# An Overview of Design Techniques for Two-Dimensional Leaky-Wave Antennas
## 1 Introduction  
Leaky-wave antennas (LWAs) are radiating devices constituted by waveguiding structures which allow for continuous power radiation while propagating along their length [1, 2, 3]. Although the working principle is rather intuitive, the design of radiating devices by partially open waveguides found its mathematical and physical rigorous foundation only in the 1950s thanks to the famous work of [PERSON] [4]. This paper, together with the works of [PERSON] and [PERSON] [5, 6], had an important impact on the development of LWAs: thanks to an efficient and elegant theory, rather simple analytical procedures were outlined for designing such radiating devices, in a period when the computational resources were very limited. Nowadays, such a straightforward theoretical approach has been experimentally verified in different scenarios and with different radiating devices [7, 8, 9, 10, 11, 12, 13, 14, 15]. Moreover, this design method is still relevant and faster than full-wave optimization procedures [16].  
It is well known that this kind of radiating devices shows many advantages since LWAs are simple and low-cost solutions able to generate highly directive beams pointing at almost arbitrary angles [17]. Moreover, they show a reconfigurability feature in terms of pointing angle since their beam naturally scans with frequency [18] or, if a fixed frequency is desired, they are able to scan electronically by incorporating tunable elements into thedesign, such as varactor diodes [19], ferrites [20], microelectromechanical systems [21] based on micromachining technologies [22], graphene [23; 24], or liquid crystals [25].  
The main critical aspects of LWAs are typically related to the following trade-offs: pattern fractional bandwidth vs. directivity (leaky-wave antennas are usually highly directive but show a narrow bandwidth), reconfigurability vs. complexity (the higher the desired reconfigurability at a fixed frequency, the more complex the device architecture), and radiation efficiency vs. aperture size (a large radiating aperture is needed to achieve a highly directive antenna) [2]. Fortunately, various solutions have been proposed in recent years to address these challenges. For instance, to improve the figure of merit defined by the product of directivity with bandwidth, Fabry-Perot cavity antennas (FPCAs) incorporating a thick partially reflecting sheet (PRS) [26; 27] or exhibiting high truncation effects [28; 29] have been developed and studied through different techniques. As concerns, instead, reconfigurable leaky-wave antennas, different technological solutions have recently been addressed from the microwave to the terahertz frequency ranges [30; 31; 32; 33; 34]. Moreover, the longstanding problem of designing directive LWAs through compact devices has been solved by exploiting tapering techniques [35; 36].  
The effectiveness and the importance of LWAs from both a theoretical and a practical viewpoint are thus clear. However, a paper reviewing simple numerical techniques for characterizing LWAs and providing design guidelines for such radiators is still missing. This work aims to fill this gap by describing, step by step, the design workflow of uniform and radially periodic two-dimensional (2-D) LWAs which represent the common architectures of more sophisticated solutions, such as holographic [37] and modulated surface antennas [38; 39; 40; 41], or even near-field focusing devices [42; 43; 44].  
This manuscript is organized as follows. In Section 2, an established classification of LWAs is presented, highlighting the need for various approaches to analyze different kinds of these radiators. In Sections 3 and 4, the numerical techniques for the analysis and design of uniform and radially periodic 2-D LWAs are presented, respectively. Conclusions are finally drawn in Section 5.