1. Introduction

Optics of modern telescopes requires mirrors with (1) high aspheric departing a few hundred microns from approximate sphere, which corresponds to $\sim 10^3$ fringes on a interferogram, (2) meter-class aperture, and (3) various shapes including positive and negative radius of curvature (RoC). Furthermore, all of the segmented telescopes and off-axis telescopes, particularly for high sensitive observations of planets and exoplanets, employ off-axis mirrors [1–6]. There is an increasing amount of demand in recent years for a measurement system which can measure these optics.

The accuracy of measurement systems must meet the requirements of the test surface. The requirement is defined with the observing wavelength and is $\sim$ 10 nm in optical telescopes, and dynamic range of few hundred millimeters is required in large optics. Requirement of sensitivity to RoC is relaxed, because the RoC error can be corrected with adjusting position of the image plane. Requirement of position accuracy in lateral direction of measurement system is also relaxed by $10^{3-5}$ comparing to vertical resolution of 10 nm because of the ratio of the asphericity and aperture size. Thanks to this, the position accuracy in lateral direction of $10^{1-3} \mu$m is required. Requirement for temperature stability of measurement system is not essential, because telescope optics are made of low thermal expansion material.

1.1 Optical method

Interferometers have been widely used as a final test of the optics and have been proved to be very powerful. However, interferometers require very-precisely-manufactured reference surface and complicated and purpose-built null optics [7] or holograms [8,9] to generate a wave front meeting to the shape of the test surface. Design and manufacturing of null optics are difficult when the asphericity gets large, and this implies that interferometers have small dynamic range and are not suitable for the measurement of high aspheric optics.

Another optical measurement method is deflectometry [10,11]. An advantage of deflectmetry is that a null optics is not required. SCOTS akin to deflectmetry was used for test of a 4.2-m off-axis parabolic mirror [5].

Although optical methods can measure large area in short time, they have the following problems: (1) being difficult or nearly impossible to measure large mirrors with a flat or convex surface which do not have the real focal point [12,13], or sub-aperture stitching technique is required [14], (2) requiring a large space comparable with the radius of curvature of the test surface, (3) requiring precise alignment process among the optics, and (4) requiring a stable environment, i.e., limited amounts of air turbulence, vibration, and ambient temperature change which produce phase shift error with changing the optical path length between the reference and test surfaces.

1.2 Swing-arm profilometer

Swing-arm profilometer, which employs the single or two-point method [15] is developed by Arizona University; they successfully measured a convex mirror [16]. A technical difficulty is, however, that the swing arm requires a highly precise rotary-bearing for the swing motion and alignment process in order for the rotating axis to run through the center of curvature of the test surface. Another downside is that the size of the measurable mirror is strictly limited according to the length of the arm.

1.3 CMM

Coordinate measuring machines (CMM) would be, ideally, desirable for measuring potentially high aspheric optics. Notable CMM for manufacturing free-form optics have been developed [17,18] (e.g., UA3P from Panasonic [19], A-Ruler from (CANON) [20], and NANOMEFOS from TNO [21]). However, their work areas were limited, and CMM also require highly stable environment.

1.4 TPM

A three-point method (TPM) [22] is similar to the spherometer, which has been widely used to measure the RoC of lenses for many years. The TPM measures the cross section by means of scanning of the second derivatives over a test surface. When we address the third or higher order structure on the test surface, the TPM fundamentally requires no reference surface and null optics in contrast to the method with interferometers. This fact suggests that the TPM has a potential to be used for a CMM for the high aspheric optics of telescopes.

The TPM consists of three distance sensors, a bench holding the sensors, and a linear guide system for scanning. The sensors are attached on the rigid bench with a separation $d$, aligned on the straight line which points the scan direction. The sensor unit is carried along the scan path with the guide system and measures the approximate second derivative $f^{\prime \prime }(x)$ along the travel direction of the test surface (Fig. 1). The approximate $f^{\prime \prime }(x)$ is given by $(S_1 + S_3-2 S_2)/d^2$ (hereinafter referred to as $f^{\prime \prime }(x)$), where $S_1$ to $S_3$ are the sensor values from the rear to the front sensors. Note that $f^{\prime \prime }(x)$ is measured along the normal direction of the guide system. The sensor train travels sequentially over the surface with a step distance being the same as the separation $d$ of the sensors. The cross-sectional shape of the surface is obtained with double integration of $f^{\prime \prime }(x)$. The exact computation is $\sum \sum (S_1 + S_3-2S_2)$.

 

Fig. 1. Schematic diagram of the conventional TPM. A set of three sensors are fixed to a rigid bench and travels along the guide system. The relative positions of the sensors are maintained during the measurement. The sensor separation $d$ defines the step distance of the scan motion.

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So far, TPMs have been used only for ground surfaces before polishing processes [23]. It is presumably because the TPMs developed so far had low spatial-sampling frequency and accuracy [24], the latter of which is mainly attributed to the noise of the distance sensor employed. In recent years, however, accurate, small, and off-the-shelf distance sensors are available.

Here we present our new CMM with the dragging TPM (DTPM), which has a potential of accurate and flexibly applicable to a wide range of high aspheric optics. In this paper, we introduce the DTPM and their general properties in Section 2. Our initial evaluation of the repeatability and accuracy of the DTPM is described in Section 3. Finally, Section4 presents a measurement of a $\phi$794 mm spherical surface with our DTPM as a demonstration.

2. Dragging three-point method (DTPM)

The TPM has been developed mainly for purposes of measuring flatness of tables for general machining tools for many years. For these purposes, a TPM unit is installed to travel on a linear guide of the tool for scan motion [15,22] (see Fig. 1). When a unit travels along the guide system over a largely curved surface, it is required that sensors in the unit have a wide measurable range or that the guide rail be curved along the test surface. The measurable range of highly precise sensors is generally limited to a few millimeters and the allowable alignment angle is also limited to a few degrees. They are critical problems to measure telescope optics, the surface of which is highly deviated from flat.

2.1 Sensor unit

We have developed a new design concept of the dragging TPM (DTPM) for the CMM and a prototype based on the design concept, where the sensor unit travel along a test surface. With this design, the problems of the conventional TPM can be overcome or circumvented. Figure 2 shows the schematic diagrams of the three-sensor and one-sensor units based on our design concept and how they work. In our design, the sensor unit is supported with three legs directly contacting on the test surface. The DTPM unit is automatically aligned with the normal of the surface regardless of the gradient, and the distance change between the sensor and test surface is maintained to be smaller than several microns. These characteristics help us to relax the requirements for the sensor and also to decrease the alignment error of sensors.

 

Fig. 2. Schematic figures of the DTPM. (a) DTPM unit of three sensors with the legs directly contacting on the test surface. Solid and dashed lines indicate the designed and actual surfaces, respectively. (b) Top view of Fig. (a). The sensors align on a straight line of the scan direction. The sensor separation $d$ corresponds to the step distance. (c) DTPM unit with a sensor. (d) Top view of Fig. (c). The two legs and the sensor align on a straight line. The sensor and the legs can be placed in any orders. (e) Top view of the DTPM unit, which is capable of scanning along a circled trajectory. The three sensors (or alternatively, two legs and one sensor) align on an arc which corresponds to the scan path.

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Figures 2(a, b) show a sensor unit of the DTPM which consists of three sensors, a rigid bench, and three legs. Because the sensors are bundled with the rigid bench, the deformation of the contact points of the legs does not produce an additional measurement error. Consequently, soft material (e.g., plastic ball) can be used as that for the contact points of the legs. Various configurations of the three legs are possible.

Figures 2(c, d) indicate a sensor unit of the DTPM with a one-sensor, a simpler design than the three-sensor one. The measurement principle by this unit is also identical to the three-sensor unit, but the sensor value of $S_1$ and $S_3$ are considered to be always zero. The relative position among the two legs and the sensor must be maintained. For this reason, the contact points of the legs are not permitted to be deformed or worn out. As a result, the material of the contact point must be hard mechanically and thermally (e.g., glass with a low thermal expansion coefficient), and the contact point might give some damage on the test surface. Any order of the sensor and two legs is acceptable.

Figure 2(e) shows a sensor unit of the DTPM for scanning along a curved line. The bias error of the sensor unit produces significant amount of measurement error. If a circle (closed) scan path is chosen, the bias error can be eliminated with a plausible assumption that the height, slope, and curvature are continuous at the connecting point of the closed path. This design would enhance the measurement accuracy. In particular, measurements with large circle paths tracing the edge of the test surface are expected to yield results with greatly improved measurement errors (Section 2.3.4).

2.2 Algorithm of DTPM

The algorithm to compute the surface figure error with DTPM is different from that with the conventional TPM (Fig. 1). Because the DTPM unit is aligned to the normal of the local surface ($\boldsymbol {n_s}$), the rolling angle of the sensor unit should be considered. $(S_1 + S_3 - 2S_2)/d^2$ represents the approximate local curvature $\rho _m(x) = \boldsymbol {n_s}\cdot \boldsymbol {n_l} \times f^{\prime \prime }(x)/(1+f^{\prime }(x)^{1.5})$, where $\boldsymbol {n_l}$ is the normal of the cross sectional curve. Now we are interested in the figure error of test surface, double integration of $\rho _m(x) - \rho _d(x)$ is used for construction of the figure error, where $\rho _d(x)$ is the local curvature of design value. The $\rho _d(x)$ should be defined along the slope of the test surface. The step distance of the scan motion is defined by the length projected on the test surface. Therefore, to design the scan path along the test surface is important. The exact computation is $\sum \sum (S_1 + S_3 - 2S_2 - Sag_d)$, where $Sag_d$ is design value of distance from the point measured by $S_2$ to the line through the points measured by $S_1$ and $S_3$, respectively. Hereinafter $S_1 + S_3 - 2S_2$ is referred to as normalized curvature.

2.3 Measurement error of the DTPM

Dominant sources of measurement error are bias error of sensor unit, random sensor noise, and alignment error of sensor unit. The bias error includes the mean value of the normalized curvatures during single scan (direct current component) and position error of the three sensors from a flat (called zero point errors).

2.3.1 Bias error

The bias error is doubly integrated and produces an measurement error of the parabolic shape in the form of $\delta _z n(n+1)/2$: sum of $n$ natural numbers, where $\delta _z$ is the bias error and $n$ is the step number. Whereas the parabolic error can be larger than 1 $\mu$m even in a case of $\delta _z \sim$ 1 nm, spherical aberration from the parabolic error is always nearly zero because the amplitude of the error is very small and the shape stays close to a sphere. For instance, an amplitude of spherical aberration of 1-m parabola with 1 $\mu$m sag is order of $10^{-9}$ mm. The bias error does not produce a big problem for optical performance of telescope optics.

The bias error produces absolute RoC error. The amplitude of the error can be suppressed to be within allowable level of 0.01% corresponding to that obtained with general interferometric tests by using a sensor unit calibrated with a conventional optical flat. Hence, the RoC error is not essential in the telescope optics except segmented mirrors in which all the segments have an identical RoC because it should be corrected by adjusting single focal position.

2.3.2 Random sensor noise

The random sensor noise is enhanced largely with the double integration process to obtain the cross-sectional shape. This error is estimated to be root sum squares of each sensor noise: a single-point measurement error of $S_1 + S_3 - 2S_2$ corresponds to $\sqrt {(\sigma _1^2 + 4\sigma _2^2 + \sigma _3^2)}$. The error propagation is given by $\sigma _R\sqrt {n(n+1)(2n+1)}$: sum of squares of $n$ natural numbers, where $\sigma _R$ is the random sensor noise. For reference, the error propagation in the two-point method is $\sigma _R\sqrt {2}n$. The measurement error is accordingly an increasing function of the scan length and sampling frequency (or inverse of $d$). Third or higher order of measurement error of $\sim$ 100 nm appears with a simulation, in the case of a single-scan measurement with 100 steps and 1-nm random sensor noise.

2.3.3 Alignment error

Pitching and piston motion errors of the sensor unit against scan direction are insensitive to the normalized curvature. Rolling motion reduces measured curvature by $\cos {\theta _R}$, where $\theta _R$ is rolling angle, and is negligible because an expected rolling angle is less than one degree, which corresponds to reducing the gain by only $10^{-4}$. However, rolling motion of the sensor unit for circled scanning path produces curvature error of $w \theta _R$, where $w$ is distance between the middle sensor and the line through the front and rear sensors. This is considerable and should be cared. Yawing motion reduces net distance between the sensor distance $d$ by $\cos {\theta _Y}$ and then reduces the gain with the same manner of the rolling motion error.

Lateral displacement errors of scanning line produce measurement error, and the magnitude of error depends on the gradient of asphericity of the optics. The position of the sensor unit with respect to the coordinates of the measurement system should be set within an accuracy of 100 $\mu$m, whereas a high apheric mirror designed with the latest technology [6] requires an accuracy of 10 $\mu$m as described in Section 1. This accuracy can be achieved by reference targets on the system. Coordinate misalignment between the optics and measurement system can be corrected by least squares method between the result and the design value.

2.3.4 Error in 2-dimensional measurement

Whereas the spherical component of the measurement error of the cross section obtained with a single scan does not need to be considered (see previous section), the spherical component of the measurement error is not negligible in the two-dimensional (2-D) measurement. The surface figure is calculated by integrating the results of multiple scans of the surface. In this integration process (hereinafter referred to as stitching process), the spherical component of measurement error is independent of those of the other scans and hence it cannot be eliminated using an arbitrary approximate sphere of the result. The spherical component associated with the bias error varies from scan to scan when the bias error arises randomly and is likely to produce mainly astigmatism. In other words, in the unlikely but ideal case where the bias error is perfectly constant, the measurement error of spherical components can be subtracted from the result using an approximate sphere of the result. The spherical errors in general cancel out each other more or less in the stitching process. The amount of the cancellation depends on the configuration of the multiple scan paths and the stitching algorithm. Roughly speaking, a more number of paths are used, more intersection points there are, and more uniform the distribution of the intersection points is, effectively better results are expected. Figure 9 shows a sample.

2.3.5 Lateral resolution

The lateral resolution of the TPM is $2d$ according to the sampling theorem. The resolution is a trade-off with the measurement error, and is inherently limited according to the physical size of the sensors. However it would not be a major issue in cases of measurements of a smooth polished surface like a mirror, which has no significant lateral structure smaller than a few millimeters scale. Furthermore, the lateral resolution can be improved with the stitching process of the multiple scan paths over the test surface.

3. Single scan measurement

To evaluate the performance of the method, we built a prototype CMM with our DTPM and executed a single-scan measurement of a flat mirror with a diameter of 300 mm with an old milling machine (Fig. 3) in a machine shop without air conditioning. In our DTPM, three sensors fixed on an invar-alloy bench with a weight of 75 g at the sensor separation $d =$ 10 mm were connected to the tool head of the machine with a thin metal plate. The mirror sits on the cushion on the table of the machine and travels, driven by automatic feed motion of the table. The lateral position was estimated from the traveling time and the velocity of the table motion.

 

Fig. 3. TPM unit and milling machine. The unit with the thin metal plate connected with the tool head of the milling machine was dragged on the flat mirror.

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The principle of the distance sensor (SI-F Keyence) is interferometry of the reflected beams between the fiber end and the test surface. The sensor had a 1-nm resolution, 5-kHz sampling frequency, 20 $\mu$m spot size, 0.05 - 1.1 mm of measurable range with 820 nm wavelength.

The measurements were conducted 27 times to evaluate repeatability of the normalized curvatures and resultant surfaces. The table velocity was 8.33 mm/s, the scan distance was 260 mm, the binning (averaging) number of the measurement points per normalized curvature was 600, and the normalized curvatures were obtained in 1-mm interval.

3.1 Result of normalized curvature

Figure 4 shows the measured normalized curvatures. The mirror has an approximately centro-symmetric shape with a normalized curvature change of 0.07 $\mu$m. Figure 5 shows the normalized curvature deviations from the average and the standard deviation (i.e., RMS) of the normalized curvatures of the 27 measurements. The RMS of the normalized curvatures was 0.6 nm at any position except at $x$ = 20 mm, which implies an excellent repeatability of the measurements. The RMS value of 10 nm at $x$ = 20 mm presumably due to some stain on the surface because that the normalized curvatures show large values at the position.

 

Fig. 4. Normalized curvatures obtained with the 27 measurements on a flat mirror as a function of the scan distance. The normalized curvature value at the start point is subtracted from each data point. Due to the excellent repeatability of the measurement, the lines are mostly indistinguishable except at $x$ = 20 mm, where fluctuations are visible.

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Fig. 5. Deviations of the normalized curvatures from their average (upper panel) and their standard deviations (lower panel) in the 27 measurements on a flat mirror as a function of the scan distance.

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3.2 Results of the surface figure

Next we doubly integrated the normalized curvatures to obtain the cross section of the flat mirror. Figure 6 shows one of the 27 measurements, together with that obtained with a Zygo interferometer (Verifire), and the difference between the two results. The repeatability of the 27 measurement is RMS = 5 nm in whole scan range but 10 nm at $x$ = 20 mm. Both the results show a double-peak curve with an amplitude of $\sim$ 500 nm. Their results were consistent with each other with a RMS of 5.5 nm and peak-to-valley (P-V) of less than 25 nm.

 

Fig. 6. Comparison between the results of the flat mirror measured with our DTPM and Zygo interferometer (left-vertical axis) and difference (right-vertical axis). Both the results were obtained with a single measurement. The spherical component was subtracted from both the results.

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The discrepancy of RMS = 5.5 nm resulted from the large single-point error at $x$ =20 and the figure error of the flat reference of the Zygo interferometer ($\lambda$/10 $\sim$ 63 nm P-V, where $\lambda$ is the wavelength of HeNe laser). Even under the poor conditions compared to environment for interferometers, the good agreement was observed. The repeatability of RMS = 5 nm of the measured surface agreed with that expected from the normalized curvature repeatability of 0.6 nm (RMS).

The single-point error produces "V-shape" measurement error with double integration process. Amplitude of the V-shape error becomes more larger at closer position of single-point error to the center of scan. In this experience, because the single-point error occured near one end of the scan, the measurement error was small.

The experiment demonstrated a potential of our DTPM, which enables us to extend a machining tool of 10-micron level accuracy to CMM for optics. This also means that the measurable area of the CMM is comparable to the working area of the machining tool and is practically not limited.

3.3 Measurement with a single-sensor unit

We built another DTPM, equipped with a single-sensor unit with two contact points made with fused silica balls as shown in Fig. 2(c) and conducted basically the same experiment, where all conditions of the measurement were the same as those with a three-sensor unit (previous subsection).

The result showed the same level of accuracy obtained with the three-sensor unit. However, after several repeated measurements, small damages were observed on the test surface, which was made of low thermal-expansion material (Clearceram-Z OHARA Inc.), and on the balls (Fig. 7). Once a damage was inflicted, the damage rapidly developed further, and as a result, the measurement accuracy became degraded rapidly. This result strongly indicates that single-sensor units are unsuitable for high-accuracy measurements of a polished surface.

 

Fig. 7. Photomicrograph of a scratched damage on a ball at a contact point.

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4. 2-D measurement

We executed a two-dimensional measurement of a mirror with the DTPM (Fig. 8). To compare a result of DTPM with an independent measurement obtained by an inteterferometer, we used a spherical mirror in hand. The mirror was a concave spherical mirror of $\phi$794 mm, RoC of 1633 mm (F1), and has a hole of $\phi$120 mm at the center. A performance for aspheric measurement of the DTPM can be estimated with a spherical mirror, because there is no big difference between aspherical and spherical surface for the DTPM measurement owing to its principle. The mirror was on a potter’s wheel on the floor with three supporting points. The experiments were carried out in a room with neither anti-vibration system nor air conditioning. We made 63 radial scans by manually rotating the wheel for every scan (Fig. 9). The length of the path was about 730 mm. The sensor unit was dragged with a linear actuator above the mirror with a thin metal sheet. The scan velocity was 20 mm/s (about 40 s for a single scan), and the sensor separation was $d =$ 10 mm.

 

Fig. 8. DTPM for the spherical mirror measurement. The mirror is on the potter’s wheel on the floor. The TPM unit is moved with a low-cost linear actuator.

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Fig. 9. Configuration of 63 scan paths. The paths are slightly deviated from the ideal regular symmetric pattern because the table was turned manually in the experiment. The rotating angle was measured by a tape scale on the side of the mirror.

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First, we evaluated the sensitivity of the method for a large scale structure (LSS). At the time of the mirror fabrication, the mirror had been supported with an ideal support system, and hence the mirror should have little figure error of the LSS. In order to produce a LSS artificially, we intentionally supported the mirror with only three points.

The RMS of the measurement normalized curvatures and cross sections calculated from every scan were 0.04 nm and 5 nm for each path, respectively. Figure 10(A) is the result of the finite element method (FEM) analysis, which clearly shows a trefoil pattern of LSS with an amplitude of 900 nm P-V induced by its own weight. Figure 10(B) is the result of this work, made with stitching of the 63 single scans by using a new stitching algorithm (in preparation).

 

Fig. 10. Results obtained with (A) the FEM analysis and (B) this work. Open circles indicate the positions of the supporting points. Spherical components are subtracted from the results. The gray scale indicates the height from −450 nm to 450 nm

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The result of the TPM is overall consistent with that of the FEM, but discrepancy is considerable at the edge of the mirror presumably because of accumulated errors in the double integration process of the random sensor noise and the bias error. In order to improve the measurement accuracy for the LSS, introduction of circular scan paths and additional straight paths having many intersections with the other scan paths at the outer region would be efficient. Combining it with a sensor unit of a large sensor separation $d$ is promising to reduce the measurement error of the LSS.

Next we discuss the sensitivity of our method for the small-scale structure (SSS). Figure 11(A) shows the result obtained with a Fizeau interferometer (performed before the previous experiment) with an ideal supporting system to avoid deformation due to its own weight. The mirror had a centro-symmetric structure with an amplitude of 30 nm and several tens millimeters periodic structure produced with rotary grinding in a pre-process of the polishing. To compare the SSS, we show an image of the SSS of the mirror in Fig. 11(B), where we subtract the LSS estimated with the FEM (Fig. 10(A)) from the result obtained in this work (Fig. 10(B)). The trefoil pattern still remains. We interpret that the residual trefoil pattern is mainly caused by the misalignment between the boundary conditions (e.g., positions and friction of the contact points) in the measurement and the FEM analysis. The method well detects the SSS. At the central region, the density of the scan path is high and the SSS is detected with a higher resolution than the theoretical lateral resolution of 20 mm in the condition. By contrast, the resolution is lower at the outer region presumably due to the lower sampling density.

 

Fig. 11. (A) Result obtained with a Fizeau interferometer. The gray scale shows the height −50 nm to 50 nm. (B) The deference between the results obtained this work and the FEM analysis (Fig. 10(B) $-$ Fig. 10(A)). The gray scale indicate the height of −200 nm to 200 nm. Open circles indicate the positions of the supporting points. Spherical components are subtracted from the results.

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6. Conclusions

We developed a new CMM for telescope optics with the DTPM. The lateral resolution of DTPM is low, however, it has many advantages, including (1) small size, (2) robustness against the environment, (3) having potential of extension of tool to CMM, and (4) low costs. We achieved with the DTPM an excellent measurement accuracy of 5.5 nm on the flat surface, even though the conditions of the experiment were poor. Our experiment showed a potential of the DTPM, with which the conventional machining tool can be applied to CMM for high aspheric optics. This also means that the measurable area of the DTPM is practically not limited. As future work, development of an improved stitching algorithm for multi-scan paths, further study for optimization of the configuration of the paths for effective data correction, and implementation of circled scan paths and correction algorithm of the gain for small-scale structure would be important to improve the accuracy further. Applications of the method to manufacturing aspheric, convex, and off-axis optics are required as the next step.