Analytical properties of the effective refractive index

The propagation of a plane wave through a periodic layered system is considered in terms of the effective parameters. The problem of introduction of effective parameters is discussed. It was demonstrated that although the effective admittance cannot be introduced, it is possible to introduce the effective refractive index, which tends toward the Rytov value when the system size increases. It was shown that the effective wave vector derivative is an analytical function of frequency. In particular, the Kramers-Kronig-like relations for real and imaginary parts of the effective wave vector derivative were obtained. The Kramers-Kronig-like relations for the effective refractive index were also considered. The results obtained numerically were proved by exact solution of Maxwell's equations in the specific case of an “equi-impedance” system.