On plasmon modes in multi-layer structures

In this paper, we consider the plasmon resonance in multi-layer structures. The conductivity problem associated with uniformly distributed background field is considered. We show that the plasmon mode is equivalent to the eigenvalue problem of a matrix, whose order is the same to the number of layers. For any number of layers, the exact characteristic polynomial is derived by a conjecture and is verified by using induction. It is shown that all the roots to the characteristic polynomial are real and exist in the span [-1, 2]. Numerical examples are presented for finding all the plasmon modes, and it is surprisingly to find out that such multi-layer structures may induce so called surface-plasmon-resonance-like band.