Expressions for reflection and transmission coefficients for one-dimensional photonic quasicrystals

We give general expressions for the light reflection and transmission coefficients for one-dimensional (1D) photonic quasicrystals in the framework of the so-called two-wave approximation. A special attention is paid to quasicrystals composed of dielectric layers, where the reflection and transmission coefficients as a function of the length of the structure take a simple form. The expression for the diffraction vectors of a 1D quasicrystal with an arbitrary number of different constituent segments is discussed as well. Introduction Quasicrystals (QCs) belong to a wide class of aperiodic systems possessing long-range order and allowing coherent Bragg diffraction of electron or electromagnetic waves [1,2]. In the case of light waves such structures are called resonant or active photonic quasicrystals if the constituting materials produce dipole-like excitations; such systems are, for instance, multiple-quantum-well structures (MQWs). The theory of the propagation of electromagnetic waves in MQWs in the framework of the two-wave approximation was developed in work [3] and then generalized on the case of dielectric contrast [4]. It should be noted that in distinction from 1D periodic systems for which one can obtain analytically exact solutions, for 1D non-periodic systems it generally appears to be impossible and one has to resort to some approximations, in particular to that stated here. In this work we theoretically study the propagation of electromagnetic waves in 1D quasiperiodic media and deterministic aperiodic structures (DAS) [3,4], in particular, in quasicrystals. Firstly, we deduce a general expression for diffraction vectors of an arbitrary 1D quasicrystal. Secondly, we consider the two-wave approximation in a general form to calculate the light reflection and transmission coefficients for a one-dimensional DAS with non-zero value of the structure factor. Thirdly, we present the expressions for the reflection and transmission coefficients in the case of an all-dielectric photonic quasicrystal; the expression for the Fourier components εG of the dielectric function of a DAS expressed by the structure factor is also given. SPbOPEN2015 IOP Publishing Journal of Physics: Conference Series 643 (2015) 012053 doi:10.1088/1742-6596/643/1/012053 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 1. The structure factor and diffraction vectors of 1D quasicrystals We use the standard definition of the structure factor [2,3]: