{y,m}^{2 }}\), where \(\overset{\rightarrow}{R}_{m}=\left[R_{x,m},R_{y,m}\right]^{T}\) is the radial displacement vector with \(R_{x,m}\), \(R_{y,m}\) being the displacements located along the horizontal and elevation axes at the detector plane, respectively. It should be noted that these variables are considered as nonzero mean Gaussian distributed random variables, i.e., \(R_{x,m}\sim N\big{(}\mu_{x,m},\sigma_{x,m}^{2}\big{)}\), \(R_{y,m}\sim N\big{(}\mu_{y,m},\sigma_{y,m}^{2}\big{)}\), where \(\mu_{x,m}\), \(\mu_{y,m}\), denote their mean values and \(\sigma_{x,m}\), \(\sigma_{y,m}\) are the jitters for horizontal and elevation displacements, respectively, [12; 28]. Here, it should be also mentioned that the approximation of Equation (5) is valid for \(w_{\delta,m}/\,r_{m}>6\), which is true in typical optical wireless links, [27; 28; 30]. Considering thus the beam width, the photodetector size(s) and the different jitter for the elevation and horizontal displacements, along with NZB, the misalignment-induced irradiance fading at each receiver aperture can be described by the suitable generalized Beckmann's distribution model [32]:  
\[ \begin{array}{l}f_{R,m}(R_{m})=\frac{R_{m}}{2\pi i\sigma_{x,m}\sigma_{y,m}} \\ \times\int\limits_{0}^{2\pi}\exp\left[-\frac{(R_{m}\cos\varphi_{m}-\mu_{x,m}) ^{2}}{2\sigma_{x,m}^{2}}-\frac{(R\sin\varphi_{m}-\mu_{y})^{2}}{2\sigma_{y}^{2} }\right]\,d\varphi_{m}\end{array}  \tag{7}\]  
where \(\varphi_{m}\) is the \(m\)-th reflected divergence angle, which stands for the increase in beam radius with distance from the modulated retroreflector [17]. It has been first shown in [30] that the [PERSON]'s distribution above can be approximated according to the modified Rayleigh distribution:  
\[f_{R,m}(R_{m})=\frac{R_{m}}{\sigma_{\text{mod},m}^{2}}\exp\left(-\frac{R_{m}^{2} }{2\sigma_{\text{mod},m}^{2}}\right),\ R_{m}\geq 0, \tag{8}\]  
where \(\sigma_{\text{mod},m}^{2}=\left(\frac{3\eta_{x,m}^{2}\sigma_{x,m}^{4}+3\eta_{y,m}^{2}\sigma_{x,m}^{4}+\sigma_{x,m}^{6}+\sigma_{y,m}^{6}}{2}\right)^{\frac{1 }{3}}\), with \(\sigma_{mod,m}\) being the joint modified jitter (standard deviation) of \(\sigma_{x,m}\) and \(\sigma_{y,m}\) for the \(m\)-th reflected transdermal path, [12, 21, 30].