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We propose a method to control and to adjust in a closed-loop a bendable x-ray mirror using displacement-measuring devices. For this purpose, the usage of capacitive and interferometric sensors is investigated and compared. We installed the sensors in a bender setup and used them to continuously measure the position and shape of the mirror in the lab. The sensors are vacuum-compatible such that the same concept can also be applied in final conditions. The measurement is used to keep the calibration of the system and to create a closed-loop control compensating for external influences: in a demonstration measurement, using a 950 mm long bendable mirror, the mirror sagitta is kept stable inside a range of 10 nm Peak-To-Valley (P-V).
© 2016 Optical Society of America
1. Introduction
The European X-Ray Free Electron Laser facility (European XFEL), under construction in Hamburg, Germany [1], is raising the requirement level of accuracy and reproducibility for reflective x-ray optics. The long beamlines, up to almost one kilometer, are very sensitive to any variation of the mirror surface: small deviations from ideal flatness will create a degradation of the beam quality and an enlargement of the focus size. A certain change of the mirror radius of curvature is relatively more important in case of long distances, producing a displacement of the focus position or a corresponding change of the beam transversal size at a given point. For this reason, the specifications of the beam transport mirrors are very tight. The mirrors are 950 mm long, to collect the full flux of the beam in various working conditions, but the required surface quality is exceptionally high: 2 nm Peak-To-Valley (P-V) of maximum surface error and a radius of curvature larger than 5640 km, corresponding to a maximum of 20 nm of sagitta on the best-fitting cylinder over the full mirror length. Even for the most sophisticated deterministic polishing methods [2–5] such requirements are difficult to achieve today. One of the limiting factors is the metrology used to tune the polishing: there are continuous efforts to push forward the precision and reliability of the metrology [6–8], approaching the limits coming from the stability of the references and the materials [9]. In the case of the European XFEL, the measurement reproducibility is on the same order of magnitude as the requirements.
To have better control over the radius of curvature of the mirrors, it is possible to use a bender setup to curve the reflecting surface shape in a certain range. Important factors to reach the desired control precision are not only the possibility to bend the surface with enough resolution but also the presence of a reliable feedback signal to compensate backlash errors and thermal drifts. Such systems are quite common in astronomical science [10] but still not fully developed in x-ray photon science. A complete system with piezo actuators to shape the mirror and corresponding strain gauges to provide a feedback signal was developed at Lawrence Livermore National Laboratory (USA) [11], but the system is quite complex and expensive when scaled to a 1 m long mirror. A simpler system, using only piezo actuators and without feedback control, was developed and characterized on a 950 mm length [12]. Another proposed method was to control a bendable mirror having a direct measurement of the surface using a Fizeau interferometer [13,14], but it is a quite expensive solution with limitations in measuring long mirrors. More often, a Fizeau interferometer is used to align and calibrate such systems [15], but it is later removed from the setup and not used as a feedback signal for the bendable mirror control.
In this paper, we introduce a simpler method to control the radius of curvature of a bendable mirror using three displacement-measuring sensors and providing a measurement of the radius of curvature of the mirror in an easy and effective way. A Fizeau interferometer is used to initially calibrate the system and to check performance using an angled cavity setup [16]. We evaluated two different possibilities for the sensors using single point capacitive sensors [17] and interferometric sensors. Depending on the setup used, the two systems have advantages and disadvantages but both can be used in the setup. Their relative performance was evaluated and a closed-loop operation was demonstrated using a previously characterized mechanical bender [18]. We think that this setup is a step forward in the development of a stable and reliable bendable system. The feedback signal would be quite impossible to obtain from the beam itself in the case of the European XFEL because of the high power involved.
2. Mechanical bender description
The simplest system to induce pure cylindrical bending on an x-ray mirror is a mechanical U-bender. The basic idea of such a bender is to clamp each end of the mirror and induce a symmetric torque with a motorized actuator. The entire system is supported through mechanical flexures, allowing the structure to follow the bending of the mirror, while the flexures deformations remain in the elastic regime. The bendable system was developed by FMB Oxford (UK) and characterized in a previous paper [18]. The torque is produced by compressing a spring against a long bar placed in between two levers that are fixed to the mirror with clamps (Fig. 1). The system we used had a 950 mm long mirror installed.
Fig. 1 Schematic of the bender. The spring is compressed and identical reaction forces are transmitted to the levers. The mirror clamps are torqued and the mirror-reflecting surface is bent (details about the actuator mechanism are removed due to property rights).
The actuator compresses a spring, producing a couple of reaction forces that are applied to the two levers. The levers create a mechanical torque on the clamps and therefore on the mirror, creating a cylindrical bending on the reflecting surface. If we consider a mirror of length L, after the bending, we obtain a cylindrical shape with radius of curvature R and corresponding sagitta s, defined in the formula [18]:
The initial sagitta s0 is in generally different from zero, due to the bender preload forces and the mirror polishing errors. This system was measured with a Fizeau interferometer using an angled incidence setup in order to measure the full mirror in an absolute and precise way [16]. It was found that the thermal drift of the device caused additional bending: the sagitta was changed by 53 nm on average with a temperature variation of 0.1°C. This variation would exceed the European XFEL requirements.To keep the drift under control, we propose to install three sensors on the back of the mirror, joined together with a long aluminum bar. The bar needs to have a reference position for the sensors and is supported with weak flexures, so it can expand against temperature without bending as well. For the present test we placed the bar on the plate with a cloth in between. The sensors were able to provide a continuous measurement of the radius of curvature, and it was possible to compensate any unwanted bending caused by temperature drift with a closed loop operation. In the next section, we describe in detail the sensors operation and compare capacitive and optical interferometric sensors.
3. Measurement of bending through displacement sensors and comparison with the Fizeau interferometer
The shape induced by the bender on the mirror surface can be very well accounted for a best-fitting parabola, as long as the radius of curvature is very large compared to the mirror length. The maximum difference between the cylinder that is theoretically produced by the bender and the best parabola is given by the term . As an example, for a 1 m long mirror with a radius of curvature larger than 200 m, the difference is less than 1 nm. So the parabola can be used to account for the produced bending.
We used a second-order polynomial with three parameters to account for the best-fitting parabola and the residual tilt, and we placed three displacement sensors on the back of the mirror. Having more than three sensors would allow us to do a best-fitting evaluation of the parabolic shape, but it would also increase complexity especially when the system is installed in ultrahigh vacuum. Having less than three sensors could work in case of particular symmetries of the system as, for example, total absence of tilt. However, in general, this assumption is not valid and therefore has to be verified.
The setup we used is pictured in Fig. 2. We installed the mirror in the mechanical bender and aligned it in between two optical flats, with a 300 mm aperture Fizeau interferometer (Table 1) illuminating the mirror and collecting the two reflections back.
Fig. 2 Measurement setup to perform a calibration of the mechanical bender (top view). The Fizeau interferometer is used in an angled setup to measure the bendable mirror surface and its radius of curvature. The three displacement-measuring sensors are installed on the back of the mirror, one corresponding to the center and the other two on opposite sides.
Table 1. Fizeau Interferometer Specifications
The bender was placed on a kinematic mount equipped with step motors, so it could be precisely aligned inside the cavity. For every bending position, we used the Fizeau interferometer to acquire a 2D height map of the surface, with a correction factor because of the angled setup, and we extracted the central profile in the longitudinal direction of the mirror. We calculated the best-fitting parabola and the corresponding radius of curvature from the profile.
The three displacement-measuring sensors were placed on the back of the mirror and connected using an aluminum bar. One sensor was placed near the mirror center, the other two were placed close to the extremities. The distance between the sensors was chosen as a compromise: the sensitivity of the measurement is higher when the distance is maximized, but, at the same time, the sensors cannot be placed at the extremities where the clamps would interfere. In our case, the sensors were distributed over a central length of d = 750 mm. The displacements measured by the sensors are named , , and , respectively. The radius of curvature is calculated geometrically with the formula [17]
and the sagitta of the mirror is derived accordingly. The sensors were installed and set to zero, so that the initial position corresponded to the nominal zero sagitta. This effect created an artificial displacement of the calibration curve, which can be corrected through the comparison with the Fizeau measurements.3.1 Capacitive sensors
We used different types of capacitive sensors, from Physik Instrumente (PI) GmbH & Co. KG (Karlsruhe, Germany) and Micro-Epsilon Messtechnik GmbH & Co. KG (Ortenburg, Germany), with similar results. The sensors are high-resolution single-point sensors (see Table 2). They work as a parallel-plate capacitor: a conductive target is placed close to the sensor and any variation in distance is converted to a variation in measured capacitance. The sensors can work after proper calibration with targets made by a semiconductor material, such as single-crystal silicon, as in our case.
Table 2. Capacitive Sensors Specifications
The maximum distance allowed between the mirror and the sensors is relatively small (Table 2). The same is true for the displacement range. Such limitations are critical when a system has to bend a lot and would need a higher range, or when the mechanical setup does not allow placing the sensors so close to the mirror. The bender setup we used has enough room to have the sensors properly installed, and we supported them by placing a bar in between the mirror and the force bar. The bar itself was placed on the main plate and not clamped, to avoid thermal stresses. In the final setup it will be built on Invar or similar low-thermal expansion material, or supported using flexures to allow a proper thermal expansion without introducing own sagitta. The sensors are ultrahigh-vacuum-compatible, a mandatory requirement for every mirror system at the European XFEL. Regarding the calibration of the capacitive sensors, this is normally done in the factory using a conductive metallic target that is electrically grounded. In the case of a semiconductor material such as mono-crystal silicon, we could experience some differences. In general, we used the factory-made calibration to perform the measurements with the capacitive sensors, and we compared the result with the corresponding Fizeau interferometer measurements to find out the differences.
To characterize the sensors output, we performed the following procedure: we activated the actuator to bend the mirror until a certain position was reached. For every position, we waited five minutes to allow the system to stabilize. A typical output of the three direct signals is reported in Fig. 3.
From the obtained signal we can see that and are different, indicating that a tilt is present on top of the bending, and making the number of three sensors indeed needed to retrieve correctly the radius of curvature and the corresponding sagitta. We calculated the sagitta for every position according to Eq. (1) and Eq. (2), using both the Fizeau interferometer and the capacitive sensors: the measurements are shown in Fig. 4. The actuator position is expressed in arbitrary units that are linked to the bender actuator, with a minimum step of 10−2 in this case.
Fig. 4 Calibration curve of the bendable mirror, obtained by the Fizeau interferometer and the three capacitive sensors, without any additional correction. The two curves differ by an offset and a proportional factor.
As we can see in Fig. 4, the two curves differ by an offset and a proportional factor. The offset results from the zeroing of the capacitive sensors at the starting position, which does not necessarily correspond to the real mirror radius. The proportional factor is related to the above-mentioned factory calibration. We can correct both factors by a comparison with the Fizeau measurement. The proportional factor found was 1.06 in this case, calculated taking the ratio between the capacitive sensors and the Fizeau measurements.
3.2 Optical sensors
In some systems, it is not possible to place the sensors so close to the mirror or to even use the same kind of sensors. We replicated the installation, placing three optical sensors that use interferometry to retrieve the displacement of a reflective target. In our case, the target surface was the back of the mirror, too poorly polished to reflect the laser beam of the sensors in a reliable way. Therefore, we used optical sensors with a small lens to create a focus on the target surface. The sensors use a near-infrared laser as a light source, and the reflectance of silicon at this wavelength is around 30% [19], strong enough to have a good signal-to-noise ratio. Details about the sensors used are found in Table 3. We examined different kinds of sensors from Attocube Systems AG to ensure the reliability of the method. Other systems could be in principle used, such as Fabry-Perot sensors, but they should allow a good range with enough accuracy without additional optical elements or special reflective coatings on the mirror back surface, which would increase risks and complexity.
Table 3. Optical Sensors Specifications
The interferometric optical sensors are intrinsically calibrated but suitable only for relative distance changes. The calibration is determined by the knowledge of the laser wavelength, stabilized using an internal iodine cell. When the sensors are switched on, a wavelength scanning procedure is able to estimate the absolute distance from the target, albeit with a very limited accuracy that is not sufficient in this application. The relative displacement measurement, on the other hand, has an accuracy comparable to the capacitive sensors.
We repeated the procedure previously performed with the capacitive sensors (Fig. 5). Similarly to previous case, we still have an offset coming from the initial zeroing of the sensors, but the proportional factor is less than 0.5%, on the same order of magnitude as the errors introduced by the Fizeau interferometry method due to the reference optics.
Fig. 5 Calibration curve of the bendable mirror, obtained by the Fizeau interferometer and the three interferometric optical sensors. In the latter case, we still have an offset, but the proportional factor is almost absent.
4. Noise level and closed-loop operation
The three sensors outputs are combined to deliver a continuous measurement of the sagitta. If the motor is kept fixed for a short time, we can evaluate the noise level of this signal, in the case of the capacitive sensors (Fig. 6) and the optical sensors (Fig. 7). We can see that, for the optical sensors, the noise level is higher, probably because of the air fluctuations and the smaller area in which the laser beam is probing the mirror. The capacitive sensor is averaging, on a bigger area, approximately the sensor dimensions, and is not influenced by the air gap. The measurements were carried out with approximately 1 Hz averaging.
When operated for a long time, the signals experience a drift in both cases, following temperature variations, as previously reported [18]. Using the sagitta measurement as feedback, it is possible to create a closed-loop operation to compensate for this effect, operating the bending actuator. We implemented this control using a particular software framework [20]. An example of closed-loop operation is reported in Fig. 8. We used the capacitive sensors for this demonstration because of their lower noise. Similar results could be obtained with the optical sensors, but with larger oscillations of the signal according to what is shown in Fig. 7.
Fig. 8 Example of a long-term measurement, with the Fizeau interferometer and the capacitive sensors measuring the mirror sagitta simultaneously. At a specified moment, the closed-loop operation mode with feedback to the capacitive sensors is activated and the sagitta is kept fixed inside the 10 nm range.
We logged various signals simultaneously: the temperature of the bender, the sagitta measured by the capacitive sensors, and the sagitta measured by the interferometer. This last measurement was done every half hour to better follow the bender behavior; the lower averaging is the reason the measurements have lower repeatability than usual but are sufficient to see the final effect. The room was not temperature-stabilized, so the temperature changed during the day. After the closed-loop activation, the sagitta was kept stable, even thought the temperature continued to change. According to the capacitive sensor measurement, the sagitta was kept stable inside a range of 10 nm Peak-To-Valley.
5. Discussion of the results and conclusions
We built a measurement system using three displacement-measuring sensors in an x-ray mechanical bender for European XFEL with the goal to control the mirror bending on the nanometer level. We compared two types of sensors: single-point capacitive sensors and interferometric optical sensors. Both systems can be calibrated using a Fizeau interferometer as a reference. Both are suitable for an ultrahigh-vacuum environment, according to the specifications and materials used by the manufacturers: the specific sensors chosen for European XFEL will be further tested by the Vacuum group. On the noise level, the capacitive sensors were better when both systems were used in air. The optical sensors had a better factory calibration: therefore, only a correct zeroing was needed. Another difference between the two systems was that the capacitive sensors delivered absolute distance measurements, retrieved even after a power shutdown of the device. The optical sensors that we examined could also deliver an absolute position, using a wavelength scanning method, but this measurement had only a limited accuracy of 10–100 microns, which would not be sufficient in our case: the device would need to recover the last measurement done before the shutdown from a logging system, losing all the changes made during the shutdown. Another element is the radiation hardness of the two devices, which has to be properly evaluated, but both systems seem to be suitable for the intended environment. From the technical point of view, the two types of sensors both delivered a reliable measurement with a 1 Hz frequency rate, with slightly better results for the capacitive sensors because of the lower noise level. The optical sensors have, in general, the advantage of allowing a much bigger distance between the sensors and the mirror.
The measured signal was successfully used in a demonstrative closed-loop operation. The measured bending was used as an encoder feedback to iteratively adjust the bender actuator to keep the radius of curvature fixed. Environmental influences, like thermal drifts, were compensated, keeping the total sagitta variation lower than 10 nm. The next step would be to test the concept under operational conditions, evaluating the performance and the reliability when combined with the other elements of the beam transport.
Acknowledgments
The authors thank FMB Oxford, Attocube and Physik Instrumente for useful suggestions about the implementation and tuning of the instrumentation used in this work. We are also grateful to Prof. Dr. Wilfried Wurth for valuable comments and discussions.
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