High-power 351 nm (3ω) laser pulses can produce damaged areas in high quality fused silica optics. Recent experiments have shown the presence of a densified layer at the bottom of damage initiation craters. We have studied the propagation of shock waves through fused silica using large-scale atomistic simulations since such shocks are expected to accompany laser energy deposition. These simulations show that the shocks induce structural transformations in the material that persist long after the shock has dissipated. Values of densification and thickness of densified layer agree with experimental observations. Moreover, our simulations give an atomistic description of the structural changes in the material due to shock waves and their relation to Raman spectra measurements.
©2001 Optical Society of America
Densification of silica glass has been observed under different experimental conditions such as neutron irradiation , static compression , or shock propagation . Fused silica is an important industrial material, particularly for optical fibers and high quality optical elements. For high power lasers, nominally transparent optical materials are required to sustain large pressures and temperatures accompanying energy absorption. During intense irradiation of fused silica optics at 3ω (351 nm), significant damage can be produced in the optics, in the form of craters. The lateral size of this damage grows exponentially with the number of pulses and limits the lifetime of the optics . It is therefore necessary to understand the underlying phenomena resulting in the modification of these materials. A possible mechanism postulated for damage initiation is the existence of localized small absorbers that heat and shock the surrounding material modifying it in a manner to make it susceptible to further damage upon re-irradiation by the next laser pulse . This hypothesis is consistent with observed behavior of bulk DKDP damage  and with that of surface damage in fused silica although it is not the only possibility.
Recent experiments by Wong et al  on damage sites have explored the morphology of the craters formed under laser fluence. These damage sites consist of a melted center region surrounded by fractured material. X-ray tomography of these damage sites has identified a layer of ~10 microns thick at the bottom of the crater that is 20% higher in density than the original silica. Moreover, Raman spectra taken at the damage sites have also shown an enhancement in the so-called D1 and D2 lines . Enhancement of the D1 and D2 lines has also been observed in silica under shock compression . In this case the intensity of the Raman lines increases as the shock pressure increases. In these experiments increases in density up to ~10% have been measured. These values of density are lower than those observed under static compression.
The presence of the D1 and D2 lines in the Raman spectra has been associated with the existence of small rings in the silica structure. Six-silicon-member rings are the most frequent ones in amorphous SiO2. Electronic structure calculations performed by Pasquarello and Car  have demonstrated that rings of size 3 and 4 are Raman active, and responsible for the appearance of these D1 and D2 lines.
We have performed molecular dynamics simulations of shock propagation in silica glass intended to aid understanding of the processes occurring in the material due to shock propagation accompanying laser energy absorption. We investigate the consequences for surrounding material in terms of changes in structure and density, and predict the response of this material under conditions that are difficult to obtain experimentally due to short time pulses and rapid changes in pressure, temperature and density.
In the next section we present the methodology used for these simulations and the characteristics of our simulated initial glass. We then present results of calculations of shock propagation and compare to experimental values of shock velocities as a function of piston velocity. We study the atomic level structural transformations occurring in the material and relate them to Raman measurements. Finally, we elucidate the implications of our work to damage observed in silica optics caused by 3ω laser irradiation with ns pulses.
2. Simulation model and sample characteristics
The interatomic potential used is that developed by Feuston and Garofalini  This potential includes two-body and three-body terms to account for the ionic and covalent components of the Si-O bonds in SiO2. This potential was fitted to reproduce the measured neutron diffraction spectra of silica glass under normal conditions of pressure and temperature. This interatomic potential has been used previously by other authors to study densification .
Our simulated amorphous silica (a-SiO2), starts from β-cristobalite. The dimensions of the initial simulation box are 71.6×71.6×716.0 Å. The total number of atoms in the simulation is 240,000. The system is melted at high temperature (7000K) using periodic boundary conditions for 25 ps. Then it is quenched to room temperature by a series of steps: from 6000K to 1000K, decreasing the temperature 1000K and relaxing at each intermediate step for 25 ps. Finally the temperature is decreased to 300K and relaxed for another 25ps. At 300K, a free surface is created along the shortest sides of the sample while periodic boundary conditions are used along the other two sides. Figure 1 shows the final simulation cell. The coordination of the silicon atoms is represented in this figure as gray tetrahedral. Yellow and green tetrahedra represent the 3-fold and 5-fold coordinated atoms respectively.
The final glass sample characteristics are computed and compared to experiments. The OSi-O bond angle distribution has a peak at ~109°, with a full-width-at-half-maximum (FWHM) of ~12°. Experimentally it is less than 14°. The Si-O-Si peak is at ~158° with a FWHM of 32°. Experimental values reveal a peak between 144° and 156°, with a FWHM of 38° . In our simulated glass 99.66% of the Silicon has coordination 4, 0.20% has coordination 3 and 0.14% has coordination 5. These numbers correspond to the entire lattice, with periodic boundary conditions in two directions and free surface in the third one. Indeed most of the 3-fold coordinated silicon atoms are located at the free surfaces, as can be observed in Figure 1. The ring size distribution peaks at 6, the sum of the ring distribution histogram being 5.93. Rings larger than 11 are not observed. The pair correlation function is also in good agreement with previous simulations using this interatomic-potential, and experimental measurements .
Using this initial sample, we study the propagation of shock waves through the material over a range of different velocities. In order to generate the shock wave we select a set of atoms at one of the free surfaces and apply a velocity to them throughout the simulation. This set of atoms will correspond to a piston traveling at a fixed velocity, Up. This method has been applied by other authors to study shock propagation through crystalline materials [13, 14].
The shock wave velocities were extracted from the velocity profiles of the atoms as a function of depth for different times. Figure 2 shows a set of velocity profiles for the cases of piston velocities of (a) 0.75 km/s and (b) 2.5km/s at different times. Observe that for intermediate times the shape of the profile does not change in time, indicating a stable shock wave front. Only at long times, when the wave reaches the surface and bounces back (reflection we see a change in the shape of the profile due to the superposition of the two waves. In order to extract the shock velocities we have computed differences between velocity profiles at different times corresponding to a value of half of the maximum particle velocity.
The values of shock wave velocities obtained from these simulations are plotted in Figure 3. We present the results of two different system sizes, one with 240,000 atoms (squares) and another one with only 96,000 atoms (filled circles), in order to check the dependence on the size of the simulation. No significant differences in the computed shock velocities were observed as a function of system size, as can be seen in Figure 3. The results of these simulations show two different regimes in the shock wave velocities. At the low piston velocities the elastic region is observed, where the shock velocities are constant with particle velocity. At about 1km/s there is a sharp decrease in shock velocity and the start of densification regime, with increasing shock velocity as the particle velocity increases.
In Figure 3 we also present the experimental results measured by Sugiura et al  in their flyer plate experiments. Our simulations are able to reproduce the transition from the elastic limit, and the agreement with the shock velocities in the densified region is remarkable. However, we are not able to reproduce the anomalous behavior observed experimentally in the elastic limit, where there is a slight increase in shock velocity as a function of particle velocity, and the measured values are below the sound velocity in silica glass.
In order to understand the structural changes occurring in the material at particle velocities between 1 km/s and 2 km/s, we have performed a detailed study of the coordination of silicon atoms and the evolution of ring distribution as the shock propagates through the material. This analysis shows that the density of five-fold coordinated silicon atoms increases dramatically in the densified silica. For the case of a 2.5 km/s shock values up to ~4.9% of 5-fold coordinated atoms are present during the first few picoseconds of shock propagation and ~1.2 % of 3-fold coordinated silicon atoms, representing 3,500% and 600% increase in concentration, respectively. Figure 4(a) shows the simulation cell after propagation of the shock wave. Observe the increase in over-coordinated silicon atoms (green) as compared to the initial lattice in Figure 1.
Our simulations show that the shocked material undergoes structural changes from the normally predominant 5–6 member ring distribution. After the shock, both the number of 3–4 member rings and the number of 7–10 member rings increase as shown in Figure 5. The small rings are more stressed than the normal structure and more prone to failure. These 3- and 4- fold rings not only have the property of being detected by Raman Spectroscopy  but they also have reduced energy barriers to ring-breaking  especially in the presence of gases (O2, N2). The larger rings represent the effect of already broken bonds. This is a type of “failed material” as has been seen in flyer plate experiments on fused silica . These larger rings may be considered microcrack precursors, i.e. broken bonds, which can coalesce into voids. Immediately behind the propagating shock, densification up to 60% is observed for the strongest shocks simulated.
In order to study the type and degree of material modification that persist long after the shock wave has passed we performed relaxation simulations of the shocked glass. This relaxation consisted on the extraction of the extra heat deposited by the shock in the material, and cooling of the configuration down to room temperature. During this relaxation the number of five-fold and three-fold coordinated silicon atoms decreases dramatically, ending up with a total of only 1.7 % of 5-fold coordinated and 0.2 % of 3-fold coordinated atoms. This still represents a significant increase (1,200 %) in the concentration of 5-fold coordinated silicon atoms over the initial value. Most of the relaxation happens quickly and the number of defects reaches a saturation value during the time scale of the simulation (33ps). Therefore we do not expect any significant changes for longer times.
Interestingly, the ring distribution after the relaxation does not show such a significant recovery. In Figure 6 we show the initial (a), after shock (b) and final (c) configurations. In this case we represent only those rings of size 3 and 4 (dark rings) and those 10 or larger (yellow rings). Observe that there is no significant change in the number of these small and large rings after relaxation. This is also clear from Figure 5, where we show the ring distribution before the shock, after the shock and after relaxation. In fact, during relaxation the number of large member rings seems to grow even more, while there is a slight decrease in the number of 3 and 4 member rings.
After relaxation some degree of densification persists. For example, in the case of a 2.5km/s shock the final density is ~20%. This densification value is in agreement with those measured experimentally under shock compression .
The thickness of the densified layer obtained from these simulations is ~300 Å, for a shock pulse of 10ps. Assuming that the thickness of the densified layer scales linearly with the duration of the shock, we can extrapolate our results to a shock pulse of 3 ns, such as those in 3ω laser damage conditions of . With this assumption our model predicts a densified layer of 9 microns, in good agreement with the measured 10 micron thick densified layer at the bottom of craters generated by laser irradiation. Moreover, the degree of densification is also in good agreement with recent experiments by Wong et al .
Our simulations also explain the presence of D1 and D2 lines in the Raman spectra by the formation of small rings that persist long after the shock (see Fig. 6). All this evidence strongly supports the hypothesis of strong shocks generated during the 3ω laser irradiation .
Furthermore, intriguing effects are observed in these simulations that could play a role in the evolution of the damage under laser irradiation. On one hand the degree of densification changes dramatically during shock propagation. Characterization of the optical properties of this highly dense material should be of great interest to this problem. On the other hand the concentration of defects also changes during the laser pulse and later relaxation of the system. A large number of 3-fold coordinated silicon atoms are observed at short time, that could be related to the presence of E’-prime centers-, and therefore couple to the laser light. Finally, the presence of high member rings could be a harbinger of ‘failed material’ and a mechanism for fast growth after subsequent laser pulses.
We have performed atomistic simulation of shock propagation in fused silica in order to understand the under-lying effects of laser damage in 3ω optics. The simulations reveal that levels of densification and thickness of the densified layers produced by shock waves are in good agreement with experimental measurements at the bottom of craters produced by UV lasers. The densification occurs by forming small 3 and 4 member rings and larger (10) member rings that persist long after the shock dissipates. This explains the presence of the D1 and D2 lines in the Raman spectra at damage sites. There is, therefore, strong evidence of the generation of shock waves during laser irradiation that result in structural changes in the un-ablated material. Our simulations also reveal changes in the material due to shock propagation that should be further investigated since they could be related to both energy absorption and material failure at longer times or subsequent laser shots.
We gratefully acknowledge helpful discussions with Dr. J. Wong, Dr. A.M. Rubenchik, Prof. S. Garofalini and Dr. Diaz de la Rubia. This work was supported by the National Ignition Facility project at LLNL. This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.
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