One-dimensional photonic crystals with two defects: An analytical approach

Abstract

In this paper, for the first time, exact analytical expressions for determination of the wavelength of defect modes (DMs) in one-dimensional (1D) photonic crystals (PCs) with two defect layers (DLs) have been obtained from the condition of total transmittance. In contrast to the approximate expressions derived earlier by other authors, exact analytical solutions depending on the reflection coefficients of the PC mirrors have been found. These equations are also an alternative to a more direct calculation of the DM wavelengths via reflection or transmission spectra of the whole structure. The main difference from this method is that the conventional method requires finding the DMs at the particular DL thicknesses while our method allows calculation of the necessary DL thicknesses for any desired DM wavelength. Also, analytical condition for the DM merging in PC with two DLs has been derived for the first time. Furthermore, the results obtained previously by other researchers concerning the merging of DMs as the number of layers in the central part of a PC with two DLs increases have been summarized and explained. Moreover, the analytical condition of the maximum amplitude of DMs was obtained for the first time. The results of this research permit the simplification of the analysis of optical sensors and filters based on 1D PCs with two DLs, as well as a more comprehensive understanding of the DM behavior within the PBGs of such structures. Such defected PCs have a variety of potential practical applications including lasers, sensors, filters and more.