2.1 Description of the MELBA experimental testbed

Reported experiments were carried out on the MELBA^[ Reference Veinhard, Bonville, Courchinoux, Parreault, Natoli and Lamaignère ^{11 ]} experimental testbed located at CEA CESTA (France). Initially, a 1053 nm laser pulse ( $1\omega$ pulse) is created in a master oscillator with an energy of the order of nJ and a pulse duration of 200 ns. A phase modulation is applied at 2 GHz to prevent stimulated Brillouin scattering during laser propagation. Then, the pulse is amplified through a fiber amplifier.

Two all-fiber Mach–Zehnder interferometers are used to shape the temporal profile of the laser pulse. The first interferometer outputs the user-defined laser pulse. The second interferometer improves the extinction ratio. The overall extinction ratio at the output of both Mach–Zehnder interferometers is 52 dB. For instance, it is possible to create flat-in-time (FIT) or Gaussian pulses of different pulse durations. In fact, the user can program any laser pulse with a temporal resolution of approximately 100 ps and a maximum pulse duration of approximately 20 ns.

To apply fast temporal modulations to the laser pulse and recreate the impact of FM-to-AM conversion on the temporal profile, an all-fiber electro-optic amplitude modulator with a 40 GHz bandwidth was set up. The amplitude modulator can be coupled to two different radio frequency (RF) drivers:

• • a low-frequency (LF) four-channel RF driver generating signals at 2, 4, 6 and 8 GHz adjustable in amplitude and phase;

• • an HF three-channel RF driver generating signals at 10, 20 and 30 GHz adjustable in amplitude and phase.

The expression of the laser power P at the output of the modulator is expressed as follows:

(1) $$\begin{align}P(t)=\frac{P_0(t)}{2}\left(1+\sin \left(\sum \limits_{k=1}^{N_{\mathrm{CH}}}\frac{V_{k}}{V_{\pi, {k}}}\pi \sin \left(2\pi {kf}_0t+{\varphi}_{{k}}\right)\right)\right),\end{align}$$

where ${N}_{\mathrm{CH}}$ is the total number of channels, which depends on the RF driver. The LF driver has ${N}_{\mathrm{CH}}=4$ while the HF driver has ${N}_{\mathrm{CH}}=3$ . In the equation, ${P}_0$ is the power at the input of the modulator, ${V}_{k}$ is the voltage input of channel $k$ and ${V}_{\pi, {k}}$ is the voltage of channel k required to induce a phase shift of $\pi$ radians. Finally, ${f}_0=2$ or 10 GHz depending on the driver and ${\varphi}_{k}$ is the phase shift applied to channel k. The idea behind using an amplitude modulator instead of directly inducing FM-to-AM conversion is twofold: better control of the temporal profile and the generation of modulation frequencies with limited harmonics^[ Reference Rothenberg, Browning and Wilcox ^{7 ]} that are adapted to measurement bandwidths. Temporal measurements at 351 nm are carried out with 40 GHz fiber-coupled photodiodes connected to a 33 GHz oscilloscope. Thus, the overall bandwidth is 33 GHz.

After applying temporal pulse shaping, the laser pulse is amplified through a regenerative amplifier to approximately 10 mJ. Then, a four-pass amplifier followed by two booster amplifiers bring the energy up to 7 J. The laser pulse is converted in the UV range at 351 nm ( $3\omega$ pulse) thanks to two lithium triborate (LBO) crystals in a type-I type-II configuration. Typical $3\omega$ energies are around 3 J. A lens focuses the laser on the silica sample under study. The sample is located at an intermediary field, so the diameter of the beam is approximately 6 mm with a top-hat spatial shape. Figure 1 represents the fluence profile of the $3\omega$ beam incident of the sample test. Many high-end diagnostics, which will be presented subsequently, allow us to properly measure all laser parameters needed for the experiments.

Figure 1 MELBA $3\omega$ spatial profile on the fused silica sample.

2.2 Studying LID on the MELBA experimental testbed

The MELBA experimental testbed is used to study LID and laser damage growth. The analysis of LID on a given test site is composed of several steps. Firstly, the test site of a superpolished synthetic fused silica sample is irradiated with the $3\omega$ beam. Then, the detection of LID is made thanks to a macroscope that is inserted after each shot. The photography taken by the macroscope is binarized. The fluence profile is determined thanks to a charge-coupled device (CCD) camera with a $1392\times 1040$ pixels large sensor and a pixel area of $6.45\;{\unicode{x3bc} \mathrm{m}}\times 6.45\;{\unicode{x3bc} \mathrm{m}}$ . The local fluence for each pixel is calculated by normalizing the integral of the image to the energy measured by diagnostics for the shot. Each damage is treated as an object and the local fluence at which it is triggered is stored as data.

With enough tested sites, we can determine damage densities $\rho$ (cm ${}^{-2}$ ) for given fluence ranges ${F}_{k}$ (J cm ${}^{-2}$ ) thanks to the following formula:

(2) $$\begin{align}\rho \left({F}_{k}\right)=\frac{N_{\mathrm{dam}}\left({F}_{k}\right)}{S_{\mathrm{px}}{N}_{\mathrm{px}}\left({F}_{k}\right)},\end{align}$$

with ${N}_{\mathrm{dam}}$ being the number of damage sites for all fluences in range ${F}_{k}$ , ${N}_{\mathrm{px}}$ being the number of illuminated CCD pixels for all fluences in the range ${F}_{k}$ and ${S}_{\mathrm{px}}$ being the surface of a pixel. A damage law is defined as being the collection of all damage densities $\rho({F}_{k})$ determined on all tested sites. Every damage law comparison is carried out on a dedicated silica sample. This allows us to compare the impact of different pulses on LID with the same sample quality.

Statistical error bars are calculated with the collected damage densities and a histogram^[ Reference Lamaignère, Bouillet, Courchinoux, Donval, Josse, Poncetta and Bercegol ^{12 ]}, for each fluence range ${F}_{k}$ . Fluence error bars are usually in the 10%–15% range^[ Reference Lamaingère, Balas, Courchinoux, Donval, Poncetta, Reyné, Bertussi and Bercegol ^{13 ]}.

2.3 Characterization of the temporal profile

Damage laws determined with different pulse shapes will be compared to each other, all else being equal. This includes the fact that pulses will have the same pulse duration ${\tau}_{\mathrm{eff}}$ . As described in the ISO standard^{[ 14 ]} for laser damage, we will use the effective pulse duration to compare different pulses. It is determined via the following formula:

(3) $$\begin{align}{\tau}_{\mathrm{eff}}=\frac{\int P(t) \mathrm{d}t}{\max P(t)}=\frac{E}{\max P(t)},\end{align}$$

where $P(t)$ is the power profile, $\max P(t)$ is the peak power reached and $E$ is the energy of the pulse. Then, the magnitude of AM is defined thanks to modulation index $\beta$ as follows:

(4) $$\begin{align}\beta =\frac{\max P(t)}{\left\langle P(t)\right\rangle }-1.\end{align}$$

For instance, a modulation index of $100\%$ means that the AMs reach two times the mean power of the pulse. A representation of temporally modulated pulses will be presented later in Section 3.1. Note that in this paper, the effective pulse duration computation does not take the AM into account. It is solely the envelope (or unmodulated pulse) that is used to determine ${\tau}_{\mathrm{eff}}$ . If we did not take this into consideration, an increase of $\beta$ would directly lead to a decrease in the effective pulse duration.

3.1 Temporal characteristics of LF and HF laser pulses

Due to the spectral broadening of laser pulses, high-power facilities such as LMJ or NIF deliver pulses that may undergo FM-to-AM conversion. This phenomenon is responsible for power modulations of different magnitudes and frequencies. As previously discussed, we can reproduce such modulations on the MELBA testbed thanks to LF and HF modulators. Several AM configurations were used to assess the impact of LF and HF modulations, as follows.

• • An FIT reference case with an effective pulse duration of 4 ns.

• • Three amplitude modulated pulses at 2 GHz with modulation indices ${\beta}_{2\mathrm{GHz}}=55\%,110\%,130\%$ . The envelope is an FIT pulse of effective pulse duration equal to 4 ns.

• • Two amplitude modulated pulses at 10 GHz with modulation indices ${\beta}_{10\mathrm{GHz}}=55\%,95\%$ . The envelope is an FIT pulse of effective pulse duration equal to 4 ns.

• • An amplitude modulated pulse at 30 GHz with a modulation index ${\beta}_{30\mathrm{GHz}}=91\%$ . The envelope is an FIT pulse of effective pulse duration equal to 4 ns.

The reference FIT pulse as well as different modulated pulses is presented in Figure 2(a). Power profiles are normalized so that the average power is equal to 1. As several sites are tested for each configuration, the calculated modulation index varies from site to site. The presented modulation index is the average modulation index for all tested sites of a given configuration. Relative standard deviations for all modulated configurations comprise between 5 and 11 percentage points.

Figure 2 (a) Examples of power profiles measured with a 33 GHz-bandwidth oscilloscope and photodiode. From left to right: FIT reference and 2, 10 and 30 GHz pulses. The modulation index is approximately equal to $100\%$ for modulated pulses. Power profiles are normalized so that the average power is equal to 1. (b) The amplitude of the Fourier transform for each power profile shown in (a).

Figure 2(a) shows the amplitude of the Fourier transform of the pulses presented in Figure 2(a). For the 2 GHz configuration, we can detect harmonics up to 8 GHz, but these harmonics are quite low when compared to the fundamental at 2 GHz. The 10 GHz configuration only shows the fundamental at 10 GHz, without any noticeable upper harmonics. Finally, the 30 GHz configuration only shows a distinctive peak at 30 GHz, as the bandwidth is limited to 33 GHz. Complementary measurements with a streak camera at $3\omega$ showed that some shots exhibited an upper harmonic at 60 GHz, but with a low peak amplitude.

3.2 Comparison of damage laws for LF and HF modulated pulses

Four different temporal profiles were considered to test the impact of LF modulations on LID: a reference case and three 2 GHz modulated pulses of increasing modulation indices ${\beta}_{2\mathrm{GHz}}=55\%,110\%,130\%$ . The silica sample used for the test was polished by the THALES-SESO company with a thickness of 10 mm. Results of these test procedures are presented in Figure 3(a). Damage laws show that LF power modulations at 2 GHz give noticeably different damage densities in the $15\text{--}30\ \mathrm{J}\;{\mathrm{cm}}^{-2}$ fluence range, well beyond error bars. Indeed, damage densities for modulated pulses are systematically above the reference FIT case. Moreover, the higher the modulation index, the greater this difference

Figure 3 (a) Damage laws for the FIT reference pulse and three 2 GHz pulses of different modulation indices. (b) Damage laws for the FIT reference case, two 10 GHz pulses with different modulation indices and a 30 GHz pulse.

is. In the worst case, we observe a difference in laser damage density between 10 and 100 times when comparing the reference FIT pulse and the 2 GHz pulse with ${\beta}_{2\mathrm{GHz}}=130\%$ .

In order to assess the impact of the frequency of AMs, tests were performed with HF AMs on another 10 mm thick silica sample polished by THALES-SESO. Again, four different configurations were tested: a smooth reference FIT power profile, two 10 GHz modulated power profiles ( ${\beta}_{10\mathrm{GHz}}=55\%$ and $\beta =95\%$ ) and a 30 GHz modulated power profile with ${\beta}_{30\mathrm{GHz}}=91\%$ . HF power modulations were achieved thanks to the HF channel modulator of the MELBA experimental testbed. Damage laws obtained for these four configurations are presented in Figure 3(b). For HF modulations, the experimental results clearly show that the damage laws are the same for a large range of frequencies (10–30 GHz) and different values of modulation indices (from $\beta =55\%$ to $\beta =95\%$ ).

A final test was carried out on a new fused silica sample, which was provided by another vendor, namely the ZYGO company. The thickness of the sample is the same as the THALES-SESO sample, namely 10 mm. A damage law for a reference FIT pulse profile was first determined. Then, we tested for two different modulation amplitudes representative of LF and HF power modulations, namely a 2 GHz modulated pulse ( $\beta =122\%$ ) and a 10 GHz modulated pulse ( $\beta =117\%$ ). The results are presented in Figure 4. We observe tendencies similar to those presented in Figure 3. Indeed, LF AMs (2 GHz) lead to higher damage densities when compared to the reference FIT case. Moreover, there is no noticeable difference between the reference and the 10 GHz damage laws. In contrast with the results presented in Figure 3, we notice that the fluence values for which we observe a difference between the reference and the 2 GHz damage laws are between $10$ and $25\ \mathrm{J}\;{\mathrm{cm}}^{-2}$ .

Figure 4 Damage laws determined on a fused silica sample polished by another vendor (reference FIT pulse, 2 GHz modulation and 10 GHz modulation).

Comparison of the size damage sites for the experiment presented in Figure 4 is shown in Figure 5. The results show no dependence of the temporal profile on damage size. This means that AMs at 2 or 10 GHz do not have an influence on laser damage size, when compared to the unmodulated reference pulse. As far as damage morphology is concerned, we did not find any difference between damage sites originating from different temporal profiles. This is coherent with work from Chambonneau et al. ^[ Reference Chambonneau, Rullier, Grua and Lamaignère ^{15 ]} and Diaz et al. ^[ Reference Diaz, Chambonneau, Courchinoux, Grua, Luce, Rullier, Natoli and Lamaignère ^{10 ]}, which showed no important damage morphology change between unmodulated and modulated $3\omega$ pulses.

Figure 5 Histogram of the damage site diameter for the reference unmodulated pulse (no mod.) and the two amplitude modulations at 2 and 10 GHz. Data were analyzed from the results of the experiment presented in Figure 4.

Moreover, work from Diaz et al. ^[ Reference Diaz, Chambonneau, Courchinoux, Grua, Luce, Rullier, Natoli and Lamaignère ^{10 ]} showed that pulses with AM gave different damage laws at $3\omega$ when compared to a smooth reference case. In their work, damage densities determined with an MLM pulse of very high peak power and high frequencies are lower than those obtained with a smooth single-longitudinal mode (SLM) pulse. This result is difficult to compare to ours, as MLM pulses vary a great deal from shot to shot and span a much greater range of frequencies^[ Reference Diaz, Courchinoux, Luce, Rouyer, Rullier, Natoli and Lamaignère ^{16 ]}. Nevertheless, it still shows that amplitude modulated pulses can lead to different damage laws.

To explain the tendencies we observed experimentally, a model of laser–matter interaction was developed to assess the impact of AMs on the LID of fused silica at 351 nm. A description of the model and the results are presented in the following section.

4.1 Description of the model

Amorphous silica can be modeled with a band structure, with a gap between the conduction and valence bands equal to 8.825 eV. This energy gap is approximately three times the energy of a UV photon at 351 nm (3.52 eV). Yet, laser damage is observed despite the transparency of amorphous silica to UV light. The triggering of laser damage in fused silica is mainly explained by single-photon absorption of the incoming laser energy by SSD. Such defects are most often considered to be either absorbing pollutant particles or micro-cracks originating from the polishing process. In the context of the LMJ facility, the quality of silica optics is such that most absorbing pollutants particles should disappear with an etching process^[ Reference Pfiffer, Cormont, Fargin, Bousquet, Dussauze, Lambert and Neauport ^{17 ]}. Thus, we will consider subsurface micro-cracks as being the sole contributor to laser damage. The model was designed similarly to the model used by Diaz et al. ^[ Reference Diaz, Chambonneau, Courchinoux, Grua, Luce, Rullier, Natoli and Lamaignère ^{10 ]} and Grua et al. ^[ Reference Grua, Lamaignère, Chambonneau, Courchinoux and Néauport ^{4 ]}. The outline representing the model is represented in Figure 6.

Figure 6 Outline of the modeling of subsurface micro-cracks. The crack is modeled by a 100 nm void surrounded by amorphous silica. Absorption of the UV laser at the SiO₂/void interface is represented by an arbitrary absorbing defect layer.

The crack is modeled by a 100 nm void surrounded by amorphous silica. At the interface, the silica can absorb part of the laser energy due to the presence of an arbitrary absorbing defect layer. Examples of such defects can be found in Ref. [Reference Skuja and Non-Crystal18]. Dimension $L$ is considered much greater than dimension $x$ , as shown on the outline in Figure 6. Thus, the model is 1D along $x$ with planar symmetry. Physical processes that may occur during laser absorption by fused silica are modeled via a 1D Lagrangian thermo-mechanical model coupled to Helmholtz equations for the propagation of electromagnetic waves in different media. The choice for the equation of state (EOS) of fused silica is detailed in Ref. [Reference Grua, Lamaignère, Chambonneau, Courchinoux and Néauport4]. The model is solved numerically thanks to the Esther^[ Reference Laffite, Baton, Combis, Clerouin, Koenig, Recoules, Rousseaux and Videau ^{19 ]} code developed by CEA, which focuses on laser–matter interaction. The key factors to consider for such a modeling are the surface absorptivity of the absorbing defect as well as the evolution of the absorptivity of fused silica as a function of the density $\rho$ and temperature $T$ of fused silica. At temperatures below 2000–3000 K, the silica typically remains isochoric and only its temperature increases with energy deposits^[ Reference Laffite, Baton, Combis, Clerouin, Koenig, Recoules, Rousseaux and Videau ^{19 ]}. We provide the numerical code with values for the imaginary refractive index ${n^{\prime\prime}}$ of silica as a function of temperature with the Urbach rule^[ Reference Grua, Lamaignère, Chambonneau, Courchinoux and Néauport ^{4 ,} Reference Saito and Ikushima ^{20 ]}:

(5) $$\begin{align}{n^{\prime\prime }}(T)=\frac{\lambda }{4\pi }{\alpha}_0\exp \left({\sigma}_0\frac{hf-{E}_\mathrm{g}}{k_\mathrm{B}T}\right),\end{align}$$

with $hf=3.52\mathrm{eV}$ being the energy of photons at 351 nm and ${E}_{\mathrm{g}}=8.825\mathrm{eV}$ being the band gap for amorphous silica. Other constants are ${\alpha}_0=4.095\times {10}^{-7}$ m ${}^{-1}$ and ${\sigma}_0=0.0455$ .

As the temperature of the fused silica $T$ rises, so does the imaginary index of refraction as $\exp \left(-1/T\right)$ . When the temperature increases beyond 2000–3000 K, fused silica tends to become strongly absorbent^[ Reference Carr, Bude and DeMange ^{21 ]}. For these high temperatures, the Drude model is preferred^[ Reference Grua, Lamaignère, Chambonneau, Courchinoux and Néauport ^{4 ]}. The expression for the permittivity for fused silica used in the code, the function of density $\rho$ and temperature $T$ are as follows:

(6) $$\begin{align}\varepsilon \left(T,\rho \right)=1+i\frac{\sigma_0\left(T,\rho \right)}{\varepsilon_0\omega }{\left(1-i\frac{\omega {m}_\mathrm{e}{M}_{\mathrm{SiO}_2}{\sigma}_0\left(T,\rho \right)}{N_\mathrm{A}\rho Z\left(T,\rho \right){e}^2}\right)}^{-1}.\end{align}$$

In the equation above, $\omega$ is the laser angular frequency, ${\varepsilon}_0=8.85\times {10}^{-12}\ {\mathrm{F\ m}}^{-1}$ is the vacuum permittivity, ${N}_\mathrm{A}$ is the Avogadro constant, ${M}_{\mathrm{SiO}_2}=60.1\ {\mathrm{g\ mol}}^{-1}$ is the molar mass of fused silica, $e=-1.60\times {10}^{-19}$ C is the elementary charge and ${m}_\mathrm{e}=9.11\times {10}^{-31}\ \mathrm{kg}$ is the electron mass. Continuous electrical conductivity ${\sigma}_0\left(T,\rho \right)$ is determined thanks to the mixing of SCAALP^[ Reference Blancard and Faussurier ^{22 ,} Reference Faussurier, Blancard, Cossé and Renaudin ^{23 ]} (self-consistent average-atom model for laboratory plasmas) data of oxygen (2 atoms) and silicon (1 atom). The ionization $Z\left(T,\rho \right)$ is used from internal data and is based on a Thomas–Fermi model with gap^[ Reference Bercegol, Grua, Hébert and Morreeuw ^{24 ]}. Heat transport is also computed from the SCAALP data of oxygen and silicon.

The initial temperature increase is exclusively driven by the surface absorptivity of the arbitrary absorbing defect. We chose an imaginary refractive index for the absorbing defect of ${10}^{-3}$ because it leads to the apparition of a laser damage for fluences between $20$ and $35\ {\mathrm{J\ cm}}^{-2}$ , depending on laser input when solving numerically. These fluences agree with the experimental fluences for which we observed LID.

The thickness of these layers is 5 nm. This choice is made because such defects are localized at the surface interface of the void and silica, so the thickness must be as low as possible. To optimize the calculation time, the meshing of silica follows a geometric progression that provides a fine meshing close to the micro-crack and a rougher meshing far from it. The temporal step for calculation is adapted to the AM frequency of the tested power profile.

The main output parameters considered to track the laser-induced damage threshold (LIDT) for different temporal configurations are the radial stress reached within the material and the temperature. An example for data output of the Esther code is presented in Figure 7. Criteria for laser damage are chosen when the maximal radial stress exceeds the tensile strength of fused silica, which is equal to $50\ \mathrm{MPa}$ . Our criteria for LIDT are chosen at $1\ \mathrm{GPa}$ . This is because we want to be sure that the damage phenomenon has occurred. In fact, the results of simulations presented in the following section show that the choice for the damage stress criteria has a weak influence for determining the LIDT, as the radial stress strongly increases with the fluence of the laser.

Figure 7 Radial stress and temperature output of the code with respect to time and the 1D parameter $x$ .

4.2 How to compare the LIDT and damage laws

On the one hand, simulations carried out with Esther code will give information about the fact that a given defect will lead to LID. The fluence at which damage occurs will be called the LIDT. On the other hand, damage laws determined experimentally give a set of damage densities as a function of fluence. Thus, the comparison of experimental and simulated results requires the possibility of defining an LIDT from a damage law. Given a damage law, the following applies.

• (1) An arbitrary damage density is fixed and defines the value for which we consider LID.

• (2) From the damage law, we define the fluence for which the associated damage density is equal to the value determined in (1). This is the LIDT.

Figure 8 represents the method used to associate an LIDT from a given damage law determined experimentally.

Figure 8 Determination of the laser-induced damage threshold (LIDT) from a damage law, given a damage density threshold defined at $100$ nb/cm ${}^2$ .

4.3 Simulations and comparison to experimental results

Pulses with AMs at 2, 10 and 30 GHz with $\beta =100\%$ were tested with our model. An additional modulated pulse at 2 GHz with $\beta =50\%$ was also used in our simulations to assess the impact of the modulation index $\beta$ . Results for the maximal radial stress reached inside the micro-crack with respect to fluence are presented in Figure 9. Simulations associated with modulated pulses are compared to a reference FIT pulse without modulations.

Figure 9 Evolution of radial stress with respect to laser fluence for different temporal modulations: 2 GHz with $\beta =50\%$ or $100\%$ , 10 GHz with $\beta =100\%$ and 30 GHz with $\beta =100\%$ . Arrows point to the LIDT obtained for the reference without modulation (no mod.) as well as modulated pulses with $\beta =100\%$ .

Firstly, we notice that modulations at 2 GHz ( $\beta =100\%$ ) give an LIDT at a lower fluence when compared to the reference FIT pulse. The LIDT associated with the 2 GHz modulated pulse with $\beta =50\%$ is lower than the reference, but this difference is smaller than in the $\beta =100\%$ case. Both of these results are in good agreement with the tendencies observed experimentally in Figures 3(a) and 4.

Concerning the modulated pulse at 10 GHz with $\beta =100\%$ , the LIDT is the same as the reference FIT case. Simulations are coherent with the experimental results obtained in Figures 3(b) and 4.

When the modulation frequency is equal to 30 GHz ( $\beta =100\%$ ), the simulated LIDT is slightly superior to the LIDT of the unmodulated reference FIT case. Nevertheless, this difference is smaller than that of the 2 GHz ( $\beta =100\%$ ) LIDT and the unmodulated reference LIDT. Thus, we can consider that the 30 GHz LIDT is closer to the reference case than the 2 GHz LIDT. This result is coherent with those obtained in Figure 3(b).

To understand the difference in simulated LIDT for modulated pulses, it is interesting to look at the evolution of thermodynamic parameters with respect to time. Figure 10 shows the evolution of temperature reached within the micro-crack for the reference FIT case (no mod.) and two modulated pulses with $\beta =100\%$ at 2 and 10 GHz. The chosen fluence for the simulation corresponds to the LIDT of the unmodulated and the 10 GHz pulse ( $25\ \mathrm{J}\;{\mathrm{cm}}^{-2}$ ).

Figure 10 Evolution of the maximal temperature reached inside the micro-crack with respect to time for the unmodulated reference pulse and two modulated pulses at 2 and 10 GHz ( $\beta =100\%$ ).

The maximal temperature reached within the micro-crack for a modulation at 2 GHz is higher than the maximal temperature in the unmodulated reference case, whereas it is the same for the 10 GHz modulation. Moreover, for time $t\le 3\ \mathrm{ns}$ , we observe that the temperature increases smoothly for the unmodulated pulse. For the 2 GHz modulation, the variation of temperature varies with a pseudo-period of $500\ \mathrm{ps}$ , which corresponds to the frequency of the AM. We notice the same behavior for the 10 GHz modulated pulse, but with a much lower amplitude. Absorption of the amplitude modulated pulses acts as a ‘low-pass’ filter, at least for frequencies equal to or higher than 10 GHz.

4.4 Discussion of the simulated results

The simulations presented in this section are in good agreement with the experimental results presented in Section 2.3, as far as qualitative behavior is concerned. In particular, to this day, it is impossible to conclude the quantitative behavior of the impact of AM on LID.

One of the main limitations is that simulations and experiments do not express LID with the same metric. Indeed, simulations only show how one particular and arbitrary SSD behaves under laser irradiation. However, global LID behavior of the optical component, which is measured as damage densities, is a function of several SSDs of different natures (NBOHC, ODC, color center, residual slurry particles from polishing processes, etc.) and with different sizes and orientation parameters (width and angle of incidence with the beam). The fact that we used a single defect in our model might not be enough to assess the impact of FM-to-AM conversion on the LID of fused silica.

Even though the silica samples that were used to carry out experimental LID tests underwent similar polishing processes, the same experiment realized on a different sample can lead to dissimilar quantitative results. This is in fact what we observed on the samples polished by the THALES-SESO and ZYGO companies. In the case of LF versus HF modulations, the same trends were observed but for different fluence ranges and different magnitudes. It is as if the nature of the SSD between the different samples exhibited a different absorption dynamic to the incident laser pulse.

An extensive study was considered in one of the papers^[ Reference Grua, Lamaignère, Chambonneau, Courchinoux and Néauport ^{4 ]} that inspired our model. It was shown that it was possible to simulate an entire damage law for fused silica at 351 nm with initial defects of different characteristics. Although we only wanted to show trends regarding the impact of temporal shapes on a single damage, similar simulations could be implemented with different temporal shapes.

Finally, a more general critique can be expressed as far as the model is concerned. Modeling of LID with our model requires data about fused silica for significant temperature $T$ and density $\rho$ ranges. The SCAALP data that we use for the plasma phase are used to compute permittivity as well as thermal conductivity. The data we use for fused silica in the plasma phase, which are based on the mix of SCAALP data for oxygen and silicon, are far from ideal and surely lack consistency.