## 3 Results
### Fundamental Limits: Equivalent Sources  
Here, we consider the equivalent sources corresponding to the patch antenna arrays radiating in free space (Figure 2). This allows us to model the case where the free-space antenna matching, efficiency, and radiated field are preserved and not modified by the phantom (scenarios 5 and 6). TPD\({}_{5}\), TPD\({}_{6}\), and Y\({}_{(6,5)}\) are calculated from Equations (8)-(15) (Figure 6a-e). Significant differences in Y\({}_{(6,5)}\) maxima and, to a smaller extent, minima between the antenna-equivalent sources are noted for d \(<\) 25 mm (Figure 6e). The TPD increases (decreases) up to (down to) 174% (39%), 342% (54%), 421% (54.7%), and 497% (54.7%) for the SPA, 2 \(\times\) 2 PAA, 2 \(\times\) 4 PAA, and 4 \(\times\) 4 PAA, respectively. This is due to the differences in the attenuation rate of the peak power density in free-space PD\({}_{\rm fs}\) of the antenna-equivalent sources (Figure 6f). Indeed, when the PD\({}_{\rm fs}\) attenuation rate is higher, Y\({}_{(6,5)}\) is lower. At d = 35 mm, \(\Upsilon\)(6,5) of all antenna-equivalent sources converges to the same oscillatory function (with relative difference \(<\) 10%) (Figure 6e). Figure 7 shows that as d increases, Y\({}_{(6,5)}\) converges to Y\({}_{(4,3)}\) as the power density in free-space decreases as 1/d\({}^{2}\) in the far-field. Maximum values of \(\overline{Y}_{(6,5)}\) are obtained for 4 \(\times\) 4 PAA (i.e., the maximum increase of 247%, 124%, 76% for \(\Delta\)d = 1 mm, 3 mm, 5 mm, respectively) (Table 3). Note that for the antenna-equivalent sources max\(\overline{Y}\) is up to 2.5 times higher compared to the plane wave with free-space losses (compare Tables 2 and 3). Figure 5: Plane-wave illumination with free-space losses without (scenario 3) and with (scenarios 4) PEC: (**a**) TPD; (**b**) TPD variations due to presence of PEC. To obtain a deeper insight into the TPD variations, we analyzed the changes in the spatial distribution of TPD for the SPA equivalent source due to the presence of PEC (Figure 8). The distribution of TPD\({}_{6}\) is affected by the presence of PEC and evolves with d. For d corresponding to the maximum TPD (i.e., 4.75 mm, 7.25 mm, 17.25 mm), the absorbed power density is concentrated around its maximum. It extends progressively over a larger surface when d approaches the value corresponding to TPD minima (i.e., 6.5 mm, 9.0 mm, 18.75 mm). When the spatial distribution of TPD is concentrated around its maxima, the spatial averaging area has a stronger impact on the mean TPD, which rapidly decreases with the averaging area (e.g., the ratio between TPD averaged over 1 cm\({}^{2}\) and 4 cm\({}^{2}\) equals to 3.26 and 1.86 for d = 4.75 and 6.5 mm, respectively). Figure 6: Equivalent sources without (scenario 5) and with (scenario 6) PEC: (**a**) peak TPD from SPA; (**b**) peak TPD from 2 × 2 PAA; (**c**) peak TPD from 2 × 4 PAA; (**d**) peak TPD from 4 × 4 PAA; (**e**) envelopes (min. and max.) of Y\({}_{(6,5)}\); (**f**) free-space peak power density. Figure 7: TPD variations due to presence of PEC: Y\({}_{(4,3)}\) and Y\({}_{(6,5)}\) for 2 × 2 PAA equivalent source. The cross-section distributions along \(x\)-axis at y = 0 mm of the x component of E\({}_{\text{Direct}}\), E\({}_{\text{Total}}\), and E\({}_{\text{R}}\) are plotted in Figure 9. The amplitude of E\({}_{\text{Total}}\) is directly related to the phase difference between E\({}_{\text{Direct}}\) and E\({}_{\text{R}}\). For d = 7.25 mm, E\({}_{\text{Direct}}\) is in phase with E\({}_{\text{R}}\) around x = 0 mm. This null phase difference evolves periodically along the \(x\)-axis resulting in either constructive or destructive interferences. Consequently, this results in an enhancement of the E\({}_{\text{Total}}\) amplitude for x \(\in\) (\(-\)5.0; 5.0) mm and in a decrease for x \(\in\) (\(-\)10; \(-\)5) U (5; 10) mm, thus explaining the higher spatial gradient of TPD in Figure 8a,c,e (middle line, scenario 6).