<title>Modeling of nanosecond-laser ablation: calculations based on a nonstationary averaging technique (spatial moments)</title>

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Semi-analytical approach to a quantitative analysis of thermal ns laser ablation is presented. It permits one to take into account: (1) Arbitrary temperature dependences of material parameters, such as the specific heat, thermal conductivity, absorptivity, absorption coefficient, etc. (2) Arbitrary temporal profiles of the laser pulse. (3) Strong (Arrhenius- type) dependence of the ablation velocity on the temperature of the ablation front, which leads to a non-steady movement of the ablation boundary during the (single) pulse. (4) Screening of the incoming radiation by the ablated products. (5) Influence of the ablation (vaporization) enthalpy on the heating process. (6) Influence of melting and/or other phase transformations. The nonlinear heat conduction equation is reduced to three ordinary differential equations which describe the evolution of the surface temperature, spatial width of the enthalpy distribution, and the ablated depth. Due to its speed and flexibility, the method provides powerful tool for the fast analysis of the experimental data. The influence of different factors onto ablation curves (ablated depth h vs. fluence (phi) ) is studied. Analytical formulas for (phi) <SUB>th</SUB> and h((phi) ) dependences are derived and discussed. The ablation curves reveal three regions of fluence: Arrhenius region, linear region, and screening region. Threshold fluence (phi) <SUB>th</SUB> and Arrhenius tails at (phi) less than (phi) <SUB>th</SUB>, are affected heavily by the temperature dependences in material parameters, surface evaporation rate, and pulse duration and shape. In contrast, the slope of the ablation curves at (phi) greater than (phi) <SUB>th</SUB>, is determined almost exclusively by the latent heat of vaporization, high temperature dependence of absorptivity, and, in the case of screening, by the absorption coefficient of the plume (alpha) <SUB>g</SUB>. In the screening region ablated depth increases logarithmically with fluence and its qualitative behavior is weakly affected by the temperature dependence in (alpha) <SUB>g</SUB> (T). Small vaporization enthalpy results in a sub-linear h((phi) ) dependence, which, nevertheless, remains faster than logarithmic. With weakly absorbing materials ablation may proceed in two significantly different regimes -- without or with ablation of the heated subsurface layer. The latter occurs at higher fluences and reveals significantly higher ablation temperatures, but is weakly reflected on the ablation curves. Calculations are performed in order to study the: (1) Influence of the duration and temporal profile of the laser pulse on the threshold fluence, (phi) <SUB>th</SUB>. This is particularly important for strong absorbers were the heat conduction determines the temperature distribution. (2) Influence of the temperature dependences in material parameters on the ablation curves (ablated depth versus laser fluence) for regimes (phi) approximately equals (phi) <SUB>th</SUB> and (phi) very much greater than (phi) <SUB>th</SUB>. (3) Consequences of shielding of the incoming radiation at high fluences. (4) Differences in ablation curves for materials with big and small ablation enthalpy (e.g., metals and polymers which ablate differences in ablation curves for materials with big and small ablation enthalpy (e.g., metals and polymers which ablate thermally). Nanosecond laser ablation has been studied for a large variety of different materials and laser wavelengths. As an illustrative example, the method is applied to the quantitative anlaysis of the single pulse ablation of polyimide Kapton <SUP>TM</SUP> H.