Scattering Resonances for Three-dimensional Subwavelength Holes

Abstract

This thesis aims to investigate scattering resonances and the field amplification at resonant frequencies for two different subwavelength structures: The first structure is a cavity with a closed bottom and width $\varepsilon$ perforated in a slab of sound hard material. The second structure is a hollow hole with both sides open with an upper and lower aperture of width $\varepsilon$, embedded in a sound hard slab. For both structures, we reformulate the boundary value problems by integral equations, and apply the asymptotic analysis and Gohberg-Sigal type theory to study the scattering resonances of the underlying differential operator. We prove the existence of scattering resonances, which are the set of complex-valued frequencies for the homogeneous problem with zero incident field, derive the asymptotic expansion of those resonances, and quantitatively analyze the field amplifications at the resonant frequencies for both cases. It is shown that the complex-valued scattering resonances attain imaginary parts of order $O(\varepsilon^2)$ and the real part of order $O(1)$. We also show that, at the resonant frequencies, the field amplification inside the cavity(hole) and in the far field is of order $O(\frac{1}{\varepsilon^2})$. This is much stronger the enhancement order in the two-dimensional subwavelength hole of the same width, which attain order $O(\frac{1}{\varepsilon}).$