Anisotropic photonic crystals: Generalized plane wave method and dispersion symmetry properties Irina A. Khromova a,b, * , Leonid A. Melnikov a a Saratov State University, Astrakhanskaya 83, Saratov 410026, Russian Federation b Public University of Navarre, Campus Arrosadia, Pamplona, Navarra E31006, Spain article info Article history: Received 29 April 2008 Received in revised form 21 July 2008 Accepted 25 July 2008 Keywords: Photonic crystal Anisotropic photonic crystal Plane wave method Band gap abstract This article presents a generalized vector plane wave expansion method, applicable to isotropic and anisotropic periodic dielectric media of arbitrary geometry and dimension. The influence of anisotropic material orientation on the symmetry properties of photonic crystal dispersion surface is discussed. It is shown that the overall Brillouin zone symmetry is formed by the intersection of the photonic crystal lattice symmetry and the symmetry determined by the anisotropic material orientation. This work explains how to define the irreducible Brillouin zone of a two-dimensional anisotropic photonic crystal and demonstrates that doing it correctly allows one to avoid erroneous results, when calculating band gap diagrams of anisotropic photonic crystals. With the help of the methods presented, the possibility of controlling the band gaps of anisotropic photonic crystals by means of external electric field is shown. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Photonic crystals (PCs), thanks to their unique properties and the wide range of application possibilities they offer, are nowadays among the most popular and studied objects in optics and photon- ics [1–5]. This work is devoted to anisotropic PCs, i.e. artificially created periodic dielectric objects with components made of anisotropic materials. PCs of this type are very interesting, as, due to their con- trollable dispersion, they can be used for creating tunable optical de- vices [6–12], in particular, devices tunable by external electric field. This paper presents the full vector plane wave expansion meth- od, suitable for eigenwave analysis of both isotropic and aniso- tropic periodic dielectric media with arbitrary geometry and dimension. It allows for tensor dielectric permittivity values and arbitrary material distributions. The main goal of using anisotropic materials within PCs lies in the idea of external band gap control. For instance, using liquid crystals (LCs) and applying external electric field one could open or close or shift the positions of band gaps of a PC. This idea is ac- tual for one-, two- and three-dimensional structures, as plenty of applications can be thought of for each of these cases. One of the most popular methods used for band gap analysis is the plane wave expansion method. It is also well known that to re- duce the calculation time only a part of the Brillouin zone can be ta- ken into account. This part is a so-called irreducible Brillouin zone, which reconstructs the whole Brillouin zone when corresponding dispersion surface symmetry group operators are applied to it. For isotropic cases the symmetry of the Brillouin zone is deter- mined only by the PC lattice symmetry, for instance, the symmetry of a two-dimensional hexagonal PC Brillouin zone is also hexagonal. However, things get more complicated in the presence of mate- rial anisotropy. This research presents a fresh view on the aniso- tropic PC dispersion symmetry problem, and it will be shown below that the symmetry of an anisotropic PC dispersion surface is determined by anisotropic material orientation with respect to PC lattice vectors and/or periodicity planes. The possibility of controlling the positions and the widths of the band gaps of anisotropic PCs by means of external electric field is also discussed in the present paper. 2. Generalized plane wave method Several groups of methods traditionally used for PCs properties calculation can be marked out: methods, dealing with integral equations [13]; methods, representing fields in PCs as superposi- tions of localized functions [14]; and those, expanding field solu- tions into plane waves [15–19]. Localized function techniques are the most suitable methods for localized modes search, for example, it is convenient to use them when analysing defect containing structures. However, this meth- od often appears to be rather time- and effort-consuming. 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.059 * Corresponding author. Address: Saratov State University, Astrakhanskaya 83, Saratov 410026, Russian Federation. E-mail address: irina.khromova@unavarra.es (I.A. Khromova). Optics Communications 281 (2008) 5458–5466 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom