  Recent progress in the fabrication of materials has made it possible to
create arbitrary non-periodic two-dimensional structures in the quantum plasmon
regime. This paves the way for exploring the plasmonic properties of electron
gases in complex geometries such as fractals. In this work, we study the
plasmonic properties of Sierpinski carpets and gaskets, two prototypical
fractals with different ramification, by fully calculating their dielectric
functions. We show that the Sierpinski carpet has a dispersion comparable to a
square lattice, but the Sierpinski gasket features highly localized plasmon
modes with a flat dispersion. This strong plasmon confinement in finitely
ramified fractals can provide a novel setting for manipulating light at the
quantum scale.
