# On the Use of Focused Incident Near-Field Beams in Microwave Imaging
## 2 Motivation
### A Synthetic Test Case  
To demonstrate the concept above, we perform the following simulation study in which the utilized incident field is a numerical model, and does not represent an actual field. (In Section 3, we will focus on simulated fields due to practical NF antennas.) In this example, we consider a two-dimensional transverse magnetic MWI. The true dielectric profile considered is shown in Figure 3a which consists of four lossless scatterers in the \(xy\) plane. It is assumed that the two light blue circular scatters with the relative permittivity of 2.0 are the OI, and the other two (the yellow and red ones) with the relative permittivity of 2.5 and 3.0 are the undesired scatterers. The true profile was created on a \(18\times 18\) cm\({}^{2}\) domain discretized into \(100\times 100\) square pulses. The frequency of operation is assumed to be 5 GHz. Two types of incident fields are considered to illuminate the imaging domain: the first one is an omnidirectional line source incident field, and the other is a focused incident field as shown in Figures 3b,c, respectively. The mathematical expression for the omnidirectional line source is \(\mathbf{E}^{\text{inc}}=\mathbb{2}\frac{1}{4j}H_{0}^{2}(k_{b}|\mathbf{r}-\mathbf{r}^{ \prime}|)\) where \(\mathbf{r}\) represents the transmitter location, \(\mathbf{r}^{\prime}\) represents the observation point within the imaging domain, and \(j^{2}=-1\). In addition, \(H_{0}^{2}\) is the zeroth order Hankel function of the second kind. The focused incident field is then _modeled_ (not, an actual field) by multiplying this omnidirectional incident field by \(\cos^{m}\psi\) where \(\psi\) is the angle between the boresight axis of the antenna and the line connecting the antenna to the observation point in the imaging domain; e.g., see [27]. (The parameter \(m\), which controls the focusing level, is set to be 300 here.) For all the antennas, the boresight axis is the line that connects the antenna to the centre of the imaging domain. We consider two types of measurement domains: (1) a full circle and (2) a circular sector. In the first one, we place 22 antennas evenly distributed on a circle of 14 cm radius around the imaging domain as shown in Figure 3d. In the second scheme, we distribute these 22 antennas on two opposite \(60^{\circ}\) sectors of the same measurement circle as shown in Figure 3e. Based on the analogy with the cylindrical NF antenna measurements noted in Section 2.3, the use of the circular sector measurement domain will not be as good as the full circular one; however, in some applications, we have to use this type of limited-view configurations as we may not have a full \(360^{\circ}\) access around the imaging domain. The irradiation of the imaging domain is performed using either the omnidirectional incident field or the focused one. In the irradiation process, when one antenna transmits, all the antennas including the transmit antenna collect the scattered field data. This scattered data is synthetically generated by a method of moment forward solver, and 3% noise is added to this data according to the formula shown in [45]. The inversion of the noisy scattered data is then performed by the multiplicative regularized Gauss-Newton inversion (MR-GNI) algorithm [16; 19; 46] on an imaging domain with the size of \(18.4\times 18.4\) cm\({}^{2}\) discretized into \(61\times 61\) cells. The inversion results for the full circular measurement domain with the omnidirectional and focused incident fields are shown in Figure 4a,b, respectively. As can be seen, both images have good reconstruction accuracy. Next, we consider the circular sector measurement domain: its inversion results for the omnidirectional and focused incident fields are shown in Figure 4c,d, respectively. As can be seen, both images are poor due to suffering from limited view angles. However, it can be noticed that the use of the focused incident field enables us to at least see the OI (i.e., the two circular scatterers).