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Abstract
Interferometric determination of sphere radii is a well known technique. To keep accuracy high and uncertainties low, a precisely controlled environment is usually necessary. Environmental changes in temperature lead to time dependent drift in important measurement parameters and to disturbed results. We present a method to minimize time dependent drift to the first order. With this method, it is either possible to reduce the uncertainties further, or to relax environmental conditions and still be able to accomplish high precision measurements. We discuss two typical measurement configurations, the associated benefits and drawbacks and some relevant error sources.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The radius of a sphere is, apart from its position, its single geometry defining property. The high symmetry of spheres makes them comparably easy to fabricate, measure and useful for many applications, such as bearings, spacers, optical surfaces or volume definitions as in the Avogadro project [1]. Interferometry is ideally suited to measure geometry deviations of spherical surfaces with highest precision down to the sub-nanometer level. The deviation from an ideal sphere can be determined using its rotational invariance, see e.g. [2], [3]. In addition, the absolute determination of the surface shape requires the measurement of the sphere radius. Standard interferometric radius measurements utilize the two so called confocal and cat’s eye positions. The confocal position is the position where the focal point of the interferometer objective (for Fizeau type objectives often referred to as "transmission sphere") coincidences with the center of the sphere to be measured (surface under test, SUT) (Fig. 1(a)).
Fig. 1. a) Measurement principle and visualisation of cat’s eye and confocal position. b) Linear fit of Zernike power term yields the exact confocal and cat’s eye position.
Under this condition, rays from the objective impinge perpendicular onto the SUT and are reflected back onto their incomming path to the objective. Shifting the SUT along the optical axis that we denote with $z$, according to standard convention in optics, we eventually reach the cat’s eye position where the focal point of the interferometer objective coincidences with the apex of the SUT. In this position the whole wavefront is reflected at that single point.
The distance along the optical axis between confocal and cat’s eye position corresponds to the radius of the SUT. This is the standard procedure of interferometric radius metrology and has been investigated by many authors. Other methods such as the recently presented wavelength shifting interferometry method by Kredba et al. [4] or the curvature sensitivity method by Psota et al. [5] rely on measurements in only one position, the confocal position - with a slight drawback on measurement accuracy. In this article we focus on the standard procedure due to its wide-spread use, its accuracy and simplicity. Selberg et al. describe the use of a direct measuring interferometer (DMI) for the distance measurement between the two positions and gives an estimation about error influences with respect to the measured sphere radius and sphere aperture [6]. He ranks the most important error sources which provides a good orientation for interferometric radius determination uncertainty. A mathematical method to correct stage errors is described by Davies et al. [7]. This theoretical discussion includes a Monte Carlo simulation to estimate the measurement uncertainty. Schmitz et al. provide a rigorous estimation of uncertainties in interferometric radius determination [8] including influences of Gaussian Beam profile and diffraction effects. They considered 13 error sources for an uncertainty analysis according to "Guide to the Expression of Uncertainty in Measurement" (GUM) [9] and NIST Technical Note 1297 "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results". Influences of the ambient air are described by Estler [10]. The described compensation for changes in refractive index of air is useful for many other interferometric measure methods as well.
An important aspect of radius metrology is the inherently time sequential measurement of the two measurement positions involved. This makes the method vulnerable to drift effects of any kind. Especially in close to production environments, where e.g. temperature stability is moderate and measurement time crucial, drifts can be a limiting factor. In this contribution, we suggest a method to mitigate drift effects and show experimental investigations on its effectiveness.
2. Influences of errors
In this section some typical dominant error mechanisms are presented, and we present ways to minimize them. There are a whole lot more error influences which have to be respected, and are described in detail in [6] –[10].
Both cat’s eye and confocal position can be determined with high reproducibility using the curvature of the object wavefront recorded by the phase measuring interferometer (PMI). Several interferograms are acquired close to cat’s eye, respectively confocal position while the SUT is moved in small steps along the optical axis. Each measured wavefront is fit with Zernike-polynomials [11] yielding a function of defocus coefficient over the z-position. Interpolation with a linear fit [12] allows to extract the z-position where the wavefront curvature changes its sign (Fig. 1(b)). This procedure relaxes the positioning requirements and requires only a precise measurement of the position.
Fig. 2 shows two configurations (a) and (b) to determine the radius of curvature of a sphere. Each configuration has its own benefits and drawbacks. In configuration (a), one beam of the DMI is directed towards the center of curvature of the SUT (1), thus the Abbe condition is fulfilled (see Section 2.1). However, the long distance between PMI and DMI reduces stability and increases potentional error influences (Section 2.2).
Fig. 2. a) Measurement configuration (a) with two stages. One beam of the distance measuring interferometer (DMI) targets the center of the sphere, thus an influence of tilt of stage 2 is minimized. b) Configuration (b) reduces the influence of drift but violates the Abbe principle.
Fig. 2(b) presents an alternative, much more compact configuration. The Abbe condition is severely violated, which requires tilt angle detection and compensation (Section 2.2).
2.1 Abbe condition
The Abbe condition states that a length measurement must take place along the line that needs to be measured. An Abbe error results if there is a lateral displacement $\Delta x$ between measurement direction, e.g. defined by the laser beam of a DMI, and the length to be measured, in the case of radius measurement the displacement vector of the SUT center along the z axis when moving from cat’s eye to confocal postion (Fig. 3). In this case a tilt $\alpha$ of the retroreflecting mirror attached to the SUT will lead to a length measuring error $\Delta z_{Abbe}$ that changes approximately linearly with $\alpha$. In all real systems, a change of $\alpha$ during movement of the SUT is unavoidable. Only if the Abbe condition is met, i.e. the DMI targets the center of the SUT, an inclination $\alpha$ does not have an effect on the result.
Fig. 3. Abbe error as a result of the combination of decentration $\Delta$x and tilt angle $\alpha$.
Minimizing the Abbe error can be done either by alignment of the appropriate measurement setup or mathematically. The mathematical correction is possible on the basis of a known angle $\alpha$ and distance $\Delta x$. Since in configuration (b) the Abbe condition can not be fullfilled, the corresponding error has to be eliminated mathematically. Here, the required angle is measured with a three beam DMI. In addition to the angle, the Abbe offsets, i.e. the x- and y-distances between DMI-axis and center of the sphere need to be determined. Note that a rotation of the three beam interferometer about its optical axis changes these values accordingly.
2.2 Mechanical errors
2.2.1 Stage torsion
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Relevant dimensions regarding stage torsion during movement are the yaw and pitch angle (see Fig. 2(b)). Since configuration (a) obeys the Abbe principle, stage torsion does not affect the measurement result in first order approximation.
In configuration (b), the yaw and pitch angles are determined by a three beam DMI. Figure 4 shows an example of yaw and pitch angles for 100 measurements in confocal and cat’s eye position.
Fig. 4. Yaw and pitch angle in configuration (b) determined by a three beam DMI. Angles vary between confocal and cat’s eye position as well between measurements. The total measurement time is 10 h 7 min, the approximate time per measurement is 6 min 10 s (mean value).
The angles vary over the measurements, which motivates the necessity to correct every single measurement. Exact knowledge of the distance between the geometrical center of the DMI beams and the center of the measured sphere (Abbe point) is necessary. Also the orientation of the coordinate system of the DMI has to match the coordinate system of the interferometer. That given, the numerical compensation for stage torsion is straightforward.
2.2.2 Interferometer torsion
Since the stiffness of real setups are limited, torsion during movement will always be present. It depends on the specific setup whether this torsion is negligible or not (for example, the measurement setup can be realized with a horizontal optical axis as in Refs. [8], [13]). In our vertical setup, in configuration (a), torison has to be considered. This torsion of configuration (a) is the difference in bending of the stage guide rails in different positions of stage 2 due to the single side bearing of the stages (Fig. 5).
Fig. 5. Example of torsion error, recorded stepwise from intial position of stage 2 to + 200 mm and back to the initial position.
The bending affects the distance of the DMI to the PMI, as well as the angle of the measurement axis of the DMI (which results in a cosine error). To determine the effect of the torsion onto the measured distance $\zeta$, it is monitored while stage 2 moves from confocal to cat’s eye position. The difference of the distance $\zeta$ measured with stage 2 in confocal and cat’s eye position is $\Delta \zeta$. To demonstrate the relevance of torsion, we show an arbitrary example of stepwise movement of stage 2 in Fig. 5. Stage 2 was moved stepwise from the initial position to a position + 200mm and back to the initial position while $\zeta$ is recorded. The direct result of this measurement is affected by drift. To compensate for that, we determine the slope of each set of measurements ($\zeta$ measured while stage 2 moves stepwise from zero to +200mm and back to zero) by a linear fit (over the whole set of data) and correct each measurement set with its slope and an offset value.
Since $\Delta \zeta$ is nonlinear and dependent on the travel range and the load of the stage, we suggest a compensation approach that is straight forward to implement: Before and after the radius measurement, a measurement of $\zeta$ is done within the travel range of the radius measurement and the same load on stage 2 as will be during measurement. For this the DMI laser beam is retroreflected by a plane mirror fixed to the PMI. The SUT has to be replaced by a dummy weight with equal weight distribution as the SUT that allows the DMI beam to pass stage 2 unblocked. The difference in $\zeta$ between zero (confocal position) and the endpoint of movement (cat’s eye position) is the compensation value which will be added to the final radius value. With an automated routine, we achieved a standard deviation of about 15 nm for this correction value.
2.3 Temperature induced errors
Of special interest are temperature induced errors, since they have a major contribution in measurement via refractive index of air and linear expansion. The response time and magnitude of the different errors due to temperature changes is quite different, due to the different mechanisms and different thermal mass of the components and environment. For example, the error due to thermal expansion is linear in temperature, whereas the error due to refractive index change is reciprocal in temperature.
2.3.1 Refractive index of air
The refractive index of air is an important dimension in interferometric measurement and is not only influenced by temperature T. Also air pressure P has a significant contribution. Other influences are humidity H and CO2 concentration, but they are negligible compared to air pressure and temperature change. The refractive index is described by the empirical Edlén formula [14]. The following formula is a converted version.
The laser wavelength of the PMI and DMI are directly affected by the refractive index of air. Figure 6 shows the variation of nair according to the Edlén formula for the four dimensions T, P, H and CO2. Within reasonable limits of the dimensions during one determination of the radius, the refractive index of air can be regarded as linear to these parameters. Therefore, the dependence of refractive index could be regarded as linear in time in an appropriate timestep.
Fig. 6. Refractive index of air as a function of humidity, CO2, air pressure and temperature according to the Edlén equation.
An example of the direct influence of the refractive index of air is the axial focus shift of the PMI. The focal point is also influenced by the refractive index of glass of the optical components, which in turn is also influenced by temperature, very slow varying and thus regarded as linear.
The overall temperature dependent error $\epsilon _{\mathrm {total}}$ is the sum of all temperature dependent error contributions $\epsilon$ (Eq. (2)). Contributors to $\epsilon _{\mathrm {total}}$ are for example the linear expansion contribututions from the setup components ($\epsilon _{\mathrm {expansion 1}}$, $\epsilon _{\mathrm {expansion 2}}$ ,…), the temperature influence to laser systems $\epsilon _{\mathrm {Laser}}$, or errors because of changig refractive index $\epsilon _{\mathrm {nAir}}$. Some of the errors also depend on other influences (e.g. air pressure p).
While in theory with this function, it would be possible to correct the measurement result at any time, it turns out that for most cases to gather and discriminate all thermal effects is at minimum a very difficult if not impossible task. However, knowledge about the evolution of the error function is required only during the relatively short measurement period between cat’s eye and confocal measurements. With the first order approximation that $\epsilon_{\textrm{total}}\;\left (T\right )$ is linear during that period, and since the evolution of the drift error in either cat’s eye or confocal position can be determined by repeated measurements, the error influence can be treated effectively without explicitly determining Eq. (2).
3. First order drift compensation
Non statistical time dependent errors are mainly induced by temperature change and the stability of the interferometer, i.e. mechanical settling or crawling effects. The magnitude of these effects typically change over a measurement series, e.g. due to friction effects in the mechanical bearings. As a consequence, even in areas with precisely controlled temperature, time dependent errors will always be present to a certain degree and affect the measurement.
An illustration of such a drift is given in the left of Fig. 7: In this example, the room temperature was poorly stabilized (a few Kelvin) and the acquisition time was not optimized, hence only 20 measurements where acquired in 5:30 hours. The circles represent the values for confocal (blue) and cat’s eye (orange) position. During the whole measurement time, the system drifts by about 6$\mu$m.
Fig. 7. Left: Absolute confocal and cat’s eye position plotted against time. Drift of the system is obvious. Right: Error by determining the radius on the basis of the next neighbours.
Radii are determined by the difference of confocal and cat’s eye of adjacent neighbours. Since the confocal and cat’s eye position can not be determined at the same time, and since environmental changes have an effect on the measurement setup over time, conditions are different for confocal and cat’s eye position. By directly taking the difference between the adjacent neighbours, an error occurs as indicated in the right subfigure of Fig. 7. Depending on whether the slope of this drift (slope between measurements of the same type, confocal respectively cat’s eye) is positive or negative, the measurement will be disturbed with a positive or negative error (measured radius too high or too low). It is simple but important to recognize, that the shorter the time delay between measurements, the smaller are the effects of a drift error.
3.1 Drift compensation
If we would be able to develop a complete mathemetical description of the sum of all time dependent errors (similar to Eq. (2)), ocurring in our system, we would have a drift function, which is a polynomial of order n. As shown above, most dominant error contributions (e.g. from change of refractive index of air or thermal expansion) behave fairly linear for small changes. Thus, removing the first order (i.e. linear) contribution from our fictitious polynomial would reduce the amount of error significantly. This is what we do in our drift compensation to the first order (of course, a drift compensation to higher orders e.g. 2nd is possible as well).
The drift function reveals by plotting positions of the same type (confocal respectively cat’s eye) against time. Clearly, if e.g. the confocal positions measured just before and after a given cat’s eye position measurement differ due to a drift in the system, the two radius values calculated from each pair of measurements will be different. Assuming a linear drift in time, we estimate the hypothetical confocal position value from the two measured values at the time of the cat’s eye measurement (see Fig. 8). Interpolation can be done between confocal positions to determine the confocal position at the time of cat’s eye measurement, or vice versa. Both methods should lead to the same result.
Fig. 8. Left: Interpolation of the confocal position at the time of the cat’s eye measurement. Right: Interpolation of cat’s eye position at the time of confocal measurement.
As stated before, the shorter the time delay between measurements, the smaller the influence of the drift function, and the better the linear approximation between measurements. Therefore, drift compensation should be combined with a reduction of acquisition time, as we illustrate with experimental data in the next section.
4. Measurement setup
The measurement setup is based on a commercially available two stage interferometer setup from Schneider Optikmaschinen GmbH & Co. KG of type ALI200 equipped with a Fisba $\mu$Phase HR2 interferometer. The location of the setup is not (as usual) a basement or ground floor, but the 1st floor in an ordinary laboratory, with only rudimentary temperature control of a few Kelvin.
Laser interferometer (PMI) A sketch of our setup is shown in Fig. 2. The granite platform is equipped with passive dampers to supress vibrations from the building. Two z-stages are availabe with a travel range of about 1000 mm and guided by guide rails made of steel and ball bearings. The platforms are driven via recirculating ball screws with dc motors. Linear encoders from Heidenhain (LS 406c) indicate the positions of the stages.
The phase measuring interferomer (PMI) is of Twyman-Green type. A six inch beam expander and a suitable objective form the object wave that is reflected by the SUT. A 6 inch objective lens with NA 0.71 was used for the measurements shown in this article. A SIOS SL 02-1 stabilized HeNe Laser is used as light source for the PMI.
Periphery For precise measurement of the distance between confocal and cat’s eye position, we use a triple beam interferometer from SIOS (SP 2000 TR). This interferometer has a sensor box to monitor environmental conditions (temperature and air pressure) and corrects the laser wavelength accordingly.
The lateral alignment of the SUT with respect to the PMI optical axis can be measured at confocal position. A decentration error can be automatically corrected using two piezo driven stages from Newport (Agilis AG-LS25 with controller AG-UC2) mounted on interferometer stage 2.
Aquisition of the interferograms, calculating and unwrapping of the phase images and fitting with Zernike polynomials is done by the interferometer software $\mathrm {\mu }$Shape from Trioptics. Measurement, control of hardware and evaluation is based on the python based open source software ITOM [15].
Interferometer setup The setup time for the interferometer depends strongly on the specific setup. Special care is necessary to align the DMI axis considering cosine error and Abbe point. A position sensitive diode (PSD, for example [16]) has proven valuable for this purpose. For our setup, the setup time for configuration (a) is about an hour. An additional hour is needed to determine the interferometer torsion before and after measurement as described in section 2.2.
Measurement in configuration (b) needs exact knowledge of the distance between DMI measurement point and the Abbe point of the SUT to compensate for the intrinsic Abbe error. This also includes the proper alignment of the DMI coordinate system to match with the coordinate system of the PMI. Altogether, like in configuration (a), about an hour is needed to setup configuration (b).
Environment As described above, environmental conditions in our laboratory are worse that typically found in high precision measurements. Vibrations from the building in 1st floor is higher than in basement, and student traffic in the laboratory lead to turbulences in air and additional vibrations from door opening etc. We treat this influences as statistical and account for that with a set of many measurements and statistic evaluation.
Climate control in the room leads to quasi-periodic steep drops in room-temperature. Figure 9 shows a record of the room temperature during one of the measurements.
Fig. 9. Top: Effect of climate control on temperature during a 15 hour measurement in summertime, showing steep temperature drops. Bottom: Focus on the three temperature drops. Linear fitting yield approximated temperature gradients of about 0.12 $^{\circ }$C / min.
The detailed graphs in the lower section of Fig. 9 show these drops and an estimation of the respective temperature gradients, which lie in the range of 0.11 $^{\circ }$C / min to 0.12 $^{\circ }$C / min.
5. Measurements
The specimen used for these experiments is a precision convex sphere with radius 20 mm made from the low thermal expansion material Zerodur. This SUT was also measured by the Physikalisch-Technische Bundesanstalt PTB [17], the national metrology institute of Germany. Comparison to this measurement is presented in the last section.
5.1 Improvements because of drift compensation
We investigate the effect of the linear drift compensation on the measurement data that was acquired under the environmental temperature condition shown in Fig. 9. This is an extreme example which is useful to understand the benefit, potential and limit of the suggested drift compensation. Figure 10 shows the raw data of the measurement row. As described above (see Fig. 1), each point in the graph is the result of a small focus series in either cat’s eye or confocal position, which takes about 1-2 min. Within this time already, the environmental changes due to the climate control directly influences the measurement of absolute positions.
Fig. 10. Absolute confocal and cat’s eye positions aquired in environmental temperature conditions shown in Fig. 9. The steep temperature drop due to the climate control directly influences the absolute positions.
Taking directly the differences of the absolute confocal and cat’s eye position RX = - {NTX - CX} and RX+1 = - {CX - NTX+1 } (see Fig. 7) would lead to RX >RX+1 or RX <RX+1, depending of the slope of the drift. As a result, the radii RX and RX+1 split into two branches, as shown in Fig. 11. In our example, all determined radii are in the range of $\pm$1.5 $\mathrm {\mu }$m with a standard deviation of $\sigma$un = 135 nm. As expected, the radii determined within the steep temperature drop show the largest deviations.
Fig. 11. Direct determination of radius without compensation. Since, under influence of drift, RX>RX+1 or RX<RX+1 (see Fig. 7), the determined radii split into two branches.
When a drift compensation to the first order (Fig. 12) is applied, the standard deviation reduces to $\sigma$Con = 21 nm in case of compensated to confocal and $\sigma$C = 28 nm in case of compensating to cat’s eye, which is a reduction by a factor of 6.4 respectively 4.8.
Fig. 12. Drift compensation to first order of measurement according to confocal and cat’s eye (Fig. 8). Left: Drift compenastion to confocal. Standard deviation is $\sigma$Con = 21 nm, Right: Drift compensation to cat’s eye. Standard deviation is $\sigma$C = 28 nm.
Though drift compensation is applied, outliers are still present in the time period of the steep temperature drop. This indicates that the behaviour of the setup is not linear enough on the time scale of a few minutes, i.e. the time span between two measurements of the same kind.
One approach to mitigate such outliers might be drift compensation to the second or higher order.
Filtering outliers based on observed drifts or based on measured rapid environmental changes which exceed the possibility of the underlying drift compensation might be an option, too.
This intrinsically opens another possibility to reduce the maginutde of outliers: improvements in hardware or measurement strategy to realize a shorter time for each measurement.
To show the effect of measurement time, we created from the dataset new datasets consisting of only every Nth data point from the original dataset. This corresponds to an N-fold increase in measurement time between two consecutive measurements for the new dataset. We created two datasets with N=2 and N=4.
As an effect of the reduced measurement frequency, the precision of the measurements worsens considerably, as shown in the boxplots of Fig. 13. In the graph the results based on standard and drift compensated evaluation are compared for the different datasets.
Fig. 13. Effect of artificial increase of mesurement time by considering only every Nth measurement. Drift compensation leads to almost the same results for full data set and half data set (deviation of median is about 1 nm, deviation of standard deviation is below 1 nm). The boxes indicate the 75 th percentile.
Drift compensation to the first order leads to almost the same results for full and half data set. The median value of the measured radius differs only by about 1 nm, and the standard deviation for both is 16 nm. For the quarter data set, the median deviates by about 4 nm from the full dataset, and the standard deviation doubles to 30 nm. Without drift compensation, the median values differ by 10 nm, depending on the data set. The standard deviations are considerably higher, with 135 nm for the original dataset, 389 nm and 929 nm for the reduced data sets.
So far, we focused on a measurement series taken under quite extreme environmental conditions and an artificial increase of measurement time. To illustrate the effect of measurement time further, Table 1 shows 3 exemplary measurement series with varying measurement times. Reducing the time by a factor of about 6 lead to a reduced standard deviation by a factor of about 8. In addition to the smaller susceptibility to drift artifacts, the overall shorter measurement time allows to increase the number of measurements in the same time, which is an statistical benefit.
Table 1. Acquired measurement sets with different average measurement time per measurement.
5.2 Measurements and configurations
With the setup configurations (a) and (b), as described above in Fig. 2, we acquired over a period of about 90 min radius measurements, with a measurement time for a single radius of about 3 min. Figure 14 shows the results of 5 measurements of configuration (a) and 4 measurements of configuration (b). The mean values of the measurements lie within $\pm$10 nm for configuration (a) and $\pm$14 nm for configuration (b).
Fig. 14. Comparison of sets of measurements from configuration (a), (b) and PTB. Error bars indicate the standard deviation. The uncertainty of PTB measurement (March 2018) is $\pm$309 nm.
Comparing the standard deviation $\sigma$ of the uncorrected radii with the drift compensated ones, we can state that drift compensation has more effect on configuration (a), where the reduction is at least a factor of 3. The reduction in configuration (b) is smaller than a factor 2 (depending on the environmental changes during measurement). A reasonable explanation might be the different stability requirements of the two configurations. In configuration (a) the overall length involved in the setup is much longer than in configuration (b), and therefore the absolute and well compensable amount of drift is higher because of higher amount of thermal expansion and longer optical pathlengths.
Regarding $\sigma$ and drift, configuration (b) has some minor benefit over configuration (a). However, we observe that the overall corrected $\sigma$ is slightly higher for configuration (b). This can be explained by the uncertainty of aligning the coordinate system of our DMI with that of the PMI. Though we took great care in determining the distance between measurement point of the DMI to the center of the SUT, there was a slight uncertainty in the angle alignment of the DMI to the PMI at the time of measurement. This highlights the sensibility of configuration (b) to the perfect alignment of the coordinate system of DMI to the one of the PMI.
To compare the results from configuration (a) with (b), results from (a) have to be corrected for the interferometer torsion effect described in section 2.2. We have determined this value to −144 nm with a standard deviation of $\sigma_{\textrm{corr}}$ = 11 nm. The mean values of both configurations lie close together with a difference of about 16 nm. Comparing these values with measurements from PTB (R = 20.047287 mm with a measurement uncertainty of $\pm$ 309 nm), then our results are well within the uncertainty of the PTB measurement.
It is important to note, that in contrast to the PTB measurements, our setup faces conditions of environments like in an office in summertime with only rudimentary temperature control. This highlights the effectiveness of the proposed drift compensation method.
6. Discussion of error influences
The relevance of drift on radius measurements is depending on the complete error budget, which needs to be considered for each setup individually, see e.g. [8]. From the presented measurements in unstable conditions we can see radius results varying up to several hundred nanometers if the drift is not corrected, see Fig. 11. This can be a dominant part of the error budget for a given SUT, see Table 2.
Table 2. Uncertainty estimation for the measured SUT with nominal radius 20 mm. Uncertainties that are denoted with "Drift compensation" are part of the error budget that is reduced using the suggested drift compensation.
According to Selberg [6], significant error influences for interferometric radius determination are Abbe error, cavity null error (finding the exact confocal position) and axial alignment error. For high NA objective lenses such as the f/number 0.7 used in the experiments shown, the cavity null error typically is less than 15 nm when the procedure described in Section 1 is used.
Table 2 shows an estimation of the main uncertainty contributions of our setup following the extensive investigation of Schmitz [8]. The table indicates which error parts are affected by the suggested drift compensation.
When measuring SUT with longer radii the cosine error increases accordingly. Due to the longer distances involved the relevance of drift increases. Longer travel ranges are also accompanied by longer positioning times. This in turn affects the effectiveness of the drift compensation, see Fig. 13. Additionally, bigger radii typically require larger f/numbers of the transmission sphere, thus sensitivity is smaller and the uncertainty of the cavity null error rises considerably, see e.g. Schmitz [8].
7. Summary
The determination of a spherical surface’s radius is an important topic in precision metrology. New applications require uncertainties much below 100 nm. The well-established interferometric determination of sphere radii in principle offers sub-nanometer resolution, but is often limited by drift effects. Therefore, high precision measurements usually require very well controlled environments to reduce thermal drifts and other influences. In this contribution, we presented a method to detect and in large parts eliminate errors due to drifts. We have shown that with this drift compensation method, precise and accurate measurements are possible even in harsh environmental conditions like production areas or laboratories without precise temperature control. We have shown that it is possible to treat environmental influences in sum as linear in an appropriate timestep and use linear interpolation to correct for it. In combination with statistical evaluation of many measurements, it allows to do precise interferometric determination of radii without the need of precise environment control, or enhance precision and accuracy to a higher level in controlled environment.
Two common setup variants have been investigated. We introduced both setup configurations and showed typical issues regarding setup and accompanied uncertainties. After taking into account also subtle error influences like a torsion of the complete interferometer setup during measurement, the results of both configurations agree well within a difference of 16 nm for the measurement of a sample with radius of curvature of 20 mm.
Under harsh environmental conditions, with temperature variation rates of >0.11$^{\circ }$C/min and other environmental fluctuations, the method reduced the standard deviation up to a factor of 6.4.
The results from measurements of both configurations, in typical environmental conditions of a laboratory with only moderate temperature control of about $\pm$1$^{\circ }$C, lie well within the stated measurement uncertainty of a reference measurement of a specimen measurement by the national metrology institute of Germany, PTB, ($\pm$ 309 nm).
Funding
European Metrology Programme for Innovation and Research (EURAMET 15SIB01: FreeFORM).
AcknowledgementS
The authors would like to express our special thanks to Michael Schulz from PTB for providing the spherical specimen and measurement results and for many fruitful discussions. Furthermore, the authors want to thank the EMPIR organization, which is jointly funden by the participating countries within EURAMET and the European Union (15SIB01: FreeFORM).
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
Data availability
The data that support the plots within this paper are available from the corresponding author upon reasonable request
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