Design and Modeling of Metamaterial Implementations for Soft and Hard Surfaces

Plane Wave Model of Metasurfaces

Any metasurface lining will necessarily be backed by the conducting horn walls. The generic metasurface structure of interest then takes on a form such as that shown in Fig. 2.2, with a patterned screen above a dielectric layer, which, in turn, is backed by a conductive ground plane. This structure can represent a variety of physical implementations, including a metallic frequency-selective-surface (FSS) type of screen [22, 23] or a wire-grid structure [24, 25]. We primarily consider FSS-like structures, as they are generally suitable to conventional printed circuit manufacturing techniques. In contrast with many FSS designs, however, the conducting vias are often critical to achieve the necessary surface impedance properties over a broad bandwidth. Moreover, they provide a conducting path that mitigates electrostatic discharge (ESD) concerns for satellite applications. Metamaterials also differ from FSS structures in that the unit cell sizes are typically a tenth of a wavelength or smaller, while FSS designs tend to have sizes on the order of half a wavelength.

Figure 2.2 Cross sections of the PEC-backed metasurface, consisting of a patterned conducting layer atop a dielectric substrate (top left), with one or more conducting vias in each unit cell. Simulations are performed with plane waves at near-grazing incidence (lower left, в₁ ^ 90°) to calculate both the TE and TM reflection coefficients, from which the surface impedances Z^TE and Z^TM are calculated. The square unit cell pattern (right) consists of an array of pixels, represented in the optimizer by either a "0" or a "1" to indicate the absence or presence of metal. The dashed line indicates a symmetry plane; the pattern on one half of the unit cell is optimized and then mirrored across the symmetry plane. The "V" indicates a typical via location.

As shown in Fig. 2.2, the metasurface has a patterned screen above a dielectric substrate with thickness t and dielectric constant ?_r The pattern is based on square unit cells with a periodicity of w in both the x- and z-directions. Contrary to a conventional FSS, the periodicity w is restricted to be much smaller than the operating wavelength, and thus the metasurface structure can be approximated by its effective surface impedances [18]. The anisotropic surface impedances are defined by the ratios of the electric and magnetic field components tangential to the boundary. As such, they provide the boundary conditions for fields outside the metasurface while accounting for power dissipation and energy storage within the metasurface structure. Surface impedances provide convenient criteria for the design and optimization of metasurfaces for arbitrary electromagnetic responses.

The anisotropic surface impedances can be determined from the reflection coefficients for plane waves at near-grazing incidence upon the metasurface as follows:

where Г denotes the reflection coefficients, R^TE and R^TM are the surface resistances, X^TE and X^TM are the surface reactances, E and H are the electric and magnetic fields, respectively, at the surface of the metallic pattern, and n denotes the transverse wave impedance, defined as:

for obliquely incident plane waves with an incident angle в₁. Although the ideal metasurface would be designed with в₁ = 90°, it is generally necessary to keep в₁ < 90° to make simulations possible, such that d₁ can approach 90° as a limit.