In laser-induced damage experiments, it is well known that the threshold fluence, Fth depends on the number N of pulses sent to a single spot on the target material. This phenomenon is called material incubation. In most cases, the threshold fluence decreases with N starting from the single-shot ablation threshold F1 and remains constant if the pulse number is larger than a critical saturation value. If the fluence is below the multiple pulse threshold F¥, ablation or damage does not occur for any number of pulses.
The first attempt to model this incubation behavior employed a simple empirical ansatz of the form Fth,N = F1 × N S-1 . Later, more elaborate models were published. The generic incubation law based on our model is of the form:
where a0 is the initial absorption coefficient of the sample, Da is a measure for the change of absorption with every laser pulse, b is a material parameter describing the efficiency with which a pulse of fluence F changes the absorption coeffiecient, while g is another material parameter, describing an efficiency of changing a critical ablation energy with every laser pulse. For details of the model, see reference /
You can download a MatLab script for fitting the equation above to your experimental data here. Unzip all files into a single directory and run the script IncubationCurveFit.m. Instructions on how the data have to be supplied are contained in the comment section of the script. Please direct questions to firstname.lastname@example.org
Single-shot ablation thresholds are often determined by measuring the crater diameter as a function of pulse fluence F, as shown in reference . Linear extrapolation of a plot of the squared diameter D² versus ln(F) yields the threshold fluence Fth at D²=0. The validity of this approach can be shown easily assuming a Gaussian excitation beam of waist w and postulating that ablation occurs within a crater of radius rc for which the local incident fluence exceeds the threshold value Fth. The advantage of this technique is that the measurements can be performed at fluences well above threshold for which the detected fingerprint signals are clearly detectable by far-field microscopy. Also, the beam radius w does not have to be known; it is obtained as slope of the regression line. As shown in reference , this method can also be used for multi-pulse damage thresholds, the linear interpolation still yields the correct threshold fluence Fth,N, however, the slope might not correspond to the correct lateral beam dimensions.
You can download a MatLab script for fitting this linear dependence to your experimental data here. Unzip all files into a single directory and run the script DsqFit.m. Instructions on how the data have to be supplied are contained in the comment section of the script. Please direct questions to email@example.com
 Y. Jee, M. F. Becker, and R. M. Walser, Journ. Opt. Soc. Am. B 5, (1988) p. 648
 J. M. Liu, Opt. Lett. 7, (1982) p. 196
 Z. Sun, M. Lenzner, W. Rudolph, Journ. Appl. Phys. 117 (2015) 073102
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