V. V. Nikolaev

Bragg reflectors for cylindrical waves

A transfer matrix method is developed to calculate the electromagnetic field in a dielectric structure with circular cylindrical symmetry. The equations for the reflection and transmission coefficients of cylindrical waves from a single cylindrical boundary between two dielectrics and from a cylindrical multilayered structure are obtained. For a single dielectric interface, enhanced reflection at small interface radii and the analogue of the Brewster effect are predicted and investigated. The design of an optimized cylindrical Bragg reflector (CBR) for cylindrical waves is proposed and its optical properties are studied. It is found that the thicknesses of the layers in the CBR must be different, to provide the adjustment of the phase of the waves, that are reflected from the interfaces at different radii.

Bandgap structure of optical Fibonacci lattices after light diffraction

The modification of the bandgap structure of optical Fibonacci lattices that arises from an increase in the system size is analyzed. It is found that there is a minimum critical size of the Fibonacci lattice necessary to form a photonic bandgap. The analysis of localized photonic modes of Fibonacci lattices shows that the removal of two layers of a lattice makes the lattice mirror-symmetric. The “method of oblique bands” is developed to analyze the bandgap structure of quasi-crystals. This method can also be applied to study two-and three-dimensional photonic and atomic quasi-crystals.

Optical eigenmodes of a cylindrical microcavity

Abstract The optical mode structure of a cylindrical microcavity has been investigated using a transfer matrix approach. We derive exact algebraic equations from which the frequencies of the optical eigenmodes of the two polarizations can be obtained, as well as approximate explicit algebraic expressions for those frequencies.

Eigenstate statistics and optical properties of one-dimensional disordered photonic crystals.

Optical eigenstates in one-dimensional disordered photonic crystals were studied. The threshold disorder level was established below which the probability of appearance of an eigenstate at the photonic bandgap center is negligible. The threshold is reached when the relative fluctuation in the optical lengths of the structure periods becomes equal to the square root of one-third of the relative bandgap width. The dependence of the ensemble-averaged structure transmission coefficient on the fluctuation of the period optical length has a break corresponding to the threshold fluctuation.

Calculation of the mode structure of multilayer optical fibers based on transfer matrices for cylindrical waves

Based on the method of transfer matrices for cylindrical waves, an analytical technique for calculating the dispersion dependences and the profiles of electromagnetic waves in multilayer and graded index optical fibers is developed.

Quantization of an electromagnetic field in two-dimensional photonic structures based on the scattering matrix formalism (S-quantization)

Based on the scattering matrix formalism, we have developed a method of quantization of an electromagnetic field in two-dimensional photonic nanostructures (S-quantization in the two-dimensional case). In this method, the fields at the boundaries of the quantization box are expanded into a Fourier series and are related with each other by the scattering matrix of the system, which is the product of matrices describing the propagation of plane waves in empty regions of the quantization box and the scattering matrix of the photonic structure (or an arbitrary inhomogeneity). The quantization condition (similarly to the onedimensional case) is formulated as follows: the eigenvalues of the scattering matrix are equal to unity, which corresponds to the fact that the set of waves that are incident on the structure (components of the expansion into the Fourier series) is equal to the set of waves that travel away from the structure (outgoing waves). The coefficients of the matrix of scattering through the inhomogeneous structure have been calculated using the following procedure: the structure is divided into parallel layers such that the permittivity in each layer varies only along the axis that is perpendicular to the layers. Using the Fourier transform, the Maxwell equations have been written in the form of a matrix that relates the Fourier components of the electric field at the boundaries of neighboring layers. The product of these matrices is the transfer matrix in the basis of the Fourier components of the electric field. Represented in a block form, it is composed by matrices that contain the reflection and transmission coefficients for the Fourier components of the field, which, in turn, constitute the scattering matrix. The developed method considerably simplifies the calculation scheme for the analysis of the behavior of the electromagnetic field in structures with a two-dimensional inhomogeneity. In addition, this method makes it possible to obviate difficulties that arise in the analysis of the Purcell effect because of the divergence of the integral describing the effective volume of the mode in open systems.