Transition Function for Closed-Form Representation of Metasurface Reactance

Metasurfaces are thin metamaterials used for manipulating propagation of plane waves and surface-waves (SWs). They can be characterized by homogenized-boundary conditions, which, in absence of losses, can be represented through an equivalent reactance. In this paper, we introduce a general representation of isotropic frequency-dependent reactance which is valid along the dispersion curve of the relevant TM SW. This representation is written in terms of a transition function derived from a manipulation of the Cardano's formula for third-degree algebraic equations. Throughout a large portion of the dispersion curve, this transition function depends on one parameter only, which is an equivalent quasi-static capacitance. Approaching the Floquet-Bloch region, where many higher order Floquet modes are excited, two additional parameters should be extracted from the full-wave data to complete the transitional representation of the reactance until the upper boundary of the Brillouin region. The final formula is valid for a generic isotropic reactance and for an anisotropic reactance when the direction of propagation is along a symmetry axis of the unit cell element.