## 4 Analytical Results  
In this section, analytical results are graphically depicted that arise from the derived analytical expressions in Equations (18) and (19). Their validity is further verified by appropriate simulations. The diameter of the receiver's aperture, \(\rho\), wasset to 0.5 mm, while the operational wavelength, \(\lambda\), wasfixed at 1100 nm, which can drastically address skin-induced attenuation due to photon absorption, as reported in [9]. Moreover, although the probability \(r\) could take numerous values, the values 0.01 and 0.1 werehere utilized, which are very common for any OWC link [42]. Furthermore, \(\eta=0.8\), \(\theta=20^{\circ}\), \(P_{s}=1\)\(\mu\)W/MHz, and \(N_{0}=\left(1.3\,\,pA/\sqrt{Hz}\right)^{2}\)[48]. Regarding pointing mismatch, varying weak to strong pointing error effects wereevaluated for two different typical transdermal link lengths, i.e., \((\delta,\,\sigma\,/\,\rho,\,\psi)=(4\,\,\text{mm},\,1.1,\,2.4),\,\,(\delta,\, \sigma\,/\,\rho,\,\psi)=(4\,\,\text{mm},\,1.5,\,1.76)\), \((\delta,\,\sigma\,/\,\rho,\,\psi)=(4\,\,\text{mm},\,2.0,\,1.32)\), \((\delta,\,\sigma\,/\,\rho,\,\psi)=(4\,\,\text{mm},\,2.3,\,1.15)\), \((\delta,\,\sigma\,/\,\rho,\,\psi)=(5\,\text{mm},\,1.1,\,2.98)\), \((\delta,\,\sigma\,/\,\rho,\,\psi)=(5\,\text{mm},\,1.5,\,2.19)\), \((\delta,\,\sigma\,/\,\rho,\,\psi)=(5\,\text{mm},\,2.0,\,1.64)\), and \((\delta,\,\sigma\,/\,\rho,\,\psi)=(5\,\text{mm},\,2.3,\,1.43)\). Note that \(\psi\) values are getting smaller as \(\sigma/\rho\) values are getting larger, especially for smaller \(\delta\) values, i.e., pointing errors are getting stronger as spatial jitter increases, as expected, especially for shorter transdermal link lengths, according to Equation (6). Indeed, the latter is valid assuming fixed \(\theta\) and \(\rho\) parameter values as mentioned above. Under these conditions, different typical TOW link configurations were investigated by means of outage capacity over a wide average SNR range, from 0 dB to 30 dB. Figure 1 illustrates the outage capacity dependence on the average electrical SNR evolution for a typical TOW link length of 5 mm with different values of normalized jitter standard deviation, provided that \(r=0.1\). We observe that higher jitter values lead to lower \(\psi\) parameter values and, thus, to stronger pointing error effects. In fact, as pointing errors effects are getting stronger, significantly lower outage capacity values are depicted, especially for lower average electrical SNR values. Figure 2 depicts the corresponding analytical results tothose of Figure 1 but with \(r=0.01\), i.e., with a smaller value of the probability \(r\). Consequently, although similar qualitative results wereobtained for the outage capacity in both cases, the decrease inthe probability value mentioned above broughtabout smaller corresponding outage capacity values. Therefore, the performance comparison between these two figures highlights the outage capacity degradation due to lower \(r\) values. This behavior is in good agreement with what happens in the wider OWC field, much less for the same probability values [42]. Additionally, the impact of pointing errors is depicted to become even more severe by decreasing this probability value, especially for lower average electrical SNR values. Figure 3 presents the corresponding analytical results tothose of Figure 2 but through a shorter transdermal link, i.e., a smaller value of parameter \(\delta\). As expected from Equation (6), shorter transdermal link distances led to larger beam waist values for a specific initial divergence angle, which resulted, in turn, in a larger amount of pointing mismatch. Indeed, for the same normalized jitter standard deviation values, we now obtained smaller corresponding \(\psi\) parameter values due to the shorter transdermal propagation distance. This translated into more important outage capacity performance degradations due to the presence of stronger pointing errors, especially for lower average electrical SNR values.