# Pulse energy scaling of femtosecond laser-generated stress waves inside fused silica

Olga Koritsoglou, Guillaume Duchateau, Olivier Utéza, Alexandros Mouskeftaras

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Olga Koritsoglou, Guillaume Duchateau, Olivier Utéza, Alexandros Mouskeftaras. Pulse energy scaling of femtosecond laser-generated stress waves inside fused silica. Optics Letters, 2025, 50 (19), pp.6113. 10.1364/OL.568702. hal-05505025

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To be published in *Optics Letters*:

**Title:** Pulse energy scaling of femtosecond laser-generated stress waves inside fused silica

**Authors:** Olga Koritsoglou,Guillaume Duchateau,Olivier Utéza,Alexandros Mouskeftaras

**Accepted:** 31 August 25

**Posted** 02 September 25

**DOI:** https://doi.org/10.1364/OL.568702

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further
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# Pulse energy scaling of femtosecond laser-generated stress waves inside fused silica

OLGA KORITSOGLOU,<fnref n="1" /> GUILLAUME DUCHATEAU,<fnref n="2" /> OLIVIER UTÉZA,<fnref n="1" /> ALEXANDROS MOUSKEFTARAS<fnref n="1" />,*

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

Femtosecond laser-generated stress waves play a crucial role in processing transparent materials. However, their dependence on laser pulse energy, a key processing parameter, remains unclear. To investigate this, we employ a pump-probe microscope-polariscope with *in situ* laser intensity imaging, complemented by multiphysics-multiscale numerical simulations. Our study examines single-shot irradiation inside fused silica using 360-fs, 1030-nm laser pulses with energies ranging from 250 nJ to several µJ. We observe a logarithmic scaling of the principal stress difference, governed by peak intensity clamping, in quantitative agreement with simulations.

Optical intensities in excess of TW/cm<fnref n="2" /> are routinely achieved with femtosecond lasers, enabling the ionization of wide-bandgap dielectric materials, such as oxide glasses and crystals (e.g., SiO₂, Al₂O₃), diamond, and other inorganic compounds (e.g., LiF). As a result, tightly focused femtosecond laser beams at sub-bandgap wavelengths can be employed for three-dimensional processing of transparent materials. Furthermore, the delivery of low pulse energies (in the nJ–µJ range) over ultrashort timescales (<1 ps) occurs well before any significant energy transfer into the lattice (>10 ps) [1], leading to unmatched precision and highly localized material modification [2].

Consequently, a wide range of three-dimensional functional micro-devices has been enabled by advanced femtosecond laser processing techniques [3–5]. Although numerous applications have already been demonstrated and some are currently industrialized [6], several challenges continue to hinder the broader adoption of this technology. In particular, a complete understanding of the link between ultrafast microscopic processes and macroscopic mechanical effects is still lacking. For instance, stress waves generated by femtosecond laser pulses can either degrade processing quality [7] or, conversely, enable the formation of unique features such as nanovoids [8]. However, their generation and scaling with respect to key processing parameters—particularly pulse energy—are still poorly understood, limiting the ability to fully harness the potential of ultrafast laser processing.

Pump-probe diagnostics are widely recognized as powerful tools for investigating ultrafast phenomena, including those involved in femtosecond laser interactions with matter [9,10]. Among these techniques, the transient lens method [11] and polariscopic analysis

[12] have been employed as optical probes in soda-lime glass, qualitatively describing the formation, propagation, and scaling of stress waves on the nanosecond timescale. In borosilicate glass, time-resolved interferometry has been used to observe various phenomena, including the generation and propagation of femtosecond laser-induced shockwaves [13]. More recently, ultrafast polarization microscopy (~ps resolution) has been applied to silica glass [14], revealing a correlation between the linear polarization of the pump laser and the ultrafast formation and orientation of anisotropic nanostructures.

One of the main challenges associated with phase-measuring diagnostics is the determination of the probe integration length, which significantly limits the extraction of quantitative information. Another major limitation stems from the fact that few experimental studies are supported by physical models and validated through numerical simulations, thereby hindering the development of robust predictive tools that integrate both modeling and experimental validation. To address these issues, we previously demonstrated the relevance of a pump-probe microscope polariscope in a perpendicular configuration for measuring time-resolved transmission and birefringence and characterize femtosecond laser-induced stress waves [15]. In parallel, we developed a multiscale, multiphysics model to support the interpretation of our experimental observations. This combined experimental-modeling approach enabled us to quantitatively determine the temporal evolution of key local material properties—namely stress, density, and temperature—at a fixed pulse energy in fused silica corresponding to approximately five times the damage threshold.

In this letter, we investigate the energy scaling of femtosecond laser-induced stress waves within the bulk of fused silica—an essential material in optical manufacturing. We begin by measuring transient probe transmission and stress-induced birefringence as a function of the incident laser pulse energy. To support our analysis, we perform in situ imaging of the pump laser fluence and provide a quantitative assessment of the optical intensity following calibration. Finally, we employ a multiscale, multiphysics model, validated against experimental results across a broad parameter space, to interpret the observed phenomena and deliver a predictive framework for stress wave generation in femtosecond laser-matter interactions.

The experimental setup used for time-resolved optical transmission and birefringence measurements has been described in detail in [15]; here, we summarize its main features. A diode-pumped Yb-doped fiber laser (Satsuma, Amplitude Laser) delivers pulses with a

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

<footnote n="2">²CEA-CESTA, 15 Avenue des Sablières, CS60001, 33116 Le Barp Cedex, France</footnote>

<footnote>*alexandros.mouskeftaras@univ-amu.fr</footnote>
maximum energy of 20 µJ, a duration of 360 fs, and a central wavelength of 1030 nm. The laser beam is split into pump and probe paths using a plate beamsplitter. The pump beam energy is adjusted using a half-wave plate and a polarizing beamsplitter, then focused at 600 µm below the surface of a fused silica sample with 10 mm sides by overfilling an aberration-corrected microscope objective (NA = 0.65). The energy of the pump pulse is varied within an interval of 250 nJ to 4 µJ. The reflected probe beam is frequency-doubled to 515 nm using a BBO crystal, and used in a Köhler-type perpendicular illumination setup to probe the irradiated volume. Imaging is performed via a 50× Mitutoyo objective (NA = 0.42) and a CMOS camera. The zero-time delay has been arbitrarily chosen as the beginning of the transmission drop due to the arrival of the pump pulse. A motorized delay line adjusts the probe delay over 5.5 ns, allowing time-resolved imaging with single-shot acquisitions at different delays while the sample is translated in the transverse plane to avoid overlap with modified surface. The system achieves femtosecond temporal and sub-micrometer (~0.6 µm) spatial resolution as defined by the Abbe criterion.

Optical transmission and birefringence measurements are performed at fixed pump-probe delays of 500 fs and 4 ns, respectively. The delay for the transmission measurements corresponds to the minimum in probe transmission observed at 1 µJ, associated with free carrier plasma formation. For birefringence measurements, the delay is selected such that the stress wave has propagated away from the focal region (over ~20 µm), allowing for clear spatial resolution. Using the photoelastic law and the procedure outlined in [15], we extract the principal stress difference as a function of pulse energy, as shown in Fig. 1(a). Considering the sensitivity and signal-to-noise ratio of our experimental setup, we were able to detect stress starting from a laser pulse energy threshold of roughly 250 nJ, corresponding to a measured value of approximately 75 MPa. A logarithmic increase in stress is observed reaching a maximum of 160 MPa at 4 µJ.

A limiting mechanism appears to prevent further buildup of stress at higher pulse energies. To investigate this behavior, we employ the previously developed multiscale model [15] to perform numerical simulations that reproduce the experimental stress evolution. The main features of the model are briefly summarized below.

The modeling describes consistently the main physical processes leading to the generation of stress waves. First, both electron and laser propagation dynamics are coupled with the Drude model to estimate the laser energy deposition into the bulk of fused silica. Electron dynamics is described with a rate equation accounting for photo- and impact ionization, and recombination. For model tractability, we assume a multiphoton dependence for the photoionization process. The laser propagation is described within the paraxial assumption. The obtained energy deposition (integrated along the propagation direction) is used as a source term for the one-dimensional axi-symmetric Esther hydrodynamic code [15] which also includes the material mechanical response. There are only two parameters which are adjusted to account for the experimentally observed evolution of the stress on the whole range of laser energy variation [0.25 - 4 µJ]: the multiphoton cross section $σ_8 = 10^{-262} m^{16}s^{-7}$ and the collision time $τ = 2$ fs. The obtained result shown in Fig. 1(a) exhibits a good agreement with the experimental behavior (except around 2 µJ which should be due to experimental uncertainties), being able to account for the scaling law. The stress

evolution is a consequence of the maximum absorbed energy densities which is shown in Fig. 1(b). For not too high amplitudes pertaining to linear regime, the stress is proportional to the temperature which is itself proportional to the absorbed energy density.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** (a) Comparison of experimental and simulated principal stress differences for a pump-prove delay of 4 ns, and (b) calculated maximum absorbed energy densities as a function of laser pulse energy. A logarithmic fit can be applied for the simulated stress in (a) yielding a coefficient of determination $R^2=0.96$.</figcaption></figure>

Although increasing the laser pulse energy may seem intuitive to increase stress, both experimental and modeling observations show that this does not lead to a proportional increase in the amplitude of the generated stress wave. This behavior can be explained by the so-called "lawn mower" model, which describes laser beam propagation in dielectrics. According to this model, nonlinear absorption mechanisms clamp the maximum intensity during propagation [16]. A similar conclusion is drawn from more recent theoretical investigations [17].

To assess the influence of this clamping effect within our experimental conditions, we imaged the laser fluence distribution using the method proposed by Xie et al. [18], allowing for a quantitative comparison. The experimental setup is shown in Fig. 2(a). A 1030-nm pump beam is focused onto the rear surface of a 0.2-mm-thick fused silica sample within the same conditions as used during stress generation. A high-magnification microscope objective
(100×, NA=0.7) is used to image this rear surface, thereby capturing the transmitted laser fluence at the geometrical focus. To ensure accuracy, the camera is synchronized with the laser to capture single-shot images, and the sample is vertically translated after each pulse energy increase to avoid previously damaged sites. Light intensity reaching the camera is attenuated using neutral density filters, and the camera gain is kept at one throughout the measurements. Careful calibration of the neutral density filters, imaging system transmission, and camera response, combined with normalization by the pulse duration, enables the conversion of the recorded camera signal into physical optical intensity units (W/cm²), as shown in the inset of Fig. 2(a). A constant sech² temporal shape is considered in the calibration process to preserve the simplicity of our approach. The extracted peak optical intensity is plotted in Fig. 2(b).

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Fluence distribution imaging and intensity calculation. (a) Experimental setup with HWP, P, ND and TL half-wave plate, polarizer, neutral density filter and tube lens respectively. (b) Peak experimental and simulated optical intensity.</figcaption></figure>

At low pulse energies, the peak intensity increases linearly up to a threshold of ~0.25 µJ (with a corresponding intensity of ~10¹³ W/cm²). A linear fit within this interval can be applied with a R² of 0.95. Beyond this point, the intensity continues to increase gradually but at a much slower rate, reaching a maximum of approximately 5×10¹³ W/cm² at around 3 µJ. As evident from Fig. 2(b), the measured optical intensity is notably lower than the theoretical vacuum intensity. For instance, at 1 µJ, the expected vacuum intensity is approximately 2×10¹⁴ W/cm², whereas the measured

value is only about 3×10¹³ W/cm², nearly an order of magnitude lower. Interestingly, the same threshold of ~10¹³ W/cm² has been predicted in the “lawn mower” model as the onset for considerable laser absorption [16].

Simulated peak intensities are plotted alongside the experimental data in Fig. 2(b) for comparison. The overall trend is reasonably well reproduced, including the onset of the characteristic "shoulder" feature. However, while the simulated peak intensity remains constant beyond this point, the experimental results show a continued slow increase. This discrepancy is likely due to the absence of nonlinear propagation effects in the simulation. Notably, around 2 µJ, the peak laser power approaches the critical power for self-focusing, which could account for the experimentally observed deviation. More refined quantitative results can be obtained by considering temporal pulse shape distortions induced by nonlinear propagation effects.

To further analyze the clamping observed in the peak intensity and gain additional insight into the underlying mechanisms, we imaged the transient plasma volume to visualize prefocal and propagation effects that may lead to intensity delocalization. The experimentally obtained optical transmission maps are shown in Fig. 3(a) for various pulse energies ranging from 0.25 µJ to 5 µJ, at a fixed pump-probe delay of 500 fs. All other experimental conditions remained similar with those used in the stress measurements.

No transmission drop was detected below approximately 0.17 µJ. At low pulse energies (e.g., 0.25 µJ), the plasma has an ellipsoidal shape, similar to that typically observed in air [19]. As the energy increases, the free-carrier plasma takes on a teardrop-like shape, with its volume extending into the prefocal region and exhibiting regions of nearly zero transmission. Similar plasma morphologies have been reported in previous studies [20] and are responsible for the asymmetrical, teardrop-like absorbed energy density map calculated with this model in our previous study [15]. We believe this prefocal plasma formation is the primary cause of the intensity clamping and the growth trend observed in Fig. 2(b), as it leads to significant beam depletion through free-carrier absorption.

An intuitive approach to assessing the stress (σ) dependence on laser pulse energy is to consider that stress is proportional to the deposited laser energy density, as previously stated. This leads to the relation σ ∝ E<sub>abs</sub>/V<sub>abs</sub>, where E<sub>abs</sub> and V<sub>abs</sub> are the absorbed energy and the absorption volume, respectively. Considering E<sub>abs</sub> = A × E<sub>0</sub>, with A denoting the absorbance and E<sub>0</sub> the incident laser energy, and further assuming that V<sub>abs</sub> scales linearly with E<sub>0</sub>, we can deduce that the stress is proportional to the absorbance. Notably, absorbance measurements under similar experimental conditions exhibit a comparable logarithmic-like scaling behavior [16], consistent with the stress measurements observed in our study. To verify the assumption of linear scaling of V<sub>abs</sub> we measure the plasma volumes, extracted from the transmission maps, as plotted in Fig. 3(b). To quantify the plasma volume, we define the plasma region as the area where transmission falls below 90%, and apply Pappus's centroid theorem to compute the volume. A nearly linear relationship is observed, with plasma volume increasing from approximately 10 µm³ to 600 µm³ across the examined pulse energy range of 0.25 µJ to 4 µJ. For comparison, plasma volumes obtained from numerical simulations (considering carrier densities exceeding 10¹⁸ cm⁻³) are plotted in Fig. 3(b), showing very good agreement with the experimental results.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** (a) Experimentally measured time-resolved transmission microscopy for a pump-probe delay of 500 fs and pump energies of 0.25 µJ – 5 µJ. (b) Volume measurements extracted from experimental data and numerical simulations.</figcaption></figure>

In this study, we have investigated the dependence of femtosecond laser-induced stress wave amplitude on pulse energy within the bulk of fused silica, using time-resolved birefringence measurements, calibrated laser fluence imaging, and numerical simulations. Our results reveal a logarithmic scaling of stress amplitude with laser pulse energy, while the peak intensity initially increases nearly linearly at low pulse energies (around 0.25 µJ), before exhibiting a clamping effect that significantly reduces the intensity compared to its vacuum equivalent.

By imaging the transmitted fluence and measuring transient plasma volumes, we identified prefocal plasma formation as a key contributor to the observed intensity clamping. These findings are well supported by our multiphysics–multiscale simulations, which reproduce both the stress evolution and the saturation-like behavior of the intensity. As a general guideline, a reduction in stress gains is expected for pulse energies exceeding 2 µJ when using a 1030-nm wavelength, 360-fs pulse duration, and a NA = 0.65 microscope objective. Recent studies have highlighted that a universal laser-induced damage criterion based solely on thermodynamic

parameters—such as deposited energy—may be insufficient [21], particularly under ultrafast irradiation conditions. These works emphasize the need to consider additional mechanisms, including mechanically driven damage, to accurately describe material response. In this context, our work provides valuable insight by quantitatively characterizing the generation and scaling of transient stress waves—an aspect often overlooked in traditional damage models.

Overall, our combined experimental and numerical approach provides a quantitative framework for understanding and predicting stress wave generation in femtosecond laser–matter interactions, with implications for optimizing ultrafast laser processing strategies in transparent materials.

**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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