On the structure of the microscopic dielectric tensor in incommensurate crystals

Spatial dispersion in incommensurately modulated insulators

“…It has been shown in [14] that the long-wavelength modulation with the wavevector q essentially affects a number of physical properties of the A 2 BX 4 crystals, thus justifying keeping the contributions from q while passing to the macroscopic parameters. Such an approach also agrees with the microscopic models which predict that the Fourier components εh i j decrease rapidly with increasing n 3 or m (see [2], chapter 4, and the proof for the IC systems [27,33]). Hence, we shall not discuss the known [13,15] procedure of macroscopic averaging of equation ( 23) but write down straightaway the mesoscopic CR for the case of a single long-wavelength periodicity (λ m = 2π/q) dominating in the structure of the IC phase:…”

“…Nonuniformity of the normal waves and a spatial dependence of their polarization state represent another difficulty in solving the 'inverse problem' of crystal optics-identification of the effects on the basis of the known normal wave polarization. According to the approach of [14,22], the difficulty might be avoided by passing from (33) to the analysis of 'effective normal waves', whose polarization does not formally depend upon the spatial coordinates but is instead determined by the phase values ϕ + qz at the boundary surfaces of a given crystal plate. Under certain circumstances, the approach may provide sample-independent results or, at least, results that are not too sensitive to the sample thickness, which are believed to be the most reasonable (see section 5.2).…”

“…denotes the phase retardation due to the linear birefringence and α ± x , α ± y are determined by the amplitudes A ± 1y and A ± 2x , which appear as the coefficients of the terms exp[±i(ϕ + qz)] in formulae (33):…”

“…From ( 34), ( 35) and (37), one can obtain the result χ y = −χ x which is well known in the gyration theory for non-modulated crystals, together with the equality θ y = θ x , which testifies that the θ parameter may indeed be treated as the uniquely defined rotation angle for the principal axes of the optical indicatrix ellipsoid. When we turn to the modulation in the IC phases, it looks also to be appropriate to neglect the terms ∼g a n, ε a n, 5 the more so since the initial wave equation is valid under the condition n n. Then formulae (33) and (36)…”