# On the Use of Focused Incident Near-Field Beams in Microwave Imaging
## NF Beams and Results
### Interrogation of Objects by the Bessel Beam Launcher
#### 3.2.2 Inversion Results  
A simple linear inversion algorithm, the same as the one used in Section 3.1.2, is employed to invert the scattered NF data. To this end, we utilize a known calibration object with a complex relative permittivity value close to that of the OL. Using this calibration object, we linearize the nonlinear inverse scattering problem, and then reconstruct the complex permittivity of the OI using one measured data point located at \((0,2\lambda,0)\), which is directly behind the OI.  
Now, let us first consider the inversion results for Cases I and II under our two configurations. As listed in Table 2, the calibration object has a relative permittivity of 1.50 and the size of \(\lambda/2\times\lambda/4\times\lambda/3\) (same size as the OI in Cases I and II). The achieved reconstructed value (real and imaginary parts) under Configuration I shows less changes compared to those under Configuration II when we go from Case I to Case II. However, as can be seen the inversion results under Configuration I are quantitatively less accurate. The same observation holds when comparing the reconstructed values for Cases III and IV under the two configurations, see Table 2. That is, under Configuration I, when we go from Case III to Case IV, the reconstructed value shows smaller changes. However, quantitatively, Configuration II yields more accurate results.  
The observation that the reconstructed values under Configuration I undergo less changes when the undesired scatterers are introduced indicates that Configuration I has provided less sensitivity to the presence of these undesired scatterers. However, it is important to discuss why the reconstructed permittivity under Configuration I is less accurate. Herein, we present a few speculations to justify this. First, the linearized inversion approach assumes that the total field in the calibration object and the OI are identical. We speculate that this assumption is more accurate for the dipole antenna due to having a broader and more uniform beam. (Note that a given inversion algorithm can be more suitable to a given measurement setup depending on its assumptions.) Second, the Bessel beam has the so-called \"self-healing\" property [50; 51], which could reduce the sensitivity of the data to the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & **Calibration Object’s Permittivity** & **Configuration I (Focused)** & **Configuration II (Non-Focused)** \\ \hline Case I & 1.50 & 1.51 –\(\beta\).11 & 2.00 –\(\beta\).13 \\ Case II & 1.50 & 1.61 –\(\beta\).01 & 2.16 –\(\beta\).130 \\ Case III & 3.50 & 3.52 –\(\beta\).16 & 3.58 –\(\beta\).063 \\ Case IV & 3.50 & 3.54 –\(\beta\).18 & 3.77 –\(\beta\).88 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The inversion results for Cases I to IV under the two configurations: with the Bessel beam (Configuration I) and the dipole antenna (Configuration II).  
Figure 15: Induced contrast sources (their magnitudes) for Cases I and II for both configurations: using the Bessel beam launcher (Configuration I), and using the dipole antenna (Configuration II). The contrast sources in each configuration have been normalized with respect to the maximum of the incident field at that configuration.  
unknown permittivity. Third, the polarization of the Bessel beam launcher used for imaging is \(\hat{y}\) and that of the dipole is \(\hat{x}\). Therefore, the induced contrast sources in the OI can be thought as some dipole antennas oriented in the \(\hat{y}\)- and \(\hat{x}\)-directions for Configurations I and II, respectively. The measured data point, located at \((0,2\lambda,0)\), will then be mainly along the axes of these dipoles in Configuration I and along the broadside of these dipoles in Configuration II. This could also result in less sensitivity for Configuration I.