Abstract
With the development of new materials and ultra-precision processing technology, the sizes of measured objects increase, and the requirements for machining accuracy and surface quality become more exacting. The traditional measurement method based on reference datum is inadequate for measuring a high-precision object when the quality of the reference datum is approximately within the same order as that of the object. Self-referenced measurement techniques provide an effective means when the direct reference-based method cannot satisfy the required measurement or calibration accuracy. This paper discusses the reconstruction algorithms for self-referenced measurement and connects lateral shearing interferometry and multi-probe error separation. In lateral shearing interferometry, the reconstruction algorithms are generally categorized into modal or zonal methods. The multi-probe error separation techniques for straightness measurement are broadly divided into two-point and three-point methods. The common features of the lateral shearing interferometry method and the multi-probe error separation method are identified. We conclude that the reconstruction principle in lateral shearing interferometry is similar to the two-point method in error separation on the condition that no yaw error exists. This similarity may provide a basis or inspiration for the development of both classes of methods.
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References
Malacara D. Optical Shop Testing. 3rd ed. New York: JohnWiley & Sons, Inc., 2007, 83, 501–503, 651–654
International Vocabulary of Metrology—Basic and General Concepts and Associated Terms. (VIM 3rd edition), JCGM 200:2012, http://www.bipm.org/en/publications/guides/#vim
Evans C J, Hocken R J, Estler W T. Self-calibration: Reversal, redundancy, error separation, and ‘absolute testing’. CIRP Annals —Manufacturing Technology, 1996, 45(2): 617–634
PHYSICS. The SID4 HR sensor. http://www.phasicscorp.com/products/wavefront-sensors/sid4-hr-wavefront-sensor.html
Korwan D. Lateral shearing interferogram analysis. Proceedings of the Society for Photo-Instrumentation Engineers, 1983, 429: 194–198
Cubalchini R. Modal wave-front estimation from phase derivative measurements. Journal of the Optical Society of America, 1979, 69 (7): 972–977
Rimmer M P, Wyant J C. Evaluation of large aberrations using a lateral-shear interferometer having variable shear. Applied Optics, 1975, 14(1): 142–150
Dai F, Tang F,Wang X, et al. Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: Comparisons of existing algorithms. Applied Optics, 2012, 51 (21): 5028–5037
Herrmann J. Cross coupling and aliasing in modal wavefront estimation. Journal of the Optical Society of America, 1981, 71(8): 989–992
Harbers G, Kunst P J, Leibbrandt G W R. Analysis of lateral shearing interferograms by use of Zernike polynomials. Applied Optics, 1996, 35(31): 6162–6172
Dai F, Tang F, Wang X, et al. Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms. Optics Express, 2012, 20(2): 1530–1544
Liu X. A polarized lateral shearing interferometer and application for on-machine form error measurement of engineering surfaces. Dissertation for the Doctoral Degree. Hong Kong: Hong Kong University of Science and Technology, 2003
Ling T, Yang Y, Yue X, et al. Common-path and compact wavefront diagnosis system based on cross grating lateral shearing interferometer. Applied Optics, 2014, 53(30): 7144–7152
Freischlad K R, Koliopoulos C L. Modal estimation of a wave front from difference measurements using the discrete Fourier transform. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 1986, 3(11): 1852–1861
Elster C, Weingärtner I. Solution to the shearing problem. Applied Optics, 1999, 38(23): 5024–5031
Flynn T J. Two-dimensional phase unwrapping with minimum weighted discontinuity. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 1997, 14(10): 2692–2701
Guo Y, Chen H, Xu J, et al. Two-dimensional wavefront reconstruction from lateral multi-shear interferograms. Optics Express, 2012, 20(14): 15723–15733
Ling T, Yang Y, Liu D, et al. General measurement of optical system aberrations with a continuously variable lateral shear ratio by a randomly encoded hybrid grating. Applied Optics, 2015, 54(30): 8913–8920
Karp J H, Chan T K, Ford J E. Integrated diffractive shearing interferometry for adaptive wavefront sensing. Applied Optics, 2008, 47(35): 6666–6674
Elster C, Weingärtner I. Exact wave-front reconstruction from two lateral shearing interferograms. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 1999, 16(9): 2281–2285
Elster C. Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears. Applied Optics, 2000, 39(29): 5353–5359
Guo Y, Xia J, Ding J. Recovery of wavefront from multi-shear interferograms with different tilts. Optics Express, 2014, 22(10): 11407–11416
Dai G M. Modified Hartmann-Shack wavefront sensing and iterative wavefront reconstruction. Proceedings of the Society for Photo-Instrumentation Engineers, Adaptive Optics in Astronomy, 1994, 2201: 562–573
Dai G M. Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 1996, 13 (6): 1218–1225
Leibbrandt G, Harbers G, Kunst P. Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer. Applied Optics, 1996, 35(31): 6151–6161
Shen W, Chang M, Wan D. Zernike polynomial fitting of lateral shearing interferometry. Optical Engineering, 1997, 36(3): 905–913
van Brug H. Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry. Applied Optics, 1997, 36(13): 2788–2790
Okuda S, Nomura T, Kamiya K, et al. High precision analysis of lateral shearing interferogram using the integration method and polynomials. Applied Optics, 2000, 39(28): 5179–5186
De Nicola S M, Ferraro P, Finizio A, et al. Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials. Proceedings of the Society for Photo- Instrumentation Engineers, Laser Optics, 2004, 5481: 27–36
Dai G M. Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier transform. Open Optics Journal, 2007, 1(1): 1–3
Saunders J B. Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer. Journal of Research of the National Bureau of Standards—B. Mathematics and Mathematical Physics, 1961, 65B(4): 239–244
Rimmer MP. Method for evaluating lateral shearing interferograms. Applied Optics, 1974, 13(3): 623–629
Hudgin R H. Wave-front reconstruction for compensated imaging. Journal of the Optical Society of America, 1977, 67(3): 375–378
Fried D L. Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements. Journal of the Optical Society of America, 1977, 67(3): 370–375
Southwell W H. Wave-front estimation from wave-front slope measurements. Journal of the Optical Society of America, 1980, 70 (8): 998–1006
Hunt B R. Matrix formulation of the reconstruction of phase values from phase differences. Journal of the Optical Society of America, 1979, 69(3): 393–399
Herrmann J. Least-square wave-front errors of minimum norm. Journal of the Optical Society of America, 1980, 70(1): 28–35
Liu X, Gao Y, Chang M. A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry. Journal of Optics A: Pure and Applied Optics, 2009, 11(4): 045702
Zou W, Zhang Z. Generalized wave-front reconstruction algorithm applied in a Shack-Hartmann test. Applied Optics, 2000, 39(2): 250–268
Zou W, Rolland J P. Iterative zonal wave-front estimation algorithm for optical testing with general shaped pupils. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 2005, 22 (5): 938–951
Yin Z. Exact wavefront recovery with tilt from lateral shear interferograms. Applied Optics, 2009, 48(14): 2760–2766
Nomura T, Okuda S, Kamiya K, et al. Improved Saunders method for the analysis of lateral shearing interferograms. Applied Optics, 2002, 41(10): 1954–1961
Yatagai T, Kanou T. Aspherical surface testing with shearing interferometer using fringe scanning detection method. Proceedings of the Society for Photo-Instrumentation Engineers, Precision Surface Metrology, 1983, 23: 136–141
Dai F, Tang F, Wang X, et al. Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 2012, 29(9): 2038–2047
Dai F, Tang F, Wang X, et al. High spatial resolution zonal wavefront reconstruction with improved initial value determination scheme for lateral shearing interferometry. Applied Optics, 2013, 52 (17): 3946–3956
Noll R J. Phase estimates from slope-type wave-front sensors. Journal of the Optical Society of America, 1978, 68(1): 139–140
Zou W, Rolland J P. Quantifications of error propagation in slopebased wavefront estimations. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 2006, 23(10): 2629–2638
Takajo H, Takahashi T. Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference. Journal of the Optical Society of America A: Optics, Image Science, and Vision, 1988, 5(11): 1818–1827
Shiozawa H, Fukutomi Y. Development of an ultra-precision 3DCMM with the repeatability of nanometer order. JSPE Publications Series, 1999, 3: 360–365 (in Japanese)
Negishi M, et al. A high-precision coordinate measurement machine for aspherical optics. JSPE Publications Series, 2000, 2000(2): 209–210 (in Japanese)
Whitehouse D J. Some theoretical aspects of error separation techniques in surface metrology. Journal of Physics E: Scientific Instruments, 1976, 9(7): 531–536
Su H, Hong M, Li Z, et al. The error analysis and online measurement of linear slide motion error in machine tools. Measurement Science and Technology, 2002, 13(6): 895–902
Kiyono S, Gao W. Profile measurement of machined surface with a new differential method. Precision Engineering, 1994, 16(3): 212–218
Li J, Zhang L, Hong M. Unified theory of error separation techniques-accordance of time and frequency methods. Acta Metrologica Sinica, 2002, 23(3): 164–166
Tanka H, Tozawa K, Sato H, et al. Application of a new straightness measurement method to large machine tool. CIRP Annals—Manufacturing Technology, 1981, 30(1): 455–459
Tozawa K, Sato H, O-hori M. A new method for the measurement of the straightness of machine tools and machined work. Journal of Mechanical Design, 1982, 104(3): 587–592
Tanaka H, Sato H. Extensive analysis and development of straightness measurement by sequential-two-point method. Journal of Engineering for Industry, 1986, 108(3): 176–182
Kiyono S, Huang P, Fukaya N. Datum introduced by software methods. In: International Conference of Advanced Mechatronics. 1989, 467–72
Kiyono S, Okuyama E. Study on measurement of surface undulation (2nd report): Feature measurement and digital filter. Journal of the Japan Society of Precision Engineering, 1988, 54(3): 513–518 (in Japanese)
Omar B A, Holloway A J, Emmony D C. Differential phase quadrature surface profiling interferometer. Applied Optics, 1990, 29(31): 4715–4719
Kiyono S, Gao W. Profile measurement of machined surface with a new differential method. Precision Engineering, 1994, 16(3): 212–218
Gao W, Kiyono S. High accuracy profile measurement of a machined surface by the combined method. Measurement, 1996, 19 (1): 55–64
Yin Z. Research on ultra-precision measuring straightness and surface micro topography analysis. Dissertation for the Doctoral Degree. Changsha: National University of Defense Technology, 2003 (in Chinese)
Tanaka H, Sato H. Extensive analysis and development of straightness measurement by sequential-two-points method. Journal of Engineering for Industry, 1986, 108(3): 176–182
Gao W, Kiyono S. On-machine profile measurement of machined surface using the combined three-point method. JSME International Journal Series C: Mechanical Systems, Machine Elements and Manufacturing, 1997, 40(2): 253–259
Gao W, Kiyono S. On-machine roundness measurement of cylindrical workpieces by the combined three-point method. Measurement, 1997, 21(4): 147–156
Gao W, Yokoyama J, Kojima H, Kiyono S. Precision measurement of cylinder straightness using a scanning multi-probe system. Precision Engineering, 2002, 26(3): 279–288
Yin Z, Li S. Exact straightness reconstruction for on-machine measuring precision workpiece. Precision Engineering, 2005, 29(4): 456–466
Li S, Tan J, Pan P. Fine sequential-three-point method for on-line measurement of the straightness of precision lathes. Proceedings of the Society for Photo-Instrumentation Engineers, Measurement Technology and Intelligent Instruments, 1993, 2101, 309–312
Su H, Hong M, Li Z, et al. The error analysis and online measurement of linear slide motion error in machine tools. Measurement Science and Technology, 2002, 13(6): 895–902
Li C, Li S, Yu J. High resolution error separation technique for insitu straightness measurement of machine tools and workpiece. Mechatronics, 1996, 6(3): 337–347
Liang J, Li S, Yang S. Problems and solving methods of on-line measuring straightness. Proceedings of the Society for Photo-Instrumentation Engineers, the International Society for Optical Engineering, 1993, 2101: 1081–1084
Fung E H K, Yang SM. An error separation technique for measuring straightness motion error of a linear slide. Measurement Science and Technology, 2000, 11(10): 1515–1521
Fung E H K, Yang S M. An approach to on-machine motion error measurement of a linear slider. Measurement, 2001, 29(1): 51–62
Yang S M, Fung E H K, Chiu W M. Uncertainty analysis of onmachine motion and profile measurement with sensor reading errors. Measurement Science and Technology, 2002, 13(12): 1937–1945
Weingärtner I, Elster C. System of four distance sensors for high accuracy measurement of topography. Precision Engineering, 2004, 28(2): 164–170
Elster C, Weingärtner I, Schulz M. Coupled distance sensor systems for high-accuracy topography measurement: Accounting for scanning stage and systematic sensor errors. Precision Engineering, 2006, 30(1): 32–38
Schulz M, Gerhardt J, Geckeler R, et al. Traceable multiple sensor system for absolute form measurement. Proceedings of the Society for Photo-Instrumentation Engineers, Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II, 2005, 5878: 58780A
Yang P, Takamura T, Takahashi S, et al. Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy. Precision Engineering, 2011, 35(4): 686–692
Yin Z, Li S, Tian F. Exact reconstruction method for on-machine measurement of profile. Precision Engineering, 2014, 38(4): 969–978
Yin Z, Li S, Chen S, et al. China Patent, CN201410533360.8, 2014-10-11
Chen F. Digital shearography: State of the art and some applications. Journal of Electronic Imaging, 2001, 10(1): 240–251
Mallick S, Robin M L. Shearing interferometry by wavefront reconstruction using a single exposure. Applied Physics Letters, 1969, 14(2): 61–62
Nakadate S. Phase shifting speckle shearing polarization interferometer using a birefringent wedge. Optics and Lasers in Engineering, 1997, 26(4–5): 331–350
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This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 51575520 and 51375488).
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Zhai, D., Chen, S., Yin, Z. et al. Review of self-referenced measurement algorithms: Bridging lateral shearing interferometry and multi-probe error separation. Front. Mech. Eng. 12, 143–157 (2017). https://doi.org/10.1007/s11465-017-0432-3
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DOI: https://doi.org/10.1007/s11465-017-0432-3