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Telecommunications and Radio Engineering
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ISSN Druckformat: 0040-2508
ISSN Online: 1943-6009

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Telecommunications and Radio Engineering

DOI: 10.1615/TelecomRadEng.v63.i9.30
pages 795-813

Band Structure Analysis of 1D and 2D Photonic Crystals by Coupled Wave Method

V. M. Fitio
Lviv Polytechnic National University 12 Stepan Bandera Str, Lviv, 79013, Ukraine
Ya. W. Bobitski
Lviv Polytechnic National University, Ukraine; and Institute of Technology, University of Rzeszow, Poland
H. P. Laba
Lviv Polytechnic National University, 12 St. Bandera Str., Lviv, 79013, Ukraine


The electromagnetic field of wave that propagates in the crystal is decomposed into plane waves in the classic method of computation of photonic crystal band structure. The problem of finding of the allowed frequencies for a wave vector in the first Brillouin zone is reduced to finding of eigenvalues and eigenvectors of the corresponding dimensional representation matrix, which provides for the required precision for finding of the allowed frequencies. Due to slow convergence of decomposition into plane waves it is necessary to solve the matrix equation of large dimension that requires considerable time of calculation. The suggested method of determination of the allowed frequencies is based on the coupled wave method (CWM) at the corresponding setting of periodic boundary conditions. This method is especially simple for 1D photonic crystals and it consists in finding the solution to the eigenvalue problem of TX = ρX type and the check-up whether the absolute value of eigenvalue p is equal to one. The dimensional representation of quadratic matrix T is 2 × 2. The eigenvalue problem of this type appears as a result of the application of the so-called T-algorithm, which is stable for 1D photonic crystal. The solution leads to the eigenvalue problem of TX = ρRX type and the check-up whether the absolute value of eigenvalue ρ is one at given frequency v and component kx of wave vector for 2D photonic crystals, too. The eigenvalue problem of type of TX = ρRX arises due to the application of stable numerical S-algorithm. Dimensional representation of matrices T and R is 2N × 2N, where N is a number of the coupled waves. Frequency will be allowed, if the absolute value of eigenvalue is one. The dimensional representation 2N × 2N of the matrix equation in the suggested method is equivalent to the dimension N2 × N2 of the classic method from the point of view of calculation accuracy. Stable S-algorithm of calculation is developed. The calculations of band structure of 1D and 2D photonic crystal of the simplest form are performed. The dependencies of calculation accuracy upon the number of the coupled waves (N varies from 1 to 29) are obtained.

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