Surface figure measurement of flat mirrors based on the subaperture stitching interferometry

Pengqian Yang123*, Stefan Hippler2, Zhaojun Yan1, Rainer Lenzen1, Wolfgang Brandner1,

Casey Deen1, Thomas Henning1, Armin Huber1, Sarah Kendrew1, Jianqiang Zhu2

1Max Planck Institute for Astronomy, Heidelberg, Germany 2Chinese Academy of Sciences, Shanghai Institute of Optics and Fine Mechanics

Shanghai 201800, China 3Graduate School of the Chinese Academy of Science, Beijing 100039, China

*Corresponding author: pengqian@mpia.de

ABSTRACT

Large flat mirrors can be characterized using a standard interferometer coupled with stitching the subaperture measurement data. Such systems can measure the global full map of the optical surface by minimizing the inconsistency of data in the adjacent regions. We present a stitching technique that makes use of a commercial phase-shifting Twyman-Green interferometer in combination with an iterative optimized stitching algorithm. The proposed method has been applied to determine the surface errors of planar mirrors with an accuracy of a few nanometers. Moreover, the effect of reference wavefront error is explored. The feasibility and the performance of the proposed system are also demonstrated, along with a detailed error analysis and experimental results. Keywords: optical testing, optical flat, subaperture interferometry

1. INTRODUCTION

Increasing accuracy requirements of optical mirrors in space-based and astronomical applications make the development of optical testing procedures for optical surfaces important. In order to extend the capability and dynamic range of a conventional phase shifting interferometer, the subaperture interferometry was first proposed to overcome the aperture size limitations of optical testing of large optics.1,2,3 The subaperture stitching interferometry can make full aperture interferometric measurements by taking several smaller subaperture measurements to cover the entire optical surface. The global stitching software is then adopted to minimize the inconsistency of data and stitch the subapture maps together. The larger field of view, higher lateral spatial frequencies and greater accuracy can be obtained by using stitching interferometry. There have been numerous publications to address the problems associated with the data combination of the suaperture data sets. The most popular method use Zernike polynomials to represent the surfaces and then a least squares fitting to obtain the coefficients of the surface under test, such as the Kwon-Thunen and the simultaneous fitting method. The detailed discussion of these two methods was given by Jensen et al.4 Both of the two methods suffer from the fitting problem when the localized irregularities exist in subaperture data, which also lead to a suppression of higher frequencies of the measurement. To overcome this shortcoming, a new algorithm, so called the discrete phase method (DPM), was proposed by Stuhlinger.5 The sub-aperture stitching described by discrete data points bases on overlaps between neighbored subapertures. With this method, the advantage of high resolution of the stitching measurement could be retained. It was further discussed by Masashi Otsubo6 and Chunyu Zhao7. So far, almost modern stitching interferometry techniques are developed on the basic philosophy of fitting the data in the overlap region. In 1996, Michael Bray built a stitching interferometer for determining the transmitted wavefront quality of large plane optics in high power laser facilities, which was believed to be the first practical stitching instrument. Chen et al. developed an iterative algorithm in the vision and robotics relying on the geometrical configuration of the test and localized data points optimization.8

6th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment, edited by Yudong Zhang, Libin Xiang, Sandy To,

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Recently, Peng Su applied Maximum-likelihood optimization to data processing of the stitching interferometry and both the reference and the test surface can be reconstructed simultaneously.9 In fact, since all the stitching methods will involve the motions of the test optics or the interferometer itself, the more important issue is how the measurement setup is realized in reality. The uncertainty of the stitching result is highly sensitive to positioning errors and the systematic error of the interferometer in individual subaperture measurements. The problem of the data stitching became the optimization problem to minimize the discrepancy in the adjacent regions accompanied with various errors. To address the above issues, we propose a novel algorithm based on a commercial phase shifting interferometer for testing flat optics, the algorithm could be also easily facilitated to test rotationally symmetric curved surface. The effect of reference wavefront error is discussed, and an experimental measurement is given to test the validity and accuracy of the algorithm. Nanometer level accuracies are routinely obtained for large planar optics by using proposed method.

2. STITCHING THEORY

2.1 Coordinate system transformation

In interferometry, the sub-interferograms acquired by the digital camera in a single exposure, can be mathematically described as ( , , )W x y φ where W denotes the jth wavefront of the subaperture,φ is the phase value relayed on the pixel ( , )x y by the imaging lenses.

Figure 1 A large flat mirror can be characterized by subaperture tests

The flow scheme of the stitching procedure is shown in Fig.1. In order to connect the data together, all of the subaperture maps have to be transformed to the global coordinate system for the purpose of stitching. For a general surface, the converting relationship is written as follows10:

[ ', ', ,1] [ , , ,1]T Tj jx y z M x y φ= ⋅

T is the matrix of coordinate transformation, the form of the matrix depends on the type of the configuration of the test, and the transformation can be the result of a multiplication of several transformation matrices. For the measurement of a plane surface, the converting matrix with respect to the 3D coordinate frame can be written as follows

1 0 0 0 cos( ) sin( ) 0 00 1 0 0 sin( ) cos( ) 0 0

( )0 0 1 0 0 0 1 0

1 0 0 0 1

M T R

tx ty tz

θ θθ θ

θ

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ⋅ = ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Where T and ( )R θ denote transformation and rotation of the coordinate system, respectively. By using the coordinate transformation method, the local subaperture phase data from the CCD camera transform to the global coordinate system for stitching.

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2.2 Iterative algorithm of stitching

After getting all the separate sub-aperture interferograms, data should be connected to attain the final figure map. In fact, there is no standard software for stitching measurement due to the different cases of the measurement. Normally, the data with piston, x-tilt, y-tilt, and power can be removed based on the conventional least squares principle by minimizing the residual phases in the overlapped regions:

2 2 2[ ( )] minj j j j jj

E a b x c y d x yφ= + + ⋅ + ⋅ + + →∑

Theφ represents the original phase data of the subaperture; while a, b, c, and d denote coefficients of piston, x tilt, y tilt, and power, respectively. To obtain a more precise stitched result, an iterative approach can be used to minimize the cost function on the basis of least squares fitting. 8,11

3. ERROR ANLYSIS

The typical error of a subaperture map can be decomposed into following parts: alignment errors, systematic errors of the interferometer itself and random noise. The presence of errors diminishes accuracy by creating the low order aberrations, which must be fitted and compensated.

3.1 Sub-aperture sampling errors

A surface figure error measurement is usually performed at the conjugate position with the interferometer in null mode. In reality, the sub-measurements are slightly shifted with respect to the designed positions because of existence of the error in the mechanical system. The positioning errors will map test surface features to incorrect positions as the translation and rotations, and significant errors can be introduced to the stitched data. A misaligned subaperture measurement with the combination of the shift along radial direction and clocking is shown in Fig 2.

Figure 2 alignment error of the subaperture

The proposed algorithm can correct small scale low order positioning error due to the misalignment, but for the aspheric surface, the more complicated optimization is needed due to imaging distortion and non-nulled problems .12

3.2 Reference wavefront errors

In either a Fizeau or a Twyman-Green interferometer, any figure defects in the reference optics or any other errors of the interferometer itself will pollute the subapeture data. The residual errors of each single test will contribute to the final stitched map in the same way. For a self-design interferometer, these system errors would be exactly estimated by using commercial lens design ray trace code, and the errors could be compensated completely. Thus, this analysis method is less useful for commercial instruments, because of the typically complex and proprietary of its imaging systems. For a commercial interferometric instrument, several methods have been developed for reference surface calibration and allowed for absolute characterization of the surface within a few nanometers.13 The estimation of residual figure errors of

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the reference surface could be generated and stored as a data file, and can be subtracted from the subsequent measurements.

Figure 3 Test results of the a flat mirror before (left) and after (right) the absolute calibration

Fig.3 shows the map of a single measurement with a wavefront error of 5.23 nm rms, and the test result after the absolute calibration of the interferometer with 3.32 nm rms is shown by using multiple azimuth position average method. In this paper, the reference wavefront is fitted by 36 terms of Zernike polynomials, which was subtracted from each subaperture measurement.

4. EXPERIMENTAL RESULTS

To demonstrate the feasibility of the proposed method, we conducted the stitching measurement on a poslished flat mirror with a diameter of 203.4 mm. For our measurement, we used the commercial Twyman-Green phase shifting interferometer from FISBA with a visible He-Ne laser with wavelength 633 nm. It has an effective test diameter of 100 mm. The translational motion of the interferometer was provided by an x-y linear translation stage. The photo of the real setup of the stitching system appears in Fig. 4.

Figure 4 Diagram of the experimental arrangement used to test the optical flat

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Errors in stitching the subaperture measurements increase as more subapertures are combined.14The ratio of the sizes of reference and test flats determines that a minimum of 9 measurements are needed to cover the whole test flat. The 9 individual sub-measurements are seen in Fig.5.

Figure 5 Result of the all 9 subaperture measurements of an optical flat surface

Figure 6 Map of phase error of stitched full aperture with proposed configurations, RMS = 8.91nm

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The interferometer was first calibrated using multiple azimuth position average method before the stitching test. The first measurement was taken in the center of the test surface, which was also the center of the global coordinate of the stitched map. Second step, the interferometer was translated to the designed position, which is shown in Fig. 2. Then the following test could be performed by rotating the test optic with respect to the right angle with the accuracy of about 0.1 deg, and the test is performed at each position until the full aperture of the surface under test is covered. Finally the subapertures are stitched together with residual Zernike polynomial reference surface removed, which gives the complete map of the test surface shown in Fig. 6.

5. CONCLUSIONS

In summary, we demonstrate a novel approach of subaperture interferometry for testing large flat mirrors. The low order misalignment aberrations, like the relative piston and tilts among the different sub-interferograms, can be described mathematically and are removed by an iterative approach based on least-squares fitting. Although the stitching algorithm presented here was primarily designed for testing flat surfaces, it is also applicable for other test setups. Nanometer level accuracies were obtained in the experiment.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge Peter Bizenberger (Max Planck Institute for Astronomy) for providing help and support with the experiments.

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