Abstract

In this thesis we examine the surface impedance of a grounded dielectric slab with a periodic perturbation of its geometry or permittivity. Using the impedance boundary condition (IBC) allows for the reduction in the number of physical structural parameters such as width, height and permittivity
to one single parameter, which is the impedance Zs. This simplifies the electromagnetic problem. We reduce a 3D IBC problem to a 2D IBC problem because it is suitable for modeling the principal planes of the 3D problem. The 3D and 2D problems are related through a Fourier transform and the 2D problem is relatively easy to analyze.
In the literature, one can find the solution for a constant IBC. However, we are interested in a general IBC field solution that allows for an arbitrary impedance. Commercial electromagnetic solvers do allow for a penetrable IBC but do not allow for an impenetrable IBC. The proposed
IBC-MoM (method of moments) is a general solution that allows for an arbitrary impenetrable IBC. We will use this solution for leaky-wave antenna design.
For an IBC to accurately model a physical electromagnetic problem, the surface impedance must match the impedance of the true physical antenna that is being modeled. In general this will not be a constant and depends on the source configuration. It is therefore imperative to consider the surface impedance variation as the excitation is changed. Using the rigorous 2D Green’s function field solution for the grounded dielectric slab, it is shown how the surface impedance varies as the excitation source location is changed. For the transmit problem (source close to the slab surface), a Sommerfeld integral is numerically evaluated in the complex w plane, which allows one to compute the exact surface impedance at any surface point, including points near the source.
The surface impedance from the exact solution is shown to agree with a numerical electromagnetic full-wave (COMSOL) solution.
For a transmitting antenna the surface impedance is a relatively complicated function, compared to the receiving case. It is therefore incorrect to assume that surface impedance is simply a number based on plane wave incidence, when considering transmitting antennas. This is illustrated by considering the surface impedance on a grounded slab, which is excited by a line source close to the slab surface. For the IBC to have the same field as the original grounded slab, the surface impedance should match the actual slab– and it is not a constant, but varies, especially near the source. A closed form analytic IBC Green’s function is not possible when the surface impedance is a function over the surface. However, we can still obtain an exact solution by using radiation integrals on the infinite aperture. We decompose the true aperture fields from the grounded slab problem into geometrical optics (GO) and the surface wave parts. This in turn allows one to compute the field solution by solving the radiation integrals over an infinite aperture. The field solution is shown to agree with COMSOL, which validates the concept.
Sinusoidal and square-wave modulations of the surface reactance are considered. An impenetrable IBC in conjunction with the MoM is used to obtain the radiation patterns for 2D leaky-wave antennas of finite extent. In order to physically realize the modulation, we start with the theory of a
grounded dielectric slab. The slab supports a surface wave and a fast spatial harmonic (in our case n = −1) will be excited via periodic perturbation of the slab along the guiding direction. First, a sinusoidal reactance modulation is analytically mapped into a slab thickness modulation. Second, a square-wave reactance modulation is mapped into a permittivitymodulation, and is realized with subwavelength perforations. The radiation patterns from the 3D physical antennas are computed with a commercial electromagnetic solver (CST) and are validated with our simplified 2D IBCMoM code. The resulting scan angles and beamwidths predicted by the CST model are found to
agree with the 2D IBC-MoM. The 2D model is shown to be useful, because it can treat not only sinusoidal and square-wave modulations but any profile. Therefore, it becomes easy to rapidly predict in 2D the key radiation characteristics (the scan direction and beamwidth) for impedance
profiles that are arbitrary. Aspects of the feed design for the 2D and 3D models are addressed to minimize contamination of the radiation pattern by direct radiation from the source.
The extreme near field on the surface is computed with CST and the results confirm that the expected surface impedances are being realized by the thickness-modulated and permittivity modulated structures.