## 1. Introduction

[2] The frequency evolution of the propagative modes of planar printed lines has received a great deal of early attention [*Wu*, 1957; *Yamashita*, 1968; *Itoh*, 1980, 1989]. Later on, this study was extended to investigate the evolution of surface as well as space leaky modes [*Menzel*, 1979; *Oliner*, 1987; *Das and Pozar*, 1991; *Tsuji et al.*, 1993; *Shigesawa et al.*, 1993; *Nghiem et al.*, 1993; *Mesa and Marqués*, 1995]. Some relevant theoretical and practical questions concerning the excitation and physical validity of surface/space-leaky modes were more recently discussed by *Di Nallo et al.* [1998], *Mesa et al.* [1999], *Jackson et al.* [2000], and further *Mesa et al.* [2002] presented an investigation on the physical and mathematical continuity between different modal solutions (proper, improper real, and leaky) in order to establish the conditions under which certain modal transitions can occur.

[3] Despite the thorough effort devoted to study the evolution and nature of the modal spectrum of printed lines, a problem not yet considered is the evolution of the modes of a planar printed line whose upper half-space is not free space but an isotropic dielectric of arbitrary permittivity ɛ_{u}. Specifically, if the permittivity of the upper half-space of a microstrip line is greater than the permittivity of the microstrip substrate (ɛ), the background grounded dielectric waveguide cannot support surface-wave (SW) modes [*Rodríguez-Berral et al.*, 2004a] but only leaky and improper real modes. This fact clearly prevents the existence of surface-wave leaky modes (SFWLM) in the line because the radiation cannot take place in the form of nonuniform SWs of the background waveguide, and hence all the leaky modes of the transmission line will present spatial radiation. Moreover, bound modes cannot occur in such line as long as the permittivity of the upper half-space is greater than the substrate permittivity. In consequence, the discrete modal spectrum for ɛ_{u} > ɛ is expected to consist only of space-wave leaky modes (SPWLM).

[4] The present work will be then devoted to study the characteristics and the evolution of the modes of a single microstrip line and a pair of coupled microstrip lines as the permittivity of the upper half-space varies from its lower value (ɛ_{0}) to values higher than the permittivity of the line substrate. In this way, this primary 2-D study will show the most relevant features of the modal spectrum and will explore the new modal transitions that may occur in this type of lines, but that could not appear in standard printed lines. Specifically, it is expected to find possible modal transitions from bound, SFWLM or real improper modes (RIM) to SPWLMs. Although the rich phenomenology of the dispersion relations to be found in the structures under study yields an inherent theoretical interest by itself, the present study also constitutes the first necessary step previous to carry out the analysis of new types of leaky-wave antennas for possible practical applications such as ground penetrating radar [*McMillan and Shuley*, 1997], hyperthermia antennas [*Dubois et al.*, 1996], and any other situation where the superstrate can be modeled as a semi-infinite cover layer.

[5] The paper is organized as follows: next section will briefly expose the mathematical aspects of the formulation of the problem and the algorithm employed to search for the modal wavenumbers. Section 3 will show and discuss the numerical results obtained for the microstrip and the pair of coupled lines. Results will be also presented for a single microstrip line with an air gap between the printed interface and the upper half-space. This latter structure can simulate the practical situation of a printed line that is not directly in contact with the medium modeled by the semi-infinite cover layer. The corresponding results will corroborate our assumption of the exclusive space-leaky nature of the modal spectrum for ɛ_{u} > ɛ, and will also show a new kind of physical modal transition from a bound mode to a SPWLM.