# Assessment of thermal effects in laser micro-machining of graphite with ultrashort pulses at high repetition rate

C.E. Cifuentes Quintal, T. Doualle, Y. Pontillon, J L Rullier, L. Gallais

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C.E. Cifuentes Quintal, T. Doualle, Y. Pontillon, J L Rullier, L. Gallais. Assessment of thermal effects in laser micro-machining of graphite with ultrashort pulses at high repetition rate. Applied physics. A, Materials science & processing, 2025, 131 (7), pp.536. 10.1007/s00339-025-08642-4. hal-05519683

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# Assessment of thermal effects in laser micro-machining of graphite with ultrashort pulses at high repetition rate

C.E. Cifuentes Quintal<fnref n="1" />,<fnref n="2" /> ID • T. Doualle<fnref n="1" /> ID • Y. Pontillon<fnref n="1" /> ID • J.L. Rullier<fnref n="3" /> • L. Gallais<fnref n="2" /> ID

Received: 13 February 2025 / Accepted: 22 May 2025 / Published online: 6 June 2025
© The Author(s) 2025

## Abstract

This work examines the thermal effects in ultrashort pulse laser micromachining of graphite at high repetition rates through a combination of experimental and numerical approaches. While several studies have explored heat accumulation in similar contexts, the integration of real-time thermal measurements with complementary simulations remains limited. Experiments were performed using a 1030 nm, 300 fs laser operating at repetition rates between 0.333 MHz and 2 MHz with varying scanning speeds (1-120 mm/s). Temperature distributions were monitored using a thermal camera to analyze heat accumulation, crater formation, and the heat-affected zone. Two numerical models were developed to study the laser-matter interaction. The two-temperature model resolves the ultrafast electron-lattice dynamics, providing insights into the ablation threshold (0.2 J/cm<fnref n="2" />) and material removal process. The continuous-wave model approximates cumulative heating effects and enables the evaluation of temperature rise and spatial thermal gradients under quasi-continuous conditions. Temperature gradients with characteristic lengths of approximately 50-250 µm were measured and reproduced by simulations. Comparisons between simulations and experimental results show good agreement, validating the ability of models to predict both transient and cumulative thermal behavior. The proposed approach provides a reliable framework for understanding thermal effects in laser micromachining of graphite and can be extended to other materials requiring precise thermal management.

**Keywords** Femtosecond laser micromachining • Heat accumulation • Graphite ablation • Heat-affected zone

## 1 Introduction

Laser micro-machining has become an essential technique in modern material processing, offering high precision, and flexibility for micro-scale applications in industrial and scientific fields. By the use of pulses with sub-picosecond to few picosecond duration the Heat-Affected Zone (HAZ) is minimized which makes them particularly effective for micromachining challenging materials, including heat-sensitive ones such as composite materials, microstructured materials, special alloys, ion-doped semiconductors, organics [1] and hard or ultra-hard materials [2].

The advent of ultrashort pulse laser with high repetition rates (MHz-GHz) and high average power (>100W) has offered new opportunities to enhance material processing efficiency [3]. However, such regimes introduce challenges related to heat accumulation. The absorbed energy is indeed not fully dissipated in the vaporized material, and the residual energy results in heating of the surrounding material. The heat accumulation, that is directly driven by pulse energy, repetition rate and scanning speed (spatial overlap), can lead to elevated surface temperatures that can alter material properties. This cumulative heating can cause localized melting [4], surface roughness modifications [5], and structural changes [6, 7]. Large thermal gradients can also lead to stress resulting in bending for thin metal sheets [8] or cracks in brittle materials, such as in ceramics [9, 10]. These effects become of considerable issue for high repetition rate lasers [11].

To overcome these detrimental effects while maintaining a high process speed, an optimization study is required in order to identify the best process parameters and find the best compromises between heat affected area and productivity. Such

<footnote>C.E. Cifuentes Quintal
Carlos-Esteban.CIFUENTESQUINTAL@cea.fr</footnote>

<footnote n="1">1 CEA, DES, IRESNE, DEC, Cadarache, F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote n="2">2 Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, France</footnote>

<footnote n="3">3 CEA, CESTA, 15 Avenue des Sablières, 33114 Le Barp, France</footnote>
task needs to be conducted for each different material as the heat accumulation depends on material properties (thermal conductivity, specific heat, phase changes, laser absorption and thermal radiation). A conventional approach is based on trial and errors. However more effective approach can be applied, such as based on sensitivity analysis with statistical methods [12], specific optimization algorithms [8] or machine learning algorithms [13, 14] to identify the most significant parameters as well as physical relations between input and output variables of complex laser processes.

Comprehensive approaches based on the study of physical process require to simulate the heat accumulation and comparing thermal simulations and experiments. These are particularly valuable for applications such as drilling, cutting, and surface structuring, where control over thermal effects is essential to achieving high-quality results. For simulations, analytical treatment of multi-pulse processing can reveal the basic mechanisms of heat accumulation and its consequence for the resulting processing quality [11]. To better take into account the complexity of the process, such as the 3-dimensional aspects or the optical and thermal nonlinearities, Finite Elements Methods (FEM) are particularly suitable [15]. For experiments, the use of thermal imaging is very appealing for such investigation as it can be used to directly resolve the thermal gradients, in situ and during the process [7, 16]. Such approach was for instance applied to optimize micro-machining of technical ceramics such as alumina [17], drilling sequences on metal sheets [7, 8]. Some limitations of this method can be the limited optical resolution on the sample (as thermal cameras operate in the mid infrared) or the difficulty in assessing the emissivity and its evolution with temperature.

While previous studies have investigated thermal effects in ultrashort pulse laser micromachining, especially at high repetition rates, the integration of real-time thermal diagnostics with complementary modeling approaches, such as the Two-Temperature Model and classical heat conduction, remains relatively underexplored. Recognizing that thermal control in the processed zone plays a key role in machining quality [18], this work seeks to contribute to this area by proposing a combined experimental-numerical framework to better understand both transient and cumulative thermal responses.

In this context our motivations lie in developing a methodology to assess the thermal effects in ultrashort pulsed laser micro-machining at high repetition rates of heat sensitive samples, with the perspective of applying laser processing to nuclear materials. By combining numerical simulations and experimental approaches, our research aims to analyze thermal dynamics and heat-affected zones, in order to guide the development of machining strategies that mitigate heat-induced effects. As a case study we have developed our methodologies on graphite: indeed in the one hand this material is used in large number of scientific and industrial

applications, and lasers are used to process it under various regimes, including long-pulse laser cutting [19], nanosecond trench ablation [20], surface structuring with ultraviolet excimer lasers [21], application to lithium-ions batteries being one typical application [22, 23], and in the other hand it has well documented properties up to high temperature [24]. Meanwhile, mechanical micro-machining has also proven to be an effective method for shaping graphite [25]. However, laser-based approaches have advantages in terms of precision, flexibility, and the ability to process without contact.

Our experiments are conducted at 1030 nm wavelength and 300 fs pulses with repetition rates up to 2 MHz, with a high resolution thermal camera coupled to a laser scanning head which provides details on the dynamics of thermal gradients. Such configuration allows for the characterization of heat accumulation, crater formation, and the heat-affected zone under different machining conditions. The experimental results are confronted with simulations that are based on the application with the FEM of the Two-Temperature Model (TTM) to capture ablation dynamics, associated to the Continuous-Wave Model (CWM) to evaluate cumulative heating effects. These models provide a detailed understanding of thermal behavior at frequencies around 1 MHz, where the short intervals between pulses limit cooling.

# 2 Experiments

## 2.1 Laser processing

For the experiments, an ytterbium femtosecond fiber laser (Satsuma HP2, 20 W/60 μJ, Amplitude) operating at a central wavelength of 1030 nm was employed. The laser was operated at repetition rates ranging from 0.333 MHz to 2 MHz, with a full-width at half-maximum (FWHM) pulse duration of 300 fs. The laser beam, collimated and directed into a 2D scanning head (ScanCube IV 14, ScanLab), was focused through an F-theta lens with a focal length of 100 mm.

The beam quality factor, $M^2$, was determined to be 1.30 before the scanning head using a 150 mm focal length lens and a beam profiler (BPRO-LP6.3, Femtoeasy). The beam caustic was analyzed after the F-theta lens to find the beam radius at the working plane. These measurements yielded a Rayleigh length, $Z_R$, of 420 µm and a waist (radius at $1/e^2$) of 13.5 µm at the working plane. A dedicated laser machining software, LaserDesk from ScanLab, was used to program the scanning head trajectories.

The scanning speed during the experiments varied between 1 mm/s and 120 mm/s to match typical values used in industrial laser processing applications ensuring sufficient temporal resolution with the thermal imaging system.
Before starting the experiments, the laser power was calibrated after the focal plane using a power meter (12A-SH-V1-RoHS, Ophir). Finally, all experiments were conducted in ambient air.

## 2.2 Thermal measurements

An infrared thermographic camera (ImageIR 9450 hp, InfraTec) was used to measure the sample temperature in situ. The camera operates within the spectral range of 1.5-5.5 µm and was calibrated in the temperature range of 20-2500°C, with a measurement accuracy of ±2% from the manufacturer. The camera was equipped with a macro lens (Magnification 1.0x) and had a working distance of 200 mm. The detector has a resolution of 1280 × 1024 pixels, with a pixel pitch of 10 µm, and the optical resolution of the lens is defined by a point spread function (PSF) of 18 µm at full width at half maximum (FWHM). The camera offers a frame rate of up to 180 Hz at full resolution.

The sample was observed through a gold mirror (PF20-03-M03 Ø2", Thorlabs), as described in Fig.1, which exhibited a nearly constant reflectance of approximately 98% within the camera's spectral range at a 45° angle of incidence. Observing the sample through the mirror restricts the depth of field on the sample due to the angle of incidence, and introduces also optical distortions. Therefore, temperature measurements were conducted over a smaller area to ensure accurate data acquisition within the focused region.

## 2.3 Samples

We have worked on synthetic graphite, specifically EDM-3 graphite, supplied by POCO Graphite Inc., which is a polycrystalline graphite produced through sintering. This material is distinguished by its ultrafine microstructure, with an average grain size smaller than 5 µm. It has a porosity of about 20%, and it is macroscopically homogeneous with a

high degree of isotropy [26]. The samples selected for this study had thicknesses ranging from 0.5 to 2 mm and diameters ranging from 2 to 20 mm.

The thermal properties of this material are well-documented up to very high temperature, the details of the thermal dependencies of conductivity, density and specific heat up to 3500°C can be found for instance in Ref.[26]. At ambient pressure graphite has a sublimation temperature that is in the range 3600-3750°C [27], meaning that very high temperature gradients can be reached during the laser process. As a result, the material transitions directly from solid to vapor without forming a liquid phase under ambient pressure, thereby avoiding the formation of a melted zone.

## 2.4 Crater characterization

Craters were measured on a Contour GT profilometer (Brucker). Surfaces of 500 x 500 µm² were scanned and some profiles crossing the maximum depths of the craters were extracted. The spatial accuracy of measurement is < 20µm and its vertical accuracy is < 2µm. An example of such analysis is shown in Fig.2 where the craters result from a repetition of hundred laser shots on a graphite sample. A different fluence was applied for each crater line made. Left part is the 3D-map of the measured area including guiding marks (dark blue), and right part shows the profile corresponding to the red line. From this measurement, we characterize each crater by its maximum depth $D_{max}$, defined with respect to the initial surface of the target, its diameter Ø, corresponding to the mean diameter of subsurface spall, and the volume V.

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Schematic representation of the experimental setup used for femtosecond laser micro-machining of graphite</figcaption></figure>

## 3 Numerical model

Heat accumulation in pulsed laser micro-machining can be studied with analytical models to describe thermal behavior during repetitive laser-material interactions and identify the main parameters that drive the effects [6, 11]. However,

<figure><img src="image_3.png" /><figcaption>**Fig. 2** 3D-map of the optical profilometer measurement (left), each horizontal line of craters corresponds to a different laser fluence at low repetition rate: from top to bottom, 4.5, 7.1, 9.5, and 11.4 J/cm². Horizontal extraction (along the red line) of depth profile (right)</figcaption></figure>
these models are often constrained by simplifying assumptions, such as constant thermal properties, that do not capture all the complexities of real-world machining scenarios. By adopting a more advanced numerical approach, it becomes possible to analyze complex configurations, including varying laser powers, scanning speeds, or material geometries, which are essential for optimizing precision in laser micromachining. In this work, two models were developed using the finite element method (COMSOL Multiphysics software) as complementary approaches to address distinct aspects of the laser-material interaction. The Two-Temperature Model (TTM) is employed to simulate laser-material interactions at the time scale of the ultrafast laser pulse, capturing the distinct thermal dynamics of the electron and lattice subsystems. In femtosecond laser ablation, rapid energy transfer from electrons to the lattice dictates the ablation process [28]. The TTM can be used to study the material removal rate and ablation depth under varying laser parameters, providing a foundation for optimizing micro-machining processes [29]. In laser machining at MHz repetition rates, the quasi-continuous delivery of energy drives the thermal response of the material outside the ablated zone. At such high frequencies, the behavior of the laser-material interaction can be approximated as continuous rather than discrete, so that a Continuous Wave Model (CWM) can be used. This transition, where high repetition rates effectively emulate steady-state heating conditions, allows for efficient modeling and prediction of thermal effects using a continuous-wave approximation [7, 15]. The use of these 2 models are motivated by the need to understand both ultrafast, localized thermal phenomena and longer-term, cumulative effects, while optimizing numerical resources for numerical simulations.

## 3.1 Two-Temperature Model for pulsed laser ablation

The governing equations of the TTM describe the evolution of the electronic temperature ($T_e$) and the lattice temperature ($T_L$) in a material irradiated by a femtosecond laser pulse. The absorbed energy is initially deposited into the electronic subsystem, which subsequently transfers heat to the lattice via electron-phonon coupling.

$$ 
\begin{aligned}
C_e \frac{\partial T_e}{\partial t} &= \nabla^2(k_e T_e) - G_{e-ph}(T_e - T_L) \\
C_L \frac{\partial T_L}{\partial t} &= \nabla^2(k_L T_L) + G_{e-ph}(T_e - T_L)
\end{aligned}
\quad (1) $$

Here, $T_e$ and $T_L$ represent the electron and lattice temperatures, respectively, while $C_e$ and $C_L$ denote their corresponding heat capacities ($Jm^{-3}K^{-1}$). Similarly, $k_e$ and $k_L$ are the thermal conductivities ($Wm^{-1}K^{-1}$). The term $G_{e-ph}$

($Wm^{-3}K^{-1}$) describes the coupling between the two subsystems.

Graphite is a highly absorbing material at the laser wavelength and the laser pulse energy is absorbed at the material's surface. Therefore, the heat source is treated as a localized boundary condition, boundary (2) in Fig. 3, modeled as an incoming heat flux at the surface. The surface heat flux is described by:

$$ S_{pulse}(r, t) = A \sqrt{\frac{4\ln(2)}{\pi \tau_p^2} \frac{2E_p}{\pi w_0^2} e^{-4\ln(2) \frac{r^2}{\tau_p^2} - 2 \frac{r^2}{w_0^2}} \quad (2) $$

where A is the absorption coefficient, $\tau_p$ is the pulse width (full width at half maximum), $E_p$ the pulse energy and $w_0$ is the beam waist (radius at $1/e^2$).

To model crater formation from the temperature rise predicted by the TTM, thermal flux loss due to ablation, $q_a$, is considered. Ablation occurs when the lattice temperature exceeds graphite's sublimation threshold, leading to material removal from the surface. Assuming that the total amount of ablated material is removed, the surface temperature remains constant just below the phase change temperature. This flux is modeled at boundary (2), following [30, 31] as a temperature-dependent heat flux activated when the lattice temperature surpasses the sublimation temperature ($T_{sub}$):

$$ q_a = h_a (T_L - T_{sub}) \quad (3) $$

The coefficient $h_a$, characterizes the rate of heat loss ($Wm^{-2}K^{-1}$) from the ablation front. A high value is applied when the lattice temperature exceeds the sublimation threshold, emulating the rapid energy dissipation associated with material ejection. This approach aligns with established methodologies in laser ablation modeling [30, 31]. In this

<figure><img src="image_4.png" /><figcaption>Fig. 3 Computational mesh and boundary conditions used in the TTM simulations for femtosecond laser ablation of graphite. The mesh is refined near the surface to capture steep thermal gradients induced by laser pulses. The boundary conditions are numerically labeled</figcaption></figure>
model, it is set to:

$$h_a = \begin{cases} 0 & \text{if } T_L \le T_{sub}, \\ 10^{10} Wm^{-2}K^{-1} & \text{if } T_L > 0. \end{cases} \quad (4)$$

The material removal rate is then calculated based on the velocity ($v_a$) of the surface front:

$$v_a = \frac{q_a}{\rho H_{sub}} \quad (5)$$

where $\rho$ the material density and $H_{sub}$ latent heat of sublimation.

Radiative heat loss ($-q$) at boundaries 2 and 3, is given by:

$$-q = \epsilon\sigma(T_{amb}^4 - T_L^4) \quad (6)$$

where $\epsilon$ is the emissivity of material, $\sigma$ is the Stefan-Boltzmann constant $T_{amb}$ is the ambient temperature, and $T_L$ is the surface lattice temperature.

Boundary (1) is defined as an axis of symmetry, as the simulations are done in an axisymmetric configuration. And, at Boundary (4), thermal isolation is imposed.

The system is initialized in thermal equilibrium, with the electron and lattice temperatures set equal to the ambient temperature:

$$T_e(t=0) = T_L(t=0) = T_{amb} = 20^\circ C \quad (7)$$

The computational mesh and boundary conditions are illustrated in Fig.3. The mesh is designed to accurately resolve both the ultrafast and long-term thermal phenomena occurring during laser-material interactions. A region extending 50 $\mu$m in depth (z-axis) from the material surface is meshed with a resolution refined according to the expected temperature gradients, enabling precise modeling of the electron-lattice energy transfer dynamics. Beyond this region, the mesh transitions to coarser elements, with sizes on the order of 100 $\mu$m to 1 mm, representing the bulk material and accounting for long-distance diffusion. The computational domain size and mesh resolution were chosen to match the dimensions of real sample, ensuring accurate representation of both localized and global thermal behavior. The values of laser parameters and material properties are given at Table 1.

Figure 4 presents the electron ($T_e$) and lattice ($T_L$) temperature profiles computed using the TTM for a fluence of 0.26 $J/cm^2$. Following the ultra-rapid energy deposition by the laser pulse, the electron temperature rises sharply, transferring energy to the lattice, which responds more gradually. As the lattice temperature reaches the sublimation threshold, it stabilizes, reflecting the onset of material ablation.

<table><caption>**Table 1** Summary of material properties and laser parameters used in the TTM simulations</caption><thead><tr><th>Property or Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Electron heat capacity, C<sub>e</sub></td><td>from [32]</td></tr><tr><td>Lattice heat capacity, C<sub>L</sub></td><td>from [24]</td></tr><tr><td>Electron thermal conductivity, k<sub>e</sub></td><td>from [32]</td></tr><tr><td>Lattice thermal conductivity, k<sub>L</sub></td><td>from [24]</td></tr><tr><td>Electron-phonon coupling constant, G<sub>e-ph</sub></td><td>from [33]</td></tr><tr><td>Absorption coefficient, A</td><td>91.5% ± 0.5% [24]</td></tr><tr><td>Pulse width (FHMW), τ<sub>p</sub></td><td>300 fs</td></tr><tr><td>Waist at 1/e<sup>2</sup>, w<sub>0</sub></td><td>13.5 μm</td></tr><tr><td>Sublimation temperature, T<sub>sub</sub></td><td>3600°C [27]</td></tr><tr><td>Latent heat of sublimation, U<sub>sub</sub></td><td>7.18x10<sup>5</sup> J/mol [34]</td></tr><tr><td>Surface emissivity, ε</td><td>0.9 [24]</td></tr></tbody></table>

## 3.2 Continuous-wave model for laser heating

The continuous-wave model is introduced to capture the quasi-continuous behavior of the high repetition rate laser processing, treating the laser as a constant heat source that moves across the material surface at a defined velocity. This approach simplifies the analysis of temperature fields while accounting for cumulative thermal behavior, enabling predictions of temperature rise, heat diffusion, and the extent of the heat-affected zone. Such modeling has proven valuable in understanding how scanning velocities, laser power, and material properties influence thermal gradients during machining [8]. The governing heat equation is expressed as:

$$C_p \rho \frac{\partial T}{\partial t} - k \nabla^2 (T) = 0 \quad (8)$$

where $\rho$ is the material density, $C_p$ is the specific heat capacity, and $k$ is the thermal conductivity [24].

<figure><img src="image_5.png" /><figcaption>**Fig. 4** Calculated electron ($T_e$) and lattice ($T_L$) temperature profiles using the TTM at 0.26 $J/cm^2$. $T_L$ stabilizes and remains constant once it reaches the sublimation temperature of graphite</figcaption></figure>
Similar to the laser source in the TTM in Section 3.1, the laser power in the CWM is assumed to be absorbed at the surface of material. The surface heat flux is modeled as a gaussian spatial distribution that moves at a constant velocity, and is described by:

$$S_{CW}(x, y, t) = A \frac{2P}{\pi w_0^2} \exp \left( -2 \frac{(x - v_x t)^2 + y^2}{w_0^2} \right) \quad (9)$$

where A denotes the absorption coefficient, $w_0$ the laser beam waist (radius at $1/e^2$), P the average power and $v_x$ the laser scanning velocity along the x-axis. The average power is defined as $P = E_p \cdot F_{rep}$ with $E_p$ is the pulse energy and $F_{rep}$ is the laser repetition rate.

The model was developed in 3D, but the boundary conditions and mesh used for the CWM remain equivalent to those described in Section 3.1. An example of the mesh is illustrated in Fig.5, which also depicts the laser beam moving along the surface of the sample, this contrasts with the TTM where the laser beam remains stationary.

# 4 Results and discussion

## 4.1 Ablation characteristics

To explore the TTM, we conducted experiments designed to study the ablation dynamics under femtosecond laser radiation. Two sets of experiments were performed: one to investigate the relationship between the number of pulses and the crater depth, and another to analyze the variation of ablation rate with fluence.

For the first study (Fig. 6), a predefined number of laser pulses was delivered to the same target location at a low repetition rate (<1 Hz) to prevent heat accumulation between successive shots. The results demonstrate a linear relation-

<figure><img src="image_6.png" /><figcaption>Fig. 5 Example of the 3D mesh used for the Continuous-Wave Model (CWM). It illustrates the laser beam moving along the surface of the sample and the applied boundary conditions</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6 Crater depth and their linear fit as a function of the number of laser pulses at 2.44 J/cm² and 4.89 J/cm²</figcaption></figure>

ship between the number of pulses and ablation depth for both fluence.

For the second study, the fluence was varied over a wider range while maintaining a fixed number of 100 pulses per site. The resulting ablation rates, including depth per pulse and volume per pulse, are presented in Fig.7. Moreover, the corresponding calculations performed using the TTM are included for comparison. Finally, the experimental results are compared with the empirical logarithmic relationship that describes the crater depth (D) as a function of fluence:

$$D = \delta \ln \left( \frac{F}{F_{th}} \right) \quad \text{if } F > F_{th} \quad (10)$$

where $\delta$ is the penetration depth and $F_{th}$ is the ablation threshold. Based on the measurements, these parameters were determined as $\delta = 0.047 \ \mu m$ and $F_{th} = 0.25 \ J/cm^2 \pm 0.03 \ J/cm^2$. Furthermore, the ablation threshold fluence calculated using the TTM is 0.20 $J/cm^2$.

<figure><img src="image_8.png" /><figcaption>Fig. 7 Variation of the ablation rate (depth and volume) as a function of fluence, along with the logarithmic fit of the depth ablation rate. The different fluences calculated by the TTM are also shown</figcaption></figure>
The measured surface roughness of the samples (RMS=0.9 µm) plays a significant role in this determination, as the first fluence value tested, $F = 0.22 J/cm^2$, did not produce a discernible crater according to profilometer measurements. However, given the roughness, it is possible that ablation depths were undetectable. Despite this limitation, the result aligns well with the calculated threshold fluence.

The ablation thresholds determined in this work are consistent with values reported in the literature for graphite under ultrashort laser pulses. For instance, previous studies on high-oriented pyrolytic graphite (HOPG) have reported thresholds ranging from 0.14 $J/cm^2$ (30 ps, 1060 nm) [35] to 0.25 $J/cm^2$ (825 nm) [36]. Other works have shown comparable values for graphite films 0.18-0.25 $J/cm^2$ [37], dry graphite 0.2 $J/cm^2$ (744 nm, 70 fs) [38] and graphite electrodes 0.107 $J/cm^2$ (1030 nm, 530 fs @ 100 kHz for 5 pulses) [39]. The slight variations can be attributed to differences in experimental setups, material properties, and laser parameters, including wavelength, pulse duration, and repetition rate.

However, deviations between experimental and simulated results become apparent at higher fluences. For instance, as shown in Fig. 7, the ablation rate deviates from the predicted logarithmic trend. Similar behavior has been reported in the literature, where deviations in ablation rate are observed under comparable conditions [36].

While the TTM provides reliable predictions at moderate fluences, several limitations must be acknowledged to explain the discrepancies at higher fluences. The model considers only thermal interactions and neglects non-thermal phenomena such as plasma shielding, dynamic variations in reflectivity and absorption, Coulomb explosion, multiphoton ionization, and electron-driven effects, which become increasingly significant at higher fluence levels. Moreover, simplifications such as constant optical and the omission of hydrodynamic motion, limit the predictive accuracy in regimes involving significant phase changes or intense energy deposition. These effects alter the laser-matter interaction and thereby influence the ablation efficiency.

To better capture these effects, future model improvements could incorporate fluence-dependent optical properties, for instance through Drude-based models accounting for free-carrier density evolution [40, 41]. Moreover, the inclusion of non-thermal mechanisms affecting energy coupling [42], could further enhance predictive accuracy under extreme irradiation conditions. Although the current model focuses intentionally on dominant thermal mechanisms for simplicity, extending the framework to account for these effects represents a promising direction for future developments.

The TTM provides reliable predictions for ablation depth and rate under femtosecond laser irradiation at moderate fluences, as demonstrated by its good agreement with experimental data. However, the observed deviations at higher fluences highlight the need for further refinements, partic-

ularly by considering more complex phenomena such as the fluence-dependent effects that may influence the ablation dynamics. The experimental threshold fluence obtained aligns well with literature values, reinforcing the accuracy of approach and its applicability to ultrashort pulse laser micro-machining of graphite.

## 4.2 Temperature distribution and heat-affected zone analysis at low fluence

In order to compare experiments and simulations for the CWM, measurements of surface temperature were conducted using the thermal camera at low fluence, meaning in a regime that do not lead to material ablation. In such a way the model's accuracy in predicting thermal behavior under different laser operating conditions can be assessed. The experiments were designed to capture the temperature dynamics in various scenarios.

As detailed in Appendix A, the numerical temperature distributions from the CWM were convolved with a Gaussian function representing the PSF of the lens system. The optical resolution of the system, is defined by the FWHM of the PSF, 18 µm. This resolution is comparable to the spatial width of the expected temperature gradients near the focal plane, which are on the order of 100 µm. Consequently, it may potentially spread and smooth the measured temperature gradients, reducing the ability to resolve fine thermal features accurately.

### 4.2.1 Stationary beam

The stationary beam case examines the thermal response of the material under a static laser spot. This scenario allows for direct evaluation of the temperature distribution and heat accumulation over time. Experiments were conducted by heating the material for 3.2 s at two powers below the ablation threshold: 180 mW (5.6 $mJ/cm^2$ @ 2 MHz) and 570 mW (18 $mJ/cm^2$ @ 2 MHz). To achieve such low fluence levels, the laser beam was deliberately defocused, resulting in waist radius ($1/e^2$) of 32 µm. These conditions were selected to avoid ablation while generating measurable thermal gradients over a spatial range of a few hundred micrometers.

The thermal camera was operating in a temperature range of 30°C to 150°C with an integration time of 130 µs and a frame rate of 1240 frames per second (FPS).

Figure 8 shows a comparison between the maximum surface temperature recorded by the thermal camera and the simulation results from the CWM under identical conditions. The close agreement demonstrates the accuracy of the model in capturing heat accumulation. Furthermore, Fig. 9 illustrates the radial temperature profile extracted from the experimental data, showcasing the lateral heat diffusion from the center of the laser spot.
<figure><img src="image_9.png" /><figcaption>**Fig. 8** Comparison of the maximum surface temperature measured by the thermal camera, simulated by the CWM, and the temperature obtained by convolving the CWM results with the camera's point spread function. The experiment was conducted with a static beam at 2 MHz using two average powers (180 mW and 570 mW) and a beam radius of 32 µm</figcaption></figure>

Finally, additional tests were conducted (Fig. 10) to examine the relationship between average power and maximum temperature at t= 3.2 s, incorporating both experimental measurements and simulations. This analysis highlights the role of average power resulting from different fluences in generating thermal gradients. The results demonstrate a clear linear relationship between average power and maximum temperature, underscoring the proportionality between the input energy and the heat accumulated within the material. These results validate the ability of CWM to correctly evaluate thermal effects in the static scenario at low fluence.

<figure><img src="image_10.png" /><figcaption>**Fig. 9** Radial temperature profile extracted from experimental measurements, CWM simulations, and the CWM results convolved with the point spread function of the camera lens in the static beam scenario. The experiment was conducted with a static beam at 2 MHz using two average powers (180 mW and 570 mW) and a beam radius of 32 µm</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 10** Relationship between average power (resulting from various fluences and repetition rates) and maximum temperature at t= 3.2 s, showing both experimental measurements and CWM simulations</figcaption></figure>

### 4.2.2 Moving beam

In the moving beam scenario, the laser was scanned across the material surface along the x-axis at a constant velocity ($v_x$= 1 mm/s), with an average power of 200 mW and a fluence of 70 $mJ/cm^2$ at 1MHz. The laser beam was at the focal plane ($w_0$=13.5 µm). The pulse overlap is calculated to be 99.99 %. The conditions for the thermal camera remained the same as in the stationary beam experiment, ensuring consistency in temperature measurements.

To better illustrate the experimental configuration, Fig. 11 presents a schematic of the moving laser beam. It includes the laser beam, scanning speed, field of view (FOV) of the thermal camera, sample surface, and the resulting temperature gradient.

Based on the frame rate of the thermal camera (1240 FPS, corresponding to a time interval of 807 µs between frames), the laser repetition rate (1 MHz) and the beam scanning speed (1 mm/s), it was estimated that the laser beam moves by 0.8 µm on the sample surface between two con-

<figure><img src="image_12.png" /><figcaption>**Fig. 11** Schematic of the moving laser beam. The diagram illustrates the laser beam scanning across the sample surface at a constant velocity, $v_x$. The field of view (FOV) of the thermal camera is limited between the two white dashed lines, capturing the temperature gradient induced by the laser beam as it moves over the sample</figcaption></figure>
secutive images. Furthermore, each frame captures the arrival of approximately 800 pulses, providing a measure of the cumulative thermal response. This analysis justifies the use of the CWM, as the high repetition rate effectively emulates a quasi-steady-state heating scenario within the timescale of the thermal camera frames.

Figure 12 presents the time evolution of the maximum surface temperature recorded during the experiment. Significant fluctuations in the maximum temperature were observed, as shown in comparison to the stationary beam case (Fig. 8), where such fluctuations are less pronounced. By repeating the experiment with 10 consecutive passes over the same area, it was observed that the fluctuations occurred consistently at the same locations. This suggests that the temperature variations are linked to the local surface roughness of the scanned area. In similar studies, the temperature variations were observed in static conditions during drilling and scanning operations [7, 8, 17]. To mitigate this variability, an average of the measured temperatures was calculated and compared to the temperatures predicted by the CWM.

The comparison reveals a good agreement between the calculated maximum temperature and the average of experimental recorded average. This consistency highlights the ability of the CWM to accurately predict the thermal response of the material under moving beam conditions. It is important to note, however, that the CWM assumes an idealized scenario, neglecting the effects of surface roughness and local variations in material properties.

In Fig. 13, the x-axis temperature gradient profiles measured at four distinct time intervals are shown, alongside the corresponding profiles calculated using the CWM. These profiles reveal the dynamic nature of the heat distribution as the laser moves. At this fluence, working at the focal plane

<figure><img src="image_13.png" /><figcaption>**Fig. 12** Time evolution of the maximum surface temperature measured during the moving beam experiments, showing experimental data and CWM simulations with and without PSF convolution. The scanning speed was 1 mm/s with 1 MHz repetition rate, 200 mW average power, and a beam radius of 13.5 µm</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 13** X-axis temperature gradient profiles measured at four distinct moments during the moving beam scenario, compared with CWM calculations. The results are adjusted by convolving with the PSF of the camera lens to account for optical effect. The scanning speed was 1 mm/s with 1 MHz repetition rate, 200 mW average power, and a beam radius of 13.5 µm</figcaption></figure>

results in temperature gradients extending no more than 100 µm, resulting in fewer than 10 pixels captured by the thermal camera. Despite this limitation, the close agreement between the experimental data and the simulated results further confirms the ability of the CWM to capture the transient thermal behavior under moving laser beam conditions.

## 4.3 Temperature distribution in the ablation regime

For the moving beam with ablation case, the laser operated at conditions producing significant ablation: an average power of 5.8 W and a fluence of 0.36 J/cm² at 1 MHz, a waist of 32 µm, and a scanning speed of 120 mm/s. The calculated overlap is 99.81 %. These parameters represent conditions for applications such as cutting or engraving graphite in our processing conditions. The experiment was repeated using two thermal camera ranges: 30-150°C and 150-300°C.

Figure 14 compares the temperature measurements recorded during the scan under two camera ranges with the results from the CWM simulations. To ensure consistency with the experimental data, the temperature limits used in the simulations were aligned with the saturation limits of the respective camera ranges. This comparison provides a clear evaluation of the model's ability to capture temperature behavior across different thermal regimes. However, due to the spatial resolution of camera, the maximum surface temperatures reached at the beam center cannot be fully resolved experimentally. The lateral temperature gradients, particularly the decay profiles perpendicular to the scanning direction, are reliably captured and can be directly compared with the simulation.
<figure><img src="image_15.png" /><figcaption>**Fig. 14** Temperature contours for the moving beam with ablation, comparing experimental data under two thermal camera ranges: a) 30-150°C and c) 150-300°C, with CWM simulations: b) and d). Model contours are truncated to match the saturation limits of the respective experiments.</figcaption></figure>

Figure 15 presents a cross-section profiles along the y-axis, illustrating the HAZ at different x-axis positions parallel to the beam’s motion. This analysis allows quantifying of thermal diffusion and the size of the HAZ.

<figure><img src="image_16.png" /><figcaption>**Fig. 15** Cross-section along the y-axis illustrating the heat-affected zone at different x-axis positions during moving beam with ablation experiments. Experimental data and CWM simulations are compared to evaluate lateral thermal diffusion. The beam was scanned at 120 mm/s, 1 MHz, with a 5.8 W average power and beam radius of 32 µm, corresponding to a fluence of 0.36 J/cm².</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>The CWM demonstrates its reliability by accurately predicting the thermal distribution under both stationary and moving beam conditions. For the stationary beam case, the comparison of maximum temperature values and radial temperature profiles highlights the model's capability to effectively simulate lateral heat diffusion, validating its predictions for non-ablation scenarios.</figcaption></figure>

In the moving beam case, the dynamic nature of the laser introduces additional complexities, such as temperature fluctuations likely caused by surface roughness. Despite this variability, the average recorded temperatures align closely with simulation results, confirming the performance of the CWM in modeling transient thermal effects.

Under significant ablation conditions, the moving beam with ablation experiments demonstrate the importance of accurate thermal modeling. By using two camera range, the experiments capture detailed thermal behavior around the ablated zone. The CWM reliably estimates the HAZ, providing valuable information on the extent of thermal diffusion and its dependence on scanning parameters.

Despite its utility in evaluating cumulative thermal effects during high-repetition-rate laser scanning, the CWM has limitations. The assumption of a continuous heat source neglects transient phenomena associated with individual
pulses. Furthermore, the impact of surface roughness and local variations in material properties was not considered, although these factors can significantly influence heat transfer and absorption, potentially explaining minor discrepancies between experimental and numerical results. Finally, the limited spatial resolution and emissivity uncertainties of the thermal imaging system introduce additional experimental errors. These limitations define the scope and validity of the CWM, emphasizing its suitability for analyzing spatial temperature distributions and cumulative heat accumulation trends under controlled conditions.

The study aims to characterize and model thermal effects under well-controlled and experimentally accessible conditions. We acknowledge that industrial processing may involve more complex and dynamic scenarios. However, our approach offers a reproducible baseline, which can be extended in future studies to broader parameter spaces including higher scanning velocities and other repetition rate regimes.

The thermal imaging approach used in this study provides sufficient spatial and temporal resolution for tracking cumulative heat accumulation. Nevertheless, further improvements could be achieved by employing imaging systems with a frame rate comparable to the laser repetition rate, allowing a more detailed capture of pulse-to-pulse temperature dynamics.

## 4.4 Energy balance

To evaluate the effective power contributing to material heating, a correction factor $\eta_{heat}$ is studied. This factor, representing the proportion of laser energy not dissipated in ablation, is similar to the approach proposed in Ref. [11]. This factor accounts for the energy partitioning between ablation and material heating. The total pulse energy ($E_{pulse}$) is distributed among three components: energy lost to ablation ($E_{ablation}$), energy required to heat the material ($E_{\Delta T}$), and residual energy contributing to cumulative heating ($E_{heat}$).

The energy lost to ablation is calculated using the ablated volume ($V$), the material density ($\rho = 1790 \text{ kg/m}^3$), and the latent heat of sublimation ($H_{sub} = 5.98 \times 10^7 \text{ J/kg}$):

$$E_{ablation} = V \cdot \rho \cdot H_{sub} \qquad (11)$$

The energy required to raise the material's temperature is estimated using the specific heat capacity ($C = 2313 \text{ J/kg/K}$), the temperature change ($\Delta T = 3600^{\circ}\text{C} - 20^{\circ}\text{C}$), and the ablated volume:

$$E_{\Delta T} = V \cdot \rho \cdot C \cdot \Delta T \qquad (12)$$

<figure><img src="image_18.png" /><figcaption>Fig. 16 Variation of energy loss ($E_{ablation} + E_{\Delta T}$) and the correction factor $\eta_{heat}$ as a function of fluence</figcaption></figure>

The correction factor, $\eta_{heat}$, is then defined as:

$$\eta_{heat} = \frac{E_{pulse} - E_{ablation} - E_{\Delta T}}{E_{pulse}} \qquad (13)$$

which indicates the fraction of energy contributing to cumulative heating. This correction factor might provide an understanding of the effect of ablation within this model, complementing thermal analysis. $\eta_{heat}$ was calculated based on the estimated ablated volume discussed in Section 4.1. In Fig. 16 the variation of the energy loss ($E_{ablation} + E_{\Delta T}$) and $\eta_{heat}$ as a function of fluence are presented.

The results show that for our experimental conditions, $\eta_{heat}$ corresponds to approximately 50-60% of the applied average power. However, when this value is incorporated into the thermal model, discrepancies arise between the simulation and the experimental measurements. These deviations suggest that the simplified energy partitioning may not fully capture the complex energy dynamics during laser-material interaction or some limitations in our model. Interestingly, better agreement between the model and experimental data is observed when the full average applied power is considered, effectively assuming $\eta_{heat} \sim 1$.

## 5 Conclusion

In this work, the thermal effects in ultrashort pulse laser micro-machining of graphite at high repetition rates were analyzed through both experimental and numerical approaches. Experiments were conducted using a femtosecond laser operating at repetition rates between 0.333 MHz and 2 MHz. The laser beam was scanned across the graphite surface at varying speeds (1-120 mm/s), and temperature distributions were monitored using a thermal camera. The experimental setup allowed for the characterization of heat accumulation,
crater formation, and the heat-affected zone under different machining conditions.

To complement the experiments, two numerical models were developed to analyze the laser-matter interaction and resulting thermal effects. The TTM focused on capturing the ultrafast electron-lattice dynamics, enabling the prediction of material ablation by resolving the temperature rise relative to the sublimation threshold. Non-thermal phenomena such as plasma formation and dynamic changes in optical properties were not considered. Comparisons with experimental results showed good agreement with the linear part of the ablation rate, validating the model's accuracy in predicting localized material removal.

Further, the CWM addressed cumulative heating effects, particularly under high repetition rates. The model produced the broader heat distribution and temperature stabilization observed in the experiments for both stationary and moving beam conditions. Thermal imaging measurements and CWM simulations close agreement, particularly in predicting temperature diffusion and heat accumulation over longer timescales, confirming its reliability in evaluating the HAZ. Minor disparities, such as slight deviations in temperature contours, were attributed to surface roughness, material heterogeneity, and limitations of the thermal imaging system under extreme conditions.

Future developments could include fluence-dependent optical properties and higher-resolution thermal measurements to extend the predictive capability of the models. The approach presented in this work offers a reliable framework for analyzing and predicting thermal effects in ultrashort pulse laser micro-machining. While the study focused on graphite, the methodologies can be extended to other materials where controlling temperature rise and heat accumulation is critical for minimizing thermal damage and optimizing machining precision.

## Appendix A: Convolution of numerical simulations with a Gaussian point spread function

In our study, the resolution of the infrared thermographic camera and the optical response of the lens system significantly affect the accuracy of the thermal measurements. The measured temperature distributions can be blurred due to the point spread function (PSF) of the lens, which represents the response of the system to a point source. This blurring effect must be taken into account to ensure that the comparison between numerical simulations and experimental results is physically accurate.

To address this, the numerical temperature distributions calculated using the Continuous-Wave Model were con-

<figure><img src="image_19.png" /><figcaption>Fig. 17 2D representation of the Point Spread Function (PSF) modeled as a Gaussian with a standard deviation of $\sigma = 7.65 \mu m$. The model accounts for the pixel size, $10 \mu m$. The color bar shows the normalized intensity values</figcaption></figure>

volved with a Gaussian function representing the PSF of the lens. The Gaussian function was selected based on the assumption that the PSF of the optical system could be approximated by an axisymmetric, two-dimensional Gaussian distribution, characterized by its standard deviation, $\sigma = 7.65 \mu m$, a value provided by the lens manufacturer, which reflects the spatial resolution of the imaging system; it was illustrated in Fig. 17. The PSF is expressed as:

$$PSF(r) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{r^2}{2\sigma^2}\right) \quad (A1)$$

The convolution operation applied to the temperature distribution can then be written as:

$$T_{cov}(r, t) = T(r, t) \star PSF(r)$$

$$= \int_{r_0}^{r} T(r', t) PSF(r - r') dr' \quad (A2)$$

Here, $T(r, t)$ is the original temperature distribution from the simulation and $T_{cov}(r, t)$ is the convolved temperature profile. This adjustment ensures that discrepancies due to the measurement setup, rather than physical thermal behavior, are minimized in our analysis.

**Author Contributions** All authors contributed to the study conception and design. Material preparation, data collection, numerical simulations, and analysis were performed by Carlos Esteban Cifuentes Quintal, Thomas Doualle, and Laurent Gallais. Jean-Luc Rullier conducted the measurements of the crater depths. Yves Pontillon contributed to discussions and provided valuable feedback on the manuscript. The first draft of the manuscript was written by Carlos Esteban Cifuentes Quintal, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
**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.

## Declarations

**Conflicts of Interest** The authors declare that there are no conflicts of interest related to this article.

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