We report a new laser datum system for precision point-by-point profilometry of large curved optical surfaces. The laser datum is sensed by a nulling quadrant photodiode mounted in a flexural system with hybrid actuators, which also carries interferometer reference optics for vertical and horizontal displacement measurement. The flexure characteristics such as cross-talk and hysteresis were investigated. The optimum environmental conditions for the active position-control were studied, and closed-loop control was modeled. The experimental results for compensation accuracy showed a repeatability of ± 4 nm rms, the compensation accuracy of 10 nm (vertical channel) and 20 nm (horizontal channel).
©2003 Optical Society of America
Beyond current 8–10m class large astronomical telescopes, there are currently at least three extremely large telescope projects proposed world-wide. They comprise a 30m class telescope (CELT ), a 50m class telescope (EURO50 ) and a 100m class telescope (OWL ). These telescopes are all based on segmented primary mirrors with 1–2m class hexagonal segments. Such telescopes pose significant technical challenges in mass production of a few hundred – if not thousand – 1–2m class precision mirrors over ~5 years. This contrasts to the ~6 month or more delivery times typical for 2m-class mirrors today. If we are to prove the feasibility of constructing such massive telescopes within a reasonable time-scale of ~15 years, prior development of efficient mass fabrication technology comprising rapid processing and testing is essential. As regards the technical requirement for testing, this is illustrated by the primary mirror segments of EURO50, which must handle a sag of ~10mm, a height accuracy of ~40nm, a lateral range of ~2m and a horizontal accuracy of ~1µm .
Interferometry has commonly been used during the fine polishing and figuring stages of large optics fabrication. Profilometry has complimentary properties to interferometry. In particular, it enables metrology of aspheric optics independent of the null lenses needed for interferometry; null lenses being a source of uncertainty and error both in their manufacture and alignment. It also handles aspherics for which there is no convenient null lens. In many cases of optical fabrication, a point-by-point measurement (rather than a continuous scan) is adequate to feed back into the polishing process, and gives information on absolute radius of curvature, which an interferometer as normally implemented does not directly provide.
One of the key factors contributing to the restricted vertical accuracy in profilometry has been the reliance on a mechanical datum (e.g. granite beam), providing an accuracy at perhaps the micron level. This is limited by effects such as hysteresis, gravity deflections under varying loads with a moving probe-assembly, and thermal distortions. In order to compensate mechanical systematic errors, a reversal method has commonly been used [5,6]. The swing-arm profilometer of Steward Observatory Mirror Laboratory  attempts to circumvent the limitations of linear mechanical datums by using purely rotational motions, which are usually more accurate. However, this is at the expense of losing absolute information on radius of curvature, which is critical for ensuring that mirror segments will all work together.
A profilometer of nanometric height accuracy over its measurement range, and reasonably fast operation, could improve optics fabrication efficiency by providing a single metrology station throughout CNC grinding, loose abrasive lapping, polishing and figuring. Accomplishing this may be divided into four principle challenges: - i) a straightness datum, ii) measurement of height with respect to the datum, iii) measurement of lateral displacement, and iv) measurement speed. In this paper we describe the use of a laser beam in free-air to define the datum.
A laser beam as a profilometric datum has been described elsewhere. The long trace profiler (LTP) [8–12] used a laser beam reflected from a flat mirror. The slope error of the carriage was subtracted from the slope directly measured on the part, giving the corrected slope of the part. Due to the limited size of the lens and detector in the instrument, surface slopes were limited to 4.2 mrad . Virdee at NPL also used a laser beam in free air as a datum for a non-contact profilometer, but his work was targeted at measuring precision flat surfaces of 300mm in diameter for international standardization of flatness .
Given that previous work focused on precision flat measurement, we have considered the application of a laser datum to measure significantly curved surfaces. The overall scheme of our profilometer test-bench is shown in Fig. 1. The lateral range is 1m, the vertical 30 mm and the target vertical accuracy was 60nm. In this instrument we were not intending to measure surface-texture, as this can be performed better by other instrumentation (e.g. a Wyko RST500 texture interferometer that can be loaded onto different areas of a large mirror). Rather, the profilometer was intended to measure global form, by taking a series of discrete measurements across the surface. This was the key to fast operation. The object was to sample horizontally at an interval smaller than half the size of the smallest sub-diameter polishing tool to be used in processing the part, in order to provide metrology data to feed back into the polishing. For a 1m part with a polishing tool as small as 5cm, there would need to be only 40 or so measurement points across a diameter. Horizontally, the maximum allowable positioning error depended on the quadratic sum of the random vertical error, and the propagation of the random horizontal error into the vertical measurement through the maximum slope of the surface. For example, a specific astronomical mirror of interest had a maximum slope of ~7 degs. A 170nm horizontal error would then lead to a 21nm vertical component which, combined with the vertical measurement error, would give an overall ~1/10 wave vertical error at the 633nm HeNe wavelength.
The laser datum system monitors the position of an interferometer component associated with vertical metrology, with respect to the centroid of the laser beam, and compensates the positional variation by a closed-loop two-axis flexure mechanism. In this paper, technical details of the laser datum system is presented and its performance studied both in simulation and experiment.
2. Laser datum system
Neville et al has previously reported the usage of quadrant detectors (QD) and a laser beam for high resolution position-monitoring [14,15]. With noise-cancellation techniques, his method showed a resolution of better than 2nm. Figure 2 shows our datum system. A stabilised He-Ne laser is located separately from the instrument and feeds via an optical fibre. The fixed horizontal laser injection module is mounted off one end of the granite beam via a Cervit block, and comprises the output end of the fibre, a spatial filter, and a beam-splitter and other components associated with the interferometric measurement of horizontal displacement of the moving carriage. The traveling laser sensor-module comprises a two-axis flexure system with hybrid actuators. This carries a quadrant photodiode to sense the laser-position, and a Cervit block supporting the reference optics for the horizontal and vertical interferometer arms. The mechanism operates through analogue nulling, i.e., the actuators translate the QD (and the interferometer optics with it) so as to equalize the left and right halves (horizontal error), and the upper and lower halves (vertical error). The heart of the vertical probe comprises a vertical contact-stylus mounted to a 10mm Invar shaft, the top of which carries a retro-reflector. After QD nulling, the vertical interferometer therefore measures the displacement of the stylus directly with respect to the laser datum.
Hybrid actuators were used because of the difficulty of achieving both nanometric precision and large dynamic range in a single actuator. The dynamic range was required to accommodate the errors of straightness of the granite beam and departures from parallelism with the laser datum. Each hybrid actuator consists of a stacked PZT and a motor mike™ (hereafter called MM). The MM has a motion range of 25 mm, but its positioning accuracy is a few micrometers at best, due to inherent backlash, friction and inertia. The PZT provided a range of 10 µm and nanometer-level positioning accuracy. Figure 3 shows the concept of the QD electronics, comprising FET-OP amps and differential amps with unity gain. The position signals from the quadrant diode are generated with simple equations as shown in Fig 3.
3. Flexure subsystem
Ideally, the position of each flexural position should not be affected by that of the other. However, as shown in Fig. 4 (a), the original flexure design caused significant cross-talk and hysteresis between two-axis movements. The horizontal flexure position was affected by 6.7% of the positional change of the vertical flexure, whereas that of the vertical flexure was influenced by about 3.7% of the horizontal flexure travel. Additionally, the motion of the vertical PZT suffered hysterisis of ~30% to the horizontal displacement. The horizontal PZT motion produced less than 0.1% hysteresis in vertical displacement.
Such cross-talk and hysteresis can be interpreted with, first, the arcuate motion of the flexure. Although this effect cannot be removed entirely from this system, precise alignment of the flexure system with respect to the datum reduced the deflection of each flexure and with it the cross-talk. The second contribution is the inherent geometry of the original flexure system as shown in Fig. 2. The horizontal PZT moves with the vertical flexure, whereas the horizontal MM is attached onto the outer frame. This problem was solved by mounting the horizontal MM on the moving platform attached to the frame of the vertical flexure. Cross-talk was then reduced to 1.1% and 1.7% of vertical and horizontal flexure movements respectively (as shown in Fig. 4 (b)), and hysteresis to <20 nm.
4. Environmental effects
Air turbulence is detrimental to defining a temporally stable line in space. Table 1 shows the displacement of the laser spot on the quadrant diode monitored under different environmental conditions. Clearly, air leakage from the air bearing affected the local environment, causing thermal fluctuations of refractive index of the air, as its exhaust temperature was measured to be some lower than ambient. Even without the air-bearing pressurized, local ‘seeing’ effects disturbed measurements. However, with a set of mechanical shields between the laser beam path and air bearing, it was demonstrated that fluctuation of the datum position could be <0.02 µm, p-v providing the air-bearing was not operated. It would in principle be possible to preheat the air-bearing supply by a corresponding 0.4°C but this has not been attempted.
Adding to the local seeing effect discussed, the thermal condition can cause a drift in the collimating lens or local bending of the structure in the vicinity of the collimator lens. For example, when looking for 60 nm accuracy of vertical measurement over a meter of travel, the laser beam should be stable to better than 0.06 µrad. Controlling or even measuring such small angular displacement in and around the collimator lens assembly is indeed a very challenging task. This problem can, in principle, be solved with another QD at the opposite end of the granite beam, monitoring any directional instability of the beam. Now, the QD sensitivity is measured to be 42 mV/µm. It follows that the full-scale QD output-signal of 5V, when digitized by a 16-bit ADC, would provide a resolution of ~1.8 nm over the total height displacement of 119 µm. Whilst this method has not yet been implemented, it is clear that the directional instability of the reference laser beam could be compensated well within the target form accuracy of 60 nm.
5. Close-loop compensation model
The sensitivity of the QD depends on the intensity and size of the laser beam. If the laser intensity were uniform over the quadrant diode, it would produce no position-dependent signal. However, the beam profile emerging from the launch optics were near-Gaussian with σ (rms radial fluctuation) = 1.1 mm, i.e., 63% of encircled energy fell within 2.2 mm in diameter, which was smaller than the 3 mm square active area of the quadrant diode. The beam size remained constant at 2.1 ± 0.1 mm over the horizontal travel range of 1m. This ensured that the quadrant diode produced a voltage proportional only to the vertical displacement of the laser beam.
Figure 5 illustrates the MM and PZT control model of the datum beam position. The selection of which actuator was energized depended on the signal from the pre-amplifier. If this signal were above a specified value, the MM was selected, and if below, the PZT. In the latter case, the system transfer function Y(s) is defined as:-
where P(s) is the disturbance function. The system loop gain G(s) is defined as:-
where α is the expansion coefficient of PZT (µm/V), β is the sensitivity coefficient (V/µm). K1, K2 and K3 take the form of an OP amp transfer function, or:-
Here Adc is the amplification of the OP amp at zero frequency, and f1 is the cut-off frequency. Taking cut-off frequencies of K1, K2, and K3 with 4Hz, 0.12 Hz, and 18.3 Hz, and considering that α is 0.14 µm/V, and β is 0.044 V/µm from experiments, the result of closed-loop model simulation is shown in Fig. 6 (a). This is the response of the system to the step function disturbance. The difference between the height of step function disturbance and the height recovered from the PZT compensation was only ~3 nm. Whilst the result implies that the system is under-damped, this simulation predicted that the positional compensation of the datum system could be accomplished well within the measurement requirement of 60 nm.
6. Compensation experiment and results
The position of the quadrant diode was artificially disturbed and then recovered by the closed loop control. Figure 6 shows the model prediction (6a) and the experiment results (6b). The datum system was disturbed vertically by 0.7 microns, before time ‘A’ in Fig. 6 (b). This generated the residual cross-talk of about 0.1–0.2 micron, as explained in Fig. 4 (b). The closed loop control commenced at ‘A’. The overshooting and oscillatory signals are shown in both channels during the position recovery, as expected from the model simulation. This transient response subsided to acceptable levels within ~1 second for both channels (the horizontal being more tolerant for modest surface-slopes).
The resulting positional accuracies are about 10 nm and 20 nm in the vertical and horizontal channels respectively. These compensation accuracies are larger than the 3 nm expected from the simulation. The discrepancy is believed to be due to the observed residual hysteresis of the demonstration system. The same test was repeated 15 times to assess its repeatability. The measured repeatability of ±4 nm rms in the vertical channel was achieved from the closed loop compensation, thereby demonstrating that the current datum system stays well within the target measurement accuracy of 60 nm.
6. Concluding remarks
Whilst previous reported work [8–13] addressed measurement of flat surfaces, the key advance of this work is a new laser datum enabling nanometric profilometry for large, curved aspheric surfaces with significant sag. The datum exhibits hysteresis of some 10–20 nm and cross-talk of ~1-2% between the two axes. The positional accuracy of the datum under active control is on the order of 10–20 nm. This corresponds to a 1 second settling time in response to an external perturbation of 0.7 micron. Such accuracy is achieved with the repeatability of about ± 4 nm rms.
Most profilometers conduct a continuous trace, whereas we have reported on a point-by-point measurement system. The ~1 sec settling time for the laser datum would introduce serious errors for a continuous trace. In contrast, it is perfectly satisfactory for a set of discrete samples, as the laser system can settle as the probe is lowered onto the surface prior to the capture of the metrology data. For a nominal 40–80 points across the surface, the down-time due to settling would then be only ~one to two minutes respectively, for the entire measurement. A 40 by 40 grid of points on an off-axis mirror segment (e.g., mounted on a rotary stage) would take ~40 mins, and an 80 by 80 <3 hours. In the context of automated polishing of large optics, this is a very useful development, particularly as texture can be measured separately by readily available commercial instrumentation.
The effect of environmental air turbulence on the datum was reduced to ~20nm by a simple polythene curtain around the table and a polythene shield covering the instrument. However, we did not overcome the deleterious effects of air-bearing leakage. Whilst controlling the input air-temperature to the bearing will undoubtedly mitigate the effect, we would recommend the use of a different linear bearing, such as a plain sliding bearing with weight-relief. In order to monitor and, if needed, to compensate the remaining directional instability of the laser reference beam, a secondary QD at the opposite end of the granite beam to the laser source is also proposed.
The successful control of the laser datum with an accuracy of 10–20 nm provides an attractive approach to profilometric referencing for 1–2m class mirrors, especially as it will provide absolute metrology of base radius-of-curvature as well as aspheric form.
The development of SPLOT was funded by the Satellite Technology Research Centre (SaTReC) in the Korea Advanced Institute of Science and Technology (KAIST) and the UK Particle Physics and Astronomy Research Council under an underpinning grant. We acknowledge that the final stage of this research, particularly manuscript preparation, was supported, in part, by Center for Space Astrophysics, under the creative research initiative program of the Ministry of Science and Technology and by Korea Institute of S&T Evaluation and Planning (Grant code M1-0206-00-0014).
References and links
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