1. Introduction

Dihydrogen phosphate crystals (KH₂PO₄ or KDP) and their deuterated analogs (KD₂PO₄ or DKDP) are widely used to perform frequency conversions of laser pulses. For instance, such crystals can be employed to produce intense nanosecond laser pulses at 3ω (a wavelength of 351 nm) in order to initiate fusion reaction [1]. However, under current operating conditions of the National Ignition Facility in the U.S.A. or the Laser MegaJoule in France, for which the laser energy density and the pulse duration are close to 10 J.cm ^-2 and 3 ns respectively, defects damaging the optical properties appear in the crystal bulk. It is now admitted that this damage is induced by precursor defects that efficiently absorb the laser energy, inducing a fast temperature rise and a subsequent shock wave [2]. In order to ensure a good working order of these large laser aperture facilities, it is crucial to manage the creation of damage. To do so, one needs to identify the nature of the precursor defects and to understand the physical mechanisms leading to damage. In order to approach more satisfactory operating conditions, i.e. to increase the LID threshold, one can employ a process consisting in pre-illuminating the crystal by a laser pulse for which the fluence is lower than that of the operating conditions [3, 4]. This process is commonly referred in the litterature to as conditioning. However, the physical mechanism induced by the conditioning remains unknown. Despite significant findings, a good comprehension of these mechanisms remains to be achieved. Furthermore, the nature of the precursor defects is still unknown. By providing simple models, this paper aims at improving the knowledge of damage and conditioning mechanisms in KDP crystals, and at providing some additional information regarding the nature of the precursor defects.

These precursor defects may be of varying nature. Since the crystal growth requires certain additives, the precursor defects may be constituted of atomic impurities, such as Fe, Cr or Si [5]. However, several experimental studies based on the correlation between Laser-Induced Damage (LID) and the concentration of impurities seem to show that this kind of defect is not involved in LID [6, 7, 8, 9]. Consequently, defects created during the crystal growth may be invoked. First, one can consider point defects, corresponding to hydrogen or oxygen atoms as interstitial atoms or vacancies in the crystalline lattice [10, 11, 12]. These can be associated with (PO₃)- [13] or (HPO₄)- groups [8, 9, 14]. Since simple considerations based on heat transfer [15, 5] have demonstrated that the precursor defect size ranges between roughly 10 nm and 100 nm in order to produce a sufficiently high temperature, only a cluster of point defects can satisfy the precursor defect size requirement. One can also envisage that, under laser exposure, the latter intrinsic defects give rise to others, for which the absorption properties are larger than the original defect. An example includes (PO₃)- units that transform into (PO₃)²- [13]. Another class of defects satisfying this size requirement corresponds to structural defects, such as mother liquid inclusions, dislocations or cracks. One can also regard LID to be due to a couple of defects for which only a cooperative mechanism permits an efficient laser energy absorption, inducing a high local temperature. It can for instance be imagined that an inclusion or a cluster of point defects produces cracks in its vicinity. Indeed, they strongly deform the surrounding lattice and, since KDP crystals exhibit a low mechanical resistance, this might lead to the creation of stress [16] and, eventually, small cracks.

In order to understand the LID origin at 3ω, a model based on the coupling of statistics and heat transfer has been developed [17]. In this modeling, the LID results from the aggregation of defects where the critical temperature is reached because of the cooperation between them. Despite the fact that this model is somewhat speculative, it renders it possible to determine several experimental trends, such as mainly the S-shape of the damage probability curves and a particular scaling law characteristic of KDP crystals linking the critical fluence, F_c, to the pulse duration, τ, as roughly F_c∝τ ^x with x≃0.35 [5, 4] while a standard value is close to 0.5 [18]. In [17], 1D and 3D heat diffusion, corresponding to planar and point defects respectively, were considered. It was shown that only planar defects were able to provide results that were in a good agreement with experimental data. It could thus be concluded from this modeling that growth bands, cracks, array of dislocations or staking faults were good candidates for explaining LID in KDP crystals. For the sake of completeness, Section 2 describes the study of 2D heat diffusion associated with linear defects. Further, in its original form, the model provided only damage probabilities. The interest in the damage density by the experimentalists has grown in the last years since it renders it possible to obtain more physical information, see for example [19, 6]. As a result, damage densities are now considered instead of probabilities, as presented in Section 3. Within this framework, an in-depth analysis of the model implications based on analytical derivations is presented.

Regarding the conditioning process, preliminary explanation attempts have been provided by Feit et al [15] who suggested that the increase in LID threshold is due to the precursor defects decreasing in size. According to Chirila et al, the conditioning may be due to a ”passivation” mechanism for which the electronic structure of the precursor defects is altered by electrons or holes produced by the laser pre-exposure. Despite that these attempts give a preliminary insight regarding the manner in which the conditioning may work, this is merely a phenomenological description of the conditioning process. Further, a detailed comparison to experimental results in order to verify and report on the reliability of the assumptions has not been carried out. Section 4 aims at introducing a modeling of the conditioning process that is based on the damage modeling assumptions. Since there exist various types of precursor defects [20] for which conditioning may differ, two physical mechanisms, corresponding to different types of precursor defects, are proposed. The first model assumes the precursor planar defects to actually be composed of point defects for which the thermally-activated migration may lead to their annihilation. Another class of defects are structural ones, and the second model relies on the fact that a rise in the temperature induces phase transition and a subsequent crystalline rearrangement. As a consequence, the defect absorption vanishes [21]. The predictions of each model are compared to a list of experimental trends, including:

• The shape of the damage density with respect to the fluence is practically unchanged by laser conditioning, the influence of the conditioning consists in shifting the whole curve to higher fluences [6].

• The conditioning efficiency can be increased by utilizing short pulses [4].

• As long as no damage appears during the conditioning, its efficiency becomes increasingly improved as the conditioning fluence is augmented [20, 6]. Further, there exists a conditioning fluence threshold below which no conditioning effect whatsoever is observed [20].

• For given conditioning pulse parameters, the conditioning efficiency increases as a function of the number of conditioning pulses [20, 22].

Finally, Section 5 provides conclusions and outlooks of the present work. For the convenience of the reader, details of certain derivations and a table describing the symbols used throughout the paper are reported in the Appendices.

2. 2D heat diffusion: study of the heating of an ensemble of linear defects

A heterogeneity can be considered as composed of an ensemble of linear defects (that can be associated with dislocations) oriented in the same direction. Since a dislocation disturbs the crystalline lattice, its absorption efficiency is larger than the one of a perfect crystal and it can be seen as a source inducing a temperature rise. The dislocation length is assumed to be larger than the thermal diffusion length given by $2\sqrt{D\tau }$ where D is the thermal diffusivity. It follows that the temperature field remains the same whatever the position along the dislocation direction. In the following, the mathematical formalism used to model the problem is briefly described, but more details are given in [17]. The temperature field in the plane perpendicular to this direction, hereafter referred to as the 𝓟-plane, and for which the characteristic dimension has been set to 1 µm which is comparable to experimental dimension [23, 24], can be obtained by solving the Fourier equation:

Here, r⃗_i refers to the position (x,y) of the dislocation i randomly distributed in the 𝓟-plane, n_ADNS is the number of dislocations in the 𝓟-plane (ADNS stands for Absorbing Defects of Nanometric Size) and A is the absorbed power per unit of volume that can be expressed as 10⁴ F/τ in units of W.cm ^-3 from empirical considerations [17]. Material physical parameters such as thermal diffusivity, D, and conductivity, λ, density, ρ, or heat capacity, C, are assumed to remain constant during the course of interaction. The function Π is defined as:

where a is the source size in the 𝓟-plane, set to 1 nm in the calculations. A general solution of Eq. (1) can be obtained [25] by summing the solutions of the Fourier equation designed for only one point source. Now, since working conditions include $a\ll \sqrt{D\tau }$ (for D_KDP=6.5×10^-7 m ².s ^-1 and τ=1 ns, $\sqrt{D\tau }\simeq 25nm$), in order to deal with simple formula allowing fast numerical calculations, the function Π(x) can be approximated by the Dirac delta function δ (r⃗). The dislocations may then be seen as a point heat source in the 𝓟-plane. The temperature rise induced by this source reads (see Appendix B for the analytical derivation):

Then, when considering a set of sources, cooperative effects lead to a higher temperature as opposed to the one induced by a single source. Further, the larger the number of ADNS involved in a temperature rise, the higher is the temperature. This is illustrated in Fig. 1 where a temperature field is associated with an ADNS distribution. Figures 1(a) and 1(b) respectively show the ADNS distribution and the resulting temperature field in a case without a big cluster. Figures 1(c) and 1(d) correspond to a case in which a cluster leading to a significant temperature rise is present (located in the circle of Fig. 1(c)). Moreover, it can be clearly seen that only clusters composed of a large number of ADNS give rise to high temperatures. It should be noted that the temperature scales differ between the two distributions. In one case, despite the appearance of clusters that significantly increase the temperature, it remained below T_c. In the second case, a cluster composed of more ADNS permitted the temperature to exceed T_c, subsequently giving rise to damage.

 

Fig. 1. A 2D temperature field in the 𝓟-plane (see text) for an ADNS distribution not leading to damage (top) and another temperature field implying a damage (bottom). The cluster leading to damage is inside the dashed red circle.

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The introduction of a damage probability by utilizing a critical temperature criterion is straightforward. For a given number of random ADNS distribution, it suffices to, at the end of the pulse, count the number of times for which the temperature is larger than the critical temperature in at least one place in the heterogeneity [17]. Figure 2 illustrates the results that can be obtained with this modeling. It shows the evolution of the damage probability as a function of the laser fluence with the following parameters: τ=3 ns, n_ADNS=2000 and 5000. The value of the parameter ξ/l has been set to 10⁵ cm ^-1 in order to find a critical fluence (defined in the modeling as the fluence giving a probability of 10% [17]) close to the experimental one. The curve shape is comparable to those provided by the experiments. Within the modeling framework, this can be understood as the temperature rise increasing with the cluster size. Since the clusters are more numerous when their sizes are small, it follows that the higher the fluence, the larger the probability. Now, in order to determine whether the linear defects may be responsible for LID in KDP, one must verify if the calculations are able to reproduce experimental data. The main parameters characterizing the LID are F_c and x. With the large value ξ/l=10⁵ cm ^-1 (compared to 10⁴ cm ^-1, obtained from empirical considerations [17]) and n_ADNS=5000 (i.e. 0.5% of defects in a heterogeneity of 1 µm), one obtains F_c=8.11 J.cm ^-2 and x=0.21 for τ=3 ns (fit as shown by the inset (a) of Fig. 2). F_c is reasonable but not the value of x. With the same value of ξ/l but with n_ADNS=2000, a reasonable value x=0.31 is obtained, however, in this case, F_c=16.51 J.cm ^-2 is too high (inset (b) of Fig. 2). Also, ξ/l has to be set to 2×10⁵ cm ^-1 in order to reproduce the experimental values of both F_c and x (F_c is here found to be 8.15 J.cm ^-2). Beyond the fact that ξ/l has a much larger value than the empirical one, it corresponds to an almost unrealistic physical process. Indeed, the temperature rise is due to the energy transfer from the free electrons (that have been produced themselves by the laser pulse [2]) to the lattice. The electronic density cannot exceed the critical plasma density given by n_c=mε ₀ ω ²/e ²≃9.1×10²¹ cm ^-3 at 3ω. By using a Drude model appropriately characterizing the electronic plasma, ξ/l can be linked to n_e as A=ξF/lτ=n_edE_c/dt, where

is the energy absorbed by one electron per unit of time, and ν_coll is the inverse of the time elapsed between two collisions of an electron on ions, commonly in the femtosecond range. In these calculations, ν ^-1 _coll was set to 3 f s [26]. It follows that ξ/l=10⁴ cm ^-1 corresponds to n_e≃8×10²¹ cm ^-3, i.e. roughly the critical plasma density. With ξ/l=2×10⁵ cm ^-1, one has n_e≃20n_c, which is physically unrealistic. These 2D heat diffusion calculations cannot be completely turned down despite the fact that slightly different parameters (a higher ν_coll for example) would provide a more realistic value of n_e. This order of magnitude calculation demonstrates that planar defects seem to provide data that are more consistent with experimental results as opposed to linear defects.

 

Fig. 2. The damage probability as a function of the laser fluence in the 2D modeling framework. The inset represents the critical laser fluence as a function of the pulse duration for (a) n_ADNS=5000 and (b) n_ADNS=2000.

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To conclude this section, an ensemble of linear defects, inducing a 2D heat transfer (that can be identified with a group of dislocations) can hardly fulfill all the conditions imposed by experimental data and physical requirements. Therefore, they appear not to be responsible for the LID in KDP crystals. Also, since 3D heat transfer calculations have already been turned down [17], focus is now placed on the 1D heat transfer associated with planar defects as the best means of explaining the LID in KDP crystals.

3. In-depth analysis of the damage density within the 1D modeling framework

It was shown that a collection of planar defects seemed to best explain the LID of KDP crystals within the present modeling framework. A study based on the damage probability, P, has been carried out [17]. Nevertheless, as of a few years, a great interest has been devoted to the experimental study of the damage density since it provides more information and avoids any dependence on the irradiated volume. Within the proposed model, the damage density, ρ_d, is simply obtained with ρ_d=Pρh, where ρh is the density of heterogeneities that are likely to cause damage. F_c is defined as the fluence required to reach a given damage density, ρ_c. It is set to 10 mm ^-3 after verification that a different value would lead to the same conclusions. Fig. 3 shows the evolution of the damage density as a function of the fluence for several values of parameters ρ_h, n_ADNS and τ with a heterogeneity of length L=10 µm [17]. The first conclusion that can be drawn is that the general shape exhibits characteristics similar to experimental results: a rapid increase at the threshold and a saturation for the highest fluences. Subsequently, the influences of the various parameters on the LID threshold were investigated. The influence of ρ_h is shown in Fig. 3(a). It appears that the larger ρh, the lower F_c. Indeed, when ρh increases, less energy is required to obtain a given number of heterogeneities for which T>T_c. The same reasoning leads to the conclusion that the larger n_ADNS, the lower F_c, as confirmed by Fig. 3(b). Concerning the influence of the pulse duration as displayed in Fig. 3(c), one finds the same behavior as the damage probability with F_c∝τ ^0.35. Actually, the scaling law exponent deviates from the expected 1/2 value (associated with only one ADNS) as a result of cooperative effects between ADNS. An analytical development based on certain assumptions provides a better insight of this fact and allows to determine an approximated expression of x as (see Appendix C):

where y(n_ADNS) is an increasing function of n_ADNS and z is a constant. Moreover, expression (5) shows that x becomes smaller as n_ADNS increases. Indeed, a large value of n_ADNS favors the cooperative effects and subsequently increases the deviation from 1/2. Also, large pulse durations provide the ADNS with more time to cooperate, thus increasing the deviation from 1/2 as shown by Eq. (5). Furthermore, it is noteworthy that the data of [27] could be better fitted with a value of x depending on the pulse duration as x=α+βlnτ (α and β are fit parameters) as opposed to with a constant value of x; a fact supporting the proposed modeling.

The behavior of F_c with respect to ρ_h and n_ADNS can be derived from analytical considerations based on probability calculations. In Appendix C, the damage density is approximated by:

where f (τ) is a function of the pulse duration and N is related to the domain length as L=Na (with a=1 nm). By solving ρ_d=ρ_c where ρ_c=ρd(F_c), an approximated expression of the LIDT can be derived:

Also, F_c is significantly reduced when n_ADNS or ρ_h increases. Furthermore, as part of the search for the nature of the precursor defects in KDP crystals, the LIDT is predicted to evolve as roughly the inverse of the logarithm of their concentration. In addition, experiments designed for measuring the LIDT variations with respect to a controlled concentration of defects should provide an answer to the question of whether the defects under investigation are responsible or not for the LID.

 

Fig. 3. The damage density as a function of the laser fluence in the 1D modeling framework. (a) The influence of the density of heterogeneities N_h with n_ADNS=100 and τ=1 ns (b) The influence of n_ADNS with Nh=10⁶ cm ^-3 and τ=1 ns (c) The influence of the pulse duration with n_ADNS=100 and Nh=10⁶ cm ^-3.

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Other means of extracting information from the experimental variations of the damage density involve considering a simple form of the theoretical expression of the damage density which only accounts for macroscopic measurable parameters and employing a fitting procedure. Since both the experimental and theoretical damage densities may suffer from artifacts at high fluences (coalescence [28] and saturation respectively), it would thus seem preferable to focus the attention on the threshold. For fluences close to F_c, the damage density can be approximated by (see derivation in Appendix C):

In Eq. (8), apart from N_ρh, all terms can be determined from experiments. Also, the value of N_ρh can be obtained from experimental data by using an adapted fitting procedure. It was verified that Eq. (8) provides a good reproduction of pure numerical calculations at the threshold.

4. Modeling of the laser conditioning

For a long time, it was experimentally demonstrated that one can increase the LIDT of KDP crystals by carrying out a laser pre-exposure. However, no clear proposal of how conditioning works has been put forward. Beyond the fact that it is of interest to understand the fundamental mechanisms underlying conditioning, from an applicative point of view, it is also important to control the influence of the laser parameters in order to optimize the conditioning protocol. Thus, in this section, two scenarios of conditioning are proposed.

4.1. Principle of the conditioning modeling

Despite that the nature of defects remains unknown, it has been determined that the LID is due to a considerable temperature rise. It is thus reasonable to expect that conditioning effects are also due to temperature rises. An increase in temperature influences a material such that it may activate the migration of defects or induce phase transition. Further, it has previously been demonstrated that planar defects seem to best explain LID in KDP crystals. For this reason, the following two scenarios of conditioning are proposed:

i. The above-mentioned planar defects are assumed to actually be composed of atomic-size defects for which displacements in the plane are governed by an Arrhenius law. It is supposed that these point defects can annihilate during their migration. Also, within this scenario, the conditioning consists in decreasing the absorption of each planar defect by decreasing the number of absorbing point defects. This scenario is hereafter referred to as Conditioning Model 1 (CM1).

ii. Here, planar defects are considered as a structural defect, and it is assumed that the conditioning laser pulse induces a phase transition (boiling) in the vicinity of the defect. Subsequently, when cooling to the room-temperature, rearrangements of the crystalline lattice remove the defect and its absorption efficiency vanishes. We here deal with a “zero-one” model. This scenario is referred to as Conditioning Model 2 (CM2).

The following section presents the details of the modeling. Within the CM1 framework, each planar defects is composed of an ensemble of point (atomic size) defects that may strongly absorb the laser energy. This results in a significant temperature rise due to cooperative heating effects. The temperature along the direction perpendicular to the plane decreases faster than exponentially [17]. Since a point defect is only able to move in a region where the temperature rise is significant (see Eq. (9) below), it can be reasonably stated that the migration takes place essentially in the plane. In the course of migration, these defects are assumed to be able to annihilate [9, 14, 13], i.e. a recombination of a pair of particles mediated by thermally activated diffusion may occur. For instance, interstitial-vacancy pairs satisfy this requirement but, atomic defects can in general migrate to a more fundamental state with a lower absorption efficiency. For the sake of simplicity, we henceforth deal with couples of interstitial and vacancy atoms. In order to numerically evaluate the absorption variation due to a laser conditioning, the following procedure was adopted. The absorption was assumed to be proportional to the number of defects pairs per unit of surface. For each time step, the probability P_m of a defect displacement from a cell (with a characteristic size of 1 nm) to an adjacent one is given by:

where E_a is the activation energy and k is Boltzmann’s constant. A jump is assumed (depending on the value of P_m) to occur every 1 ps [9], with the same probability in each direction of the plane. After a time step, it is verified whether the interstitial atom has met or not a vacancy. If so, the number of pairs decreases by one. If no meeting occurs, the migration can go on. This cycle is performed as long as the temperature is such that P_m(t) is larger than one percent of P_m(t=τ), which is the maximum value of the probability (for nanosecond conditioning pulses, a significative migration takes place up to 10 ns after the pulse has been switched off). After the migration has stopped, the remaining (non-annihilated) defect pairs are counted and the new absorption of a plane is given by α_cond=n_rα ₀/n ₀, where n_r is the density of the remaining pairs, n ₀ is the density of the initial pairs, and α ₀ is the plane absorption before the conditioning. Various details concerning the numerical implementation of the algorithm permitting to optimize the calculations are given in Appendix D.

For CM1, in order to speed up the calculations, other reasonable assumptions have been made. Also, periodic conditions have been used to carry out the migration process. The planar defect absorption efficiency was assumed constant in the course of interaction whereas a few recombinations may occur. Nevertheless, the migration becomes really effective at the end of the laser pulse and after a certain time when the temperature is the largest and the probability, accordingly, is the highest. Furthermore, a homogeneous heating of the plane was assumed, i.e. T(x,y, t)=T(t) in the calculations.

Concerning CM2, since it was shown that the defects might have a planar geometry and, because one often deals with rapidly grown crystals, structural defects represent good candidates for explaining LID. When illuminated by a laser pre-exposure, the absorption properties of a structural defect may lead to a significant local temperature rise and subsequently to a phase transition in its surroundings. The phase transition of interest is the boiling, as it is the most suited for removing structural defects [29]. Since this phase transition takes place in a confined medium where a pressure of tens of GPa can be reached [2], a higher temperature than ambient is required for the boiling point, T_bp [30, 31]. Under ambient conditions, the boiling temperature is close to 673 K, however, in the calculations, T_bp is set to roughly 2000 K in order to take into account the latter fact. Now, depending on the structural defect thickness, e, a minimum quantity of matter has to be heated up to T_bp in order to remove this defect. Since a 1D heat diffusion is considered, this quantity of matter is proportional to the distance, d, from the structural defect for which T ≥ T_bp; at least for an instant (since one has a propagating heat front). Therefore, in the modeling, the conditioning succeeds if d≥β_e, where β is a constant set to 5 and e is assumed to range randomly between 0.1 nm and 2 nm [32]. For a given structural defect, provided that the latter inequality is satisfied, the absorption then is assumed to be zero. It is worth noting that any reasonable variation of these parameters does not lead to a change in the main results presented hereafter.

At this point, on the basis of both conditioning models, it is possible to predict the influence of a pre-exposure on the damage density produced by a testing pulse. The numerical simulation mimics the experimental protocol: for a given ADNS distribution with the absorption efficiency as defined previously, a conditioning is simulated by first calculating the laser-induced temperature field and, then, evaluating the new absorption efficiency of each ADNS. It is noteworthy that each ADNS is subjected to a specific temperature (depending on whether cooperative effects take place or not), and as a result, each ADNS has its own new absorption efficiency. With these absorption efficiencies, it is possible to carry out the standard damage testing in order to obtain the damage density. Since the influence of the conditioning is to decrease the absorption efficiency, it is clear that the LIDT will increase with such a treatment. Once the values of the LIDT before and after conditioning have been evaluated, hereafter referred to as F ^(bc) _c and F ^(ac) _c, respectively, one can define the conditioning gain as:

for which the values are larger than unity. Here a ratio is deliberately considered because it is less sensitive to the value of the modeling parameters (compared to an absolute value of the fluence or the damage density after conditioning). Moreover, g depends on the testing pulse duration, τ, the conditioning pulse duration, τ_cond, as well as the conditioning laser fluence.

4.2. Damage densities

By applying the above-mentioned procedure, one can evaluate the damage density after conditioning. An illustration of the results that can be obtained for certain values of the CM parameters is given by Fig. 4. Calculations have been performed with constant values of the laser pulse parameters, i.e. τ_test=3 ns, F_cond=3 J.cm ^-2 and τ_cond=2 ns. A value of τ_cond shorter than τ_test was deliberately chosen due to the conditioning being all the more efficient when τ_cond is short. This is further demonstrated in the following section.

Figure 4(a) shows the predictions of CM1 with E_a=0.9 eV and 1.2 eV, and n ₀=6.×10¹² cm ^-2 and 1.2×10¹³ cm ^-2 (corresponding to a few hundreds of point defects in 100 nm×100 nm in size plane). As expected, regardless of the CM1 parameter values, the LIDT could be increased by performing a conditioning procedure. Further, the larger the Ea, the lower the LIDT since the probability (9) is a decreasing function of E_a. Indeed, as this probability is lowered, less annihilations occur and the absorption thus decreases to a lesser extent. Concerning the influence of n ₀, it appears that the LIDT increases along with n ₀. For the largest value of n_2D, the probability of a moving interstitial to meet a vacancy is at a maximum. This results in a large number of annihilations and subsequently to a low absorption efficiency. The results of Fig. 4(a) exhibit a particular trend with regard to the shape of the damage densities with respect to the fluence: after conditioning, the damage density slope displays a significant increase. This is due to the fact that the biggest clusters are the most sensitive to the conditioning (they exhibit the highest temperatures due to the cooperative effects [17]). Experimentally, this fact has not been observed at 3ω.

 

Fig. 4. The damage density as a function of the laser fluence for (a) an unconditioned and a conditioned crystal within the CM1 modeling framework and (b) a conditioned crystal within the CM2 modeling framework.

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Figure 4(b) presents the results of CM2 obtained with c=5 and 15, and T_bp=1600 K and 2000 K. As CM1, CM2 predicts an increase of the LIDT. For c=5, the damage density slope is significantly increased by applying CM2 (compared to an unconditioned crystal). This, again, is due to the conditioning being more efficient for larger clusters. In the boiling temperature range of interest (i.e. T_bp around 2000 K), for c=5, the influence of T_bp principally consists in shifting the damage density curve. For a given T_bp, in addition to the displacement of the curve to lower fluences, the increase of c leads to a decrease of the slope. In that case, for the biggest clusters, the matter has to be distributed among the ADNS (forming the cluster), which results in an effective distance that may be shorter than the smaller clusters. In fact, numerical calculations (that provide the average number of ADNS related to every value of d) shows that all cluster sizes are conditioned for c=5, whereas the biggest clusters are less conditioned for c=15. There are thus two influences that offset each other, which results in a slope that does not evolve much between an unconditioned and a conditioned crystal. Also, with CM2, one can find a set of parameters that almost allows the conservation of the damage density slope after conditioning.

4.3. Influence of the laser pulse parameters

From a general point of view, the conditioning gain depends on the pulse duration of both the testing and the conditioning laser pulses, as well as the fluence of the conditioning pulse. Let us first focus on the influence of the pulses duration. The conditioning gain with respect to τ_test and τ_cond provided by CM1 and CM2 is given in Figs. 5(a) and 5(b) respectively. For both models, F_cond=5 J.cm ^-2. The CM1 parameters are given by E_a=1.2 eV and n ₀=1.2×10¹³ cm ^-2, and those of CM2 are c=6 and T_bp=2000 K. These parameters have been chosen in order to obtain comparable gains for both models. From both graphs, due to the iso-gain curves being mainly horizontal, it can be deduced that the gain depends more on τ_cond than on τ_test. In order to better appreciate the evolution of the gain as a function of τ_cond, a cut of Figs. 5(a) and 5(b) has been plotted in Fig. 5(c) for τ_test=3 ns. This is the pulse duration of interest for the laser facilities presented in the introduction. For both CM’s, it appears clear that the conditioning efficiency is better for shorter values of τ_cond. Moreover, Fig. 5(c) shows that CM2 gives rise to the largest gain variations with respect to τ_cond. For both CM’s, it is worth noting that for the shortest conditioning pulses (sub-nanosecond duration), the temperature of the biggest clusters may exceed T_c. It follows that the conditioning may produce damage.

Next, it is investigated how the gain evolves with respect to the conditioning pulse duration and fluence. Again for τ_test=3 ns, Figs. 6(a) and 6(b) portray this gain evolution predicted by the CM1 and the CM2, respectively. In order to obtain a better insight of the gain evolution as a function of F_cond, a cut of Fig. 6(a) and Fig. 6(b) has been reported to Fig. 6(c), with τ_cond=500 ps and τ_cond=3 ns. Both models exhibit the same trend: the conditioning gain depends more on F_cond than on τ_cond. Indeed, the conditioning efficiency is directly related to the temperature reached by each cluster during the conditioning. This temperature is proportional to Fcond whereas it evolves slowly with respect to τ_cond (as τ-0.3 [17, 26]). The gain shape with respect to Fcond is given in Fig. 6(c). The general behavior of all curves implies that the gain remains close to unity up to a certain fluence, after which there is a linear increase. The curves thus exhibit a conditioning fluence threshold above of which the conditioning becomes efficient. The existence of such a threshold is linked to the fact that a minimum temperature has to be reached in order to modify the absorption properties of the ADNS. Within CM1, the probability Pm of defect displacement, and the subsequent possibility of annihilation, become non-negligible for kT≥E_a. Within CM2, the absorption efficiency of an ADNS vanishes only if T≥T_bp in the area surrounding the structural defect. Concerning the behavior of g for fluences larger than the above-mentioned threshold, it can be easily understood by the statement that the temperature is proportional to the fluence. Further, from Fig. 6(c), one notices that the longer τ_cond, the higher is the conditioning threshold. In addition, one finds again that CM2 is more sensitive to τ_cond than CM1.

 

Fig. 5. The LIDT gain obtained by conditioning with respect to τ_test and τ_cond within (a) CM1 and (b) CM2. For both graphs F_cond=5 J.cm ^-2. (c) A cut of (a) and (b) for τ_test=3 ns.

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4.4. Influence of the number of pre-exposures

In the previous section, it was seen that utilizing a conditioning procedure comprising several laser pulse pre-exposures is the best way to optimize the crystal performances. It is thus interesting to consider how the conditioning gain evolves as a function of the number, N_p, of laser pre-exposures.

Within the CM1, for a given laser pre-exposure, the initial positions of defect couples correspond to the final positions of the previous laser pulse pre-exposure. The details of an optimized implementation are given in Appendix D. After each shot, annihilations occur. However they are fewer and fewer since the absorption and thus also the temperature are increasingly low. Therefore, the gain is expected to increase as a function of the number of pre-exposures, however with smaller and smaller variations. This is confirmed by Fig. 7, portraying this behavior as numerically provided by CM1 with various pulse parameters, i.e. F_cond=2 J.cm ^-2 and 3 J.cm ^-2, and τ_cond=500 ps and 3 ns. Between two sucessive laser exposures, the fluence and duration remain unchanged and it is assumed that the temperature has enough time to decrease back to room temperature. This behavior is further confirmed by a numerical benchmark and an analytical derivation specifying the gain evolution as close to a logarithmic increase (see Appendix D). The inset of Fig. 7 represents the gain with a logarithmic scale on the horizontal axis and allows to convince of this fact. More precisely, regarding the numerical benchmark, the migration of the couples of defects has been fully simulated (without optimization) in a single plane. This calculation also provides the same logarithmic behavior as the one obtained with CM1. Further, the point defects migration and annihilation in a 3D space has been simulated and again, the calculations show a logarithmic-like evolution of the absorption (assumed to be proportional to the density of non-annihilated point defects) as a function of the number of pre-exposures, closely related to the ones obtained previously. It follows that, in this case, the space dimensionality does not permit a discrimination of the defect geometry (plane or sphere).

 

Fig. 6. The LIDT gain obtained by conditioning with respect to τ_cond and F_cond within (a) CM1 and (b) CM2. For both graphs τ_test=3 ns. (c) Cut of (a) and (b) for τ_cond=500 ps and 3 ns.

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Concerning CM2, since an ADNS has been conditioned (with the first pre-exposure), it no longer contributes to the temperature rise for the following pre-exposure. Also, the temperature rise induced by the second pre-exposure is lower than the rise of the first whatever the considered ADNS. As a result, the ADNS that have not been conditioned by the first pre-exposure cannot be conditioned by the second. It also turns out that CM2 does not provide any variation of the gain with respect to the number of pre-exposures for N_p>1.

4.5. Discussion

In order to put forward a conditioning model, this section first proposes a comparison of the results of CM1 and CM2 to available experimental data. The influence of the conditioning on the shape of the damage density with respect to the fluence, the influence of the conditioning pulse duration, the influence of the conditioning fluence, and the evolution of the gain as a function of the number of laser pre-exposures are also examined.

From previous works [6, 4], it seems that the shape of the damage density with respect to the fluence remains unchanged after a laser conditioning at 3ω. More precisely, the influence of the conditioning consists mainly in shifting the whole curve to larger fluence values. Section 4.2 showed that CM1 always leads to a modification of the damage density shape whereas, within CM2, it is possible to find parameters that allow the conservation of this shape, the conditioning simulation mainly leading to a shift of the curve under these conditions. Therefore, concerning the influence of the conditioning on the damage density shape, CM2 displays a better capacity for mimicking the experimental results.

 

Fig. 7. The conditioning gain as a function of the number of laser pre-exposures. The inset shows the case F_cond=2 J.cm ^-2 and τ_cond=3 ns with a logarithmic scale on the horizontal axis.

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Regarding the influence of the conditioning pulse duration on the gain, both models predict an increase in the conditioning efficiency as the pulse length decreases under conditions where F_cond is a constant not depending on τ_cond. This trends is in agreement with the experimental data [4]. Nevertheless, it is worth noting that a closer comparison to the experimental protocol should have used a ramp with increasing fluences; the highest fluence being an increasing function of τ_cond [4]. One should however keep in mind that the study was performed by varying physical parameters one by one in order to reach a good insight of the physical mechanisms. Further, the choice not to strictly follow the experimental protocol was based on the fact that it was not possible to accurately reproduce all the experimental conditions, such as for instance an installation change required to obtain large variations of τ_cond [33]. In the performed calculations, the introduction of F_cond increasing with respect to τ_cond would have been to decrease the slope of g. Going further in the comparison to experiments, the agreement is decent as long as τ_cond is not too short (no less than roughly 800 ps according to [4]). For shorter conditioning pulses, the trend is reversed: the shorter the pulse, the lower the gain. In such a case of very short pulses, one may expect that new physical effects are involved. For instance, this trend could be reproduced by CM1 by invoking the saturation of the migration probability. Indeed, this probability becomes close to unity and remains unchanged as the temperature goes up. Also, for increasingly short pulses, inducing higher and higher temperatures, the time allowed for migration vanishes whereas the probability ceases to increase. This results in shorter defect displacements and subsequently less annihilations, consequently leading to a lower gain. The only difference between CM1 and CM2 lies in the variations of the gain as a function of the conditioning pulse duration: CM2 exhibits larger variations than CM1. However, due to all the uncertainties (theoretical and experimental alike), the discrepancy is not enough to discriminate one model as opposed to the other by comparing to the experimental data.

Both conditioning models predict that a minimum conditioning fluence, F^(m)_cond, is required to initiate an improvement of the crystal resistance. For F_cond<F^(m)_cond, F_c remains unchanged (i.e. F ^(ac)_c=F^(bc)_c). Again, this fact has been experimentally observed [20], showing that a certain temperature is required to initiate a conditioning process. This also validates the proposed models based on temperature-driven mechanisms. Further, above F^(m)_cond, for an increasing conditioning fluence, the gain is also augmented, which is in good agreement with the experimental observations. It follows that the higher the fluence, the better the gain. But it has been numerically observed that the critical temperature can be reached for a too high fluence, thus leading to damage. In that case, the beneficial effect of the conditioning is lost. This inconvenience can be eliminated by using several conditioning pulses of increasing fluence. Under such conditions, the first conditioning pulse with the lowest fluence passives only the biggest clusters. Subsequently, for an increasing fluence, smaller and smaller clusters are treated. This procedure also allows the use of a final high fluence, F^max_cond, which gives rise to a decent gain, but without creating any damage. This protocol is also used experimentally and is called a conditioning ramp [4]. Moreover, the higher the F^max_cond, the larger g, as also observed elsewhere [6]. Nevertheless, there exists an experimental limit for F^max_cond [6] that may be due to the fact that another kind of defect begins to be excited.

The evolution of the gain as a function of the number of laser pre-exposures is the best way to test the CM’s since they predict very different trends. Within CM2, the gain does not evolve whereas CM1 predicts an increase. The latter phenomenology was observed in [20]: the damage density decreases as a function of the number of pre-exposures. Since the damage density after the conditioning and the conditioning gain are closely related, it can be concluded that CM1 accounts well for the influence of the number of pre-exposures. The comparison can be taken further by analyzing the variations of the conditioning efficiency with respect to the number of pre-exposures. According to [20], due to the damage density evolving almost linearly with a horizontal logarithmic scale, it was concluded that it exhibits a behavior close to a logarithmic evolution with respect to the number of laser pre-exposures. For fused silica, Suprasil and BK7, it is worth noting that a logarithmic behavior of F^(ac) _c, i.e. of the conditioning gain, has been also experimentally observed [34, 35, 36]. As shown in Section 4.4, CM1 appeared to also be able to reproduce this particular logarithmic-like behavior. Further, within this horizontal logarithmic scale, the slope of g depends on the conditioning fluence (see Appendix D), as it has been observed [22] for λ=532 nm and λ=355 nm. Furthermore, the higher F_cond, the larger the variations in g with respect to N (i.e. the larger the variations of the damage density). It is important to stress that this logarithmic-like behavior arises from the fact that an Arrhenius law was employed (see Appendix D). Another law would provide another type of behavior. Since this law is representative of the temperature-driven defect migration process, it follows that an agreement with the experimental data represents a strong evidence that the defects responsible for the LIDT in KDP are in fact clusters of absorbing point defects.

Nevertheless, a model based solely on this migration mechanism is not able to reproduce all the experimental trends, such as the evolution of the damage density shape after conditioning. It has been seen that the other model succeeds in this prediction. These results thus suggest that the precursor defects may have an extremely complex nature, combining the defect natures of both models. Also, only the association of a 2D structural defect with point defects could be able to reproduce all the experimental results that have been previously mentioned.

Now, one can wonder about the physical nature of the defects which can be associated with the above-mentioned point and structural defects. With regard to the point defects, couples of interstitials atoms and vacancies correspond well to the idea of them migrating and annihilating when an interstitial atom meets a vacancy. In that case, the absorption associated with these couples becomes zero after the annihilation. Concerning the nature of the considered atom, oxygen is believed to be a good candidate. Indeed, an oxygen vacancy introduces states located in the band gap [10, 11, 12] that enhance the absorption. In addition, these states allow an explanation for the particular experimental evolution of F_c as a function of the photon energy exhibiting successive plateaus [37]. The planar structural defects may be associated with cracks. Indeed, they exhibit the required absorption efficiency, since the breaking of periodic conditions first introduces states in the band gap in the vicinity of the crack surface [21], after which electric field enhancements can facilitate the free electron production [38, 39, 21] leading to the absorbing plasma. Furthermore, a crack may contain impurities that re-enhance the previous influences.

5. Conclusion

A model was developed based on statistics and heat transfer permitting to evaluate the temperature field induced by a distribution of defects in a heterogeneity. A damage is due to the presence of a cluster in which cooperative effects lead to a dramatic temperature rise. Within this modeling framework, the heat diffusion can be allowed in one, two or three spatial dimensions, and the corresponding defects are planes, lines or points respectively. The present work completed the study of the influence of the defect geometry by performing 2D calculations. The comparison of 1D, 2D and 3D calculations to experimental data such as the value of the scaling law exponent, led to the conclusion that the 1D framework was the best suited to explain laser-induced damage in KDP crystals. The defects responsible for laser-induced damage in KDP were thus believed to be a collection of planar defects. Within this framework, it was possible to predict the damage density as a function of the laser fluence. An in-depth analysis allowed the derivation of simple analytical expressions of the scaling law exponent, the critical fluence and the damage density with respect to several physical parameters. From this, additional information concerning the nature of the precursor defects, such as their density, could be obtained based on experimental data.

Two models of conditioning were developed based on temperature-driven mechanisms. Both models assume that the conditioning results from a decrease in the defects’ absorption. For this, the first model assumes that the absorption decrease is due to the annihilation of point defects that can migrate in the above-mentioned planar geometry. The second model assimilates the planar defects to structural ones, the heating of which might lead to crystalline rearrangements and to a subsequent significant decrease in absorption. Both models account for particular experimental trends. More precisely, these models reproduce the fact that the conditioning gain is better for shorter conditioning pulse durations. Moreover, there exists a minimum conditioning fluence required for initiating a conditioning. Furthermore, when a conditioning ramp is used, the higher the final fluence, the better the conditioning efficiency. Regarding the evolution of the gain with respect to the number of laser pre-exposures, the modeling based on the migration of point defects renders it possible to reproduce a logarithmic-like increase that is experimentally observed. Concerning the zero-one model based on the structural change, it fails completely in this prediction. Therefore, this particular result suggests that the planar defects are composed of at least point defects such as couples of interstitial atoms and vacancies.

In order to reproduce all the experimental facts, the use of both conditioning models is required. It follows that both the associated defect natures have to be taken into consideration. This suggests that the precursor defects actually have a complex nature in the sense that they associate planar structural defects and point defects. This study thus leads to the conclusion that a one-hundred-nanometer crack associated with couples of interstitial oxygen and vacancies may be the precursor defect responsible for laser-induced damage in KDP crystals.

Appendix

A. List of parameters and variables

View Table

B. Analytical solution of the 2D diffusion equation with a point source

In the following, we solve the Fourier equation with a point source in 2D space:

(11)$$\genfrac{}{}{0.1ex}{}{\partial T}{\partial t}=D\mathrm{\Delta }T+\genfrac{}{}{0.1ex}{}{E\delta \left(\overrightarrow{r}\right)F\left(t\right)}{\rho C}$$

where D, ρ and C are the thermal diffusivity, the density and the specific heat capacity respectively. E has the dimension of an energy per unit surface and F(t) is the temporal profile of the laser pulse. In the case where F(t)=δ (t), by using Fourier and Laplace transformations, it can be shown that [25]:

(12)$${\theta }_{2\mathit{D}}(\overrightarrow{r},s)=T-T_0=\genfrac{}{}{0.1ex}{}{E}{2\pi {\lambda }_{\mathit{KDP}}}{K}_0\left(r\sqrt{s⁄D}\right)$$

where K ₀ is the modified Bessel function (see e.g. [40]). Now, instead of a temporal Dirac source, we assume a time-dependent rectangular laser pulse shape such that:

(13)$$\{\begin{array}{cc}\multicolumn{1}{c}{F\left(t\right)=\gamma }& \multicolumn{1}{c}{\mathrm{if}0\le t\le \tau }\\ \multicolumn{1}{c}{F\left(t\right)=0}& \multicolumn{1}{c}{\mathrm{elsewhere}}\end{array}$$

Using convolution properties of Laplace transformation, the new temperature rise reads:

(14)$${\theta }_{2D}(\overrightarrow{r},s)=\genfrac{}{}{0.1ex}{}{E}{2\pi {\lambda }_{\mathit{KDP}}}{K}_0\left(r\sqrt{s⁄D}\right)\genfrac{}{}{0.1ex}{}{1-e^{-\tau s}}{s}$$

where (1–e−^τs)/s is simply the Laplace transformation of the rectangular laser pulse given by Eq. (13). The latter expression can be written in the following form:

(15)$${\theta }_{2D}(\overrightarrow{r},s)=\zeta (\overrightarrow{r},s)-e^{-\tau s}\zeta (\overrightarrow{r},s)$$

with

(16)$$\zeta (x,s)=\genfrac{}{}{0.1ex}{}{E}{2\pi {\lambda }_{\mathit{KDP}}\tau }{K}_0\left(r\sqrt{s⁄D}\right)⁄s$$

We are only interested in the temperature at t=τ, thus implying that the last term of Eq. (15) can be omitted (since it contributes only for t>τ). Finally, the temperature induced by a 2D point source reads:

(17)$${\theta }_{2D}(\overrightarrow{r},t)=\genfrac{}{}{0.1ex}{}{A{a}^2}{4{\lambda }_{\mathit{KDP}}}{\int }_{r^2⁄4\mathit{Dt}}^{\infty }\mathit{du}\exp \left(-u\right)⁄u$$

where E has been set to πa ². The integral in Eq. (17) is evaluated numerically.

C. Analytical derivations in the 1D framework

C.1. Evaluation of the scaling law exponent

In order to analytically evaluate the scaling law exponent x linking the critical fluence F_c to the laser pulse duration τ as F_c∝τ ^x, one considers two critical fluences F _c1 and F _c2 associated with the pulse durations τ ₁ and τ ₂ respectively. Consequently, x reads:

(18)$$x=\genfrac{}{}{0.1ex}{}{\mathrm{}\ln (F_{c1}⁄F_{c2})}{\mathrm{}\ln ({\tau }_1⁄{\tau }_2)}$$

In order to simplify the problem, we assume that a compact cluster of planar defects has the same temperature evolution as a single plane, i.e. T=T ₀ +αF/τ ^1/2 [25] where α is a constant. We further assume that the pulse of duration τ ₂ probes essentially this compact cluster (located at x=0) whereas the pulse of duration τ ₁ also probes all the ADNS close to the cluster which cooperate for the temperature increase at x=0. We thus suppose that τ ₂≤a few ns [41] and τ ₁≫τ ₂. Under such conditions, we have:

(19)$$T_2\left(x=0,t={\tau }_2\right)=T_0+\alpha \genfrac{}{}{0.1ex}{}{F_2}{{\tau }_2^{1⁄2}}$$

and

(20)$$T_1\left(x=0,t={\tau }_1\right)=T_0+\alpha \genfrac{}{}{0.1ex}{}{F_1}{{\tau }_1^{1⁄2}}+\genfrac{}{}{0.1ex}{}{n_{\mathit{coop}}}{4\sqrt{D{\tau }_1}}{\int }_{-2\sqrt{D{\tau }_1}}^{2\sqrt{D{\tau }_1}}\mathit{dx}{\theta }_{1D}(x,{\tau }_1)$$

Since we integrate up to a distance $2\sqrt{D{\tau }_1}$ from the cluster, we approximate θ _1D(x,τ ₁) by αFτ ^1/2 exp(−x ²/4Dτ) (the term that principally contributes to the integral). Further, $\sqrt{\mathit{D\tau }}$, where n ₀ is a constant proportional to n_ADNS. Simple algebra leads to:

(21)$$T_1\left(x=0,t={\tau }_1\right)=T_0+\alpha \genfrac{}{}{0.1ex}{}{F_1}{{\tau }_1^{1⁄2}}+\beta F_1{n}_0$$

where β is a constant. We then obtain:

(22)$$x=\genfrac{}{}{0.1ex}{}{\ln \left(\genfrac{}{}{0.1ex}{}{{\tau }_1^{1⁄2}}{{\tau }_2^{1⁄2}\left(\alpha +\beta n_0{\tau }_1^{1⁄2}\right)}\right)}{\ln ({\tau }_1⁄{\tau }_2)}=\genfrac{}{}{0.1ex}{}{1}{2}-\genfrac{}{}{0.1ex}{}{\ln \left(\alpha +\beta n_0{\tau }_1^{1⁄2}\right)}{\ln ({\tau }_1⁄{\tau }_2)}$$

For a given ratio r=τ ₁/τ ₂ and under the assumption of a sufficiently large value of τ ₁ in order to obtain the inequality β n ₀ τ ^1/2 ₁≫α, we finally have:

(23)$$x\simeq \genfrac{}{}{0.1ex}{}{1}{2}-\genfrac{}{}{0.1ex}{}{\ln \left(\beta n_0\right)}{\ln r}-\genfrac{}{}{0.1ex}{}{\ln {\tau }_1}{2\ln r}$$

Despite that this expression has been established based on several assumptions, it allows us to shed light on the behavior of x with respect to n_ADNS and τ : the larger n_ADNS, the lower x and the longer the pulse duration, the lower x as we can simply obtain with a physical feeling.

C.2. Damage density and LIDT

Sections 2 and 3 demonstrated that the LID is due to clustering of planar defects distributed along a direction perpendicular to the planes. Also, the damage probability corresponds to the probability of the cluster having a spatial random distribution of the defects. This probability is thus given by the ratio between the number of configurations with a cluster and the total number of configurations (number of ways to distribute n_ADNS in N places). The latter number is simply given by $C_N^{n_{\mathit{ADNS}}}$ where C^p_n=n!/p!(n–p)! is the binomial coefficient. The number of configurations with a cluster of size n_d (all ADNS are assumed to be adjacent) is given by $(N-n_{\mathit{ADNS}})C_{N-n_d}^{n_{\mathit{ADNS}}-n_d}$, as long as n_ADNS≪N. Straightforward calculations then lead to:

(24)$$P\simeq \genfrac{}{}{0.1ex}{}{\left(N-n_{\mathit{ADNS}}\right)\left[n_{\mathit{ADNS}}\left(n_{\mathit{ADNS}}-1\right)\dots \left(n_{\mathit{ADNS}}-n_d-1\right)\right]}{N\left(N-1\right)\dots \left(N-n_d-1\right)}$$

In the case where n_ADNS≫n_d (which represents a correct assumption in our calculations [17]), we obtain:

(25)$$P\simeq \left(N-n_{\mathit{ADNS}}\right){\left(\genfrac{}{}{0.1ex}{}{n_{\mathit{ADNS}}}{N}\right)}^{n_d}$$

Now, as long as the cluster size is small compared to the thermal diffusion length, the temperature is proportional to the number of ADNS composing the cluster multiplied by the fluence. We can therefore write n_d=f (τ)T_c/F where f (τ) is a function of the pulselength. Finally, the damage density reads:

(26)$${\rho }_d\simeq {\rho }_h\left(N-n_{\mathit{ADNS}}\right){\left(\genfrac{}{}{0.1ex}{}{n_{\mathit{ADNS}}}{N}\right)}^{f\left(\tau \right)T_c⁄F}$$

We define the LIDT F_c as ρ_d(F_c)=ρ_c where ρ_c can be set to any arbitrary value. By taking the logarithm of Eq. (26), simple algebra leads to:

(27)$$F_c=f\left(\tau \right)T_c\genfrac{}{}{0.1ex}{}{\ln (n_{\mathit{ADNS}}⁄N)}{\ln ({\rho }_c⁄N{\rho }_h)}$$

C.3. Damage density at the threshold

In order to approximate the damage density evolution with respect to the fluence at the threshold, we can develop ρ_d as a Taylor expansion:

(28)$${\rho }_d{\left(F\right)]}_{F=F_c}={\rho }_d\left(F_c\right)+{\genfrac{}{}{0.1ex}{}{\partial {\rho }_d}{\partial F}]}_{F=F_c}\genfrac{}{}{0.1ex}{}{\left(F-F_c\right)}{1!}+$$
$${\genfrac{}{}{0.1ex}{}{{\partial }^2{\rho }_d}{\partial F^2}]}_{F=F_c}\genfrac{}{}{0.1ex}{}{{\left(F-F_c\right)}^2}{2!}+\cdots +{\genfrac{}{}{0.1ex}{}{{\partial }^n{\rho }_d}{\partial F^n}]}_{F=F_c}\genfrac{}{}{0.1ex}{}{{\left(F-F_c\right)}^n}{n!}+\cdots$$

Since n_ADNS≪N, we have ‖lnⁿ(n_ADNS/N)‖≫‖ln^n-1(n_ADNS/N)‖. By using Eq. (26), the derivative at the order n of ρ_d can thus be approximated by:

(29)$$\genfrac{}{}{0.1ex}{}{{\partial }^n{\rho }_d}{\partial F^n}\simeq \left(-\genfrac{}{}{0.1ex}{}{f\left(\tau \right)T_c}{F^2}\mathrm{}\ln (n_{\mathit{ADNS}}⁄N)\right){\rho }_d$$

It follows that:

(30)$${\rho }_d{\left(F\right)]}_{F=F_c}\simeq {\rho }_c\underset{n=0}{\overset{\infty }{\mathrm{\Sigma }}}\left(-\genfrac{}{}{0.1ex}{}{f\left(\tau \right)T_c}{F_c^2}\ln (n_{\mathit{ADNS}}⁄N)\right)\genfrac{}{}{0.1ex}{}{{\left(F-F_c\right)}^n}{n!}$$

(31)$$\simeq {\rho }_c\exp \left\{-\genfrac{}{}{0.1ex}{}{\beta \left(\tau \right)T_c\left(F-F_c\right)}{F_c^2}\ln (n_{\mathit{ADNS}}⁄N)\right\}$$

Finally, by using the expression (27) of F_c, we obtain an approximated form for the evolution of ρ_d at the threshold:

(32)$${\rho }_d\left(F\right)\simeq {\rho }_c{\left(\genfrac{}{}{0.1ex}{}{N{\rho }_h}{{\rho }_c}\right)}^{\left(F-F_c\right)⁄F_c}$$

Due to the exponential nature of ρ_d(F) and because of the particular form of its derivatives, expression (32) can be shown to be simply the tangent of lnρ_d(F) at F=F_c.

D. Conditioning modeling

D.1. Details regarding the numerical implementation of the conditioning

We define the normalized absorption efficiency, α, as:

(33)$$\alpha =1-\genfrac{}{}{0.1ex}{}{n_a}{n_i}$$

where n_i is the initial density of couples of defects and n_a stands for the number of annihilations per unit of surface of the plane. Also, if no annihilations occur, n_a=0 and α=1 remain unchanged. In a case where all the couples of defects annihilate, n_a=n_i and subsequently α=0 subsequently. When setting α_N to be the absorption efficiency after N_p laser pre-exposures, it is straightforward that:

(34)$${\alpha }_{N_p}=1-\genfrac{}{}{0.1ex}{}{n_a^{\left(1\right)}+n_a^{\left(2\right)}+\cdots +n_a^{\left(N_p\right)}}{n_i}$$

Here, n^(Np)_a is the number of annihilations due to the N_pth laser pre-exposure. If we define the absorption efficiency α^(Np)=1-n^(Np)_a/n_i, then, Eq. (34) can be rewritten as:

(35)$${\alpha }_{N_p}={\alpha }^{\left(1\right)}+\left({\alpha }^{\left(2\right)}-1\right)+\left({\alpha }^{\left(3\right)}-1\right)+\cdots +\left({\alpha }^{\left(N_p\right)}-1\right)$$

In practice, we obtain a new absorption efficiency with the relation of recurrence α _Np+1=αN_p+(α ^(N_p)-1) where αN_p is known from the previous calculation and α^(N_p) is determined as described below. α ^(N_p) is a function of the number of displacements, dN_p, of each interstitial atom induced by the N _pth shot. We have tabulated the evolution of n^(Np)_a for any number of displacements. Then α^(Np) is obtained from these tabulated values for a known value of dN_p that is simply given by the number of time steps weighted by the displacement probability, i.e. ∑_i P(t=t_i). Here P(t=t_i) is the probability given by Eq. (9) and t_i is a multiple of 1 ps (see Section 4.1).

D.2. Analytical analysis of the evolution of the gain as a function of the number of pre-exposures

From the last paragraph, it is straightforward that:

(36)$${\alpha }_{N_p+1}={\alpha }_{N_p}-\genfrac{}{}{0.1ex}{}{n_a^{\left(N_p+1\right)}}{n_i}$$

Since n^(Np)_a is roughly proportional to the length covered by a migrating defect, provided that not too many annihilations have occurred, this length being itself proportional to the probability (9) of defect displacement, we deduce that n^(Np) _a is proportional to exp(-E_a/kT) where T is the temperature induced by the N_pth laser pre-exposure. Within our modeling, this temperature is proportional to α_Np. Thus Eq. (36) transforms into:

(37)$${\alpha }_{N_p+1}={\alpha }_{N_p}-\beta \exp (-\gamma ⁄{\alpha }_{N_p})$$

where β is a constant and γ∝Ea/kF_cond. The equivalent continuous form of Eq. (37) is:

(38)$$\genfrac{}{}{0.1ex}{}{d\alpha }{d{N}_p}=-\beta \exp (-\gamma ⁄\alpha )$$

Since F_c∝α ^-1 and using the definition of the gain (10), we deduce that g=α ^-1. It follows that the differential equation governing the variation of the gain with respect to N_p reads:

(39)$$\genfrac{}{}{0.1ex}{}{\mathit{dg}}{d{N}_p}=\beta g^2\exp \left(-\gamma g\right)$$

To the best of our knowledge, this differential equation does not have a solution. Nevertheless, we can obtain some information concerning the behavior of g. First, the second term of Eq. (39) is positive, implying that g is an increasing function. Secondly, the asymptotic behavior of g when N_p→∞ is imposed by the exponential function. Also, for the large values of N_p, Eq. (39) can be approximated by g′=β exp(-γg), for which the solution is simply a logarithmic function. Also, we can expect that the solution of (39) is close to a logarithmic function. Indeed, a numerical resolution of Eq. (39) leads to the mentioned logarithmic behavior of g. Finally, since γ is inversely proportional to F_cond, a higher F_cond gives rise to larger variations of g with respect to N_p.

Acknowledgments

Anthony Dyan is acknowledged for fruitful discussions, as are all members of the Groupe de Travail Endommagement Laser (GTEL), especially Pierre Grua, Jean-Pierre Morreeuw and Hervé Bercegol. Nicolas Mallejac has also contributed with helpful input. Many thanks are due to the MP2 group of the Institut Fresnel in Marseille. The AlphaScience company is acknowledged for its english corrections of the present manuscript.

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32. If several ADNS contribute to the determination of d, e.g. n, then d transforms into d/n in our calculations.

33. The variation of the gain with respect to τ_cond can be increased by setting the modeling parameters to values differing from the ones used.

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41. The equivalent distance of about 100 nm corresponding to the mean distance between two ADNS with n_adns=100 and n=10000, which represent standard values in our calculations.