Analytical Solutions of a Growth Model for a Melt Region Induced by a Focused Laser Beam

We consider processes in which a focused laser beam is used to induce the melting of silicium. The first goal of this paper is to propose a simple three-dimensional (3D) model of this melting process. Our model is partly based on an energy balance equation. This model leads to a nontrivial ODE describing the evolution in time of the dimension of the melt region. The second goal of this paper is to obtain approximate analytical solutions of this ODE. After using basic solution methods, we propose an original geometrical method to derive asymptotic solutions for time → ∞. These solutions turn out to be the most useful for the description of this process.