## 2 Materials and Methods
### Simulated Point Spread Function  
The PSF describes the response of an imaging system to a point source or to a point object, thus it can be seen as the impulse response of the system. From a practical point of view, the PSF gives a direct measure of the quality of an imaging system, being strictly related to the spatial resolution.  
Two full-wave simulations (using the Ansys Electronics Desktop software) were done to determine the simulated PSF of the proposed mm-wave imaging system. Both the simulated full-wave models included 28 rectangular waveguides (working in the frequency range [26.5-40] GHz) in a linear configuration, in air (_Case 1_) and in a dielectric medium with the same dielectric properties of human breast _ex vivo_ tissues (_Case 2_).  
The antennas used in the simulated scenarios are common open ended WR28 waveguides. These have the same upper working frequency of the antennas used for the experimental setup (PNR180), but a higher lower working frequency (26.5 GHz instead of 18 GHz). Although this simulated setup works at frequencies not identical to those of the experimental setup, it is useful to provide an estimation of the PSF, while maintaining the computational effort within the limits of the available workstations (to deliver accurate, simulated, results for the image formation algorithm, the working bandwidth is discretized with a fine step of 50 MHz, and all points are calculated independently, thus the WR28 antennas are more convenient, to this aim, than PNR180 antennas). The distance between phase centers of adjacent radiators is equal to half wavelength at 30 GHz in each corresponding medium. In both cases, the target was a sphere with the same average dielectric properties of human neoplastic tissues (\(\epsilon^{\prime}\)@30 GHz = 19 and \(\epsilon^{\prime}\)@30 GHz = 20, [8]), with a diameter of 2 mm (i.e., point-like, in accordance with the guidelines published in [32] for the evaluation of the PSF for a radar system), and placed at 4 cm depth from the center of the array. In these simulations, the final dimension of the array was 13.5 cm in air and 7.8 cm in fat.  
The main-lobe width along the lateral and range directions of the simulated PSFs in both cases, i.e., in air and in fat, are summarized in Table 1, where also the comparison with the theoretical spatial resolution (computed according to [31; 33]) is shown. In Figure 5a,b the DAS image for the simulations corresponding to _Case 1_ and _Case 2_ are shown, respectively. Results show that, in both cases, the theoretical and the simulated resolutions are comparable, considering that the formulas used to predict the theoretical values, while useful to provide an estimation, make use of assumptions that cannot be rigorously applied to the cases under analysis. First, the formulas are derived under far-field approximations, which is not the case, and the lateral resolution is determined according to the lateral dimension of the array, which is not immediately defined according to simple geometrical consideration because of the lossy medium and, again, the near-field propagation. In any case, the plane-wave approximation to derive the order of magnitude of the achieved resolution was used because it can greatly simplify the study of the problem and the interpretation of the results, without compromising the veracity of the analysis itself.