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Free-Form Surface Reconstruction Method from Second-Derivative Data

형상이차미분을 이용한 자유곡면 형상복원법

  • Kim, Byoung Chang (Department of Mechanical Engineering, Kyungnam University) ;
  • Kim, DaeWook (2College of Optical Science, University of Arizona) ;
  • Kim, GeonHee (Center for Analytical Instrumentation Development, Korea Basic Science Institute)
  • Received : 2014.08.11
  • Accepted : 2014.09.30
  • Published : 2014.10.25

Abstract

We present an algorithm for surface reconstruction from the second-derivative data for free-form aspherics, which uses a subaperture scanning system that measures the local surface profile and determines the three second-derivative values at those local sampling points across the free-form surface. The three second-derivative data were integrated to get a map of x- and y-slopes, which went through a second Southwell integration step to reconstruct the surface profile. A synthetic free-form surface 200 mm in diameter was simulated. The simulation results show that the reconstruction error is 19 nm RMS residual difference. Finally, the sensitivity to noise is diagnosed for second-derivative Gaussian random noise with a signal to noise ratio (SNR) of 16, the simulation results proving that the suggested method is robust to noise.

일련의 국부영역으로부터 이차미분값을 획득하여 전체 형상을 복원하는 측정법을 제안하였다. 측정시 기울기가 제거된 국부형상에 대해 곡률과 이차미분이 동일시 되는 점을 이용하여, 3개의 이차미분값으로부터 직교하는 2방향을 따라 3차원형상을 복원할 수 있는 알고리즘을 구현하였다. 임의로 발생시킨 Zernike다항식의 계수로 자유곡면형상을 생성시키고, 구현된 알고리즘을 적용함으로써 검증과정을 수행하였다. 적용한 결과 최대 0.8 mm Sag를 갖는 직경 200 mm영역의 자유곡면형상에 대해 RMS 19 nm 형상복원오차를 갖고 복원됨을 확인하였다. 측정오차에 대한 복원오차 민감도를 진단하기 위해 SNR(Signal-to-Noise Ratio) 16의 가우시언 랜덤 노이즈를 부여한 후, 복원되는 형상의 오차를 진단한 결과, 197 nm의 형상복원오차가 발생함을 확인하였다.

Keywords

References

  1. B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, "3-D shape measurement using curvature data," Proc. SPIE 7389, 73892H-8 (2009).
  2. P. E. Glenn, "Angstrom level profilometry for submillimeter to meter scale surface errors," in Advanced Optical Manufacturing and Testing, G. M. Sanger, P. B. Reid, and L. R. Baker, eds., Proc. SPIE 1333, 326-336 (1990).
  3. P. E. Glenn, "Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique," in Advanced Optical Manufacturing and Testing II, V. J. D. D. V. M. ed., Proc. SPIE 1531, 61-64 (1992).
  4. P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, "A facility for the curvature-based measurement of the nanotopography of complex surfaces," in Optical Devices and Diagnostics in Materials Science, D. L. Andrew, T. Asakura, S. Jutamulia, W. P. Kirk, M. G. Lagally, R. B. Lal, and J. D. Trolinger eds., Proc. SPIE 4098, 94-101 (2000).
  5. I. Weingartner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, "Measurement of steep aspheres: A step forward to nanometer accuracy," in Optical Metrology for the Semiconductor, Optical, and Data Storage Industries II, A. Duparre and B. Singh, eds., Proc. SPIE 4449, 195-204 (2001).
  6. M. Schulz, "Topography measurement by a reliable large-area curvature sensor," Optik 112, 86-90 (2001). https://doi.org/10.1078/0030-4026-00015
  7. M. Schulz and I. Weingartner, "Measurement of steep aspheres by curvature scanning: An uncertainty budget," Proc. 2nd Euspen International Conference, 478-481 (2001).
  8. C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingartner, "Reconstructing surface profiles from curvature measurements," Optik 113, 154-158 (2002). https://doi.org/10.1078/0030-4026-00138
  9. M. Schulz, R. D. Geckeler, and J. Illemann, "High accuracy form measurement of large optical surfaces," in Recent Developments in Traceable Dimensional Measurements II, J. E. Decker and N. Brown, eds., Proc. SPIE 5190, 211-219 (2003).
  10. B. C. Kim, T. Saiag, Q. Wang, J. Soons, R. S. Polvani, and U. Griesmann, "The geometry measuring machine (GEMM) project at NIST," in Free-Form Optics: Design, Fabrication, Metrology, Assembly, ASPE 2004 Winter Topical Meeting (North Carolina, USA, 2004), pp. 108-111.
  11. U. Griesmann, N. Machkour-Deshayes, J. Soons, B. C. Kim, Q. Wang, J. R. Stoup, and L. Assoufld, "Uncertainties in aspheric profile measurements with the geometry measuring machine at NIST," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II, A. Duparre, B. Singh, Z. Gu, eds., Proc. SPIE 5878, 112-124 (2005).
  12. Y. K. Kwon, B. C. Kim, D. H. Wang, Y. W. Lee, H. S. Yang, and H. G. Rhee, "Aspheric measurement based on the curvature sensing method," Proc. SPIE 6671, 667119-667126 (2007).
  13. D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, "Algorithm for surface reconstruction from curvature data for freeform aspheric," Optical Manufacturing and Testing X, Proc. SPIE 8838, 88380B1-9 (2013).
  14. W. H. Southwell, "Wavefront estimation from wave-front slope measurements," JOSA 70, 998-1006 (1980). https://doi.org/10.1364/JOSA.70.000998

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