The optical properties of a stack of metamaterial-based cholesteric liquid crystal (CLC) layers and isotropic medium layers are investigated. The problem is solved by a modification of Ambartsumian's layer addition method. CLCs with two types of chiral nihility are defined. The peculiarities of the reflection spectra of this system are investigated and it is shown that the reflection spectra of the stacks of CLC layers of these two types differ from each other. Besides, in contrast to the single CLC layer case, these systems have multiple photonic band gaps. There are two types of such gaps: those selective with respect to polarization of the incident light and nonselective ones. It is shown that the system eigenpolarizations mainly coincide with the quasi-orthogonal, quasi-circular polarizations for normally incident light, except the regions of diffraction reflection selective with respect to the polarization of incident light. The influence of the CLC sublayer thicknesses, the incidence angle, the local dielectric (magnetic) anisotropy of the CLC layers, and the refractive indices and thicknesses of the isotropic media layers on the reflection spectra and other optical characteristics of the system is investigated.
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