Effect of damage incubation in the laser grooving of sapphire

Ultrafast pulsed-laser ablation of materials is a rapidly developing field, seeing increased demand across a broad spectrum of industrial applications. While many of the applications are adaptations of traditional machining techniques, such as cutting and drilling, an increasing amount of research has focused on novel processes with few conventional parallels, such as micro/nanostructuring and material deposition.^1,2 In such contexts, laser processing is rapidly evolving from being simply an option, to being an essential means to an end.

This increased connection to the cutting-edge gives laser processing technology an important role as a potential enabler of future industrial technologies. An example would be the manufacturing of vertical interconnect access (VIA) structures on substrates, such as in integrated circuits (ICs). These structures link substrate structures vertically, unlocking a third dimension for potential use; this, for example, allows for the continuation of densification of ICs beyond 2D resolution limits.³ The high precision and potential scalability of speed in ultrafast laser drilling makes it a strong contender for industrial implementation of VIA processing.

The flip side of this relationship to the cutting-edge is that the limitations of laser ablation technologies often translate to potential bottlenecks in the realization of these applications. Not only are these limitations restricted to the domain of what can or cannot be done but also to the concepts of efficiency and controllability, important to actual implementation. Of increasing value then is the knowledge of how things occur, i.e., the link between application and fundamental physics, as concrete indicators of what can and should be expected.

The physics underlying ultrafast laser ablation of materials, however, is undoubtedly difficult. The essence of its difficulty lies in the multiscaled nature of the problem. The reaction has processes with typical temporal and spatial scales differing by several orders of magnitude, making a complete description of the phenomenon, from material excitation to destruction, a formidable task. That is not to say efforts have not succeeded in reproducing key aspects.^5–10 To simplify the problem, many of these fundamental studies focus on single-pulse ablation, where the reaction between a single pulse and a pristine material surface is studied. In dielectrics, for example, paired with conventional probing techniques, studies have furthered the understanding of ablation thresholds and morphology, as well as the microscopic electronic excitation processes involved.^5,7

An important link required to bridge these fundamental studies to actual application is the expansion of the problem from single- to multiple-pulse ablation situations. Various changes induced by previous pulses not only cause deviations from a simple linear addition of single-pulse results but also yields exotic results not possible with single pulse irradiation. A corresponding example would be heat accumulation due to continued irradiation, which, when properly tuned, was demonstrated in the literature to improve ablation efficiency and cause less thermal damage to the surrounding lattice than what a simple compound value would suggest.¹¹ 

A representative scenario for a typical multiple-pulse ablation process and ideal for study is that of laser direct writing, where the laser pulse is scanned along a trajectory on the sample surface. Through this process, shallow grooves, often used in scribe-and-break process of wafers, can be created, while repeated application can lead to direct laser cutting. In the case of moderate overlap, it has been shown that a simple addition of single pulse ablation is rather effective in modeling groove depths.^12,13 For further overlap, when the aspect ratio of grooved holes becomes greater than one, propagation effects of the laser interacting with the crater profile have been shown to lead to further tapering of the crater profile, as well as to the creation of heat and electronically induced defect structures surrounding the processed region.^14,15 However, in all of these studies, they have yet to address changes in the intrinsic material properties and its effects.

Damage incubation is a widely observed phenomenon in laser ablation where preablation modifications of the irradiated material alter the light absorption characteristic for subsequent pulses. It is observed regardless of the specifics of the material and manifests in metallic, semiconducting, and dielectric materials.¹⁶ Most prominently, damage incubation is used to explain the observed decrease in the laser fluence required for laser-induced damage of materials when irradiated by multiple pulses in contrast to that for a single pulse.^17,18,16 Physically, the formation of semipermanent electronic defect states, most often color centers, is believed to be responsible for damage incubation in dielectrics.^19,20,7 Such defects provide subbandgap levels which serve as easily excitable sources for subsequent electronic excitations. These slight changes in optical property are magnified in the highly nonlinear excitation process of ablation, leading to macroscopic changes in not only the ablation threshold, but also the morphology of observed processed regions.²¹ Despite the prominent role that incubation has been shown to qualitatively play in multiple-pulse ablation situations, only a few studies have been done to quantify the effects of damage incubation, and particularly so in processing conditions (above the damage threshold).

In this paper, we focus on quantifying the effects of damage incubation on multiple-pulse ablative behavior in sapphire using a picosecond laser, currently one of the most widely applied lasers for ultrafast laser micromachining. We choose sapphire for its industrial relevance, as well as the fact that it is a material known to show strong damage incubation characteristics.^21,22 The effects of incubation should become most pronounced at a moderate amount of incident pulses when defect concentrations have not saturated, and in situations where other effects, such as defocusing and material recast, are comparatively small. Laser grooving of shallow trenches is just such a case. Here, first, we create surface grooves of varying pulse energy and spatial overlap on sapphire samples and identify characteristic features of incubation through depth profile measurements. We then propose a generic model to incorporate the effects of incubation on the absorption characteristic of an arbitrary material. The simplicity of the model allows us to arrive at analytic asymptotic solutions displaying simple scaling behaviors in regard to the interpulse overlap and pulse energies, i.e., the incident energy density. Lastly, we, demonstrate how pulse-to-pulse absorption changes caused by incubation are dominant for sapphire; by proper rescaling according to our model predictions, we succeed in deriving a universal curve for ablated depths in regard to the incident energy density.

It is often difficult to explore a certain facet regarding ablation effects systematically, as many competing processes occur with different governing physics. To overcome these difficulties, it is essential to tailor a purpose-made experiment in order to isolate the physics under study: that is, in our case, incubation effects. In this section, we first introduce our experimental setup and our reasoning for such design, and later discuss the potential effects of other competing processes.

Differences in ablation processes originating from incubation effects should be most pronounced with differing pulse overlaps. In order to systematically and stably create single scan grooves at different pulse spatial overlap, a helical drilling setup is utilized, as shown in Fig. 1.^23–26 In this setup, a rotating Dove prism rotates the laser path into a circular spatial trajectory; the angular speed is equal to twice the mechanical rotation speed of the Dove prism. By focusing this laser onto the sample surface, it is possible to drill circular paths onto the sample surface. In our experiment, we utilize this setup to systematically alter the overlap of subsequent pulses in an easy to characterize, compact circular track on the sample surface. A wedge plate is also present before the Dove prism system, and by adjusting its angle, the rotational radius of the laser beam (or helical radius: $rh$⁠) at the focus can be tuned. Various angular speed and trajectory radii can be achieved by changing the rotation speed of the Dove prism and the position of the wedge plate, respectively. From the circumference $2πrh$⁠, laser pulse repetition rate $f$⁠, and the time it takes for the laser to make one rotation $Trot$⁠, it is possible to calculate the average number of pulses per unit length $ppl=f⋅Trot/2πrh$ along the ablated grooves. Another measure of the pulse overlap, valid when adjacent pulse centers are less than a diameter apart, is the degree of overlap $η$ defined as

where $w$ is the beam radius. The degree of overlap is a dimensionless parameter (often expressed as a percent value), which describes the distance between pulse centers linearly; it takes a value of 1 when pulses coincide and 0 when they are exactly a diameter apart. Spacing beyond this yields unintuitive negative values, but as the current scope of the paper is for incubation-regime situations with well-overlapped pulses, we will use this analytically simple expression.

Another factor in varying incubation effect behavior is believed to be the laser pulse energy. This can be controlled by an integrated power attenuator in the laser system. The laser system used in the experiment is a 515 nm central wavelength picosecond laser system (TRUMPF, TruMicro 5270) with a pulse duration of 7 ps. The maximum pulse energy is 150 $μ$J, and the laser is used at 50 kHz for the experiments. On–off switching of the laser is controlled by an external computer.

The rest of the ablation setup is standard. The rotating laser beam from the helical drilling setup is focused onto the sample surface by a 60 mm focal length processing lens equipped with a custom nozzle to allow for the on-axis application of processing gasses. In the experiment, compressed air at 2 bars pressure is used to prevent contamination of the optical system from processing debris. The focus position is adjusted by moving the processing lens mounted on a linear stage. Real-time monitoring of the sample is possible from a focus-calibrated, coaxial camera illuminated by a red LED. The laser spot size at the focus is determined in situ before each experiment by an ablated area extrapolation scheme often used to determine laser damage thresholds.²⁷ Such measurements yielded a slightly elliptical laser spot, with major and minor axis radii of 9.9 and 9.3 $μ$m, corresponding to a Rayleigh length of around 300 $μ$m. When a radius value is required in the analysis, the average of the two values is used. At the same time, the damage threshold of our current setup was measured to be at a pulse energy of around 3.7 $μ$J, corresponding to approximately 2.6 J/cm$2$⁠.

Using this setup, we ablate a sapphire plate, with the c-plane oriented on the surface (⁠$≥$99.99% purity, 2-in. diameter; Siegert Wafer). The thickness of the wafer is approximately 430 $μ$m, with both front- and back-side polishing. The wafer is attached to a motorized xy-stage by a custom vacuum suction holder to ensure high mechanical stability during processing.

The actual processing conditions are as follows. The rotation speed of the Dove prism is varied from 120 revolutions per second (rps) to 10 rps in increments of 10 rps. To increase the range of overlaps available, processing is done at two helical radii of 31 $μ$m and 65 $μ$m. These values are measured after the experiment from the damage tracks on the sample surface. At each rps, in order to restrict the ablation to a single revolution, the laser is programmed to turn on for 1/(2 · rps) s, which is the amount of time it takes for the laser to make one optical revolution. Due to latency in the laser switching, the laser is not at full power for the duration of the programmed time, but slightly lower directly after starting and before stopping. These portions are omitted from analysis. Each ablated groove in the experiment is done at new locations on the sample surface. Grooves are translated at least 50 $μ$m away edge-to-edge to reduce cross-contamination effects, such as debris and thermal-induced stress accumulation.

After the experiment, samples undergo ultrasonic cleaning in an ethanol bath for 30 min to remove residual debris. After samples are cleaned, their morphologies are observed under a laser scanning microscope (LSM; Keyence, VK9700). With this, the height profile can be measured with submicrometer vertical resolution. Due to decreasing brightness in high aspect ratio holes, the range of the depth observable by the LSM is limited to around 20 $μ$m. Only holes where the whole hole is within resolution are analyzed, which consequently restricts the analysis to situations where all grooves are an order of magnitude shallower than the Rayleigh length of the processing beam. Hence, defocusing effects are ignored in the analysis.

In quantitative analysis, we focus on the maximum depth of the ablated grooves. The reasoning for this, as opposed to analyzing the volume or whole cross section, is mainly to avoid effects of changing groove shapes. The grooves are roughly U-shaped in our experiments; the light in the groove center sees a relatively flat surface. This allows us to ignore differences caused by an angled projection of the incident fluence, and diffraction effects, which are more pronounced at curved (at scales relative to the wavelength) surfaces. Furthermore, we would like to avoid the chipping and/or recast that is predominantly seen at the crater edges, which is mostly irrelevant to the physics under discussion.

Lastly, we would like to comment on heat accumulation and plasma shielding as possible competing factors affecting morphology. Wide-spread heat accumulation effects, as in dependence of ablation on the locality of the groove trajectory, are seen to have little significance in our geometries, as the two tested helical radii conditions yielded similar results, as will be shown in Sec. III. While it is difficult to completely rule out more local heat accumulation, we gauged its significance by conducting multiple-rotation experiments, where two to five beam rotations were done on the same trajectory. We found that as long as the total incident pulse numbers were similar (for example, a single rotation compared to two rotations at twice the speed), the morphologies were so as well. If local, subsequent-pulse heat accumulation were significant, a more dramatic change should be observable, as two orders of magnitude of time are different for the pulse-to-pulse interval seen by the material (during which heat dissipation can occur) in the two cases. Interpulse plasma shielding is mitigated in our experiment because (a) the laser, not the sample, is scanned (i.e., the laser spatially avoids developed clouds), (b) we employ processing gas to create airflow away from the processed region, and (c) the shallow nature of the grooves helps prevent local plasma confinement. Furthermore, ultrafast laser-induced plasmas usually decay to densities below $1018$ cm$−3$ at submicrosecond time scales, which is the repetition rate of our laser system.²⁸ Even if we assume a linear buildup, it would require close to 1000 pulses (99.9% overlap) for the plasma frequency of this plasma to approach the laser wavelength, whereas in the current experiment the overlaps are far lower. Experimentally, the aforementioned multiple-rotation experiments also help rule out interpulse plasma shielding as a significant player; plasma shielding should also be dependent on the spatial order in which grooves are ablated, which was not the case in our experiment. As a final comment, it should also be noted that fixing the repetition rate of the laser is also conscientious, as both these competing processes are heavily dependent on interpulse delays. This underlies our choice to vary the helical drilling conditions as opposed to the laser repetition rate to achieve varying pulse overlaps.

The deepest depth of the grooves as a function of $ppl$ is shown in Fig. 2(a). Corresponding values of $η$ are shown on the upper scale of the figure. It can be seen that the helical radius has little influence on the total achieved depth, apart from some discrepancy for the 10 $μ$J data near 1.5 $ppl$⁠, for reasons explained later.

It is clear from the graph that there are two distinct regimes of ablation in sapphire. These are a shallow ablation regime characterized by a smooth morphology [Fig. 2(b)] and a deep ablation regime characterized by a fractured morphology [Fig. 2(c)]. These two regimes are what is referred to as “gentle” and “strong” ablation in the literature.^21,29 Gentle ablation, often associated with melting and vaporization, or in the case of ultrashort pulses, Coulomb explosion,²⁹ is favored in our experiment when pulse energies are low and overlap between pulses is small. On the other hand, strong ablation, associated with strongly thermal processes such as material phase explosion,³⁰ takes place at high pulse energies and when there is considerable overlap (exceeding 90%) between pulses. A crucial factor in the transition from gentle to strong ablation is believed to be damage incubation.²¹ An increase in energy absorption through incubation allows for a later pulse to gain the critical energy needed to initiate strong ablation processes. The probabilistic nature of the defect accumulation is most pronounced in the transition regime [the sudden jump in the depths in Fig. 2(a)], which we believe to be the origin of the discrepancy for the 10 $μ$J data in Fig. 2(a). In this regime, grooves with mixed strong and gentle ablation regimes could also be observed.

In order to understand these results in a unified matter, we model the essence of the laser grooving setup with newly incorporated incubation effects. For our model, we think of an infinite train of pulses traveling along a one-dimensional path with spacing $d$⁠, as seen in Fig. 3(a). The circular nature of our experimental trajectory can be largely ignored, as the helical radius is much larger than the beam diameter, and the start and stop points, where orders of the pulses become discontinuous, are omitted from the analysis. We focus on a certain point, coinciding to the center of the 0-th pulse, as indicated by the orange arrow in Fig. 3(a), and calculate behaviors of the ablated depths depending on the interpulse spacing and pulse energies at this point. We do this by first calculating the absorbed total energy at this point, and then assume that this energy and ablated depth have a one-to-one correspondence. By rescaling experimental data in terms of this absorbed energy, we should be able to derive a single universal curve describing the depth as a function of the scaled irradiated energy.

It should be noted that the relationship between absorbed energy and ablated depth is not a trivial matter. In the case of sapphire, it has been shown that ablated depth to pulse energy relationships based on Lambert-Beer type (logarithmic) penetration depths of the local incident fluence fail to explain experimental results.³¹ In addition, most models deal with single pulse ablation depths, and no models exist to our knowledge which deal with damage incubative situations, where the correspondence between incident pulse energies and ablated depths varies by an order of magnitude even for the same pulse energy, as evidenced in our data. As the essence of incubation effects is believed to reside in the nonlinearity of the absorption, not in the penetration depths of absorbed energies, we simplify the connection between ablated depth and absorbed energy density to a simple linear relation (hence dealing with the total absorbed energy) and assume that key nonlinearities reside in how much energy is absorbed.

In our model, we restrict discussions to pulses with energies moderately above the damage threshold, a regime in which, while initial processes are highly nonlinear, the majority of the energy absorption is done nearly metallically by the excited free-electron plasma via inverse bremsstrahlung.⁷ Thus, we postulate an effective absorptivity $an$ for the $n$-th incident pulse with incident local fluence $Fn$ and assume that the material absorbs energy $Fn⋅an$ of the pulse; we also incorporate the effects of reflection and transmission within this parameter. We include incubation effects into $an$ through the following equation:

Here, $as$ is some saturation absorptivity, $Fn$ is the local incident fluence of the $n$-th pulse, $F0$ is the peak fluence (i.e., the peak fluence of the 0-th Gaussian pulse), and $β$ is an incubation strength factor. In the case of constant fluence, this yields an asymptotic increase of the absorption to $as$⁠; $β$ determines how fast the effective absorptivity approaches this asymptotic value. As we are only concerned with scaling relationships and not on absolute values, these parameters are taken with arbitrary units. It is also experimentally known that the incubation behavior which causes a gentle-to-strong ablation transition in sapphire has a weak dependence on the pulse energy for picosecond lasers, depending more on the number of deposited pulses.³² Thus, the second term is normalized by $F0$ to reflect this weak pulse energy dependence. In general, changes in linear reflection are negligible, as refractive index changes due to incubation are only of the order of $10−4$ for transparent dielectrics at typical processing wavelengths.³³ Additional terms reflecting the effects of surface quality on the effective absorptivity may at times be appropriate, as it has been shown to cause a factor decrease in the damage threshold of sapphire.³⁴ However, the incubation process in sapphire, which, for example, drives the gentle to strong ablation phase change, is not driven by this process, as the surface roughness has been shown to change little during the gentle ablation phase.³² 

With this model, the total amount of energy absorbed at the monitored point can be expressed as

We expect the depth to have a one-to-one correspondence with this total value. We take the summation from $n=−∞$ to $n=∞$ to make the problem analytically easier to solve, although the pulse number is finite in reality. This treatment is valid because pulses with large n contribute little to the total sum derived by a more rigorous summation of fluences above a threshold value. We use spatial Gaussian pulses $Fn=F0exp⁡[−2(nd/w)2]$⁠, where $w$ is the $1/e2$ beam intensity radius. For simplicity, we rescale distance in terms of $w$⁠, i.e., we set $w=1$ and make $d$ unitless.

Equation (3) allows us to calculate $Etot$ analytically for two limiting cases. The first one is the “quasistatic absorption limit,” where changes in $an$ are minimal near $n=0$⁠. This occurs when incubation is either very strong (⁠$a0≈as$⁠) or very weak (⁠$as≈a+∞$⁠). In such cases, we can set $an=constant$⁠, and the total absorbed energy simply becomes a discrete summation along a Gaussian profile. Consequently,

where

is the Jacobi theta function of the third kind.³⁵ While the above is a rigorous equality, one can transform the theta function (a summation of Gaussians) in Eq. (4) as

where we have utilized Poisson’s summation formula.³⁶ When $d≪π/(22)$ (or $e−π2/(4d2)≪1$⁠), as is the case for our experimental conditions, the summation of the exponential on the right-hand side of the equation quickly decays to zero for nonzero $m$⁠, and we see the value become proportional to $1/d$⁠. The total absorbed energy in Eq. (4) can then be seen to scale as $F0/d$⁠.

The second limiting case is the “accumulative absorption limit,” when the strong absorption change near $n=0$ takes place [in cases where $as≫a0$ in Eq. (2)],

By employing the Jacobi theta function and approximation for $d≪π/(22)$ as before,

As the first term dominates for small $d$⁠, total absorbed energy which scales as approximately $F0/d2$⁠.

In Fig. 3(b), calculations for the two analytical results (red and blue) and a numerical calculation for a case in between (green; parameters $a0=0.01$⁠, $as=1$⁠, $β=0.016$⁠) are shown, where the total absorbed energy at $d=0.01$ is normalized to 1. We can clearly see the $1/d$ and $1/d2$ dependencies (dashed lines) for the quasistatic absorption limit and the accumulative absorption limit, respectively. Discrepancies at higher $d$ originate from the small $d$ approximation employed during the derivation of the scaling relationship. For parameters not at the two limits, the absorption scaling characteristic behaves in the middle of the two, bounded by the two limiting cases.

To analyze our experimental data in the context of our newly formed model, we first scale our ablated depth results according to the quasistatic scaling behavior, which is physically equivalent to plotting ablated depth against the incident energy linear density, effectively ignoring incubation effects. The result of this rescaling, conducted by dividing the pulse energy by the interpulse spacing, is shown in Fig. 4(a). It can be seen that the results are unsatisfactory in achieving agreement among different pulse energies, with a highly pulse energy dependent discrepancy between data with the same $Ep/d$ ratio.

Next, we scale the data according to the accumulative absorption scaling prediction of $Ep/d2$⁠. Results of such rescaling, conducted by now dividing the pulse energy by the square of the interpulse distance, are shown in Fig. 4(b). Especially when compared with quasistatic scaling, great agreement is observed with the experimental results in the range of 20–50 $μ$J. The result is consistent with the strong incubation effects seen in laser processing of sapphire. Because the absorption characteristics of the material change dynamically in the accumulative absorption situation, it is better suited to model the evolving ablation behavior in sapphire. A noticeable deviation is seen for the 10 $μ$J data from the other data, possibly attributed to a couple of factors, such as the relatively low pulse energy in disagreement with our initial assumption of high pulse energy, or to the fact that the high pulse numbers saturates incubation, i.e., a more proper fitting with a nonlimiting case (of currently an unknown analytical form) may then yield even better agreement. The accuracy of the depth scaling despite the highly simplified arguments regarding energy absorption mechanisms and the subsequent material ablation phenomenon displays just how dominant incubation effects are in comparison to other factors in this shallow grooving regime.

From a microscopic standpoint of the physics involved, the relevant defects associated with sapphire are believed to be neutral and charged F-centers: oxygen vacancies in the sapphire lattice with one (F$+$-center) or two (F-center) trapped electrons. Creation of such defects may be facilitated by vacancy-interstitial pair formation or else by the emission of certain atom types from the material surface.^5,37 The formation of color centers in sapphire has traditionally been studied under pulsed-electron and neutron irradiation³⁸ but has recently been directly observed in ultrashort pulse-laser processed sapphire by photoluminescence excitation.²² Electrons captured in F-centers have relative energies 6–7 eV below the conduction band minimum³⁹ and are easier to excite than electrons in the valence band (requiring around 9 eV), considering the multiphoton nature of the initial excitation in the ultrashort pulsed-laser excitation. This slight difference in optical absorption is believed to be amplified by avalanche ionization mechanisms, which is highly sensitive to seed electron density and allowing the damaged material to absorb more of the incident energy. The $1/d2$ dependence of the accumulative absorption case can be thought of as two increasing mechanisms working in tandem: a $1/d$ contribution from increasing the energy density and a $1/d$ contribution from increasing defect concentrations. To incorporate these effects microscopically, past models have relied on a rate-equation approach to characterize changes in the threshold for low-energy pulses, usually postulating defect densities below 0.1% of the atomic density or of the order of $1021$ cm$−3$ for the maximum defect densities.⁴⁰ It should be stressed that while more quantitative arguments may require a similar approach, our model captures macroscopic behaviors without depending on the knowledge of the multitude of nontrivial parameters and their functions required for a strict microscopic treatment.

A last remark on time scales should be made. Through interferometry⁴¹ and double-pulse measurements,⁴² the free-carrier-like response of excited electrons in sapphire has been shown to have lifetimes in the subnanosecond to single-nanosecond range. Additionally, from photoluminescence excitation measurements in Ref. 43, luminescence lifetimes of F-centers were determined to be 35 ms, while lifetimes of F$+$-centers were shorter than 7 ns. The 50 kHz laser used in the current experiment has a 20 $μ$s interpulse spacing, and thus, residual electron excitation is only affected by residual F-center excited electrons. As this lifetime is long, and provided the ignored heat accumulation effects can still be ignored, similar scaling would be expected for similar kHz to MHz lasers used in typical ultrashort pulsed-laser processing. Outside this range, different transient interpulse absorption characteristics will come into play and the current results will not trivially hold. For example, at GHz repetition rates, it should be possible to re-excite “free” electrons before significant relaxation, which will make subsequent energy absorption notably different.

In this work, we have focused on qualifying the effects of damage incubation on the ablated morphology of a material, in this case, sapphire, and perform a purpose-made experiment to best isolate its effects on ablated morphology. We see clear morphological evidence of incubation-induced changes in the morphology and depth dependence, which is nonlinear to the incident total energy density. In order to explain these experimental results with sapphire, a material known to have very strong incubation effects, we create a phenomenological model considering damage incubation and subsequent energy absorption changes. We derive analytical expressions which, for certain common conditions, reduces to simple scaling relationships explaining ablated depths in terms of the incident pulse energy and overlap. We finally explain the observed ablated depth behavior in the laser grooving of sapphire for varying pulse energies and overlaps in the context of our model.

The arguments used to derive the new scaling relationship are highly general, and thus should also hold validity for other materials. While beyond the scope of the current paper, it may be possible to create a simple “incubation parameter,” to properly describe the functional form of the nonlimiting behaviors, thus allowing similarly general arguments for a broader class of materials not just behaving in either of the two limiting regimes. It may also be interesting to link the relationship between the traditional incubation parameter for damage incubation below the single pulse threshold with that of the current damage incubation above threshold. Related to the above, the exploration of the physical mechanisms of the microscopic processes governing incubation and how they manifest to the observed macroscopic consequences may lead to enhanced understanding of ultrafast ablation dynamics, leading to greater process control.

This work not only helps in furthering our understanding of energy absorption mechanisms in the multiple-pulse regimes for future modeling but also provides a highly simple tool to analyze and organize experimental results.

This work was supported by the Photon Frontier Network Program and the Leading Graduates Schools Program, Advanced Leading Graduate Course for Photon Science (ALPS), both funded by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, and the Center of Innovation Program funded by the Japan Science and Technology Agency (JST). The authors would like to thank the German Research Foundation DFG for the kind support within the Cluster of Excellence “Internet of Production” - Project-ID: 390621612.