Abstract
Femtosecond lasers have proven to be a highly effective tool for surface micro- and nanomachining, offering high precision through minimal heat-affected zones compared to longer-pulse lasers. Moreover, recent advancements in burst technology show great promise for enhancing both efficiency and precision. This study investigates the dependence of fluence and number of sub-pulses on the ablation depth and diameter of 316 L Stainless Steel with a MHz intraburst repetition rate. An extended two-temperature model, incorporating the heat diffusion between sub-pulses, is introduced to predict ablation depth and diameter for varying number of sub-pulses. Simulated results are compared with experimental data, showing a good qualitative agreement between them. The findings reveal that burst mode can significantly improve ablation rates by mitigating efficiency loses at high fluence and provide insights into how to achieve the optimal parameters for its application.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-37443-9.
Keywords: Ultrafast Laser-Metal interaction, Femtosecond laser, Burst mode, Stainless steel, Two-Temperature model.
Subject terms: Engineering, Optics and photonics, Physics
Introduction
The use of ultrafast lasers has significantly increased in recent years for micro- and nanomachining of materials, due to their versatility to work with a wide range of materials, as well as their high precision, achieved through the generation of a small heat-affected zone (HAZ) relative to longer-pulsed lasers1. However, from an industrial perspective, a high throughput is one of the most important factors, an area where ultrafast lasers still offer potential for improvement. Currently, industrial lasers are achieving peak powers around 1 kW, along with repetition rates ranging from hundreds of kHz to MHz. While increasing the repetition rate does allow for a linear increase in throughput, raising the power leads to a significant decline in ablation efficiency2. In recent times, burst mode has become a popular choice in microfabrication3. This burst mode enables controlling the energy deposition of each pulse by splitting it into several sub-pulses with an intraburst frequency 
in the MHz to GHz range, see Fig. 1.
Fig. 1.
Schematic representation of two pulses separated in time by a repetition frequency f. Within each pulse, different burst configurations are illustrated in different colours depending on the number of sub-pulses (n). The temporal separation between sub-pulses is denoted as 
.
With this new strategy, the severe efficiency drop observed at higher laser powers is mitigated thanks to the redistribution of the total energy among the different sub-pulses, allowing the efficiency to remain stable while increasing both power and number of sub-pulses.
As a result, the throughput scales linearly with the power4. Moreover, this approach not only maintains the efficiency, but even enhances it when the optimal parameters are applied for certain materials, such as copper and silicon5–7. For stainless steel, a substantial number of experimental measurements have been conducted2,8,9. However, there is a lack of studies incorporating theoretical modelling and simulations of the process, despite their importance for understanding the underlying interaction mechanisms and to predict optimal processing conditions.
In the ablation process, due to the elevated temperatures reached by the material at thermodynamic equilibrium, part of the ejected material transitions into a plasma state. For frequencies in the range of hundreds of kHz, this plasma does not affect the subsequent pulse as it has been completely dissipated. However, when using burst mode, it is important to consider the plasma effect, as part of the laser radiation is shielded by the plasma and does not reach the material underneath10. Several authors have proposed models to consider how the plasma plume affects material absorption depending on the fluence and the time of the subsequent sub-pulse11–13. In addition, experimental studies varying the intraburst time have led to the identification of three regimes14,15. In the first regime, for intraburst time shorter than 
, the energy between sub-pulses arrives during electron-phonon interaction, meaning the sub-pulses can be considered as a single pulse, with the energy being the sum of the individual sub-pulses. In this case, dividing the energy into sub-pulses has an effect very similar to that of delivering a single pulse with the same total energy16. Therefore, in this regime, a drop in efficiency will also be observed when high energies are used. In the second regime, from picoseconds up to nanoseconds, the second sub-pulse arrives while the plasma plume is still present, causing efficiency to decrease as the plasma forms, reaching a minimum when the plume is at its peak, and then increasing as the plasma dissipates. Finally, in the third regime, for several nanoseconds, the sub-pulse reaches the material after the plasma plume has dissipated, allowing each sub-pulse to be treated individually. This shows that by selecting an intraburst time within the third regime, it is possible to transition from the logarithmic regime observed when fluence is increased to a linear regime as the one observed when increasing the number of pulses17. In this work, we present an extended two-temperature model that accounts for burst mode operation on the third regime. This model dynamically transitions from the two-temperature model to the heat diffusion equation once thermodynamic equilibrium is reached between the electron and lattice subsystems, significantly reducing simulation time. The simulations have been validated experimentally, and based on the combined insights, two analytical models were developed: one based on the well-known logarithmic relationship between fluence and ablation depth, and a novel model describing the transition between a linear and logarithmic regimes, yielding even more accurate results. The combination of simulation, experimental and modelling, enhances the fundamental understanding of burst-mode ablation by linking fluence and sub-pulse number for the optimization of the ablation efficiency. These findings provide a foundation that can be adapted to optimize processing rates in future industrial applications.
Numerical modelling
In the present work, the simulations are based on our own previously developed Two-Temperature Model, detailed in earlier studies for both aluminum17,18 and stainless steel19. All the parameters used in this work are identical to these employed in19, where the temperature dependencies of the material properties were adopted from the study by Metzner20. Based on this established framework, we extend the model to incorporate burst mode effects. Given that the laser radiation is going to be divided into multiple sub-pulses, it is crucial to model this accurately, so we have considered the laser pulse absorption Q, which can be further divided into the following components:
 |
1 |
Where 
represents the spatial term of the laser:
 |
2 |
Here, R refers to the reflectance of the metal and 
is the effective penetration depth. 
, given by the equation 
, represents the laser fluence F of a burst divided by the number of sub-pulses n. In this case, we have considered all the pulses uniform. 
is the radius of the laser beam at the focal plane and 
considers the defocusing effect due to the ablation21. Finally, in Eq. (
, the term 
refers to the material surface. It should be noted that for the first sub-pulse, it will be 0, but for consecutive sub-pulses, it will evolve according to the ablation profile. This profile evolves as the ion lattice temperature reaches 
times the critical temperature, at which the spallation and phase explosion take place22. In the case of stainless steel, 
23.
It should be noted that Eq. (
does not account for any plasma shielding, as this study focuses on improving ablation efficiency in the third regime. For stainless steel, this regime starts from the first nanoseconds onward14. In our simulation, the interpulse delay is set to 24 ns, a value chosen to match the actual pulse separation of our experimental laser system. This choice ensures that the model accurately reflects the operating conditions where plasma shielding is negligible.
The temporal part 
models the pulse as a sum of gaussian sub-pulses:
 |
3 |
Here, 
denotes the full width at half maximum of the pulse duration, the sub-pulses are separated by 
, and each sub-pulse j is centred at 
.
The computations were performed using a custom Python code, which has been optimized with the numba library24. Furthermore, in simulations involving multiple pulses with frequencies on the order of kHz, it is assumed that the not ablated material returns to ambient temperature before the arrival of the next pulse25. However, in the case of the burst mode, where the time between sub-pulses ranges from picoseconds to nanoseconds, this assumption is no longer valid. Given the current timestep, calculating the temperature evolution up to the arrival of the subsequent sub-pulse becomes computationally prohibitive. Therefore, the two-temperature model is transitioned to a single-temperature heat diffusion equation once both subsystems reach thermodynamic equilibrium, approximately 15 picoseconds after the laser-material interaction26:
 |
4 |
Here 
and 
and 
is the temperature of the material.
The mesh limits are set to 
and 
For the two-temperature model, a fine mesh is used with spatial steps 
, 
, and a time step 
. When transitioning to the heat diffusion model, the mesh is coarsened by increasing 
by a factor of two and 
by a factor of three through cubic interpolation. This mesh coarsening enables the time step to be extended to 
. This interpolation can be implemented due to the geometry of the problem and the fact that the mesh resolution is less critical compared to the mesh required for the TTM ablation dynamics. As a result, the heat transfer problem between pulses becomes computationally manageable. This coarser mesh is utilized until the simulation reaches a time 
, at which point the finer mesh is reconstructed, and the TTM is re-applied for subsequent calculations. The mesh configuration during the different stages of the simulation is illustrated in Fig. 2.
Fig. 2.
Schematic representation of the mesh used, considering the two-temperature model or heat diffusion, as well as the finer or coarser meshing depending on time.
Experimental work
To validate the proposed model, experimental measurements were performed under single-pulse conditions, producing individual ablation craters while varying the fluence and the number of sub-pulses within each pulse. The various conditions employed in the experiments are summarized in Table 1, with each value across columns is used in combination with every value in the other columns. An example of the generated ablation craters is shown in Fig. 3.
Table 1.
Laser conditions.
| Parameter | Fluence (J/cm2) | Sub-Pulses |
|---|---|---|
| Values | 0.22, 0.32, 0.43, 0.65, 0.75, 1.29, 1.62, 2.48,3.23, 4.96, 6.36 | 1,2,3,4,5 |
Fig. 3.
Optical microscopy images of single-pulse ablation craters for a fluence of 
and three sub-pulses: (a) 50x magnification and (b) 150x magnification.
Since we aim to examine exclusively the dependency on fluence and the number of sub-pulses within a singular pulse, it is imperative to minimize the effect of surface irregularities on the area to be ablated. This also ensures that measurements obtained at low fluence levels are more accurate and representative of the laser ablation. To effectively minimize surface irregularities, we utilized 316 L stainless steel that had been previously polished with SiC paper, achieving a surface roughness with an 
As laser source, we employed a diode-pumped ultrafast laser system (Amplitude Satsuma HP) characterized by a pulse duration FWHM 
, a spot size 
, an infrared wavelength 
, a maximum average power 
at a repetition rate of 
Also, it incorporates MHz burst mode, with an intraburst time of 
. For stainless steel, other works have proved that this delay is sufficient for the complete removal of the plasma plume2,14.
A Lasea LS-LAB micromachining station, equipped with various modules to adjust the power, frequency, spot size, and number of sub-pulses, was used to achieve the conditions presented in Table 1. Before and after laser processing, the polished steel sample was cleaned in an ultrasonic bath with acetone for two minutes, followed by an ultrasonic bath in ethanol, also for two minutes.
Furthermore, due to the limited resolution of the available laser camera, changes in the focus position smaller than 100 
cannot be visually distinguished. Therefore, the initial focus was adjusted using the 
Z-resolution provided by the camera. Subsequently, the experiments were repeated on clean areas of the material, performing Z-displacements of 
(both upward and downward) with respect to the focal plane. This offset is smaller than the Rayleigh length of our system, which is approximately 
, thus ensuring that the beam remained within the effective focal region. Furthermore, for each Z position and laser condition, 25 replicas were produced to ensure repeatability, minimize possible defects, and enable subsequent statistical analysis. The results presented correspond to the measurements taken at the Z position that yielded the greatest depth and the smallest diameter, ensuring that the sample was as close as possible to the focal point.
To characterize the ablation craters, consisting in the measurements of diameter and depth, a Sensofar S Neox optical profilometer was employed using the CSI (Coherence Scanning Interferometry) technique with a Nikon DI 50x objective, providing a height resolution of 
. To ensure reliable measurements, from the 25 replicates of each condition, the 5 ablation craters with the fewest defects were selected for diameter and depth analysis. This selection enables a statistical study, enhancing measurement robustness and providing a confidence interval.
Results and discussion
Simulation
The simulations conducted are based on the model presented in Sect. 2. To maintain consistency and ensure the comparability of the results, the fluences specified in Table 1 have been applied. Additionally, for certain conditions, higher number of sub-pulses beyond these listed in the table were also simulated to further investigate their effects on ablation.
Figure 4 illustrates the temperature evolution for a total fluence of 
across different number of sub-pulses. Focusing on Fig. 4. (a), the highest temperature is observed for a single pulse, which is consistent with the fact that the fluence of that sub-pulse, 
, is the largest. However, to achieve ablation, it is sufficient to exceed the ablation temperature 
. In this scenario, a significant portion of the energy is lost as it is not used for ablation but instead heats the already ablated material. As the number of pulses increases, we observe a decreased value of the temperature, as the fluence per sub-pulse is divided by the total number of sub-pulses.
Fig. 4.
Evolution of the maximum temperature as a function of time for different number of sub-pulses, each corresponding to a total fluence of 
. (a) Evolution during the first sub-pulse. In the legend, 
denotes the electron temperature and 
the lattice (phonon) temperature, (b) Evolution spanning the total burst time. The inset illustrates the temperature diffusion between the first two sub-pulses.
.
If we now examine Fig. 4. (b), we can observe that after the initial 
and the removal of the material, the maximum temperature decays exponentially until the arrival of next pulse. Additionally, it can also be seen that the higher the energy of the preceding sub-pulse, the higher the material temperature will be just before the arrival of the next sub-pulse. Therefore, if the fluence is sufficiently high for ablation, it appears more efficient to divide the energy into sub-pulses, as this reduces the potential for heat accumulation and allows for greater ablation, since the energy from subsequent sub-pulses is absorbed by the newly surface of the material.
Figure 5 presents the variation of depth with the number of sub-pulses and fluence sub-pulse. To gain a better understanding of these relationships, in addition to the experimental conditions listed in Table 1, simulations were also carried out for up to 10 and 15 sub-pulses at fluences of 
and 
. Figure 5. (a) shows that, for a low fluence, increasing the number of sub-pulses is not effective, as the fluence per sub-pulse becomes too small, approaching or even falling below the threshold fluence, resulting in either no ablation or less ablation compared to a single sub-pulse. However, as the total fluence increases, using a higher number of sub-pulses leads to an increase in depth. Nevertheless, it is important to monitor the sub-pulse fluence 
, as this approach provides a clear understanding of the underlying process and helps reducing the number of dependencies involved. In Fig. 5. (b), we can observe that the curves for the different fluences follow the same trend. When fluence per sub-pulse is below the threshold fluence, no ablation occurs. However, as we increase 
, the depth increases until it reaches a maximum, which occurs approximately at:
 |
5 |
Fig. 5.
(a) Simulated depth as a function of different number of sub-pulses for different fluences, the secondary axis represents the fluence used to plot the grey dashed line, which denotes the optimal fluence, calculated according to Eqs. 7–8. (b) Simulated depth as a function of the fluence per sub-pulse for different fluences, threshold and optimal sub-pulse fluences are marked with red and grey lines.
So, the optimum number of sub-pulses based on the total fluence is given by:
 |
6 |
This value is related to the limit where the linear approximation of depth holds before transitioning to a logarithmic regime. In our case, this value is approximately 
, which is very similar to the value that also has been.
experimentally determined in other studies, such as Neuenschwander et al.2 that obtains 
for stainless steel or Hodgson et al.8, who report a value of 
for steel with a single shot. In this latter study, it is noteworthy that the intraburst frequency is lower than 
, which leads to a decrease in ablation efficiency as the number of pulses increases, likely due to plasma shielding. At higher fluences than this optimal value, the depth begins to decrease again as we increase the sub-pulse fluence, as we enter the logarithmic regime of the approximation, approaching a saturation value. It is also worth noting that the greater the available total fluence, the greater the depth will be, as increasing the number of sub-pulses at a fixed 
close to the optimal value causes the depth to increase almost linearly. Moreover, it is easier to achieve a sub-pulse fluence close to the desired value of approximately 
, when both the total fluence and the number of sub-pulses are high. In such cases, dividing a high total fluence by a high number of sub-pulses results in smaller incremental changes, whereas with a low total fluence the increments become larger.
Experimental
The experimental characterization results for the ablation depth and diameter under the conditions proposed in Table 1 are presented in Fig. 6. where the error bars correspond to the statistical analysis of the measurements. Focusing on the depth results, Fig. 6. (a) shows that, for a single shot, the depth evolution follows a nearly logarithmic trend27:
 |
7 |
Fig. 6.
Experimental depth (a), and diameter (b), as a function of sub-pulse fluence for different number of sub-pulses.
However, as the number of sub-pulses increases, the depth grows almost linearly with the pulse count. For instance, if we consider a fluence per sub-pulse around 
, the values are shown in the Table 2. Thus, we observe that the depth increases in almost linear manner as the number of sub-pulses increases while keeping the fluence per sub-pulse constant. It is important to note that maintaining 
constant requires increasing the total fluence proportionally to the number of sub-pulses.
Table 2.
Depths for different sub-pulses at a sub-pulse fluence of around 
.
| Sub-Pulses | Fsub (J/cm2) | Depth (nm) |
|---|---|---|
| 1 | 1.29 | 34  9 |
| 2 | 1.24 | 54  6 |
| 3 | 1.08 | 82  3 |
| 4 | 1.24 | 111  8 |
| 5 | 1.27 | 157  22 |
However, this is not usually a limitation, as Fig. 5. (b) shows that the optimal fluence is around 
, while nowadays, lasers can operate at fluences on the order of 
. On the other hand, for the study of the diameter, for just one shot we can fit the squared diameter using28:
 |
8 |
If we use this equation, we could say that keeping the fluence constant results in the same diameter value. However, we need to consider the sub-pulses, and as we are in a regime where they are sufficiently separated, they should be considered independent. Therefore, it is necessary to use the fluence per sub-pulse instead of the total fluence:
 |
9 |
By analysing Fig. 6. (b), when the fluence per sub-pulse is considered instead of the total fluence, the diameter exhibits a high degree of similarity for the same 
. As the fluence increases, it appears that with a higher number of sub-pulses, the diameters are slightly larger compared to a lower number of sub-pulses. This could be due to the experimental set up. All the processes started by fixing the z-position and carefully focusing them for the mono-burst, without varying the z during the process, as the sample was assumed to be completely flat. However, small changes in the height can lead to these changes in diameter, with the smallest diameter corresponding to a single shot. On the other hand, it could be attributed to the incubation effect29:
 |
10 |
Where N is the number of pulses. In our previous study19, we obtained an incubation factor of 
. However, using this value and introducing the sub-pulses as N in Eq.(
, the threshold fluence value decreases below 
when more than one sub-pulse is employed. However, ablation does not occur with a 
. Therefore, it seems that incubation with the number of sub-pulses is not the same as that observed with the number of pulses. Gaudiuso et al.30 propose the existence of an incubation dependent on both the number of pulses 
and the number of sub-pulses 
. As the number of sub-pulses increases, the factor 
decays, leading to a decrease in the threshold fluence. However, in this study, the time between sub-pulses is 
. Therefore, a more detailed study in MHz burst regime is necessary to assess whether the number of sub-pulses plays a role in the incubation effect.
Comparison
For the model validation the simulations results have been compared with the experimental value. Regarding the depth, as shown in Fig. 7. (a), there is a good agreement between the model results and the experimental data across the entire range of studied conditions. Therefore, the proposed model, which accounts for heat diffusion between pulses, is well validated based on the obtained results. With burst mode, we observe that increasing the number of sub-pulses shifts this logarithmic regime towards higher fluences, effectively avoiding the saturation of the logarithmic regime, where the depth barely increases despite increasing the fluence. For the diameters, observing Fig. 7. (b), we can see that at low fluences, the results show very good agreement. However, as the fluence increases, the model seems to overestimate the actual value.
Fig. 7.
Comparison of simulation (diamonds) and experimental (dots) for ablation as a function of total fluence. (a) Depth. (b) Diameter.
This may be due to the spot size used in our model, with a value of 
. However, in our study for aluminium with a single pulse18, the experimental spot size obtained using Liu’s method was around 
. Furthermore, adjusting the single-pulse values in our study conducted for steel19, the spot size was 
. Therefore, the value of 10 microns used in the simulation might be slightly overestimating the actual value of the laser spot.
Analytical models
The proposed numerical modelling provides detailed insights into the laser-metal interaction, and the experimental results validate its prediction. However, these calculations are computationally demanding and time-consuming, making them less practical for quick estimations. In contrast, an analytical model can offer a more intuitive understanding of the key parameters, providing a faster and more accessible approach to describe the laser ablation. Therefore, we have developed two analytical models to provide a more accessible framework for predicting the depth of the ablation based on the fluence and the number of sub-pulses. The first model is a simpler approach which provides a quick estimation, though less accurate compared to experimental data. The second model, more complex in nature, yields result that align more closely with the experimental measurements, offering a higher degree of predictability. These models facilitate the selection of optimal processing conditions.
Logarithmic model
As seen in Fig. 6. (a), we observe a nearly linear increase in depth as the number of sub-pulses increases. Therefore, we can modify the Eq. (
to account for this linearity with respect to the number of sub-pulses:
 |
11 |
Here, we can see that we are improving the ablation depth, though not in a purely manner when considering the fluence and sub-pulses, as the term n also appears in the denominator of the logarithm. However, if we consider the fluence per sub-pulse, where the term n in the logarithm is embedded, we can indeed consider this linear relationship of depth with the number of sub-pulses. It is worth noting that when the value inside the logarithm of 
equals 
, the logarithm becomes zero, thus preventing ablation. Besides, as the value of 
can be less than 
, to ensure physical consistency, we set the depth to zero rather than allowing it to take on a negative value. Furthermore, we can determine the number of sub-pulses at which the ablation is maximized for a given fluence by differentiation of the function:
 |
12 |
By setting this value to zero to find the maximum, we obtain 
, and by regrouping, we arrive at:
 |
13 |
From this equation, we can see that sub-pulse fluence value that maximized the ablation depth is constant with respect to the threshold fluence. This result was also predicted in the simulations, as shown in Fig. 5. (b). However, in the simulations, the optimal value was around 
, and using this model, the value is 
, closer to the value reported by2. Moreover, in the study by Le et al.31, where a constant incident fluence of 
was used and the number of sub-pulses was varied between 1 and 400, the authors found that the optimal number of sub-pulses was 50, which also corresponds to a sub-pulse fluence of 
.
This result suggests that our model may be extrapolated to conditions that are more relevant for industrial applications, involving higher power levels, where a larger number of sub-pulses can be employed.
Linear & logarithmic model
Returning to Eq.(
, if we expand the logarithm 
as a Taylor series around 
, then, 
, using this in our equation:
 |
14 |
If we compare the simulated and experimental values for a single pulse from our previous study19, using the values of 
and 
obtained in the simulation, we can see in Fig. 8. that this linear approximation can be considered valid for the experimental values up to approximately 
, as the relative error between the logarithmic and linear models is similar in this range. Beyond this range, the linear approximation becomes invalid and cannot be used, as the error increases significantly.
Fig. 8.
Comparison between linear and logarithmic fits for the simulation and experimental single-shot data. The inset shows the relative error for experimental measurements between both methods up to a fluence of 
.
Therefore, we can consider a first linear approximation up to fluences around 
, and then change to a logarithmic approach:
 |
15 |
Here, we can distinguish the linear part:
 |
16 |
And the logarithmic:
 |
17 |
Here, the factor 
is introduced to ensure continuity between both functions at 
Furthermore, to also ensure differentiable transition, a sigmoidal function 
is employed:
 |
18 |
Where k controls the smoothness of the transition. Additionally, in in this model it is also assumed that if the sub-pulse fluence drops below 
, the logarithmic value is set to zero, preventing it from becoming negative.
In this model, solving the derivative to find the maximum leads to a transcendental equation in which the optimal fluence also depends on the introduced parameter 
, a more detailed analysis is provided in the supplementary data. We have used the value 
, determined by minimizing the squared errors. With this value, solving the derivative yields an optimal fluence per sub-pulse:
 |
19 |
In this case, we obtain a value 
, which is an intermediate value between the simulation and the purely logarithmic model.
The comparison between the approximation using Eq. (
and Eq. (
is shown in Fig. 9. To enhance clarity, the data has been split into two separate graphs, preventing excessive compactness and enabling a clearer visualization of the trends. We observe that both models predict the behaviour of the experimental findings, with depth increasing as both fluence and the number of sub-pulses increase. However, this depth reaches a maximum close to 
before decreasing again, eventually approaching zero when the sub-pulse fluence falls below the threshold fluence. Additionally, it is noted that the model which uses the linear approach up to 
provides a better fit to the experimental results. For instance, if we observe the fluence of 
, we can see that the linear & logarithmic model recreates better the number of sub-pulses where the ablation returns to zero, while the logarithmic model overestimates it. Given the consistency observed between the experimental results and the proposed models, it is evident that both models offer valuable insights into the behaviour of ablation. While the purely logarithmic model provides a lower accuracy, it can be advantageous for quick assessments due to its simplicity. On the other hand, for a more comprehensive analysis, the linear & logarithmic model yields a better result. To the best of the author’s knowledge, this is the first presentation of these models, which serve as a basis for further exploration and refinement of burst mode processing.
Fig. 9.
Comparison of depth vs. sub-pulses for different fluences between experimental results and linear & logarithmic approximation and just logarithmic approximation. The graph is divided into two plots to improve clarity in the visualization.
Conclusions
In this study, we have investigated the ablation of 316 L stainless steel with ultrashort laser pulses and its dependence with the fluence and the number of sub-pulses in the MHz burst regime using the two-temperature model, where the plasma plume has already been considered dissipated. Through the evolution and optimization of the TTM, we were able to model and simulate the temperature evolution as a function of the number of sub-pulses and laser fluence. By understanding this temperature evolution, the ablation profile can be determined under different laser conditions. As shown in Fig. 5. the results demonstrate a dual dependency of ablation depth on both fluence and the number of sub-pulses. However, by considering the fluence per sub-pulse, this relationship can be simplified to a single dependence. Specifically, when 
is below the threshold fluence, no ablation occurs, however, as the 
increases, the ablation depth also grows until it reaches a maximum. The maximum ablation depth using the two-temperature model is achieved at approximately 
, and beyond this point, further increases in 
do not result in a deeper ablation due to each sub-pulse operates within the logarithmic saturation regime described by Eq. (
. Therefore, in this case, applying additional sub-pulses is more effective than increasing the fluence. This means that we could improve the ablation depth by converting the logarithmic trend of the fluence into a linear one by using the number of sub-pulses. From the experimental results, it is evident that increasing the number of sub-pulses is an effective strategy to improve the depth of ablation when the fluence is significantly higher than the threshold fluence, since dividing it between each sub-pulse increases the depth in a nearly linear manner. Concerning the diameter, it can be observed that, for a given fluence, as the number of sub-pulses increases, the diameter decreases. However, using the fluence per sub-pulse instead of the total fluence, the diameters match almost exactly, regardless of the number of sub-pulses. This suggests that there is no clear incubation effect dependent on the number of sub-pulses, although further testing is required. From the comparison, we can see that our TTM is consistent with the experimental results and can be used to predict the ablation depth and diameter values. Additionally, two analytical models for the depth have been presented: the first, which maintains a purely logarithmic regime irrespective of fluence, is simpler and provides a quick estimation of the optimal fluence at 
, making it suitable for rapid assessments. The second model, while more complex, provides a better agreement with experimental results, offering a finer level of accuracy in predicting the ablation depth and the optimal fluence, 
. Therefore, by using this approach, the ablation depth for the whole range of fluences and sub-pulses can be predicted, as well as the optimal processing conditions based on the fluence and the number of available sub-pulses.
However, it is important to acknowledge the limitations of our simulation and the presented models. One key constraint is that the intraburst time interval cannot be shorter than 24 nanoseconds, which may limit the applicability of these findings to regimes with even higher repetition rates or in high-aspect ratio ablation, such as drilling, where saturation in the ablation process will appear32. Additionally, at higher fluences per sub-pulse, changes in the ablation mechanisms may occur, potentially affecting the accuracy of the proposed models33. Despite these limitations, the proposed models provide a reliable framework for predicting ablation depth and diameter across a wide range of fluences and sub-pulses. The agreement between our TTM simulations, analytical models, and experimental results highlights its robustness for the whole conditions studied and potential applicability in optimizing ultrafast laser processing strategies.
Supplementary Information
Below is the link to the electronic supplementary material.
Author contributions
L.O: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing –original draft, Visualization. S.M.O: Writing – review&editing, Supervision, Resources, Project administration. A.R: Writing – review&editing, Validation, Supervision. M.G: Writing – review&editing, Validation, Supervision, Resources, Investigation. I.A: Writing – review&editing, Validation, Supervision, Resources. E.C: Writing – review&editing, Validation, Supervision, Methodology, Formal analysis, Conceptualization.
Funding
This work has been sponsored by Horizon Europe Digital, Industry and Space Programme under Grant No. 101091623 BILASURF, and Horizon Europe Programe under Grant No.101086227 L4DNANO.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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This section collects any data citations, data availability statements, or supplementary materials included in this article.
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Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.