# On the Use of Focused Incident Near-Field Beams in Microwave Imaging
## NF Beams and Results
### Interrogation of Objects by the NF Plate
#### 3.1.2 Inversion Results  
Now, let us consider inverting the scattered NF data collected using the NF plate setup. The choice of the inversion algorithm depends on many parameters including the number of measured NF data points. In MWI, it is often desirable to choose a nonlinear inversion algorithm (e.g., contrast source inversion or Gauss-Newton inversion methods) in which the Ol's dielectric profile is iteratively reconstructed. However, these nonlinear inversion algorithms are better suited when the OI is illuminated from different angles of incidence. (That is why we used a nonlinear inversion algorithm in Section 2.4.) As in Section 3.1.1, we have only illuminated the OI from one direction, the choice of nonlinear inversion algorithms is not appropriate. Therefore, we use a linear inversion algorithm in order to reconstruct the complex permittivity of the OI. In our linearized inversion approach, we assume that the scattered NF data is related to the unknown dielectric contrast \(\chi\) linearly. That is, \(\mathbf{E}^{scat}=\mathcal{L}(\chi)\) where \(\mathcal{L}\) is a linear operator. As can be seen from Equation (2), this linear operator depends on both the green's function of the system and the total field induced in the object. To determine \(\mathcal{L}\), we use a calibration object similar to the method presented in [7]. Herein, for simplicity, we reconstruct one relative complex permittivity value for the OI. In addition, we limit the data collection scheme to only one data point behind the centre of the OI on the receiver domain. Thus, the operator \(\mathcal{L}\) simply becomes a complex number which is multiplied by the contrast to output the scattered NF value at the single measurement point. In other words, \(\mathcal{L}\) becomes the slope of the line that linearizes the relation between the scattered NF data and the contrast around the contrast of the calibration object. Denoting the dielectric contrast of the calibration object as \(\chi^{c}\), \(\mathcal{L}\) simply becomes \(\mathbf{E}^{scat}(\chi^{c})/\chi^{c}\) where \(\mathbf{E}^{scat}(\chi^{c})\) is the scattered NF data due to the known calibration object. Once this \(\mathcal{L}\) is found, the unknown contrast can be found as \(\chi=\mathbf{E}^{scat}/\mathcal{L}\).  
Now, consider the reconstruction of the permittivity value of the OI for the MWI setup shown in Figure 5 under our two configurations (with and without the NF plate) for case studies I and II. We consider the same dielectric boxes (with the size of \(\lambda/20\times\lambda/10\times\lambda/10\)) as presented earlier in Figure 5 as the OI and undesired scatterer, but now with a relative permittivity of 1.70, instead of 1.50. (As will be seen, the permittivity of 1.50 will be used as one of the calibration objects.) The inversion results for Configurations I and II for two different calibration objects (having the same size as the OI) are reported in Table 1. As can be seen, the inversion results for Cases I and II under the focused incident field illumination (Configuration I) do not change as much as those under Configuration II. (For example, compare the change from 1.90 - \(j\)0.29 to 1.89 - \(j\)0.32 with the stronger change from 1.74 - \(j\)0.04 to 1.58 - \(j\)0.25.) This demonstrates that Configuration I is less sensitive to the presence of  
Figure 8: Induced contrast sources (their magnitudes) for Cases I and IV for both configurations: with the NF plate (Configuration I), and without the NF plate (Configuration II). The contrast sources in each configuration have been normalized with respect to the maximum of the incident field at that configuration.  
the undesired scatterer. It should also be noted that the inversion result of Configuration II are more accurate. We speculate that this is due to the fact that the total fields in the calibration object and that in the OI are closer to each other for the non-focused incident beam, thus, making the utilized linear inversion algorithm more suitable for Configuration II.