© 2003 OSA/BGPP 2003
MD20
A comparative study of numerical methods for the calculation of the birefringence of UV-illuminated fibers N. Belhadj1, K. Dossou2, X. Daxhelet3, S. LaRochelle1, S. Lacroix3, and M. Fontaine4 Canadian Institute for Photonic Innovations (CIPI) (1) Département de génie électrique et de génie informatique, Université Laval, Québec QC, Canada G1K 7P4
[email protected] (2) Département de mathématique et de statistique, Université Laval, Québec QC, Canada G1K 7P4 (3)Département de génie physique, École Polytechnique de Montréal, Montréal QC, Canada H3C 3A7
[email protected] (4) Département d'informatique, Université du Québec en Outaouais, C.P. 1250, succ. B, Hull QC, Canada J8X 3X7
Abstract: We compare the effective indices and the birefringence of a fiber with an asymmetric transverse profile, calculated by three numerical methods, one vectorial and two scalar formulations with polarization corrections. 2003 Optical Society of America
OCIS codes: (000.4430) Numerical approximation and analysis, (060.2340) Fiber optics components.
1. Introduction: Writing of photo-induced gratings is usually performed by side exposure of germanosilicate fibers to UV-laser radiation. Due to the strong absorption of the UV-light across the photosensitive core, this index change occurs in an asymmetric way [1] which modifies the birefringence of the fiber by adding a form birefringence. Several undesirable properties result from this birefringence including increased polarization dependent loss (PDL) and polarization mode dispersion (PMD). The photo-induced birefringence is particularly detrimental to the performance of chirped fiber Bragg gratings in which low birefringence can induce a significant amount of differential group delay (DGD) [2]. A reliable analysis of low birefringence waveguides requires a highly accurate effective index computation. Recently, two numerical methods were proposed to evaluate the magnitude of the form birefringence due to the UV side-exposure of optical fibers. In both studies, an exponential decay of the photo-induced index change was assumed across the fiber core [3,4]. The first method is based on a scalar formulation of the wave equation associated to the perturbation method. It is referred in the following as the scalar method (SM). The second method is a vectorial formulation of a finite element method (VFEM). In this paper, we introduce higher order polynomials in the vectorial finite element method to improve its accuracy. We also consider a third method based on a finite difference formulation with polarization corrections (SFDPC). The results obtained with all the numerical methods are compared for various asymmetries of the index profile. 2. The UV-induced refractive index profile: Before UV-illumination, the optical fiber used for calculation has a standard circular step-index profile with a core radius ρ=4.15 µm, a core index n1=1.4493, and a cladding index n2=1.444. At a free space wavelength λ=1530 nm, the effective index for this fiber is analytically calculated to be neff=1.4463861518422. After exposure, the refractive index change is assumed to have an exponential decay across the fiber core [3,4]. This index profile is characterized by two parameters, the peak refractive index change on the exposed side, noted δnp, and the asymmetry coefficient 2α. Τhe UV-induced index change is thus written δn 2 ( x, y) = δn 2p exp − 2α ( x + ρ 2 − y 2 ) for 0≤ r ≤ ρ
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and δn 2 ( x, y) = 0 elsewhere. Here, (x,y) are the Cartesians coordinates in the fiber transverse plane with its origin on the fiber axis and r = x 2 + y 2 . The UV exposure is therefore incident along the x-axis (from x
