# On the Use of Focused Incident Near-Field Beams in Microwave Imaging
## 2 Motivation
### Suppressing Undesired Scattering Effects  
Based on the above discussion, it can be understood that the contrast sources induced in the OI can be considered as the _cause_ for the scattered field data, similar to the current distribution of an antenna being the cause for its radiation pattern. More specifically, the scattered field at a given receiver due to a given transmitter can be thought as the weighted summation of all the contrast sources induced at different locations within the OI's geometrical domain. (This is parallel to the concept of antenna arrays in which the field at a given location in space is the weighted summation of the effects of all the single antenna elements.) This can be better understood by observing the so-called  
Figure 1: Demonstration of the relation between the MWI and inverse source problems. (**a**) represents the MWI problem when the OI is interrogated by the transmitter \(Tx1\) (red antenna) while the other antennas (grey antennas) act as receivers; (**b**) represents the equivalent inverse source problem for (**a**) in which \(Tx1\) has been replaced by contrast sources in the geometrical domain of the OI; similarly (**c**,**d**) demonstrate the same concept but for a different transmitter: \(Tx2\). data equation_[41] which maps the contrast sources from the imaging domain \(D\) to the scattered field on the measurement domain \(S\) for the 2D scalar problem  
\[\mathbf{E}^{\text{scat}}(\mathbf{r}\in S)=k_{b}^{2}\int_{D}g(\mathbf{r}\in S,\mathbf{r}^{\prime} \in D)\;\mathbf{w}(\mathbf{r}^{\prime}\in D)\;d\mathbf{r}^{\prime}. \tag{2}\]  
The above equation indicates that the scattered field at the receiver location \(\mathbf{r}\) is affected by all the contrast sources weighted by the green's function \(g\) of the background medium and the background wavenumber \(k_{b}\). Therefore, one way to not _see_ a specific part of the imaging domain, e.g., an undesired scatterer, is to ensure that the contrast sources induced in the undesired scatterer are small, and are ideally zero. Due to the fact that the dielectric profile within the imaging domain is unknown, we are not able to fully control the distribution of contrast sources within the imaging domain. However, since the contrast sources are affected by the incident NF distribution, \(\mathbf{E}^{\text{inc}}\), through the so-called _domain equation_[41],  
\[\mathbf{w}(\mathbf{r}\in D)=\chi(\mathbf{r}\in D)\mathbf{E}^{\text{inc}}(\mathbf{r}\in D)+k_{b}^{2 }\int_{D}\delta(\mathbf{r}\in D,\mathbf{r}^{\prime}\in D)\;\mathbf{w}(\mathbf{r}^{\prime}\in D )\;d\mathbf{r}^{\prime}, \tag{3}\]  
we can use the incident NF distribution to _partially_ control the distribution of the contrast sources. The idea to be pursued here has been demonstrated in Figure 2 where a focused incident NF in Figure 2a and a non-focused one in Figure 2b irradiate three distinct dielectric objects, with the central green one represents the OI, and the other two black ones represent the undesired scatterers. Due to the fact that the incident NF in Figure 2a is more focused toward the OI, the contrast sources within the undesired scatterers are more _likely_ to be weaker in Figure 2a, and subsequently, the scattered data collected by the grey receiver in Figure 2a is more likely to contain less information about the undesired scatterers as compared to the grey receiver shown in Figure 2b. Therefore, if the main purpose is to see the OI (central object), Figure 2a offers an advantageous measurement scenario as the received signal is mainly due to the OI. We will investigate this further in Section 3 of this paper. (This is similar to our approach toward justifying an appropriate incident field for 2D transverse magnetic inversion [32].)