# Lifespan prediction procedure of volume nanogratings imprinted by femtosecond laser in optical glasses

Qiong Xie, Maxime Cavillon, Bertrand Poumellec, Matthieu Lancry

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Qiong Xie, Maxime Cavillon, Bertrand Poumellec, Matthieu Lancry. Lifespan prediction procedure of volume nanogratings imprinted by femtosecond laser in optical glasses. Ceramics International, 2024, 50 (11), pp.20303-20312. 10.1016/j.ceramint.2024.03.153. hal-05466515

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# Lifespan prediction procedure of volume nanogratings imprinted by femtosecond laser in optical glasses

Qiong Xie<sup>a</sup>, Maxime Cavillon<sup>a</sup>, Bertrand Poumellec<sup>a</sup>, Matthieu Lancry<sup>a, *</sup>

<sup>a</sup> Institut de Chimie Moléculaire et des Matériaux D'Orsay (ICMMO), Université Paris-Saclay, CNRS, Orsay, 91405, France

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## ARTICLE INFO

Handling Editor: Dr P. Vincenzini

**Keywords:**
- Nanogratings
- Nanopores
- Oxide glasses
- Thermal aging
- Master curve
- Viscosity

## ABSTRACT

Volume nanogratings imprinted by infrared femtosecond laser in oxide glasses exhibit a characteristic birefringent signature, which translates into measurable retardance. Upon thermal annealing, such signature is progressively erased, typical of nanograting erasure. In this work, we propose a procedure to predict the lifespan of nanogratings by two main approaches: 1/ numerical modeling of optical retardance ageing using the Rayleigh-Plesset equation, and 2/exploiting VAREPA (VAriable REaction PAthways) framework fed by simulated ageing data. By considering experimental time – temperature annealing conditions, the modeled retardance is gathered as a function of demarcation energy to build a so-called Master Curve and then compared to accelerated ageing experiments. The erasure constant rate $k_0$ can be determined for 8 commercial optical glasses. Based on a distributed Rayleigh-Plesset model, $k_0$ and activation energy distribution are linked to glass viscosity and its temperature dependency. Finally, we discussed the restrictions on VAREPA application for an accurate lifetime prediction. This work provides guidelines for the future development of nanogratings based devices and applications, including optical data storage, birefringent devices, and optical sensors, through a judicious choice of glass composition and associated properties.

## 1. Introduction

Femtosecond laser direct writing (FLDW) enables both local and volumetric transformations of transparent materials, such as optical glasses [1]. Among the possible modifications to be imprinted inside glass, volume nanogratings (NGs), also labelled Type II in the literature, are of prime interest [2,3]. These structures are composed of self-organized pseudo-periodic nanolayers, and their orientation can be controlled by light polarization [2]. In oxide glasses, they generally take the form of porous nanolayers [4] composed of nanopores (typ. few tens of nm in size), but other NGs can be composed of alternate layers of crystalline and amorphous phases [5]. Porous NGs lamellar structuring induces a strong birefringence optical response from negative index variation [6], and can even lead to optical chirality [7]. Consequently, they have drawn attention for applications/fields including birefringent optical devices [8], 3D geometric phase optics [9], optical data storage [10,11], microfluidic [12], structural health monitoring and high-temperature sensing [13–15].

From a practical standpoint, the lifespan of a given device composed of NGs must be anticipated, principally considering a time-temperature

couple. Expected lifespans can be very different depending on the application, for example i) 25 years at 50 °C for telecommunication [16, 17] ii), few years at 400 °C–600 °C for the oil or gas drilling industry [18], or 600 °C–800 °C in new nuclear reactors generation cooled with liquid sodium [19] and Tokamak [20], iii) higher temperatures for aeronautics instruments of aircraft engines (typ. 800–1600 °C) [21–24], and iv) 600 °C–1100 °C for the recycling of precious metals or metal-based manufacturing processes [13,25–28]. In all these applications, it is mandatory to have temperature and/or strain sensors that ideally must be stabilized (i.e., without unpredicted drifts of their monitored response) [14]. This requires the predictions of the correct stabilization conditions for related components. However, when considering glass as the device material, difficulties arise from its intrinsic disordered and amorphous nature. Therefore, to model the relaxation process(es) in disordered media, several approaches have been proposed, where an extended report can be found in Ref. [29]. This disorder impacts the various steps of a given physicochemical reaction leading to a progressive relaxation of the observed quantity (e.g., laser induced refractive index modification [30,31]). For instance, if atomic hopping is involved in a reaction where a given modification is erased to

<footnote>* Corresponding author.
E-mail address: matthieu.lancry@universite-paris-saclay.fr (M. Lancry).</footnote>
return to an “initial state”, the disorder will factor into the hopping distance or frequency. Hence, the resulting activation energy of the given reaction will be distributed due to the multiplicity of transition state configurations. While the approach is based on several previous works in glasses, e.g., Kohlrausch in 1847 for the residual electric discharge in Leyde bottle [32], W.A. Yager in 1936 for the relaxation function in dielectrics [33], V. Vand (1943) [34] or W. Primak (1955) [35], the aforementioned aspects were first introduced in optical fibers by Lemaire et al., in 1984 [36] to predict hydrogen-induced loss. Then Erdogan et al. developed in 1994 a comprehensive formalism on thermal decay of UV-induced fiber Bragg gratings [30]. Later in 1997, Kannan et al. [37] refined this framework to improve the reliability of fiber gratings lifetime predictions. In 1998 and based on these previous works, Poumellec generalized the approach by introducing the VAREPA (VAriable REaction PAthways) framework, [38,39]. Under conditions that must be validated beforehand, it becomes possible to predict the lifespan of optical components which are based on a property variation, including the ubiquitous refractive index changes [16,39], or retardance (proportional to birefringence), characteristic of nanogratings imprinted in silica, or silicate glasses [40].

In this paper, we investigate the applicability of VAREPA to predict the lifespan of nanogratings inscribed by femtosecond laser inside a wide variety of oxide glasses. First, we remind the key hypotheses of VAREPA and its experimental applicability for the prediction of nano-gratings erasure at high temperature in silica. Second, a model of nano-gratings aging considering the evolution of nanopores as a function of temperature is considered using the Rayleigh-Plesset equation [41,42]. Third, based on a distributed Rayleigh-Plesset model, we rationalize the applicability of VAREPA using modeled thermal ageing data. Then, we compare experimental data to the simulated Master Curves (MC) and related activation energies in 8 commercial glasses, extending our work much beyond silica. Finally, we exploited these results by comparing lifetime predictions at high temperature using both approaches namely VAREPA and based on the Rayleigh-Plesset equation.

## 2. Preliminary hypotheses and experimental details

**Preliminary hypothesis on the investigation of nanograting thermal stability:** In most applications, the exploited property of NGs is their form birefringence, resulting to an optical retardance that can be optically measured. Form birefringence is the dominating factor to the overall birefringence response [7,43], and in this work, we solely focus on this contribution. It is worth mentioning that other contributions can arise from point defects, stress fields, interlayers densification, or any kind of birefringent contribution [43], but this aspect goes beyond the framework of this study that is focused on the “ultimate” thermal stability of nanogratings [43]. Especially, we work on “burnt-in” samples from which defects [43], silicon [44] and residual stress have been erased. Our previous work done in a multitude of glass compositions [45] reveals that it is experimentally relevant to assume that linear birefringence LB (t, T) is solely related to the kinetics of nanopores erasure at high temperatures (T > 0.9 T_a, T_a being the glass annealing temperature). In addition, we assume also that retardance erasure is mainly due to nanopores erasure and not to any laser track length variation, nanoplane distortion, dispersion of the nanopores, or any other change. Alternatively, the presence of refractory oxide nanocrystals will be instead something that could likely leads to some deviation from this model like quartz formation in silica [40]. However, for all the chosen compositions in this paper, we did not find any nanocrystals formation (including post laser irradiation and thermal treatments) and such behavior has neither been reported as well in the literature contrarily to what could happen in SiO₂–Na₂O [46], GeO₂–Na₂O [47], Dy₂O₃–Al₂O₃ [48], SiO₂–Al₂O₃ [49] or LNS glasses [50] in specific laser conditions.

**Accelerated ageing data:** To investigate the lifespan of NGs in oxide glasses, optical retardance (R) is monitored both experimentally and theoretically/numerically. In both cases, R is defined as the product of

the linear birefringence (LB) with the thickness (l) of the birefringent object. From an experimental point of view, R is monitored by optical microscopy using the Sénarmont method after isochronal thermal annealing for NGs with a time step Δt = 30 min [45].

**Relationship between nanogratings and the modeled optical properties:** To numerically compute the optical retardance R from the NGs structure (i. e., arising from its form birefringence), a series of expressions, given in Ref. [51], and also provided in the Appendix, are employed. In short, NGs in the investigated glasses are all composed of a subwavelength assembly of porous nanolayers with a significant refractive index contrast [4]. The LB can be calculated from knowing this opto-geometric morphology, along with the refractive indices of the porous nanolayers and in between those layers (e.g., considering pristine glass) [6,52].

## 3. Analysis of the experimental retardance data with VAREPA framework

From both modeled or experimental accelerated ageing data (e.g., using isothermal or isochronal annealing experiments), the VAREPA framework can be applied to predict the lifetime of an optical property [38] as it was extensively done in the literature for UV written fiber Bragg gratings [31,53,54], providing that several assumptions are fulfilled. This allows to find the distribution function of the activation energy by providing only one constant: the erasure rate $k_0$ of the main erasure reaction, that is assumed to be the nanopores erasure. In simple cases, this is determined by a simple variable change from (t, T) to an activation energy, noted $E_d$. Then one can use this distribution function to predict the lifetime at any temperature and it is also possible determine some suitable annealing conditions for extending a lifetime [38]. In this work, we focus on the evolution of $R_{norm}$ (t, T). There are two groups of assumptions that should be validated in order to use this framework.

### 3.1. First group of assumptions (on the reaction)

a) Only one elementary reaction is involved in the process. The reaction pathway is symbolically written as $B \rightarrow A$, and B and A are respectively attached to some configurations in the nanostructured and rejuvenated glass. In our case, we will assume that the measured or simulated $R_{norm}$ is originated from a single type of contribution through the form birefringence, e.g., the change of nanopores radius in the porous nanolayers. We will thus neglect others processes such as Si formation or point defects as mentioned above if a suitable annealing (burnt-in) erased them.

b) The process is an Arrhenius based process with a reaction constant k ($E, T$) that is thermally activated. $k(E, T) = k_0e^{-E_a/k_B T}$ where $k_B$ is Boltzman constant (eV/K), $k_0$ is a pre-exponential factor ($s^{-1}$), $E_a$ the activation energy (eV), and T the temperature (K).

c) $E_a$ is distributed here (not a single value) in the case of nanopores erasure, arising from the disordered nature of glass as it is the case for radiation-induced loss, recovery of photo- or radio-darkening [55], stress relaxation [56,57], viscosity [58] or UV-induced refractive index changes [38].

### 3.2. Second group of assumptions (on the process)

We assume that $R_{norm}$ is formed by the superposition of elementary retardance contributions that we call $R_{elem}$. We consider that the erasure mechanism of the elementary reaction with activation energy $E_a$ follows a first order reaction kinetics, and this in all investigated oxide glasses. As a result, $\frac{dR_{elem}}{dt} = -k(E, T)R_{elem}$ and so we assume that $R_{elem}$ (t, T) can be written as follow: $R_{elem}(E_a, t, T) = e^{-k_0 t \cdot e^{-E_a/k_B T}}$. Then, with the assumption of additivity of the elementary contributions, we get $R_{norm}(t, T) = \int_{0}^{\infty} R_{elem}(E_a, t, T)g(E_a)dE_a$ where $g(E_a)$ is the distribution
of the activation energy ($\int_{0}^{\infty} g(E_a)dE_a = 1$). This integral is solvable assuming that $R_{elem}(E_a, t, T)$ can be assimilated to a rapid varying function at a specific energy called demarcation energy, $E_d = k_B T \ln(k_0 t)$, associated to any given $T$ and $t$ couple (a Vand cutting energy [34,59]), and corresponds to the boundary between an achieved and a non-achieved reaction. The consequence of this last assumption is that $R_{norm}(t, T)$ becomes $R_{norm}(E_d) = \int_{0}^{E_d} g(E_a) dE_a$ only dependent on $E_d$ whatever $T$ or $t$ [40,57] provided the distribution is independent on temperature or time. With a proper choice of $k_0$, isotherms or isochrons can be collapsed into a unique curve, the so-called Master Curve (MC), here describing $R_{norm}(E_d)$. Note that the shape of the distribution function $g(E)$ can be obtained from the differentiation of the MC against $E_d$. It is important to note that several reasons may lead to the impossibility to find a MC, e.g., if there are multiple reactions with additional rate constants or distributions associated. Finally, on the premise of meeting a single function gathering all isotherms, the master curve plot (i.e., the lifetime function) can be used to predict the NGs lifetime at various temperatures.

To summarize, in the following we suppose that the measured optical retardance $R_{norm}(t, T)$ of nanogratings during an annealing treatment is mostly proportional to one “chemical species” B included in the nanogratings namely the population of nanopores. In this Arrhenius first order erasure reaction (e.g., nanopores erasure in ta viscous fluid), the activation energy will be distributed, as it is the case for viscoelastic relaxation or stress relaxation mechanisms in glasses. This overall approach was successfully applied in silica to a many isochronal data made by Wang et al. [40] proving that VAREPA can be exploited for porous nanogratings. In the next sections after introducing Rayleigh-Plesset equation and developing a new distributed Rayleigh-Plesset model, we will investigate if VAREPA can be reliably applied on a set of data obtained using Rayleigh-Plesset and which are the restrictions on its application for an accurate lifetime prediction.

### 4. Rayleigh-Plesset modelling of nanopores erasure kinetics

In the following calculations of the optical retardance as a function of annealing conditions in a context of thermal ageing, the Rayleigh-Plesset (R–P) equation is employed as initially suggested by Rudenko [60]. The Rayleigh-Plesset (R–P) equation was used to accurately simulate the evolution of nanopores erasure $R_{pore}(t, T)$ over time ($t$) and temperature ($T$), and then compared to isochronal annealing experiments. This equation describes the evolution of a spherical bubble inside an incompressible Newtonian fluid, and it can be expressed as follows:

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

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 time (in s); $\eta(T)$ is the glass viscosity (in Pa·s), and $\sigma$ is the surface energy (around 0.3 J m⁻² for silica [61], whose temperature dependence remains within <5% and does not change lifetime prediction significantly). Note that the non-sphericity of the nanopores, which are oblate [62], does not change the physical observation and thus that nanopores are considered spherical. Initial conditions are $R_{pore}$ (at $t=0$) = $R_{ini}$ i.e. the initial nanopore size and $\frac{dR_{pore}}{dt} = 0$ at $t=0$.

In the R–P equation, the second derivative and the square of the first derivative have very small contributions numerically in our conditions and are therefore neglected, i.e., slow dynamics so no inertial pressure. $\Delta P$ is small enough and thus set to be null in the erasure case. As the glass viscosity $\eta(T)$ is the main driven force in this process, this gives a simplified R–P equation: $\frac{dR_{pore}}{dt} = \frac{d(R_{pore} - R_{ini})}{dt} = -\frac{\sigma}{2\eta}$. It leads to $\frac{R_{pore}(t, T)}{R_{ini}(0.20\cdot C)} = 1 - \frac{\sigma\cdot t}{2\eta(T)R_{ini}}$ which can be also deduced from Ref. [45].

Then, and as described in Appendix, by using the effective medium

theory using the Maxwell-Garnet equation [63], an average refractive index of the porous nanolayer can be calculated based on nanopore radius $R_{pore}(t, T)$ introduced in a volumic filling factor (FF), i.e., the volume fraction occupied by the nanopores inside the nanolayers. We assume also that non-sphericity contribution to the birefringence is negligible. Finally, $LB$ can be calculated using the so-called form birefringence model [51,52]. In these calculations, $R_{norm}(t, T)$ or $LB_{norm}(t, T)$ are normalized by the initial value before annealing $R_{norm}(0, T)$ and $LB_{norm}(0, T)$ respectively. In other words, $R_{norm}(t, T) = f(1 - \frac{\sigma\cdot t}{2\eta(T)R_{ini}})$ with $f(t=0, \forall T) = 1$ and $f(t > \frac{2\eta(T)R_{ini}}{\sigma}, T) = 0$. Recently, the thermal stability of form birefringence was quantitatively correlated to the one of nanograting’s porosity-filling factor, which has been demonstrated recently in various oxide glasses including the ones of this study [45,51]. Their erasure kinetic, ascertained by isochronal retardance measurements and attributed to the collapse of nanopores, was well predicted from the Rayleigh-Plesset equation.

### 5. A distributed Rayleigh-Plesset model and its impact on the erasure process

In the following we will now develop a “distributed Rayleigh-Plesset” model by considering elasticity that is usually not negligible in the solids. At the beginning of the discussion, the variation of the apparent activation energy of the dynamical viscosity (macroscopic) with the temperature arises from a distribution of activation energy ($E_i$) at the microscopic level. The relevant elementary process is the change of distance ($l$) between two groups of atoms under a stress $\varphi$ that is partly considered in the R–P model, it is the deformation: $\varepsilon = \ln(\frac{l}{l_0})$, $l_0$ is the distance at rest. Due to the large number of structural configurations in the glass, the way to respond to the stress solicitation leads to consider a distribution of the activation energy according to a distribution function $g(E_i)$ normalized on all possible values (ca. 0 to $\infty$). Usually, one can simply use Maxwell viscoelastic model that works well for soft solids (e. g., in the vicinity of softening temperature) but here we used Kelvin-Voigt visco-elastic model allowing for example to model creep and deformation behavior at low loading. So, the relation between the stress and the deformation takes the following form:

$$ \varphi = G \cdot \varepsilon_i(E_i, t, T) + \eta(E_i, T) \frac{d\varepsilon_i(E_i, t, T)}{dt} \text{ and } \eta(E_i, T) = \eta_0 \exp\left(\frac{E_i}{k_B T}\right), \text{ where } G \text{ is the shear modulus. This gives:} $$

$$ \frac{d\varepsilon_i(E_i, t, T)}{dt} = - \frac{G}{\eta(E_i, T)} \cdot \varepsilon_i(E_i, t, T) + \frac{\varphi}{\eta(E_i, T)} $$

As $t$ is independent of $T$ and $E_i$, the differential equation can be solved when considering a constant stress $\varphi$ suddenly applied and we obtain:

$$ \varepsilon_i(E_i, t, T) = \frac{\varphi}{G} + \left(\varepsilon_i^0 - \frac{\varphi}{G}\right) \exp\left(-\frac{G}{\eta(E_i, T)}t\right) $$

We note that when $G$ is small in front of $\frac{\eta(E_i, T)}{t}$, $\varepsilon_i(E_i, t, T) = \varepsilon_i^0 + \frac{\varphi}{\eta(E_i, T)}t$. Also note that for a solid, $G$ is never null. For obtaining the macroscopic value considering the glass disorder, we must make the sum over all configurations, on all values of $E_i$. We get:

$$ \varepsilon(t, T) = \int_{0}^{\infty} \left( \varepsilon_{i}^{0} - \frac{\varphi}{G} \right) \exp \left[ -\frac{Gt}{\eta_{0}} \exp \left( -\frac{E_{i}}{k_{B}T} \right) \right] g(E_{i}) dE_{i} + \frac{\varphi}{G} $$

The first term of this solution can be treated as in the frame of VAREPA as the double exponential function can be assimilated to a Heaviside function of $E_i$. The cutting energy defines a demarcation energy. $E_d = k_B T \ln(k_0 t)$ with $k_0 = \frac{G}{\eta_0}$. Note this is also the constant rate, $k_0$, we obtained using a simple Maxwell model when considering that $G$ is small enough (e.g., in the vicinity of melting temperature). This part will be just a function of $E_d$ after expressing $t$ into $E$. Note that $\frac{1}{\eta(T)} = \int_{0}^{\infty} \frac{1}{\eta_0} \exp\left(-\frac{E_i}{k_B T}\right) g(E_i) dE_i$. This defines the “average viscosity”.
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Plotting of $dx/dt_{eff}$ versus $x$, (b) plotting of $x$ versus effective time, $t_{eff}$, for various values of the shear stress $G$ ($G = 0$ is for a liquid); for $G < e\sigma/2R_{ini}$ we have a monotonous behavior. The situation described here corresponds to high temperatures much beyond $T_g$ when $G$ decreases strongly.</figcaption></figure>

Let us now make the remark that the nanopore radius and their eventual deformation (non-sphericity mentioned above) are quantities of the same nature and thus similar law can be applied. Now, we consider thus just one nanopore and let us apply what we showed above. Following the R-P equation and its simplification, the elementary reaction including now a visco-elastic term by taking into account that $\epsilon_i(E_i, t, T) = \ln(R_i(E_i, t, T)/R_{ini})$ and $\phi_i = -\frac{\sigma}{2R_i(E_i, t, T)}$ the interfacial term (with the orientation convention at the interface) and $\sigma$ the surface tension, we get:

$$ \frac{dR_i(E_i, t, T)}{dt} = -\frac{G}{\eta(E_i, T)} R_i(E_i, t, T) \ln\left(\frac{R_i(E_i, t, T)}{R_0}\right) + \frac{\phi R_i(E_i, t, T)}{\eta(E_i, T)} - \frac{G}{\eta(E_i, T)} R_i(E_i, t, T) \ln\left(\frac{R_i(E_i, t, T)}{R_{ini}}\right) - \frac{\sigma}{2\eta(E_i, T)} $$

Without $\phi$, the differential equation is solvable but not with it. What we must know is if the solution is a rapid varying function with $E_i$ or not. We can rewrite the equation as the following posing $x_i(E_i, t, T) = \frac{R_i(E_i, t, T)}{R_{ini}}$.

$$ \frac{dx_i(E_i, t, T)}{dt} = -\frac{G}{\eta(E_i, T)} x_i(E_i, t, T) \ln x_i(E_i, t, T) - \frac{\sigma}{2\eta(E_i, T)R_{ini}} $$

A plot of $dx/dt$ versus $x$ reveals a monotonous behavior according to time and the ratio as shown in Fig. 1. When $2GR_{ini}/e\sigma < 1$ there is a monotonous behavior (as for $G = 0$, for a liquid). However, when $2GR_{ini}/e\sigma > 1$, the nanopores do not disappear and can even grow before to be erased. This gives an aspect of the nanopore resistance, i.e., it is necessary to heat above the glass-transition temperature $T_g$ and then the viscosity determines the temporal behavior. Note the interfacial energy is not obviously increasing on heating. At high temperatures, there is no simple solution for $x$, but assuming $G$ small enough in front of $e\sigma/2R_{ini}$ we can write:

$$ x_i(E_i, t, T) = 1 - \frac{\sigma t}{2\eta(E_i, T)R_{ini}} - \frac{G}{\eta(E_i, T)} \int_{x(0)}^{x(t)} \frac{x \ln(x)}{dx/dt} dx $$

The numerical analysis shows that this expression leads to a rapid varying function of $x_i(E_i, t, T)$ with the energy $E_i$ when $G$ is large enough in front of $\frac{\sigma}{2R_{ini}}$ although smaller than $e\sigma/2R_{ini}$. In that case, the average variation of normalized nanopore radius that is obtained by summation on activation energy is $x(t, T) = \int_{0}^{\infty} x_i(E_i, t, T)g(E_i)dE_i = \int_{E_d}^{\infty} g(E_i)dE_i$ with a demarcation energy conventionally taken for $x_i = 0.5$, this a good

approximation and the resulting distribution is not dependent of $t$ or $T$. Note that when $G$ decreases strongly much beyond $T_g$, the approximation is worse. So, the pseudo demarcation energy obtained in the frame of the conventional definition is $E_d(t, T) = k_B T \ln[\frac{\sigma t}{2\eta_0 R_{ini}}]$. Here we can state $\frac{\sigma}{2\eta_0 R_{ini}} = k_0$, that is also the reaction rate constant we can obtain using Maxwell model if $G$ is large enough in front of $\frac{\sigma}{2R_{ini}}$ (e.g. in the vicinity of $T_g$). As the normalized retardance is obtained by $x(t, T)^6$, the same conclusion applied and the expressions of the pseudo-distributions are:

$$ \text{For isochrones, } \frac{\partial R_{norm}(E_d, t)}{\partial E_d} = \frac{Bln10}{0, k_0 t e^{-\frac{E_d}{k_B} \ln(k_0 t) - T_0}} > 1 \\ \frac{6Bln10}{k_B \ln(k_0 t)} \left[ 1 - e^{-\frac{Bln10}{E_d/k_B \ln(k_0 t) - T_0}} \right]^5 \frac{k_0 t e^{-\frac{Bln10}{E_d/k_B \ln(k_0 t) - T_0}}}{(E_d/k_B \ln(k_0 t) - T_0)^2}, \text{ otherwise} $$

$$ \text{For Isotherms, } \frac{\partial R_{norm}(E_d, T)}{\partial E_d} = \left\{ \begin{array}{ll} 0, & e^{\frac{E_d}{k_B T}} \frac{Bln10}{T-T_0} > 1 \\ \frac{6}{k_B T} \left(1 - e^{\frac{E_d}{k_B T} \frac{Bln10}{T-T_0}}\right)^5 e^{\frac{E_d}{k_B T} \frac{Bln10}{T-T_0}}, & \text{otherwise} \end{array} \right. $$

A priori, we have thus to make the difference between $k_{0\_isotherms}$ and $k_{0\_isochrones}$. The plot of these $T$ dependent distribution functions in the case of SuprasilCG is shown in Fig. 2. We note that the energy range is similar and some variation with either the isochronal time or the isothermal temperature but if the variations are moderate and thus are weakly detectable on the experimental data. This dispersion arises from the dispersion of activation energy on the viscosity in the model. The relations between the position of the maximum and the viscosity at the same $T$ can be obtained. Through derivation of the viscosity law as a function of $T$, the activation energy in the viscosity is $E_{visco} = \frac{k_B T^2 Bln10}{(T-T_0)^2}$. For isotherms: $E_{isoT} = \frac{k_B T Bln10}{T-T_0} - 1.8 k_B T$. As seen in Fig. 2b, the shift remains quite small, 0.1–0.2 eV with respect to the 6 eV. So, the position of the pseudo master curve at $R_{norm} = 0.5$ is almost coincident with the activation energy of the viscosity at the operating $T$. So, from that point, the VAREPA framework can now be tentatively applied to data obtained through Rayleigh-Plesset data thus we can make the link not only between the matter and $k_0$ but also between the distribution and the temperature dependence of viscosity.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Plotting of the T dependent distribution functions for SuprasilCG silica. (a) is for isochronal experiments, while (b) is for isothermal ones.</figcaption></figure>

# 6. Applying VAREPA approach to data provided by Rayleigh-Plesset model

## 6.1. Building the master curve from Rayleigh-Plesset simulated $R_{norm}$ values

Normalized retardance values, $R_{norm}(t,T)$, were calculated based on the Rayleigh-Plesset equation (see Section 4), for a series of isothermal annealing conditions. Fig. 3 shows such predicted $R_{norm}$ values in the case of SuprasilCG silica for several temperatures ranging from 1100 °C to 1350 °C for a total time duration of 24 h (plotted according to log time in Fig. 3a), and according to an activation energy defined as $k_B T \ln(t)$ in Fig. 3b. It is worth pointing out that during the numerical simulation, the effect of temperature was introduced by varying the viscosity $\eta(T)$. The viscosity was calculated using Vogel-Fulcher-Tammann (VFT) law taking the form $\log(\eta, Pa \cdot s) = A - 1 + \frac{B}{T - T_0}$, and all parameters (A, B and $T_0$) are available in Ref. [45]. Here, the initial pore size was set to 30 nm for SuprasilCG, which is a typical value that was obtained from electron microscopy imaging analysis from Ref. [45]. As expected, $R_{norm}(t,T)$ decreases faster for the highest temperatures, for which the viscosity is lower. Indeed, a lower viscosity facilitates the collapse of the nanopores, hence the decrease of $R_{pore}$ and FF, and ultimately $R_{norm}$. If all the curves follow the same trend, they appear shifted when plotted as a function of $k_B T \ln(t)$ as seen in Fig. 3b. The shift is $k_B T \ln(k_0)$ and one can tentatively try to gather the isotherms into a single curve by identifying the $k_0$ value. To do this, the $k_0$ value is progressively varied using least squares optimization until the $\chi^2$ value reaches a minimum. Here, the optimal/best curves overlapping is found for $k_0 = 7.45 \times 10^{16} \text{ s}^{-1}$ ($\ln k_0 = 38.85$ where $\chi^2$ is minimized). The results are provided in Fig. 4, with the merged curve identified as a Master Curve, and the

optimization shown in the inset. From the inset, the uncertainty on $k_0$ can be calculated in the range of $\ln k_0 = 38.8 \pm 0.8$ with a 90% confidence interval. So, from that point, it appears that VAREPA framework may be applied on Rayleigh-Plesset data but one must keep in mind the restrictions mentioned above, in particular the T dependence of the distribution.

## 6.2. Comparing the modeled master curve to experimental ageing data

VAREPA framework was now applied for 8 oxide glasses namely Schott Borofloat 33, BK7 and AF32, Corning ULE, EagleXG and 7059, Sodalime, and Heraeus Suprasil CG whose viscosity laws and compositions can be found in Ref. [45]. Here we have fixed the nanopores radius ($R_{ini} = 30–50 \text{ nm}$) [2,3] and the surface tension ($\sigma = 0.3 \text{ J m}^{-2}$) for all glasses and we have simply adjusted $\eta(T)$. Following this procedure, it was possible to simulate thermal aging data using Rayleigh-Plesset model to feed coherently VAREPA, so we obtain a Master Curve and the corresponding $k_0$ for all investigated glasses. A set of specific examples is shown in Fig. 3a with Borofloat 33, ULE and SuprasilCG. Following this work, and to validate these predicted lifetime curves, step isochronal annealing experiments were performed to verify if the datapoint would align on Master Curve. Since the experimental data were obtained from step isochronal annealing, one must first evaluate if the isochronal datapoints are independent. The achieved condition for independency of the datapoints is $(\Delta t \cdot k_0)^{-(\Delta T/T_{max})} \ll 1$ [54], where $\Delta t$ is the time step (30 min) of the isochronal annealing experiment; $\Delta T$ is the temperature step separating two datapoints; $T_{max}$ is the maximum temperature during the isochronal experiment. In this work, this criterion is mostly satisfied for all data.

In Fig. 5a, the experimental $R_{norm}$ ($\Delta t = 30 \text{ min}, T$) resulting from step-isochronal annealing as a function of $E_d$ are drawn and compared to

<figure><img src="image_5.png" /><figcaption>Fig. 3. (a) Calculated $R_{norm}$ as a function of time for a series of isotherms. (b) Identical $R_{norm}$ data but plotted as a function of $k_B T \ln(t)$. Data were calculated for SuprasilCG silica glass with an initial pore diameter $R_{ini}$ of 30 nm and $\sigma = 0.3 \text{ J m}^{-2}$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Master curve of simulated data in Fig. 1(b) with an optimized *k*₀ value. Note that *E*₉ for *R*ₙorm = 0.5 is around 5.8 eV for SiO₂. The inset shows the least squares method to obtain the best *k*₀. All data are for SuprasilCG silica.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** (a) The MC curves of Borofloat 33, ULE, and Suprasil CG glasses, along with experimental data obtained from step isochronal thermal annealing. (b) The predicted erasure *E*<sub>d</sub> values of different oxide glasses as a function of experimental one taken at *R*<sub>norm</sub> = 0.5. The error bar is from the confidence interval of *k*<sub>0</sub>.</figcaption></figure>

the theoretical master curves within the whole *E*<sub>d</sub> interval for those three
glasses. It appears there that experimental results are matching the
predicted master curves deduced from the Rayleigh-Plesset modeling
within the error bar from the confidence interval of *k*<sub>0</sub>. To generalize the
comparison between predicted and the experimental master curve for all
glasses, we tentatively compared the *E*<sub>d</sub> values (predicted and experi-
mental ones) for multiple degrees of erasure. As an example, in Fig. 5b
the predicted versus experimental *E*<sub>d</sub> values for 8 oxide glasses are dis-
played, taking *R*<sub>norm</sub> = 0.5 (corresponding to 50% erasure). As can be
shown in the figure, experimental and simulated results are in excellent
agreement. Note that this degree of erasure was validated from 20% up

to 99% (not presented here), strengthening our approach. The quite
good agreement between model and experimental data may suggest that
the glass viscosity η(T) has not been impacted significantly by the laser
writing process and the formation of nanogratings while Frenkel defects
and eventually Si formation happen at the nanopores interface due to
the SiO₂ reduction as discussed in section 5.

7. Lifetime prediction

Using VAREPA framework, we show above that R<sub>norm</sub>(t, T) can be
expressed as a function of the unique variable called demarcation energy
$E_d$. The curve $R_{norm}(E_d)$ is called Master Curve. In simple cases, it is unique whatever the $(t, T)$ couple may be but here we show that the distribution is $T$ or $t$ dependent. So, we can note that $T$ is not equivalent to $\ln(t)$ in $E_d$ expression and thus isochronal ageing data are not equivalent to isothermal ageing data for establishing MC. Anyway, for specific using conditions ($t_{use}$, $T_{use}$), we can determine $k_0$ values for any glasses and thus the corresponding $E_d$ limit. Then the MC plot allows the user to predict the optical component lifetime, providing that the anticipated conditions of use $E_d = f(t_{use}, T_{use})$ correspond to a point on the MC that has been actually sampled during the annealing experiments. Let us go on further to exemplify such prediction in all investigated optical glasses. There are two solutions: one is to fit the Master Curve with a polynomial function and the second one is to fit a known expression. Since the life function is now available, it is possible to calculate the lifetime for either isothermal or isochronal working conditions, e.g. $T = 600$ °C and 30 years, by using the optimized $k_0$ value (determined following the above procedure) As a results, for predictions shown in Fig. 6, $R_{norm}$ will decrease to 0.95 at 600 °C for AF32 glass and 30 years' duration or 42% erasure in Eagle XG glass while there is no significant erasure for silica in these conditions. Among the investigated glasses, Suprasil silica glass (above hundred million years) has the longest lifetime, whereas BK7 (around 10 min) holds the shortest one. In addition, it is also possible to increase the lifetime of these components eventually. The principle is based on the distribution of activation energies and thus nanopores stability at various sites. A burning step during the process suppresses the less stable sites, which cannot be achieved with a simple exponential kinetic. This widespread method has been proven to be effective for enhancing the stability of index changes in UV written Fiber Bragg Gratings [39] but also in fs-laser induced FBG [40]. Of interest, our approach based on nanopores erasure kinetics is quite general and one can predict others optical properties related to nanopores by linking the adequate optical model to the nanopores population. For example, one can predict not only nanopores birefringence or nanogratings form birefringence but also light scattering or anisotropic light scattering dependence on time and temperature. Another possibility would be to predict refractive index changes using effective medium theory or eventually more complex objects such as the spectral properties of FBG for various sensing applications.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** The predicted lifetime of normalized retardance for isothermal annealing at 600 °C in all investigated glasses. Dots are for predictions made with VAREPA approach whereas full lines are for predictions made using the Rayleigh-Plesset equation.</figcaption></figure>

## 8. Discussion

Within this work, a distributed Rayleigh-Plesset model was developed to make the link with VAREPA framework and then to obtain master curves using a database made on simulated R–P data. The validity of the approach was further confirmed using step isochronal annealing experiments. From that point we performed some examples of lifetime prediction according to ($t_{use}$, $T_{use}$) based on a rationale extrapolation made within VAREPA that was successfully compared to Rayleigh-Plesset predictions.

The main driving force is the interfacial force, but the viscosity is the limiting process. It is commonly assumed that shear viscosity is a thermally activated process. Viscous flow follows an Arrhenius law in the high and low (typ. below the strain point) temperature limit regimes with constant activation energy in a Frenkel-Andrade model [64]. However, between high and low temperature regimes, the activation energy of flow changes and the viscosity cannot be described simply using the Arrhenius-type behavior. For example, the activation energy of viscosity varies with temperature [65–68] and there is a need to introduce a distribution of activation energy [58,69–71]. More specifically, in the intermediate $T$ range between strain and melting points, the VFT model [72,73] is commonly used especially in the industry, which is derivate from free volume theory, and can be written as $\log(\eta, Pa \cdot s) = A - 1 + \frac{B}{T-T_0}$, where A and B are temperature independent and $T_0$ is the extrapolated temperature where viscosity tends to infinity. This empirical equation of viscosity converts to $\eta(T) = \eta_0 e^{\frac{B}{T-T_0}}$ where $\eta_0 = 10^{A-1}$. Developing the distributed Rayleigh-Plesset model, we have identified above that $k_0$ takes the following form $k_0 = \frac{\sigma}{2n_0R_{min}}$ and we have extracted the expression of the master curve and its differentiation (i.e., the distribution) that is not solely proportional to $E_d$ but also to the working temperature (for isotherms). The origin of this $T$ dependence is simply related to the chosen expression of the glass viscosity $\eta(T)$ and inherent to its non-Arrhenius behavior. In addition, in the literature, the apparent activation energy that can be derived from $\ln \eta$ vs $1/T$ can be found around 4.0 eV for $T > 2000$ °C (up to 2400 °C for Ref. [68]) and around 8.1 eV for $T = 1400$ °C [67,74–76] for silica. This agrees with recent results from Shao et al. in the 1000–1200 °C temperature range [65]. For sodalime silicate, the apparent activation energy lied around 4–6 eV for temperatures within $T_g - 1.4T_g$ [77]. These values agree with activation energies deduced within our framework, which originated from the viscosity. Note that the irradiated glass structure is slightly different with Frenkel defects, silicon formation [78] at the interfaces due to the $SiO_2$ reduction and a larger fictive temperature after a first pulse resulting in a smaller viscosity with different $\eta_0$ and activation energy. However, within the logic of the prediction procedure, there is no importance excepted that the proof of the model is more difficult.

In the intermediate temperature range (between strain and melting temperatures), Adam-Gibbs (AG) [79] and Avramov and Milchev (AM) [70] developed viscosity models based on a different theory which fits well viscosity measurements like the VFT model (relative to an Arrhenius law). Of interest here, AM viscosity model assumes that the magnitude of activation energy varies due to the disordered nature of glass. Consequently, the distribution function of the activation energy is determined by its configurational entropy $S_{conf}(T)$ [58]. The viscosity can be written as $\ln \eta(T) = A_{AM} + 2, 3(12, 5 - A_{AM})\left(\frac{T_g}{T}\right)^\alpha$ where the glass-transition temperature ($T_g$) is defined as $\ln \eta(T), Pa \cdot s = 12.5$ and $\alpha$ is the Avramov fragility parameter, that is related to the distribution width. In this model the resulting activation energy distribution is thus a $T$ dependent function as we discussed above. However, if the working conditions ($t_{use}$, $T_{use}$) fall down within a reasonable temperature range, e. g., below or around the strain point, the viscosity can thus be approximated by an Arrhenius law (i.e. the Frenkel-Andrade model [64]). As a result, the Master Curve and the underlying distribution $g(E_d)$ will be independent of the temperature within this restricted range of
<figure><img src="image_9.png" /><figcaption>Fig. 7. Relationship between lnk₀ and Angell glass fragility m in the investigated glasses. k₀ values were determined by feeding VAREEPA framework using simulated RP data and confirmed through step isochronal annealing experiments.</figcaption></figure>

temperatures.

Trying to bring some roadmap for engineers, one can compare k₀ values (obtained using VAREPA being fed by Rayleigh-Plesset data) to the Angell fragility, m, deduces from the slope of the glass viscosity-temperature curve when T approach T_g, thus defined as m = $\frac{\partial \log \eta(T)}{\partial \ln T} |_{T=T_g}$ [80]. It reflects the strong or fragile behavior of the glass material. From the AM expression we obtain m = α(1.25–A<sub>AM</sub>), where A<sub>AM</sub> = ln(η₀) which is thus link with the above-suggested expression of k₀ = $\frac{\sigma}{2\eta_0 R_{in}}$. Illustrating this purpose, Fig. 7 shows ln(k₀) as a function of the fragility m revealing a linear correlation for all investigated oxide glasses. Note that here we assume surface tension σ remains similar (around 0.3 J m⁻² for all glasses) and the same initial pore size (around 30 nm) in all investigated oxide glasses. As a guideline, the erasure constant rate k₀ is much smaller for “strong glasses” like SuprasilCG and ULE whereas k₀ own higher values (faster nanopores erasure) for “fragile glasses” such as BK7 and Soda-lime glass while alumino-borosilicate remain in the intermediate range.

## 9. Conclusion

In conclusion, the VAREPA framework was applied to determine the lifetime of IR-fs Type II modifications (i.e., nanogratings) written in 8

commercial optical glasses. Providing several assumptions are fulfilled, this Master Curve could effectively be obtained in all investigated oxide glasses and was shown to originate from the distribution of the glass viscosity thanks to a distributed Rayleigh-Plesset model. The approach was further strengthened by comparing the Master Curves to accelerated aging experiments through the monitoring of nanogratings birefringence signature. In addition, the log of the erasure constant rate, ln(k₀), was shown to linearly scale up together with the Angell’s glass fragility. These conclusions are expected to provide guidelines for the future exploitation of nanogratings [2,3] and related devices such optical data storage, birefringent devices or sensors written in wide range of oxide glasses.

## 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.

## Disclosures

The authors declare no conflicts of interest.

## Author contributions

Conceptualization, M.C. and M.L.; Funding acquisition, M.L.; Investigation, Q.X.; Methodology, Q.X., M.C., B.P. and M.L.; Project administration, M.L.; Resources, M.L.; Supervision, M.L. and M.C.; Validation, Q.X.; Visualization, Q.X. and M.C.; Writing—original draft, Q.X.; Writing—review & editing, Q.X., M.C., B.P. and M.L. All authors have read and agreed to the published version of the manuscript.

## 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.

## Acknowledgments

Qiong Xie is grateful to Jiafeng Lu for the help of Matlab coding.

## Appendix

The form birefringence related to the so-called nanogratings is quantitatively correlated to the porosity-filling factor of these nanostructures has been demonstrated by our group recently [51]. The nanopore size diameter from the R–P equation needs to be converted into a normalized retardance (R) value. In the first step, the Maxwell-Garnet equation is first used to calculate the average refractive index of the porous nanolayer (n<sub>p</sub>):

$$n_{pl} = \sqrt{\frac{n_G^2 + \frac{(1+2FF)}{3}(n_{pore}^2 - n_G^2)}{n_G^2 + \frac{(1-FF)}{3}(n_{pore}^2 - n_G^2)}} \quad (A1)$$

n<sub>G</sub> is the glass refractive index (taken as n<sub>G</sub> = 1.4599, index of silica at 551 nm), n<sub>pore</sub> is the nanopore refractive index (taken equal to 1). Another important factor is the filling factor (FF) which is defined as the proportion of the nanopores’ volume taken in a unit volume of porous nanolayer (V), so it is obviously related to the number of nanopores (N) with an averaged nanopore radius (R<sub>pore</sub>) and can be calculated using:

$$FF = N \frac{4\pi R_{pore}^3}{3V} \quad (A2)$$

The birefringence LB can be determined by the difference between the refractive indices of the ordinary axis (n<sub>o</sub>) and the extraordinary axis (n<sub>e</sub>). The following equation illustrates the detail:
$$ LB = n_o - n_e = \sqrt{\left[1 - \frac{\delta}{\Lambda}\right] n_G^2 + \frac{\delta}{\Lambda} n_{pl}^2} - \sqrt{\left[\frac{1 - \frac{\delta}{\Lambda}}{n_G^2} + \frac{\delta}{n_{pl}^2}\right]} \quad (A3) $$

$n_G$ represents the refractive index between the porous nanolayers, $\Lambda$ is the average spacing between nanolayers, $\delta$ is the porous nanolayer thickness, and $(\Lambda - \delta)$ is the interlayer thickness. With the birefringence $LB$ calculated, the retardance $R$ can be worked out by the expression $R = LB \times l$, where $LB$ is the birefringence and $l$ is the length of the birefringent object being measured. We assume the $R_{norm}(t, T)$ and $LB_{norm}(t, T)$ are normalized by the initial value $R_{norm}(0, T)$ and $LB_{norm}(0, T)$ respectively. They change square of FF and the sixth power of $R_{pore}$.

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