  We investigated on the interface modes in a heterostructure consisting of a
semi-infinite metallic layer and a semi-infinite Fibonacci quasi-periodic
structure. Various properties of the interface modes, such as their spatial
localizations, self-similarities, and multifractal properties are studied. The
interface modes decay exponentially in different ways and the modes in the
lower stable gap possess highest spatial localization. A localization index is
introduced to understand the localization properties of the interface modes. We
found that the localization index of the interface modes in the upper stable
gap will converge to two slightly different constants according to the parity
of the Fibonacci generation. In addition, the localization-delocalization
transition is also found in the interface modes of the transient gap.
