Several theorems for reflection and transmission coefficients of plane wave incidence on planar multilayer metamaterial structures

Consider an arbitrarily polarised plane wave obliquely incident on a planar multilayer structure composed of a combination of natural materials and metamaterials, then the following three new theorems are proved in this paper. Theorem 1: Consider a planar multilayer structure made of a combination of common materials and metamaterials situated between two half spaces composed of lossless media. Now each layer is filled by its dual media according to the interchanges DPS ↔ DNG and ENG ↔ MNG. Then, the reflection (R) and transmission (T) coefficients from the structure become the complex conjugates of their counterparts. Consequently, the reflected power and transmitted power from the structure are the same for the two dual cases. Theorem 2: If the interchanges DPS ↔ DNG and ENG ↔ MNG are made in all the layers except in the half spaces on the two sides of the multilayer structure, then the reflection coefficients become complex conjugates and the reflected power remains the same. Theorem 3: If a planar multilayer structure is backed by a perfect electric conductor and the media interchanges DPS ↔ DNG and ENG ↔ MNG are made in the layers, then the reflection coefficients of the two dual structures become complex conjugates of each other, and the reflected powers are equal. Finally, several examples and applications with dispersion are included.