Modeling the longitudinal temperature evolution of a Chirped fiber Bragg grating submitted to temperature gradients

The chromatic dispersion of optical fiber is a limiting factor in the increase of optical communications transmission bitrates. Chirped Fiber Bragg Gratings submitted to temperature gradients can provide tuneable dispersion compensation, being the tuneability achieved by an appropriate control of the applied temperature gradient. In this work we propose a numerical model that describes heat transfer mechanisms along the fiber grating and its surroundings to obtain a temperature distribution along the fiber grating longitudinal axis. This model, which enables the subsequent modeling of the grating dispersion, demonstrated that the temperature distribution in a steady state is linear. The subsequent modeling of the fiber grating chromatic dispersion agreed with experimental results. Introduction Chromatic dispersion is a physical limitation which is responsible for the temporal broadening of optical binary pulses, and had become a limiting factor in the increase of optical communications transmission bit-rates. Although several dispersion compensation techniques have proved to be successful, the long-term stability remains to be evident. In addition to this, the use of reconfigurable optical path in modern networks constrains the communication systems to provide tuneable compensation. Among the available chromatic dispersion compensators, the Chirped Fiber Bragg Grating (CFBG) are valuable because they are compact, low-lossy, polarization insensitive and do not generate nonlinear perturbations. The CFBG are made by exposing laterally the core of a single mode optical fiber to a high intensity ultraviolet light pattern. This exposure will produce permanent changes in the optical fiber refractive index, imposing a periodic (for FBG) or aperiodic (for CFBG) variation of the light phase that propagates through it [1]. At the Bragg condition, resonance is observed for the counter-propagated mode, resulting in high intensity reflection spectrum. Since the CFBG dispersion is related to its bandwidth, a perturbation like heating that produces an alteration at the grating optical period (perturbation period and/or refractive index) can be deliberately used to tune its dispersion. In one approach, a CFBG with a longitudinal thermal gradient produces a controllable dispersion enabling a tunable compensation scheme [2, 3]. The present work presents a way to evaluate the temperature distribution of a CFBG submitted to a temperature gradient over its longitudinal length. This estimation is very important because it allow us to determine the dispersion provided by the CFBG for an applied temperature gradient. To fulfill this task, we propose a model that takes into account the heat exchange mechanisms inside the CFBG and its surroundings (conduction, convection and radiation) when a steady state is attained. Materials Science Forum Vol. 553 (2007) pp 106-111 online at http://www.scientific.net © (2007) Trans Tech Publications, Switzerland Online available since 2007/Aug/15 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-17/04/08,14:41:03) The CFBG dispersion is computed subsequently based on the previous heat exchange study and the model is verified with experimental results. For a matter of simplicity, the CFBG will be considered as a cylinder made of Silica with its extremity in thermal contact to heat reservoirs at fixed temperature and its lateral area in thermal contact with its surrounding air which is also at a fixed temperature. Modeling the CFBG heat transfer The previously referred situation was considered to be equivalent to a glass bar whose extremities are at fixed temperatures (T1 and T2) but whose length is in contact with the environment at a given temperature, Te. The bar is considered to be in thermal contact with a heat reservoir at each end at temperatures T1 and T2, respectively. Heat flows through the bar due to conduction mechanisms. The local temperature at each bar position depends on this flow but also on the convection and radiation exchange with the environment. The proposed problem is to determine how the temperature will vary along the bar at any instant of time, T(x,t).The heat diffusion equation that describes such behavior can be expressed by the heat diffusion equation with Newton cooling in the following way [4]: ∂T(x, t)/∂t = D ∂T(x, t)/∂x h [T(x, t) Te]. (1) where D is the heat diffusion constant given by Eq. 2: D = Kbar/(cbar.ρbar). (2) being Kbar the thermal conductivity of the bar, cbar the bar specific heat and ρbar the bar density. h is a constant which comprehends the heat transfer by convection and radiation. It is quite difficult to attain an analytical expression for the convection heat transfer coefficient, since it relies on the system geometry. Usually, it is derived empirically, resorting to dimensional analysis and experimental data. In our situation, the heat power transferred from the fibre to its surrounding air by free convection is given by Eq. 3 [5]: ∂q/∂t = [T(x, t) – Te].π Lbar Kair Nu. (3) where Lbar , Kair and Nu are the fiber length, the air thermal conductivity and the Nusselt number respectively. The crucial point in applying the formula above is the correct formulation of the Nusselt number. This dimensionless quantity includes effects from the flow around the fibre and heat transfer. Bearing in mind that the power transferred by convection is also given by Eq. 4, the heat transfer coefficient is provided by Eq. 5: ∂q/∂t = m.cbar.∂T(x,t)/∂t. (4) hcon = (π.Lbar.Kair.Nu)/(cbar.m). (5) The derivation of the radiation heat transfer coefficient is based on an analogous procedure. We will first consider radiated heat power given by the Stefan-Boltzmann equation for a grey body. Prad = σ.A.e.[T (x, t) – Te ]. (6) Being σ, A and e the Stefan-Boltzmann constant, the irradiative area (corresponding to the lateral area of a 125 μm diameter and 3 cm long cylinder), and the emissivity (taken to be 0.5), respectively. Materials Science Forum Vol. 553 107