## 3 Results
### Fundamental Limits: Plane-Wave Illumination  
First, we assess the TPD changes due to presence of a PEC layer in front of the skin model for plane-wave illumination (scenarios 1 and 2). To this end, \(\text{TPD}_{1}\), \(\text{TPD}_{2}\), and \(\text{Y}_{(2,1)}\) are calculated using (3), (4), (7), and (15) for \(\text{E}_{0}=10\) V/m and \(\text{R}_{1}=-0.59+\text{j}0.16\) (Figure 4).  
Figure 4 shows that the TPD at the surface of the phantom is strongly altered by the presence of PEC (increase up to 574% and decrease down to 61.7%). \(\mathrm{Y}_{(2,1)}\) can be expressed as:  
\[\mathrm{Y}_{(2,1)}=\frac{\mathrm{TPD}_{2}}{\mathrm{TPD}_{1}}=\frac{\left| \mathrm{T}_{1}\frac{1}{1+\mathrm{e}^{-\mathrm{j}2\mathrm{d}_{0}\mathrm{R}_{1}} }\mathrm{E}_{\mathrm{inc}}\right|^{2}}{\left|\mathrm{T}_{1}\mathrm{E}_{ \mathrm{inc}}\right|^{2}}=\left|\frac{1}{1+\mathrm{e}^{-\mathrm{j}2\mathrm{d} _{0}\mathrm{R}_{1}}}\right|^{2}. \tag{16}\]  
The positions of \(\mathrm{Y}_{(2,1)}\) maxima and minima (\(\mathrm{d}_{\mathrm{max}}^{\mathrm{Y}_{(2,1)}}\) and \(\mathrm{d}_{\mathrm{min}}^{\mathrm{Y}_{(2,1)}}\)) depend on the phase of \(\mathrm{R}_{1}\) and given by:  
\[\mathrm{d}_{\mathrm{min}}^{\mathrm{Y}_{(21)}}=\frac{\left|\mathrm{R}_{1}- \pi+\pi\mathrm{n}\right.}{2\mathrm{k}_{0}}\, \tag{17}\]  
\[\mathrm{d}_{\mathrm{max}}^{\mathrm{Y}_{(21)}}=\frac{\left|\mathrm{R}_{1}- \pi+2\pi\mathrm{n}\right.}{2\mathrm{k}_{0}}\, \tag{18}\]  
where n is an integer number, \(\left|\mathrm{R}_{1}\right.\) is the phase shift introduced by the phantom interface to the reflected plane-wave. The fundamental limits of \(\mathrm{Y}_{(2,1)}\) can be found by replacing (17) and (18) in (16):  
\[\frac{1}{\left|1+\left|\mathrm{R}_{1}\right|\right|^{2}}\leq\mathrm{Y}_{(2,1)} \leq\frac{1}{\left|1-\left|\mathrm{R}_{1}\right|\right|^{2}}. \tag{19}\]  
Equation (19) shows that the higher the magnitude of the phantom reflection coefficient, the higher the TPD variations. For example, for a wet skin model, the variations of TPD are more pronounced (increase up to 629% and decrease down to 62.3%, respectively). Note that these variations are also age-dependent as the tissue properties evolve with age [29]. In particular, for 5 year old children the enhancement increases to 640%. For the sake of brevity, in the rest of the paper, the analysis for wet skin and age-dependent effects will be omitted (except Section 3.3). Table 1 provides the maximum and minimum \(\overline{\mathrm{Y}}_{(\mathrm{m},\mathrm{n})}\). The results show that the average TPD increases due to the presence of PEC (roughly a 60% increase for \(\Delta\mathrm{d}=5\) mm).  
Next, the free-space losses are taken into account in the analysis (scenarios 3 and 4). TPD\({}_{3}\), TPD\({}_{4}\), and \(\mathrm{Y}_{(4,3)}\) are calculated using (5), (6), (7), and (15) (Figure 5).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\Delta\mathrm{d}\) (mm) & **1** & **3** & **5** \\ \hline \(\mathrm{max}\overline{\mathrm{Y}}_{(2,1)}\) & 3.21 & 2.14 & 1.61 \\ \(\mathrm{min}\overline{\mathrm{Y}}_{(2,1)}\) & 0.43 & 1.40 & 1.59 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average \(\overline{\mathrm{Y}}_{(2,1)}\) over \(\Delta\mathrm{d}\).  
Figure 4: Plane-wave illumination without (scenario 1) and with (scenario 2) PEC: (**a**) transmitted power density (TPD); (**b**) TPD variations due to presence of PEC.  
TPD\({}_{4}\) demonstrates a damped oscillatory behavior around TPD\({}_{3}\) (increase up to 80% and decrease down to 28%). Due to the free-space loss, the oscillation amplitude of Y\({}_{(4,3)}\) is lower compared to Y\({}_{(2,1)}\). The averaged TPD over distance increases due to the presence of PEC (i.e., maximum increase of 41%, 15%, 5% for \(\Delta\)d = 1 mm, 3 mm, 5 mm, respectively) (Table 2).