6120 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 11, NOVEMBER 2017 Communication Beamwidth Properties of Endﬁre 1-D Leaky-Wave Antennas Walter Fuscaldo, David R. Jackson, and Alessandro Galli Abstract— In this communication, the beam properties of one-dimensional leaky-wave antennas (1-D LWAs) are explored when the LWA scanning approaches endﬁre. By increasing the phase constant beyond the ordinary endﬁre condition, smaller beamwidths can be produced at the expense of increased sidelobe level (SLL). This tradeoff is quantiﬁed here for the ﬁrst time for endﬁre 1-D LWAs. Simple CAD formulas are also proposed that describe accurately the beamwidth and SLL for 1-D LWAs operating at endﬁre. This communication is conducted for different radiation efﬁciencies covering most practical ranges. A closed-form expression for the beamwdith of a 1-D LWA with an inﬁnite aperture is also given, in which case there are no sidelobes. These results allow for a better understanding and efﬁcient design of LWAs radiating at endﬁre. Index Terms—Beamwidth, endﬁre antennas, Hansen-Woodyard condi- tion (HW condition), leaky-wave antennas (LWAs), sidelobe level (SLL). I. I NTRODUCTION AND MOTIVATION It is well known that for continuous antenna arrays, as well as for 1-D LWAs, the pointing angle θ 0 and the phase constant β are related through β = k 0 cos θ 0 , where k 0 is the wavenumber in vacuum. As a matter of fact, this relation is exact as long as β ≤ k 0 . When β ≥ k 0 the beam points at endﬁre (θ 0 = 0). Hansen and Woodyard [1] have shown that a particular value of β allows for improving the directivity at endﬁre for a linear array. In particular, they found that the optimum condition for maximizing directivity at endﬁre is given by the so- called Hansen-Woodyard condition (HW condition) [1] β = k 0 + 2.94/L (1) where L is the length of the antenna (see Fig. 1). Since β = k 0 (i.e., θ 0 = 0) corresponds to the ordinary endﬁre condition (OE condition), it is seen that the phase shift (β ) given by β = 2.94/L (2) represents the perturbation of the OE condition to achieve maximum directivity at endﬁre for antenna arrays. This condition applies only for antenna arrays with a constant aperture distribution. However, it has recently been demonstrated [2] that a modiﬁed version of (2) can suitably be applied to endﬁre 1-D LWAs (see Fig. 1 for an example), which are equivalent to antenna arrays with an exponentially decaying aperture distribution of the form exp(- jk z z ) where k z = β - j α is the complex longitudinal wavenumber, and α is the attenuation constant (or leakage constant ). Manuscript received February 8, 2017; revised August 1, 2017; accepted August 26, 2017. Date of publication September 1, 2017; date of current version October 27, 2017. (Corresponding author: Walter Fuscaldo.) W. Fuscaldo is with the Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, 00184 Rome, Italy, and also with the Institut d’Électronique et de Télécommunications de Rennes, UMR CNRS, 6164 Rennes, France, and also with the Université de Rennes 1, 35042 Rennes, France (e-mail: fuscaldo@diet.uniroma1.it). D. R. Jackson is with the Department of Electrical and Computer Engineer- ing, University of Houston, Houston, TX 77204-4005 USA. A. Galli is with the Department of Information Engineering, Electronics and Telecommunications, Sapienza University of Rome, 00184 Rome, Italy. Color versions of one or more of the ﬁgures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identiﬁer 10.1109/TAP.2017.2748363 Fig. 1. Example of a 1-D LWA. A leaky mode is excited at the source location z = 0. The metallic top wall of a rectangular waveguide is replaced by a partially reﬂecting screen to allow the propagating mode to leak out along the z -axis. The antenna is terminated with a matched load at z = L . This new condition is known as the modiﬁed HW condition (mHW condition) and it states that β = τ/L (3) where τ is a parameter that depends on the radiation efﬁciency e r of the antenna due to load termination [2]. The above condition gives the phase constant necessary for achiev- ing the maximum directivity from a 1-D LWA. This does not imply, however, a minimum beamwidth. In this communication, we explore the beamwidth properties of a 1-D LWA when the beam is scanned beyond OE (β > k 0 ) showing that beamwidths smaller than in the HW condition are possible, but at the expense of an increased sidelobe level (SLL). This tradeoff is quantiﬁed, and simple CAD formulas are introduced for the beamwidth and the SLL in terms of the phase constant and the radiation efﬁciency of the antenna. These formulas are accurate for the range of radiation efﬁciencies usually encountered in LWA design, where the efﬁciency e r (due to a load termination) is less than 95%. The nature of the beam changes when the aperture becomes very large, corresponding to very high radiation efﬁciencies. A separate formula, which is exact, is derived for the beamwidth in the inﬁnite aperture case, when the beam is scanned to or beyond OE. In this case there are no sidelobes, and the beamwidth is only a function of the phase and attenuation constants. The formulas derived here assume a single leaky wave on the aperture, with a complex wavenumber. In a practical LWA, there will usually be some amount of space-wave radiation or mode coupling that takes place as the beam is scanned to and beyond endﬁre, since the leaky mode is then entering into the “spectral gap” region where it begins to lose physical signiﬁcance [3]. The degree to which this happens is structure speciﬁc, however, and such effects should be considered on a case-by-case basis. An example of this is discussed in [4]. It is worth mentioning here that general beamwidth formulas for 1-D unidirectional LWAs have recently appeared in [5]. However, in that work, exact and approximate analytical expressions are 0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.