1. Introduction

The laser induced damage of fused silica optics used in large high power laser facilities has been the subject of extensive studies. Recent studies suggest that the subsurface damages in the tens of microns subsurface damage layer produced by grinding or polishing process are primarily responsible for initiating the laser induced surface damage of fused silica optics exposed to 355 nm ultraviolet wavelengths [1–5]. The grinding process can be thought of as the repeated indentation of mechanically loaded hard indenters (abrasives) sliding on the optics surface, as a result, generation of high density fractured subsurface damages will be inevitable [6–8]. Fortunately, the following optical polishing process could remove the grinding induced subsurface damages. But the optical polishing process would also generate some subsurface damages especially in large aperture (~400 mm) fused silica optics because of the direct contact among the polishing pad, particles and optics. These polishing induced subsurface damages are hard to be detected and removed by the final precision polishing process (such as ion beam figuring and magnetorheological finishing) considering of the poor material removal depth and removal efficiency and are detrimental for high power laser optics.

Furthermore, several studies focus on understanding the complex mechanical interactions that occur during the polishing process to predict the distribution of the polishing induced surface/subsurface damages (mostly identified as scratches and dots). Suratwala et al. have investigated the crack depth and the number density and length of scratches on fused silica formed during polishing resulting from the addition of rogue particles in the base slurry. They use the viscoelastic model to characterize the embedding of particles into pad, which could obtain the length of a scratch under different parameters [9]. But the model could not explain the influence of polishing parameters on the depth distribution and density of subsurface damages. Cheng et al. have experimentally investigated the influence of pad material, particle size and pressure on the subsurface damages depth in the computer controlled small tool polishing process [10]. Liao et al. have developed a model for the interaction of the optics, impurity particle and pad during the polishing process to illustrate the scratch formation process [11]. But the model can only predict the normal load applied on a single particle, and could not characterize the subsurface damages distribution (such as depth distribution and density of subsurface damages) under different polishing parameters. Thus, more and deeper investigations are still needed to understand the numerical correlations between polishing parameters (such as particle size distribution, applied load and pad material) and the density and depth distribution of polishing induced subsurface damages.

In addition, the previous researches mainly focus on the polishing induced fractured subsurface damages [9–11]. But unlike the grinding process, the polishing process usually use much smaller particles (about 1 μm for polishing while more than 10 μm for grinding) and softer pads (pitch or polyurethane pad for polishing while steel or copper pad for grinding), which finally results in sharp decrease of the load per particle possibly below the fracture initiation load [12–14]. So the material removal mode and subsurface damages induced in the polishing process should be much different from the grinding process [14,15]. Shen et al. have revealed that most of the polishing particles would abrasion fused silica optics in the plastic deformation region by using an atomic force microscope (AFM) to mimic the mechanical interactions of polishing particles during optical polishing [16,17], which implies that most of the polishing induced subsurface damages should be plastic damages in theoretical. Some previous studies also show that the polishing induced subsurface damages contain not only a few fractured damages but also high density plastic damages and polishing with nanometer sized particles could obtain optics without any fractured subsurface damages [18,19]. Recent studies indicate that the plastic subsurface damages could also initiate laser induced damages [20,21]. But the influence of polishing parameters on the plastic subsurface damages distribution and laser induced damage performance has not been revealed yet.

In the present work, the plastic subsurface damages distribution of fused silica optics polished with different pads are characterized by AFM and Nomarski microscopy. The elastic interaction model is applied for describing the interaction of the optics, particles and pad during optical polishing process to calculate the load per particle. Then, the plastic indentation model and wear relationships during optical polishing process are used to establish the numerical correlations between polishing parameters and subsurface damages distribution (include the depth distribution, density and kinds of subsurface damages). Finally, the influence of the plastic subsurface damages induced by different polishing parameters on the laser induced damage performance is analysed. This research is meaningful for further establishing the quantitative relationships between polishing parameters, subsurface damages distribution and laser induced damage performance, and finally to improve and predict the laser induced damage performance and lifetime of 3ω fused silica optics used in large high power laser facilities.

2. Experiments and results

2.1 Sample preparation methods

Five 50 mm diameter, 5 mm thick commercially available initial polished UV-grade fused silica (Corn 7980) samples (labeled A, B, C, D and E) were manufactured by different procedures. Firstly, all samples were etched in HF based acid (1% hydrofluoric acid and 15% ammonium fluoride) with megasonic agitation with a constant material removal of 10 μm per side. Then, all samples were polished with ZrO₂ (Universal Photonics NTD. ZQS) with different polishing pads (as shown in Table 1). All the polishing processes were performed in the same conditions such as pressure and rotation speed. The pressure was set to be about 20 g/cm² during polishing. Finally, all samples were etched in HF based acid (1% hydrofluoric acid and 15% ammonium fluoride with megasonic agitation) with the same etching time (~6 minute) to get a constant material removal of about 100 nm per side, which could remove the redeposition layer and expose the subsurface damages induced during the polishing process [22].

Table 1. Sample preparation methods.

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2.2 The size distribution of the polishing slurry

The size distribution of the commercially available polishing slurry was measured by a Zetasizer Nano ZS nano-particle analyzer. Figure 1 presents the particle size distribution of polishing slurry. The particle size distribution of ZQS ZrO₂ is from 615 to 2305 nm, and the average particle size is 1193 nm.

 

Fig. 1 The particle size distribution of polishing slurry.

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2.3 Elastic modulus of polishing pads

The elastic modulus of different polishing pads was tested by the SII DMS/6100 dynamic thermomechanical analysis (DMA). DMA is used to measure the mechanical and viscoelastic properties of many kinds of materials such as thermoplastics, elastomers, ceramics and polymers (such as polishing cloth and polyurethane pad used in the above polishing experiments). During the DMA test, the sample is subjected to a periodic stress in tension or compression mode, and the storage modulus (represent the elastic modulus) and loss modulus (represent the viscous property) as a function of temperature (in our experiments is 21 degrees centigrade) could be directly obtained. The measured elastic modulus of different polishing pads at room temperature is shown in Table 2. It indicates that different polishing pads have much different elastic modulus.

Table 2. Elastic modulus of different polishing pads at room temperature (21 degrees centigrade).

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2.4 The subsurface damages distribution

The atomic force microscope (AFM) and Nomarski microscope were used to measure the subsurface damages distribution. Figures 2(a)-2(e) presents the AFM images (50 μm × 50 μm) of all samples with the same color scale (−10 nm~10 nm). Figure 3 shows the depth profile of the AFM images in Figs. 2(a)-2(e). Table 3 presents the surface roughness (Rq), the maximum subsurface damage depth (Rmax) and the subsurface damages area percentage for all samples measured by AFM. The surface area where is lower than −2 nm in AFM images is used to estimate the subsurface damages area percentage (which usually represents the subsurface damages density) in different samples. All samples have many subsurface damages such as scratches and dots (the depth is from several nanometers to tens of nanometers) with much different density. Considering the fractured subsurface damages always have the depth of several microns, the much shallower subsurface damages observed by AFM should be plastic damages. It can be found that with the increased elastic modulus of polishing pad, the subsurface damages area percentage, the maximum subsurface damage depth and the surface roughness are rapidly increased (as shown in Table 3).

 

Fig. 2 The AFM images of (a) sample A, (b) sample B, (c) sample C, (d) sample D and (e) sample E with the same color scale.

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Fig. 3 The depth profile of the AFM images.

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Table 3. Measured subsurface damage area percentage, maximum subsurface damage depth and surface roughness for all samples.

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Considering that the AFM images could only show the subsurface damages distribution in small area, the Nomarski microscopy was used to reveal the subsurface damages distribution in larger area. Figures 4(a)-4(e) shows the Nomarski images (500 μm × 500 μm) of different samples. It shows the similar subsurface damages distribution to the AFM images. There is nearly no subsurface damages can be found in sample A while many plastic scratches could be found in samples B, C, D and E. A few fractured scratches (marked by the white arrows in Figs. 4(d) and 4(e)) are only found in samples D and E. The subsurface damages density increases rapidly with the increased pad elastic modulus and most of the polishing induced subsurface damages are plastic scratches (plastic scratch should have no brittle fracture but just plastic modification on the surface and usually presents as a smooth continuous scratch) [9].

 

Fig. 4 The surface morphologies of (a) sample A, (b) sample B, (c) sample C, (d) sample D and (e) sample E observed by Nomarski microscopy.

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2.5 The laser induced damage performance

A tripled Nd:YAG laser was used at a wavelength of 355 nm in our laser induced damage test equipment. The pulse is a single longitudinal mode with about 10 ns (FWHM). During the test, the beam was focused on the exit surface to provide a near Gaussian beam with a diameter of about 0.35 mm at 1/e² of maximum intensity. Fluence fluctuations have a standard deviation of about 10%. The online microscope has the resolution of about 10 μm. The damage threshold was obtained using the standard “1-on-1” test mode. Raster scan damage test was used to detect the damage density and the scan area is about 10 cm².

Figure 5 shows the damage probability curve of different samples, the solid lines are the linear fitting results. Figure 6 shows the zero probability damage threshold and damage density at the fluence of 40 J/cm² for all samples. It reveals that the damage threshold generally decreases and the damage density increases with the increased pad elastic modulus.

 

Fig. 5 The damage probability curves of different samples, the solid lines are the linear fitting curves.

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Fig. 6 The zero probability damage threshold and damage density at the fluence of 40 J/cm² for different samples.

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3 Discussion

3.1 Relationships governing subsurface damages formation

Before introduction of the theoretical model, some key assumptions are used to simplify the analysis. First, all the polishing particles are assumed to be spherical and distributed at the optics–pad interface as a single layer of particles. Second, only elastic contact (the viscoelasticity and plastic deformation is ignored) is supposed to happen between particles and pad. Third, the hydrodynamic fluid forces are not considered here. Fourth, the mechanical properties modification of optics and pad surface caused by hydration, densification and ageing during the polishing process is not considered. Fifth, the influence of sliding or rolling of particles on the bear load per particle is neglected. However, it is believed that the following model could still largely describe and predict the bear load per particle and scratches initiation with these assumptions.

From Hertzian contact mechanics, the bear load $F(r)$ on each particle with radius r is given by Eq. (1) [14,24,25]:

where ${\delta }_p$ is the penetration depth of the particle into the polishing pad, $E_{sp}$ is referred to the efficient elastic modulus of contacting between the particle and polishing pad. Since the polishing pad is normally much softer than the optics and particles, the $E_{sp}$ can be obtained by Eq. (2) [24]:

where $E_p$ and ${\upsilon }_p$ are the elastic modulus and Poisson's ratio of the polishing pad, and the subscript ‘p’ represents the pad, the subscript ‘o’ represents the optics. Since the penetration depth of the particle into the optics ${\delta }_o$ is much smaller than ${\delta }_p$ and r, the penetration depth ${\delta }_p$ can be calculated by Eq. (3):

where g is the equilibrium gap (or slurry film gap) at the optics-pad interface [11,25]. Substituting Eqs. (2) and (3) into Eq. (1) yields Eq. (4):

we assume that the polishing particles will abrasion fused silica optics in the plastic deformation region during optical polishing [16,17]. Then the plastic indentation depth ${\delta }_o$ of the particle into the optics can be calculated by Eq. (5) [14,24]:

where $H_o$ is the hardness of optics. Substituting Eq. (4) into Eq. (5) yields Eq. (6):

Note that the particles will be loaded and indent into optics only when the particle diameter (2r) is larger than the equilibrium gap g, which means only a fraction of the polishing particles (the active particles) will participate in the polishing process. Once the equilibrium gap g is determined, the bear load $F(r)$ on each particle and plastic indentation depth ${\delta }_o$ of the particle into the optics is known. Note that the equilibrium gap g used here is a compositive value which is determined by many factors such as the pad micro-topography (the distribution of asperities and pores) and could not be directly measured as the slurry film thickness [25,26]. To determine g, a load balance is used for the whole polishing system given by the following equation:

where $F_0$ is the total applied load, $f_{po}$ is the fraction of the optics area making contact with the polishing pad, $S_o$ is the area of the optics, $N_p$ is the areal number density of particles present at the interface and $f(r)$ is the fractional number of particles with radius r.

3.2 The calculated subsurface damages distribution

In our experiments, all the polishing processes have been performed using the same slurry and equal size optics, thus the $N_p$ and $S_o$ is constant in all polishing process. We assume that the polishing process using five different pads have the same $f_{po}$, then with the same total applied load $F_0$, we could deduce that ${\displaystyle {\sum }_{r}f(r)F(r)}$ is equal for all samples. Because the areal number density $N_p$ and the contact area fraction $f_{po}$ are unknown in our experiments, the equilibrium gap g in the polishing process cannot be calculated directly. But in AFM experiments, we get the maximum subsurface damage depth (Rmax), so we could use Eq. (6) to calculate the equilibrium gap g in the polishing process. As in the polishing process of sample A, the largest particle has the diameter of 2305 nm, and the maximum subsurface damage depth (Rmax) is 8.0 nm. We could get the equilibrium gap g for sample A is 840 nm through Eq. (6). Considering that ${\displaystyle {\sum }_{r}f(r)F(r)}$ is equal for all samples, so the equilibrium gap g for all samples (as shown in Table 4) could be calculated by Eqs. (4) and (7). Then, the maximum subsurface damage depth (Rmax) in all samples (as shown in Table 4) can also be calculated using Eq. (6).

Table 4. Calculated results in all samples.

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With the calculated equilibrium gap g, the bear load $F(r)$ and indentation depth ${\delta }_o$ into optics for different particles in different polishing processes can be calculated (as shown in Fig. 7 and Fig. 8). It can be found that when the pad elastic modulus increase, the equilibrium gap g also increase and the larger particles will bear larger load and thus create deeper scratches. It is because that when the pad elastic modulus increases, the particles with same diameter will bear larger load with the same equilibrium gap g from Eq. (4). But the total applied load $F_0$ in Eq. (7) is constant, so the equilibrium gap g must increase to decrease the bear load $F(r)$ of single particle and the fraction of particles being loaded (as shown in Table 4 and Fig. 7).

 

Fig. 7 The calculated bear load $F(r)$ of different particles in different polishing processes.

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Fig. 8 The calculated indentation depth ${\delta }_o$ into optics for different particles in different polishing processes.

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Because the fraction number ($f(r)$) of particles with different diameter and the depth (${\delta }_o$) of the scratches generated by different particles is also known (as shown in Fig. 1 and Fig. 8), the instantaneous fraction number ($f_{ins}({\delta }_o)$) of scratches with different depth ${\delta }_o$ generated during a certain polishing time could be obtained (as shown in Fig. 9). Considering that with the increased polishing time and material removal ($\Delta$), the previously generated scratches will be shortening continuously and new scratches will be generated. The final fraction number ($f_{fin}({\delta }_o)$) of scratches (as shown in Fig. 10) with different depth ${\delta }_o$ can be obtained by Eq. (8) if hardness micro-modification and plastic flow of optics surface is neglected [12]:

 

Fig. 9 The calculated instantaneous fraction number ($f_{ins}({\delta }_o)$) of scratches with different depth generated during a certain polishing time.

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Fig. 10 The calculated final fraction number ($f_{fin}({\delta }_o)$) of scratches with different depth generated during different polishing process.

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Normally scratches generated by spherical particles show inverted triangle shape in the depth direction (as shown in Figs. 11(a)-11(d)). Figure 11(a) is a AFM image of sample D. Figure 11(b) is the depth profile of the white line in Fig. 11(a). Figure 11(c) is the enlarged 3D picture of the red square in Fig. 11(a). Figure 11(d) is the side view picture of Fig. 11(c). The included angle ($\theta$) of the inverted triangle for the scratch in Fig. 11(b) is about 170°. Then the cumulative depth distribution ($S(\delta )$) of the scratch with depth ${\delta }_o$ and included angle ($\theta$) can be obtained by Eq. (9):

 

Fig. 11 (a) The AFM image of sample D. (b) The depth profile of the white line in Fig. 11(a). (c) The enlarged 3D picture of the red square in Fig. 11(a). (d) The side view picture of Fig. 11(c).

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Where L is the scratch length. If we assume that all the scratches have the same length and included angle ($\theta =170°$) for simplicity, the only difference is the depth. Then we could obtain the final cumulative depth distribution ($S_{cum}(\delta )$) of scratches generated during the polishing process by Eq. (10) (as shown in Fig. 12):

 

Fig. 12 Calculated final cumulative depth distribution ($S_{cum}(\delta )$) of scratches generated during different polishing process.

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Then, the final depth distribution ($S_{cum}(\delta )$) of scratches generated during the polishing process can be calculated by Eq. (11):

The average scratches depth ${\delta }_{ave}$ (which is considered to be the zero height surface in AFM images) can be obtained by Eq. (12):

With the known final depth distribution ($S_{dep}(\delta )$) of scratches, the surface roughness (Rq) can be obtained by Eq. (13) (as shown in Table 4):

We sum the scratch area where is 2 nm deeper than the average scratches depth ${\delta }_{ave}$ as the final subsurface damage area percentage (as shown in Table 4). The result is much different from AFM measured date in Table 3. Considering that the soft hydrated layer formed during optical polishing would possibly cover the shallow scratches through plastic flow. We sum the scratch area deeper than 5 nm as the final subsurface damage area percentage (as shown in Table 4), which is much more consistent with the AFM measured results.

We should note that the above model mainly describes the distribution of polishing induced plastic subsurface damages because only the plastic indentation model is used (as shown in Eq. (5)) to calculate the subsurface damages depth. But the model could also be enlarged to describe the polishing induced fractured subsurface damages when the critical fracture initiation load and fractured indentation model are considered in the calculations.

3.3 Types of polishing-induced subsurface damages

Previous reports indicate that the critical load needed to observe plastic deformation is about 1.0 μN and the critical load needed to observe fractured deformation is about 0.1 N for fused silica glass [12,16]. In Fig. 7, almost all the bear load of polishing particles is between 1 μN and 0.1 N and the largest bear load of polishing particle is about 3 mN which is about two orders of magnitude lower than the critical fracture initiation load. So the model indicates that the micron sized particles will abrasion fused silica optics in plastic deformation region and most of the generated subsurface damages should be plastic damages. The model is also consistent with the experimental results as shown in Figs. 4(a)-4(e), in which nearly all of the scratches are continuous and smooth without brittle fracture (in most cases are plastic scratches). But in Figs. 4(d) and 4(e), there are a few fractured scratches exist in samples D and E. Considering that the bear load of polishing particle increases rapidly with the increased particle size (as shown in Fig. 7), the few fractured scratches are possibly generated by some extremely large interfused impurity particles coming from dust in the air, debris of the pad or agglomeration of polishing particles.

3.4 Initiation of fractured subsurface damages

If we assume that 0.01% impurity particles with diameter of 10 μm have been induced in the polishing slurry, according to Eqs. (4) and (7), the equilibrium gap g will basically keep the previous value in Table 4. Then the calculated bear load of the impurity particle in all samples is 0.013 N (in sample A), 0.023 N (in sample B), 0.055 N (in sample C), 0.087 N (in sample D) and 0.265 N (in sample E), respectively. It reveals that the bear load of the impurity particle increases rapidly with the increased pad elastic modulus. The bear load of the impurity particle in samples D and E has almost reached the critical fracture initiation load (~0.1 N), which would possibly generate fractured scratches in samples D and E. So using soft polishing pads such as polishing cloth with lower elastic modulus is prominent for restricting the generation of fractures. Note that the largest bear load of the slurry particle (0.4 mN in sample A, 0.6 mN in sample B, 1.0 mN in sample C, 1.4 mN in sample D and 3.0 mN in sample E) is much lower than the fracture initiation load and is about two orders of magnitude lower than that of the impurity particle, which indicates that control of the interfused large impurity particles is critical for obtaining fracture free fused silica optics.

3.5 Influence of polishing parameters on plastic subsurface damages distribution

In Fig. 7 and Fig. 8, the bear load and indentation depth (the depth of the initiated subsurface damages) increase rapidly (nearly exponentially) with the increased particle size for all samples. It indicates that the deepest plastic subsurface damages are induced by the largest particles in the tail end of the particle size distribution. If we assume that only the depth of the polishing induced plastic subsurface damage is deeper than 5 nm (marked by the gray area in Fig. 8), the subsurface damage could be residual and observed on the optics surface after the polishing process considering the complex plastic flow effect (Note that the boundary between observable and unobservable plastic subsurface damages should exist and the value of 5 nm used here is rough estimated according to AFM results). The smallest particle size needed to produce observable plastic subsurface damages (depth deeper than 5 nm) in samples A, B, C, D and E is 1840, 1710, 1663, 1672 and 1736 nm, respectively. It indicates that only 5.4%, 11.6%, 14.3%, 13.6% and 8.9% of the polishing particles in the tail end distribution could initiate observable plastic subsurface damages in samples A, B, C, D and E. So a few largest particles in the tail end of the particle size distribution will mainly determine the final depth distribution and density of the plastic subsurface damages. In other words, control of the few largest particles in the polishing slurry could greatly decrease the depth and density of the plastic subsurface damages.

In Table 4, the smallest diameter of loaded particles (same to the equilibrium gap g) increase and the fraction of loaded particles decrease rapidly with the increased pad elastic modulus. It indicates that the few largest particles in the tail end of the particle size distribution will bear much larger load (as shown in Fig. 7) and induce deeper plastic subsurface damages (as shown in Fig. 8) with larger pad elastic modulus. Besides, the fraction (92% for sample A, 80% for sample B, 61% for sample C, 50% for sample D and 26% for sample E) of unobservable plastic subsurface damages (depth lower than 5 nm, marked by the gray area in Fig. 9) in all of the instantaneous initiated subsurface damages decreases rapidly with the increased pad elastic modulus. So when polishing pad with larger pad elastic modulus is used, more and more loaded particles participating in polishing will generate observable plastic subsurface damages during the polishing process. Finally, the optics surface will have more and deeper plastic subsurface damages (as shown in Figs. 2(a)-2(e)), which presents as larger maximum subsurface damage depth, subsurface damage area percentage and surface roughness (as shown in Table 3 and Table 4). Comparing sample A with sample E, the maximum subsurface damage depth, subsurface damage area percentage and surface roughness of sample A is one order of magnitude lower than that of sample E. So using soft polishing pads such as polishing cloth with lower elastic modulus could substantially decrease the depth and density of polishing induced plastic subsurface damages.

3.6 Comparison with experimental results

Comparing Table 4 with Table 3, the calculated maximum subsurface damage depth, subsurface damage area percentage and surface roughness has the same trend with the measured results, in which the maximum subsurface damage depth, subsurface damage area percentage and surface roughness increase rapidly with the increased pad elastic modulus. However, there still has divergence between the theoretical model and experiments. It is believed that two main factors would possibly cause the divergence. First, the optical polishing process is a very complex chemical mechanical process. The main polishing process consists of formation of a soft hydrated layer and then abrasion (or removal) of the hydrated layer by abrasive particles [27,28]. So when the particles generate scratches, the plastic flow of the hydrated layer will fill in and cover the scratches, which changes the depth distribution of the scratches and cause the divergence. Second, the particle size distribution especially the tail end distribution is hard to be measured correctly because of the measurement accuracy and aggregation of the particles. But in this model, the few largest particles in the tail end of the particle size distribution will mainly decide the depth distribution and density of the subsurface damages. As a result, this factor would also cause the divergence.

3.7 Correlation of laser induced damage performance with subsurface damages distribution and pad elastic modulus

To analyse the influence of polishing induced plastic subsurface damages distribution on the laser induced damage performance, the Rmax, Rq and subsurface damage area percentage versus laser induced damage threshold and damage density are shown in Fig. 13 and Fig. 14. It could be found that the laser induced damage threshold is inversely proportional to the logarithm of the Rmax, Rq and subsurface damage area percentage while the logarithm of the laser induced damage density is proportional to Rmax, Rq and subsurface damage area percentage. The larger pad elastic modulus will make the optics have more and deeper plastic subsurface damages (which presents as larger Rmax, Rq and subsurface damage area percentage) and finally show lower laser induced damage threshold and higher damage density. The results indicate that the laser induced damage performance of polished optics has direct correlation with the plastic subsurface damages distribution and pad elastic modulus. Optimizing the polishing parameters (such as using soft polishing pad with lower elastic modulus) to obtain optics with fewer and shallower plastic subsurface damages is important and useful for improving the laser induced damage performance.

 

Fig. 13 Rmax, Rq and subsurface damage area percentage versus laser-induced damage threshold for all samples.

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Fig. 14 Rmax, Rq and subsurface damage area percentage versus laser-induced damage density for all samples.

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Besides, the damage probability curves (as shown in Fig. 5) of polished optics present larger slope and lower damage threshold with the increased pad elastic modulus. It is also correlated with the distribution of the polishing induced plastic subsurface damages in different samples. If we consider that only one kind of damage precursor (the plastic subsurface damages) is distributed with densities $d_i$ and damage threshold $T_i$ on the optics surface. The damage probability $P(F)$ as a function of laser fluence F when a gaussian beam (D is the diameter at e⁻²) is used can be calculated by Eq. (14) [29]:

where $S_i$ is the effective spot size for $F\ge T_i$ and $S=\pi D^2$ is the gaussian beam spot size. From Eq. (14), we could find that with the higher density $d_i$ of damage precursors (or plastic subsurface damages), the lower laser fluence F is needed to get the same damage probability, which would result in a larger slope of the damage probability curve. Considering the larger pad elastic modulus will induce more and deeper plastic subsurface damages. The deeper plastic subsurface damages would possibly have severer easily damaged nano-cracks which would lower the damage threshold of the subsurface damages. The larger pad elastic modulus will make the damage probability curve of polished optics show larger slope and lower damage threshold.

4. Conclusion

The plastic subsurface damages distribution and their influence on the laser induced damage performance of fused silica optics polished with different pads are investigated. The elastic interaction model, plastic indentation model and wear relationships are combined together to establish the numerical relationship between polishing parameters and subsurface damages distribution in different polishing process, which is consistent with the experimental results. It reveals that: (1) the micron sized polishing particles will abrasion fused silica optics mostly in the plastic deformation region during optical polishing process and most of the polishing induced subsurface damages are plastic damages; (2) a few largest polishing particles in the tail end of the particle size distribution mainly decide the final depth distribution and density of the polishing induced plastic subsurface damages; (3) the larger pad elastic modulus will make the few largest polishing particles in the tail end of the particle size distribution bear much larger load and generate larger proportion of observable plastic subsurface damages during optical polishing; (4) using polishing pad with lower elastic modulus is prominent for restricting the generation of fractures and could substantially decrease the depth and density of plastic subsurface damages; (5) The laser induced damage performance of fused silica optics shows tight correlation with plastic subsurface damages distribution and pad elastic modulus, the larger pad elastic modulus would make the polished optics show lower damage threshold, higher damage density and larger slope damage probability curve.