Evaluation of the deformation value of an optical flat under gravity

Figures 3 and 4 respectively show a photograph and a cross-sectional view of the reference flat. The reference flat is 350 mm in diameter and 100 mm in thickness, and is made of synthetic fused silica. It is mounted in an aluminum ring with 54 supporting points as shown in figure 4. The deformation value of the reference flat under the force of gravity was calculated by FEM analysis [17]. The deformation value of the reference flat was 23 nm in the case of a 300 mm diameter. A correction map of the reference flat was made from the results of the three-flat test and FEM analysis.

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To verify the FEM analysis results, the deformation value of the reference flat was experimentally measured. Figure 5 shows a schematic view of the experimental measurement. First, the surface profile of the reference flat is measured with respect to another optical flat before the reference flat is mounted in the aluminum ring. Then, the same measurement is carried out after the reference flat is mounted. Finally, the difference between the two results gives the deformation value of the reference flat under the force of gravity.

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Figure 6 shows a comparison of the results by FEM analysis (solid line) and experimental measurement (dotted line). The conditions of the FEM analysis and experimental measurement are reported in detail in the literature [17]. The results show good agreement, but there is a peak-to-valley (P–V) difference of a few nanometers. The difference between experimental and calculated results is a major source of uncertainty of the Fizeau flatness interferometer. To reduce the uncertainty of the Fizeau flatness interferometer, it is necessary to improve the evaluation of the reference flat under the force of gravity.

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We developed the SDP to evaluate the deformation value of the reference flat. The SDP does not require a reference flat and can directly measure the surface profile.

Figures 7 and 8 show a schematic view and a photograph of the developed SDP, respectively. The SDP is constructed using an autocollimator and a pentamirror fixed on an air slide table. The travel distance of the air slide table is 1.2 m. The measurement beam of the autocollimator is bent by the pentamirror so that it illuminates a specimen. The optical system of the SDP is aligned according to the literature [18–20] of an alignment procedure for a deflectometry system. The size of the measurement beam is $\varnothing$34 mm. An aperture is set in front of the specimen for increasing the special frequency. Local slope angles at each position of the specimen are measured by scanning the pentamirror. The pentamirror is effective for minimizing the moving error during scanning, and the influence of the moving error on the local slope angle measurement is negligible [18–20]. The surface profile F(x_i) of the specimen is obtained by integrating the local slope angles as follows:

$$\begin{equation} f( {x_i }) = 0\quad ( {i = 1})\end{equation} \tag{ 1 }$$

$$\begin{eqnarray} &&\fl f( {x_i }) = f( {x_{i - 1}})+ \{ {f'( {x_i }) - f'( {x_1 })}\}( {x_i - x_{i - 1} })\nonumber\\ &&( {i = 2,3, \ldots ,N}), \end{eqnarray} \tag{ 2 }$$

where i is a measurement number, N is the total number of measurement points, f(x_i) is a profile,  f'(x_i) is a measured local slope angle and x_i is measurement position. The measured slope angle f(x₁) of the first measurement position x₁ can be selected as an arbitrary angle. The numerical integration of equations (1) and (2) is calculated such that the measured angle of the first measurement position is zero. A dc component a₀ of the measured slope angle f'(x_i) is a line g(x_i) = a₀x_i + b. Then, the least-squares line g(x_i) is calculated from the profile f(x_i). By subtracting g(x_i) from f(x_i), we can obtain the final profile F(x_i) as follows:

$$\begin{equation} F( {x_i }) = f( {x_i }) - g( {x_i }).\end{equation} \tag{ 3 }$$

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We used a commercial autocollimator (MÖLLER-WEDEL OPTICAL GmbH, ELCOMAT 3000) of which the total measurement range and the resolution are ±1000 arcsec (5.1 mrad) and 0.001 arcsec (4.8 nrad), respectively. The measurement accuracy of the autocollimator is the most important aspect of the SDP based on the angle measurement. We calibrated the autocollimator using a high-accuracy angle index table at NMIJ. The index table is controlled in arbitrary angle positions around 360° using a rotary encoder of which the total number of graduation lines and the electric interpolator are 225 000 and 2048 times division, respectively; therefore, the index angle resolution is 0.0028 arcsec (13.6 nrad). The servo-control stability is about ±1 pulse (±0.0028 arcsec). Figure 9 shows a photograph of a calibration setup of the autocollimator. We set a highly reflective mirror on the index table. The distance between the autocollimator and the mirror is 160 mm. The autocollimator is calibrated by measuring the rotation angle of the mirror. NMIJ is making an uncertainty budget for measuring a rotation angle, though the measurement accuracy is less than 0.005 arcsec. The measurement system of a rotation angle is reported in detail in the literature [21]. The evaluation results of the autocollimator using the high index table are described in the following sections.

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3.2.1. Random error

Figure 10 shows the short-term stability of the autocollimator, when the mirror is controlled at an angular position. The sampling rate and the number of measurement points were 25 Hz and 100 points, respectively. The standard deviation σ_rand was 0.008 arcsec (38.9 nrad).

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As a surface profile is calculated by integrating the local slope angles at each measurement position, the random error is also integrated. The standard uncertainty u_rand of the random error is given by

$$\begin{equation} u_{{\rm rand}} = \sigma _{{\rm rand}} \cdot \sqrt N {\rm d}x, \end{equation} \tag{ 4 }$$

where N and dx are the total number of measurement points and the scanning pitch, respectively. When N and dx are 300 points and 1 mm, respectively, the standard uncertainty is 0.67 nm. The final profile F(x_i) is obtained by subtracting a dc component of the measured angle slope f'(x_i) from equation (3); therefore, a dc component of an integrated random error is also subtracted. Equation (4) is overestimated but the expected value of the random error is zero and the overestimated value is very small.

3.2.2. Systematic error

The reference flat of the Fizeau flatness interferometer is approximated by the quadratic curve of the concave surface. The maximum profile deviation (sag value) and the maximum deviation of the local slope angle over the measurement range of 300 mm are 23 nm and ±0.06 arcsec, respectively. Thus, the measurement range of the autocollimator is very small. Figure 11 shows the calibration curve of the autocollimator. The range and pitch of the calibration angle were ±5 arcsec and 0.05 arcsec, respectively. The maximum deviation e_a of the angle over the range of calibration (P–V value) was 0.032 arcsec (155.1 nrad). The measurement range of the autocollimator varies with the surface profile of the specimen. We assume that the standard deviation of the systematic error is set to be the uniform distribution of the maximum angle deviation of the calibration curve, then the standard uncertainty u_sys of the systematic error is given by

$$\begin{equation} u_{{\rm sys}} = \frac{{e_{\rm a} }}{{2\sqrt 3 }} \cdot \sqrt N\, {\rm d}x.\end{equation} \tag{ 5 }$$

N and dx are 300 points and 1 mm, respectively, and the standard uncertainty is 0.78 nm.

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The literature [11, 22–24] reported that the systematic error of an autocollimator is changed by measurement conditions: distance between an autocollimator and a mirror, aperture size and reflectance. The evaluation results in figures 10 and 11 were only one condition. We are estimating an uncertainty budget of the SDP from a number of measurement conditions.

To verify the FEM analysis results, we measured an optical flat using the Fizeau flatness interferometer and the SDP. Figure 12 shows a photograph of the optical flat used for comparison, which is 340 mm in diameter and 70 mm in thickness, and is made of the ultralow-expansion ceramic NEXCERA™. A mask of size 300 mm × 30 mm is placed on the optical flat. The comparison measurement is performed along the center line of the mask.

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The optical flat was set up on the rotary stage of the Fizeau flatness interferometer. The flatness of the optical flat within the mask area was measured. After the measurement, the optical flat was rotated 30°. Then, the flatness measurement was performed again. As a result, the optical flat was set up in 12 different orientations and measured in each orientation. Figure 13 shows the measurement results for each orientation. The orthogonal coordinate XY is the CCD camera coordinate of the Fizeau flatness interferometer and all orientation results have a common coordinate. The pixel size is 1024 × 1024 and the lateral size is about 0.34 mm/pixel. Figure 14 shows the center line of the mask, which was extracted from the measurement results for each orientation. The P–V value and the root mean square (RMS) value for the average curve of the measured profiles were 32.7 nm and 10.8 nm, respectively. The standard deviation of the P–V value and the RMS value for each orientation were 0.46 nm and 0.11 nm, respectively. Figure 15 shows the difference curves from the average center line curve for each orientation. The profiles for each orientation are in good agreement, and the dispersion excluding the spike noise was about ±1 nm. This result indicates the reliability of the correction map for the reference flat of the Fizeau flatness interferometer obtained by the three-flat test. However, this result does not validate the correction map for the deformation under the force of gravity obtained by FEM analysis. This is because of the axisymmetrically deformed profile. The correction map for the deformation is an axisymmetrical profile. When a profile of an optical flat with respect to a different line of the reference flat is measured, the amount of correction for a deformation of the reference flat is not changed. Thus, the correction map for the absolute deformation value of the reference flat cannot be validated from the results in figures 13–15.

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The profile of the optical flat was measured using the SDP. The aperture size was $\varnothing$8 mm and the scanning pitch was 1 mm. To avoid drift, we adopted a measurement strategy [25] in which the measurement was performed in four directions: forward, backward, backward and forward. Figure 16 shows the measured local slope angle. The measurement range of the autocollimator was about 0.4 arcsec. The profile was calculated by integrating the curve obtained by averaging the four obtained measurement curves. Figure 17 shows the calculated profile. There are 10 profiles, that is, the total number of measurements was 40. The P–V value and the RMS value for the average curve of the measured profiles were 32.2 nm and 10.5 nm, respectively. The standard deviation of the P–V value and the RMS value for ten profiles were 0.62 nm and 0.15 nm, respectively. Figure 18 shows the difference curves from the average profile curve. The repeatability was ±1.3 nm.

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Figure 19 shows the measurement results obtained by the two different methods. The P–V values for the Fizeau flatness interferometer and SDP were 32.7 nm and 32.2 nm, respectively. The RMS values for the Fizeau flatness interferometer and SDP were 10.8 nm and 10.5 nm, respectively. Figure 20 shows the difference curve between the Fizeau flatness interferometer and the SDP. The measurement results are in good agreement, and the P–V value and the RMS value for the difference curve were 3.3 nm and 0.60 nm, respectively. For the Fizeau flatness interferometer, when the correction map for the deformation value of the reference flat under the force of gravity calculated by the FEM analysis is incorrect, the difference curve has a quadratic component. As a result, the quadratic component of the difference curve was less than 1 nm P–V. The comparison results validate the correction map for the reference flat obtained using the Fizeau flatness interferometer by the three-flat test and FEM analysis.

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