Absolute surface metrology by rotational averaging in oblique incidence interferometry Weihao Lin, Yumei He, Li Song, Hongxin Luo, and Jie Wang* Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201204, China *Corresponding author: [email protected] Received 4 March 2014; revised 15 April 2014; accepted 20 April 2014; posted 23 April 2014 (Doc. ID 207510); published 22 May 2014
A modified method for measuring the absolute figure of a large optical flat surface in synchrotron radiation by a small aperture interferometer is presented. The method consists of two procedures: the first step is oblique incidence measurement; the second is multiple rotating measurements. This simple method is described in terms of functions that are symmetric or antisymmetric with respect to reflections at the vertical axis. Absolute deviations of a large flat surface could be obtained when mirror antisymmetric errors are removed by N-position rotational averaging. Formulas are derived for measuring the absolute surface errors of a rectangle flat, and experiments on high-accuracy rectangle flats are performed to verify the method. Finally, uncertainty analysis is carried out in detail. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.4800) Optical standards and testing; (120.6650) Surface measurements, figure; (340.6720) Synchrotron radiation. http://dx.doi.org/10.1364/AO.53.003370
1. Introduction
The rapid development of dedicated synchrotron radiation sources and free electron laser facilities places an unprecedented high-precision demand on optical manufacturing and metrology. Surface figure deviation has been a major consideration in the use of grazing incidence optics in synchrotron beam line instrumentation [1]. Generally, optical surfaces used to reflect x rays at extreme grazing incidence angles have a long tangential dimension of up to 1 m or more. Owing to the nature of grazing incidence x-ray optics, conventional interferometric techniques for surface figure measurement are not easily employed in the absolute testing. The best known surface figure measurement method of large synchrotron beam line optics is the scanning pentaprism test [2–5]. The test typically uses a laser scanning technique, which obtains the
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surface slope from the deflection angle of the reflected laser beam. The surface figure is then calculated by slope integration. The representative instrument is the long trace profiler [6–9]. This method offers a good solution but is time consuming and often marred by a few inaccuracies due to the vibration and mechanical movement of the scanning elements. Besides, the scanning systems usually yield absolute profiles only along certain lines over the surface of interest. Interferometry is the common method of flatness measurement of optical surfaces due to its speed and accuracy. However, if the size of the optics is larger than 150 mm, the use of a commercial standard 150 mm interferometer will become difficult. The subaperture stitching technique stitches together many smaller measurements [10,11]; however, this requires a large translation stage and it is difficult to stitch many measurements on very flat optics without introducing errors. The Ritchey–Common test permits for testing of large flats in a diverging spherical beam [12]. It requires a large spherical mirror and creates a large air path.
The test is not widely used, owing in part to difficulties in data reduction. The Skip-Flat test, which uses a flat mirror as the reflective element, is a variation of the Ritchey–Common configuration [13,14]. This test uses an interferometer with a collimated output that is much smaller than the test surface. The collimated beam, reflected from the test surface at an oblique angle, is returned with a flat mirror to provide a narrow profile. For most grazing incidence x-ray optics in which the width is shorter while the length is longer than the aperture size of the interferometer, the full aperture test can be done, which avoids stitching. At large incidence angle, the surfaces under test appear smoother, so the Skip-Flat test is a good approach for the measurement of rough surfaces. However, to test high-precision flats, the accuracy of this method is usually limited by the reference surface and the transmission surface, of which the normal quality is about λ∕20 PV (about 6 nm rms), so it necessitates the use of absolute testing techniques to reach subnanometer accuracy in large surface figure measurement. A demonstration on synthetic data of absolute flatness measurement using oblique incidence setup and iterative algorithm has been reported recently, showing the capability of the approach to achieve an accuracy on the order of nanometer rms [15]. As to interferometric metrology, three-flat tests are the archetypes of measurement procedures to separate the test surface errors from the reference surface errors [16–33]. Several ingenious techniques have been devised to obtain an absolute two-dimensional (2D) map of the surface under test. Typical methods are Zernike polynomial fitting [18], even and odd function decomposition [20,27], and solutions based on rotation symmetry [30] and mirror symmetry [31,32]. However, in all three-flat methods, the reference flat surface is dismantled from the interferometer and is calibrated by procedures that should use combinations of pairs of three surfaces, including the reference flat surface and the other two flat surfaces. On account of difficulties in setting the reference flat to the interferometer exactly and quickly, these methods are not easily used. Furthermore, Zernike polynomials are not appropriate for wavefront analysis of noncircular pupils, such as rectangular, elliptical, and annular pupils, due to their lack of orthogonality over such pupils [34]. In the method described in [20], it is assumed that all flats can be expressed as the sum of four components using the symmetry properties of the odd and the even functions. However, this method requires that all interferograms maintain twofold mirror symmetry in the uniform Cartesian coordinate system, which introduces some difficulties in calibrating noncircular flats. Therefore, many of the previously mentioned methods are not very suitable for absolute flatness measurement of long grazing incidence optics with rectangular shape. In this work, we present a simple method for measuring an absolute figure of the large optical flat
surface in synchrotron radiation by a small aperture Fizeau interferometer. The oblique incidence interferometry combined with multiple rotating measurements at N equally spaced positions is performed to determine the absolute flatness of the overall 2D large surface. The advantage of the proposed solution is that the reference flat stays in the same position throughout the testing procedure, while in three-flat methods, the reference flat is dismantled. In addition, as a consequence of the high reflectivity of the x-ray mirrors, at visible wavelengths, uncoated transmission and reference flats could be used in oblique incidence measurement and the fringe contrast would generally be very good. 2. Principle
Before describing the solution of large optical surface testing, properties of mirror symmetry [31,32] and rotation symmetry [23,24,30] through an N-position average are briefly reviewed owing to the pivotal importance of these concepts to our method. For any given wavefront Wx; y, it can be written as a sum of a mirror symmetric component W e x; y and a antisymmetric component W o x; y. They are constructed under reflections at the y axis, which are defined as 1 W e x; y Wx; y W−x; y; 2 1 W o x; y Wx; y − W−x; y: 2
(1)
As to the solution based on rotation symmetry, the wavefront Wx; y can be decomposed into the rotational invariant component W R x; y and rotational variant component Ωx; y. W R x; y, also called rotation average, is defined as ! N−1 1X k·ΔΦ W x; y lim Wx; y : N→∞ N k0 R
(2)
Here, the operator •ΔΦ is used to indicate a rotation by an angle ΔΦ 2π∕N. The rotational variant wavefront component Ωx; y is the difference between Wx; y and W R x; y. The definition of the rotation average implies that Ω has a rotation average of zero: ΩR W − W R R W R − W R 0:
(3)
Applying Eqs. (2) and (3) to Eq. (1) results in W e R W R ;
W o R 0:
(4)
We are shown by Eq. (4) that the rotational invariant part of the wavefront is entirely contained in the mirror symmetric component, and the rotational average of the mirror antisymmetric component is zero. To obtain the interferometry from the oblique incidence configuration with a Fizeau interferometer, one more reference flat is required to return back 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
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the beam reflected by the long test surface. We name A, B, C for the transmission flat, reference flat, and the long test flat, respectively. For each surface, a Cartesian coordinate system is considered, with the origin at the center and the z axis coincident with the external normal. The method can be illustrated in Fig. 1. The first step is to make a measurement in the oblique incidence configuration as described in Fig. 1(a), in which the test flat C is inserted with its external normal at an angle α to the optical axis of the interferometer. Then, rotating measurements of N equally spaced positions between the transmission flat and reference flat are performed, as shown in Fig. 1(b). Note that the initial position of the reference flat in rotating measurements is coincident with the first step in the oblique incidence test. According to the Cartesian coordinate system of the interferometer defined in Fig. 1, the results of the measurements can be written as W 1 x; y W A −x; y W B −x; y x ;y ; 2 cos α · W C cos α
W e1 x; y W o1 x; y W eA x; y − W oA x; y W eB x; y − W oB x; y 2 cos α x x ; y W oC ;y ; · W eC cos α cos α W e2 x; y W o2 x; y W eA x; y − W oA x; y W eB x; y W oB x; y;
W 2 x; y W A −x; y W B x; y; W 3 x; y W A −x; y W B x; yR :
The negative sign for the x coordinate accounts for the flip of the flat in the x axis. Here, we consider that the flatness error of the surface is half of the wavefront error in normal reflection. Owing to the oblique incidence configuration, spatial resolution in the horizontal direction is impacted by a scale 1∕ cos α as shown in Eq. (5). Prior to processing, interferograms from experiments are prepared by removing pistons and tilts in the least-squares sense. All wavefronts in Eq. (5) can be split into their mirror symmetric and antisymmetric components according to Eq. (1), and the antisymmetric component of wavefront W B is zero under rotational averaging, as described in Eq. (4). Therefore, the expression of Eq. (5) can be rewritten as
(5)
Here, W i x; y, i 1, 2, 3, defines the measurement of the relative wavefront errors. Note that W 3 x; y shows the average result of N-position rotating measurements. For simplicity, signs are taken to be positive. W A x; y, W B x; y, and W C x; y stand for wavefronts normally reflected by the transmission flat, reference flat, and the long test flat, respectively.
W e3 x; y W o3 x; y W eA x; y − W oA x; y W B x; yR 0:
(6)
Here, antisymmetric components can be independently extracted from Eq. (6), as given in the matrix form 2
W o1 x; y
3
2
−1 −1 2 cos α
6 o 7 6 W 2 x; y 7 6 4 5 4 −1 o W 3 x; y −1
1
0
0
0
32 76 54
W oA x; y W oB x; y
3 7 5:
W oC x∕ cos α; y (7)
Where we invert the matrix, the solution for antisymmetric components is 2
W oA x; y
3
7 6 o 7 6 W B x; y 5 4 W oC x∕ cos α; y 2 0 6 4 0
0
−1
1
−1
32
W o1 x; y
3
76 W o x; y 7 5: 54 2 o W 3 x; y 1∕2 cos α 1∕2 cos α −1∕ cos α
Fig. 1. Schematic representation of the absolute flatness testing for long optics. (a) Configuration of oblique incidence measurement in top view. The large surface under test is arranged by its external normal at an angle α to the optical axis of the interferometer. (b) N-position rotating measurements between transmission flat and reference flat in side view. The reference flat is rotated by an angle ΔΦ 2π∕N in N equal steps about the optical axis. The coordinate system of the interferometer is indicated with bold arrows. 3372
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(8) According to Eq. (6), the mirror symmetric component of the large surface can be solved as x 1 ;y W e1 x; y − W e2 x; y: cos α 2 cos α
W eC
(9)
Finally, two parts of the wavefront in Eqs. (8) and (9) can be added to get the solution for the deviation of the large surface: x x x WC ;y W eC ;y W oC ;y cos α cos α cos α e o e o W x; y W 1 x; y − W 2 x;y W 2 x;y − 2W o3 x;y 1 2 cos α W 1 x; y − W e2 x; y W o2 x;y − 2W o3 x;y : (10) 2 cos α
The differential of Eq. (13) follows: − sin α · dα
Please note that the shape of the wavefront W 1 is rectangle while W 2 and W 3 are circles. Therefore, wavefronts W 2 and W 3 should be sheared to the same data form of W 1 based on uniform space coordinates. In addition, most x-ray mirrors are made today with extremely smooth surfaces whose slope errors are below 1 μrad rms and roughness errors less than 3 Å rms. In this case, it is sufficient to use the bicubic spline interpolation to reconstruct the complete surface figure [35], with negligible influence on the final accuracy achieved. The corresponding simulation results are displayed in Section 4. 3. Uncertainty Analysis
There are a number of contributions to the uncertainty of any interferometric measurement of a surface figure. Here, we analyze the combined uncertainty of our method from the systematic part and the random part. Systematic errors are mainly introduced by the finite number of N equally spaced positions taken for the average during rotating measurements and the oblique incidence angle. The random error represents the measurement reproducibility error, which is mainly due to the changes in the environment during the long period of measurements (air, temperature, vibrations). Now we estimate the measurement uncertainty by these components as mentioned previously.
dW C
dW f 1 − cos α · dLC − · WC: 2 cos α LC
ΔW C ≈
ΔW f 1 − cos α · ΔLC − · W C; 2 cos α LC
Wf dW f sin α · dα · W f ; (12) dW C d 2 cos α 2 cos α 2 cos2 α
where the oblique incidence angle α can be determined by the ratio of protection length LC measured on a charge-coupled device (CCD) detector to real length L of the long surface. It is given by cos α
LC : L
(13)
(16)
where ΔW f is dependent on the finite rotating number. We define ΔΩR as representing the approximate residual error, which the rotational variant component of wavefront W B is not completely removed by the N-position average. Rotational variant component Ω can be expressed as a Fourier series according to its periodicity of 2π: Ωr; θ
∞ X an · cos nθ bn · sin nθ;
(17)
n1
where rotating a wavefront to N equally spaced positions and averaging could remove rotational variant terms of all angular orders except those that are integer multiples of N, which is described briefly in [23]. Therefore, mirror antisymmetric components in the fourth expression of Eq. (6) are revised as (18)
Substituting Eq. (18) into Eq. (11), we obtain ΔW f 2 · ΔΩR :
(11)
Substituting Eq. (11) into Eq. (10), we get the differential simultaneously on both sides of Eq. (10):
(15)
If we assume that dW f ΔW f and dLC ΔLC , then dW C ≈ ΔW C . It is obvious that ΔLC depends on the resolution of the CCD. Therefore, Eq. (15) is approximated as
W o3 −W oA ΔΩR :
For the sake of simplicity, we define W f , representing the numerator of the right-hand side in Eq. (10), as
(14)
For simplicity, here we assume that dL dLC . Using W f 2 cos α · W C and Eq. (14), then Eq. (12) can be solved as
A. Evaluation of the Uncertainty Associated with Systematic Errors
W f W 1 − W e2 W o2 − 2W o3 :
L · dLC − LC · dL : L2
(19)
Applying Eq. (19) to Eq. (16), the systematic error is evaluated as ΔW C ≈
ΔΩR 1 − cos α · ΔLC − · WC: cos α LC
(20)
In addition, if we assume ΔS for the error of the bicubic spline interpolation used to reconstruct the complete surface figure, the final systematic error is expressed as ΔW C
ΔΩR 1 − cos α · ΔLC − · W C ΔS; cos α LC
1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
(21)
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where we see that the systematic uncertainty is mainly determined by the finite rotating average error, the oblique incidence angle, the resolution of the CCD, and the quantity of the long surface itself. Generally, LC is a constant denoting the aperture of the interferometer because we usually make full use of aperture to obtain as much information as possible. B. Evaluation of the Uncertainty Caused by Random Errors
If we assume δ0 for the standard deviation of random errors in all wavefront measurements which are uncorrelated, the uncertainty of each item in Eq. (10) can be calculated according to the knowledge of uncertainty propagation [36] as 1 δ2 W e2 δ2 W o2 δ20 ; 2 1 1 2 δ : δ2 W o3 δ2 W 3 2 2N 0
(22)
Therefore, the combined standard uncertainty δr of random errors is given by r 1 1 · 1 · δ0 : (23) δr p N 2 cos α The result has shown that the combined standard uncertainty of the measurement of the long optical flat surface is inversely proportional to cos α. In addition, increasing the rotating number N is also essential for achieving a low uncertainty caused by random errors. 4. Numerical Simulations
In order to validate our method, we have generated three synthetic surfaces using 36 random Zernike polynomials with zero piston and tilt, and flatness of λ∕20 PV. Two of them were circular surfaces of 400 pixels in diameter, while another surface, representing the long flat under test, was rectangular with 2664 pixels in the tangential dimension. Then we have computed the corresponding interferograms using Eq. (5) by choosing 82° for the oblique incidence angle. The rotational average was approximated with 12-position averaging. After processing as previously described, we have compared the reconstructed surface with the surface generated initially. The difference between initial and final maps is 1.877 × 10−5 λ rms, as shown in Fig. 2 (λ 632.8 nm). The residual errors in the final results, from a systematic standpoint, mainly have two sources, including the finite rotating number and the influence of the oblique incidence angle. Here, we have run a number of simulations in order to get an impression on the order of magnitude for the systematic error by each influence, respectively. The effect of the finite rotating number on the rotating average error ΔΩR is displayed in Fig. 3. We could see that the flatness of the reference flat, which is rotated in our 3374
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Fig. 2. Simulation of the absolute oblique incidence test at an 82° angle with 12-position averaging. Top row: simulated long flat wavefront; middle row: reconstructed long flat wavefront; bottom row: difference between the simulated wavefront and the reconstructed wavefront by enlarging 500×.
method, has an influence on choosing the rotating number advisability according to the required accuracy. As 36 terms of Zernike polynomials are up to a maximum azimuthal frequency of 5θ, the result satisfies our expectation that having N ≥ 6 rotational average measurements could remove rotational variant terms mostly. Note that the operation of the wavefront rotating in software necessarily involves interpolation errors that would not arise in the case in which a physical part is rotated about the optical axis. The result reveals that the rotating average error decreases to 7.682 × 10−6 λ rms when the flatness of the transmission flat is λ∕20 PV and the rotating number is selected as 12. Next, we have simulated the influence of the oblique incidence angle on the final residual errors. It is clear that the systematic uncertainty increases if the oblique incidence angle becomes large, seeing the first two items of Eq. (21). The third item, which means interpolation errors, is also associated with the oblique incidence angle for the data compression scaling. The overall figure is reconstructed by the bicubic spline interpolation in our method due to the oblique incidence test, which leads to data compression in the horizontal direction. Now we have changed the oblique incidence angle for different
Fig. 3. Decrease of rotating average error rms by N rotational position averaging with different flatness of the reference flat B which is rotated.
Fig. 4. Relative interpolation error due to oblique incidence angle for different flatness of the long surface under test.
flatness of the long surface to observe interpolation errors. The simulated results are shown in Fig. 4. Clearly, compared to the respective original wavefront errors, relative interpolation rms value errors are consistent for different flatnesses of the long surface. The relative interpolation error is 0.06% when the oblique incidence angle works at 82°. Furthermore, we see that the relative interpolation error is below 0.1% if the oblique incidence angle works under 84°. Please note that the precondition of this result is that the testing surface is the smooth surface without acute local defect. 5. Experiment Results and Discussion
We demonstrated the absolute oblique incidence test with a Zygo GPI-XP phase-shifting interferometer working at λ 632.8 nm. The camera is a 640 × 480 pixel CCD detector with 370 μm resolution. The reference and transmission flats are both made of 150 mm diameter fused silica with about λ∕20 PV. In order to validate our method, we chose a rectangular plane mirror (single crystal silicon, size 245 × 50 × 40 mm, Rhodium-coated) produced by SESO to perform as the long surface. The temperature of the optical laboratory was kept at 23 0.5°C. The experiment was carried out according to the configuration described in Fig. 1. The first step was oblique incidence measurement, and the second was multiple rotating measurements. During all the measurements, the transmission flat A remained attached to the Fizeau interferometer, so we can assume that its spatial position was fixed. A fiducial crosshair generated by the interferometer software MetroPro was used to mark the center of flat A, and an auxiliary physical crosshair was used to mark the center of reference B, so as to guarantee the spatial coincidence. The misalignment was controlled to be within two pixels. All the experiment data were exported from the Zygo interferometer and processed by our algorithm procedure according to Eq. (10). Since the wavefront arrays were stored in matrix form, it was easy to perform the reflection operation by flipping the matrix left/right to get the mirror symmetric and antisymmetric components. Then we used the bicubic spline interpolation to reconstruct the complete surface figure.
Fig. 5. Measurement results of the flatness of the rectangular plane at oblique incidence angles of 38.9°, 61.0°, 72.8°, and 81.7°. To the left are the relative test results, and to the right are the corresponding reconstructed absolute results with 24-position averaging.
For the purpose of analyzing the influences of oblique incidence angle and rotating number for measurement accuracy, we selected a 120 × 30 mm active area of the rectangular plane and carried out the measurements at four different oblique incidence angles with varied rotating numbers. Measurement results of the flatness of the rectangular plane at oblique incidence angles of 38.9°, 61.0°, 72.8°, and 81.7° were shown in Fig. 5. We displayed simultaneously the oblique incidence relative test results and reconstructed absolute results after removing figure errors of the reference and transmission flats with 24-position rotational averaging. It is clear that the difference between the two measurement results becomes more evident when the oblique incidence angle increases, which highlights the necessity of our method to obtain the absolute figure errors of high-precision flats from large oblique incidence tests. Taking the 81.7° oblique incidence test for example, the figure errors of the rectangular plane measured from the relative interferometric test are 56.92 nm PV and 7.24 nm rms, while we calculate 50.29 nm PV and 6.35 nm rms by 24 rotational
Fig. 6. Comparison of reconstructed oblique incidence test results from oblique incidence angle of 81.7° with the normal incidence absolute result. Top row: reconstructed relative flat figure; middle row: reconstructed absolute flat figure with 24-position averaging; bottom row: absolute flat figure obtained from the normal incidence measurements using modified Griesmann’s approach. 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
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Fig. 7. Differences of measured figure results of rectangular plane by oblique incidence and absolute normal incidence methods. To the left are the differences of the relative oblique incidence result and normal incidence result, and to the right are the differences of the absolute oblique incidence result and normal incidence result.
averaging in oblique incidence interferometry. To compare the oblique incidence test results, we calibrated the absolute figure of the rectangular plane in the active area within the aperture of interferometer, using a modified three-flat test approach based on a simple mirror symmetry described by Griesmann [31]. The comparison results are shown in Fig. 6 and the difference maps are exhibited in Fig. 7. Absolute flat figure errors obtained by modified Griesmann’s approach, in which the rotating number was selected as 24 as well, are 51.82 nm PV and 6.02 nm rms. We find out that the results of the absolute oblique incidence test and modified threeflat test coincide well with each other both in the surface map and in numerical value, as it can be seen that the pixel-to-pixel difference between the two methods is 0.69 nm rms (Fig. 7). By comparison, our method has reduced the difference in rms to
and rotating numbers as displayed in Table 1. It is found that the results gradually approximate to the true flatness errors as we have increased rotating numbers and tested at lower oblique incidence angles, which agrees well with the simulation results as mentioned above. Here we have referred to modified three-flat test results as the true flatness errors of the rectangular plane. According to the uncertainty analysis described in Section 3, we have estimated the final standard uncertainty of the absolute oblique incidence measurement working at 81.7° with 12-position rotational averaging. First, we consider the reproducibility error, which fundamentally limits a test setup’s ability to make reliable measurements. Experiments were implemented 50 times continuously and results were averaged every 16 measurements. The statistical analysis of results has got a reproducibility error δ0 of 0.00081λ (0.51 nm) rms. Then, the standard random uncertainty δr of the absolute oblique incidence test could be calculated according to Eq. (23): r 1 · δ 2.6 nm: δr p · 1 12 0 2 cos 81.7° 1
Next, we estimate the standard systematic uncertainty δs based on Eq. (21). From simulation results, the standard deviation of 12-position rotating average error δΔΩR is 7.682 × 10−6 λ rms and the relative bicubic spline interpolation error is 0.06%. In addition, δWc , standing for the standard deviation of flatness errors of the long rectangular plane, is 6.37 nm rms, which is measured at 81.7° with 12-position rotational averaging. Therefore, we can calculate the standard systematic uncertainty δs as
s 2 2 δΔΩR 1 − cos 81.7° × 0.37 2 · δWc 0.06% · δWc 0.036 nm: δs cos 81.7° 150
approximately one fourth that of the conventional relative oblique incidence test. Moreover, we have comprehensively listed results of flatness errors of the rectangular plane solved by the absolute oblique incidence test with different oblique incidence angles Table 1.
Angle/deg
3376
(25)
Then, the combined standard uncertainty of the absolute oblique incidence measurement is q δ δ2s δ2r 2.6 nm:
(26)
PV and RMS Values of Flatness Errors of the Rectangular Plane by Absolute Oblique Incidence Test with Different Oblique Incidence Angles and Rotating Numbers
N3
38.9 61.0 72.8 81.7
(24)
N4
N6
N 12
N 24
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
52.86 52.96 53.72 54.01
6.22 6.43 6.50 6.83
52.77 52.62 52.89 53.34
6.18 6.31 6.37 6.60
52.38 52.57 52.65 52.80
6.08 6.19 6.25 6.43
52.25 51.94 52.03 52.17
6.04 6.15 6.20 6.37
51.94 51.65 52.30 50.29
6.01 6.15 6.19 6.35
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We can see that the random error is the dominating error source compared with systematic errors. Thus, higher accuracy is possible with more care and lower environment noise. 6. Conclusions
We have presented a simple method for measuring the absolute figure of the large optical flat surface in synchrotron radiation by small-aperture Fizeau interferometer. Proposed as an alternative to overcome the aperture limitation, the oblique incidence interferometry combined with multiple rotating measurements at N equally spaced positions is performed to determine absolute flatness of the overall 2D large surface. Unlike the typical three-flat test, it is convenient in that the reference flat in our method is not required to be dismantled throughout the testing procedure. The uncertainty analysis associated with systematic and random errors has been given in detail. Moreover, numerical simulations have been provided for quantifying the order of magnitude for systematic errors by the finite rotating number and the oblique incidence angle. It is pointed out that higher accuracy can be achieved if we increase rotating numbers and test at a lower oblique incidence angle, as long as we could make full use of interferometer aperture. Experiments on high-accuracy rectangle flats have been carried out at four different oblique incidence angles with varied rotating numbers, checking the results available against the prediction given by simulations. In addition, from comparison with modified three-flat test results, the difference between the two methods is estimated to be better than 0.7 nm rms. An uncertainty estimate for the absolute oblique incidence measurement at 81.7° with 12-position rotational averaging has also been worked out. The systematic uncertainty is calculated to be 0.036 nm rms, while the random uncertainty based on reproducibility is 2.6 nm rms. Cleary, the random error relating to environment stability is the dominating error source compared with systematic errors. As compared to other methods of large surface figure measurement, the absolute oblique incidence test is particularly simple. This approach could be implemented to facilitate the task of absolute flatness measurement of large synchrotron beam line optics. The authors acknowledge support from the National Natural Science Foundation of China (Grant No. 11105215) and the Major Program for Fundamental Research of the Chinese Academy of Sciences, China (Grant No. Y228011061). References 1. L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001). 2. R. D. Geckeler, “Optimal use of pentaprisms in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115–125 (2007).
3. P. C. V. Mallik, C. Y. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007). 4. J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466–8474 (2007). 5. G. Ehret, M. Schulz, M. Stavridis, and C. Elster, “Deflectometric systems for absolute flatness measurements at PTB,” Meas. Sci. Technol. 23, 094007 (2012). 6. S. N. Qian, P. Takacs, G. Sostero, and D. Cocco, “Portable long trace profiler: concept and solution,” Rev. Sci. Instrum. 72, 3198–3204 (2001). 7. F. Siewert, J. Buchheim, and T. Zeschke, “Characterization and calibration of 2nd generation slope measuring profiler,” Nucl. Instrum. Methods Phys. Res. A 616, 119–127 (2010). 8. S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, F. Siewert, and T. Zeschke, “Optimal alignment of mirror-based pentaprisms for scanning deflectometric devices,” Opt. Eng. 50, 073602 (2011). 9. L. Assoufid, N. Brown, D. Crews, J. Sullivan, M. Erdmann, J. Qian, P. Jemian, V. V. Yashchuk, P. Z. Takacs, N. A. Artemiev, D. J. Merthe, W. R. McKinney, F. Siewert, and T. Zeschke, “Development of a high-performance gantry system for a new generation of optical slope measuring profilers,” Nucl. Instrum. Methods Phys. Res. A 710, 31–36 (2013). 10. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33, 608–613 (1994). 11. P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Opt. 49, 21–31 (2010). 12. S. Han, E. Novak, and M. Schurig, “Application of RitcheyCommon test in large flat measurements,” Proc. SPIE 4399, 131–136 (2001). 13. C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and ‘absolute testing’,” CIRP Ann 45, 617–634 (1996). 14. S. Han, E. Novak, and J. Sullivan, “Application of grazing incidence interferometer to rough surface measurements,” Proc. SPIE 5144, 391 (2003). 15. M. Vannoni, “Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data,” Opt. Express 22, 3538–3546 (2014). 16. G. D. Dew, “Measurement of optical flatness,” J. Sci. Instrum. 43, 409–415 (1966). 17. G. Schulz and J. Schwider, “Precise measurement of planness,” Appl. Opt. 6, 1077–1084 (1967). 18. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 234379 (1984). 19. G. Schulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992). 20. C. Y. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993). 21. K. E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437–2446 (1994). 22. J. Grzanna, “Absolute tesing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33, 6654–6661 (1994). 23. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996). 24. W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multiposition form error metrology,” Precis. Eng. 21, 72–82 (1997). 25. P. Hariharan, “Interferometric testing of optical surfaces: Absolute measurements of flatness,” Opt. Eng. 36, 2478–2481 (1997). 26. R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, and B. Loucks, “Measurements of the LIGO pathfinder optics,” Proc. SPIE 3134, 95–111 (1997). 27. R. E. Parks, L. Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998). 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
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28. V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38, 2018–2027 (1999). 29. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637–1648 (2001). 30. M. F. Kuchel, “A new approach to solve the three flat problem,” Optik 112, 381–391 (2001). 31. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856–5865 (2006).
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32. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601 (2007). 33. M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15, 6809–6816 (2007). 34. G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. 47, 3433– 3445 (2008). 35. W. H. Press, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007), Chap. 3, pp. 120–124. 36. “OIML Guide to the expression of uncertainty in measurement,” (ISO, 1995).
A modified method for measuring the absolute figure of a large optical flat surface in synchrotron radiation by a small aperture interferometer is presented. The method consists of two procedures: the first step is oblique incidence measurement; the second is multiple rotating measurements. This simple method is described in terms of functions that are symmetric or antisymmetric with respect to reflections at the vertical axis. Absolute deviations of a large flat surface could be obtained when mirror antisymmetric errors are removed by N-position rotational averaging. Formulas are derived for measuring the absolute surface errors of a rectangle flat, and experiments on high-accuracy rectangle flats are performed to verify the method. Finally, uncertainty analysis is carried out in detail. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.4800) Optical standards and testing; (120.6650) Surface measurements, figure; (340.6720) Synchrotron radiation. http://dx.doi.org/10.1364/AO.53.003370
1. Introduction
The rapid development of dedicated synchrotron radiation sources and free electron laser facilities places an unprecedented high-precision demand on optical manufacturing and metrology. Surface figure deviation has been a major consideration in the use of grazing incidence optics in synchrotron beam line instrumentation [1]. Generally, optical surfaces used to reflect x rays at extreme grazing incidence angles have a long tangential dimension of up to 1 m or more. Owing to the nature of grazing incidence x-ray optics, conventional interferometric techniques for surface figure measurement are not easily employed in the absolute testing. The best known surface figure measurement method of large synchrotron beam line optics is the scanning pentaprism test [2–5]. The test typically uses a laser scanning technique, which obtains the
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surface slope from the deflection angle of the reflected laser beam. The surface figure is then calculated by slope integration. The representative instrument is the long trace profiler [6–9]. This method offers a good solution but is time consuming and often marred by a few inaccuracies due to the vibration and mechanical movement of the scanning elements. Besides, the scanning systems usually yield absolute profiles only along certain lines over the surface of interest. Interferometry is the common method of flatness measurement of optical surfaces due to its speed and accuracy. However, if the size of the optics is larger than 150 mm, the use of a commercial standard 150 mm interferometer will become difficult. The subaperture stitching technique stitches together many smaller measurements [10,11]; however, this requires a large translation stage and it is difficult to stitch many measurements on very flat optics without introducing errors. The Ritchey–Common test permits for testing of large flats in a diverging spherical beam [12]. It requires a large spherical mirror and creates a large air path.
The test is not widely used, owing in part to difficulties in data reduction. The Skip-Flat test, which uses a flat mirror as the reflective element, is a variation of the Ritchey–Common configuration [13,14]. This test uses an interferometer with a collimated output that is much smaller than the test surface. The collimated beam, reflected from the test surface at an oblique angle, is returned with a flat mirror to provide a narrow profile. For most grazing incidence x-ray optics in which the width is shorter while the length is longer than the aperture size of the interferometer, the full aperture test can be done, which avoids stitching. At large incidence angle, the surfaces under test appear smoother, so the Skip-Flat test is a good approach for the measurement of rough surfaces. However, to test high-precision flats, the accuracy of this method is usually limited by the reference surface and the transmission surface, of which the normal quality is about λ∕20 PV (about 6 nm rms), so it necessitates the use of absolute testing techniques to reach subnanometer accuracy in large surface figure measurement. A demonstration on synthetic data of absolute flatness measurement using oblique incidence setup and iterative algorithm has been reported recently, showing the capability of the approach to achieve an accuracy on the order of nanometer rms [15]. As to interferometric metrology, three-flat tests are the archetypes of measurement procedures to separate the test surface errors from the reference surface errors [16–33]. Several ingenious techniques have been devised to obtain an absolute two-dimensional (2D) map of the surface under test. Typical methods are Zernike polynomial fitting [18], even and odd function decomposition [20,27], and solutions based on rotation symmetry [30] and mirror symmetry [31,32]. However, in all three-flat methods, the reference flat surface is dismantled from the interferometer and is calibrated by procedures that should use combinations of pairs of three surfaces, including the reference flat surface and the other two flat surfaces. On account of difficulties in setting the reference flat to the interferometer exactly and quickly, these methods are not easily used. Furthermore, Zernike polynomials are not appropriate for wavefront analysis of noncircular pupils, such as rectangular, elliptical, and annular pupils, due to their lack of orthogonality over such pupils [34]. In the method described in [20], it is assumed that all flats can be expressed as the sum of four components using the symmetry properties of the odd and the even functions. However, this method requires that all interferograms maintain twofold mirror symmetry in the uniform Cartesian coordinate system, which introduces some difficulties in calibrating noncircular flats. Therefore, many of the previously mentioned methods are not very suitable for absolute flatness measurement of long grazing incidence optics with rectangular shape. In this work, we present a simple method for measuring an absolute figure of the large optical flat
surface in synchrotron radiation by a small aperture Fizeau interferometer. The oblique incidence interferometry combined with multiple rotating measurements at N equally spaced positions is performed to determine the absolute flatness of the overall 2D large surface. The advantage of the proposed solution is that the reference flat stays in the same position throughout the testing procedure, while in three-flat methods, the reference flat is dismantled. In addition, as a consequence of the high reflectivity of the x-ray mirrors, at visible wavelengths, uncoated transmission and reference flats could be used in oblique incidence measurement and the fringe contrast would generally be very good. 2. Principle
Before describing the solution of large optical surface testing, properties of mirror symmetry [31,32] and rotation symmetry [23,24,30] through an N-position average are briefly reviewed owing to the pivotal importance of these concepts to our method. For any given wavefront Wx; y, it can be written as a sum of a mirror symmetric component W e x; y and a antisymmetric component W o x; y. They are constructed under reflections at the y axis, which are defined as 1 W e x; y Wx; y W−x; y; 2 1 W o x; y Wx; y − W−x; y: 2
(1)
As to the solution based on rotation symmetry, the wavefront Wx; y can be decomposed into the rotational invariant component W R x; y and rotational variant component Ωx; y. W R x; y, also called rotation average, is defined as ! N−1 1X k·ΔΦ W x; y lim Wx; y : N→∞ N k0 R
(2)
Here, the operator •ΔΦ is used to indicate a rotation by an angle ΔΦ 2π∕N. The rotational variant wavefront component Ωx; y is the difference between Wx; y and W R x; y. The definition of the rotation average implies that Ω has a rotation average of zero: ΩR W − W R R W R − W R 0:
(3)
Applying Eqs. (2) and (3) to Eq. (1) results in W e R W R ;
W o R 0:
(4)
We are shown by Eq. (4) that the rotational invariant part of the wavefront is entirely contained in the mirror symmetric component, and the rotational average of the mirror antisymmetric component is zero. To obtain the interferometry from the oblique incidence configuration with a Fizeau interferometer, one more reference flat is required to return back 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
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the beam reflected by the long test surface. We name A, B, C for the transmission flat, reference flat, and the long test flat, respectively. For each surface, a Cartesian coordinate system is considered, with the origin at the center and the z axis coincident with the external normal. The method can be illustrated in Fig. 1. The first step is to make a measurement in the oblique incidence configuration as described in Fig. 1(a), in which the test flat C is inserted with its external normal at an angle α to the optical axis of the interferometer. Then, rotating measurements of N equally spaced positions between the transmission flat and reference flat are performed, as shown in Fig. 1(b). Note that the initial position of the reference flat in rotating measurements is coincident with the first step in the oblique incidence test. According to the Cartesian coordinate system of the interferometer defined in Fig. 1, the results of the measurements can be written as W 1 x; y W A −x; y W B −x; y x ;y ; 2 cos α · W C cos α
W e1 x; y W o1 x; y W eA x; y − W oA x; y W eB x; y − W oB x; y 2 cos α x x ; y W oC ;y ; · W eC cos α cos α W e2 x; y W o2 x; y W eA x; y − W oA x; y W eB x; y W oB x; y;
W 2 x; y W A −x; y W B x; y; W 3 x; y W A −x; y W B x; yR :
The negative sign for the x coordinate accounts for the flip of the flat in the x axis. Here, we consider that the flatness error of the surface is half of the wavefront error in normal reflection. Owing to the oblique incidence configuration, spatial resolution in the horizontal direction is impacted by a scale 1∕ cos α as shown in Eq. (5). Prior to processing, interferograms from experiments are prepared by removing pistons and tilts in the least-squares sense. All wavefronts in Eq. (5) can be split into their mirror symmetric and antisymmetric components according to Eq. (1), and the antisymmetric component of wavefront W B is zero under rotational averaging, as described in Eq. (4). Therefore, the expression of Eq. (5) can be rewritten as
(5)
Here, W i x; y, i 1, 2, 3, defines the measurement of the relative wavefront errors. Note that W 3 x; y shows the average result of N-position rotating measurements. For simplicity, signs are taken to be positive. W A x; y, W B x; y, and W C x; y stand for wavefronts normally reflected by the transmission flat, reference flat, and the long test flat, respectively.
W e3 x; y W o3 x; y W eA x; y − W oA x; y W B x; yR 0:
(6)
Here, antisymmetric components can be independently extracted from Eq. (6), as given in the matrix form 2
W o1 x; y
3
2
−1 −1 2 cos α
6 o 7 6 W 2 x; y 7 6 4 5 4 −1 o W 3 x; y −1
1
0
0
0
32 76 54
W oA x; y W oB x; y
3 7 5:
W oC x∕ cos α; y (7)
Where we invert the matrix, the solution for antisymmetric components is 2
W oA x; y
3
7 6 o 7 6 W B x; y 5 4 W oC x∕ cos α; y 2 0 6 4 0
0
−1
1
−1
32
W o1 x; y
3
76 W o x; y 7 5: 54 2 o W 3 x; y 1∕2 cos α 1∕2 cos α −1∕ cos α
Fig. 1. Schematic representation of the absolute flatness testing for long optics. (a) Configuration of oblique incidence measurement in top view. The large surface under test is arranged by its external normal at an angle α to the optical axis of the interferometer. (b) N-position rotating measurements between transmission flat and reference flat in side view. The reference flat is rotated by an angle ΔΦ 2π∕N in N equal steps about the optical axis. The coordinate system of the interferometer is indicated with bold arrows. 3372
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(8) According to Eq. (6), the mirror symmetric component of the large surface can be solved as x 1 ;y W e1 x; y − W e2 x; y: cos α 2 cos α
W eC
(9)
Finally, two parts of the wavefront in Eqs. (8) and (9) can be added to get the solution for the deviation of the large surface: x x x WC ;y W eC ;y W oC ;y cos α cos α cos α e o e o W x; y W 1 x; y − W 2 x;y W 2 x;y − 2W o3 x;y 1 2 cos α W 1 x; y − W e2 x; y W o2 x;y − 2W o3 x;y : (10) 2 cos α
The differential of Eq. (13) follows: − sin α · dα
Please note that the shape of the wavefront W 1 is rectangle while W 2 and W 3 are circles. Therefore, wavefronts W 2 and W 3 should be sheared to the same data form of W 1 based on uniform space coordinates. In addition, most x-ray mirrors are made today with extremely smooth surfaces whose slope errors are below 1 μrad rms and roughness errors less than 3 Å rms. In this case, it is sufficient to use the bicubic spline interpolation to reconstruct the complete surface figure [35], with negligible influence on the final accuracy achieved. The corresponding simulation results are displayed in Section 4. 3. Uncertainty Analysis
There are a number of contributions to the uncertainty of any interferometric measurement of a surface figure. Here, we analyze the combined uncertainty of our method from the systematic part and the random part. Systematic errors are mainly introduced by the finite number of N equally spaced positions taken for the average during rotating measurements and the oblique incidence angle. The random error represents the measurement reproducibility error, which is mainly due to the changes in the environment during the long period of measurements (air, temperature, vibrations). Now we estimate the measurement uncertainty by these components as mentioned previously.
dW C
dW f 1 − cos α · dLC − · WC: 2 cos α LC
ΔW C ≈
ΔW f 1 − cos α · ΔLC − · W C; 2 cos α LC
Wf dW f sin α · dα · W f ; (12) dW C d 2 cos α 2 cos α 2 cos2 α
where the oblique incidence angle α can be determined by the ratio of protection length LC measured on a charge-coupled device (CCD) detector to real length L of the long surface. It is given by cos α
LC : L
(13)
(16)
where ΔW f is dependent on the finite rotating number. We define ΔΩR as representing the approximate residual error, which the rotational variant component of wavefront W B is not completely removed by the N-position average. Rotational variant component Ω can be expressed as a Fourier series according to its periodicity of 2π: Ωr; θ
∞ X an · cos nθ bn · sin nθ;
(17)
n1
where rotating a wavefront to N equally spaced positions and averaging could remove rotational variant terms of all angular orders except those that are integer multiples of N, which is described briefly in [23]. Therefore, mirror antisymmetric components in the fourth expression of Eq. (6) are revised as (18)
Substituting Eq. (18) into Eq. (11), we obtain ΔW f 2 · ΔΩR :
(11)
Substituting Eq. (11) into Eq. (10), we get the differential simultaneously on both sides of Eq. (10):
(15)
If we assume that dW f ΔW f and dLC ΔLC , then dW C ≈ ΔW C . It is obvious that ΔLC depends on the resolution of the CCD. Therefore, Eq. (15) is approximated as
W o3 −W oA ΔΩR :
For the sake of simplicity, we define W f , representing the numerator of the right-hand side in Eq. (10), as
(14)
For simplicity, here we assume that dL dLC . Using W f 2 cos α · W C and Eq. (14), then Eq. (12) can be solved as
A. Evaluation of the Uncertainty Associated with Systematic Errors
W f W 1 − W e2 W o2 − 2W o3 :
L · dLC − LC · dL : L2
(19)
Applying Eq. (19) to Eq. (16), the systematic error is evaluated as ΔW C ≈
ΔΩR 1 − cos α · ΔLC − · WC: cos α LC
(20)
In addition, if we assume ΔS for the error of the bicubic spline interpolation used to reconstruct the complete surface figure, the final systematic error is expressed as ΔW C
ΔΩR 1 − cos α · ΔLC − · W C ΔS; cos α LC
1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
(21)
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where we see that the systematic uncertainty is mainly determined by the finite rotating average error, the oblique incidence angle, the resolution of the CCD, and the quantity of the long surface itself. Generally, LC is a constant denoting the aperture of the interferometer because we usually make full use of aperture to obtain as much information as possible. B. Evaluation of the Uncertainty Caused by Random Errors
If we assume δ0 for the standard deviation of random errors in all wavefront measurements which are uncorrelated, the uncertainty of each item in Eq. (10) can be calculated according to the knowledge of uncertainty propagation [36] as 1 δ2 W e2 δ2 W o2 δ20 ; 2 1 1 2 δ : δ2 W o3 δ2 W 3 2 2N 0
(22)
Therefore, the combined standard uncertainty δr of random errors is given by r 1 1 · 1 · δ0 : (23) δr p N 2 cos α The result has shown that the combined standard uncertainty of the measurement of the long optical flat surface is inversely proportional to cos α. In addition, increasing the rotating number N is also essential for achieving a low uncertainty caused by random errors. 4. Numerical Simulations
In order to validate our method, we have generated three synthetic surfaces using 36 random Zernike polynomials with zero piston and tilt, and flatness of λ∕20 PV. Two of them were circular surfaces of 400 pixels in diameter, while another surface, representing the long flat under test, was rectangular with 2664 pixels in the tangential dimension. Then we have computed the corresponding interferograms using Eq. (5) by choosing 82° for the oblique incidence angle. The rotational average was approximated with 12-position averaging. After processing as previously described, we have compared the reconstructed surface with the surface generated initially. The difference between initial and final maps is 1.877 × 10−5 λ rms, as shown in Fig. 2 (λ 632.8 nm). The residual errors in the final results, from a systematic standpoint, mainly have two sources, including the finite rotating number and the influence of the oblique incidence angle. Here, we have run a number of simulations in order to get an impression on the order of magnitude for the systematic error by each influence, respectively. The effect of the finite rotating number on the rotating average error ΔΩR is displayed in Fig. 3. We could see that the flatness of the reference flat, which is rotated in our 3374
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Fig. 2. Simulation of the absolute oblique incidence test at an 82° angle with 12-position averaging. Top row: simulated long flat wavefront; middle row: reconstructed long flat wavefront; bottom row: difference between the simulated wavefront and the reconstructed wavefront by enlarging 500×.
method, has an influence on choosing the rotating number advisability according to the required accuracy. As 36 terms of Zernike polynomials are up to a maximum azimuthal frequency of 5θ, the result satisfies our expectation that having N ≥ 6 rotational average measurements could remove rotational variant terms mostly. Note that the operation of the wavefront rotating in software necessarily involves interpolation errors that would not arise in the case in which a physical part is rotated about the optical axis. The result reveals that the rotating average error decreases to 7.682 × 10−6 λ rms when the flatness of the transmission flat is λ∕20 PV and the rotating number is selected as 12. Next, we have simulated the influence of the oblique incidence angle on the final residual errors. It is clear that the systematic uncertainty increases if the oblique incidence angle becomes large, seeing the first two items of Eq. (21). The third item, which means interpolation errors, is also associated with the oblique incidence angle for the data compression scaling. The overall figure is reconstructed by the bicubic spline interpolation in our method due to the oblique incidence test, which leads to data compression in the horizontal direction. Now we have changed the oblique incidence angle for different
Fig. 3. Decrease of rotating average error rms by N rotational position averaging with different flatness of the reference flat B which is rotated.
Fig. 4. Relative interpolation error due to oblique incidence angle for different flatness of the long surface under test.
flatness of the long surface to observe interpolation errors. The simulated results are shown in Fig. 4. Clearly, compared to the respective original wavefront errors, relative interpolation rms value errors are consistent for different flatnesses of the long surface. The relative interpolation error is 0.06% when the oblique incidence angle works at 82°. Furthermore, we see that the relative interpolation error is below 0.1% if the oblique incidence angle works under 84°. Please note that the precondition of this result is that the testing surface is the smooth surface without acute local defect. 5. Experiment Results and Discussion
We demonstrated the absolute oblique incidence test with a Zygo GPI-XP phase-shifting interferometer working at λ 632.8 nm. The camera is a 640 × 480 pixel CCD detector with 370 μm resolution. The reference and transmission flats are both made of 150 mm diameter fused silica with about λ∕20 PV. In order to validate our method, we chose a rectangular plane mirror (single crystal silicon, size 245 × 50 × 40 mm, Rhodium-coated) produced by SESO to perform as the long surface. The temperature of the optical laboratory was kept at 23 0.5°C. The experiment was carried out according to the configuration described in Fig. 1. The first step was oblique incidence measurement, and the second was multiple rotating measurements. During all the measurements, the transmission flat A remained attached to the Fizeau interferometer, so we can assume that its spatial position was fixed. A fiducial crosshair generated by the interferometer software MetroPro was used to mark the center of flat A, and an auxiliary physical crosshair was used to mark the center of reference B, so as to guarantee the spatial coincidence. The misalignment was controlled to be within two pixels. All the experiment data were exported from the Zygo interferometer and processed by our algorithm procedure according to Eq. (10). Since the wavefront arrays were stored in matrix form, it was easy to perform the reflection operation by flipping the matrix left/right to get the mirror symmetric and antisymmetric components. Then we used the bicubic spline interpolation to reconstruct the complete surface figure.
Fig. 5. Measurement results of the flatness of the rectangular plane at oblique incidence angles of 38.9°, 61.0°, 72.8°, and 81.7°. To the left are the relative test results, and to the right are the corresponding reconstructed absolute results with 24-position averaging.
For the purpose of analyzing the influences of oblique incidence angle and rotating number for measurement accuracy, we selected a 120 × 30 mm active area of the rectangular plane and carried out the measurements at four different oblique incidence angles with varied rotating numbers. Measurement results of the flatness of the rectangular plane at oblique incidence angles of 38.9°, 61.0°, 72.8°, and 81.7° were shown in Fig. 5. We displayed simultaneously the oblique incidence relative test results and reconstructed absolute results after removing figure errors of the reference and transmission flats with 24-position rotational averaging. It is clear that the difference between the two measurement results becomes more evident when the oblique incidence angle increases, which highlights the necessity of our method to obtain the absolute figure errors of high-precision flats from large oblique incidence tests. Taking the 81.7° oblique incidence test for example, the figure errors of the rectangular plane measured from the relative interferometric test are 56.92 nm PV and 7.24 nm rms, while we calculate 50.29 nm PV and 6.35 nm rms by 24 rotational
Fig. 6. Comparison of reconstructed oblique incidence test results from oblique incidence angle of 81.7° with the normal incidence absolute result. Top row: reconstructed relative flat figure; middle row: reconstructed absolute flat figure with 24-position averaging; bottom row: absolute flat figure obtained from the normal incidence measurements using modified Griesmann’s approach. 1 June 2014 / Vol. 53, No. 16 / APPLIED OPTICS
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Fig. 7. Differences of measured figure results of rectangular plane by oblique incidence and absolute normal incidence methods. To the left are the differences of the relative oblique incidence result and normal incidence result, and to the right are the differences of the absolute oblique incidence result and normal incidence result.
averaging in oblique incidence interferometry. To compare the oblique incidence test results, we calibrated the absolute figure of the rectangular plane in the active area within the aperture of interferometer, using a modified three-flat test approach based on a simple mirror symmetry described by Griesmann [31]. The comparison results are shown in Fig. 6 and the difference maps are exhibited in Fig. 7. Absolute flat figure errors obtained by modified Griesmann’s approach, in which the rotating number was selected as 24 as well, are 51.82 nm PV and 6.02 nm rms. We find out that the results of the absolute oblique incidence test and modified threeflat test coincide well with each other both in the surface map and in numerical value, as it can be seen that the pixel-to-pixel difference between the two methods is 0.69 nm rms (Fig. 7). By comparison, our method has reduced the difference in rms to
and rotating numbers as displayed in Table 1. It is found that the results gradually approximate to the true flatness errors as we have increased rotating numbers and tested at lower oblique incidence angles, which agrees well with the simulation results as mentioned above. Here we have referred to modified three-flat test results as the true flatness errors of the rectangular plane. According to the uncertainty analysis described in Section 3, we have estimated the final standard uncertainty of the absolute oblique incidence measurement working at 81.7° with 12-position rotational averaging. First, we consider the reproducibility error, which fundamentally limits a test setup’s ability to make reliable measurements. Experiments were implemented 50 times continuously and results were averaged every 16 measurements. The statistical analysis of results has got a reproducibility error δ0 of 0.00081λ (0.51 nm) rms. Then, the standard random uncertainty δr of the absolute oblique incidence test could be calculated according to Eq. (23): r 1 · δ 2.6 nm: δr p · 1 12 0 2 cos 81.7° 1
Next, we estimate the standard systematic uncertainty δs based on Eq. (21). From simulation results, the standard deviation of 12-position rotating average error δΔΩR is 7.682 × 10−6 λ rms and the relative bicubic spline interpolation error is 0.06%. In addition, δWc , standing for the standard deviation of flatness errors of the long rectangular plane, is 6.37 nm rms, which is measured at 81.7° with 12-position rotational averaging. Therefore, we can calculate the standard systematic uncertainty δs as
s 2 2 δΔΩR 1 − cos 81.7° × 0.37 2 · δWc 0.06% · δWc 0.036 nm: δs cos 81.7° 150
approximately one fourth that of the conventional relative oblique incidence test. Moreover, we have comprehensively listed results of flatness errors of the rectangular plane solved by the absolute oblique incidence test with different oblique incidence angles Table 1.
Angle/deg
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(25)
Then, the combined standard uncertainty of the absolute oblique incidence measurement is q δ δ2s δ2r 2.6 nm:
(26)
PV and RMS Values of Flatness Errors of the Rectangular Plane by Absolute Oblique Incidence Test with Different Oblique Incidence Angles and Rotating Numbers
N3
38.9 61.0 72.8 81.7
(24)
N4
N6
N 12
N 24
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
PV/nm
rms/nm
52.86 52.96 53.72 54.01
6.22 6.43 6.50 6.83
52.77 52.62 52.89 53.34
6.18 6.31 6.37 6.60
52.38 52.57 52.65 52.80
6.08 6.19 6.25 6.43
52.25 51.94 52.03 52.17
6.04 6.15 6.20 6.37
51.94 51.65 52.30 50.29
6.01 6.15 6.19 6.35
APPLIED OPTICS / Vol. 53, No. 16 / 1 June 2014
We can see that the random error is the dominating error source compared with systematic errors. Thus, higher accuracy is possible with more care and lower environment noise. 6. Conclusions
We have presented a simple method for measuring the absolute figure of the large optical flat surface in synchrotron radiation by small-aperture Fizeau interferometer. Proposed as an alternative to overcome the aperture limitation, the oblique incidence interferometry combined with multiple rotating measurements at N equally spaced positions is performed to determine absolute flatness of the overall 2D large surface. Unlike the typical three-flat test, it is convenient in that the reference flat in our method is not required to be dismantled throughout the testing procedure. The uncertainty analysis associated with systematic and random errors has been given in detail. Moreover, numerical simulations have been provided for quantifying the order of magnitude for systematic errors by the finite rotating number and the oblique incidence angle. It is pointed out that higher accuracy can be achieved if we increase rotating numbers and test at a lower oblique incidence angle, as long as we could make full use of interferometer aperture. Experiments on high-accuracy rectangle flats have been carried out at four different oblique incidence angles with varied rotating numbers, checking the results available against the prediction given by simulations. In addition, from comparison with modified three-flat test results, the difference between the two methods is estimated to be better than 0.7 nm rms. An uncertainty estimate for the absolute oblique incidence measurement at 81.7° with 12-position rotational averaging has also been worked out. The systematic uncertainty is calculated to be 0.036 nm rms, while the random uncertainty based on reproducibility is 2.6 nm rms. Cleary, the random error relating to environment stability is the dominating error source compared with systematic errors. As compared to other methods of large surface figure measurement, the absolute oblique incidence test is particularly simple. This approach could be implemented to facilitate the task of absolute flatness measurement of large synchrotron beam line optics. The authors acknowledge support from the National Natural Science Foundation of China (Grant No. 11105215) and the Major Program for Fundamental Research of the Chinese Academy of Sciences, China (Grant No. Y228011061). References 1. L. Assoufid, O. Hignette, M. Howells, S. Irick, H. Lammert, and P. Takacs, “Future metrology needs for synchrotron radiation grazing-incidence optics,” Nucl. Instrum. Methods Phys. Res. A 467, 267–270 (2001). 2. R. D. Geckeler, “Optimal use of pentaprisms in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115–125 (2007).
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