Abstract

In this paper, a novel optical encoder enabling the simultaneous measurement of displacement and the position of precision stages is presented. The encoder is composed of an improved single-track scale grating and a compact two-probe reading head. In the scale grating, multiple reference codes are physically superimposed onto the incremental grooves, in contrast to conventional designs, where an additional track is necessary. The distribution of the reference codes follows a specific mathematical algorithm. For the reading head, a two-probe structure is designed to identify the discrete reference codes by means of the superimposition of the codes with a stationary mask and to read the continuous incremental grooves by means of a grating interferometry, respectively. A prototype encoder was designed, constructed and evaluated, and experimental results show that the distance code precision achieved is 0.5 μm, while the linearity error of the linear displacement measurement is less than 0.06%.

© 2016 Optical Society of America

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  25. A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
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2016 (2)

2015 (3)

2014 (2)

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

2013 (2)

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

2010 (2)

W. Gao and A. Kimura, “A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating,” CIRP Ann. 59(1), 505–508 (2010).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

2008 (1)

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

2007 (1)

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

2005 (1)

2004 (1)

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

2003 (1)

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Alonso, J.

J. Saez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method,” Opt. Express 13(1), 195–201 (2005).
[Crossref] [PubMed]

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135.
[Crossref]

Arai, Y.

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Bernabeu, E.

J. Saez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method,” Opt. Express 13(1), 195–201 (2005).
[Crossref] [PubMed]

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135.
[Crossref]

Bosse, H.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Brueck, S. R. J.

S. R. J. Brueck, “Optical and Interferometric Lithography-Nanotechnology Enablers,” in Proceedings of the IEEE (IEEE, 2005) 93(10), pp. 1704–1721.

Cai, Y.

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

Chen, W.

Chen, Y.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Dejima, S.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Delbressine, F.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Dian, S.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Estler, W. T.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Fujita, K.

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Gao, W.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

W. Gao and A. Kimura, “A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating,” CIRP Ann. 59(1), 505–508 (2010).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Haitjema, H.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Hsieh, H. L.

Ishimura, S.

Ito, S.

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

Ito, T.

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

Katakura, K.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Kikuchi, K.

Kim, W.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Kimura, A.

W. Gao and A. Kimura, “A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating,” CIRP Ann. 59(1), 505–508 (2010).
[Crossref]

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

Kiyono, S.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Knapp, W.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Kunzmann, H.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Li, K.

Li, X.

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Liu, Z.

Lu, X.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Matsuzoe, Y.

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Muto, H.

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Nakayama, T.

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Niu, J.

Rosa-Zurera, M.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135.
[Crossref]

Saez-Landete, J.

J. Saez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method,” Opt. Express 13(1), 195–201 (2005).
[Crossref] [PubMed]

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135.
[Crossref]

Salcedo-Sanz, S.

J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135.
[Crossref]

Schmitt, R.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Schwenke, H.

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Shimizu, Y.

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

X. Li, W. Gao, H. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Tomita, Y.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Tsuji, N.

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Wang, L.

Wang, S.

Weckenmann, A.

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

Yanai, H.

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

Yoshizawa, T.

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Zeng, L.

S. Wang and L. Zeng, “Analysis and minimization of spacing error of holographic gratings recorded with spherical collimation lenses,” Opt. Express 23(5), 5532–5546 (2015).
[Crossref] [PubMed]

A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010).
[Crossref]

Zhang, Y.

CIRP Ann. (5)

W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

W. Gao and A. Kimura, “A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating,” CIRP Ann. 59(1), 505–508 (2010).
[Crossref]

Int. J. Precis. Eng. Manuf. (1)

X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013).
[Crossref]

Opt. Eng. (2)

X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014).
[Crossref]

Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003).
[Crossref]

Opt. Express (5)

Precis. Eng. (3)

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[Crossref]

W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004).
[Crossref]

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[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Operational principle of two-probe absolute linear encoder.

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Fig. 2
Fig. 2 Optical configuration of the reference assembly.

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Fig. 3
Fig. 3 Schematic of the single-track scale grating and location of the multiple reference codes.

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Fig. 4
Fig. 4 Expanded design of the displacement assembly.

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Fig. 5
Fig. 5 Code of the reference mark (a) and the simulated signal (b).

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Fig. 6
Fig. 6 Optical microscope image of the fabricated reference mask and its sectional images tested by an AFM (a) and picture of the scale grating with reference code, the optical microscope image and the AFM images of the reference code (b).

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Fig. 7
Fig. 7 Prototype encoder (a) and experimental setup (b).

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Fig. 8
Fig. 8 The experiment reference pulse signal.

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Fig. 9
Fig. 9 Reference positioning accuracy.

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Fig. 10
Fig. 10 X-directional outputs and nonlinear error components.

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Equations (14)

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t ( x ) = ∑ i = 1 n a i × r e c t ( x − i × b b ) , 0 ≤ x ≤ 3 D
I ( x , z ) = | F − 1 { F { t ( x ) } ⋅ exp ( i 2 π z 1 / λ 2 − v 2 ) } | 2
f ( x ) = { 1 , 0 ≤ x < D 1 − t ( x ) , D ≤ x < 2 D 1 , 2 D ≤ x ≤ 3 D
I f 1 ( x 0 , z ) = I ( x , z ) ⋅ f ( x + x 0 ) , 0 ≤ x ≤ D , 0 ≤ x 0 ≤ 2 D
I f 2 ( x 0 , z ) = | F − 1 { F { I f 1 ( x 0 , z ) } ⋅ exp ( i 2 π z 1 / λ 2 − v 2 ) } | 2
I m ( x 0 , z ) = I f 2 ( x 0 , z ) ⋅ t ( x )
P O = P I I − ( x 1 − x 2 − L k )
Δ X = 1 2 ⋅ g 2 π ⋅ { arc tan ( S X + 1 S X + 1 ' ) − arc tan ( S X − 1 S X − 1 ' ) }
Δ Z = 1 2 ⋅ 1 1 + cos θ ⋅ λ 2 π { arc tan ( S X + 1 S X + 1 ' ) + arc tan ( S X − 1 S X − 1 ' ) }
S X + 1 = I X + 1 ( 0 0 ) − I X + 1 ( 180 0 ) I X + 1 ( 0 0 ) + I X + 1 ( 180 0 )
S X + 1 ' = I X + 1 ( 90 0 ) − I X + 1 ( 270 0 ) I X + 1 ( 90 0 ) + I X + 1 ( 270 0 )
S X − 1 = I X − 1 ( 0 0 ) − I X − 1 ( 180 0 ) I X − 1 ( 0 0 ) + I X − 1 ( 180 0 )
S X − 1 ' = I X − 1 ( 90 0 ) − I X − 1 ( 270 0 ) I X − 1 ( 90 0 ) + I X − 1 ( 270 0 )
r = a 1 + a 2 2

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