Integral and integrodifferential equations for quasiperiodic structures

Abstract

Periodic structures (PSs) are interesting for many categories and fields of physics and chemistry, so correspondingly there is tremendous number of works devoted to their analysis. PSs describe different physical phenomena [1]. Especially it is concerning towards following areas: crystalline semi-conductors and metals in solid-state and semiconductor physics [2], slow-wave structures (SWS) and periodic waveguides in electromagnetics and electronics [3], crystals in crystal optics [4], photonic crystals (PC) [5,6], periodic metamaterials (artificial media) [7] in electrodynamics, optics, and acoustoelectronics, X-ray and particle diffraction by crystals. So, recently the considerable interest to PPS arose in different branch of knowledge (electronics, slow-wave, electrodynamic PC and metamaterials structures, photonics, optics, nanostructure physics). It is caused by the development of nanotechnology and also by such circumstance, that PS is the mathematical abstraction, and all real structures are the PPSs. The violation of periodicity arises owing to number of circumstance. The main is the finiteness of all real structures. This principal factor is essential here when the number of periods is not large. Its influence is sharply decreasing when the number of periods increases over a certain value, and the PPS is inherently similar (in the internal regions) to PS one. The periodicity-breaking factors also are the nonstationarity (aperiodicity) in time, the loss and amplification (generation). Notice, that in finite passive (lossy) or active PC and SWS structures the eigenfrequencies are complex. It means the time aperiodicity i.e. the nonstationarity.