# Modeling nanogratings erasure at high repetition rate in commercial optical glasses

Qiong Xie, Maxime Cavillon, Matthieu Lancry

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Qiong Xie, Maxime Cavillon, Matthieu Lancry. Modeling nanogratings erasure at high repetition rate in commercial optical glasses. Ceramics International, 2024, 50 (23), pp.49157-49164. 10.1016/j.ceramint.2024.09.257. hal-05466529

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# Modeling nanogratings erasure at high repetition rate in commercial optical glasses

Qiong Xie, Maxime Cavillon, Matthieu Lancry*

Institut de Chimie Moléculaire et des Matériaux d'Orsay (ICMMO), Université Paris-Saclay, CNRS, Orsay, 91405, France

## ARTICLE INFO

Handling Editor: P. Vincenzini

**Keywords:**

Nanogratings

Nanopores

Erasure

Commercial glasses

Energy deposition

## ABSTRACT

Volume nanogratings (NGs) imprinted by infrared femtosecond laser in commercial optical glasses take the form of orientable subwavelength birefringent nanostructures, being composed of an assembly of nanopores. The existence of NGs strongly depends on the laser parameters and glass composition. Therefore, in this work, we tentatively model the erasure threshold of NGs in a pulse energy - repetition rate processing window. For this purpose, we combine i) a heat diffusion model to simulate the thermal treatment experienced by the glass upon laser irradiation, and ii) the Rayleigh-Plesset equation to take into account the evolution of a nanopore size during laser processing. We first determine a criterion for nanopores erasure, falling within a typical characteristic time of few tens of ns, for which the cooling of the last laser pulse absorbed by the material is progressively cooled. Then, considering a multipulse regime, and the dependence of the deposited pulse number on the thermal treatment experienced by the glass, the modeled NGs erasure threshold follows the experimental trend. Finally, by using a steady state regime for various repetition rates and adjusting the energy deposition (absorption coefficient or beam waist) as a function of the pulse energy, the NGs existence window can perfectly match the experimental values.

## 1. Introduction

Transparent materials, such as optical glasses, can undergo localized and in volume (3D) modifications using femtosecond laser direct writing (FLDW). Volume nanogratings (NGs), also referred as Type II modifications in the literature [1], are one of the most interesting transformations to imprint inside oxide glasses. The arrangement of these subwavelength nanostructures can be controlled by light polarization state and orientation [2], yielding to a pseudo-periodic array of nanolayers. In most oxide glasses like ubiquitous SiO₂ or GeO₂ [3], they usually take the form of porous nanolayers made of oblate nanopores whose sizes typically range within a few tens of nanometers. However, other types of NGs can be induced, made of alternating layers of crystalline and amorphous phases [4,5]. The porous NGs possess a significant birefringent optical response [6] due to i) strong refractive index contrast between the nanolayers and their surrounding, and ii) associated significant stress [7], and can even result to optical chirality [8]. Consequently, these developments have gained interest in several applied areas such as birefringent optical devices, optics based on 3D geometric phase, storage of optical data, microfluidics, structural health

monitoring, or sensing at high temperatures [9,10].

The formation of porous NGs by FLDW is suggested to originate through a plasma-mediated nanocavitation process, a “light-forced organization” [11]. Starting from some nanoscale inhomogeneities seeding the process at the femtosecond timescale, nanoplasma hot spots are formed, followed by a localized heat distribution due to electron-phonon energy transfers. Along the formation and erasure of NGs, the heat cooling profile (a temperature-time couple) assists this nanocavitation process to imprint NGs. This results in boundaries defining the existence of NGs under specific pulse energy, duration or repetition rate irradiation conditions [12].

In the literature, it was clearly reported in Refs. [13–15] that heat accumulation, arising when pulse period is too short to remove the heat from the previously deposited pulses (i.e., at high repetition rate), limits or even hinders the formation of this nanoporous phase and thus nanogratings in various oxide glasses such as SiO₂, GeO₂, or B33 (Borofloat glass). In Ref. [15], it was found that above a fixed apparent power of 0.23 W, nanogratings cannot be observed. Later on, a NG formation threshold using a cavitation approach ($T_{cav} \approx 2000$ K) in fused silica was defined [16]. However the upper temperature also limits the

<figure><img src="image_2.png" /></figure>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: matthieu.lancry@universite-paris-saclay.fr (M. Lancry).</footnote>
NGs survival in oxide glasses [17]. If the temperature is too high, mechanisms including boiling, rapid nucleation and growth, or spinodal decomposition, may occur. Based on work from [16], when the Péclet number is above 1, typically corresponding to temperature higher than 3000 K for SiO₂, a rapid viscous growth rate condition yields to “un-stable” nanopores. This approach was recently generalized to a wide range of oxide glasses. In a first approximation the estimated window of NGs existence within a pulse energy (Eₚ) – Repetition Rate (RR) domain was defined between Tₘₐₓ (η = 10³ Pa·s, the working point of the glass) and Tₛₐₗt (η = 10⁶.⁶ Pa·s, the softening point of the glass) [18]. Here, η is the glass viscosity. The upper temperature limit of the NGs survival intimately relates to the temperature dependent glass viscosity, η(T). However, this is in fact a time dependent problem and this upper temperature depends on the heating conditions and the heating cooling profile [17].

In this paper, we first exploit a heat diffusion model combined with Rayleigh-Plesset (R-P) equation to simulate the evolution of nanopores composing the NGs over time and temperature during a given laser cooling profile. We establish some objective criteria to determine the energy threshold corresponding to the closure of the NGs processing window. Then, we consider a multipulse approach and the dependence of the heat accumulation at steady state for various repetition rates in the erasure of nanopores. We finally simulate the maximum deposited energy beyond which NGs erase along the laser cooling profile, for a wide range of commercial optical glasses and in comparison to available experimental data.

## 2. Modeling approach

### 2.1. Temperature profile T (r,t)

The classical heat diffusion model is taken from Refs. [19,20] where more details can be found and compared to the related literature. First, the heat deposited locally by the laser inside materials follows Fourier's law. As a first approximation we did not consider the geometrical changes of the focal zone (beam waist) by the pulse energy, for instance arising from self-focusing and electron plasma defocusing. Considering the spatial temperature distribution induced by fs laser irradiation in a static configuration (i.e., no beam scanning), the equation takes the following form:

$$ \frac{\partial T(r, t)}{\partial t} - D_{th} \cdot \Delta T(r, t) = \frac{Q(r, t)}{\rho C_p} \quad (\text{Eq. 1}) $$

<table><caption>**Table 1**</caption><caption>Glass samples, molar composition, density, heat capacity, refractive index at 1030 nm, diffusivity and viscosity parameters [17].</caption><thead><tr><th>Glass Samples</th><th>Molar Composition</th><th>Density ρ (kg/m3)</th><th>Heat capacity C<sub>p</sub> (J/(kg·K))</th><th>Refractive index n</th><th>Diffusivity D<sub>th</sub> (10<sup>-7</sup>)</th><th>Viscosity Parameters (A; B; T<sub>0</sub>)<sup>a</sup></th></tr></thead><tbody><tr><td>BK7</td><td>69.13 SiO<sub>2</sub>, 10.75 B<sub>2</sub>O<sub>3</sub>, 3.07 BaO, 10.40 Na<sub>2</sub>O, 6.29 K<sub>2</sub>O, others &lt;1</td><td>2510</td><td>858</td><td>1.5168</td><td>5.17</td><td>(-2.475; 4677; 527.8)</td></tr><tr><td>Soda lime</td><td>72.6 SiO<sub>2</sub>, 13 Na<sub>2</sub>O, 8.8 CaO, 4.3 MgO, 0.6 Al<sub>2</sub>O<sub>3</sub>, 0.3 K<sub>2</sub>O, 0.2 SO<sub>3</sub>, 0.1 Fe<sub>2</sub>O<sub>3</sub></td><td>2500</td><td>750</td><td>1.52</td><td>5.33</td><td>(-1.741; 4309; 537.7)</td></tr><tr><td>Borofloat 33</td><td>81 SiO<sub>2</sub>, 2 Al<sub>2</sub>O<sub>3</sub>, 13 B<sub>2</sub>O<sub>3</sub>, 4 Na<sub>2</sub>O/K<sub>2</sub>O</td><td>2230</td><td>830</td><td>1.4714</td><td>6.48</td><td>(-1.681; 6578; 384.8)</td></tr><tr><td>7059</td><td>63 SiO<sub>2</sub>, 8.5 Al<sub>2</sub>O<sub>3</sub>, 16 B<sub>2</sub>O<sub>3</sub>, 12.5 BaO</td><td>2760</td><td>750</td><td>1.534</td><td>4.83</td><td>(-19.81; 34 510; -140)</td></tr><tr><td>AF32</td><td>66.43 SiO<sub>2</sub>, 11.28 Al<sub>2</sub>O<sub>3</sub>, 10.73 B<sub>2</sub>O<sub>3</sub>, 5.30 CaO, 4.63 MgO, 1.36 BaO, others &lt;1</td><td>2430</td><td>800</td><td>1.5099</td><td>5.97</td><td>(-4.062; 8876; 480.8)</td></tr><tr><td>Eagle XG</td><td>65.71 SiO<sub>2</sub>, 11.10 Al<sub>2</sub>O<sub>3</sub>, 11.65 B<sub>2</sub>O<sub>3</sub>, 8.64 CaO, 2.28 MgO, others &lt;1</td><td>2380</td><td>768</td><td>1.5099</td><td>5.96</td><td>(-6.584; 13 170; 319.4)</td></tr><tr><td>ULE</td><td>94.25 SiO<sub>2</sub>, 5.75 TiO<sub>2</sub></td><td>2210</td><td>767</td><td>1.4828</td><td>7.14</td><td>(-21.47; 90 920; -1365)</td></tr><tr><td>Suprasil CG</td><td>100 SiO<sub>2</sub></td><td>2200</td><td>703</td><td>1.444</td><td>8.92</td><td>(-9.5; 34 200; -127)</td></tr></tbody></table>

Here ρ is density, Cₚ is specific heat capacity, κ is the heat diffusivity, and the thermal diffusivity can be written as Dₜh = κ/ρCₚ. For simplification in the resolution of the Fourier equation, we restricted ourselves assuming that the physical properties of the material are independent of the temperature, but this does not prevent the possibility of physical de-ductions as commented and reviewed in Table 1 of Ref. [19]. Furthermore, r and t are the radial distance away from the focus center, and the laboratory time, respectively.

About the heat source Q(r, t), we suppose the laser beam takes up a Gaussian shape as:

$$ Q(r, t) = \frac{AE_p}{\pi^2 \omega^3} \cdot e^{-\frac{r^2}{\omega^2}} \cdot f(t) \quad (\text{Eq. 2}) $$

Then we can use a Dirac combdefined as $f(t) = \sum_{0}^{N_p} \delta(t - N_p \cdot \tau_p)$ to consider the multipulse regime at various repetition rates RR. Here we define $\tau_p = 1/R$ as the pulse period, Nₚ is the pulse number, Eₚ is the pulse energy. Finally, A is the absorbed fraction of the pulse energy, which was taken independent of Eₚ and RR as a first hypothesis, but will be adjusted later on.

Therefore, the temperature rise can be expressed as follow $T(r, 0) - T_{room} = \frac{1}{\rho C_p} \int_{pulse} Q(r, t) \cdot dt = T_0 \cdot e^{-\frac{r^2}{\omega^2}}$ where $T_0$ is the maximum temperature at the focus region center, which can be simply written as $T_0 = \frac{AE_p}{\pi^2 \rho C_p \omega^3}$. After hundreds of fs (i.e., the pulse duration), the heat source is null, and one can use the following expression demonstrated in Ref. [19]:

$$ T(r, t) = T_0 \frac{\omega^3}{(\omega^2 + 4D_{th} \cdot t)^{\frac{3}{2}}} \cdot e^{-\frac{r^2}{(\omega^2 + 4D_{th} \cdot t)^2}} + T_{room} \quad (\text{Eq. 3}) $$

This equation describes the temperature distribution after one pulse irradiation. Tₚₐₗt is the ambient temperature taken herein as 20 °C. In this equation, $\omega^3 = \frac{\omega_{th}^2}{2} \cdot \frac{\omega_z^2}{2}$ where $\omega_z$ is the typical beam length along the laser propagation, z, taken as 30 µm based on experiments. $\omega_{th}$ is the beam waist defined as $\omega_{th} = \frac{2\pi n}{\pi N_\lambda}$; n is the wavelength of the laser pulses (1030 nm); numerical aperture (Nₐ) is 0.6 for the laser beam; n is the refractive index of each oxide glass, taken at 20 °C since it is weakly dependent on temperature.

<footnote>a These parameters provide a value of log (η) with η in P (dPa·s), and T in Kelvin.</footnote>
## 2.2. Multipulse regime and steady state condition

In the case of the absorption of $N_p$ pulses by the glass material, we easily obtain the evolution of the distribution considering the linearity of the differential equation and making up the sum of the solution for one impulsion but shifted in time $\tau_p$, the pulse period [19]. In some conditions, this can result in a pulse-to-pulse heat accumulation, especially at high RR. It is thus necessary to define the steady state where the temperature distribution reaches saturation and this is related to the ratio $R_t = \tau_p/\tau_d$ between $\tau_p$ and the thermal diffusion time noted $\tau_d = \omega_g^2/4D_{bh}$ [19]. When $R_t$ values are large, corresponding to low RR, less $N_p$ are needed to reach a steady state. The contribution of an increase of temperature induced by consecutive pulses overlapping in time (heat accumulation) is weak or even negligible when $R_t > 10$ [19]. However, considering high RR values (low $R_t$), this pulse-to-pulse temperature increase is significant, and accompanied by a larger $N_p$ to reach steady state. Within our conditions for RR ranging from 1 to 10 000 kHz, we can estimate that the steady state can be effectively reached for $N_p = 10 000$ pulses for all investigated glasses [19]. The simulation of thermal part is a finite-difference heating model under Matlab [19].

## 2.3. Erasure of nanopores using Rayleigh-Plesset equation

Using the heat diffusion model, one can deduce the temperature profile $T(r, t)$ of the last pulse for a given set of laser parameters ($E_p$ and RR). In the following, the temperature distribution $T(r, t)$ is taken at $r = 0$ µm, that is, at the center of the beam, where the temperature is the highest [20]. From that point, we combine such heating-cooling profile $T(0, t)$ with the well-known Rayleigh-Plesset (R-P) equation to simulate the evolution of nanopores (their erasure), that are composing the NGs [21]. The R-P equation was initially suggested and employed by Rudenko et al. to model the evolution of a spherical bubble inside an incompressible Newtonian fluid [22], and is expressed as:

$$ \frac{\Delta P}{\rho} = R_{pore} \frac{d^2 R_{pore}}{dt^2} + \frac{3}{2} \left( \frac{d R_{pore}}{dt} \right)^2 + \frac{4\eta(T)}{\rho R_{pore}} \frac{d R_{pore}}{dt} + \frac{2\sigma}{\rho R_{pore}} \quad (Eq. 4) $$

Here, $\Delta P$ is the pressure difference (in Pa) between the inside of the nanopore (supposed spherical) and far away from it; $\rho$ is the glass density (in kg·m⁻³); $R_{pore}$ the radius (in m) of the spherical nanopore; $t$ is the cooling time (in s); $\eta(T)$ is the glass viscosity (in Pa·s), and $\sigma$ is the surface energy (0.3 J m⁻² for silica over a wide temperature range [23]). Note that the non-sphericity of the experimentally observed nanopores, which are oblate [11], does not change the physical interpretation and thus that nanopores are considered spherical. The viscosity and its dependence with respect to temperature can be taken into account in the model by using a Vogel-Tammann-Fulcher (VTF) law in the form $\log(\eta, Pa \cdot s) = A - 1 + B/(T - T_0)$. Initial conditions are $R_{pore}$ (at $t=0$) = $R_{ini}$, that is, the initial nanopore radius before any heat treatment and $dR_{pore}/dt = 0$ m·s⁻¹ at $t=0$ s.

In our conditions, the second derivative and the square of the first derivative from the R-P equation have very small contributions numerically in our conditions and can be therefore neglected i.e., slow dynamics so no inertial pressure. In the following, we consider that $\Delta P$ is small enough and thus set to be null in the erasure case. Indeed, the pressure inside the pores is experimentally "low", as can be deduced from the characteristic $O_2$ Raman peak centered around $1555$ cm⁻¹ [24], i.e., which is a "near to ambient" pressure range [25]. The second argument is that under stochiometric silica (even if $SiO_2 \cdot x$ with $x > 0.1$) or ODC defects measured in VUV-UV [26] should yield a strong visible absorption (the irradiated region should turn black), which is not the case experimentally. These arguments validate that only a very small fraction of $O_2$ is dissociated [27]. Now, we can consider two thermal treatment conditions: under low temperature/long times like isochronal annealing or high temperature/short time like laser heating-cooling

process [17]). The Rayleigh-Plesset equation considers the evolution of a spherical pore in a viscous medium. From this view, no difference is expected based on the two annealing processes since the glass viscosity, of upmost importance as demonstrated in Ref. [18], is a smooth decreasing function of temperature. Now, if from the "slow" isochronal process the $O_2$ can progressively recombine, one can wonder if, under high temperature conditions where the $O_2$ might still be present inside the pores, the collapse would still be possible. By using the simplified RP equation (Eq. 5), and assuming a high temperature (about 2900 °C for silica, where viscosity reaches 1 Pa·s (i.e., $\log(\eta) = 0$)), this gives an erasure rate of -0.15 µm/µs if a $\Delta P = 0$ Pa (no pressure difference) and -0.1438 µm/µs if a $\Delta P = 1$ MPa (which is already 10 times the atmospheric pressure). Now, the low pressure $O_2$ could be easily compressed and there is no reason to prevent the pore reduction. So using the above mentioned hypotheses, as the glass viscosity $\eta(T)$ and surface tension dictate the dynamics of the process, the simplified R-P equation form is used [17]:

$$ \frac{dR_{pore}}{dt} = \frac{d(R_{pore} - R_{ini})}{dt} = - \frac{\sigma}{2\eta} \quad (Eq. 5) $$

To further illustrate this purpose, we can observe in Fig. 1 a typical heating-cooling profile for a single pulse modeled in $SiO_2$ (black symbols) and the related nanopore erasure $R_{pore}$ (t) for different values of $R_{ini}$. First, the equivalence of the two equations is demonstrated in Fig. 1, showing a similar nanopore erasure rate using either Eq. (4) or Eq. (5). Secondly, the nanopore size diminution is the same considering different initial nanopore radius thus leading here to a full erasure of small nanopores while big ones are partially erased. In addition, it should be highlighted that the normalized retardance ($R_t$) is a strong function of the pore radius, $R_t^6$ [28]. This means that even a 50 % decrease of the pore size yields to >98 % reduction of the overall retardance, and, consequently, to a closing of the "Type II window".

## 3. Experimental database to support modeling

We use several commercial oxide glasses (Schott BK7, Borofloat 33 and AF32, Corning ULE, Eagle XG and 7059, Heraeus Suprasil CG and Marienfield Superior soda lime) to perform both experiments and the numerical simulations. The thermal properties of the glasses are provided from the glass manufacturers and summarized in Table 1. Additionally, the viscosity parameters deduced for VTF fit, namely (A, B and $T_0$) are also available in Table 1. After laser writing experiments

<figure><img src="image_3.png" /><figcaption>Fig. 1. Different nanopore diameters erased by the same heat cooling curve $T(0, t)$ in SuprasilCG glass using both the full R-P equation (in solid line) and its simplified version (hollow symbols). The corresponding parameters in heat diffusion model are: $\omega_{th} = 1.58$ µm; $N_p = 1$; $A = 0.25$; $E_p = 1$ µJ; $SiO_2$ glass.</figcaption></figure>
described in Ref. [14], we have quantified the optical retardance (related to the birefringence of the Type II structures). The presence of nanogratings was ascertained through polarization dependent form birefringence measurements and by direct observations using scanning electron microscopy in all glasses, which further allow to determine the typical pore size for each glass composition [29]. From that point, their NGs (Type II) processing window in a $E_p$-RR domain have been reported in our previous work [14].

In the following, we first determine the criterion for determining the NGs erasure from the thermal modeling point of view. Then, we model nanopores erasure according to absorbed pulse energy at different RR. For each glass, we will first model the heating-cooling profile for a fixed absorption coefficient, and we consider various pulses numbers up to reaching the steady state. Finally in a second attempt, we introduce an adjustment within the model considering the absorption coefficient being function of the pulse number and thus on RR within our experimental conditions.

## 4. Results and discussion

Nanopores composing NGs are the key point here since they are at the origin of the ultimate nanogratings thermal stability. It is worth mentioning that other contributions to birefringence (hence retardance) can arise from point defects, stress fields, interlayers densification, etc. [27], but this aspect goes beyond the framework of this study that is focused on the “ultimate” erasure of nanopores. As shown in the literature, nanopores typically erase within 30 min at ~1.1$T_a$ where $T_a$ stands for the glass annealing temperature (when $log(\eta, \text{Pa.s}) = 12$). However, we are interested here to model the survival of nanopores upon fs-laser direct writing in a $E_p$-RR domain where temperatures exceed 2000 °C [16].

### 4.1. Case of when $R_r > 10$

Let us first consider the case of most oxide glasses within which porous NGs (excluding partly crystallized ones) are imprinted. Once the oxide glass has been decomposed through a plasma-mediated nano-cavitation process [11,16], one can observe the nanopores few 100's of ps after the end of a pulse. From that point, there is a thermalization of the glass network, and nanopores can be partly erased along the subsequent heat cooling profile, until the next pulse through which they can be eventually regenerated, reshaped or even erased. Following this view, in a first example (Fig. 2) we will only consider the last pulse, i.e., the last cooling curve in our model, which corresponds to experimental conditions where $R_r$ is big enough (typ. >10) in such that there is no appreciable heat accumulation (e.g. $R_r = 140$ at 10 kHz in $SiO_2$). In Fig. 3 we will consider the multipulse regime that may lead to a heat accumulation thus affecting both the shape of the spatial heat distribution (that becomes Lorentzian [19] and the cooling curve $T(t)$ as well). Note that for all glasses investigated, although the value of $D_{th}$ (hence $R_r$), can be slightly different, the reasoning remains the same.

However here, one first need to define an erasure criterion to compare modeled and experimental values. So, we choose that the observable measurand, namely the optical retardance $R$, is no more measurable below our detection limit and estimated to 1 nm. Then we translate this criterion in term of nanopores size by means of the form birefringence model which makes the link between nanopores size, filling factor and the birefringence or retardance amplitude. So we define the $P$ factor, $P = R_{\text{pore}}(t,T)/R_{\text{ini}}$, corresponding to the fraction of nanopore erasure, and expressed in percent. Typically, 80 % reduction of the nanopore diameter corresponds to 99.99 % decrease of the related optical retardance (Ret) value

In the following part, we take the situation of a 50 nm nanopore. Then using the simplified R-P equation, we can compute the time required to partly or fully erase such a nanopore along a given temperature profile $T(r,t)$ taken at the center of the pulse, $r = 0$ µm. On the

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** The curves of the erasure time with different erasure percentage (nanopore diameter is 50 nm as an example) versus the heat cooling curves of one pulse for 1 % and 100 % erasure of the nanopore in Suprasil CG glass. The corresponding parameters in heat diffusion model are: $\omega_{th} = 1.58$ µm; $N_p = 1$; $A = 0.05$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** The curves of the erasure time with different erasure percentage (initial nanopore diameter is set as 50 nm) versus the heat cooling curves of the last pulse with variable RR from 10 kHz to 1000 kHz for 100 % erasure of the nanopore in Suprasil CG glass. The corresponding parameter in heat diffusion model is: $\omega_{th} = 1.58$ µm; at steady state [19].</figcaption></figure>

one hand, the erasure time of different $P$ values can be calculated by the equation $\tau_e = \frac{2\eta R_{\text{ini}} P}{\sigma} = \frac{2 \times 10^{A-1} \cdot \frac{B}{T-T_0} \times R_{\text{ini}} \times P}{\sigma}$ [17]. As previously mentioned, here we keep $\sigma$ term constant, which is a reasonable assumption for this work [17].

The results for various $P$ values from 1 to 100 % are summarized in Fig. 2 for silica (Suprasil CG). Whatever the chosen fraction $P$, the intuitive trend that the pore will collapse faster when brought to higher temperature is verified. Now, let us see on the graph the thermal cooling for a single pulse in silica glass which corresponds to the case where $R_r$ is big enough so no heat accumulation is present (e.g. $R_r = 14$ for Suprasil CG at 100 kHz). These two heat cooling curves are calculated by the above-mentioned heat diffusion model. We thus let the heat cooling curves touch the curves of erasure time by adjusting the $E_p$. Note that the
cooling curves of one pulse are shifted along T, but its temporal shape
remains unchanged. The minimum absorbed energy to start erasing the
nanopore (P = 1 %) or fully erase it (P = 100 %) can be deduced once the
heat cooling curve and the erasure one touch each others. In Fig. 2, the
minimum absorbed energy for P = 1 % and P = 100 % are 4.3 × 10³ J/
cm³ and 5.3 × 10³ J/cm³, respectively. Since the results are provided for
Np = 1, the RR has no effect in this first simulation. The characteristic
erasure times (initially defined in Ref. [17]) are typ. ~8 ns and ~10 ns
for P = 1 % and P = 100 %, respectively.

## 4.2. The effect of the repetition rate when $R_τ < 10$ (heat accumulation)

The objective now is to compute the value of the temperature T (r,t) when N<sub>p</sub> ≫ 1. We will show how the cooling profile changes with the RR according to heat accumulation (hence R<sub>τ</sub>), i.e., when T at the end of the period cumulates with the T increase induced by the absorption of the successive pulse. So, in the following curves, we will consider the situation where the steady state (typ. we chose N<sub>p</sub> = 10,000 in our conditions [19]) has been reached for a given practical number of pulses. Finally, as in previous case we will consider only the cooling profile of the last pulse at the center, since as the gaussian beam is moving away and the maximum temperature is decreasing. To some extent, one possible view is that it takes more time to create a nanopore (multipulses) whereas a single pulse (i.e. here the last pulse) is enough to erase it.

Fig. 3 shows the heat cooling curves of the last pulse after the steady state regime is reached, for multiple RR values from 10 to 1000 kHz. In addition, we add the R-P based erasure time curves for P = 100 % (full erasure). First, one can see that minimum deposited energy necessary to fully erase a given nanopore ranges typ. from 2 up to 5 × 10³ J/cm³. Second, the shape of the heat cooling curves of the last pulse at steady state are different (e.g., the slope around the touching point) because of heat accumulation impacted by RR. Finally, the characteristic erasure time proportionally varies from ~10 ns to ~29 ns for 10 and 1000 kHz respectively.

One first comment is that this erasure time is much smaller than a
typical diffusion time (~0.69 µs in SiO<sub>2</sub>). This very short characteristic
time arises from the cooling temperature profile where the temperature
drastically decreases within a fraction of µs but the key driving force, the
viscosity, varies itself in a log scale versus T. With such a characteristic
time, it is shown that temperatures around 2700 °C and 3300 °C are
required to collapse the nanopore in SuprasilCG. Similarly, this erasure
temperature remains quite similar whatever the RR and this is true for
each glass. For example, it is around 3100 °C in Suprasil CG glass;
2870 °C in Borofloat33 glass; 2080 °C in AF32 glass; 1890 °C in Eagle XG
glass; 1700 °C in BK7 glass; 1350 °C in 7059 glass. A second comment is
that such calculations imply some very low viscosity values (i.e., high
temperatures. So we can determine a viscosity “limit” of log (η, Pa·s) ~
-0.75 for nanopores erasure within such timescale. However, one must
bear in mind that uncertainties can be quite large (typ. Δlog (η, Pa·s) ±
1) using viscosity fits instead of experimental data.

## 4.3. Experiment vs. modeling comparison of nanopores erasure in the $E_p$-RR domain

In Fig. 4, the experimental curve of E<sub>p</sub> versus RR is taken from
Ref. [14], and corresponds to the upper threshold of the NGs window in
SuprasilCG glass, for which there is no more form birefringence asso-
ciated with a very low retardance (only residual stress). To set a value of
the absorption coefficient A in these conditions, we use the experimental
curve, at 10 kHz, where no heat accumulation is present. As mentioned
above, the erasure criterion of the full erasure is around 80 % erasure of
the nanopore diameter. In the following A was then fixed to 0.2 and the
beam size ω<sub>th</sub> was kept constant as well. When N<sub>p</sub> = 1, there is obviously
no heat accumulation, and the modeled curve is thus flat according to
RR. As N<sub>p</sub> increases, the curves of modeled E<sub>p</sub> versus RR start to drop in

<figure><img src="image_6.png" /><figcaption>Fig. 4. The modeled and experimental E<sub>p</sub> versus RR erasure curves with variable N<sub>p</sub> at A = 0.2 for Suprasil CG glass. N<sub>p</sub> = 10 000 corresponds to the steady state. The inset shows the measured ω<sub>th</sub> (dots) from the laser and ω<sub>th</sub> values used for the modeling (stars symbols) versus experimental E<sub>p</sub>.</figcaption></figure>

the high RR domain, getting closer to the experimental observations. Based on previous modeling, we can estimate that the temperature for P = 80 % in this condition needs to reach around 3100 °C whatever RR and N<sub>p</sub>. Ultimately when reaching the steady state (e.g. when N<sub>p</sub> = 100 000), the modeled curve exhibits a quite similar trend, but the match is still not perfect. The same result stands for all investigated chemical compositions. To refine a bit more the model in silica, we used a varying value of the beam waist ω<sub>th</sub> according to the pulse energy E<sub>p</sub>. We have thus measured the experimental beam size vs E<sub>p</sub> within Type II regime and at low repetition rate, so we have no heat accumulation. The beam size was defined as the width over which we can effectively measure nanogratings by SEM as seen in the inset of Fig. 4. Following that experimental curve, we have then adjusted ω<sub>th</sub> values used in the thermal model, at the steady state, to model the nanopores erasure energy. The resulting curve (purple line in Fig. 4) provides a quite good agreement with the experimental measurements.

To go one step further and generalize to other glasses, we introduced
such dependence in the energy deposition in J/cm³ according to the
repetition rate for all glasses. Indeed, if A or ω<sub>th</sub> are varied with respect
to E<sub>p</sub> at various RR (i.e., serving as an adjustment variable), a better
match to the experimental conditions can be obtained. So, Fig. 5 shows
the curves of modeled E<sub>p</sub> versus RR at steady state with variable A values
in 8 commercial oxide glasses. For each curve, A was adjusted according
to the experimental E<sub>p</sub> at the corresponding RR. As expected, the
modeled curves can be perfectly matched to the experimental ones. The
overall modeled trend is that one need to increase the apparent ab-
sorption coefficient A when increasing RR.

This result is not obvious and there are few aspects to consider there. Indeed, one must keep in mind that behind this apparent parameter, there are in fact changes of both the E<sub>p</sub> and the pulses number N<sub>p</sub> that can both affect the energy deposition in different ways including a change of ω<sub>th</sub>, which usually increases with E<sub>p</sub> (see inset of Fig. 4).

i/It is well known that the fraction of the incident energy effectively absorbed by the material is a function of the incident pulse energy, the pulse duration, and the material chemical composition. Absorption primarily occurs through nonlinear absorption namely multiphoton ionization and then avalanche occurs as shown by plasma measurements at short time scales [30,31]. In addition, plasma reflection or scattering is also expected at high energies, thus reducing the effective absorption coefficient. Finally at higher energies, some intensity clamping further happen, e.g. 60 TW/cm² in similar experimental conditions [32]. The origin of the clamping is related to the shielding of the field on its way towards the focus by the generated plasma [33]. As a result, the
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** The modeled and experimental $E_p$ (for nanogratings erasure) versus RR at steady state in different oxide glasses. The values of absorption coefficients are written in each legend and decided by experimental $E_p$ from 10 kHz to 500 kHz (e.g. $0.050 \rightarrow 0.167$ in Silica SuprasilCG). For each RR we have chosen the minimum absorption coefficient necessary to obtain nanopores full erasure at the corresponding experimental energy $E_p$.</figcaption></figure>

absorption coefficient initially increases with $E_p$ and then tends to saturate. In silica the absorption of a single laser pulse within the nanogratings regime range from about 2 % to 25 % depending on pulse energy and duration [34]. For example, at 500 kHz, 20 % pulse energy was absorbed for pulse duration of 500 fs and an energy of 1 uJ, a value typically in a reasonable agreement with our modeling.

ii/A second point is to take into account the higher number of pulses seen by the material at high repetition rates. Indeed, in agreement with the literature, the measured thresholds decrease when the number of pulses increases. Such “incubation effects” were already observed nearly two decades ago by Schäffer [35] leading to a decrease in the modification threshold of various oxide glasses when increasing the number of pulses, but also in surface experiments [36,37]. In the literature, it has been demonstrated [38] that the origin of such incubation can be attributed to i) the accumulation of point defects which results in a more efficient absorption when increasing $N_p$ ii) it introduces new possibilities of energy deposition in the glass network, e.g. based on a coupling of electrons trapped in the lattice. This accumulation of points defects is proportional to the initial defect concentration (thus chemical composition), the increment in the number of pulses, and also depends on the number of atomic sites likely to be affected during the heat accumulation process. In the authors’ view, this aspect is expected to play a rather minor role compared to the previous and following discussed contributions.

iii/When considering the energy deposition in volume, not only the absorption coefficient but also the beam waist and its energy dependence are crucial information. At first self-focusing phenomenon occurs when the power of the laser beam is greater than the critical power (related to the nonlinear refractive index). Note that for tight focusing conditions the lens-focusing is dominating and the Kerr nonlinearity could be partly neglected, and for loose focusing the self-focusing is dominating. The effective width at the focus in the material is given by Refs. [35,39,40] but this does not include the plasma defocusing effect which depends on the excited electron density. To quantitatively estimate the beam size evolution with $E_p$, one can use the width of Type II nanogratings since their nanostructure is achieved only by the laser light. In Lei et al. [41], the beam width increases already above 0.2 µJ although the excited electron density is of the order of January 0, $10^{21}/\text{cm}^3$. In silica, scanning electron microscope (SEM) micrographs

reveal that for a beam size around 1.5 µm (0.6 NA focusing) at low energy, there is monotonous growth of the beam size and a trend to saturation resulting in a 3 µm width around 1 µJ/pulse [42]. Based on our work [43–46], we note that the dependence of the beam waist with $E_p$ is not very sensitive to the glass composition within silicate glasses family. So, the numerical modeling of nanopores erasure, which reveals an apparent increase of absorption coefficient at higher RR is not necessarily true. This is simply an “image” of the effective volume energy density that is necessary to erase nanopores in these conditions of $E_p$-RR. One should rather consider the changes of the beam waist, which is much smaller in the case of erasure threshold at high RR (i.e. erasure energy <0.5 µJ) compared to low repetition rate (e.g. high energy of a few uJ/pulse). Such changes of the beam size by a factor 2 are more than enough to explain our observations. However, here the balance between Kerr self-focusing and plasma defocusing is determined by the effective pulse length undergoing ionization at nonlinear focus, which is difficult to predict without heavy computations, or robust knowledge of many parameters involved in the process, usually approximated, or simply ignored.

## 5. Conclusion

In this work, using a Fourier heat diffusion model combined with the Rayleigh-Plesset equation, we simulate the nanopore time-temperature evolution along the laser heat cooling profile. We show the modeled $E_p$ versus RR curves at a steady state in 8 commercial oxide glasses. The curves correspond to the erasure of nanopores and is defined by the “touching point” between the T(t) heat cooling profile and characteristic erasure time deduced from Rayleigh-Plesset equation. Additionally, we have observed that the temporal shape of the cooling curves of the last pulse at a steady state is dependent on RR, which can be attributed to heat accumulation. As the pulse number increases, the modeled curves of $E_p$ versus RR become closer to experimental data for a fixed absorption coefficient. However, to obtain a perfect match with experimental data, the energy deposition was adjusted according to RR which could be likely attributed to changes in the beam waist with increasing energy due to plasma defocusing. Future work would be to refine this approach by considering the shape of fs-laser focus, making differences between transversal radius and longitudinal light distribution to deduce how the
laser track changes over time. Another interesting point is the temper-
ature dependence of the physical–chemical parameters that would need
finite element calculation. Our approach allows us to choose the most
representative parameters for applying such a calculation. In addition, it
could be interesting to take into account the variations of the focal
volume considering nonlinear phenomena such as Kerr focusing, plasma
defocusing etc.

Funding

This research is funded by Agence Nationale de la Recherche (ANR), FLAG-IR Project, award number ANR-18-CE08-0004-01; and REFRAC-TEMP project, award number ANR-22-CE08-0001-01. Qiong Xie acknowledges the China Scholarship Council (CSC) for the funding of her PhD fellowship, NO.202007040018.

CRediT authorship contribution statement

**Qiong Xie:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation. **Maxime Cavillon:** Writing – review & editing, Visualization, Supervision, Software, Methodology, Funding acquisition, Formal analysis, Conceptualization. **Matthieu Lancry:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Appendices

All the simulations use the software Matlab (R2019b version). The
simulation code could be available upon reasonable request to the
authors.

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