Stress manipulated coating for fabricating lightweight X-ray telescope mirrors

1. Introduction

The next generation of major US X-ray missions has a default goal to get an effective area at least 10 times larger than the Chandra X-ray Observatory (CXO), a similar angular resolution (i.e., <1 arc-second), and lower cost and mass per effective area [1–5]. The effective area of the Chandra High Resolution X-ray Mirror Assembly (CHXMA) was, however, about 400 cm² [6]. The effective area per unit mass was about 0.5 cm²/kg. Thus, a boost of about 10 times in area requires so much costly smooth (< 0.5 nm) surface as to require a replication process to bring down the mirror fabrication cost. Furthermore, the low ratio of effective area to mass of the CHXMA translates into an unacceptably high launch cost, even if the fabrication costs for Chandra-like mirrors could be reduced significantly. Hence, thin and lightweight replicated optics are called for.

Therefore, since the inception of the Con-X and Gen-X concepts in the early 2000’s, the problem of how to increase the mirror area yet not increase the total mission cost above that of the CXO has been well known [7–13].

As a result, several post mirror replication technologies have been explored [14–20]. The concepts have all been based on the idea that because the replication process would not produce a high quality mirror figure, post corrections to the mirror figure will bring the mirror quality to within specifications of figure accuracy and mass/unit area.

In order to address the requirement of a 10x effective area of the CXO with CXO angular resolution, we have been developing new figure correction processes. We based our new concept on the observation that the optical reflection layers on an X-ray reflector distort the figure of the thin walled substrate due to stresses generated in the coating process [20]. Therefore, we undertook a set of experiments whereby we locally controlled the stress of the applied coating.

In this paper, we report the first results of putting into practice our idea of a spatially controlled stress coating applied by a DC magnetron sputtering system. In particular, we report on the repeatability and stability of our process.

2. Process

Our concept is to modify the stress during deposition by controlling a voltage bias between the sample holder (Fig.1(e)) and the mask (Fig. 1(c)). In this manner, we can produce figure correction with stress by synchronizing the bias voltage with the location of a translation stage. Our process of changing the bias while applying the coating on the back side of the mirror is different from the published work by Chan et al. (2012). They used a monolithic multilayer deposition over the entire test mirror. As noted by these authors, “deposition of such bi-layer on full-size mirror did not yield undistorted mirrors yet.”[20]

 

Fig. 1 Schematic of the modified DC magnetron deposition system with a glass sample (D). A mask with a slit (C) is fixed under the sputtering source (A) to restrict the flow of coating material. The sample holder (E) is put on a translation stage (B) which is mounted below the mask.

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3. Process design

We show in Fig. 1 the design of the coating setup with a 2 inch diameter sputter Cr source (Fig. 1(a)), produced by AJA Co. We selected Cr as the coating material firstly because it is low cost and common in the lab. Additionally, Cr coatings can produce a wide range of stresses from tensile to compressive with the variation of negative bias voltage [21], and also the adhesive force between the Cr and the substrate is good as an ideal binding layer [20]. The translation stage used (Fig. 1(b)) was manufactured by Zaber (T-LSM050-SV). The stage has a motion range of 50 mm and an adjustable motion speed as high as 12 mm/sec.

In the experiments we report here, we used a combination of a 4 mm width slit on the mask (Fig. 1(c)), and fixed the motion speed to 0.022 mm/sec, i.e., a travel time over the 4 mm slit width of 3 min. We set the target distance at 90 mm, and the mask (Fig. 1(c)) to sample (Fig. 1(d)) gap to 2 mm. Thus, with the small mask-substrate gap, the coating area was well defined by the slit location.

Further, we connected the sample holder (Fig. 1(e)) with the bias power supply to modulate the bias manually between 0 V and −100 V. The sample used (Fig. 1(d)) was a 20 mm x 5 mm x 0.2 mm glass strip (D263), which we judged appropriate for proof of concept. During the coating process, we applied a constant power of 100 W to the sputter target, while we set the pressure of Argon gas to 3 ± 0.1 mTorr. With these settings, we applied a non-uniform stress Cr layer 100 nm, uniform in thickness, to create a desired local stress distribution.

4. Repeatability test

4.1 The metrology

Before advancing to making improvements to the figure, we judged it was necessary to demonstrate the repeatability of our process by coating five samples. Therefore, we made easily measureable shape deformations.

In order to describe the shape analytically, we approximated both the original and the final shape by a spherical profile. In this case, then the difference of curvature (hereafter, DOC) follows Stoney’s Equation [22]:

For X-ray optics, our interest is the slope change of the figure, as the deviation of the X-rays is affected by the reflecting grazing angle. Therefore, we convert the DOC to an angle Δθ by Eq. (2). Δθ is the slope change. As defined in Eq. (1), R_raw and R_coated are the radii of curvature before and after coating, respectively. Δh is the displacement corresponding to the slope change, and l is the length over which Δθ occurs. For an l = 20 mm long piece, the slope change of angle 0.0004 radians corresponds to a DOC of 0.08/m.

Note that our current work was focused on improving the 1-D surface profile along the length of the sample because the slope error parallel to the optical axis dominates the point spread function of an X-ray mirror. Thus, the 1-D measurements reported here were made with Veeco Dektak 150 along the centerline of the length of our glass strips.

4.2 Details of the setup

The primary purpose of this work is to report on a new method of adjusting a 1-D surface profile. However for completeness, we report here that a critical part of the sputtering system was the Al tape shown in Fig. 2. Although the required electrical connection may be obvious to some, it may not be obvious to all. To make this necessary electrical connection, we used 0.1 mm thick Al tape to overlap the sample at one side in order to establish electrical conductivity. At the same time, we made the Al tape narrow enough (1 mm) so as not to introduce a measurable shape change to the sample. We put the glass strip on a 50 mm x 50 mm x 2 mm Al plate outside of the coating chamber. After we bonded the sample to the Al plate with the tape, we placed the combination onto a round sample holder. This design allowed us to apply a bias voltage to the surface of the sample through the wire shown in Fig. 2.

 

Fig. 2 Setup of the glass sample on the sample holder.

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In order to demonstrate that the Al tape plays a crucial role in this process, we coated 5 samples with and 5 samples without the Al tape. We found that the variation of the DOC for the two groups of samples was quite different (Fig. 3) for the two setups shown in Fig. 4. For the Al taped samples, the variation of the DOC is less than 0.6% (red line), whereas without Al we found variations of over 200% (green line). We attribute the difference of the DOC between the two cases to the Al tape providing the necessary additional electrical conductivity.

 

Fig. 3 Comparison of the repeatability test results between different sample setups. Red: result of the samples with Al tape overlap. Green: result without Al tape overlap.

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Fig. 4 Transparencies of the coated samples under light. (A) sample without overlapping when coated. A blue arrow in image (A) marks a white stripe, see text. (B) sample with Al tape as part of the electrical connection.

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To understand why the Al tape connection made a significant difference, we took advantage of the coating being semi-transparent. We placed the samples coated with and without the Al connection under light to observe the coating thickness. Figure 4(a) shows an obvious white stripe where as in Fig. 4(b), no such stripe is visible. From the brightness variations in Fig. 4(a), we estimate that the transmittance varies roughly from 5% to 35% along the length of the sample. We explain this effect as being due to static charge buildup when there was no Al tape connection. The connection between two metal conductors – coating Cr and sample holder – could attenuate the space charge near the sample surface. Thus, the observable white stripe bleaching in coating might result from the mask window, and the bias at the surface deviated from the setting on the bias power supply.

The result of coating without the Al tape results in a pattern similar to the one shown in Fig. 4(a), and Fig. 3 shows the relatively large variations in the DOC without Al tape (green line) versus that with Al tape (red line). Since the measured length of these samples was 13 mm, a Δh of 0.1 μm corresponds to 0.1E-6/13E-3 = 7E-6 radians or about 1.6 arc-seconds. This value of 1.6 arc-seconds is close to our target of 1 arc-second. Overall, we estimate that our procedure with the Al tape reduces unwanted deformations on our glass samples by a factor of over 160.

5. Stability test

As our target is a stable figure, after carrying out a shape modification, we searched for changes in shape over time. Therefore, we have begun monitoring the shape of our test samples by continuously measuring the DOCs shown in Fig. 3 (Red line). We show the current results in Fig. 5.

 

Fig. 5 Evolution of five samples’ differences of curvatures (DOCs) over 14 weeks. See text for details. Curvatures before (1/R_raw) and after (1/R_coated) coating are measured by Veeco Dektak 150 stylus profiler. Left: shape evaluation of each sample. Right: the average and the standard deviations of 5 samples at various time periods.

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We created stability tests whereby first, the five samples coated via Al tape were annealed (70 °C, 5 hrs) within hours of the initial coating to accelerate the stress relaxation, similar to what was reported in Chalifoux et al. (2013) [14]. Next, we kept the samples under vacuum (3 mTorr) to prevent oxidization, and then monitored the DOCs of the 5 separate samples as shown in Fig. 5 (Left). We exposed each sample to air for only about 1 hr per measurement. It is not clear how much of an effect air exposure has on our coating and in the future we will explore a method of protecting the coating to eliminate air exposure.

Before and after annealing, we found a measurable variation in the DOCs of the individual samples. We hypothesize that the variations may be due to the temperature gradient of our annealing system. Thus, in the future we will develop a more sophisticated annealing system to improve the uniformity of temperature. As a comparison, we will also monitor the stability of samples without annealing.

With the annealing system we have in place, we monitored the DOCs over the course of 14 weeks. We found that the DOCs of the individual samples changed within a small range of ≤2.5%.

In order to visualize this small effect, we have plotted on an expanded scale, the average value of the DOC versus time in Fig. 5 (Right). Here, the error bars denote the variation in the values of the DOCs of the 5 samples. From the data in Fig. 5, we conclude that the stress modified shape can be locked in for periods as long as 14 weeks after the deposition. Then, referring back to Section 4, the variation in the average DOC over time shown in Fig. 5 corresponds to a Δh of about 0.004 mm/m or 4E-6 radians or about 1 arc-second. Also as noted above, this value is close to the goal of maintaining a slope stability of about 1 arc-second (4.85E-6 radians).

6. Calibration process

Having shown that the technique has promise, we now discuss how to use voltage bias to bring about a specific controlled shape change. We therefore carried out a series of tests on 7 more samples, and coated these samples with the Al tape setup described in Section 4.2. We made the coating as uniform as possible (to better than 2% based on our transparency measurements) coated with biases of −42 V, −46 V, −50 V, −54 V, −58 V, −62.5 V and −100 V. The resulting DOCs are shown in Fig. 6.

 

Fig. 6 Samples’ differences of curvatures (DOCs) versus the bias voltages.

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There it can be seen that with increasing negative bias, the DOC varies from negative to positive, which means the stress can be manipulated from tensile to compressive. We can combine the dependence of DOC on bias shown in Fig. 6 with the repeatability tests we have done. From this work, we conclude that in 1-D we can bring about the desired and stable shape changes on 200 μm thick Schott D263 glass.

7. Preliminary results of stress manipulation

In our proof of concept, we produced a predictable 1-D shape change to compare with our data used as input for a finite element analysis (FEA). We remind the reader that the goal of this test was not to correct a 1-D figure, but rather to demonstrate that with initial metrology of a figure, we can design a coating process to correct the figure.

Then, in our simple test, for the first half of the coating we set the bias to −42 V, which produced a DOC of about −0.071/m. In the second half, we manually switched the bias to −58 V to produce a DOC of about 0.07/m. Figure 7 shows the height converted to a sinusoidal profile. The purple curve is the initial surface profile.

 

Fig. 7 Measured surface profile before coating (Purple line), after coating (Cyan line) and FEA simulation (Orange line).

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We used our results displayed in Fig. 6 as the data for an FEA model built in COMSOL Multiphysics to calculate a theoretical smooth surface profile. The FEA model used in COMSOL was set up as a 0.2 mm flat glass substrate with a 100 nm Cr coating on the top. Then we divided the chromium layer into two regions for different stress setups. Considering the raw glass substrate has a default convex curvature of 69 m as the purple line shows in Fig. 7, we calculated the corresponding initial compressive stress which is 88.9 MPa by Eq. (1). In addition, considering that the produced DOC of 0.07/m corresponds to a stress of 430 MPa, we determined that the stress on the left side of the sample was set to be a tensile stress of 341.1 MPa (=430 MPa – 88.9 MPa). On the right side, the stress was set as a compressive one of 518.9 MPa (=430 MPa + 88.9 MPa).

The orange line in Fig. 7 is the FEA result, which is consistent with the measured one. This is a promising first step in carrying out more complex shape changes to correct rather than to simply modify a shape.

8. Numerical optimization of the stress distribution and the fabrication result

Although Stoney’s equation can predict the stress in coating on a spherical surface by means of the difference of curvature (DOC), it is not applicable for aspherical surfaces. In fact, typical mirror shells for X-ray telescopes are designed to be parabolic and hyperbolic. Thus, we developed a numerical optimization process to derive a desired stress distribution in coating which could reshape the mirrors to the targeted aspherical profile. Since the attempt to correct the shape via stress control in coating is a new approach, only the long dimension of the mirror is being corrected in this paper. The extension of this work to the 2-D situation will be of future work.

Our idea is to replace the continuous stress distribution with finely spaced discrete steps. As is shown in Fig. 8, we assume the stress within each region is uniform to correct the local shape. The combination of these discrete stresses will lead to a designed global change of the surface due to the compliance and continuity of the surface. Therefore to carry out the calculation we used discrete stresses as parameters that are adjusted by an algorithm to produce the desired result.

 

Fig. 8 Sketch of the stress discretization. Note that the stressed layer is coated on the back side (non-reflection) of the mirror.

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Note that the discrete number N is determined by the frequency of the waviness of the initial mirror sample. In this paper, N was fixed at 10 because the current waviness of our sample was of such a low frequency that 10 distinguished regions were enough to compensate the shape distortion.

The diagram of the optimization code is shown in Fig. 9.

 

Fig. 9 Diagram of optimization process to calculate the desired stress distribution. The initial parameters are produced by a theoretical model. Then, they are input into an FEA model which could calculate the current surface profile and output the difference from a desired one by means of a merit function (MF). The FEA model is connected with a numerical optimization software (ISIGHT) so that an iteration loop is established to minimize the MF by optimizing the parameters

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As with nearly all fitting procedures, convergence requires that the initial guess not be too far away from a best fit. In order to have a good starting point for the best fit, we derived a model of required stresses that was based on the geometry, the material and the initial profile of a mirror. Here we found it is sufficient to assume a uniform stress as the initial conditions. Then, we calculated the initial parameters for our fit based on the difference of curvature (DOC) between the measured initial surface and the target profile.

In Fig. 9, we show the results of the iterations based on computations using a combination of ABAQUS^{(FEA software) and ISIGHT(Optimization software). The commercial finite element code ABAQUS was selected due to its seamless integration with the commercial optimization software ISIGHT, needed for an iterative process for our process. We calculated surface profile compared with the target by means of a merit function (MF), which is defined as follows:}

is the discrete location, and h_current(x_i) is the calculated height while h_target(x_i) stands for the target profile. M is the total number of the discrete locations on the surface, and MF means the standard deviation between the calculated profile and the target. In order to determine the optimum of spatially distributed stresses, an ISIGHT algorithm was used to find the optimum of the parameters by minimizing the MF. In particular, a present Hook-Jeeves algorithm was used due to its efficiency.

As a proof of concept, we carried out the optimization on a physical sample, which was the glass substrate with an intrinsic curvature. Our goal was to make it flatter. Assuming the effect of stress on part curvature is reversible, our approach is to start with an FEA model with a flat surface, then optimize the stress distribution such that the deformable shape conforms to the initial measured curved surface. Then we reverse the stress sign to be used in the coating process to bring the initially curved surface to the flat one.

In Fig. 10(a) we show the initial surface in FEA as the gray line and the resulting surface profile after stress correction in FEA is shown in the red line. Note that the corrected surface (red line) matches with the measured original surface profile of the sample (blue line) extremely well. This demonstrates that the process was successful. Figure 10(b) shows the optimized stress distribution.

 

Fig. 10 An example of the surface correction on a glass strip. (A) the optimized surface profile (red line) matches the measured profile. (B) the optimized stress distribution. In this case, the stresses are compressive (C) the evolution of the merit function versus the iteration steps. (D) the profile after coating (red line) versus that before coating (blue line).

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Figure 10(c) illustrates the evolution of the merit function during the iteration process. Figure 10D shows real fabrication results which are measured surfaces before coating (blue) and after coating (red). The difference of the coated surface from the flat one is about 0.1 μm, which is consistent with the result shown in Fig. 7. The measured result demonstrates the success of the process, i.e., with the combination of the optimization and the stress manipulated coating we can improve the surface of a glass strip by a factor of 10.

9. Future applications on a full-size mirror

Up to now in this paper we have discussed the 1-D results based on coating relatively small strips with 0.2 mm thickness. We now extend our work to demonstrate the feasibility of applying the stress control method to full-sized X-ray telescope mirror segments by using data from a prototype X-ray telescope mirror shell. In Fig. 11 we show a residual error map of an X-ray telescope mirror segment measured by the INAF Astronomical Observatory of Brera, Italy [23]. The dimension of the slumped glass (D263T) mirror is 200 mm x 200 mm, 0.4 mm thick.

 

Fig. 11 Surface height error map of an X-ray telescope mirror. Data provided by INAF Astronomical Observatory of Brera. Modeling was done along a 1-D profile in the direction of the red arrow.

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In the above figure, the maximum residual error is about 60 μm relative to the curvature along the X axis. However, the error along the incident direction is the most important for image quality, and thus we carried out modeling along a 1-D profile in the Y axis direction at the center of the mirror (red arrow in Fig. 11). We modeled a stress manipulated coating on the back (non-reflecting) side of the INAF mirror shell to correct the 1-D profile. For completeness, we modeled mirrors with 0.2 mm, and 0.8 mm thickness as well as the actual INAF mirror thickness of 0.4 mm.

As we show in Fig. 12(a), the optimized surface profiles with the different thicknesses are similar. However, the amplitudes of the optimized stresses are different. When the thickness is 0.2 mm, the average stress to be added in the coating is within Mega Pascal level, which is one order smaller than that in Fig. 10(b). In section 4, we demonstrated repeatability with a precision in the DOC of 0.00053/m, which corresponds to ~3 MPa. Therefore, the average optimized amplitude of the stresses in Fig. 12(b) is close to the repeatability precision achievable on a 0.2 mm thickness mirror. However, the easy solution is to use a thicker mirror shell.

 

Fig. 12 Optimized stress distributions and corresponding surface profile. Note that the negative stress is corresponding to tensile.

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As a comparison, mirrors with the thickness of 0.4 mm (Fig. 12(c)) and 0.8 mm (Fig. 12(d)) require much higher stresses to balance the surface profile. Especially in case of the 0.8 mm thick mirror, the average amplitude to be added in the coating is over 150 MPa, which is still quite achievable from the current coating system. However, a 0.4 mm thick mirror could also be selected for future X-ray telescope missions. In this case, our current process is slightly sensitive. The strategy to reduce the sensitivity of the method described here is to reduce the thickness of the coating or use a different coating material.

In addition to the sensitivity, the problem of how to extend the process to produce 2-D correction is important. In future work, we plan to use a 2-D translation stage which should address this issue.

As for the stress calculations, we plan in the future to use discrete stress distribution by applying a mesh to the 2-D surface with an optimization strategy similar to what has been described in this work. However, the deformation in 2-D is more complicated as we demonstrated in Fig. 13. Since the interferometer’s field of view is small (1.4 mm by 1.0 mm), for simplicity in this demonstration, we only measured the region at the center.

 

Fig. 13 2-D scan on a sample before (A) and after (B) coating. 1-D scan derived from the cross hair section of the sample (C) horizontally and (D) vertically. Note: The coating stress is uniform.

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After coating, the DOC along the horizontal (x) line (long side direction of the strip) is 0.0667/m, which is different from the vertical (y) line of 0.0535/m. This could be due to the anisotropy of the stress [24] or the asymmetric geometry of the substrate [25]. Based on the results, we suggest that our initial guess for a best fit optimization will require a more complicated model than the uniform stress we used as an initial guess for our 1-D stress optimization calculations.

10. Conclusions

In summary, we described a stress manipulated coating process on the non-reflection side to reshape the surface profile for lightweight X-ray telescope mirrors. For the coating process itself, we found that the use of a conductive tape was necessary. We performed a stress-stability test by monitoring the DOCs of five coated samples for 14 weeks, which is equivalent to slope stabilities of about 1 arc-second. Furthermore, we have shown that it is possible to calibrate the process and to carry out a controlled shape change. We therefore conclude the process described has performed well enough to warrant further investigation for use in the fabrication of affordable lightweight optics for future 1 arc-second resolution X-ray observatories.

In addition, we have developed an optimization strategy to calculate the stress distributions for a target profile. We have proved that the optimization method is efficient. Currently, the combination of the optimization strategy and the coating process could improve the mirror surface by at least a factor of 10.

In future work we plan to improve the stability, if necessary, since at this point the apparent small shape changes may be due to measurement errors. However, annealing, what coating materials we use, and what substrates we use are all things that could also be modified to improve stability.

We further plan to extend our process to producing shape modification of the type desired for X-ray optics, i.e., reducing the slope errors of a 2-D sample.

Acknowledgments

The authors appreciate William W. Zhang in NASA Goddard Space Flight Center (GSFC) for providing us glass substrates and the translation stage. The authors also thank Giovanni Pareschi and Marta Civitani at INAF Astronomical Observatory of Brera in Italy for providing the error map of mirror shell. This work was supported in part by NASA Grant No. NNX11AG05G-000005.

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