Calculation of dispersion characteristics of microstructured optical fibers by the finite element method

Abstract

ABSTRACT Chromatic dispersion characteristics of dispersion compensating fibers with arbitrary shapes and index profiles are calculated by using a finite element method. A vector finite element method with mixed interpolation type triangular elements implemented. The approach uses optimized way of global sparse system assembling which is considerably reduces calculation time. Different types of mesh formation examined for better resolve of the dielectric interface boundaries. Features of the algorithm are demonstrated by examples of calculation for microstructured fibers. The correctness of assessments confirmed by comparison with those of known solutions and experimental data. Keywords: Finite element method, chromatic di spersion, dispersion compensating fibers, microstructured optical fibers 1. INTRODUCTION Continuous growth for transmission links bandwidth of telecommunication networks determines the interest for the study of chromatic dispersion characteristics of optical fibers. Control of chromatic dispersion in photonic crystal fibers (PCFs) is a very important problem for practical applications to optical communication systems and dispersion compensation and have been one of the most interes ting recent developments in fiber optics. PCFs are generally made from regular lattices of cylindrical inclusions, for example, air holes running parallel to the fiber length, in a dielectric matrix. The core usually consist of a defect of the lattice, which can be an inclusion of a different type or size or, in bulk core, a missing inclusion surrounded by multiple air holes. PCFs are divided into two different kinds of fibers. The first one, index-guiding PCF, guides light by total internal reflection between a solid core and a cladding region with multiple air-holes [1, 2] (Index-guiding PCFs, also called holey fibers (HFs) or microstructured optical fibers (MOFs)). On the other hand, the second one uses a perfectly periodic structure exhibiting a photonic band-gap (PBG) effect at the operating wavelength to guide light in a low index core-region [3, 4]. Index-guiding PCFs possess the especially attractive property of great controllability in chromatic dispersion by varying the hole diameter and hole-to-hole spacing. Interesting disp ersion properties, indicate that MOFs may be good candidates for dispersion management in optical communication systems. Very accurate and flexible numerical anal ysis and design techniques are required to deal with the increasing complexity of modern optical waveguides such as MOFs. The vector finite element method (FEM) is widely recognized as a very powerful technique for the analysis of optical waveguides with arbitrary shapes, index profiles, nonlinearities, and anisotropies. The most serious problem associated with vector finite elements is the appearance of spurious solutions. This attributed to improper modeling of the discontinuities in the field compone nts by node elements across dielectric boundaries. A numerous papers were devoted to eliminate the appearance of spurious solutions [5-7]. The penalty function method has been used to cure this problem [5, 6], but in this technique an arbitrary positive constant, called the penalty coefficient, is involved and the accuracy of solutions depends on its magnitude. Use of penalty term does not actually eliminate the spurious modes, rather it pushes the spurious modes out of the range of interest. Note that the penalty function method cannot provide a direct solution for the propagation constant and that an extra stage of iteration may be needed if the solution required fo r a particular wavelength. The use of vector finite element approach with mixed interpolation type triangular elements avoids spurious solutions and provides a direct solution for propagation constants