# Leaky Wave Generation Through a Phased-Patch Array
## 1 Introduction  
It has been demonstrated that inhomogeneous waves can enable deeper penetration in lossy media. Such waves can often be approximated by the field produced by a leaky-wave antenna. However, in specific applications--such as hyperthermia treatment, where the radiating environment (i.e., biological tissues) varies from patient to patient--generating a precise field configuration requires tuning that is challenging to achieve with classical leaky-wave antennas, particularly under deep penetration constraints [1]. In such cases, an alternative approach that maintains the desired field characteristics while offering greater flexibility may yield better results. This can be achieved through the use of a phased array. Leaky waves are the only inhomogeneous waves capable of delivering power in a lossless medium, as they are not bounded to a surface, as happens with surface waves, plasmonic polaritons, or others [2]. As with surface waves, on the other hand, leaky waves are waves that appear only in the presence of a discontinuity or an asymmetry, in general [3]. Leaky-wave antennas are well-known artificial structures used to generate near-field radiation that closely approximates a leaky wave, and they are traditionally employed to produce a leaky-wave field. In this article, we aim to replace the leaky-wave antenna with a phased array. As is well known in the literature, the field produced by a continuous aperture antenna can be approximated by that of an array by sampling the currents on the antenna aperture [4]. As stated in [4], the accuracy of this approximation increases as the distance between the array elements decreases. In the limit, the two fields become identical, though this is clearly impractical. When the spacing between elements islarge, the fields between the two antennas may not match well and direct sampling may not provide good results. Solutions to this problem have been proposed in the literature, including root-matching methods and perturbation techniques [4; 5], as well as more recent integrated sampling methods [4; 6]. The need for these methods highlights the complexity of achieving a sufficiently accurate approximation of the field produced by an antenna array compared to that of a continuous antenna. For this reason, this paper investigated whether, in the case considered, simple sampling at a distance of \(\lambda/2\) is sufficient for the application proposed in [7], by first comparing the far fields and then the near fields produced by the two antennas. This study represents an initial step in that direction. In the literature, many LWAs structures have been proposed, to address aspects such as miniaturisation, directivity, full scanning, and stop-band suppression [8], but they were not strictly relevant to the goal of this article. For this work, therefore, we chose a simple LWA--the Menzel antenna--due to its specific attenuation and phase vector properties [9]. A practical way to realise a leaky-wave antenna (LWA) is to consider a travelling-wave structure with several asymmetries properly designed to implement a given value for the attenuation and propagation constants [10; 11; 12]. In the literature, a well-known interpretation of the leaky-wave radiation phenomenon also explains the presence of a shadow cone where the field behaves improperly, as shown in Figure 1[11]:  
Inside the dielectric, as is well known from the electromagnetic theory [13; 14; 15], whenever the field reaches the dielectric-air interface, it is partially reflected and transmitted. The reflection and transmission coefficients, also known as Fresnel coefficients, depend on the angle of incidence, the electromagnetic characteristics of the two media involved, and on the polarization (TE or TM) of the considered field [13; 14; 15]. What is important is that at each incidence a leakage occurs, generating a propagating field in the air that moves away from the interface. The ratio of this field with respect to the impinging one is, by definition, the transmission coefficient [13; 14; 15].