# Pulse duration dependence of photoinduced carrier density in ultrashort laser-excited wide-bandgap solids

Alexandros Mouskeftaras, Allan Bildé, Sergey Klimentov, Stéphane Guizard

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Alexandros Mouskeftaras, Allan Bildé, Sergey Klimentov, Stéphane Guizard. Pulse duration dependence of photoinduced carrier density in ultrashort laser-excited wide-bandgap solids. Optics Express, 2025, 33 (6), pp.12901. 10.1364/OE.539349. hal-05222508

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# Pulse duration dependence of photoinduced carrier density in ultrashort laser-excited wide-bandgap solids

ALEXANDROS MOUSKEFTARAS,<fnref n="1" />,<fnref n="4" /> ID ALLAN BILDÉ,<fnref n="2" />
SERGEY KLIMENTOV,<fnref n="3" /> AND STÉPHANE GUIZARD<fnref n="2" />,<fnref n="5" />

**Abstract:** Using Fourier transform interferometry, we measure the variation in the dielectric constant induced by a single laser pulse at intensities corresponding to damage threshold, for pulse durations ranging from 100 fs to 1.5 ps, in three different wide-band gap dielectrics: α-SiO₂, α-Al₂O₃, and MgO. The experimental results show that the excitation density decreases significantly for increasing pulse duration. A numerical model that incorporates the main physical mechanisms: photoexcitation, carrier heating, impact ionization and carrier trapping is proposed to interpret the experiment. A successful comparison with experimental data yields quantitative information about the elementary processes involved in the interaction.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

It is now public knowledge that short laser pulses can be successfully used to modify the properties of transparent dielectrics in the bulk, or for surface micromachining. For instance in silica, the ability to create 3D modulations of the refractive index [1,2] or birefringence linked to laser-induced nanostructuring [3] opens up important applications in the domain of optical components or fibers. These perspectives motivate a move towards a better control of the processing, aiming in particular to achieve the highest possible spatial resolution, and lead to the use of complex laser pulses with tailored spatial [4] and temporal profiles [5]. To predict and model the propagation of such complex ultrashort light pulses and their interaction with matter, a thorough understanding of the elementary mechanisms occurring during the pulse is essential. Despite significant effort by the community for over twenty years, there is still a need to clarify the role of the different physical processes and their dependence on key parameters, such as the laser pulse duration [6–8].

A model which is widely accepted [9] describes the sequence of stages leading to optical breakdown as follows. The first step is field ionization leading to injection of carriers in the conduction band (CB). These excited carriers are further “heated” within the CB, which can be described as a series of inverse bremsstrahlung collisions, or by a sequence of resonantly enhanced single or multiphoton transitions [10]. Then, inelastic collisions between valence band (VB) and CB electrons with sufficient kinetic energy (impact ionization) increases the density of excited carriers; the repetition of this sequence of laser-induced carrier heating and impact ionization results in an electronic avalanche, supposed to be responsible for the breakdown/damage of the solid [11]. This model was initially developed to interpret the evolution of optical breakdown

<footnote n="1">¹Aix-Marseille Université, CNRS, LP3, UMR7341, 13009 Marseille, France</footnote>

<footnote n="2">²Laboratoire Interactions Dynamiques, Lasers, CEA-DRF-IRAMIS, Université Paris –Saclay, 91191 Gif Sur Yvette, France</footnote>

<footnote n="3">³Vladimir State University, 60000 Vladimir, Russia</footnote>

<footnote n="4">⁴alexandros.mouskeftaras@univ-amu.fr</footnote>

<footnote n="5">⁵stephane.guizard@cea.fr</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Experimental setup for Fourier transform interferometry measurements. C1, C2: compressors, L₁, L₂, L₃: convex lenses, CCD: Charged-Coupled Device detector, LD: laser diode. An onset with the spatially-resolved spectrum is shown.</figcaption></figure>

Finally, the image of the front surface of the sample is formed with these two beams at the entrance of a spectrometer. This imaging is set in a way that the wave fronts passing through the entrance slit of the spectrometer correspond to an unperturbed part of the beam for the first pulse – the “reference pulse”, and the perturbed part of the beam crossing the excited region of the sample for the second pulse. The interference pattern formed at the exit of the monochromator is recorded and analyzed with a Fourier transform algorithm extracting the phase shift and the absorption or transmittivity, both directly linked to the refractive index change induced by the pump laser.

As usual for high energy Ti-Sa lasers with chirped pulse amplification scheme emitting at 790 nm, the pulse is temporally stretched before amplification. In our system, the laser beam is separated in two parts, each with its own compressor system. This allows us to maintain the shortest pulse for the probe, while the duration of the pump pulse can be adjusted between 100 fs and several picoseconds (FWHM) by changing the distance between the gratings of the compressor. The pump pulse, with variable duration, is mildly focused on the front surface of the sample at normal incidence, into a typical beam diameter of 45 µm (FWHM). The probe beam, of about 600 µm in diameter, penetrates the surface at an angle of 20°.

The samples are 1 mm thick, high purity (impurities ≤1 ppm) crystal wafers, optically polished on both sides, and mounted on a three-axis motorized stage. In all cases the sample is translated after each laser shot, so that only single shot damage is monitored on a fresh sample area. Finally, the sample is illuminated with a UV laser diode and the magnified image of the surface is visualized with a charge-coupled device (CCD) camera. The OBT is defined as the lowest intensity for which a permanent modification of the image contrast is induced by a single pulse.
3. Results

The phase shift and transmission have been recorded as a function of time delay between the pump and the probe pulse. These measurements for three different single crystal samples, SiO2, Al2O3 and MgO, are reported in Fig. 2.

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Phase shift and transmission factor measured in SiO2, Al2O3 and MgO, as a function of time, for a pump pulse duration of 100 fs. Black filled squares represent experimentally measured phase shifts, while red circles denote the transmission factors. The corresponding numerical simulations are shown as continuous lines.</figcaption></figure>

The initial positive phase shift due to Kerr effect lasts as long as the pump and the probe pulses overlap in the sample and is observed in all three crystals used. Then, the phase shift becomes negative, while the transmitted intensity decreases. This is due to the presence of photo-excited carriers in the CB. Finally, in SiO2 only, an electron and a hole are self-trapped [16], which leads to a final positive phase shift and a complete recovery of the transmissivity. We have repeated these measurements in all three samples, for different pulse durations ranging from 100 fs to 1.6 ps, in each case at an intensity corresponding to the single-shot laser damage threshold reported in Fig. 3.

The results of this set of measurements are summarized in Fig. 4, where we have plotted the transmission factor and phase shift value when a plateau is reached, i.e. at least 2 ps after the pump pulse. Note that for SiO2, this phase shift is positive, again due to the self-trapping of excited carriers, while the transmission remains at 100%, since the absorption band of self-trapped excitons (STE) is in the UV, far from our probe wavelength [16]. The evolution is the same in all materials: the signal due to the excited carriers decreases when the pulse duration increases. In other words, the longer the pulse duration, the smaller the density of excited carriers needed to
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Optical breakdown thresholds as a function of pulse duration and material. Black, red and blue symbols are used for SiO₂, Al₂O₃ and MgO respectively. This threshold corresponds to the minimum intensity for which a single shot leads to a permanent modification of the surface.</figcaption></figure>

reach optical breakdown. Therefore, we can directly conclude that a “critical” excitation density cannot be used as a criterion to determine the OBT.

In order to simulate and deeper interpret these experimental data, we have developed a numerical model, including the main physical mechanisms involved in the interaction: non-linear

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Phase shift and transmission change induced by a pump pulse, at intensities corresponding to single-shot, optical breakdown threshold, in Al₂O₃, MgO and SiO₂, as a function of pulse duration.</figcaption></figure>
excitation from VB to CB, heating of carriers by successive single-photon absorption in the CB and in the case of SiO₂ only [17], impact ionization and self-trapping. Following a simplified model initially proposed by B. Rethfeld [18], the conduction band is represented by a set of discrete energy levels separated by one photon energy (1.55 eV for a 790 nm beam), and the evolution of the population occupying each of these levels is governed by a set of rate equations. Thus, a single level accounts for the VB, the conduction band is represented by the set of $k = \left[ \left( 1 + \frac{m^*}{m} \right) \frac{E_g}{\hbar v} + 2 \right]$ levels, where $v$ is the pump optical frequency and $h$, the Planck constant, $\lfloor \rfloor$ stands for the integer part, $m^*$ represents the reduced effective mass of the electrons and $E_g$ the bandgap energy. In most cases the excitation density is very low (<1%) and reaches 10% for the shortest pulse duration (100 fs). Only in the latter case the hypothesis of a constant gap could be questioned. However, the good agreement obtained for all pulse durations led us to keep this hypothesis in all cases, and finally $E_g$ is considered to be constant. The population of each level is computed at every step of the simulation by numerically solving the rate equations:

$$ \frac{\partial N_1}{\partial t} = N_v \sigma_n I^n - \sigma \frac{I}{\hbar \omega} N_1 + 2 \tilde{\alpha} \frac{N_v}{N_{v,i}} N_k - \frac{N_1}{\tau_{tr}} \\ \frac{\partial N_i}{\partial t} = \sigma \frac{I}{\hbar \omega} (N_{i-1} - N_i) - \frac{N_i}{\tau_{tr}} \qquad (1) $$

$$ \frac{\partial N_k}{\partial t} = \sigma \frac{I}{\hbar \omega} N_{k-1} - \tilde{\alpha} \frac{N_v}{N_{v,i}} N_k - \frac{N_k}{\tau_{tr}} $$

where $N_{v,i}$ is the valence band electronic density at rest, $N_v$ the valence band electronic density at a given time and $N_i$ the electronic density of level $i$. The multiphoton interband ionization is modeled by the first term in Eq. (1), where $\sigma_n$ is the corresponding cross section (in cm²n/(s.Wn)), and $I$ is the intensity of the pump pulse (in W/cm²). This non-linear term is of 6th order in SiO₂ and Al₂O₃ and 5th order in MgO, since the absorption of respectively 6 photons and 5 photons is required to ensure the transition from VB to CB. A CB electron can absorb a photon and reach a higher level in this band. This heating phenomenon is modeled by the second term in Eq. (1), $\sigma$ being the single-photon intraband absorption cross-section (in cm²). When an electron with high energy collides with an electron in the VB, it can transfer its energy to the latter and promote it to the CB. This phenomenon, known as impact ionization, is represented by the third term in Eq. (1), $\tilde{\alpha}$ being the impact ionization parameter. In SiO₂, electrons are trapped at an energy level within the bandgap, which is what the last term in Eq. (1) accounts for, $\tau_{tr}$ being the characteristic trapping time. No other electronic relaxation mechanism is considered given the picosecond timescales investigated in this manuscript and the long electron recombination times for Al₂O₃ (100 ps) [19,20] and MgO (50 ps) [20]. Those processes have a direct impact on the refractive index of the sample which is simulated using the Drude model:

$$ n^2 - 1 = \frac{e^2}{m\epsilon_0} \left( \frac{N_v f_{12}}{\omega_g^2 - \omega^2 - i \frac{\omega}{\tau_{12}}} + \frac{N_{tr} f_{tr}}{\omega_{tr}^2 - \omega^2 - i \frac{\omega}{\tau_{tr}}} \right) - \frac{e^2}{m^* \epsilon_0} \frac{N_c f_c}{\omega^2 + i \frac{\omega}{\tau_{coll}}} + \chi_{3,eff} E^2 \quad (2) $$

On the right side, the first three terms respectively account for the electrons in the VB, the trap level and the conduction band. Here, $\omega_g$ is defined by, $\omega_g = E_g/\hbar$ and $\omega_{tr}$ the absorption band frequency for STE [16]. $h/\tau_{12}$ is the width of the VB→CB transition, $\tau_{coll}$ the characteristic collision time of an electron with other particles or quasi-particles (phonons, ions, electrons, etc.). $f_{12}$, $f_{tr}$ and $f_c$ are respectively the oscillator strengths of valence band, trapped and CB electrons. Their values are estimated by accounting for the refractive index of non-irradiated material ($n_0$), ratio of photoionized carriers being trapped in STE and sum of transition probabilities for
CB electrons. The last term in Eq. (4) accounts for the Kerr effect, $\chi_{3,eff}$ being the third-order effective susceptibility of the sample and E the electric field associated with the pump pulse. The simulation also considers the evolution of the pump pulse intensity, due to multiphoton absorption and absorption by excited carriers. Phase transitions, that usually take place on time scales above 10 ps, -i.e. long after the shorter time investigated in the present study, are not considered. Thus, at each time step we calculate the population of every electronic energy level, the intensity of the pump pulse and the modification of the refractive index, in two dimensions, assuming an axial symmetry. From this set of data, and considering the angle between the two beams, the modification of the phase shift ($\Delta\Phi=2\pi L/\lambda \times (n-n_0)$ with L the integration length) and the transmission factor ($\exp(-2L\omega/c \times \text{Im}(n))$) are calculated. Their temporal evolution as a function of pump-probe delay, as well as their dependence on pump pulse duration, are displayed as full lines, respectively in Fig. 2 and Fig. 4. A satisfactory agreement is obtained in all cases. Due to the large number of parameters, it is possible to use different sets of values to fit a curve for particular pulse duration. Of course, the requirements are much higher when the fitting of the data obtained for all pulse durations must be optimized. The set of parameters leading to the best results is given in Table 1.

<table><caption>Table 1. Simulation parameters.</caption><thead><tr><th></th><th>SiO<sub>2</sub></th><th>Al<sub>2</sub>O<sub>3</sub></th><th>MgO</th></tr></thead><tbody><tr><td>E<sub>gap</sub> (eV)</td><td>9 [21]</td><td>8.8 [22]</td><td>7.7 [23]</td></tr><tr><td>N<sub>v,i</sub> (cm<sup>-3</sup>)</td><td>2.2 10<sup>22</sup> [24]</td><td>2.1 10<sup>22</sup> [19]</td><td>8.3 10<sup>22</sup></td></tr><tr><td>σ<sub>n</sub> (m<sup>2n</sup>.s<sup>-1</sup>.W<sup>-n</sup>)</td><td>7.4 10<sup>-95</sup></td><td>1.7 10<sup>-93</sup></td><td>9.5 10<sup>-77</sup></td></tr><tr><td>σ (cm<sup>2</sup>)</td><td>7.5 10<sup>-18</sup></td><td>9.4 10<sup>-19</sup></td><td>1.1 10<sup>-17</sup></td></tr><tr><td>ᾱ (s<sup>-1</sup>)</td><td>3.0 10<sup>13</sup></td><td>0</td><td>0</td></tr><tr><td>m<sup>*</sup>/M<sub>e</sub></td><td>0.5 [24]</td><td>0.75 [19]</td><td>0.85</td></tr><tr><td>f<sub>12</sub></td><td>2.88</td><td>5.44</td><td>0.99</td></tr><tr><td>f<sub>CB</sub></td><td>1</td><td>1</td><td>1</td></tr><tr><td>f<sub>tr</sub></td><td>1.12</td><td>0</td><td>0</td></tr><tr><td>τ<sub>coll</sub> (fs)</td><td>0.15</td><td>3.1 10<sup>-2</sup></td><td>0.5</td></tr><tr><td>τ<sub>tr</sub> (fs)</td><td>150 [24]</td><td>/</td><td>/</td></tr><tr><td>ħω<sub>tr</sub> (eV)</td><td>5.2 [24]</td><td>/</td><td>/</td></tr><tr><td>X<sub>3,eff</sub> (m<sup>-2</sup>.V<sup>-2</sup>)</td><td>5.6 10<sup>-23</sup> [25]</td><td>6.0 10<sup>-23</sup> [26]</td><td>1.2 10<sup>-22</sup> [27]</td></tr></tbody></table>

From the simulation, we can also deduce the excited carrier density and its spatial distribution along the propagation axis. These data are reported in Fig. 5, for the different laser pulse durations and intensities corresponding to the measurements of Fig. 2 and Fig. 4. As already stated, a considerable variation is observed in both the excitation density and its evolution as a function of depth. For the shortest pulse, the excited carrier density drops down dramatically within a thin layer behind the surface, whereas for longer pulses, this damping effect is negligible, giving a smooth density profile. In the simple simulation used here, non-linear propagation effects (self-focusing [28], etc.) are not taken into account. We suspect that they could have a noticeable influence on the pump pulse propagation, only for the shortest pulse. However, we consider in this work the optical breakdown at the surface, and we expect that if the inclusion of propagation effect can influence the excitation density profile, this will have minor influence, if any, on the one calculated at the surface.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Calculated excitation density profiles as a function of depth in SiO₂, Al₂O₃ and MgO at intensities corresponding to breakdown threshold for 5 different pulse durations.</figcaption></figure>

From our model we can also extract the amount of the deposited energy density as a function of depth. This is calculated as follows:

$$E_{abs} = \int_{t}^{t} \left( \frac{2\omega}{c} Im(n)I \right) dt \quad (3)$$

These results are plotted in Fig. 6. Similar trends as for the excited electron density are observed. Interestingly, deposited energy densities in the order of ~kJ/cm³ are only observed for the shortest pulse duration of 100 fs indicating that a thermal criterion based on fusion also fails to reproduce experiments.
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Calculated absorbed energy density as a function of depth in SiO₂, Al₂O₃ and MgO at intensities corresponding to breakdown threshold for 5 different pulse durations.</figcaption></figure>

**4. Discussion**

The measured laser-induced damage fluence thresholds (Fig. 3) lie within a few ~J/cm² range. Previously reported damage thresholds for SiO₂ [7,29] and Al₂O₃ [30] lie within the same range for similar pulse durations. As the pulse duration increases, higher laser fluence is needed for photoionization to be an efficiently contributing mechanism. This explains qualitatively the increase in the damage thresholds as a function of pulse duration. We expect for longer pulse to retrieve the square root dependence of the thresholds with pulse durations [11]. Interestingly, the threshold for MgO is scaling at a slower rate than for the other two crystals. This could be due to an extrinsic origin such as a higher defect content.

By successfully comparing numerical simulations with experimental results, we derived a set of parameters (see Table 1) that effectively reproduces the laser-matter interaction within the considered regimes. We start the discussion section by commenting on this set of parameters. Multiphoton absorption cross section values vary greatly in literature and the order of magnitude is given by the empirical expression of Agostini and Petite [31] $\sigma_6 = 4.3 \ 10^{-63} \pm 10 \ s^{-1}(\text{cm}^2\text{W}^{-1})^6$ and $\sigma_5 = 10^{-50} \pm 8 \ s^{-1}(\text{cm}^2\text{W}^{-1})^5$ for 6-photon (SiO₂ and Al₂O₃) and 5-photon absorption (MgO)
respectively. In SiO₂ for example, a lower limit of $\sigma_{6,min} = 2.7 \ 10^{-74} \pm 0.9 \ s^{-1} (\mathrm{cm}^2 \mathrm{W}^{-1})^6$ [32] and an upper limit of $\sigma_{6,max} = 9.6 \ 10^{-70} \ s^{-1} (\mathrm{cm}^2 \mathrm{W}^{-1})^6$ [33] have been reported in literature highlighting the difficulty in its precise determination. The large imprecision is due to the fact that this parameter is fitted among many others unknown in models used to correctly reproduce experimental observables such as the damage/ablation threshold. Considering this, the fact that we correctly reproduce the magnitude of the phase shift and absorption measured for different pulse duration gives good confidence in the value that we obtain.

Next, an important parameter describing the interaction is the collision time used in the Drude model. In SiO₂, collision times in the 0.5-1.5 fs [24,34,35] range have been reported previously for similar experimental conditions for a pulse duration of 100 fs in SiO₂. Our value of 0.15 fs falls on the low end of this interval but this could be explained by the fact that this is a phenomenological parameter, valid for all pulse durations, and therefore valid for low excitation density level where electron-phonon collisions with a characteristic time of ~0.1 fs [36] dominate. Similar results are obtained in the case of MgO where comparison with literature is more difficult due to lack of studies. In Al₂O₃, a very low characteristic collision time is reported (0.03 fs), similar to one reported in our previous study [19].

Last, we comment on the value of linear absorption cross-sections. In literature, they vary from $10^{-19} \mathrm{cm}^2$ [36,37] to $10^{-16}-10^{-17} \mathrm{cm}^2$ [10] depending on whether indirect or direct transitions are considered in the conduction band. Our findings lie within the range of reported values.

In the present work we consider optical breakdown as any visible permanent modification of the surface. It has been shown, that two types of surface modification can be distinguished: ablation – i.e., formation of craters with depths in the range of hundreds of nm, or much more shallow craters or damage, which can be observed usually for longer pulse, for instance 400 fs and above [8]. If a more sensitive method of surface characterization is applied, this different threshold could even be observed for 30 fs pulses [7]. The intensity difference between these two thresholds is on the order of 60% [8,38]. Using such increased threshold would dramatically increase the corresponding excitation density and deposited energy obtained from the model.

Concerning the mechanism leading to optical breakdown, preexisting defect or impurities play a role that strongly depends on the pulse duration. In the conclusion of their study of the role of defects in laser damaging of fused silica and silica coatings, Laurence et al [39] estimate that linear absorption by impurities is the initial excitation mechanism, and that defect and surface inhomogeneities play a role in the damage onset, even for the shortest pulses used in their experiments, i.e., 1 ps.

Optical glasses are fragile materials, and numerous defects like cracks, scratches, pits can be mechanically introduced by polishing. It has been shown recently that point defects, are associated with such imperfections [40]. These point defects, like for instance E' centers, have energy levels in the band gap, increasing the laser absorption efficiency, thus sharply decreasing the laser-induced damage threshold.

We could also observe the influence of such mechanically induced defect as presented in Fig. 7. In this figure, a single 5-ps laser pulse at a fluence of 9 J/cm² produced a high density of nanometric pits. What is striking is the alignment of these pits on a defect line probably originating from the polishing process. In our opinion, this indicates that extrinsic properties of the material surface accounting for different types of defects should be considered if we wish to develop a universal laser damage model. In the present form of our model, these defect states are not accounted for. It could be possible to include the influence of preexisting absorption center in the model, for example by adding a certain density of point defects that can linearly absorb light. However, this would add another parameter to the model, and we have preferred to keep it simple, as the multiplication of adjustable parameters would reduce its applicability.

Finally, we would like to discuss the limitations of our model and ways to improve it, at the expense of complexity. First, we consider only multiphoton absorption as the sole mechanism
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Damage observed on SiO₂ with a single 5-ps laser pulse at 9 J/cm².</figcaption></figure>

for photoionization of valence electrons. Although, this holds true for moderate intensities (and therefore longer pulses) it may fail at high intensities (~10¹³ W/cm² and therefore the shortest pulses) where tunneling ionization may have a non-negligible contribution. A Keldysh photoionization rate [41] could provide a more complete description of the photoionization in that case. Next, the Drude model is used to describe the time-varying dielectric function. Its applicability can be questioned when an important fraction of the valence band electrons is photoionized, distorting the band structure of the solid. This has an immediate effect on the effective electron mass that accounts for the band structure and the effective bandgap energy determining the multiphoton absorption order. Next, the validity of the approximation of a constant collision characteristic time can be questioned when different pulse durations are considered. An effective collision characteristic time accounts for electron-phonon, electron-electron, electron-ion and electron-neutral collisions that depend on the excitation density and the electron temperature. An improved model could consider the harmonic mean between these different components [7]. Last, a criterion based on mechanically driven damage (by laser-induced stress waves) has been suggested [42]. A recently proposed multiscale-multiphysics model [43] could be used to account for such modifications.

**5. Conclusion**

To conclude, we have shown that time-resolved experimental investigations bring valuable information about the physical mechanisms occurring during optical breakdown of wide-band gap dielectrics. For instance, the excitation density needed to reach optical breakdown decays significantly as the pulse duration varies from 100 fs to 1.6 ps. The critical density is almost never reached for pulses down to 100 fs. Thus, it is possible to achieve a given optical modification
under significantly different experimental conditions. This has already been demonstrated by the use of a double pulse sequence [44]. By tuning the pulse duration, the plasma density can be adjusted. This adjustment can reduce propagation effects due to plasma screening for instance, or due to nonlinear propagation associated with the high peak power of the shortest pulses. Such effects have been observed; for example, the optimum pulse duration was found to be 650 fs for the inscription of gratings in the bulk of SiO₂ [45]. Finally, calculations of deposited energy density demonstrate that a thermal criterion is only valid for the shortest pulse duration (100 fs). Damage morphology for longer pulses indicates that for longer pulses defect contributions should be accounted for along with intrinsic processes leading to laser damage. Any model dealing with laser damage in a broad pulse duration range should include the influence of such defects, as it is clear that purely intrinsic laser-induced breakdown occurs only for the shortest pulses, not exceeding a few hundred of fs.

**Funding.** The project received funding from the Agence Nationale de la Recherche (ANR-19-CE30-015-01) TOCYDYS project.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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