Theory of Seamless-Scanning Periodic Leaky-Wave Antennas based on $\mathcal{PT}$-Symmetry with Time to Space Mapping

TL;DR

This paper develops a comprehensive electromagnetic theory of periodic leaky-wave antennas (P-LWAs) using $ ext{PT}$-symmetry and time-to-space mapping, addressing the broadside issue and providing insights into their spectral and scattering properties.

Contribution

It introduces a $ ext{PT}$-symmetry framework for P-LWAs, linking their spectral features to electromagnetic conditions and deriving their scattering parameters with implications for antenna design.

Findings

P-LWAs are $ ext{PT}$-symmetric systems with characteristic Riemann surfaces.

The broadside issue relates to the $ ext{PT}$-symmetric double pitchfork spectrum.

Matching occurs only at one end of the P-LWA structure.

Abstract

Periodic Leaky-Wave Antennas (P-LWA) offer highly directive and space-scanning radiation. Unfortunately, they have been plagued by the ``broadside issue'', characterized by a degradation in gain when the antenna's main beam is steered across broadside. While this issue has been addressed by circuit and network approaches, a related fundamental and general electromagnetic theory has been lacking. This paper fills this gap. We first show that a P-LWA is a $\mathcal{PT}$-symmetric system, whose even- and odd-mode coupling in the complex space of temporal frequencies leads to the characteristic pair of two-sheet Riemann surfaces. We observe that the branch cuts of these surfaces, which form the well-known $\mathcal{PT}$-symmetric double pitchfork spectrum, correspond to the ``balanced frequency'' condition, while the branch point at the junction of the pitchforks, is an exceptional point that corresponds to the ``$Q$-balanced'' condition, two conditions that where previously shown to be the conditions for eliminating the broadside issue. In order to acquire an independent and rigorous interpretation of this spectrum, we further transform the coupled complex temporal eigenfrequencies into complex spatial eigenfrequencies. We identify the resulting spatial frequencies as coupled forward-backward modes, with frequency-independent imaginary parts (leakage factors), a condition for P-LWA equalization across broadside. Finally, we derive the scattering parameters of the P-LWA and show that matching is achieved only at one of the two ends of the P-LWA structure. This work both provides a solid foundation to the theory of P-LWAs and represents an original contribution to the field of $\mathcal{PT}$-symmetry.