A Finite Difference Method for Studying Thermal Deformation in 3D Double-Layered Micro-Structures Exposed to Ultrashort-Pulsed Lasers

Ultrashort-pulsed lasers have been attracting worldwide interest in science and engineering. Studying the thermal deformation induced by ultrashort-pulsed lasers is important for preventing thermal damage. This article presents a new numerical method for studying thermal deformation in 3D double-layered film micro-structures exposed to ultrashort-pulsed lasers. The method is demonstrated by investigating thermal deformations in a 3D double-layered thin film and a 3D double-layered sphere, respectively. INTRODUCTION Ultrashort-pulsed lasers have been attracting worldwide interest in science and engineering, because their pulse durations are of the order of sub-picoseconds to femtoseconds and because they possess exclusive capabilities of limiting the undesirable spread of the thermal process zone in the heated sample [1]. The success of using high-energy ultrashort-pulsed lasers in real applications relies on three factors [1]: (1) well characterized pulse width, intensity and experimental techniques; (2) reliable microscale heat transfer models; and (3) prevention of thermal damage. Up to date, a number of models that focus on heat transfer in the context of ultrashort-pulsed lasers have been developed. However, only a few mathematical models for studying thermal deformation induced by ultrashort-pulsed lasers have been developed [1-7]. Tzou and his colleagues [1] presented a onedimensional model in a double-layered thin film. The model was solved using a differential-difference approach. Chen and his colleagues [5] considered a two-dimensional axisymmetric cylindrical thin film and proposed an explicit finite difference method by adding an artificial viscosity term to eliminate numerical oscillations. Dai and his colleagues [2, 4] developed a new method for studying thermal deformation in 2D thin films exposed to ultrashort-pulsed lasers. The method was obtained based on the parabolic two-step heat transport equations and implicit finite difference schemes on a staggered grid. It accounts for the coupling effect between lattice temperature and strain rate, as well as for the hot-electron blast effect in momentum transfer. Numerical results show that there are no numerical oscillations in the solution. Unfortunately, when applied to a 3D thin film case, they found that the nonphysical oscillations appeared again in the normal stress in the thickness direction. Recently, Dai and his colleagues [8, 9] have improved their previous method by developing a fourth order compact finite difference scheme for solving the dynamic equations of motion. Results show that the non-physical oscillations disappear. In this article, we extend this method to study thermal deformation *Address correspondence to this author at the Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA; Tel: 1-318-257-3301; Fax: 1-318-257-2562; E-mail: [email protected] in 3D double-layered metal thin films and micro spheres exposed to ultrashort-pulsed lasers. Layered metal thin films are considered because they are widely used in engineering applications due to the fact that a single metal layer often cannot satisfy all mechanical, thermal and electronic requirements, while micro spheres are of interest related to micro resonators in optical applications, such as ultra-lowthreshold lasing, sensing, optoelectronic microdevices, cavity quantum electrodynamics and their potential in quantum information processing. This research provides a numerical method for studying thermal deformations induced by ultrashort-pulsed lasers when layered micro-structures are considered. MATHMATICAL MODEL Consider a 3D double-layered thin film in Cartesian coordinates, which is exposed to an ultrashort-pulsed laser, as shown in Fig. (1a). The governing equations for studying thermal deformation in the thin film can be expressed as follows: (1) Dynamic Equations of Motion [1, 2, 5, 9] , 2 ) ( ) ( ) ( ) ( ) ( ) (