Introduction

Nanosecond laser-induced damage (LID) remains an issue for the development of high-power laser systems. At the National Ignition Facility (NIF), nanosecond laser beams at 1053-nm wavelength (1 ω) laser frequency are converted to 527 nm (2 ω) and 351 nm (3 ω). Since frequency conversion efficient required for ignition experiments is typically only 55 to 65%, the fused silica gratings that separate the unconverted 1 ω and 2 ω beams are simultaneously irradiated by the main 3 ω beam and also the 1 ω and 2 ω residual ones. It is well known that there is stronger absorption of laser energy by major nano-metric impurities on the subsurface of fused silica at 3 ω irradiation compared to 1 ω or 2 ω irradiation, so the short wavelength is easier to cause damage to the optical material than long wavelength beam [1–3]. In the case of the damage performance for simultaneous multi-wavelength irradiation, some studies have been carried out on potassium hydrogen phosphate (KH2PO4 or KDP) [4–9], optical coatings [10–14], fused silica [15–17], etc. Even if the laser-induced damage threshold (LIDT) and damage probability for fused silica is well characterized in the mono-wavelength configuration, few studies have been conducted in multi-wavelength configurations.

Under the calculation of previous work [1, 18, 19], considering a uniform intensity of irradiation, thermal models coupled to statistics were proposed based on damage initiation by heating of size-distributed impurities. Because of the uncertainty about the distribution of impurities, it may be located in different laser intensity within an irradiation area for the Gaussian distribution of irradiation intensity. In the present study, a model considering the Gaussian distribution of irradiation intensity has been presented to obtain precise calculation result about laser damage probability in section 2. In section 3, the laser damage probability curves have been measured with different mono-wavelength or multi-wavelength configurations. Then, considering two major impurities (Fe and CeO2) from polishing process measured by inductively coupled plasma optical emission spectrometry (ICP-OES), the theoretical calculation has been applied to fit the experimental results in order to indentify the distribution of potential precursors initiating damage and the results are discussed.

Model for Simulating Damage Probability

Impurities detected by ICP-OES for the samples used in the experiment include the major polishing and cleaning components (Ce from CeO2 and Al from Al2O3) and other metals (Fe, Cu, Cr, Ni, etc.) induced by the polishing steps or earlier grinding steps. There is a strong absorption for CeO2 at 3 ω [19], but it is nonabsorbent material at 1 ω and 2 ω. The major metallic nano-particles can absorb laser pulses at all three wavelengths [2, 25]. Thus, the damage precursors can be classified as short-wavelength precursors (SWPs) absorbing 3 ω laser and Multi-wavelength precursors (MWPs) absorbing 1 ω, 2 ω, and 3 ω lasers. CeO2 and Fe impurities were chosen in order to calculate laser-induced damage probability at different wavelength configurations because of their representative absorptivity as SWPs and MWPs and excellent thermal conductivity especially for Fe particles [18]. The material thermal and optical parameters used in our calculation have been given in Table 1 [21]. It is assumed that all parameters are independent of the temperature, because large variations in thermal properties of inclusion only weakly affect temperature predictions by the inclusion-dominated model of laser-induced damage [20].

Table 1 Main parameters used in calculation
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The aim of the model is to simulate damage probability curves by calculating the fluence necessary to initiate damage as a function of precursor properties. It is assumed that the damage is initiated by laser absorption in nano-particles. As the dual-wavelength pulses present simultaneously, the total absorbed power can be expressed as Q = (3/4a)·(α m I m + α n I n) (I m,n = I 0exp(−2r 2/w 2)•exp.(−4τ 2/t 2), m, n = 1 ω, 2 ω or 3 ω), where a is the radius of the particle and I m,n is the intensity (It also represents mono-wavelength absorption at I m = 0 or I n = 0). α m,n is the absorptivity which can be calculated with Mie theory [22–24]. τ is the pulse width at 1/e and w is the radius of spot at 1/e 2. The absorptivity of Fe and CeO2 inclusions as a function of particle diameter (D) under the irradiation of different wavelength laser has been presented in Fig. 1.

Fig. 1
figure 1

The absorptivity of Fe and CeO2 inclusions calculated at 1 ω, 2 ω, and 3 ω

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We can see from Fig. 1 that the absorptivity of Fe particles at 3 ω is larger than that of 1 ω and 2 ω. The absorptivity of CeO2 particles at 3 ω is similar to that of Fe particles at 1 ω.

Considering the absorptivity of inclusions, the temperature evolution in the inclusion and the surrounding material can be calculated by solving the equation of heat conduction [25]:

$$ \frac{1}{D_k}\frac{\partial {\theta}_k}{\partial t}=\frac{1}{r^2}\left({r}^2\frac{\partial {\theta}_k}{\partial r}\right)+\frac{Q}{C_k}\phi \left( a- r\right) $$
(1)

where θ k  = T−T 0 is the excess of the temperature T at the heated location over the initial temperature T 0 and a is the radius of inclusion. C k and D k respectively represent thermal conductivity and thermal diffusivity. k has two values: i corresponds to the inclusion and h to the surrounding material. The critical temperature initiating damage is set to 2300 K in all of calculation which is relatively low compared to the one measured in the real process (reach 10,000 K) [26], since it is based on the fact that above a critical temperature a thermal runway process takes place [27, 28]. The critical intensity I c required to reach critical temperature can be obtained by calculating the heating of the inclusion as a function of the intensity. Furthermore, by iterating this calculation for different absorber size, the critical intensity I c can be plotted as a function of the inclusion size as shown in Fig. 2.

Fig. 2
figure 2

Critical intensity calculated as a function of particle diameter for Fe and CeO2 inclusions

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We can see from Fig. 2 that the larger particles will initiate damage at lower laser fluence (as predicted by the absorptivity of nano-particles shown in Fig. 1). Fe particles require higher intensity to initiate damage at 1 ω but lower intensity to create damage at 3 ω. CeO2 particles require higher intensity to initiate damage at 3 ω because of its poor absorptivity and weak thermal conductivity compared to Fe particles.

The power law distribution is assumed in our model since this type of variation is typically found for clusters [29], which can be expressed as [30]

$$ \rho (a)=\frac{\left(1-\gamma \right){d}_0}{a_{\max}^{1-\gamma}-{a}_{\min}^{1-\gamma}}{a}^{-\gamma} $$
(2)

where d 0 is the density of defects per unit of surface. With the knowledge of the critical intensity (critical intensity as a function of size in Fig. 2), the defect size distribution, and the Gaussian beam profile, the number of defects △N located under the laser spot area △S = (πw 2/2)•ln(I g+1/I g) whose intensity is between I g and I g+1 (described in Fig. 3)

$$ \Delta N=\Delta S{\int}_{a_{g+1}}^{a_{\max }}\rho (a) da=\frac{\pi {w}^2}{2} \ln \left(\frac{I_{g+1}}{I_g}\right)\frac{d_0\left({a}_{\max}^{1-\gamma}-{a}_{g+1}^{1-\gamma}\right)}{a_{\max}^{1-\gamma}-{a}_{\min}^{1-\gamma}} $$
(3)
Fig. 3
figure 3

Impurity distribution in the spot area

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where Ig can be writen as

$$ {I}_g={I}_0 \exp \left(-\frac{4{t}^2}{\tau^2}\right) \exp \left(-\frac{2{r_g}^2}{w^2}\right)\kern0.5em {r}_g= g\frac{2 w}{M}, g=0,1,2\cdot \cdot \cdot M $$
(4)

where the simulation region is gridded with a uniform grid △r = 2w/M (M is set to 200 in calculation). Thus, the damage probability P(F) at the fluence F can classically be given by [1]

$$ P(F)=1- \exp \left[- N(F)\right]=1- \exp \left(-\sum_0^M\varDelta N\right) $$
(5)

where the N(F) is the number of defects under the laser spot that can initiate damage at fluence F.

Experiment and Analysis

This is achieved by measuring the probability of damage sites created in the surface of fused silica under simultaneous exposure to the harmonics of a nanosecond-Nd:YAG laser at multi-wavelength configurations with different fluence combinations and without delay times. Figure 4 shows an illustration of the experimental arrangement for multi-wavelength laser damage test. The generation of 2 ω light is first ensured by doubling the fundamental light of a Nd:glass laser. Tripling is then achieved combining the unconverted 1 ω and 2 ω beams. The pulse durations (at 1/e) of the fundamental, second, and third harmonics are 12, 10, and 8 ns, respectively, measured by a fast photodiode. The temporal overlap of the laser pulses at each harmonic was achieved by keeping the distances of the beam paths the same. The combination of a half-wave plate and Brewster polarizer (56.6°) with high damage resistance is used to control the energy of each harmonic beam. The beams with different wavelength configurations are then aligned to copropagate and focused by cylindrical lenses (f = 1000 mm) to the surface of the sample. Galilean telescopes are placed in each beam path to correct for differences of the index of refraction of the cylindrical lens at each wavelength in order to focus each beam at the same location on the surface of the sample. The spatial profiles of each beam are measured using a beam profiling system linked to a CCD camera with a spatial resolution of 5 μm, which can also be used to observe the overlap of different harmonics. The diameters of focus spot (at 1/e 2) are 335 μm at 1 ω, 210 μm at 2 ω, and 112 μm at 3 ω.

Fig. 4
figure 4

Experimental setup for multi-wavelength laser damage test

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The fused silica (Corning HPFS 7980) samples were polished by cerium oxide slurries. The surface roughness of the polished samples was 0.834 ± 0.048 nm. The depths of impurities embedded in these samples were 107.4 to 121.2 nm, which were measured with secondary ion mass spectroanalyzer method (SIMS). The size of samples was 50 mm × 50 mm horizontal area and 5 mm thick. Experimental data were collected by irradiating up to 50 virgin sites on the rear surface of a single sample with individual pulses at each mono-wavelength or multi-wavelength configurations. The percentage of damage sites was plotted vs fluence to obtain a damage probability s curve. Applying our model with Fe and CeO2 defects fits the experimental data in order to check the validity of this assumption. There is a good agreement between the experimental data and theoretical curves as shown in Fig. 5, when considering an appropriate size distribution parameters for Fe particles (d 0 = 5 × 104 cm−2, γ = 3, and D ≤ 11 nm) and CeO2 particles (d 0 = 2 × 104 cm−2, γ = 4, and D ≤ 25 nm). This interesting point is that the model can describe the observed wavelength dependence since the data obtained at the three different wavelengths can be fitted with the calculation result based on the same size distribution.

Fig. 5
figure 5

The laser-induced damage probability under respective exposure to 1 ω, 2 ω, and 3 ω pulses

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As seen in Fig. 5, the fluences at 0% damage probability are 10.1, 20.8, and 79.3 J/cm2, respectively, for 3 ω, 2 ω, and 1 ω laser. The fluences of 3 ω, 2 ω, and 1 ω pulses at 50% damage probability are 19.5, 42.5, and 144.8 J/cm2, respectively. The damage threshold (defined as 0% damage probability) decreases from 1 ω to 3 ω lasers, since the considered nano-particles (Fe and CeO2 particles) have stronger absorption at lower laser frequencies. It can be deduced from the absorptivity of damage precursors that the damage by individual 1 ω or 2 ω pulses arises from MWPs but the damage by individual 3 ω pulse arises from both SWPs and MWPs.

Subsequently, the damage performance under simultaneous exposure to dual-wavelength pulses was quantified in order to understand influence of the laser wavelength on the LID mechanisms. The laser damage tests were performed according to a one on one procedure using synchronous dual-wavelength pulses. Figure 6 represents the laser-induced damage probability initiated by 1 ω pulse coupled with a constant fluence 2 ω or 3 ω pulses.

Fig. 6
figure 6

Laser damage probability resulting from constant 2 ω and 3 ω fluences vs 1 ω fluence

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The fluences at 0 and 50% damage probability from 1 ω and 3 ω (fixed fluence at 10 J/cm2) dual-wavelength exposures are respectively 33 and 27% lower than that of only 1 ω fluence. The fluences at 0 and 50% damage probability from 1 ω and 2 ω (fixed fluence at 20 J/cm2) dual-wavelength exposures are respectively 37 and 43% lower than that of only 1 ω fluence. The laser damage threshold from 1 ω pulse combined 2 ω pulse is lower than that of 1 ω pulse combined 3 ω pulse. The damage by individual 1 ω pulse arises from MWPs because of their absorptivity for 1 ω laser. The damage by 1 ω pulse combined 2 ω or 3 ω pulses is initiated by MWPs which jointly absorb the dual-wavelength (1 ω, 2 ω or 1 ω, 3 ω) pulses. The SWPs will not affect the damage probability at the dual-wavelength pulses since they have no absorption at 1 ω and 2 ω lasers and less absorption at 10 J/cm2–3 ω laser not enough to cause damage.

Figure 7 illustrates the laser-induced damage probability by the dual-wavelength configuration at 2 ω pulse accompanying 3 ω or 1 ω pulse with a constant fluence. The fluences at 0 and 50% damage probability induced by the 2 ω and 3 ω (fixed fluence at 10 J/cm2) dual-wavelength configuration are respectively 22 and 14% lower than the only 3 ω fluence. The fluences at 0 and 50% damage probability induced by the 2 ω and 1 ω (fixed fluence at 75 J/cm2) dual-wavelength configuration are respectively 64 and 40% lower than the only 3 ω fluence. The laser damage threshold from 2 ω pulse combined 1 ω pulse is lower than that of 2 ω pulse combined 3 ω pulse. As described in the case of 1 ω pulse combined 2 ω or 3 ω pulse, the damage by 2 ω pulse combined 1 ω or 3 ω pulse is also initiated by MWPs (not by SWPs) which jointly absorb the dual-wavelength (1 ω, 2 ω or 2 ω, 3 ω) pulses.

Fig. 7
figure 7

Laser damage probability resulting from constant 1 ω and 3 ω fluences vs 2 ω fluence

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Figure 8 shows the damage behavior under simultaneous exposure to the combination of constant 1 ω fluence (just below the 0% damage probability at mono-wavelength damage) and different 3 ω fluences. The fluences at 0 and 50% damage probability from simultaneous 3 ω and 2 ω (fixed fluence at 20 J/cm2) exposures are respectively 24 and 13% lower than that of individual 3 ω exposure. The fluences at 0 and 50% damage probability from simultaneous 3 ω and 1 ω (fixed fluence at 75 J/cm2) exposures are respectively 32 and 29% lower than that of individual 3 ω exposure. The laser damage threshold from 3 ω pulse combined 1 ω pulse is lower than that of 3 ω pulse combined 2 ω pulse. The damage by 3 ω pulse combined 1 ω or 2 ω pulses is initiated by both the SWPs and MWPs. Thus, the SWPs can absorb 3 ω pulse but the MWPs jointly absorb the dual-wavelength (1 ω, 3 ω or 2 ω, 3 ω) pulses. The total number of defects under the laser spot is the sum of SWPs and MWPs.

Fig. 8
figure 8

Laser damage probability resulting from constant 1 ω and 2 ω fluences vs 3 ω fluence

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Conclusion

A model considering the Gaussian distribution of irradiation intensity has been proposed in order to precisely calculate laser damage probability. Theoretical curves of laser damage probability have a good agreement with experimental results measured with different wavelength configurations, when the fit parameters are chosen in our simulation. Consequently, the size distribution of nano-absorbers on the surface of optical materials can be obtained according to the simulation method. The results demonstrate that the damage by individual 1 ω or 2 ω pulses arises from MWPs because of their absorptivity for 1 ω or 2 ω lasers. The damage by individual 3 ω pulse arises from both SWPs and MWPs. The damage by 1 ω pulse combined a constant fluence 2 ω or 3 ω pulse and 2 ω pulse combined a constant fluence 1 ω or 3 ω pulse is initiated by MWPs which jointly absorb the dual-wavelength pulses. The damage by 3 ω pulse combined a constant fluence 1 ω or 2 ω pulses is initiated by both SWPs and MWPs.