Numerical modeling of laser ablation of materials

In this paper, we report a numerical simulation of laser ablation of a material by ultrashort laser pulses. The thermal mechanism of laser ablation is described in terms of a one-dimensional nonstationary heat conduction equation in a coordinate system associated with a moving evaporation front. The laser action is taken into account through the functions of the source in the thermal conductivity equation that determine the coordinate and time dependence of the laser source. For a given dose of irradiation of the sample, the profiles of the sample temperature at different times, the dynamics of the displacement of the sample boundary due to evaporation, the velocity of this boundary, and the temperature of the sample at the moving boundary are obtained. The dependence of the maximum temperature on the sample surface and the thickness of the ablation layer on the radiation dose of the incident laser pulse is obtained. Numerical calculations were performed using the finite difference method. The obtained results agree with the results of other works obtained by their authors.


Introduction
In recent years, pulsed laser ablation [1]- [3] (any process of laser-stimulated removal of matter, including the emission of electrons) of various materials has attracted more and more interest from the point of view of fundamental study of processes in matter under extreme conditions of ultrafast energy supply. This implies constructing a new physical theory describing strongly nonlinear effects.
For a detailed analysis of the processes in the experiment, it is required to measure various characteristics of the ablation processes with pico-and femtosecond time resolution, which in itself is a rather difficult task. Therefore, the problem of mathematical modeling of physical phenomena in this area becomes extremely urgent.
To describe the dynamics of fast processes in a substance, the method of molecular dynamics (MD) can be used [4]. MD is quite effective for microscopic analysis of the mechanisms of melting and evaporation under overheating conditions both in the bulk of the target [5] and for a system with a free surface [6]. The emergence and propagation of pressure waves generated by laser radiation [7], [8], as well as the dynamics of laser ablation [9], is well modeled using MD.
In this paper, we consider continuous methods (various modifications of the heat equation) for modeling the effect of laser radiation on matter.
The evaporation process is mathematically described within the framework of the boundary value problem of thermal conductivity for a condensed medium in a coordinate system associated with a moving solid-vapor interface or a melt-vapor interface on which evaporation occurs. If we do not take into account the lateral removal of the laser radiation energy due to thermal conductivity, which is valid under the strict condition 0 ≫ √ , where is the duration of the laser beam exposure to the material, is the thermal conductivity, 0 is the radius of the overheating spot, then the problem of the motion of the evaporation boundary can be considered within the framework of the one-dimensional model [3]. In Ref. [10], the primary results of numerical simulation of ablation of materials were published. In this paper, the required work is presented in a more extended form.

Setting of the problem
Numerical modeling of laser ablation of materials was carried out based on the heat conduction equation written in a moving coordinate system associated with the evaporation front, with initial and boundary conditions [2]: where ( ), ( ), ( ) are the specific heat, thermal conductivity and density of the material at the temperature ( , ), ℎ( ), respectively is the depth of the crater on the surface of the sample at time , is the maximum distance, ( ) is the velocity of the boundary displacement due to evaporation, is the specific heat of sublimation. The source function ( , ) has the form [2] ( , ) = 1 ( ) 2 ( ), Here 0 is the laser intensity, ( ) is the reflection coefficient of the laser from the sample surface, , are the absorption coefficients of the laser pulse in the sample material and in the vapor, respectively. The irradiation dose Φ, the intensity of the source 0 and the temporal form of the source ( ) are related by the relation: Here the source function has a factorized form, as in the work [11], when the material is affected by a pulsed beam of charged particles rather than by a laser pulse. In general, the heat capacity, thermal conductivity, and density of the material depend on temperature. In a particular case, the dependence of some parameters of the sample material can be neglected. In this work, the temperature dependence of the density of the sample material and the laser reflection coefficient is neglected.

Discussion of numerical results
In Ref. [2], problem (1)-(4) was solved by the method of moments for a polyimide material. In our work, this problem was solved using the finite difference explicit scheme [12]. The temporal shape of the source ( ), the temperature dependence of the boundary motion velocity due to evaporation ( ), the specific heat ( ) and the thermal conductivity ( ) are taken for the polyimide material similar as in Ref.  Figure 1 shows plots of these dependencies. Figure 2 shows the temperature profiles of a polyimide sample at different times: = ⋅ 5 ; = 1.10, the dynamics of the sample boundary motion due to evaporation, the velocity of this boundary motion, and the sample temperature at the moving boundary = ℎ( ), when exposed to energy fluence Φ = 10 3 / 2 with parameters = 0.93, = 4.25 ⋅ 10 7 m −1 ( = 0.45 ), = 5 ⋅ 10 5 / , = 1420 / 3 . Figure 3 shows the dependencies of the maximal temperature at the sample surface

Conclusion
For a given dose of the sample irradiation, the profiles of the sample temperature at different times, the dynamics of the displacement of the sample boundary due to evaporation, the velocity of this boundary and the temperature of the sample at the moving boundary were obtained. The dependencies of the temperature maximum on the sample surface and the thickness of the ablation layer on the radiation dose of the incident laser pulse are determined.