Simple models for laser-induced damage and conditioning of potassium dihydrogen phosphate crystals...

Simple models for laser-induced damageand conditioning of potassium

dihydrogen phosphate crystals bynanosecond pulses

Guillaume DuchateauCommissariat a l’Energie Atomique, Centre d’Etudes du Ripault, BP16, 37260 Monts, France

[email protected]

Abstract: When potassium dihydrogen phosphate crystals (KH2PO4 orKDP) are illuminated by multi-gigawatt nanosecond pulses, damages mayappear in the crystal bulk. One can increase damage resistance through aconditioning that consists in carrying out a laser pre-exposure of the crystal.The present paper addresses the modeling of laser-induced damage andconditioning of KDP crystals. The method is based on heating a distributionof defects, the cooperation of which may lead to a dramatic temperaturerise. In a previous investigation [Opt. Express 15, 4557-4576 (2007)], cal-culations were performed for cases where the heat diffusion was permittedin one and three spatial dimensions, corresponding respectively to planarand point defects. For the sake of completeness, the present study involvesthe 2D heat diffusion that is associated with linear defects. A comparison toexperimental data leads to the conclusion that 1D calculations are the mostappropriate for describing the laser-induced damage in KDP. Within thisframework, the evolution of the damage density is given as a function ofthe laser energy density and an in-depth analysis of the results is providedbased on simple analytical expressions that can be used for experimentaldesign. Regarding the conditioning, assuming that it is due to a decreasein the defect absorption efficiency, two scenarios associated with variousdefect natures are proposed and these account for certain of the observedexperimental facts. For instance, in order to improve the crystal resistanceto damage, one needs to use a conditioning pulse duration shorter than thetesting pulse. Also, a conditioning scenario based on the migration of point(atomic-size) defects allows the reproduction of a logarithmic-like evolutionof the conditioning gain with respect to the number of laser pre-exposures.Moreover, this study aims at refining the knowledge regarding the precursordefects responsible for the laser-induced damage in KDP crystals. Withinthe presented modeling, the best candidate permitting the reproductionof major experimental facts is comprised of a collection of one-hundred-nanometer structural defects associated with point defects as for instancecracks and couples of oxygen interstitials and vacancies.

© 2009 Optical Society of America

OCIS codes: (140.3330) Laser damage; (140.3390) Laser materials processing; (320.4240)Nanosecond phenomena

#108732 - $15.00 USD Received 16 Mar 2009; revised 30 May 2009; accepted 2 Jun 2009; published 8 Jun 2009

(C) 2009 OSA 22 June 2009 / Vol. 17, No. 13 / OPTICS EXPRESS 10434

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and n = 10000, which represent standard values in our calculations.

1. Introduction

Dihydrogen phosphate crystals (KH2PO4 or KDP) and their deuterated analogs (KD2PO4 orDKDP) are widely used to perform frequency conversions of laser pulses. For instance, suchcrystals can be employed to produce intense nanosecond laser pulses at 3ω (a wavelength of351 nm) in order to initiate fusion reaction [1]. However, under current operating conditionsof the National Ignition Facility in the U.S.A. or the Laser MegaJoule in France, for whichthe 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 thisdamage is induced by precursor defects that efficiently absorb the laser energy, inducing a fasttemperature rise and a subsequent shock wave [2]. In order to ensure a good working orderof these large laser aperture facilities, it is crucial to manage the creation of damage. To doso, one needs to identify the nature of the precursor defects and to understand the physicalmechanisms 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 thecrystal 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 physicalmechanism induced by the conditioning remains unknown. Despite significant findings, a goodcomprehension of these mechanisms remains to be achieved. Furthermore, the nature of theprecursor defects is still unknown. By providing simple models, this paper aims at improvingthe knowledge of damage and conditioning mechanisms in KDP crystals, and at providing someadditional information regarding the nature of the precursor defects.

These precursor defects may be of varying nature. Since the crystal growth requires certainadditives, 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 notinvolved in LID [6, 7, 8, 9]. Consequently, defects created during the crystal growth may beinvoked. First, one can consider point defects, corresponding to hydrogen or oxygen atoms asinterstitial 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 and100 nm in order to produce a sufficiently high temperature, only a cluster of point defects cansatisfy 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 largerthan 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, suchas mother liquid inclusions, dislocations or cracks. One can also regard LID to be due to acouple of defects for which only a cooperative mechanism permits an efficient laser energyabsorption, inducing a high local temperature. It can for instance be imagined that an inclusionor a cluster of point defects produces cracks in its vicinity. Indeed, they strongly deform thesurrounding lattice and, since KDP crystals exhibit a low mechanical resistance, this mightlead 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 andheat transfer has been developed [17]. In this modeling, the LID results from the aggregationof 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 determineseveral experimental trends, such as mainly the S-shape of the damage probability curves anda particular scaling law characteristic of KDP crystals linking the critical fluence, Fc, to thepulse duration, τ , as roughly Fc ∝ τx 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 werein a good agreement with experimental data. It could thus be concluded from this modelingthat growth bands, cracks, array of dislocations or staking faults were good candidates forexplaining LID in KDP crystals. For the sake of completeness, Section 2 describes the studyof 2D heat diffusion associated with linear defects. Further, in its original form, the modelprovided only damage probabilities. The interest in the damage density by the experimentalistshas grown in the last years since it renders it possible to obtain more physical information, seefor 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 implicationsbased on analytical derivations is presented.

Regarding the conditioning process, preliminary explanation attempts have been providedby Feit et al [15] who suggested that the increase in LID threshold is due to the precursordefects decreasing in size. According to Chirila et al, the conditioning may be due to a ”pas-sivation” mechanism for which the electronic structure of the precursor defects is altered byelectrons or holes produced by the laser pre-exposure. Despite that these attempts give a pre-liminary insight regarding the manner in which the conditioning may work, this is merely aphenomenological description of the conditioning process. Further, a detailed comparison toexperimental results in order to verify and report on the reliability of the assumptions has notbeen carried out. Section 4 aims at introducing a modeling of the conditioning process that isbased on the damage modeling assumptions. Since there exist various types of precursor defects[20] for which conditioning may differ, two physical mechanisms, corresponding to differenttypes of precursor defects, are proposed. The first model assumes the precursor planar defectsto actually be composed of point defects for which the thermally-activated migration may leadto 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 crystallinerearrangement. As a consequence, the defect absorption vanishes [21]. The predictions of eachmodel are compared to a list of experimental trends, including:

• The shape of the damage density with respect to the fluence is practically unchanged bylaser conditioning, the influence of the conditioning consists in shifting the whole curveto 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 increas-ingly improved as the conditioning fluence is augmented [20, 6]. Further, there existsa conditioning fluence threshold below which no conditioning effect whatsoever is ob-served [20].

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

Finally, Section 5 provides conclusions and outlooks of the present work. For the conve-nience of the reader, details of certain derivations and a table describing the symbols usedthroughout 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 beassociated with dislocations) oriented in the same direction. Since a dislocation disturbs thecrystalline lattice, its absorption efficiency is larger than the one of a perfect crystal and itcan be seen as a source inducing a temperature rise. The dislocation length is assumed to belarger than the thermal diffusion length given by 2

√Dτ where D is the thermal diffusivity. It

follows that the temperature field remains the same whatever the position along the dislocationdirection. In the following, the mathematical formalism used to model the problem is brieflydescribed, but more details are given in [17]. The temperature field in the plane perpendicular tothis direction, hereafter referred to as the P-plane, and for which the characteristic dimensionhas been set to 1 μm which is comparable to experimental dimension [23, 24], can be obtainedby solving the Fourier equation:

∂T∂ t

= DΔT +A

ρC

nADNS

∑i=0

Π(�r−�ri) (1)

Here,�ri refers to the position (x,y) of the dislocation i randomly distributed in the P-plane,nADNS is the number of dislocations in the P-plane (ADNS stands for Absorbing Defects ofNanometric Size) and A is the absorbed power per unit of volume that can be expressed as104F/τ in units of W.cm−3 from empirical considerations [17]. Material physical parameterssuch as thermal diffusivity, D, and conductivity, λ , density, ρ , or heat capacity, C, are assumedto remain constant during the course of interaction. The function Π is defined as:

{Π(�r−�ri) = 1/a if x ∈ [xi −a/2;xi +a/2] and y ∈ [yi −a/2;yi +a/2]Π(�r−�ri) = 0 elsewhere

(2)

where a is the source size in the P-plane, set to 1 nm in the calculations. A general solutionof Eq. (1) can be obtained [25] by summing the solutions of the Fourier equation designed foronly one point source. Now, since working conditions include a � √

Dτ (for DKDP = 6.5×10−7 m2.s−1 and τ = 1 ns,

√Dτ � 25 nm), in order to deal with simple formula allowing fast



numerical calculations, the function Π(x) can be approximated by the Dirac delta function δ (�r).The dislocations may then be seen as a point heat source in the P-plane. The temperature riseinduced by this source reads (see Appendix B for the analytical derivation):

θ2D(�r, t) =Aa2

4λKDP

∫ ∞

r2/4Dtduexp(−u)/u (3)

Then, when considering a set of sources, cooperative effects lead to a higher temperature as op-posed to the one induced by a single source. Further, the larger the number of ADNS involvedin a temperature rise, the higher is the temperature. This is illustrated in Fig. 1 where a tempera-ture field is associated with an ADNS distribution. Figures 1(a) and 1(b) respectively show theADNS distribution and the resulting temperature field in a case without a big cluster. Figures1(c) and 1(d) correspond to a case in which a cluster leading to a significant temperature riseis present (located in the circle of Fig. 1(c)). Moreover, it can be clearly seen that only clusterscomposed of a large number of ADNS give rise to high temperatures. It should be noted thatthe temperature scales differ between the two distributions. In one case, despite the appearanceof clusters that significantly increase the temperature, it remained below Tc. In the second case,a cluster composed of more ADNS permitted the temperature to exceed Tc, subsequently givingrise to damage.

Fig. 1. A 2D temperature field in the P-plane (see text) for an ADNS distribution notleading to damage (top) and another temperature field implying a damage (bottom). Thecluster leading to damage is inside the dashed red circle.

The introduction of a damage probability by utilizing a critical temperature criterion isstraightforward. For a given number of random ADNS distribution, it suffices to, at the endof the pulse, count the number of times for which the temperature is larger than the critical tem-perature in at least one place in the heterogeneity [17]. Figure 2 illustrates the results that canbe obtained with this modeling. It shows the evolution of the damage probability as a functionof the laser fluence with the following parameters: τ = 3 ns, nADNS = 2000 and 5000. The valueof the parameter ξ/l has been set to 105 cm−1 in order to find a critical fluence (defined in themodeling 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 frame-work, this can be understood as the temperature rise increasing with the cluster size. Since theclusters are more numerous when their sizes are small, it follows that the higher the fluence, thelarger the probability. Now, in order to determine whether the linear defects may be responsiblefor LID in KDP, one must verify if the calculations are able to reproduce experimental data.The main parameters characterizing the LID are Fc and x. With the large value ξ/l = 105 cm−1

(compared to 104 cm−1, obtained from empirical considerations [17]) and nADNS = 5000 (i.e.0.5% of defects in a heterogeneity of 1 μm), one obtains Fc = 8.11 J.cm−2 and x = 0.21 forτ = 3 ns (fit as shown by the inset (a) of Fig. 2). Fc is reasonable but not the value of x. Withthe same value of ξ/l but with nADNS = 2000, a reasonable value x = 0.31 is obtained, how-ever, in this case, Fc = 16.51 J.cm−2 is too high (inset (b) of Fig. 2). Also, ξ/l has to be set to2×105 cm−1 in order to reproduce the experimental values of both Fc and x (Fc is here foundto be 8.15 J.cm−2). Beyond the fact that ξ/l has a much larger value than the empirical one, itcorresponds to an almost unrealistic physical process. Indeed, the temperature rise is due to theenergy 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 bync = mε0ω2/e2 � 9.1×1021 cm−3 at 3ω . By using a Drude model appropriately characterizingthe electronic plasma, ξ/l can be linked to ne as A = ξ F/lτ = nedEc/dt, where

dEc

dt=

e2

ε0m× νcoll

c(ω2 +ν2coll)

× Fτ

(4)

is the energy absorbed by one electron per unit of time, and νcoll is the inverse of the timeelapsed 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 tone � 8×1021 cm−3, i.e. roughly the critical plasma density. With ξ/l = 2×105 cm−1, one hasne � 20nc, which is physically unrealistic. These 2D heat diffusion calculations cannot be com-pletely turned down despite the fact that slightly different parameters (a higher νcoll for example)would provide a more realistic value of ne. This order of magnitude calculation demonstratesthat planar defects seem to provide data that are more consistent with experimental results asopposed to linear defects.

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

To conclude this section, an ensemble of linear defects, inducing a 2D heat transfer (thatcan 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 forthe LID in KDP crystals. Also, since 3D heat transfer calculations have already been turneddown [17], focus is now placed on the 1D heat transfer associated with planar defects as thebest 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 crys-tals within the present modeling framework. A study based on the damage probability, P, hasbeen carried out [17]. Nevertheless, as of a few years, a great interest has been devoted to theexperimental study of the damage density since it provides more information and avoids anydependence on the irradiated volume. Within the proposed model, the damage density, ρd , issimply obtained with ρd = Pρh, where ρh is the density of heterogeneities that are likely tocause damage. Fc is defined as the fluence required to reach a given damage density, ρc. It isset 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 ofparameters ρh, nADNS and τ with a heterogeneity of length L = 10 μm [17]. The first conclusionthat can be drawn is that the general shape exhibits characteristics similar to experimental re-sults: 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 influenceof ρh is shown in Fig. 3(a). It appears that the larger ρh, the lower Fc. Indeed, when ρh increases,less energy is required to obtain a given number of heterogeneities for which T > Tc. The samereasoning leads to the conclusion that the larger nADNS, the lower Fc, as confirmed by Fig. 3(b).Concerning the influence of the pulse duration as displayed in Fig. 3(c), one finds the samebehavior as the damage probability with Fc ∝ τ0.35. Actually, the scaling law exponent deviatesfrom the expected 1/2 value (associated with only one ADNS) as a result of cooperative effectsbetween ADNS. An analytical development based on certain assumptions provides a better in-sight of this fact and allows to determine an approximated expression of x as (see AppendixC):

x � 12− y(nADNS)− z lnτ (5)

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

The behavior of Fc with respect to ρh and nADNS can be derived from analytical considerationsbased on probability calculations. In Appendix C, the damage density is approximated by:

ρd = ρh(N −nADNS −1)(nADNS

N

) f (τ)Tc/F(6)

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 ρd = ρc where ρc = ρd(Fc), an approximated expression of theLIDT can be derived:

Fc = f (τ)Tcln(nADNS/N)ln(ρc/Nρh)

(7)

Also, Fc is significantly reduced when nADNS or ρh increases. Furthermore, as part of the searchfor 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 designedfor measuring the LIDT variations with respect to a controlled concentration of defects shouldprovide an answer to the question of whether the defects under investigation are responsible ornot 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 Nh with nADNS = 100 and τ = 1 ns (b) Theinfluence of nADNS with Nh = 106 cm−3 and τ = 1 ns (c) The influence of the pulse durationwith nADNS = 100 and Nh = 106 cm−3.

Other means of extracting information from the experimental variations of the damage den-sity involve considering a simple form of the theoretical expression of the damage densitywhich only accounts for macroscopic measurable parameters and employing a fitting proce-dure. Since both the experimental and theoretical damage densities may suffer from artifactsat high fluences (coalescence [28] and saturation respectively), it would thus seem preferableto focus the attention on the threshold. For fluences close to Fc, the damage density can beapproximated by (see derivation in Appendix C):

ρd(F) � ρc

(Nρh

ρc

)(F−Fc)/Fc

(8)

In Eq. (8), apart from Nρh, all terms can be determined from experiments. Also, the value ofNρh can be obtained from experimental data by using an adapted fitting procedure. It was veri-fied 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 KDPcrystals by carrying out a laser pre-exposure. However, no clear proposal of how conditioningworks has been put forward. Beyond the fact that it is of interest to understand the fundamentalmechanisms underlying conditioning, from an applicative point of view, it is also importantto 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 isdue to a considerable temperature rise. It is thus reasonable to expect that conditioning effectsare also due to temperature rises. An increase in temperature influences a material such that itmay activate the migration of defects or induce phase transition. Further, it has previously beendemonstrated that planar defects seem to best explain LID in KDP crystals. For this reason, thefollowing two scenarios of conditioning are proposed:

i. The above-mentioned planar defects are assumed to actually be composed of atomic-sizedefects for which displacements in the plane are governed by an Arrhenius law. It is supposedthat these point defects can annihilate during their migration. Also, within this scenario, theconditioning consists in decreasing the absorption of each planar defect by decreasing the num-ber 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 con-ditioning laser pulse induces a phase transition (boiling) in the vicinity of the defect. Subse-quently, when cooling to the room-temperature, rearrangements of the crystalline lattice removethe defect and its absorption efficiency vanishes. We here deal with a “zero-one” model. Thisscenario is referred to as Conditioning Model 2 (CM2).

The following section presents the details of the modeling. Within the CM1 framework, eachplanar defects is composed of an ensemble of point (atomic size) defects that may stronglyabsorb the laser energy. This results in a significant temperature rise due to cooperative heatingeffects. The temperature along the direction perpendicular to the plane decreases faster thanexponentially [17]. Since a point defect is only able to move in a region where the temperaturerise is significant (see Eq. (9) below), it can be reasonably stated that the migration takes placeessentially in the plane. In the course of migration, these defects are assumed to be able toannihilate [9, 14, 13], i.e. a recombination of a pair of particles mediated by thermally activateddiffusion may occur. For instance, interstitial-vacancy pairs satisfy this requirement but, atomicdefects 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. Inorder to numerically evaluate the absorption variation due to a laser conditioning, the followingprocedure was adopted. The absorption was assumed to be proportional to the number of defectspairs per unit of surface. For each time step, the probability Pm of a defect displacement from acell (with a characteristic size of 1 nm) to an adjacent one is given by:

Pm(t) = exp

(− Ea

kT (t)

)(9)

where Ea is the activation energy and k is Boltzmann’s constant. A jump is assumed (dependingon the value of Pm) to occur every 1 ps [9], with the same probability in each direction of theplane. After a time step, it is verified whether the interstitial atom has met or not a vacancy. Ifso, the number of pairs decreases by one. If no meeting occurs, the migration can go on. Thiscycle is performed as long as the temperature is such that Pm(t) is larger than one percent of



Pm(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 themigration has stopped, the remaining (non-annihilated) defect pairs are counted and the newabsorption of a plane is given by αcond = nrα0/n0, where nr is the density of the remainingpairs, n0 is the density of the initial pairs, and α0 is the plane absorption before the condition-ing. Various details concerning the numerical implementation of the algorithm permitting tooptimize the calculations are given in Appendix D.

For CM1, in order to speed up the calculations, other reasonable assumptions have beenmade. Also, periodic conditions have been used to carry out the migration process. The planardefect absorption efficiency was assumed constant in the course of interaction whereas a fewrecombinations may occur. Nevertheless, the migration becomes really effective at the end ofthe 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, be-cause one often deals with rapidly grown crystals, structural defects represent good candidatesfor explaining LID. When illuminated by a laser pre-exposure, the absorption properties of astructural defect may lead to a significant local temperature rise and subsequently to a phasetransition in its surroundings. The phase transition of interest is the boiling, as it is the mostsuited for removing structural defects [29]. Since this phase transition takes place in a confinedmedium where a pressure of tens of GPa can be reached [2], a higher temperature than ambientis required for the boiling point, Tbp [30, 31]. Under ambient conditions, the boiling tempera-ture is close to 673 K, however, in the calculations, Tbp is set to roughly 2000 K in order to takeinto account the latter fact. Now, depending on the structural defect thickness, e, a minimumquantity of matter has to be heated up to Tbp in order to remove this defect. Since a 1D heatdiffusion is considered, this quantity of matter is proportional to the distance, d, from the struc-tural defect for which T ≥ Tbp; at least for an instant (since one has a propagating heat front).Therefore, in the modeling, the conditioning succeeds if d ≥ βe, where β is a constant set to5 and e is assumed to range randomly between 0.1 nm and 2 nm [32]. For a given structuraldefect, 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 inthe main results presented hereafter.

At this point, on the basis of both conditioning models, it is possible to predict the influenceof a pre-exposure on the damage density produced by a testing pulse. The numerical simulationmimics the experimental protocol: for a given ADNS distribution with the absorption efficiencyas defined previously, a conditioning is simulated by first calculating the laser-induced tempera-ture field and, then, evaluating the new absorption efficiency of each ADNS. It is noteworthythat each ADNS is subjected to a specific temperature (depending on whether cooperative ef-fects take place or not), and as a result, each ADNS has its own new absorption efficiency. Withthese absorption efficiencies, it is possible to carry out the standard damage testing in order toobtain the damage density. Since the influence of the conditioning is to decrease the absorp-tion 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:

g =F(ac)

c

F(bc)c

(10)

for which the values are larger than unity. Here a ratio is deliberately considered because it isless 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 pulseduration, τ , the conditioning pulse duration, τcond , as well as the conditioning laser fluence.

4.2. Damage densities

By applying the above-mentioned procedure, one can evaluate the damage density after condi-tioning. An illustration of the results that can be obtained for certain values of the CM parame-ters is given by Fig. 4. Calculations have been performed with constant values of the laser pulseparameters, i.e. τtest = 3 ns, Fcond = 3 J.cm−2 and τcond = 2 ns. A value of τcond shorter thanτtest was deliberately chosen due to the conditioning being all the more efficient when τcond isshort. This is further demonstrated in the following section.

Figure 4(a) shows the predictions of CM1 with Ea = 0.9 eV and 1.2 eV , and n0 =6.× 1012 cm−2 and 1.2× 1013 cm−2 (corresponding to a few hundreds of point defects in100 nm× 100 nm in size plane). As expected, regardless of the CM1 parameter values, theLIDT 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 Ea. Indeed, as this prob-ability is lowered, less annihilations occur and the absorption thus decreases to a lesser extent.Concerning the influence of n0, it appears that the LIDT increases along with n0. For the largestvalue of n2D, the probability of a moving interstitial to meet a vacancy is at a maximum. Thisresults in a large number of annihilations and subsequently to a low absorption efficiency. Theresults of Fig. 4(a) exhibit a particular trend with regard to the shape of the damage densitieswith respect to the fluence: after conditioning, the damage density slope displays a significantincrease. This is due to the fact that the biggest clusters are the most sensitive to the condition-ing (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 anda conditioned crystal within the CM1 modeling framework and (b) a conditioned crystalwithin the CM2 modeling framework.

Figure 4(b) presents the results of CM2 obtained with c = 5 and 15, and Tbp = 1600 K and2000 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 temperaturerange of interest (i.e. Tbp around 2000 K), for c = 5, the influence of Tbp principally consistsin shifting the damage density curve. For a given Tbp, in addition to the displacement of thecurve to lower fluences, the increase of c leads to a decrease of the slope. In that case, for thebiggest clusters, the matter has to be distributed among the ADNS (forming the cluster), whichresults in an effective distance that may be shorter than the smaller clusters. In fact, numericalcalculations (that provide the average number of ADNS related to every value of d) shows thatall cluster sizes are conditioned for c = 5, whereas the biggest clusters are less conditioned forc = 15. There are thus two influences that offset each other, which results in a slope that doesnot evolve much between an unconditioned and a conditioned crystal. Also, with CM2, one canfind a set of parameters that almost allows the conservation of the damage density slope afterconditioning.

4.3. Influence of the laser pulse parameters

From a general point of view, the conditioning gain depends on the pulse duration of both thetesting and the conditioning laser pulses, as well as the fluence of the conditioning pulse. Let usfirst focus on the influence of the pulses duration. The conditioning gain with respect to τtest andτcond provided by CM1 and CM2 is given in Figs. 5(a) and 5(b) respectively. For both models,Fcond = 5 J.cm−2. The CM1 parameters are given by Ea = 1.2 eV and n0 = 1.2× 1013 cm−2,and those of CM2 are c = 6 and Tbp = 2000 K. These parameters have been chosen in order toobtain comparable gains for both models. From both graphs, due to the iso-gain curves beingmainly horizontal, it can be deduced that the gain depends more on τcond than on τtest . In orderto better appreciate the evolution of the gain as a function of τcond , a cut of Figs. 5(a) and 5(b)has been plotted in Fig. 5(c) for τtest = 3 ns. This is the pulse duration of interest for the laserfacilities presented in the introduction. For both CM’s, it appears clear that the conditioningefficiency is better for shorter values of τcond . Moreover, Fig. 5(c) shows that CM2 gives rise tothe largest gain variations with respect to τcond . For both CM’s, it is worth noting that for theshortest conditioning pulses (sub-nanosecond duration), the temperature of the biggest clustersmay exceed Tc. It follows that the conditioning may produce damage.

Next, it is investigated how the gain evolves with respect to the conditioning pulse durationand fluence. Again for τtest = 3 ns, Figs. 6(a) and 6(b) portray this gain evolution predicted bythe CM1 and the CM2, respectively. In order to obtain a better insight of the gain evolutionas a function of Fcond , a cut of Fig. 6(a) and Fig. 6(b) has been reported to Fig. 6(c), withτcond = 500 ps and τcond = 3 ns. Both models exhibit the same trend: the conditioning gaindepends more on Fcond than on τcond . Indeed, the conditioning efficiency is directly related to thetemperature reached by each cluster during the conditioning. This temperature is proportionalto Fcond whereas it evolves slowly with respect to τcond (as τ−0.3 [17, 26]). The gain shapewith respect to Fcond is given in Fig. 6(c). The general behavior of all curves implies that thegain remains close to unity up to a certain fluence, after which there is a linear increase. Thecurves thus exhibit a conditioning fluence threshold above of which the conditioning becomesefficient. The existence of such a threshold is linked to the fact that a minimum temperaturehas to be reached in order to modify the absorption properties of the ADNS. Within CM1, theprobability Pm of defect displacement, and the subsequent possibility of annihilation, becomenon-negligible for kT ≥Ea. Within CM2, the absorption efficiency of an ADNS vanishes only ifT ≥ Tbp in the area surrounding the structural defect. Concerning the behavior of g for fluenceslarger than the above-mentioned threshold, it can be easily understood by the statement that thetemperature is proportional to the fluence. Further, from Fig. 6(c), one notices that the longerτcond , the higher is the conditioning threshold. In addition, one finds again that CM2 is more



Fig. 5. The LIDT gain obtained by conditioning with respect to τtest and τcond within (a)CM1 and (b) CM2. For both graphs Fcond = 5 J.cm−2. (c) A cut of (a) and (b) for τtest =3 ns.

sensitive to τcond than CM1.

4.4. Influence of the number of pre-exposures

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

Within the CM1, for a given laser pre-exposure, the initial positions of defect couples corre-spond to the final positions of the previous laser pulse pre-exposure. The details of an optimizedimplementation are given in Appendix D. After each shot, annihilations occur. However theyare 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, how-ever with smaller and smaller variations. This is confirmed by Fig. 7, portraying this behavioras numerically provided by CM1 with various pulse parameters, i.e. Fcond = 2 J.cm−2 and3 J.cm−2, and τcond = 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 decreaseback to room temperature. This behavior is further confirmed by a numerical benchmark andan analytical derivation specifying the gain evolution as close to a logarithmic increase (seeAppendix D). The inset of Fig. 7 represents the gain with a logarithmic scale on the horizontalaxis and allows to convince of this fact. More precisely, regarding the numerical benchmark, themigration of the couples of defects has been fully simulated (without optimization) in a singleplane. This calculation also provides the same logarithmic behavior as the one obtained withCM1. Further, the point defects migration and annihilation in a 3D space has been simulatedand again, the calculations show a logarithmic-like evolution of the absorption (assumed to beproportional 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 spacedimensionality does not permit a discrimination of the defect geometry (plane or sphere).

Fig. 6. The LIDT gain obtained by conditioning with respect to τcond and Fcond within (a)CM1 and (b) CM2. For both graphs τtest = 3 ns. (c) Cut of (a) and (b) for τcond = 500 psand 3 ns.

Concerning CM2, since an ADNS has been conditioned (with the first pre-exposure), it nolonger contributes to the temperature rise for the following pre-exposure. Also, the temperaturerise induced by the second pre-exposure is lower than the rise of the first whatever the consid-ered ADNS. As a result, the ADNS that have not been conditioned by the first pre-exposurecannot be conditioned by the second. It also turns out that CM2 does not provide any variationof the gain with respect to the number of pre-exposures for Np > 1.



4.5. Discussion

In order to put forward a conditioning model, this section first proposes a comparison of theresults of CM1 and CM2 to available experimental data. The influence of the conditioning onthe shape of the damage density with respect to the fluence, the influence of the conditioningpulse duration, the influence of the conditioning fluence, and the evolution of the gain as afunction 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 thefluence remains unchanged after a laser conditioning at 3ω . More precisely, the influence of theconditioning consists mainly in shifting the whole curve to larger fluence values. Section 4.2showed that CM1 always leads to a modification of the damage density shape whereas, withinCM2, it is possible to find parameters that allow the conservation of this shape, the conditioningsimulation mainly leading to a shift of the curve under these conditions. Therefore, concerningthe influence of the conditioning on the damage density shape, CM2 displays a better capacityfor mimicking the experimental results.

Fig. 7. The conditioning gain as a function of the number of laser pre-exposures. The insetshows the case Fcond = 2 J.cm−2 and τcond = 3 ns with a logarithmic scale on the horizontalaxis.

Regarding the influence of the conditioning pulse duration on the gain, both models predictan increase in the conditioning efficiency as the pulse length decreases under conditions whereFcond is a constant not depending on τcond . This trends is in agreement with the experimentaldata [4]. Nevertheless, it is worth noting that a closer comparison to the experimental protocolshould have used a ramp with increasing fluences; the highest fluence being an increasing func-tion of τcond [4]. One should however keep in mind that the study was performed by varyingphysical 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 itwas not possible to accurately reproduce all the experimental conditions, such as for instancean installation change required to obtain large variations of τcond [33]. In the performed calcu-lations, the introduction of Fcond increasing with respect to τcond would have been to decreasethe slope of g. Going further in the comparison to experiments, the agreement is decent as longas τcond is not too short (no less than roughly 800 ps according to [4]). For shorter conditioningpulses, the trend is reversed: the shorter the pulse, the lower the gain. In such a case of veryshort pulses, one may expect that new physical effects are involved. For instance, this trendcould 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 defectdisplacements and subsequently less annihilations, consequently leading to a lower gain. Theonly difference between CM1 and CM2 lies in the variations of the gain as a function of theconditioning pulse duration: CM2 exhibits larger variations than CM1. However, due to all theuncertainties (theoretical and experimental alike), the discrepancy is not enough to discriminateone 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 Fcond < F(m)cond , Fc remains unchanged

(i.e. F(ac)c = F(bc)

c ). Again, this fact has been experimentally observed [20], showing that a cer-tain 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 con-

ditioning fluence, the gain is also augmented, which is in good agreement with the experimentalobservations. It follows that the higher the fluence, the better the gain. But it has been numeri-cally observed that the critical temperature can be reached for a too high fluence, thus leading todamage. In that case, the beneficial effect of the conditioning is lost. This inconvenience can beeliminated by using several conditioning pulses of increasing fluence. Under such conditions,the first conditioning pulse with the lowest fluence passives only the biggest clusters. Subse-quently, for an increasing fluence, smaller and smaller clusters are treated. This procedure alsoallows the use of a final high fluence, Fmax

cond , which gives rise to a decent gain, but without cre-ating any damage. This protocol is also used experimentally and is called a conditioning ramp[4]. Moreover, the higher the Fmax

cond , the larger g, as also observed elsewhere [6]. Nevertheless,there exists an experimental limit for Fmax

cond [6] that may be due to the fact that another kind ofdefect begins to be excited.

The evolution of the gain as a function of the number of laser pre-exposures is the best wayto test the CM’s since they predict very different trends. Within CM2, the gain does not evolvewhereas CM1 predicts an increase. The latter phenomenology was observed in [20]: the damagedensity decreases as a function of the number of pre-exposures. Since the damage density afterthe conditioning and the conditioning gain are closely related, it can be concluded that CM1accounts well for the influence of the number of pre-exposures. The comparison can be takenfurther by analyzing the variations of the conditioning efficiency with respect to the number ofpre-exposures. According to [20], due to the damage density evolving almost linearly with ahorizontal logarithmic scale, it was concluded that it exhibits a behavior close to a logarithmicevolution 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 toreproduce this particular logarithmic-like behavior. Further, within this horizontal logarithmicscale, the slope of g depends on the conditioning fluence (see Appendix D), as it has beenobserved [22] for λ = 532 nm and λ = 355 nm. Furthermore, the higher Fcond , the larger thevariations in g with respect to N (i.e. the larger the variations of the damage density). It isimportant to stress that this logarithmic-like behavior arises from the fact that an Arrhenius lawwas employed (see Appendix D). Another law would provide another type of behavior. Sincethis law is representative of the temperature-driven defect migration process, it follows that anagreement with the experimental data represents a strong evidence that the defects responsiblefor 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 allthe 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 thatthe 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 beable 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 withthe above-mentioned point and structural defects. With regard to the point defects, couples ofinterstitials atoms and vacancies correspond well to the idea of them migrating and annihilatingwhen an interstitial atom meets a vacancy. In that case, the absorption associated with thesecouples 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 locatedin the band gap [10, 11, 12] that enhance the absorption. In addition, these states allow anexplanation for the particular experimental evolution of Fc as a function of the photon energyexhibiting successive plateaus [37]. The planar structural defects may be associated with cracks.Indeed, they exhibit the required absorption efficiency, since the breaking of periodic conditionsfirst introduces states in the band gap in the vicinity of the crack surface [21], after whichelectric field enhancements can facilitate the free electron production [38, 39, 21] leading to theabsorbing plasma. Furthermore, a crack may contain impurities that re-enhance the previousinfluences.

5. Conclusion

A model was developed based on statistics and heat transfer permitting to evaluate the tem-perature field induced by a distribution of defects in a heterogeneity. A damage is due to thepresence of a cluster in which cooperative effects lead to a dramatic temperature rise. Withinthis modeling framework, the heat diffusion can be allowed in one, two or three spatial dimen-sions, and the corresponding defects are planes, lines or points respectively. The present workcompleted 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 thescaling law exponent, led to the conclusion that the 1D framework was the best suited to ex-plain laser-induced damage in KDP crystals. The defects responsible for laser-induced damagein KDP were thus believed to be a collection of planar defects. Within this framework, it waspossible to predict the damage density as a function of the laser fluence. An in-depth analysisallowed the derivation of simple analytical expressions of the scaling law exponent, the criticalfluence and the damage density with respect to several physical parameters. From this, addi-tional information concerning the nature of the precursor defects, such as their density, couldbe obtained based on experimental data.

Two models of conditioning were developed based on temperature-driven mechanisms. Bothmodels 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 defectsthat can migrate in the above-mentioned planar geometry. The second model assimilates theplanar defects to structural ones, the heating of which might lead to crystalline rearrangementsand to a subsequent significant decrease in absorption. Both models account for particular ex-perimental trends. More precisely, these models reproduce the fact that the conditioning gain isbetter for shorter conditioning pulse durations. Moreover, there exists a minimum conditioningfluence 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 thegain with respect to the number of laser pre-exposures, the modeling based on the migration ofpoint defects renders it possible to reproduce a logarithmic-like increase that is experimentallyobserved. Concerning the zero-one model based on the structural change, it fails completely inthis prediction. Therefore, this particular result suggests that the planar defects are composedof 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 re-



quired. 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 as-sociate planar structural defects and point defects. This study thus leads to the conclusion that aone-hundred-nanometer crack associated with couples of interstitial oxygen and vacancies maybe the precursor defect responsible for laser-induced damage in KDP crystals.

Appendix

A. List of parameters and variables

Description of variables and parameters Symbol ValueCritical temperature that induces a damage Tc 5000 KCritical density threshold ρc 10 mm−3

Heterogeneity density ρh [105;107] cm−3

Heterogeneity size (1D modeling) Na 10000×1 nm = 10 μmNumber of ADNS per heterogeneity nADNS 100Room-temperature T0 300 KBoiling temperature of KDP Tbp 2000 KLaser fluence F a few J.cm−2

Conditioning laser fluence Fcond a few J.cm−2

Laser-induced damage threshold Fc a few J.cm−2

Testing pulse duration τtest [500 ps ; 5 ns ]Conditioning pulse duration τcond [500 ps ; 5 ns ]Number of laser pre-exposures Np [1;1000]Thermal diffusivity of KDP D 6.5×10−7 m2.s−1

Thermal conductivity of KDP λKDP 1.3 W.K−1.m−1

Laser absorption efficiency ξ/l [104,106] cm−1

CM2 conditioning criterion c [5,15]Crack thickness for CM2 e [0.1 ; 2 nm]Barrier energy for CM1 Ea about 1 eVLIDT gain obtained by conditioning g [1, 3]

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:

∂T∂ t

= DΔT +Eδ (�r)F(t)

ρC(11)

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

θ2D(�r,s) = T −T0 =E

2πλKDP

K0

(r√

s/D)

(12)

where K0 is the modified Bessel function (see e.g. [40]). Now, instead of a temporal Diracsource, we assume a time-dependent rectangular laser pulse shape such that:

{F(t) = γ if 0 ≤ t ≤ τF(t) = 0 elsewhere

(13)



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

θ2D(�r,s) =E

2πλKDP

K0

(r√

s/D) 1− e−τs

s(14)

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

θ2D(�r,s) = ζ (�r,s)− e−τsζ (�r,s) (15)

with

ζ (x,s) =E

2πλKDPτK0

(r√

s/D)

/s (16)

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 2Dpoint source reads:

θ2D(�r, t) =Aa2

4λKDP

∫ ∞

r2/4Dtduexp(−u)/u (17)

where E has been set to πa2. 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 Fc to thelaser pulse duration τ as Fc ∝ τx, one considers two critical fluences Fc1 and Fc2 associated withthe pulse durations τ1 and τ2 respectively. Consequently, x reads:

x =ln(Fc1/Fc2)ln(τ1/τ2)

(18)

In order to simplify the problem, we assume that a compact cluster of planar defects has thesame temperature evolution as a single plane, i.e. T = T0 +αF/τ1/2 [25] where α is a constant.We further assume that the pulse of duration τ2 probes essentially this compact cluster (locatedat x = 0) whereas the pulse of duration τ1 also probes all the ADNS close to the cluster whichcooperate for the temperature increase at x = 0. We thus suppose that τ2 ≤ a few ns [41] andτ1 � τ2. Under such conditions, we have:

T2(x = 0, t = τ2) = T0 +αF2

τ1/22

(19)

and

T1(x = 0, t = τ1) = T0 +αF1

τ1/21

+ncoop

4√

Dτ1

∫ 2√

Dτ1

−2√

Dτ1

dxθ1D(x,τ1) (20)

Since we integrate up to a distance 2√

Dτ1 from the cluster, we approximate θ1D(x,τ1) byαFτ1/2 exp(−x2/4Dτ) (the term that principally contributes to the integral). Further, ncoop =n0√

Dτ , where n0 is a constant proportional to nADNS. Simple algebra leads to:

T1(x = 0, t = τ1) = T0 +αF1

τ1/21

+βF1n0 (21)



where β is a constant. We then obtain:

x =ln

(τ1/21

τ1/22 (α+βn0τ1/2

1 )

)

ln(τ1/τ2)=

12− ln(α +βn0τ1/2

1 )ln(τ1/τ2)

(22)

For a given ratio r = τ1/τ2 and under the assumption of a sufficiently large value of τ1 in order

to obtain the inequality βn0τ1/21 � α , we finally have:

x � 12− ln(βn0)

lnr− lnτ1

2 lnr(23)

Despite that this expression has been established based on several assumptions, it allows us toshed light on the behavior of x with respect to nADNS and τ: the larger nADNS, the lower x and thelonger 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 distributedalong a direction perpendicular to the planes. Also, the damage probability corresponds to theprobability of the cluster having a spatial random distribution of the defects. This probabilityis thus given by the ratio between the number of configurations with a cluster and the totalnumber of configurations (number of ways to distribute nADNS in N places). The latter numberis simply given by CnADNS

N where Cpn = n!/p!(n− p)! is the binomial coefficient. The number

of configurations with a cluster of size nd (all ADNS are assumed to be adjacent) is given by(N −nADNS)C

nADNS−ndN−nd

, as long as nADNS � N. Straightforward calculations then lead to:

P � (N −nADNS)[nADNS(nADNS −1)...(nADNS −nd −1)]N(N −1)...(N −nd −1)

(24)

In the case where nADNS � nd (which represents a correct assumption in our calculations [17]),we obtain:

P � (N −nADNS)(nADNS

N

)nd(25)

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

ρd � ρh(N −nADNS)(nADNS

N

) f (τ)Tc/F(26)

We define the LIDT Fc as ρd(Fc) = ρc where ρc can be set to any arbitrary value. By taking thelogarithm of Eq. (26), simple algebra leads to:

Fc = f (τ)Tcln(nADNS/N)ln(ρc/Nρh)

(27)

C.3. Damage density at the threshold

In order to approximate the damage density evolution with respect to the fluence at the thresh-old, we can develop ρd as a Taylor expansion:

ρd(F)]F=Fc= ρd(Fc)+

∂ρd

∂F

]F=Fc

(F −Fc)1!

+ (28)

∂ 2ρd

∂F2

]F=Fc

(F −Fc)2

2!+ ...+

∂ nρd

∂Fn

]F=Fc

(F −Fc)n

n!+ ...



Since nADNS �N, we have ‖ lnn(nADNS/N)‖� ‖ lnn−1(nADNS/N)‖. By using Eq. (26), the deriva-tive at the order n of ρd can thus be approximated by:

∂ nρd

∂Fn �(− f (τ)Tc

F2 ln(nADNS/N))

ρd (29)

It follows that:

ρd(F)]F=Fc� ρc

∞

∑n=0

(− f (τ)Tc

F2c

ln(nADNS/N))

(F −Fc)n

n!(30)

� ρc exp

{−β (τ)Tc(F −Fc)

F2c

ln(nADNS/N)}

(31)

Finally, by using the expression (27) of Fc, we obtain an approximated form for the evolutionof ρd at the threshold:

ρd(F) � ρc

(Nρh

ρc

)(F−Fc)/Fc

(32)

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

D. Conditioning modeling

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

We define the normalized absorption efficiency, α , as:

α = 1− na

ni(33)

where ni is the initial density of couples of defects and na stands for the number of annihilationsper unit of surface of the plane. Also, if no annihilations occur, na = 0 and α = 1 remainunchanged. In a case where all the couples of defects annihilate, na = ni and subsequently α = 0subsequently. When setting αN to be the absorption efficiency after Np laser pre-exposures, itis straightforward that:

αNp = 1− n(1)a +n(2)

a + ...+n(Np)a

ni(34)

Here, n(Np)a is the number of annihilations due to the Npth laser pre-exposure. If we define the

absorption efficiency α(Np) = 1−n(Np)a /ni, then, Eq. (34) can be rewritten as:

αNp = α(1) + (α(2)−1)+(α(3)−1)+ ...+(α(Np) −1) (35)

In practice, we obtain a new absorption efficiency with the relation of recurrence αNp+1 =αNp +(α(Np) − 1) where αNp is known from the previous calculation and α(Np) is determined

as described below. α(Np) is a function of the number of displacements, dNp , of each interstitial

atom induced by the Npth 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 dNp

that is simply given by the number of time steps weighted by the displacement probability, i.e.∑i P(t = ti). Here P(t = ti) is the probability given by Eq. (9) and ti is a multiple of 1 ps (seeSection 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:

αNp+1 = αNp −n

(Np+1)a

ni(36)

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(−Ea/kT ) where T is

the temperature induced by the Npth laser pre-exposure. Within our modeling, this temperatureis proportional to αNp . Thus Eq. (36) transforms into:

αNp+1 = αNp −β exp(−γ/αNp) (37)

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

dαdNp

= −β exp(−γ/α) (38)

Since Fc ∝ α−1 and using the definition of the gain (10), we deduce that g = α−1. It followsthat the differential equation governing the variation of the gain with respect to Np reads:

dgdNp

= βg2 exp(−γg) (39)

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 ofg when Np → ∞ is imposed by the exponential function. Also, for the large values of Np, Eq.(39) can be approximated by g′ = β exp(−γg), for which the solution is simply a logarithmicfunction. 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 Fcond , a higher Fcond gives rise to larger variations of g withrespect to Np.

Acknowledgments

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



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