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Abstract
The mechanical stress in thin films can have a deleterious effect on the quality of optical components by deforming the underlying substrate. In addition, the substrate might be deformed by gravity or stress induced by mounting. We suggest a method to compensate the substrate deformation by laser generated tensile stresses in the backside of the substrate. We show results for irradiation of the borosilicate glass Schott D263M with an ArF excimer laser. We measured the integrated stress in dependence of the laser fluence and corrected a sample for the deformation by a chromium coating. We show that also antibiaxial plane stress components can be induced. For precise corrections a scheme for stabilization of the generated surface stresses still needs to be developed.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thin films play a significant role in optical technologies, as they can be used to alter the optical properties of a surface. In general, thin films exhibit tensile or compressive mechanical stress after deposition [1], which can lead to a significant deformation of the underlying substrate [2–4]. The curvature $k$ of a substrate of thickness $t_s$ due to a film of thickness $t_f \ll t_s$ is described by the Stoney equation [5]
if the deflection of the substrate is much smaller than $t_s$ [6]. In Eq. (1) $R$ is the radius of curvature of the substrate, $\nu _s$ is the Poisson’s number of the substrate material, $E_s$ is its Young’s modulus and $\sigma$ is the film stress. The substrate can also be deformed by gravity and the optical mount [3].To avoid substrate deformation by a stressed film, a thick substrate can be used, as can be seen from the inverse square dependence of $k$ on the substrate thickness $t_s$ in Eq. (1). However, there are fields of application where the usage of thin substrates is highly beneficial, like, for example, in space applications [7]. Another common approach is to counter balance the film stress by deposition of another stressed film on the same or the backside of the substrate [3,8], which can be a difficult task due to reproducibility of the film parameters and geometric constraints [9]. In some cases the film stress can be reduced by tuning of the deposition parameters or annealing of the sample after deposition [2,3,9], but by this also other parameters of the film might be changed [2].
Due to these issues, alternative methods for figure correction of substrates have been developed [10–15], of which many are based on a locally varied compensating stress on the backside of the substrate. If the film exhibits a tensile stress, also a tensile stress on the backside of the substrate is needed for compensation. This can be accomplished by femtosecond laser focusing inside the substrate material [11] or by deposition of a stressed film on the backside, which is then structured by laser ablation [16] or lithography. As the former method is based on small stress centers inside the material, it is a slow process in the case of large areas. The latter method is based on at least a deposition process and a structuring process, although an annealing process in between for stabilization of the film stress might also be necessary. In case of lithography the structuring process is rather complicated involving many different steps [13].
It has been reported that metal foils and slides of fused silica can be remotely bent by irradiation with the ultraviolet light of excimer lasers [17,18]. Here we suggest a method based on short time melting of the backside surface of a borosilicate glass substrate with the ultraviolet light of an ArF excimer laser. By this a tensile stress can be generated at the surface. The high pulse energy and the large beam profile of the laser allow the processing of a large area with only a few laser pulses. Besides equibiaxial stress components, also antibiaxial stress components can be induced, which was shown to increase the scope of correctable deformations in figure correction methods [19]. The value of the generated surface stress can be controlled by the laser fluence or by the local density of the laser spots. By the latter approach we successfully compensate for the deformation generated by a deposited chromium film.
2. Experimental
Our samples were $\sim 210\,\mu \textrm {m}$ thick quadratic cover glasses of a borosilicate glass (Schott D263M) with a lateral size of $22\,\textrm {mm}$, which have been cleaned in acetone, deionised water and isopropyl alcohol in an ultrasonic bath before use. Some of the samples have been etched in a solution of $5\, \textrm{wt}\%$ of potassium hydroxide in deionised water for $100\,\textrm {min}$ in an ultrasonic bath at a temperature of $\sim 80\,^\circ \textrm {C}$.
The laser source was an ArF excimer laser (Lambda Physik, LPX Pro) with a wavelength of $193\,\textrm {nm}$ and pulse duration of $\sim 20\,\textrm {ns}$. The optical setup (Fig. 1(a)) consisted of a mask with a quadratic aperture or a line pattern, which was illuminated by the laser light and projected onto the sample surface by virtue of a plano-convex lens of fused silica with a focal length of $100\,\textrm {mm}$. The image of the mask was demagnified by a factor of 10, sharply bounded and had by experience a rather homogeneous fluence profile. For each sample one of the two sides was irradiated with a pattern of the laser spots by automated movement of the sample inside the image plane. For analysis of the induced surface stress a homogeneous pattern, as shown in Fig. 1(b), was used. And among the samples the fluence and the number of pulses have been varied. For figure correction the local density of the laser spots was varied. The fluence was calculated by dividing the measured pulse energy by the area of the laser spot. In case of the line pattern a duty cycle of 0.5 was assumed for fluence calculation.
Fig. 1. (a) Schematic of the optical setup used for irradiation of the samples. A mask is imaged onto the sample surface with a demagnification of 10. (b) For analysis of the induced surface stress by irradiation, the quadratic laser spot with edge length $a$ was scanned over the whole sample area with a period of $p$ in both directions. (c) Schematic sketch of a sample during measurement of the height profiles. The measurement traces are shown as solid lines on the sample surface. For measurement the sample was positioned onto three steel balls
For determination of the curvature change of the samples by irradiation or etching, two height profiles of the non irradiated surface have been measured before and afterwards by tactile profilometry (Bruker Dektak XT-A) with a tip radius of $2\,\mu \textrm {m}$ and a contact force of $30$ or $40\,\mu \textrm {N}$. For the measurement the samples were reproducibly positioned onto three steel balls (Fig. 1(c)). The two measurement traces were oriented orthogonally two each other, parallel to two edges of the sample and centered on the surface. Three dimensional height maps have been achieved by stitching of parallel height profiles. For each height map we measured 21 parallel height profiles, rotated the sample $90^\circ$ around the axis orthogonal to the surface and measured another 21 parallel profiles. As the relative height difference between parallel profiles cannot be conserved, for each set of parallel profiles we used one profile from the orthogonal set to correct the height. Therefore we achieved two height maps for every processing step. These two height maps have been compressed to 21x21 pixels each and combined by taking the arithmetic mean. The tactile profilometer was also used for analysis of the irradiated surface.
For figure correction a few samples haven been coated by a thin film of chromium with a thickness of $\sim 90\,\textrm {nm}$ by electron beam evaporation of chromium pellets at a base pressure of $\sim 5\cdot 10^{-6}\,\textrm {mbar}$. Before and after deposition and after irradiation height maps of the coated surface have been measured.
3. Results and discussion
3.1 Example for deformation due to irradiation
Figure 2 shows an image of the surface of a sample after irradiation with quadratic spots with $a=200\,\mu \textrm {m}$ and $p=220\,\mu \textrm {m}$ (see Fig. 1(b)) at a fluence of $2.0(1)\,\textrm {J/cm}^2$ and one pulse per position.
Fig. 2. A microscope image of the irradiated surface of a sample for irradiation with one pulse per position and a fluence of $2.0(1)\,\textrm {J/cm}^2$. The edge length of the quadratic laser spot is $a=200\,\mu \textrm {m}$ and the period of the pattern is $p=220\,\mu \textrm {m}$. The image has been taken with differential interference contrast.
Figure 3 shows an example for a sample irradiated with the same pattern but with a fluence of $360(30)\,\textrm {mJ/cm}^2$ and one pulse per position. In Fig. 3(a) the height profiles of the pristine sample, after irradiation and after etching in KOH solution are shown. The concave profile of the pristine sample is flattened by irradiation. But after etching the profile of the pristine sample is retrieved. The profiles haven been smoothed by a gaussian filter and the curvature profiles have been calculated. By subtraction of the curvature profile of the pristine sample, the profiles of curvature change (Fig. 3(b)) are obtained. It can be seen that irradiation leads to a homogeneous curvature change, which is characterized by the average value in the region between $2.5\,\textrm {mm}$ and $17.5\,\textrm {mm}$. The same procedure has been repeated for the profiles in orthogonal measurement direction and the average curvature change of both directions is $\bar {k}=-0.259(1)\,\textrm {m}^{-1}$ after irradiation and $\bar {k}=-0.0003(4)\,\textrm {m}^{-1}$ after etching.
Fig. 3. The results of the profile measurement in one direction for a sample irradiated with quadratic spots at a fluence of $360(30)\,\textrm {mJ/cm}^2$ and 1 pulse per position. (a) The leveled height profiles of the pristine sample, the sample after irradiation and after a subsequent etch step in KOH solution. Please note that the first and the latter profiles lie on top of each other. (b) The profiles of curvature change with respect to the curvature of the pristine sample for the same processing steps. The diverging values at the boundarys of the curvature profile are a result of smoothing of the height profiles before curvature calculation.
3.2 Integrated stress in dependence of fluence and number of pulses
The average curvature change $\bar {k}$ per sample, calculated as explained in section 3.1, has been used to calculate the stress $N$ integrated over the thickness of the stressed surface layer by the Stoney Eq. (1)
Figure 4 shows the integrated stress $N$ plotted against the laser fluence for irradiation with one pulse per position, a spot size of $a=200\,\mu \textrm {m}$ and a period of the irradiation pattern of $p=220\,\mu \textrm {m}$. There is a step-like dependence on the fluence (blue dots). For fluences below about $100\,\textrm {mJ/cm}^2$ no curvature change and hence no integrated stress can be observed. Then there is a transition region between about $100$ and $500\,\textrm {mJ/cm}^2$, inside which the integrated stress rises to a value of about $250\,\textrm {N/m}$, which corresponds to a tensile stress in the irradiated surface. Above $500\,\textrm {mJ/cm}^2$ the integrated stress stays roughly at this position up to about $2\,\textrm {J/cm}^2$, which is the highest fluence we used. As is shown for four of the irradiated samples (orange squares), the above mentioned etching process in KOH solution reduces the integrated stress to zero. Thus the height profiles relax to the form before irradiation, as shown in Fig. 3(a). If the same etching process is applied to the samples prior to irradiation (green triangles), the integrated stress follows a similiar dependence on the fluence as without etching.
Fig. 4. The measured integrated stress in dependence of the average fluence of the laser spot. Four of the samples haven been etched in a KOH solution after irradiation. By this the effective integrated stress has been removed. Another six samples have been etched before irradiation. They still follow the general trend after they have been irradiated.
Our interpretation of this behavior is based on the one by Konovalov and Herman [18]: By irradiation with a fluence above $100\,\textrm {mJ/cm}^2$ the absorbed energy of the laser light leads to melting of a thin surface layer and by resolidification this layer develops a tensile stress due to thermal contraction. With higher fluences the thickness of the molten layer increases, and thus a higher integrated stress develops. The thickness of the molten layer is limited by removal of material by ablation, so that the integrated stress saturates above a certain fluence value. By the etching process the stressed material layer is removed, while etching before irradiation should not effect the induced stress in the surface layer.
This interpretation is supported by the measured height profiles on the irradiated side of the sample (Fig. 5). For the sample in Fig. 4 irradiated with $360(30)\,\textrm {mJ/cm}^2$ and all samples irradiated with a higher fluence, the height of the irradiated areas decreased with respect to the non irradiated areas (Fig. 5(b)). But for the samples irradiated between $150(10)\,\textrm {mJ/cm}^2$ and $250(20)\,\textrm {mJ/cm}^2$ we observed an increase in surface height due to irradiation (Fig. 5(a)). We therefore assume that in the latter case only melting of the surface without significant ablation took place, while in the former case the surface height was reduced by material removal due to ablation. Thus the ablation threshold fluence lies inside the transition region in Fig. 4 and the onset of surface melting lies roughly at the same fluence at which a curvature change can be detected.
Fig. 5. Height profiles of the irradiated side of two samples. Both samples have been irradiated with quadratic spots with an edge length of $a=200\,\mu \textrm {m}$, which were positioned on a grid with a period of $p=220\,\mu \textrm {m}$. On each position only one laser pulse was applied. They differ in the average fluence: (a) $250(20)\,\textrm {mJ/cm}^2$ and (b) $360(30)\,\textrm {mJ/cm}^2$. As can be seen by comparison of the irradiated and non irradiated regions, in (a) the surface height has been increased by irradiation while in (b) it decreased. The rough surface in the non irradiated areas in (b) is probably due to debris produced by the ablation process.
Our interpretation is further supported by the observation that there is no pronounced trend of the integrated stress in dependence of the number of pulses (not shown). This is understandable, as each subsequent pulse induces the same thickness of molten material.
3.3 Method for figure correction
We suggest to use these results for figure correction of substrates made out of borosilicate glass by irradiation of the backside of the substrate. As a proof of principle, we compensated the deformation induced by a film of $\sim 90\,\textrm {nm}$ chromium by irradiation of the non coated side of the sample. Figure 6(a) shows the difference in surface height between the coated and pristine surface. The tensile stress of the chromium coating leads to a bowl shaped deformation. The rms value is $3.76\,\mu \textrm {m}$ and the peak-to-valley deformation is $16.1\,\mu \textrm {m}$.
Fig. 6. As a proof of principle the deformation of a sample due to $90\,\textrm {nm}$ of Cr has been corrected. (a) The height difference before and after the deposition. (b) The calculated average curvature change of the sample due to the coating. (c) The height difference before deposition and after figure correction. Please note that the vertical scale has been changed with regard to (a). (d) The change of curvature induced by the figure correction. For better comparison the sign of the curvature has been changed to match the sign in (b)
As the height map is basically build out of a web of height profiles (s. Sect. 2) we calculated the curvature change along those lines and for every knot of the web the arithmetic mean of the curvature change in the two orthogonal directions. This is shown in Fig. 6(b). There are two areas where the curvature change is much smaller than in the rest of the sample. These areas were intentionally covered during deposition by the clamps holding the sample.
We used the Stoney Eq. (2) locally to calculate the integrated stress necessary to compensate for the film stress. Although this approach is a rough approximation that neglects any non-local effects [19], it suffices for our purpose of proof of principle. The backside of the sample has been irradiated with a fixed fluence of $600\,\textrm {mJ/cm}^2$ and one pulse per irradiated laser spot, and the local integrated stress has been adjusted by the local density of the $a=200\,\mu \textrm {m}$ large laser spots, which is the number of equally sized and non overlapping spots per $\textrm {mm}^2$. The local spot density corresponds to the factor $A_{\textrm {tot}}/A_{\textrm {irr}}$ in Eq. (2). This is illustrated in Fig. 7, where a small area of the irradiated surface is shown, which consists of irradiated and non irradiated areas.
Fig. 7. A microscope image of the irradiated surface of a sample after figure correction. The integrated stress has been adjusted by local variation of the density of the quadratic laser spot with edge length $a=200\,\mu \textrm {m}$. Therefore irradiated squares and non irradiated squares can be seen. In the irradiated squares the fluence was $600\,\textrm {mJ/cm}^2$ at one laser pulse. The image has been taken with differential interference contrast.
The deformation of the sample after irradiation is shown in Fig. 6(c). The rms value has been improved by a factor of 14 to $0.26\,\mu \textrm {m}$ and the peak-to-valley deformation has been improved by a factor of 10 to $1.6\,\mu \textrm {m}$. Figure 6(d) shows the average curvature change (inverse sign for better comparison) due to the irradiation. By comparison with Fig. 6(b) it can seen, that the average curvature change by the coating has been reproduced quite well. A small deviation can be observed in the regions where no film was deposited, because also in this regions a small integrated stress has been generated by irradiation. A non-local model for calculation of the irradiation pattern might have been able to avoid this.
3.4 Stability of the curvature change at ambient conditions
We measured the stability of the curvature change for storage at ambient conditions. The results are shown in Fig. 8, where the normalized curvature change is plotted against the storage time at room temperature and ambient atmosphere for two different fluences. It can be seen that the normalized curvature change decreases with time. This decrease is significant faster for a fluence of $250\,\textrm {mJ/cm}^2$ compared to $1\,\textrm {J/cm}^2$. One possible explanation is that the surface stress relaxes, but further studies are necessary. However, we note that the decrease in curvature change decelerates with time and that the decrease is slower at higher fluences. Therefore the issue of stability might be solved by use of a high laser fluence and stabilization of the surface stress by thermal treatment after the irradiation.
Fig. 8. The measured curvature change with respect to the pristine sample normalized by the value measured directly after irradiation plotted against the storage time under ambient conditions. The samples have been irradiated with one laser pulse per position at $250\,\textrm {mJ/cm}^2$ and at $1\,\textrm {J/cm}^2$, respectively. The errorbars are displayed ten times bigger than the real value for better visibility.
3.5 Anisotropic curvature change by structured irradiation
In figure correction it is highly desirable to be able to induce not only equibiaxial surface stresses, but also antibiaxial stress components, as this increases the range of correctable deformations [19]. As an approach to achieve antibiaxial stress components we changed the mask in the optical setup to a line pattern with a period of $80\,\mu \textrm {m}$ and a duty factor of 0.5, so that the image on the sample surface is a line pattern with a period of $\sim 8\,\mu \textrm {m}$. We found that irradiation of the whole sample surface with this laser spot with $a \sim 500\,\mu \textrm {m}$ and $p=520\,\mu \textrm {m}$ at high fluences and many pulses per position leads to an anisotropic curvature change. An example is shown in Fig. 9 where the profiles of curvature change orthogonal and parallel to the lines are plotted in Fig. 9(b). Both are constant in the central $15\,\textrm {mm}$ of the sample surface. It can be seen that the curvature change is highly anisotropic with a ratio of $k_s/k_p = 0.48$ of the average curvature $k_s$ and $k_p$ orthogonal and parallel to the lines, respectively. Therefore, we conclude that an antibiaxial stress component has been induced into the sample surface.
Fig. 9. (a) Differential interference contrast image of the surface of a sample after irradiation with a pattern of spots with a line substructure. The edge length of the spots and the period of the pattern were $a \sim 500\,\mu \textrm {m}$ and $p=520\,\mu \textrm {m}$, respectively. At each position 20 pulses of an estimated average fluence in the trenches of $1.3\,\textrm {J/cm}^2$ have been used. (b) Profiles of curvature change with respect to the pristine sample after irradiation with the pattern shown in (a). The profiles have been measured orthogonal and parallel to the line orientation through the center point of the sample surface.
The mechanism behind the antibiaxial stress component is still unclear. In Fig. 10(a) the ratio $k_s/k_p$ of the curvature changes is shown as a function of the number of pulses per position for four different estimated fluences (duty cycle of 0.5 assumed for calculation). This data implies that the ratio $k_s/k_p$ decreases with increasing number of pulses and increasing fluence.
Fig. 10. (a) The ratio $k_s/k_p$ of the curvature change $k_s$ and $k_p$ orthogonal and parallel to the line pattern, respectively, plotted against the number of pulses per position for seven different samples at four different fluences. For better visibility the errorbars are displayed ten times bigger than the real value. (b) The average of the integrated stresses in both directions for the same samples as in (a) plotted against the estimated fluence on the lines. For calculation of the integrated stress a homogeneous fluence profile without a line pattern has been assumed.
Figure 11 shows microscopic images of the line structures for three samples irradiated at $0.9\,\textrm {J/cm}^2$ or $1.3\,\textrm {J/cm}^2$ and 10 or 20 pulses. The trenches and ridges can be well distinguished. We measured the trench width at ten different positions and calculated the average for each of the samples. These values are also given in the figure. For these samples the trench width lies between $3.8(3)\,\mu \textrm {m}$ and $4.6(1)\,\mu \textrm {m}$. A slight increase of the trench width with increasing fluence and pulse number can be noticed.
Fig. 11. Microscope brightfield images of the line structure on the surface of three samples after irradiation. For each sample the estimated average fluence in the trenches, the number of pulses per position, the curvature ratio $k_s/k_p$ and the width of the trenches is given. The latter is the average of ten measured values. The measurement bar illustrates the principle of measurement of the trench width.
It has been reported that in the case of a prestressed and structured film the anisotropic curvature change of the underlying substrate is controlled by the aspect ratio (height divided by width) of the line structure of the film [20]. As we expect the aspect ratio of our line structures to increase with increasing pulse number and fluence, the dependence of the ratio $k_s/k_p$ in Fig. 10(a) is in qualitative agreement with the observation in [20]. However, in [20] only the ridges of the film structure are prestressed. But in our case the fluence in the trenches of the line structure has been higher than on the ridges. Therefore, it is interesting to analyze if the laser affected zone is limited to the trenches of the line structure or extends significantly to the ridges as well.
Since the size of the laser affected zone influences the total stress [21], we calculated the average of the integrated stresses in the directions orthogonal and parallel to the lines by use of Eq. (2) and the average curvature change $\bar {k}=(k_s+k_p)/2$. For the ratio $A_{\mathrm {tot}}/A_{\mathrm {irr}}$ we assumed a homogeneous fluence distribution inside the laser spot. This is, we neglected the line substructure of the laser spot. These values of the integrated stress plotted against the estimated average fluence on the lines of the line pattern are shown in Fig. 10(b). The integrated stress increases with increasing fluence and increasing pulse number and saturates at a value of $\sim 240\,\textrm {N/m}$, which agrees well with the saturation value given in Fig. 4. This behavior allows for two alternative interpretations: Either the integrated stress in the trenches is for some reason double the upper limit given in Fig. 4 or the upper limit value of integrated stress given in Fig. 4 has been induced in the trenches and on the ridges. The latter interpretation seems to be more appropriate to us, since it is not in contradiction to the data shown in Fig. 4. Thus, for low ratios of $k_s/k_p$, or correspondingly for high pulse numbers and high fluences, the laser affected zone (area of solidified melt) is the whole area of the $a \sim 500\,\mu \textrm {m}$ sized laser spot and is not limited to the trenches of the line substructure of the spot. This is a plausible result, because resolidified molten material on the ridges of a line structure, formed by excimer laser ablation of a borosilicate glass surface, has been observed before [22].
If the above reasoning holds, there is still a major difference in between our experiments and the experiments in [20] in that in our case there is also a stress induced in the trenches of the lines. Thus, further analysis is necessary to verify the above reasoning and to understand the influence of differences between the lines studied in [20] and our line patterns.
4. Conclusion
We suggested a method for figure correction of substrates of borosilicate glass by excimer laser irradiation with a homogeneous beam profile. We showed that the induced integrated stress, which controls the curvature, can be adjusted by the laser fluence. Alternatively the integrated stress can be controlled by the local density of the irradiated area at a constant fluence. By use of the latter approach we corrected the deformation of a cover glass induced by a thin film of chromium. One advantage of our method is that it is based on a homogeneous beam profile. Therefore the calibration curve in Fig. 4 is independent of the area of the laser spot. Another advantage is, that due to the high pulse energy of the excimer laser a large area can in principle be processed with only one laser pulse. As an example, a fluence of $1\,\textrm {J/cm}^2$ on an area of $1\,\textrm {cm}^2$ can in principle be reached. We also showed that for more complex corrections antibiaxial stress components can be generated by irradiation with a line pattern. Unfortunately, the induced integrated stress is not stable over time. Therefore further investigation of the relaxation mechanism and process engineering is necessary in order to reach highest precision. A major drawback of the method is that only tensile surface stresses can be generated. This could be circumvented by a compressively stressed backside coating, which is ablated to control the integrated stress [14]. Thereby compressive and tensile integrated stress could be adjusted in one processing step.
Funding
European Regional Development Fund (ZW 6-85005827).
Acknowledgments
The authors would like to thank L. J. Richter for deposition of the chromium coatings.
Disclosures
The authors institution has filed a patent application that is related to the content of this publication.
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