CLC number: TH744.3; TH822
On-line Access: 2024-08-27
Received: 2023-10-17
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Peng-cheng Hu, Di Chang, Jiu-bin Tan, Rui-tao Yang, Hong-xing Yang, Hai-jin Fu. Displacement measuring grating interferometer: a review[J]. Frontiers of Information Technology & Electronic Engineering, 2019, 20(5): 631-654.
@article{title="Displacement measuring grating interferometer: a review",
author="Peng-cheng Hu, Di Chang, Jiu-bin Tan, Rui-tao Yang, Hong-xing Yang, Hai-jin Fu",
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pages="631-654",
year="2019",
publisher="Zhejiang University Press & Springer",
doi="10.1631/FITEE.1800708"
}
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Abstract: A grating interferometer, called the “optical encoder,” is a commonly used tool for precise displacement measurements. In contrast to a laser interferometer, a grating interferometer is insensitive to the air refractive index and can be easily applied to multi-degree-of-freedom measurements, which has made it an extensively researched and widely used device. Classified based on the measuring principle and optical configuration, a grating interferometer experiences three distinct stages of development: homodyne, heterodyne, and spatially separated heterodyne. Compared with the former two, the spatially separated heterodyne grating interferometer could achieve a better resolution with a feature of eliminating periodic nonlinear errors. Meanwhile, numerous structures of grating interferometers with a high optical fold factor, a large measurement range, good usability, and multi-degree-of-freedom measurements have been investigated. The development of incremental displacement measuring grating interferometers achieved in recent years is summarized in detail, and studies on error analysis of a grating interferometer are briefly introduced.
[1]Akiyama K, Iwaoka H, 1986. High Resolution Digital Diffraction Grating Scale Encoder. US Patent US4629886A.
[2]Asano S, Goto T, Tanimura H, et al., 2001. Development of a new measurement system for straightness error by a heterodyne interferometer with a grating. Proc 10th Initiatives of Precision Engineering at the Beginning of a Millennium, p.799-803.
[3]Badami VG, de Groot PJ, 2013. Displacement measuring interferometry. In: Harding K (Ed.), Handbook of Optical Dimensional Metrology. CRC Press, Boca Raton, USA, p.157-238.
[4]Badaroglu M, Ng K, Salmani M, et al., 2014. More Moore landscape for system readiness—ITRS2.0 requirements. IEEE 32nd Int Conf on Computer Design, p.147-152.
[5]Bae EW, Kim JA, Kim SH, 2011. Multi-degree-of-freedom displacement measurement system for milli-structures. Meas Sci Technol, 12:1495-1502.
[6]Bai Y, Hu PC, Lu YF, et al., 2017. A six-axis heterodyne interferometer system for Joule balance. IEEE Trans Instrum Meas, 66(6):1579-1585.
[7]Bi JL, Liu SB, 2016. Precision measurement of displacement with two quasi-orthogonal signals for linear diffraction grating interferometric sensors. Proc 6th Int Conf on Machinery, Materials, Environment, Biotechnology and Computer, p.25-29.
[8]Cai HJ, 2015. Research on Picometer Resolution Phase Subdivision for Heterodyne Laser Interferometer Signals. MS Thesis, Harbin Institute of Technology, Harbin, China (in Chinese).
[9]Chang D, Xing X, Hu P, et al., 2019. Double-diffracted spatially separated heterodyne grating interferometer and analysis on its alignment tolerance. Appl Sci, 9(2): Article 263.
[10]Cheng F, Fan KC, 2011. Linear diffraction grating interferometer with high alignment tolerance and high accuracy. Appl Opt, 50(22):4550-4556.
[11]Cheng F, Fan KC, Fei YT, 2009a. An improved design of the linear diffraction grating interferometer. Proc Asian Symp on Precision Engineering and Nanotechnology, p.146-149.
[12]Cheng F, Fei YT, Fan KC, 2009b. New method on real-time signal correction and subdivision for grating-based nanometrology. Proc 4th Int Symp on Advanced Optical Manufacturing and Testing Technologies: Design, Manufacturing, and Testing of Micro- and Nano-Optical Devices and Systems, No. 728403.
[13]Cho A, 2016. The long road to LIGO. Science, 353(6299): 534-535.
[14]Chu XC, Lü HB, Zhao SH, 2008. Research on long-range grating interferometry with nanometer resolution. Meas Sci Technol, 19(1):017001.
[15]Chung YC, Fan KC, Lee BC, 2011. Development of a novel planar encoder for 2D displacement measurement in nanometer resolution and accuracy. Proc 9th World Congress on Intelligent Control and Automation, p.449-453.
[16]Cloud G, 2005. Optical methods in experimental mechanics, part 16: the optical Doppler effect. Exp Tech, 29(2):17-19.
[17]Cosijns SJAG, Jansen MJ, 2014. Advanced optical incremental sensors: encoders and interferometers. In: Nihtianov S, Luque A (Eds.), Smart Sensors and MEMS: Intelligent Devices and Microsystems for Industrial Applications. Woodhead Publishing, Oxford, UK, p.230-277.
[18]Damm C, Peschel T, Risse S, et al., 2001. Wafer stage assembly for ion projection lithography. Microelectron Eng, 57-58:181-185.
[19]Deck LL, de Groot PJ, Schroeder M, 2015. Interferometric Encoder Systems. US Patent US898 869 0B2.
[20]de Groot PJ, Schroeder M, 2012. Interferometric Heterodyne Optical Encoder System. US Patent US2012/0194824.
[21]de Groot PJ, Badami VG, Liesener J, 2016. Concepts and geometries for the next generation of precision heterodyne optical encoders. Proc 31st Annual Meeting of the American Society for Precision Engineering, p.146-149.
[22]Deng JJ, Yan XN, Wei CL, et al., 2018. Eightfold optical encoder with high-density grating. Appl Opt, 57(10): 2366-2375.
[23]Dobosz M, 1999. High-resolution laser linear encoder with numerical error compensation. Opt Eng, 38(6):968-973.
[24]Dr. Johannes Heidenhain GmbH, 2018. Exposed linear encoders. https://www.heidenhain.com/fileadmin/pdb/media/img/208960-2E_Exposed_Linear_Encoders_en.pdf [Accessed on Aug. 15, 2018].
[25]Ellis JD, 2014. Field Guide to Displacement Measuring Interferometry. SPIE Press, Bellingham, UK.
[26]Fan KC, Cheng F, 2010. LDGI signal subdivision by soft computing for nanomeasurement. Proc 6th Int Symp on Precision Engineering Measurements and Instrumentation, No. 75440I.
[27]Fan KC, Liu YS, Chen YJ, et al., 2006. A linear diffraction grating interferometer with high accuracy. Proc 3rd Int Symp on Precision Mechanical Measurements, No. 628008.
[28]Fan KC, Lai ZF, Wu P, et al., 2007. A displacement spindle in a micro/nano level. Meas Sci Technol, 18(6):1710-1717.
[29]Fan KC, Miao JW, Gong W, et al., 2011a. High precision measurement system based on coplanar XY-stage. Proc 7th Int Symp on Precision Engineering Measurement and Instrumentation, No. 832119.
[30]Fan KC, Lee BC, Chung YC, 2011b. A planar laser diffraction encoder in Littrow configuration for 2D nanometric positioning. Int J Autom Smart Technol, 1(2):93-99.
[31]Fan KC, Liao BH, Chung YC, et al., 2012a. Displacement measurement of planar stage by diffraction planar encoder in nanometer resolution. Proc IEEE Int Instrumentation and Measurement Technology Conf, p.894-897.
[32]Fan KC, Zhang YL, Miao JW, et al., 2012b. Error compensation of grating interferometer due to angular error of linear stage. Proc IEEE/ASME Int Conf on advanced Intelligent Mechatronics, p.428-431.
[33]Fan KC, Wang HY, Yang HW, et al., 2014. Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines. Adv Opt Technol, 3(4): 375-386.
[34]Feng C, Kajima M, Gonda S, et al., 2012. Accurate measurement of orthogonality of equal-period, two-dimensional gratings by an interferometric method. Metrologia, 49: 236-244.
[35]Feng C, Zeng LJ, Wang SW, 2013. Heterodyne planar grating encoder with high alignment tolerance, especially insensitivity to grating tilts. Proc 8th Int Symp on Precision Engineering Measurement and Instrumentation, No. 87593L.
[36]Fu HJ, Ji RD, Hu PC, et al., 2018a. Measurement method for nonlinearity in heterodyne laser interferometers based on double-channel quadrature demodulation. Sensors, 18(9): Article 2768.
[37]Fu HJ, Wang Y, Hu PC, et al., 2018b. Nonlinear errors resulting from ghost reflection and its coupling with optical mixing in heterodyne laser interferometers. Sensors, 18(3): Article 758.
[38]Gao W, 2010. Precision Nanometrology. Springer-Verlag London Limited, London, UK, p.69-140.
[39]Gao W, Kimura A, 2007. A three-axis displacement sensor with nanometric resolution. CIRP Ann, 56(1):529-532.
[40]Gao W, Kimura A, 2010. A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating. CIRP Ann, 59:505-508.
[41]Gao W, Saito Y, Muto H, et al., 2011. A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann, 60:515-518.
[42]Gao ZY, Hu JC, Zhu Y, et al., 2013. A new 6-degree-of-freedom measurement method of X-Y stages based on additional information. Prec Eng, 37(3):606-620.
[43]Gao W, Kim SW, Bosse H, et al., 2015. Measurement technologies for precision positioning. CIRP Ann, 64(2):773-796.
[44]Goßler S, Bertolini A, Born M, et al., 2010. The AEI 10m prototype interferometer. Class Quant Grav, 27(8): Article 080423.
[45]Guan J, Köchert P, Weichert C, et al., 2017. A differential interferometric heterodyne encoder with 30 picometer periodic nonlinearity and sub-nanometer stability. Prec Eng, 50:114-118.
[46]Guo DM, Wang M, 2015. Note: design of a laser feedback interferometer with double diffraction system. Rev Sci Instrum, 86(9):096111.
[47]Guo DM, Wang M, Hao H, 2015. Displacement measurement using a laser feedback grating interferometer. Appl Opt, 54(31):9320-9325.
[48]Guo DM, Wang M, Hao H, 2016. Self-mixing grating interferometer: theoretical analysis and experimental observations. Proc Interferometry XVIII, No. 996019.
[49]Guo DM, Shi L, Yu YG, et al., 2017. Micro-displacement reconstruction using a laser self-mixing grating interferometer with multiple-diffraction. Opt Exp, 25(25): 31394-31406.
[50]Heilmann RK, Chen CG, Konkola PT, et al., 2004. Dimensional metrology for nanometer-scale science and engineering: towards sub-nanometer accurate encoders. Nanotechnology, 15(10):S504-S511.
[51]Holzapfel W, 2008. Advancements in displacement metrology based on encoder systems. Proc 23rd Annual Meeting of the American Society for Precision Engineering, p.1-4.
[52]Hori Y, Gonda S, Bitou Y, et al., 2018. Periodic error evaluation system for linear encoders using a homodyne laser interferometer with 10 picometer uncertainty. Prec Eng, 51:388-392.
[53]Hosono K, Kim W, Kimura A, et al., 2011. Surface encoders for a mosaic scale grating. Int J Autom Technol, 5(2):91-96.
[54]Hossein GM, Lin MJ, Horng JB, et al., 2016. Critical femtosecond laser parameters for the fabrication of optimal reflecting diffraction grating on Invar36. Opt Laser Eng, 81:97-102.
[55]Hou WM, Zhang YB, Hu HJ, 2009. A simple technique for eliminating the nonlinearity of a heterodyne interferometer. Meas Sci Technol, 20(10):105303.
[56]Hsieh HL, Chen W, 2016. Heterodyne Wollaston laser encoder for measurement of in-plane displacement. Opt Exp, 24(8):8693-8707.
[57]Hsieh HL, Pan SW, 2013. Three-degree-of-freedom displacement measurement using grating-based heterodyne interferometry. Appl Opt, 52(27):6840-6848.
[58]Hsieh HL, Pan SW, 2015. Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements. Opt Exp, 23(3):2451-2465.
[59]Hsieh HL, Lee JY, Wu WT, et al., 2010. Quasi-common-optical-path heterodyne grating interferometer for displacement measurement. Meas Sci Technol, 21(11): 115304.
[60]Hsieh HL, Chen JC, Lerondel G, et al., 2011. Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer. Opt Exp, 19(10):9770-9782.
[61]Hsieh HL, Lee JY, Chung YC, 2014. Wavelength-modulated heterodyne grating shearing interferometry for precise displacement measurement. Adv Opt Technol, 3(4):395-400.
[62]Hsu CC, Wu CC, Lee JY, et al., 2008. Reflection type heterodyne grating interferometry for in-plane displacement measurement. Opt Commun, 281(9):2582-2589.
[63]Hsu CC, Lee JY, Wu CC, 2009. Compensated laser encoder with symmetric and quasi-common-path heterodyne interferometry. Optical Measurement Systems for Industrial Inspection, No. 73891M.
[64]Hsu CC, Sung PP, Lin ZR, et al., 2013. Prototype of a compact displacement sensor with a holographic diffraction grating. Opt Laser Technol, 48:200-205.
[65]Hsu CC, Chen H, Chiang CW, et al., 2017. Dual displacement resolution encoder by integrating single holographic grating sensor and heterodyne interferometry. Opt Exp, 25(24):30189-30202.
[66]Hsu CC, Chen H, Tseng HY, et al., 2018. High displacement resolution encoder by using triple grating combination interferometer. Opt Laser Technol, 105:221-228.
[67]Hu PC, Pollinger F, Meiners-Hagen K, et al., 2010. Fine correction of nonlinearity in homodyne interferometry. 6th Int Symp on Precision Engineering Measurements and Instrumentation, No. 75444E.
[68]Hu PC, Tan JB, Chen P, 2014. Double Frequency Laser Grating Interference Three-dimensional Measurement Method and System with Optical Aliasing Resistance. CN Patent CN103 604 376A (in Chinese).
[69]Hu PC, Zhu JH, Guo XB, et al., 2015. Compensation for the variable cyclic error in homodyne laser interferometers. Sensors, 15:3090-3106.
[70]Huang HL, Liu CH, Jywe WY, et al., 2007. Development of a three-degree-of-freedom laser linear encoder for error measurement of a high precision stage. Rev Sci Instrum, 78(6):066103.
[71]Ito S, Aihara R, Kim WJ, et al., 2014. Three-axis vibration measurement by using a grating-interferometric vibrometer. Adv Opt Technol, 3(4):435-440.
[72]Jakštas A, Kaušinis S, Flügge J, 2005. Investigation of calibration facilities of precision line scales. Mechanika, 53(3):62-67.
[73]Jang YS, Kim SW, 2017. Compensation of the refractive index of air in laser interferometer for distance measurement: a review. Int J Prec Eng Manuf, 18(12):1881-1890.
[74]Jiang ML, Li FP, Wang XD, 2011. Mini-nano-displacement measurement with double diffraction grating. Adv Mater Res, 230-232:1159-1163.
[75]Jiang ML, Li FP, Wang XD, 2012. Nano-displacement measurement with grating interference. Appl Mech Mater, 103: 35-40.
[76]Jiang ML, Li HF, Wang XD, et al., 2013. Research status and developing trends of grating nanometer measuring technology. 6th Int Symp on Precision Mechanical Measurements, No. 891631.
[77]Jourlin Y, Jay J, Parriaux O, 2002. Compact diffractive interferometric displacement sensor in reflection. Prec Eng, 26:1-6.
[78]Kao CF, Chang CC, Lu MH, 2005a. Double-diffraction planar encoder by conjugate optics. Opt Eng, 44(2):023603.
[79]Kao CF, Lu SH, Lu MH, 2005b. High resolution planar encoder by retro-reflection. Rev Sci Instrum, 76(8):085110.
[80]Kao CF, Lu SH, Shen HM, et al., 2008. Diffractive laser encoder with a grating in Littrow configuration. Jpn J Appl Phys, 47(3):1833-1837.
[81]Karhade OG, Degertekin FL, Kurfess TR, 2008. SOI-based micro scanning grating interferometers: device characterization, control and demonstration of parallel operation. J Microm Micr, 18(4):045007.
[82]Kilby J, 2000. The integrated circuit’s early history. Proc IEEE, 88(1):109-111.
[83]Kim JA, Kim KC, Bae EW, et al., 2000. Six-degree-of-freedom displacement measurement system using a diffraction grating. Rev Sci Instrum, 71(8):3214-3219.
[84]Kim JA, Bae EW, Kim SH, et al., 2002. Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets. Prec Eng, 26(1):99-104.
[85]Kim P, Kim D, You K, 2012. Adaptive compensation for the nonlinearity error in a heterodyne interferometer. J Korean Phys Soc, 61(11):1759-1765.
[86]Kimura A, Gao W, Arai Y, et al., 2010a. Design and construction of a two-degree-of-freedom linear encoder for nanometric measurement of stage position and straightness. Prec Eng, 34(1):145-155.
[87]Kimura A, Gao W, Zeng LJ, 2010b. Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder. Meas Sci Technol, 21:054005.
[88]Kimura A, Hosono K, Kim WJ, et al., 2011. A two-degree-of-freedom linear encoder with a mosaic scale grating. Int J Nanomanuf, 7(1):73-91.
[89]Kimura A, Gao W, Kim WJ, et al., 2012. A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement. Prec Eng, 36(4): 576-585.
[90]Kunzmann H, Pfeifer T, Flügge J, 1993. Scales vs. laser interferometers performance and comparison of two measuring systems. CIRP Ann, 42(2):753-767.
[91]Kwan YBP, 2011. Lithographic Apparatus, Device Manufacturing Method and Device Manufactured Thereby. US Patent US794 039 2.
[92]Lan HB, Ding YC, Liu HZ, et al., 2007. Review of the wafer stage for nanoimprint lithography. Microelectron Eng, 84(4):684-688.
[93]Lazar J, Číp O, Čížek M, et al., 2010. Multiaxis interferometric system for positioning in nanometrology. Proc 9th WSEAS Int Conf on Microelectronics, Nanoelectronics, Optoelectronics, p.92-95.
[94]Lee CB, Lee SK, 2013. Multi-degree-of-freedom motion error measurement in an ultraprecision machine using laser encoder. Rev J Mech Sci Technol, 27(1):141-152.
[95]Lee CB, Kim GH, Lee SK, 2011. Design and construction of a single unit multi-function optical encoder for a six-degree-of-freedom motion error measurement in an ultraprecision linear stage. Meas Sci Technol, 22(10): 105901.
[96]Lee CB, Kim GH, Lee SK, 2012. Uncertainty investigation of grating interferometry in six degree-of-freedom motion error measurements. Int J Prec Eng Manuf, 13(9):1509-1515.
[97]Lee CK, Wu CC, Chen SJ, et al., 2004. Design and construction of linear laser encoders that possess high tolerance of mechanical runout. Appl Opt, 43(31):5754-5762.
[98]Lee JY, Lu MP, 2011. Optical heterodyne grating shearing interferometry for long-range positioning applications. Opt Commun, 284(1):857-862.
[99]Lee JY, Jiang GA, 2013. Displacement measurement using a wavelength phase-shifting grating interferometer. Opt Exp, 21(21):25553-25564.
[100]Lee JY, Chen HY, Hsu CC, et al., 2007. Optical heterodyne grating interferometry for displacement measurement with subnanometric resolution. Sensor Actuat A, 137(1): 185-191.
[101]Lee JY, Hsieh HL, Lerondel C, et al., 2011. Heterodyne grating interferometer based on a quasi-common-optical path configuration for a two-degrees-of-freedom straightness measurement. Appl Opt, 50(9):1272-1279.
[102]Lee TH, 2007. The (pre-) history of the integrated circuit: a random walk. IEEE SSCS News, 12(2):16-22.
[103]Li XH, Gao W, Muto H, et al., 2013. A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage. Prec Eng, 37(3):771-781.
[104]Li XH, Shimizu Y, Ito T, et al., 2014. Measurement of six-degree-of-freedom planar motions by using a multiprobe surface encoder. Opt Eng, 53(12):122405.
[105]Li Q, Liu X, Zhao L, et al., 2017. A novel vibration sensor based on phase grating interferometry. Appl Phys B, 123(5): Article 162.
[106]Liesener J, 2013. Compact Encoder for Interferometric Encoder System. US Patent US201 311 406 2A1.
[107]Lin CB, Yan SH, Wei C, et al., 2013. Optimized design and error analysis of optical system for heterodyne grating interferometry. Int Conf on Optical Instruments and Technology: Optoelectronic Measurement Technology and Systems, No. 90460C.
[108]Lin CB, Yan SH, Du ZG, et al., 2015a. High-efficiency gold-coated cross-grating for heterodyne grating interferometer with improved signal contrast and optical subdivision. Opt Commun, 339:86-93.
[109]Lin CB, Yan SH, Du ZG, et al., 2015b. Symmetrical short-period and high signal-to-noise ratio heterodyne grating interferometer. Chin Opt Lett, 13(10):100501.
[110]Lin CB, Yan SH, Ding D, et al., 2018. Two-dimensional diagonal-based heterodyne grating interferometer with enhanced signal-to-noise ratio and optical subdivision. Opt Eng, 57(6):064102.
[111]Lin DJ, Jiang H, Yin C, 2000. Analysis of nonlinearity in a high-resolution grating interferometer. Opt Laser Technol, 32(2):95-99.
[112]Lin J, Guan J, Wen F, et al., 2015. Grating encoder for wide range three-axis displacement measurement. 9th Int Symp on Precision Engineering Measurement and Instrumentation, No. 944602.
[113]Lin J, Guan J, Wen F, et al., 2017. High-resolution and wide range displacement measurement based on planar grating. Opt Commun, 404:132-138.
[114]Liu CH, Cheng CH, 2012. Development of a grating based multi-degree-of-freedom laser linear encoder using diffracted light. Sens Actuat A, 181:87-93.
[115]Liu CH, Jywe WY, Tzeng SC, 2004. Simple three-dimensional laser angle sensor for three-dimensional small-angle measurement. Appl Opt, 43(14):2840-2845.
[116]Liu CH, Huang HL, Lee HW, 2009. Five-degrees-of-freedom diffractive laser encoder. Appl Opt, 48(14):2767-2777.
[117]Lu YC, Zhou CH, Li S, et al., 2016a. Study of a grating interferometer with high optical subdivision technique. Holography, Diffractive Optics, and Applications VII, No. 1002214.
[118]Lu YC, Wei CL, Jia W, et al., 2016b. Two-degree-freedom displacement measurement based on a short period grating in symmetric Littrow configuration. Opt Commun, 380:382-386.
[119]Lu YX, Qi XD, Li XT, et al., 2016. Removal of all mosaic grating errors in a single-interferometer system by a phase-difference reference window. Appl Opt, 55(28): 7997-8002.
[120]Lu ZG, Wei PP, Wang CQ, et al., 2016. Two-degree-of-freedom displacement measurement system based on double diffraction gratings. Meas Sci Technol, 27(7): 074012.
[121]Lv Q, Liu ZW, Wang W, et al., 2018. Simple and compact grating-based heterodyne interferometer with the Littrow configuration for high-accuracy and long-range measurement of two-dimensional displacement. Appl Opt, 57(31):9455-9463.
[122]Makinouchi S, Watanabe A, Takasaki M, et al., 2011. An evaluation of a modulated laser encoder. Prec Eng, 35(2): 302-308.
[123]Olayee S, Firoozjah AG, 2015. Modeling the nonlinearity of polarizing beam splitters in nano-displacement measurement of laser encoder using Jones matrix analysis. Proc 13th Int Conf on Telecommunications, p.1-6.
[124]Olayee S, Firoozjah AG, Naraghi A, 2017. Mathematically modeling of imperfect polarized laser beam in laser encoders for automotive applications. Proc 5th Int Conf on Control, Instrumentation, and Automation, p.44-48.
[125]Pan SW, Hsieh HL, Wang WC, 2013. 6-DOF displacement and angle measurements using heterodyne laser encoder. Instrumentation, Metrology, and Standards for Nanomanufacturing, Optics, and Semiconductors VII, No. 881909.
[126]Post D, 1971. Moiré fringe multiplication with a nonsymmetrical doubly blazed reference grating. Appl Opt, 10(4): 901-907.
[127]Prince J, 2011. Compact Littrow Encoder. US Patent US786 433 6.
[128]Ramm P, Klumpp A, Weber J, et al., 2010. 3D system-on-chip technologies for more than Moore systems. Michrosyst Technol, 16(7):1051-1055.
[129]Renkens MJM, Struycken AM, Kok RJ, et al., 2006. Lithographic Apparatus, Measurement System, and Device Manufacturing Method. US Patent US710 272 9.
[130]Richardson C, Tsuriya M, Fu H, 2017. Technology roadmap overviews and future direction through technology gaps. Int Conf on Electronics Packaging, p.35-40.
[131]Riemer O, 2011. Advances in ultra precision manufacturing. Proc Int Symp on Ultraprecision Engineering and Nanotechnology, p.1-6
[132]Ronchi V, 1964. Forty years of history of a grating interferometer. Appl Opt, 3(4):437-451.
[133]Saito Y, Arai Y, Gao W, 2009. Detection of three-axis angles by an optical sensor. Sensors Actuat A, 150(2):175-183.
[134]Schmidt RHM, 2012. Ultra-precision engineering in lithographic exposure equipment for the semiconductor industry. Phil Trans R Soc A, 370(1973):3950-3972.
[135]Schwenke H, Neuschaefer-Rube U, Pfeifer T, et al., 2002. Optical methods for dimensional metrology in production engineering. CIRP Ann, 51(2):685-699.
[136]Shang P, Xia HJ, Fei YT, 2016. High-resolution diffraction grating interferometric transducer of linear displacements. 7th Int Symp on Precision Mechanical Measurements, No. 99031I.
[137]Shankar A, Tanwar LS, Sirohi RS, 2007. High resolution and low run-out errors in displacement measuring grating interferometer using natural interference zone. J Opt, 36(3): 111-122.
[138]Shibazaki Y, Kohno H, Hamatani M, 2009. An innovative platform for high-throughput, high-accuracy lithography using a single wafer stage. Proc SPIE, Optical Microlithography XXII, No. 72741I.
[139]Shimizu Y, Kim WJ, Ito S, et al., 2012. Form error characterization of reflective-type gratings. Key Eng Mater, 523-524:859-864.
[140]Shimizu Y, Ito T, Li X, et al., 2014. Design and testing of a four-probe optical sensor head for three-axis surface encoder with a mosaic scale grating. Meas Sci Technol, 25(9):094002.
[141]Shin D, Kim B, 2010. An optical encoder using micromachined two gratings with phase. Proc ASME Int Mechanical Engineering Congress & Exposition, p.479-483.
[142]Shin D, Kim B, 2011. A laser interferometer encoder with two micromachined gratings generating phase-shifted quadrature. J Micromech Microeng, 21(8):085036.
[143]Shishova MV, Odinokov SB, Lushnikov DS, et al., 2017. Mathematical modeling of signal transfer process into optical system of a linear displacement encoder. Proc Eng, 201:623-629.
[144]Shu XM, Zuo Y, Xu XB, 2010. Error correction technology of the length grating measuring system. Proc 6th Int Symp on Precision Engineering Measurements and Instrumentation, No. 75442G.
[145]Šiaudinytė L, Molnar G, Köning R, et al., 2018. Multi-dimensional grating interferometer based on fibre-fed measurement heads arranged in Littrow configuration. Meas Sci Technol, 29(5):054007.
[146]Tan JB, Hu PC, Xing X, 2014. Double Frequency Laser Grating Interference Two-dimensional Measurement Method. CN Patent CN103 604 375A (in Chinese).
[147]Tao Z, Tan JB, Cui JW, 2015. Linear response, multi-order grating interferometry using a reversal shearing imaging system. Opt Lett, 40(19):4552-4555.
[148]Teimel A, 1991. Technology and application of grating interferometers in high-precision measurement. Progress in Precision Engineering. Springer-Verlag, Berlin, Germany, p.15-30.
[149]Wang GC, Xie XD, Yan SH, 2010. Influence of non-ideal performance of lasers on displacement precision in single grating heterodyne interferometry. Proc 5th Int Symp on Advanced Optical Manufacturing and Testing Technologies, No. 76562X.
[150]Wang GC, Yan SH, Yang DX, et al., 2011. An interference signal processing method for displacement measurement by dual wavelength and single grating. 7th Int Symp on Precision Engineering Measurement and Instrumentation, No. 83211A.
[151]Wang GC, Yan SH, Zhou WH, et al., 2012. Dynamic tracking down-conversion signal processing method based on reference signal for grating heterodyne interferometer. Opt Eng, 51(8):081512.
[152]Wang J, 2017. Research on 3-DOF Grating Interference Measurement Model of Mask Table. MS Thesis, Harbin Institute of Technology, Harbin, China (in Chinese).
[153]Wang LJ, Zhang M, Zhu Y, et al., 2014. A novel heterodyne grating interferometer system for in-plane and out-of-plane displacement measurement with nanometer resolution. Proc 29th Annual Meeting of the American Society for Precision Engineering, p.173-177.
[154]Wang XZ, Dong XH, Guo J, et al., 2004. Two-dimensional displacement sensing using a cross diffraction grating scheme. J Opt A Pure Appl Opt, 6(1):106-111.
[155]Wei CH, Yan SH, Lin CB, et al., 2015a. Compact grating displacement measurement system with a 3×3 coupler. Chin Opt Lett, 13(5):051301.
[156]Wei CH, Yan SH, Lin CB, et al., 2015b. Signal processing for single grating displacement measurement based on 3×3 coupler. Proc 9th Int Symp on Precision Engineering Measurement and Instrumentation, No. 944625.
[157]Wei PP, Lu X, Qiao DC, et al., 2018. Two-dimensional displacement measurement based on two parallel gratings. Rev Sci Instrum, 89(6):065105.
[158]Williams RP, Hord SK, Hall NA, 2017. Optically read displacement detection using phase-modulated diffraction gratings with reduced zeroth-order reflections. Appl Phys Lett, 110(15):151104.
[159]Wise S, Quetschke V, Deshpande AJ, et al., 2005. Phase effects in the diffraction of light: beyond the grating equation. Phys Rev Lett, 95(1):013901.
[160]Wu CC, Chen YC, Lee CK, et al., 1999. Design verifications of a linear laser encoder with high head-to-scale tolerance. Current Developments in Optical Design and Optical Engineering VIII, p.73-82.
[161]Wu CC, Chang CC, Kao CF, et al., 2003. Novel planar laser encoder system for two-dimensional positioning with ultra high head-to-scale alignment tolerance. Current Developments in Lens Design and Optical Engineering IV, p.55-63.
[162]Wu CC, Wu WJ, Pan ZS, et al., 2007. Laser linear encoder with both high fabrication and head-to-scale tolerances. Appl Opt, 46(16):3169-3176.
[163]Wu CC, Hsu CC, Lee JY, et al., 2008. Optical heterodyne laser encoder for in-plane nanopositioning. Interferometry XIV: Techniques and Analysis, No. 70631A.
[164]Wu CC, Cheng CY, Yang ZY, 2010. Optical homodyne common-path grating interferometer with subnanometer displacement resolution. Interferometry XV: Applications, No. 779105.
[165]Wu CC, Hsu CC, Lee JY, et al., 2011. Common-path laser encoder for nanopositioning in long travel range. Proc SICE Annual Conf, p.817-820.
[166]Wu CC, Yang JS, Cheng CY, et al., 2013a. Common-path laser encoder. Sens Actuat A, 189:86-92.
[167]Wu CC, Chen YZ, Liao CH, 2013b. Common-path laser planar encoder. Opt Exp, 21(16):18872-18883.
[168]Wu CC, Hsu CC, Lee JY, et al., 2013c. Heterodyne common-path grating interferometer with Littrow configuration. Opt Exp, 21(11):13322-13332.
[169]Wu CC, Hsu CC, Lee JY, et al., 2013d. Littrow-type self-aligned laser encoder with high tolerance using double diffractions. Opt Commun, 297:89-97.
[170]Xia HJ, Fei YT, 2010. Precise stage design with planar diffraction grating interferometer. Proc 6th Int Symp on Precision Engineering Measurements and Instrumentation, No. 754411.
[171]Xie JD, Yan LP, Chen BY, et al., 2017. Iterative compensation of nonlinear error of heterodyne interferometer. Opt Exp, 25(4):4470-4482.
[172]Xing X, Chang D, Hu PC, et al., 2017. Spatially separated heterodyne grating interferometer for eliminating periodic nonlinear errors. Opt Exp, 25(25):31384-31393.
[173]Yan SH, Wang GC, Lin CB, et al., 2015. Displacement measurement by single-grating heterodyne interferometry. Proc 11th Conf on Lasers and Electro-Optics Pacific Rim, p.1-2.
[174]Ye GY, Fan SJ, Liu HZ, et al., 2014. Design of a precise and robust linearized converter for optical encoders using a ratiometric technique. Meas Sci Technol, 25(12):125003.
[175]Yu HY, Liu HZ, Li X, et al., 2015. Calibration of non-contact incremental linear encoders using a macro–micro dual-drive high-precision comparator. Meas Sci Technol, 26(9): Article 095103.
[176]Yuan JL, Lyu BH, Hang W, et al., 2017. Review on the progress of ultra-precision machining technologies. Front Mech Eng, 12(2):158-180.
[177]Zhang M, Zhu Y, Wang LJ, et al., 2016. Two-DOF Heterodyne Grating Interferometer Displacement Measurement System. US Patent US013 890 3A1.
[178]Zhang M, Zhu Y, Ni C, et al., 2017. Two-DOF Heterodyne Grating Interferometer Displacement Measurement System. CN Patent CN106 289 068A (in Chinese).
[179]Zhao B, Wang L, Xu ME, et al., 2015. A displacement measuring system based on grating double diffraction. Proc 9th Int Symp on Precision Engineering Measurement and Instrumentation, No. 94464J.
[180]Zhao JL, 2016. Key Technology of Signal Processing in Heterodyne Laser Interferometry with Pico Meter Resolution. MS Thesis, Harbin Institute of Technology, Harbin, China (in Chinese).
[181]Zhao SS, Hou CL, Bai J, et al., 2011. Nanometer-scale displacement sensor based on phase-sensitive diffraction grating. Appl Opt, 50(10):1413-1416.
[182]Zherdev AY, Odinokov SB, Lushnikov DS, et al., 2017. Optical position encoder on four-section diffraction grating. Proc SPIE, Holography: Advances and Modern Trends, No. 102331I.
[183]Zhu Y, Zhang M, Wang LJ, et al., 2014. Dual-Frequency Grating Interferometer Displacement Measurement System. WO Patent WO201 407 180 6A1.
[184]Zhu Y, Zhang M, Wang LJ, et al., 2018. Dual-frequency Grating Interferometer Displacement Measurement System. US Patent US988 556 B2.
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