## Abstract

Relationships between subsurface damage (SSD) depth and peak to valley surface roughness (*Rt*) have been widely studied and present a major interest for an easy assessment of the SSD depth. We look at the relation between SSD depth and other surface roughness parameters using the Abbott-Firestone curve on a large campaign of grinding tests (with different abrasive grain size, grinding speed and grinding mode). The results reveal that Abbott-Firestone parameters are better for an assessment of SSD depth and that relationships between SSD depth and surface roughness are not universal but depend on the grinding process.

© 2013 Optical Society of America

## 1. Introduction

Performances of high energy laser facilities such as Laser Megajoule (LMJ) or National Ignition Facility (NIF) are typically limited by laser induced optical damage of optical components located in their final assembly. When exposed to high fluences in the ultraviolet range, fused silica optics can exhibit damage that growth exponentially under subsequent shots. Then lifetime of these optical components falls drastically. Early works have shown that small cracks located under the surface of the optic can notably be precursors of laser damage [1]. This subsurface damage (SSD) is mainly due to the manufacturing steps of the optics and the brittle nature of fused silica. During the grinding the diamond grains present in tools, act indeed as indenter that embrittle locally the material [2]. The reduction of SSD introduced during the polishing process is consequently a major objective to produce high damage threshold optics. The knowledge of the SSD depth permits to remove the SSD during each polishing process of an optic.

In the present paper, we focus on SSD measurement of ground or fine grounded surfaces produced by fixed abrasive diamond grinding or loose abrasive lapping. The characterization of SSD is the subject of many studies and several methods have been developed to measure it. Firstly destructive methods have been studied. The principle consists generally in achieving either a local wear (ball dimpling [3] or MRF dimpling [4]) or a global wear (e.g. taper polishing or MRF taper [5]) on the piece to reveal the micro-cracks whose depth can then be measured by conventional microscopic observation. Another method consists of following, the evolution of peak to valley surface roughness, during successive etchings of the optic in a HF bath, the SSD depth then corresponds to the maxima of the peak to valley roughness [6]. Secondly non-destructive methods have been elaborated to characterize the SSD as confocal fluorescence microscopy [7] or light scattering techniques [8, 9]. Although these methods are non-destructive, they are quite complicated and time consuming to implement in an industrial process e.g. between each grinding steps. That is why many studies tried to assess an empirical relationship between the depth of SSD and the surface roughness, which is straightforward to measure even with portable devices.

First, Preston [10] reported that the depth of subsurface defects is more than three times higher than the Peak-to-Valley roughness. The scaling factor observed between SSD depth and *Rt* roughness are equal to 3.9 for Aleinikov [11], 6.4 for Hed [12], 1.4 for Randi [13] and 9.1 for Miller [14]. The origin of the discrepancy in the proportional factor is not easy to find. These differences can be due to various preparation methods and measuring principles used for both surface roughness and SSD [6] or to the variety of grinding condition used. In a previous work we investigate the Abbott-Firestone curve [15], or bearing area curve which describes the surface texture of an object, as a tool to estimate SSD depth. The parameters of Abbott-Firestone allow assessing the average depth and the fraction of valleys in the roughness profile. We demonstrated on D126 and D64 diamond ground surface that Abbott-Firestone curve parameter *Mr2* could be an alternative to assess the SSD depth [16].

When previous work was established on few diamond grinding experiments, we investigate here the relationships between several surface roughness parameters (including the peak to valley surface roughness *Rt*, the mean surface roughness *Ra*, the root mean square surface roughness *Rq* and the Abbott-Firestone parameters) and the SSD depth taking into account several grinding experiments with different modes of abrasion (bound abrasive or loose abrasive) and with varying abrasive grain size, and cutting parameters. We determine the best indicator of the SSD depth and calculate the precision of the relationships between the SSD depth and the surface roughness parameter considered. We also study the robustness of the indicator when grinding parameters change.

In the first section, the sample preparation and associated characterization are briefly described. Then we explain the method for the comparison of the different relationships between the SSD depth and the different surface roughness parameters. Results are finally presented and discussed in the later section.

## 2. Experiments and methods

#### 2.1 Sample surface preparation

Two types of materials were used for this study: square-shaped fused silica glass samples (HOQ from Heraeus, 100x100mm^{2} x 20mm thick) were used for bound abrasive grinding tests, and Round-shaped Corning 7980 fused silica samples (50 mm diameter and 10 mm thick) were used for loose abrasive grinding tests.

For bound abrasive grinding we used an OPTOTECH SMP500-2C grinder machine. In order to have a good repeatability of the process we control the spindle temperature of the machine. Additionally a first roughing operation was performed on each sample with a D126 grit metallic bond diamond wheel and the following process parameters: wheel speed = 1500 rev/min, workpiece rotation speed = −16 rev/min, feed rate = 0.15 mm/min and depth of cut = 0.2 mm. This roughing operation was done in order to have the same reference state for each sample that guarantees a depth of SSD smaller than 100 microns prior to the grinding experiments. Then all these experiments were done with a depth of cut greater than or equal to 100 µm. The grinding wheels were dressed using an alumina dressing stick for 20 seconds every five grinding tests (correspond to every 15min of grinding).

Three sample batches (labeled A, B and C) were manufactured using a simplified grinding mode as shown in Fig. 1(a).The samples were ground by blocking the rotation axis of the workpiece table and by translating the grinding wheel along the horizontal axis. This configuration is used to simplify the kinematics in regard to the industrial grinding mode (Fig. 1(b)) where the grinding wheel and the sample rotate oppositely and the feed is along the vertical axis. This non-conventional kinematics for the optics industry was developed in order to be able to measure the grinding forces and moments applied to the sample with a dynamometer fixed under the part. This study of the relationship between SSD and grinding forces will be part of another contribution. Batch A (respectively B and C), refers to the milling operation using a rough D126 grit (respectively intermediate D91 and D64 grit) metallic bond diamond wheel. In batch A (respectively C) 5 samples (respectively 3) are grinded with the same conditions to analyze the repeatability of the process. Varying processing parameters were tested with three values considered for each of the following: cutting speed, feed rate and depth of cut.

A fourth batch (D) was manufactured with the industrial configuration (Fig. 1(b)) using a finish D25 grit resin bond diamond wheel. In this batch three samples are grinded with the same parameters.

A Logitech PM5 single side lapping and polishing machine with a cast iron lapping plate was used for loose abrasive grinding operation. A fifth batch (E) was manufactured using varying processing parameters with different rotation speeds, loads, abrasive materials, abrasive sizes and slurry concentrations.

The grinding parameters used for each sample are summarized in Table 1 and Table 2. .

#### 2.2 Sample characterization

To characterize the precision of the grinding process, the surface roughness for each of the grinded surface was measured using a MITUTOYO (SJ-201) contact stylus profilometer, equipped with an inductive probe with 2μm radius and 90° angle. Each profile is filtered with a Gaussian filter (and a double Gaussian filter for the calculation of Abbott-Firestone curve) according to the norm ISO 13565 [17]. The cut-off length for the filter is 0.8 mm.

According to the normalized procedure [16], scans of 4mm length were performed, on 16 areas uniformly distributed on the surface of each sample [6]. For each sample, several parameters are calculated from the measured profile (except for *Rtmax* all these parameters are mean values obtained on all 16 scans):

- - the peak to valley surface roughness
*Rt*, - - the mean surface roughness
*Ra*, - - the RMS surface roughness
*Rq*, - - the kurtosis surface roughness
*Rku*characteristic of the narrowness of the height distribution, - - the surface roughness
*δ*(moment of order 4) defined by Suratwala [6], - - the peak to valley surface roughness
*Rtmax*(maximum value obtained on all 16 scans), - - the Abbott-Firestone parameters (
*Rk*,*Rpk*,*Rvk*,*Mr1*,*Mr2*).

These parameters come from the Abbott-Firestone curve (example shown on Fig. 2) which is built by computing the cumulative probability density function of the surface profile's height by integrating the profile trace.

The cumulative probability is divided in 3 areas by means of the tangent with the smallest slope that fit 40% of the curve. The intersection of the curve and the previous tangent define *Rk* the roughness core profile and *Mr1*, *Mr2* which are linked respectively to the percentage of peak and valley. *Rpk* and *Rvk* are defined by right-angled triangles with the same area than curve’s areas above and below the tangent. They characterize respectively the reduced peak height and the reduced valley depth. More information on Abbott-Firestone curve is presented in our previous work [16].

SSD depth is measured by means of acid etching (80% HF, 20% HNO3); it consists in following the
evolution of surface roughness *Rt* during successive acid etching.
*Rt* is measured after each etching step and the maximal value of *Rt*
among the whole set of measurements is equal to the SSD depth as described elsewhere [6]. All the results are reported in Table 3.

#### 2.3 Method to research the best indicator of the SSD depth

For each *R**, one of the surface roughness parameters studied, one searches the best linear relationship between the measured SSD depth (*SSD _{meas}*) and

*R**that fit the data of the whole batch of experiments. The found relationship present the following form:

For every test *i*, with that best fitting relationship, an estimated SSD depth (*SSD ^{i}_{calc}*) is calculated from the roughness parameter measured

*R*:

^{i}_{meas}Then, for every surface roughness parameter and for every test, the error (in percent) between the measured SSD and the estimated SSD depth is calculated with the following formula:

Finally the mean and the maximum of the error are calculated for every surface roughness parameters studied (for *i* = 1 to n). The best indicator of the SSD depth is determined by the surface roughness parameter that gives the minimal error. The same error calculation is done on the data of Miller [14] and compared with our results.

In a second step, to check if indicators of SSD depth depend on the grinding parameters, the same calculi are done on three reduced sets of experiments:

- - Batches A to C represent rough and intermediate grinding
- - Batch D represents finishing
- - Batch E represents loose abrasive grinding

Then the best indicators for each reduced set are compared with each other.

## 3. Results and discussions

#### 3.1 Repeatability of measurements

First we check the repeatability of our grinding experiments. Figure 3 presents for each set of tests using the same operating conditions in batch A, C and D the scattering of the measured SSD depth and surface roughness parameters (5 tests for batch A, 3 for batch C and D). This scattering, given in percent, is expressed by the standard deviation of the set of measures divided by the mean values of the set.

For batches A and C, which correspond to rough and intermediate grinding tests, the variability of the SSD depth measurements is less than three percent. For the surface roughness parameters the scattering is between five and ten percent. This corresponds to the precision of the device used in the study, as measured within ten tests on the same spot. Therefore the grinding experiments prove to be highly repeatable. For the finishing process (batch D), the repeatability is not quite as good. We observe a much larger variability of SSD depths (up to 35%) and surface roughness measurements. The increasing load on the spindle seen during this set of grinding experiments could be responsible for that scattering. This load increase could be the consequence of several phenomena: glazing of the tool, loss of the grinding wheel abrasive power, thermal expansion of the spindle... In fact, due to very low feed rate for finishing (5 µm/min), the grinding test duration approaches half an hour and the thermal expansion during the grinding is not negligible (up to 5° measured during tests) despite the thermal control of the cutting fluid which flows through the spindle. Another possible explanation of this data scattering could come from the SSD depth measurement technique retained. With the small scratch generated by the finishing wheel, both spatial resolution and small field of measurement of our profilometer based method could be problematic. These peculiar measured roughness profiles usually exhibit a smooth profile with some random and rare deep cracks. With this low fraction of cracks the measure of 16 scans could be not statistically sufficient to obtain a surface roughness representative of the sample surface. Improving field of measurement by performing 2D line scans on a large surface or switching to white light interferometry [4] could possibly reduced this discrepancy.

#### 3.2 Best SSD indicator for the whole tests

Figure 4 presents indicators of SSD depth for a wide
range of grinding parameters (all the sample batches used). For each surface roughness parameter, it
shows the mean error (Fig. 4(a)) and the maximal error (Fig. 4(b)) between the measured SSD and the estimated SSD as
presented in §2.3. The graph shows that the parameter *Mr2* minimizes the mean
and the maximal error for the estimation of SSD depth. The figure shows also that using
*Mr2* for the estimation of SSD depth conducts to less mean error than using
*Rtmax* (30% for *Mr2* against 40% for *Rtmax*) and
less maximal error (190% for *Mr2* against 230% for *Rtmax*). Despite
the relatively high mean error on the estimated SSD depth (30%) Fig.
5, in which is plotted the SSD depth as function of *Mr2* for all the tests,
clearly shows a relationship between SSD depth and this surface roughness parameter. Such error can
be explained by the scattering of experimental data, which is expected because of the probabilistic
behavior of the brittle fracture mechanism than form the SSD [18].

We have also compared our results to the one of Miller [11]. Miller have studied the relationship between *Rtmax* and SSD depth in grinding for a wide range of abrasive grain sizes (7µm to sand blast) and different abrasive grinding modes (bound or loose abrasive grinding). We used his results to find the best fitting linear relationship between *Rtmax* and SSD depth and plot the error of the estimated SSD in Fig. 4. Using *Rtmax* parameter, the result shows similar maximal error for Miller’s data or ours and a ten percent higher mean error for Miller’s data. But the two sets of experiments are fairly comparable in this regard.

All these results demonstrate that, for the grinding and lapping of fused silica, the *Mr2* roughness parameter is a slightly better indicator of the SSD depth than *Rtmax* which was used in past studies. The following relationship between SSD depth and *Mr2* is found to best fit the experimental data for a wide range of grinding parameters (linear regression on Fig. 5):

*Mr2* being linked to the fraction of valley in the measured profile, SSD depth could be linked to the fraction of valley.

#### 3.3 Best SSD indicator for subsets of samples

To check the stability of the SSD depth indicators the same indicator investigation is done on reduced sets of tests. Figure 6 presents the mean and maximal errors of the estimated SSD depth with a fit done on the following three reduced subsets of experiments:

- - first a subset constructed with all rough and intermediate grinding tests (batch A-C),
- - second a subset constructed with only the finishing tests (batch D),
- - and finally a subset constructed with the lapping tests (batch E).

To keep this figure readable, only the best indicators of SSD depth for each subset were shown and compared with *Rtmax* and *Mr2*.

This figure shows that the best indicator of the SSD depth for the rough and intermediate grinding experiments is the *Rvk* parameter (mean error = 13%, maximal error = 30%). Other parameters like, *Rtmax*, *Rt* and *Mr2* lead to slightly higher error on the estimated SSD depth (3% more). For the finishing experiments only, *Rku* proves to be the best indicator of SSD depth, while all the other roughness parameters cannot compete. But due to large scattering of the experimental data the mean error found is in the range of 30%. Then, *Rt* proves to be the best indicator for the lapping experiments, (mean error = 16%, maximal error = 35%) while *Rtmax* and *Rvk* lead to slightly higher error and *Mr2* leads to very high error (> 40% in mean). Therefore these results indicate that the relationship between SSD depth and the surface roughness depends on the grinding process. There is no universal indicator of the SSD depth. The best indicator is different for rough grinding, finishing, and lapping (only *Rku* relevant for finishing, *Mr2* not relevant for lapping). Further investigations will be needed to explain the mechanicals and physicals differences between each grinding configuration that leads to these differences in the SSD formation and in the surface roughness after grinding.

Except for finishing, the strength of the relationship found between SSD depth and the best roughness parameter is much higher when considering a subset of experiments than the whole set of tests. For rough and intermediate grinding the best indicator (*Rvk*) leads to a mean error of 13% and a maximal error of 30%. For all tests, including grinding, finishing and lapping, the best indicator (*Mr2*) leads to more than two time higher mean error (30%) and to more than six times higher maximal error (190%). Thus defining the SSD depth indicator in function of the grinding process is much more precise and leads to smaller safety factor needed to account for the scattering of the relationship between SSD depth and surface roughness.

## 4. Conclusion

Characterization of the SSD is a crucial step to determine the optimum grinding parameters. We evidence that for various fused silica grinded or lapped samples, the *Mr2* parameter, based on the Abbott-Firestone curve, is a slightly better indicator of the SSD depth than the peak to valley roughness (*Rt*) used in past studies. With a fairly large safety factor, it can be used for quickly assess the depth of SSD, regardless of grinding parameters. Further studies shall be made to check if it can be extended to other glass materials. We also highlight that there is no universal indicator of the SSD depth. The relationship between SSD depth and surface roughness after grinding depends on the grinding process (loose or bound abrasive grinding, rough or finish grinding). Much better indicators of the SSD depth can be found considering only one grinding process (*Rt* for lapping, *Rvk* for rough grinding and *Rku* for finishing) with a resulting mean error on SSD depth assessment divide by up to two. Such a set of indicators for each step of the grinding process can be used for a better assessment and an easy monitoring of the SSD depth during the process.

## Acknowledgments

This work is supported by the Conseil Régional d’Aquitaine and is performed in the framework of the EFESO 2 project.

The authors thank Denis Penninckx from CEA for his advice on the experimental data analysis.

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