The dielectric permittivity tensor for incommensurately modulated phases in insulating crystals is considered. The main attention is paid to the spatial dispersion giving rise to optical gyration effects and the influence of the modulation phase on those effects. General properties of the dielectric tensor are analysed in relation to the lattice and incommensurate superlattice periodicity requirements, the Onsager principle and the condition of absence of the radiation losses in the optical medium. The structure of the microscopic components of the tensor and a macroscopic averaging procedure for the plane-wave modulation region are discussed. The possibility of existence of an anti-Hermitian part in the dielectric tensor of lossless incommensurate crystals originating from the modulation-induced spatial dispersion is revealed. The performed analysis shows a necessity for introduction into the constitutive equation of the term including spatial derivatives of the optical activity tensor.

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Subject classification: 64.70.Rh; Crystal optical properties of insulating materials possessing incommensurately modulated phases are studied in the framework of a microscopic model. Quantum-mechanical expressions for the microscopic dielectric permittivity tensor are derived. It is shown that the contributions from the long-wavelength reciprocal lattice vectors with the lowest microscopic indices should be taken into account in the optical response of the incommensurate crystals. This justifies, from the standpoint of microscopic theory, a mesoscopic approach to the problem of light propagation in incommensurate crystals adopted in a number of earlier studies. The mesoscopic tensors associated with the first-order spatial dispersion are obtained and analyzed in detail. The optical gyration is revealed to be described by the dielectric tensor components linear in both the light wave vector and the mesoscopic incommensurate modulation wave vector. The latter contributions correspond to the gyration mechanism related to the mesoscopic inhomogeneity of the optical medium under study.

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