^{*}

In the present work the widths of layers constituting the non-ideal superlattice are much bigger then the characteristic scales of space dispersion. In such a case the contribution of individual layers to gyrotropy can be regarded as independed. Thus the corresponding optical quantities can be expressed through the layers’ gyrotropic characteristics. This approach is applied to calculate the specific rotation angle of plane of polarization of light propagating through a nonideal 1D-superlattice, which varies in composition as well as in layers’ width. We carry out numerical calculation of the frequency dispersion of optical activity of a non-ideal superlattice, which includes impurity layers with point defects.

The procedures used to calculate the transmission coefficients and refractive indices for light developed in [

The problem of finding polariton modes (necessary for calculating gyrotropic characteristics) in spatially dispersed superlattices remains as yet unsolved. At the same time it allows for an approximate treatment at an assumption that the widths of the layers comprising a multilayer much exceed the characteristic scales of space dispersion. In such a case the contribution of individual layers to gyrotropy can be regarded as independent. With regard to the natural optical activity (NOA) this means that a sole knowledge of the layerwise specific angles of rotation (—is the frequency of light,—is the number of the elementary cell in a one-dimensional superlattice,—is the number of the layer in the given cell) and the concentration of impurity layers (if any) is sufficient for finding the total specific angle of rotation. In paper [_{2}-liquid-crystal multilayer. For a specified microscopic expression for the proposed approach enables to identify the dominating experimentally relevant mechanisms of gyrotropy as well as to determine the connection of the given function to the micro-characteristics of the medium (such as dipole, quadrupole, magneto-dipole moments of the structural units etc.) and to find the corresponding frequency characteristics. Of special interest are the superlattices comprised by macroscopically homogeneous systems, which include point defects. In such a case is a function not only of the concentration of impurity layers, but of the concentration of point defects as well. In the present work we carry out numerical calculation of the frequency dispersion of optical activity of a non-ideal superlattice, which includes impurity molecular layers. This extends the capabilities of modeling of composite materials so that their gyrotropic properties and parameters of disordering can be continuously varied.

The above-mentioned approach [4,5] gives the following expression for the angle of optical rotation in an imperfect topologically ordered one-dimensional superlattice composed of N unit cells:

We assume N to be big enough to be able to do configuration averaging. In (1) and are the configuration-dependent specific angle of the light propagation plane rotation and thickness of the α-th layer of the n-th unit cell; σ is the number of unit-cell layers.

According to the general principles of the physics of disordered systems, the rotation angle measured experimentally should be equal to, where is the configuration averaging operator [

Below we consider propagation of linearly polarized light through an imperfect molecular superlattice with two elements-layers per cell, one layer being a mixed molecular crystal and the second being an orientationally disordered crystal. In such a case the expression for the specific angle of optical rotation as it follows from

[

where—specific rotation angles for mixed and orientationally disordered molecular crystals;—thicknesses of first and second layers of corresponding perfect superlattice;—concentrations of isotopic impurities and orientationally disordered molecules,—concentrations of foreign layers in relevant sublattices with thickness differing from those of the perfect system. Here each of the summands has the sense of the rotation angle (by a corresponding sublattice) per one unit cell.

In the most general case, the specific rotation angle for a multicomponent topologically ordered impurity (including both mixed and orientationally disordered) molecular crystal with a primitive lattice in the exciton region of the spectrum is, within the framework of single-level model, of the form:

Here—volume of the unit cell of molecular crystal, r—number of molecular groups, each relating to a definite -th or -th type of molecules; . Quantities and are:

,(4)

In formulae (4), (5)—Levy-Civita tensor, ,—molecular excitation energies,—matrix elements of the dipole-moment operators, —matrix elements of operators derived from operators of molecular currents of the -th and -th molecular types, respectively. Matrix is locator function and matrix is proportional to matrix of the resonance intermolecular interaction (figuring in exciton Hamiltonian [

Let us consider the propagation of electromagnetic radiation in an imperfect SiO_{2}/liquid crystal layered system. The results of modeling the concentration dependence of the specific angle of rotation for this non-ideal superstructure presented in _{2}) and (for model liquid crystal sublattice). The thickness of base-substance layer are denoted by and, respectively, and are those of the impurity (index Т stands for variation of impurity layers in thickness).

For non-ideal superlattice, which includes impurity

molecular crystal layers with point defects sublattice, we use values in mononodal approximation [

The character of the frequency-concentration dependence of rotator power is specified by the relative position of exciton energies and by values of parameters,.

In particular calculations it is considered that for a mixed molecular crystal it can be, within a good accuracy, assumed [

, (as a consequence, all

). For an orientationally disordered molecular crystal [

, , whereas for the orientationally disordered one—

, , and, . Here, the numerical simulation was done for the following values of concentration of heterogeneous superlattice layers:, and for relative thicknesses (). The concentration dependences of the specific rotation angle of light are graphically shown in Figures

2-4 in relative units, where (see

(3)).

The frequency dependences of the specific rotation angle for specific values of concentration and is illustrated in Figures 4 and 5. It is seen that changes in concentration with constant result in mutual removal of resonance frequencies (

The present microscopic consideration of dispersion of the optical activity of imperfect 1D-multilayer material is helpful for a numerical simulation of the frequencyconcentration dependence of the specific rotation angle for molecular superlattices in the exciton region of the spectrum.

The model was a two-sublattice system with parameters typical of orientationally disordered and mixed molecular crystals. Gyrotropy features due to disorder types have been studied. An 1D-superlattice with layers containing point defects has been investigated. This creates additional possibilities for simulation of optically active multilayer composite materials.

The authors are grateful to Dr. Esfir Shtaerman for his help in numerical simulation. This work is supported by Project “Dynamical and static properties of complex low-dimensional systems in external fields” of National Academy of Sciences of Ukraine.