1. Introduction

Impurities, pittings, scratches and cracks of the optical components can be induced by the force loaded by the indenter, the chemical process and/or the thermal field in manufacturing processes which include cutting, grinding, lapping and polishing [1, 2]. Second, these damages can also be caused by the hard particles like dust when the optical components are exposed in the atmosphere [3]. Finally, they can also occur due to the interaction between the strong illumination light and the glass in the imaging process [4]. It has been observed that cracks largely influences the optical intensity and distribution of the scattering light, limiting the imaging quality of the optical systems, and severely limits the laser damage threshold, reducing the lifetime of the optical components [1, 5–7]. Cracks are mainly produced near the surface of optical element by the loaded force, and the cracks beneath the surface are called subsurface damages (SSD).

To characterize SSD, the three layer model can be used to describe the general properties of SSD [3, 8, 9]. It was then found that each layer is of different properties after different manufacturing processes. The results can be regarded as a guide for the later manufacturing process. For example, after grinding, the large damages like scratches locate in the upper layer, the micro defects tend to be in the middle area, and the lowest layer is just the elastic deformation [8, 9]. For the optical component through grinding, lapping and polishing, the SSD can be also divided into three layers, including polished layer, defect layer and the deformed layer [3]. The SSD is expected to locate within the first two layers.

To measure the actual distribution of SSD both destructive and non-destructive methods have been proposed [10–22]. One usually used destructive approach is first to perform magnetorheological finishing (MRF) and to observe SSD through a microscope [10-11]. The method has the advantage that the MRF would not introduce any extra damage to the optical component and has the high measurement accuracy. However, as a destructive method, the operations involved in the method are complex. At the same time, the high-precision profiler is expensive which should be employed to detect the finished profile in this method.

The main non-destructive methods are various imaging techniques, including electron microscopy [12-13], confocal microscopy [14-15], total internal reflection microscopy [16-17], quasi-Brewster angle technique [18], X-ray diffraction method [19-20], Raman spectroscopy [21] and digital holographic microscopy [22]. X-ray diffraction method [19-20] and Raman spectroscopy [21] can be used to detect the residual stress distribution. The drawback is that for the same optical element different kinds of X-ray may lead to different images of stress distributions because of their different penetration capabilities. In Raman spectroscopy the imaging depth is limited due to the weak laser light.

With transmission illumination, the quasi-Brewster angle technique [18], total internal reflection microscopy [16-17] and digital holographic microscopy [22] can image the SSD over all the depth. However, in the first two methods, the SSD within the imaging depth is projected onto a two dimensional plane and detected by a charge coupled device (CCD) (for example). So they have a low depth resolution. The three dimensional information about SSD such as the shapes and sizes can be obtained by using digital holographic microscopy with one exposure. However, its longitudinal resolution is limited.

The confocal microscopy has both high longitudinal and lateral resolutions and can resolve the SSD at the level of ~200nm [23]. However, the mechanical scanning is required to realize the three dimensional reconstruction, increasing the cost and system complexity. The electron microscopy also has the similar limitation. Only the microstructure in the shallow region can be visualized, even though the electron microscopy [12-13] has the ultrahigh resolution at the 0.1nm level.

The optical coherence tomography (OCT) is a recently developed technique and has the capability of generating depth-resolved images [24, 25]. Demos et al. [26] obtained the image of Hertzian cracks in the silicon nitride ball by using the OCT system with the 10-μm axial resolution and 20-μm lateral resolution. Due to its low resolution the SSD at the micron level cannot be resolved. However, for the high-precision manufacturing of optical elements, the SSD with the size less than 2μm and over a range of 200μm beneath the surface is expected to be measured.

Spectral domain OCT (SDOCT) is a new implementation of OCT concept and has the capability of imaging the structures within the whole depth each reconstruction [27]. In our design to realize the measurement of SSD with the size less than 2μm and over a depth range of 200μm, the objective lens with a large numerical aperture (NA) in the sample arm, a spectrometer with high spectral resolution in the detection arm and the supercontinuum laser with the ultra-wide band should be employed. Limitations of our SDOCT system are that the transversal resolution may decrease quickly with the increase of imaging depth due to the large NA used and that the large dispersion mismatch between the glass and air may lead to the quick decrease of axial resolution [28].

In this work, due to the advantage of the strong surface reflection of the optical component a self-referenced SDOCT system was set up to quantitatively measure the SSD within it. The resolutions less than 2μm and the imaging depth over 200μm were realized. By using this system the three dimensional images of SSD within eight pieces of BK7 glass samples ground for different grinding times were reconstructed from which the quantitative information of SSD can be obtained. To compare the measured maximum depths with those calculated by the theoretical and empirical formulas we first summarized all the formulas proposed so far for predicting the maximum depth and the application conditions of different formulas were discussed. Addition to the maximum depth the boundaries of three layers can also be clearly seen from the cross sectional image, and the properties of SSD in each layer can be found. From the en-face images other important parameters, including the cluster depth and the density, can be estimated accurately. All these results are helpful for providing a new standard in which the quality of optical components in the manufacturing process can be evaluated by the sizes of the SSD and their distributions in optical elements.

2. Maximum depth of SSD

During the manufacturing processes one purpose of each step is to remove the material containing SSD produced in the previous steps. So the maximum depth of SSD is the most important parameter in the manufacture. It is evident that the maximum depth of SSD greatly depends on both the physical characteristics of the optical element and the manufacturing conditions. Many efforts have been devoted to theoretical analysis and experimental measurements of SSD. In the following, several theoretical formulas derived under assumed manufacturing conditions and various empirical formulas for different specific manufacturing processes are analyzed and compared.

2.1 Theoretical formulas

There exist two main types of SSD, median cracks and lateral cracks. The median cracks are generated by the interactive effect of the elastic and plastic deformations, while the lateral cracks might be caused at any moment in the manufacturing processes, even when the indenter is removed. The trajectories of median cracks are almost vertical to the surface and trajectories of lateral cracks are parallel to the surface, suggesting that the maximum depth of the median crack will be larger than that of the lateral crack. Thus here the maximum depth of the median crack is regarded as the maximum depth of SSD.

The maximum depth of SSD is dependent on the indenters used. Under the assumption that the load is only along the vertical direction and the cracks are well-developed [1], the theoretical formulas for predicting the maximum depth of SSD were derived (see Table 1). The results show that the maximum depth is a function of the parameters that represent the influences of the materials and the manufacturing techniques, where $\Theta$ denotes the product of all the factors before SR, $\kappa$is the correction factor of median crack depth and usually is equal to 2.23,$m$represents the dimensionless constant with a value ranging from 1/3 to 1/2, ${\alpha }_k$and$\beta$ are determined by the value of $m$, $\psi$is the sharpness angle of indenter, $K_r$is the fracture toughness, E denotes the elastic modulus, H is the hardness, P is the load, R is the radius of the blunt indenter. For the specific material and indenters used, the formulas reveal that the maximum depth has a power law dependence on the load. For the sharp indenter, the power is 2/3 [1, 29], and for the blunt indenter, the power is 8/9 [1, 30]. Other parameters like the sharpness angle of the indenter are also included in the formulas. However, in practice the exact values of some parameters are not easy to be determined, so it is hard to calculate the accurate maximum depth through these formulas.

Table 1. The theoretical formulas for the prediction of the maximum depth of the SSD

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Because the SR of the optical elements can be measured directly [25–27], it is expected that if the relationship between the depth of SSD and SR can be found it will allow us to estimate the maximum depth of SSD indirectly. The first effort to this goal is the work contributed by Lambropoulos et al. [29, 31]. For the sharp and blunt indenters, by assuming that the maximum depth of lateral cracks is equal to the peak-to-valley (PV) value, Lambropoulos et al. [29, 31] derived the theoretical relationships between the maximum depth of SSD and SR. In their analysis, the PV value was regarded as a measure of the SR. However, one limitation of their result is that the formula contained the load variable which has the significant influence on the value of the maximum depth but it is difficult to accurately measure in actual manufacturing conditions.

To avoid this difficulty, by calculating the ratio of the SSD over SR, Wang et al. [32] successfully removed the load variable from Lambropoulos’ formula (see the formulas in the second column in Table 1). The result shows that the maximum depth of the SSD is proportional to the 4/3 power of the SR for both sharp and blunt indenters. At the same time, the maximum depth is also affected by other parameters such as the elastic modulus and the hardness of the glass. For the given material and manufacturing method, all these parameters have fixed values and their influences can be effectively represented by a factor $\Theta$. In this case, the maximum depth can be determined if one knows the SR and the value of $\Theta$. However, for a piece of K9 glass after grinding and lapping with the sharp indenter, the theoretical values of the coefficients $\Theta$are 2.51 and 3.2, respectively. The corresponding measured values, however, are 3.18 and 3.6 respectively. The fact means that the actual values of the $\Theta$are significantly dependent on the manufacturing processes. Thus it is necessary to measure the values of $\Theta$for each manufacturing method.

2.2 Empirical formulas

As mentioned above, the maximum depth of the SSD is proportional to the 4/3 power of the SR. However, the large differences between the theoretical values of the coefficients and the measured ones suggest that there may exist other parameters which also influence the maximum depth. A straightforward way is to directly measure the value of the coefficient. A great deal of experimental data shows that, as a first order approximation, the maximum depth of SSD was proportional to the PV value. So if one can determine the ratio of the maximum depth and SR, the maximum depth can be estimated by measuring the SR for each manufacturing method.

Table 2 summarizes the values of this ratio for different materials and manufacturing methods. It can be seen that the ratio changes greatly for different methods, ranging from 1.4 to 9. For single crystalline optical materials (e.g. silicon) and optical glasses (e.g. BK7) with deterministically microground and lapped respectively, the corresponding coefficients are small, ranging from 1.4 to 2 [33, 34]. For the fused silica after loose abrasive lapping and for the brittle materials after rolling, sliding and wheel grinding the values vary from 2.8 to 4 [35, 36]. The similar range of 3.7~4.1 can be used for the optical glasses and ceramics ground by the sand or the loose abrasive [37–39]. Also the relatively large values were obtained in some situations [31, 40–42]. The fixed-abrasive pellet lapping method and the single pellet grinding method were respectively applied to the BK7 glass and the fused silica, and then the coefficients of these materials measured range from 5.1 to 8.7 [40,41], while by using diamond wheel grinding and loose abrasive lapping the ratio is 9 [31, 42].

Table 2. The linear relationship between the maximum depth of SSD and the SR

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Note that the maximum depth is also affected by the sizes of the abrasive particles. For different materials and the manufacturing methods, the maximum depth can be non-linearly and linearly related to the sizes of the abrasive particles. Table 3 summaries some results. For samples made of different brittle materials such as fused glasses and polycrystalline ceramics and ground by different manufacturing methods, the maximum depth is the power function of the abrasive size, ranging from 0.3L^0.68 to 2L^0.85 [43]. For materials like silicon wafers, there exist some linear relationships between the maximum depth and the sizes of the abrasive particles. The ratios, of course, change with the specific material and the method used [31, 44, 45].

Table 3. The empirical relationships between the maximum depth of SSD and the abrasive size L

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3. Experimental system description

The schematic of our self-referenced SDOCT is shown in Fig. 1. The system was illuminated by the supercontinuum light source (Fianuim SC400-2). The light from the source was coupled into the single fiber and then was divided into two beams by the 50:50 fiber coupler, but only one beam was used for the later interference measurements. The divergent light emitting from the output of the coupler was collimated by the lens L1 before impinging on the scanning mirrors. By changing the scanning angle of the scanning mirror, different lateral positions can be imaged. Finally the collimated beam was focused onto the sample by an objective lens L2 with the NA of 0.4. In order to guarantee that the reference light is strong enough the focus of L2 should be located close to the surface of the glass. The interferometric signal returning from the probe arm was delivered to the spectrometer where the light was collimated by L3 with f = 30mm before being dispersed by the transparent grating (Wasatch 1002-1) and focused by L4 to the linear CCD (e2v EM4).

 

Fig. 1 Schematic diagram of the self-referenced SDOCT system. L1-L4: lenses; CCD: charge coupled device.

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It is usually assumed that one A-scan is focused on one point at the sample surface. In fact the sample is illuminated by a light spot with a finite diameter. In our self-referenced SDOCT system, the diffraction-limited light spot is with a diameter of less than 2μm. Furthermore, to avoid the strong surface reflection and to ensure the large intensity of light backscattered from the damage area the sample light beam is focused to a point located beneath the surface, resulting in an increased illuminated area at the sample surface in which both the damage and non-damage areas are contained for the damage with a size at the micron level. Part of the incident light is reflected from the non-damage surface and acts as the reference beam.

One limitation of our self-referenced method is that when the surface damage is larger than the illumination spot, there is a risk of not capturing the on-surface damage in the reconstructed image because in this case both the sample and the reference light are the same.

The lateral and axial resolutions measured were less than 2μm, estimated by measuring the USAF1951 resolution target and the normalized intensity curve of the reflective surface respectively. And the sample containing the TiO2 particles was made by ourselves. First the TiO2 particles were mixed with the colloid. After 24 hours the colloid solidifies it can be visualized to determine the imaging depth which was over 200μm.

4. Experimental results and discussion

4.1 The maximum depth of SSD after grinding

We made eight pieces of BK7 glass elements ground for different times (See Table 4). SRs of eight samples were then measured by a Zygo interferometer. It can be found that the value of SR tends to decrease with the increase of grinding time except 0.039μm. The phenomenon is reasonable because the same manufacturing method has been employed to produce the SSD for each sample, ensuring the distribution of SSD similar. So the glass sample ground for a longer time should have a smoother surface. Based on the measured SRs, the maximum depths were calculated by using the theoretical and empirical formulas [29, 31–42].

Table 4. Maximum depths of SSD with the increase of grinding time

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Note that the values of parameters in the theoretical formulas in Table 1 should be determined before the theoretical maximum depth is computed. The mechanical properties of the BK7 glass are the elastic modulus$E=81GPa$, the hardness$H=7.2GPa$ and the fracture toughness$K_r=0.82MPa$ respectively. In the Lambropoulos’s analysis [29] the dimensionless constant $m=1/3$ is more appropriate, while $\kappa$is related to the indentation coefficients for the elastic stress field and residual stress field and is equal to 2.23 [32, 46]. Mahmoud et al. [47] measured the sharpness angles of diamond grains which are in the range of $\psi ={46}^{\circ }-{82}^{\circ }$ and $\psi ={62}^{\circ }-{89}^{\circ }$with the rake face and rake ridge considered as the cutting edges, respectively. Because the maximum depth is inversely proportional to the sharpness angle, $\psi ={46}^{\circ }$was chosen to calculate the maximum value. According to these values the factor $\Theta$ can be calculated and is equal to 2.5. As mentioned above once the $\Theta$ and the SR are known the theoretical maximum depth can be computed. The theoretical maximum depths calculated are shown in the fourth row of Table 4.

Note also that the empirical maximum depths were calculated based on the ratios of the measured maximum depth and SR in Table 2. However, the values range from 1.4 to 9 because the samples were fabricated under different manufacturing conditions and measured by different methods. It is hard to infer what the ratio of the measured maximum depth to SR in our experiment should be. So we used the ratio range 1.4~9 to compute empirical maximum depths, shown in the fifth row in Table 4.

In order to compare the actual maximum depths with the ones calculated by theoretical and empirical formulas, the self-referenced SDOCT system was applied to estimate the maximum depth of each sample. In the practical measurement the SSD within a local area was detected, and this area is consisted of many points each of which has a depth. So the maximum depth of the local area can be determined by comparing the depths of all the points. To find the maximum depth of the sample the five fixed local areas of each sample, the central part and the four sides, were chosen to be measured. Then each local area was measured more than ten times and the average maximum depth of each area was computed. The maximum among the five average values is the maximum depth of the sample shown in Table 4.

To accurately obtain the maximum depth the actual depth represented by one pixel should be measured. In our work one piece of the cover glass was first measured by the white light interferometer (VEECO NT9100) and its average thickness measured was 179μm. Then this cover glass was imaged by our self-referenced SDOCT system and its two surfaces were reconstructed. In this case the intensity curves of two surfaces can be plotted, and the pixels corresponding to peaks of two curves can be found. The difference between the two pixels represents the thickness of the cover glass. Therefore, the actual depth represented by one pixel can be acquired by the ratio of 179μm to the pixel difference. When the SSD image was reconstructed, the binary image should be obtained by choosing the proper threshold from which the pixels corresponding to the edge of SSD and the pixels corresponding to the surface can be found. Then the difference of pixels representing the maximum depth of SSD can also be obtained. Therefore, the actual depth of SSD can be calculated by multiplying the pixel difference with the actual depth represented by one pixel, shown in the sixth row in Table 4.

It can be seen from Table 4 that the theoretical value of each sample is in the corresponding range calculated by the empirical formula. However, the ranges of the maximum depth calculated with empirical formulas are very large and change with the samples, especially for Sample 1 and Sample 2. It is evident that for the rough surface the measured maximum depths are similar with the maximum empirical ones but a little larger than the theoretical ones. For example, the measured maximum depths are 57.53μm and 58.16μm for Sample 1 and Sample 2 respectively, correspondingly the maximum empirical values are 57.87μm and 57.60μm respectively and the theoretical values are 38.45μm and 38.27μm respectively. It can also be seen that the measured maximum depths are much larger than the theoretical and empirical ones for the smooth surface. For instance, for Sample 6, Sample 7 and Sample 8 the actual values are 22.47μm, 28.36μm and 12.58μm respectively, while the ones calculated by formulas are less than 1μm. This may suggest that the theoretical and empirical methods are more suitable for optical components with relatively rough surfaces. For the optical element with a smooth surface several damages might totally locate below the surface. Thus its actual maximum depth only can be determined by using the imaging techniques. Note that the big difference between the measured maximum depths and the theoretical values can be attributed to the fact that the theoretical formula was derived under the condition that only one single indenter is employed in the manufacturing process, whereas in practice numerous diamond grains are used to grind the samples, which is really a complicated process.

Note also that the abrasive particles with the average size of 10μm were employed to grind the samples. According to the empirical relationship, 0.3L^0.68-2L^0.85, the maximum depth calculated varies from 1.44μm to 14.16μm. It can be found that the measured maximum depth for Sample 8 is in this range. The maximum depths of Sample1~Sample7 are larger than 14.16μm because the amount of materials containing SSD generated in the previous steps is not totally removed. Therefore the formula for brittle materials is valid for the situation where SSD is produced by the 10μm abrasives themselves.

4.2 Comparison with the three layer model

Traditionally, a microscope with a high magnification is used to realize the high lateral resolution. Because of the absence of the depth-resolved imaging capability, conventional microscopes are not able to determine the depth of the SSD within the sample. So the maximum depth is determined by first making a dimple by the MRF in the sample and then the microscope should be employed to obtain a series of en-face images along the depth direction of the dimple. Until the depth where the image contains no SSD the measurement is finished. With these images areas and numbers of damages at each depth can be estimated which can be used as a measure of the acceptance of the optical elements manufactured. In addition to be destructive, it is evident that this method is not reliable to detect the cracks located in the deeper region.

As mentioned above, the structure of SSD can be roughly described by the three layer model. In this model, for the optical elements after being ground, lapped and polished, the polished layer is defined as the first layer with a thickness of less than 1μm and the second layer is the defect layer with a thickness of less than 100μm. For the ground samples, the first layer consists of scratches and cracks and the second layer mainly contains micro-cracks. The thicknesses and the characteristics of the cracks within each layer, of course, are dependent on manufacturing methods chosen.

It is clear that it is not easy to investigate the layered properties of the SSD with the traditional methods. On the other hand, our self-referenced SDOCT system can generate the cross sectional images of the SSD over the whole depth each of which consists of 1000 A-scans, making it possible to quantitatively determine the thickness of each layer, the sizes and shapes of the cracks within each layer. And then the parameters for quantitatively evaluating the quality of the elements can be calculated.

Figure 2 shows the typical cross sectional images of the SSD. Figures 2(e)-2(g) are magnified versions of the parts denoted by the red rectangular square in Fig. 2(a) to 2(d), respectively. Note that the three layer structure can be seen clearly from the images, and the boundary of the first two layer can be determined by the irregular curves in the cross sectional images indicated by the red dash lines. It can also be determined by the damage density of SSD, for example, the depth at which the damage density decreases to a half of that at the surface can be regarded as the boundary.

 

Fig. 2 Cross-sectional images of SSD. (a)-(d) are the typical distributions and (e)-(h) are the magnified pictures corresponding to the red dashed boxes.

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The actual layer thickness of our ground sample can be determined. As shown in Fig. 2(e)-2(h) the first layers mostly contain the micro lateral cracks and impurities the thicknesses of which are less than 10μm, with 6.5μm, 5.0μm, 7.3μm and 6.9μm respectively. However, the median cracks are dominant in the second layer and the maximum depths of second layers depend on the maximum depth of SSD, varying from 28.1μm to 59.2μm. These results are also in agreement with the theory that the defects are supposed to be within 100μm. Here the deep regions without cracks are regarded as the third layer within which elastic deformation occurs and cannot be detected by the present system. Note that usually only the first two layers of the SSD are concerned because the damages might develop in the chemical or thermal or radial environment, but the elastic deformation will not.

With the help of the cross sectional images, we are able to determine the properties of the cracks within each layer. First of all, because of the strong surface reflection this layer looks very bright. The sizes of lateral cracks and impurities are on the order of 1~2μm. It is observed that the cracks in the second layer tend to have complex morphologies. Here the bright areas are locations where the cracks occurred. As shown in Fig. 2(e) a crack with a width of 69.8μm looks like a network consisted of defects with different shapes and sizes. The median crack develops to 24.9μm, and then the discrete small cracks hide in the deeper region with the maximum depth of 59.2μm. Similarly the damage in Fig. 2(f) has the lateral dimension of 27.5μm and it propagated to 56.8μm. Note that the micro crack is so small that its path is not clear and complicated. As displayed in Fig. 2(g) the SSD with the maximum depth of 41.2μm is a group of small cracks. They irregularly suspend within the glass and the smallest length is about 0.9μm, while the crack with the width of 20.0μm is like an inverted triangle. The crack indicated by the green dashed circle in the Fig. 2(h) is the combination of two median cracks and the one indicated by the blue dashed circle looks like a diamond beginning from the surface. Note that the size almost remains unchanged before suddenly increasing to the diameter of around 10μm. The crack extending to the maximum depth of 28.1μm is not a single one but a bunch of some discrete micro defects with several microns. The width of the crack close to the surface is 25.1μm below which some damages with the sizes of less than 5μm occur.

Note that the interference spectrum directly detected by CCD consists of three terms, the reference intensity, the sample autocorrelation term and the useful mutual interference signal. The reference intensity is so strong that it may reduce the quality of the image. To eliminate it the mean value of each A-scan spectrum signal should be calculated and then is subtracted from each A-scan spectrum signal before the latter numerical operation. The sample autocorrelation term can be neglected because its value is much smaller than that of the useful mutual interference signal which is the multiplication of the reference light with the sample light. After removing the reference intensity the surface damage and the shallow-depth damage can be reconstructed by performing inverse Fourier transform. In addition, the surface damage is located only on the surface, while the shallow-depth damage may be totally within the glass. Thus they can be easily recognized from cross sectional images.

4.3 Cluster depths, defect numbers and damage densities

As discussed above, the maximum depth of SSD is an important parameter during the manufacturing processes for determining thicknesses of the materials to be removed in the next steps. To evaluate the quality of the optical components, in addition to the remained SSD, the cluster depth, the defect number and the damage density are also of great importance. Here the cluster depth is defined as the depth within which the majority damage occurs. In some cases, it may be enough to evaluate the quality of the elements by only estimating the cluster depth. Note that the SSD is not uniformly distributed over the areas beneath the surface. So to accurately characterize SSD, the damage density is also needed. Here the damage density is defined as the ratio of the damage area to the whole field of view and is used as a measure for the acceptance of the elements.

To calculate the values of the cluster depth and the damage density, first the three dimensional images of samples were reconstructed from which the en-face images at different depth were generated. Evaluating the quality of surfaces and sub-surfaces of optical components is one of the most important ways to estimate the quality of optical components. To observe the distribution of SSD within each sample the surface of each sample should be a reference plane. So the en face planes of samples at the same depth were chosen to study the influence of grinding time on the SSD distribution. Figures 3(a) and 3(d) show the typical reconstructed B-scan images of Sample 1 and Sample 5, respectively and Figs. 3(b) and 3(e) display the corresponding en-face images at the depth of 7.34μm which are indicated by the red dashed lines in Figs. 3(a) and 3(d).

 

Fig. 3 Reconstructed images of Sample 1 and Sample 5. (a) and (d) are the three dimensional images, (b) and (e) show the en-face images, and (c) and (f) present the corresponding binary images.

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Before computing the parameters we applied the binarization operation to the en-face images, as shown in Fig. 3(c) and 3(f). The bright areas represent the damages and the dark areas are the non-damaged ones. It can also be seen that the sizes of most damages in Fig. 3(c) are much larger than those in Fig. 3(f), indicating that the larger damages tend to locate near the surface. Because the grinding time is longer for Sample 5 the majority of large cracks close to the surface produced in the last step have been removed successfully.

Due to the irregularity of SSD we regarded a connected area as one damage. The total number of the pixels contained in the damage is regarded as the damage area. So the damage density is the ratio of the damage area to the total pixels of the image. The results computed are summarized in Table 5. Close to the surface at the depth of 7.3μm the number of SSD of Sample 1 is 3267 which nearly doubles that of Sample 5, while the damage density almost triples with 20.32% and 7.43% respectively. The number of SSD remains at the thousand level till 14.7μm, whilst the value of Sample 5 changes to 391 at 11.0μm. The corresponding damage densities decrease to 5.36% and 4.90% respectively. Then they largely reduce to 1.66% and 1.20% at the depths of 22.0μm and 18.4μm respectively which are regarded as the cluster depths. In the deeper region the micro cracks in Sample 1 fluctuate from 0.013% to 0.081% and no damages occur when the depth is larger than 66.1μm. However, the maximum depth of Sample 5 is about 36.7μm at which the damage density turns to 0.

Table 5. Numbers of SSD, damage areas and damage density at different depths

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With the help of the cross sectional images and these estimating parameters the three layers of ground samples can be defined as follows: the first layer is characterized by its high density of SSD and its thickness is determined by identifying the boundary between the high density region and low density regions. It can also be estimated by choosing an appropriate threshold. For example, the depth at which the damage density decreases to a half of that at the surface can be regarded as the boundary. The second layer is characterized by the SSD with a length of more than 10 microns and the thickness of the second layer corresponds to the maximum depth of SSD. The third layer is the dark region without SSD.

Note that the values of the parameters, including sizes, shapes, cluster depth and damage density, are very useful in evaluating the performances of optical elements manufactured. For example, it was found that the laser-induced damage threshold (LIDT) is dependent on the sizes of the cracks. Camp et al. studied the LIDT of the fused silica with cracks illuminated by the light with a wavelength of 355nm, and the LIDTs were 8-21J∕cm² and 3-10J∕cm² when the lengths of cracks were 2-10μm and 20-40μm, respectively, showing that the value of the LIDT decreases with the increase of the length of SSD. As an approximation, Fig. 4 depicts the change of the LIDT with the size of the crack. Here for the cracks with the sizes of 2µm, 10µm, 20µm and 40µm, the corresponding values of the LIDT are 21 J∕cm², 8 J∕cm², 10 J∕cm² and 3 J∕cm², respectively. It can be seen that once a crack occurs the LIDT may decrease a lot. When the value of LIDT reduces to less than 7 J∕cm² the slope of the curve decreases slightly slow. The length of crack corresponding to this LIDT value is about 20μm. When the size of the crack is very large the LIDT decreases slowly with the increase of the crack length because it is already so small that the optical component can be easily broken.

 

Fig. 4 Change of LIDT with the lengths of cracks

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Note also that the relationship between the LIDT and the crack size suggests a new way for evaluating the quality of optical components in the manufacturing process by the sizes of SSD. For the comparatively low light power, ranging from 10mW to 100mW, if the radius of the collimated light beam is about 2mm, the corresponding power density varies from 0.08 J∕cm²~0.8 J∕cm². In this case the optical components with the cracks with a length of more than 40μm can be accepted. For the imaging system illuminated by the light with the light power ranging from 1W to 10W the power density calculated is in the range of 8 J∕cm²~80 J∕cm². The allowable sizes of cracks are about 10μm ~20μm and less than 2μm, respectively. Another important application is the strong laser with the light power over 1KW, even 1MW, the power density can reach 8 KJ∕cm² and 8 M J∕cm². This result suggests that in the strong laser application the sizes of cracks might be at the nm level.

On the other hand, for evaluating the LIDT the microscopes are usually employed to observe the damage condition from the en-face image before and after the irradiation. The cracks with big changes along the lateral direction could be observed clearly in the en-face image. However, as mentioned above, cracks in the deeper region are mainly the median cracks which are more likely to develop along the axial direction. In this case no changes might be seen in the en-face image, indicating that the LIDT measured might be larger than the actual value. Thus with our self-referenced SDOCT system the small changes in any direction could be easily visualized, which provides a more accurate way for estimating the effect of SSD on LIDT. Furthermore, the LIDT is a very complex parameter which is dependent on the physical properties of material itself. In fact the characteristic of material is closely related to sizes, shapes, depths and damage density of SSD. Due to the absence of the quantitative knowledge about the cracks within the glass, few investigations have been conducted about the relationship between these factors and LIDT. Therefore the quantitative measurement of SSD by using our self-referenced SDOCT system may promote the research in this area.

5. Conclusion

In our work, a self-referenced SDOCT system has been set up by making use of the strong surface reflection of the optical element. With our system the three dimensional structure of SSD within optical components can be measured, from which the parameters of SSD, including the maximum depth, the cluster depth, the shape, the size and the damage density, can be obtained. All these parameters are very helpful for quantitatively evaluating the quality of the optical elements, and very useful to provide a new standard in which the quality of optical components in the manufacturing process should also be evaluated by these parameters of SSD.

To compare the actual maximum depths with the ones computed by other approaches first the theoretical relationships derived with the ideal assumptions and the empirical formulas for different manufacturing methods were discussed and summarized in the tables. By using these formulas the maximum depths of SSD within eight samples were calculated and compared with the measured values. It was found that the empirical relationship, 0.3L^0.68-2L^0.85, is reliable for the situation where SSD is produced by the abrasives themselves, and other theoretical and empirical methods are more suitable for optical components with relatively rough surfaces at several microns. Note that for the high precision manufacturing the smooth surfaces of optical components are required. In this case our self- referenced SDOCT system is able to provide the actual values of maximum depths.

In addition, the three layer structure of the ground sample can be clearly identified from the cross sectional images, which is in agreement with the three layer model proposed by Wely et al. [3, 8-9]. Then the thickness of each layer, the sizes and shapes of the cracks within each layer were analyzed.

LIDT is an important parameter to evaluate the lifetime of optical components which is dependent on the physical properties of material itself. We utilized the data available in Ref. 48 to approximately plot the change of LIDT with the length of SSD, which may provide a new way for evaluating the quality of optical components in the manufacturing process by the sizes of SSD. Furthermore, if the power density of the light incident on optical components in the optical imaging system is known, the corresponding maximum size of SSD allowed in this component can be predicted. On the other hand, the LIDTs of optical elements can be found with their maximum depths of SSD measured by our self-referenced SDOCT system. However, due to the absence of the quantitative knowledge about SSD few investigations have been conducted about the relationship between LIDT and the sizes of cracks. In this case the quantitative measurement of SSD by using our self-referenced SDOCT system may provide a new guidance for the study of LIDT.