Norm of iegnfunction of one-dimension photonic crystal

Abstract

Relevance. In recent decades (about the 90-s ХХ century) there has been rapid development of photonic. Thus, to arise scientific interest to optic range of electromagnetic radiation. Currently, the diffraction problem about scattering electromagnetic waves on such object as photonic crystal is impotent problem. As well known, this problem can be reduced to a solution of wave equation. The need to calculate the norm iegnfunction spectral iegnfunction Sturm-Liouville problem, however, to arise in the transition from one complete orthogonal system to another complete orthogonal system of functions by separating variables method, correspondingly, for a wave equation solving. The purpose of the work. We indicate a direct approach to calculating of the norm of iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal (a direct approach to calculating of the norm that is presuppose a direct integration); and propose a methodologically different approach, which is based on the marginal transition in the scalar product, which accordingly sets this norm. Materials and methods. Taking the limit in calculation the norm of the iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal encounters difficulties, associated with the emergence of species uncertainty . Such infinitive investigates by the Lopital's rule. In turn, Lopital's rule entails the need to find a derivative of solution of spectral equation by a spectral param. In this way we have to face the solution a linear inhomogeneous differential equation 2-nd order. Results. We propose a methodic of calculating of norm of iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal. Conclusion. Unlike the direct approach, proposed methodic to make it possible to understand the character of dependencies the required norm of iegnfunction itself (ending expression containing the iegnfunction itself). Further work in this direction of development of this approach may be aimed at simplifying the final expression for the norm.