Abstract

A measurement system to simultaneously measure six degree-of-freedom (6DOF) geometric errors is proposed. The measurement method is based on a combination of mono-frequency laser interferometry and laser fiber collimation. A simpler and more integrated optical configuration is designed. To compensate for the measurement errors introduced by error crosstalk, element fabrication error, laser beam drift, and nonparallelism of two measurement beam, a unified measurement model, which can improve the measurement accuracy, is deduced and established using the ray-tracing method. A numerical simulation using the optical design software Zemax is conducted, and the results verify the correctness of the model. Several experiments are performed to demonstrate the feasibility and effectiveness of the proposed system and measurement model.

© 2017 Optical Society of America

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References

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  1. M. A. V. Chapman, R. Fergusson-Kelly, and W. Lee, “Interferometric straightness measurement and application to moving table machines,” (Renishaw technical white paper), http://www.renishaw.com/media/pdf/en/c5d16ced95164dd7936f75b0f664f8ba.pdf
  2. M. A. V. Chapman, R. Fergusson-Kelly, A. Holloway, D. Lock, and W. Lee, “Interferometric angle measurement and the hardware options available from Renishaw,” (Renishaw technical white paper), http://www.renishaw.com/media/pdf/en/f6fc86acb84b42d28fdef091c69884ab.pdf
  3. A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part one: linear positional errors,” J. Mater. Process. Technol. 105(3), 394–406 (2000).
    [Crossref]
  4. A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part two: angular errors,” J. Mater. Process. Technol. 105(3), 407–420 (2000).
    [Crossref]
  5. S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).
  6. J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).
  7. P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
    [Crossref]
  8. C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
    [Crossref]
  9. K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
    [Crossref]
  10. K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
    [Crossref]
  11. Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Precis. Eng. Manuf. 9(4), 26–31 (2008).
  12. F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013).
    [Crossref] [PubMed]
  13. C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
    [Crossref] [PubMed]
  14. C. Cui, Q. Feng, and B. Zhang, “Compensation for straightness measurement systematic errors in six degree-of-freedom motion error simultaneous measurement system,” Appl. Opt. 54(11), 3122–3131 (2015).
    [Crossref] [PubMed]
  15. X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
    [Crossref] [PubMed]
  16. B. Chen, B. Xu, L. Yan, E. Zhang, and Y. Liu, “Laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters,” Opt. Express 23(7), 9052–9073 (2015).
    [Crossref] [PubMed]
  17. E. Zhang, Q. Hao, B. Chen, L. Yan, and Y. Liu, “Laser heterodyne interferometer for simultaneous measuring displacement and angle based on the Faraday effect,” Opt. Express 22(21), 25587–25598 (2014).
    [Crossref] [PubMed]
  18. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
    [Crossref]
  19. W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
    [Crossref]
  20. H. L. Hsieh and S. W. Pan, “Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements,” Opt. Express 23(3), 2451–2465 (2015).
    [Crossref] [PubMed]
  21. SIOS technical white paper http://www.sios-de.com/wp-content/uploads/2016/02/MI_engl_2015.pdf

2016 (2)

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

C. Cui, Q. Feng, B. Zhang, and Y. Zhao, “System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser,” Opt. Express 24(6), 6735–6748 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (1)

2013 (1)

2011 (1)

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

2008 (1)

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Precis. Eng. Manuf. 9(4), 26–31 (2008).

2006 (1)

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

2005 (1)

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

2000 (3)

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part one: linear positional errors,” J. Mater. Process. Technol. 105(3), 394–406 (2000).
[Crossref]

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part two: angular errors,” J. Mater. Process. Technol. 105(3), 407–420 (2000).
[Crossref]

1998 (1)

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

1995 (1)

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

1994 (1)

S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).

1992 (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Arai, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

Bin, Z.

Chen, B.

Chen, M. J.

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Cui, C.

Cuifang, K.

Cunxing, C.

Ellis, J. D.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Ertekin, Y. M.

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part one: linear positional errors,” J. Mater. Process. Technol. 105(3), 394–406 (2000).
[Crossref]

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part two: angular errors,” J. Mater. Process. Technol. 105(3), 407–420 (2000).
[Crossref]

Fan, K. C.

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Feng, Q.

Fenglin, Y.

Gao, W.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

Gillmer, S. R.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Hao, Q.

Hsieh, H. L.

Hsu, C. C.

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

Hsu, T. H.

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

Huang, P. S.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Huang, W. M.

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Imai, N.

S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).

Jywe, W. Y.

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

Kiyono, S.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

Kuang, C.

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Precis. Eng. Manuf. 9(4), 26–31 (2008).

Lee, H. S.

S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).

Liu, C. H.

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

Liu, Y.

Muto, H.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

Ni, J.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Okafor, A. C.

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part one: linear positional errors,” J. Mater. Process. Technol. 105(3), 394–406 (2000).
[Crossref]

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part two: angular errors,” J. Mater. Process. Technol. 105(3), 407–420 (2000).
[Crossref]

Pan, S. W.

Park, C. H.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

Qibo, F.

Saito, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

Shibuya, A.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

Shimizu, S.

S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).

Shimizu, Y.

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

Woody, S. C.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Wu, S. M.

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Xu, B.

Yan, L.

Yu, X.

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

Yusheng, Z.

Zhang, B.

Zhang, E.

Zhao, Y.

Appl. Opt. (1)

CIRP Ann-Manuf, Techn. (1)

W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Ann-Manuf, Techn. 60(1), 515–518 (2011).
[Crossref]

Int. J. Jpn. Soc. Precis. Eng. (1)

S. Shimizu, H. S. Lee, and N. Imai, “Simultaneous measuring method of table motion errors in 6 degrees of freedom,” Int. J. Jpn. Soc. Precis. Eng. 28(3), 273–274 (1994).

Int. J. Mach. Tools Manuf. (2)

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

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

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Precis. Eng. Manuf. 9(4), 26–31 (2008).

J. Eng. Ind. (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

J. Mater. Process. Technol. (2)

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part one: linear positional errors,” J. Mater. Process. Technol. 105(3), 394–406 (2000).
[Crossref]

A. C. Okafor and Y. M. Ertekin, “Vertical machining center accuracy characterization using laser interferometer, part two: angular errors,” J. Mater. Process. Technol. 105(3), 407–420 (2000).
[Crossref]

Opt. Express (5)

Precis. Eng. (2)

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 97–103 (2006).
[Crossref]

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

Rev. Sci. Instrum. (2)

X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, “Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology,” Rev. Sci. Instrum. 87(6), 065109 (2016).
[Crossref] [PubMed]

C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, “Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage,” Rev. Sci. Instrum. 76(5), 055110 (2005).
[Crossref]

Other (3)

M. A. V. Chapman, R. Fergusson-Kelly, and W. Lee, “Interferometric straightness measurement and application to moving table machines,” (Renishaw technical white paper), http://www.renishaw.com/media/pdf/en/c5d16ced95164dd7936f75b0f664f8ba.pdf

M. A. V. Chapman, R. Fergusson-Kelly, A. Holloway, D. Lock, and W. Lee, “Interferometric angle measurement and the hardware options available from Renishaw,” (Renishaw technical white paper), http://www.renishaw.com/media/pdf/en/f6fc86acb84b42d28fdef091c69884ab.pdf

SIOS technical white paper http://www.sios-de.com/wp-content/uploads/2016/02/MI_engl_2015.pdf

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Figures (10)

Fig. 1
Fig. 1 Schematic of the measurement system.
Fig. 2
Fig. 2 Basic idea of the measurement model establishment.
Fig. 3
Fig. 3 Coordinate establishment. (a) Ideal corner-cube retroreflector. (b) Retroreflector with fabrication errors. (c) Actual measurement coordinate system
Fig. 4
Fig. 4 The model and simulation of corner-cube retroreflector in Zemax. (a) An ideal corner-cube retroreflector. (b) The spot position obtained using the footprint analysis.
Fig. 5
Fig. 5 Simulation result in the presence of 20 arcsec in pitch.
Fig. 6
Fig. 6 Comparison between the ZEMAX simulation results and the model calculations results.
Fig. 7
Fig. 7 Simulation result.
Fig. 8
Fig. 8 Comparison between model calculation and Zemax simulation. (a) Influence on the horizontal straightness. (b) Influence on the vertical straightness.
Fig. 9
Fig. 9 Stability test results.
Fig. 10
Fig. 10 Repeatability and comparison results: Repeatability results of position error using (a) the proposed system and (b) the interferometer, and (c) comparison result of position error. Repeatability results of horizontal straightness using (d) the proposed system and (e) the interferometer, and (f) comparison result of horizontal straightness. Repeatability results of vertical straightness using (g) the proposed system and (h) the interferometer, and (i) comparison result of vertical straightness. Repeatability results of yaw using (j) the proposed system and (k) the photoelectric autocollimator, and (l) comparison result of yaw. Repeatability results of pitch using (m) the proposed system and (n) the photoelectric autocollimator, and (o) comparison result of pitch. Repeatability results of roll using (p) the proposed system and (q) the electronic level, and (r) comparison result of roll.

Tables (1)

Tables Icon

Table 1 The result of simulation and calculation in different measured distance

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

incidentplane: n 0i i = [ 3 3 3 3 3 3 0 ] T reflectingsurfaceAOC: n 1 i = [ 0100 ] T reflectingsurfaceAOB: n 2 i = [ 0010 ] T . reflectingsurfaceBOC: n 3 i = [ 1000 ] T emergentplane: n 0e i = [ 3 3 3 3 3 3 0 ] T
surfaceAOC: n 1 e = [ β 13 1 0 0 ] T surfaceAOB: n 2 e = [ β 23 β 21 1 0 ] T .
T 1 =[ 3 3 3 3 3 3 0 2 2 2 2 0 D 6 6 6 6 6 3 H 0 0 0 1 ].
T=[ 1 ε zx ε yx δ xx ε zx 1 ε xx δ yx ε yx ε xx 1 δ zx 0 0 0 1 ].
incidentplane: n 0i =T T 1 n 0i i = [ 1 ε zx ε yx 0 ] T surfaceAOC: n 1 =T T 1 n 1 e =[ 6 6 ε yx 2 2 ε zx + 3 3 3 3 β 13 6 6 ε xx + 3 3 ε zx + 2 2 + 2 2 β 13 2 2 ε xx 3 3 ε yx 6 6 + 6 6 β 13 0 ]. surfaceAOB: n 2 =T T 1 n 2 e =[ 6 3 ε yx + 3 3 + 3 3 β 23 3 3 β 21 6 3 ε xx + 3 3 ε zx 2 2 β 23 2 2 β 21 3 3 ε yx + 6 3 6 6 β 23 + 6 6 β 21 0 ] surfaceBOC: n 3 =T T 1 n 3 i =[ 6 6 ε yx + 2 2 ε zx + 3 3 6 6 ε xx + 3 3 ε zx 2 2 2 2 ε xx 3 3 ε yx 6 6 0 ] emergentplane: n 0e =T T 1 n 0e i = [ 1 ε zx ε yx 0 ] T
A' = n n' [ A N ( A N ) ] N 1 ( n n' ) 2 + ( n n' ) 2 ( A N ) 2 ,
R=[ 12 N x N x 2 N x N y 2 N x N z 2 N x N y 12 N y N y 2 N y N z 2 N x N z 2 N y N z 12 N z N z ],
{ x QD1 =A y QD1 =( 2DB )+2 δ yx 2H ε xx +( ε zx + σ H )2h( 1 1 n ) +( 2 6 3 ( β 13 ) RR1 + 6 3 ( β 23 ) RR1 6 3 ( β 21 ) RR1 )h +( 2 3 3 ( β 13 ) RR1 + 2 3 3 ( β 23 ) RR1 2 3 3 ( β 21 ) RR1 )( HC ) σ H 2An( 2 6 3 ( β 13 ) RR1 6 3 ( β 23 ) RR1 + 6 3 ( β 21 ) RR1 )( Ah ) z QD1 =( 2HC )+2 δ zx +2D ε xx ( ε yx σ V )2h( 1 1 n ) +( 2 ( β 23 ) RR1 + 2 ( β 21 ) RR1 )h +( 2 3 3 ( β 13 ) RR1 2 3 3 ( β 23 ) RR1 + 2 3 3 ( β 21 ) RR1 )( DB ) σ V 2A+n( 2 ( β 23 ) RR1 + 2 ( β 21 ) RR1 )( Ah ) ,
{ δ yx-1 = y QD1 2 +H ε xx ε zx h( 1 1 n ) ( 2 6 3 ( β 13 ) RR1 + 6 3 ( β 23 ) RR1 6 3 ( β 21 ) RR1 ) h 2 ( 2 3 3 ( β 13 ) RR1 + 2 3 3 ( β 23 ) RR1 2 3 3 ( β 21 ) RR1 ) ( HC ) 2 +n( 2 6 3 ( β 13 ) RR1 6 3 ( β 23 ) RR1 + 6 3 ( β 21 ) RR1 ) ( Ah ) 2 + σ H A σ H h( 1 1 n ) δ zx-1 = z QD1 2 D ε xx + ε yx h( 1 1 n ) ( 2 ( β 23 ) RR1 + 2 ( β 21 ) RR1 ) h 2 ( 2 3 3 ( β 13 ) RR1 2 3 3 ( β 23 ) RR1 + 2 3 3 ( β 21 ) RR1 ) ( DB ) 2 n( 2 ( β 23 ) RR1 + 2 ( β 21 ) RR1 ) ( Ah ) 2 + σ V A σ V h( 1 1 n ) .
{ δ yx part1 =H ε xx ε zx h( 1 1 n ) δ zx part1 =D ε xx + ε yx h( 1 1 n ) .
{ δ yx part2 =( 2 6 3 β 13 + 6 3 β 23 6 3 β 21 ) h 2 ( 2 3 3 β 13 + 2 3 3 β 23 2 3 3 β 21 ) ( HC ) 2 +n( 2 6 3 β 13 6 3 β 23 + 6 3 β 21 ) ( Ah ) 2 , δ zx part2 =( 2 β 23 + 2 β 21 ) h 2 ( 2 3 3 β 13 2 3 3 β 23 + 2 3 3 β 21 ) ( DB ) 2 n( 2 β 23 + 2 β 21 ) ( Ah ) 2
{ δ yx part3 = σ H A σ H h( 1 1 n ) δ zx part3 = σ V A σ V h( 1 1 n ) ,
{ δ yx2 = y QD2 2 +H ε xx ε zx h( 1 1 n ) ( 2 6 3 ( β 13 ) RR2 + 6 3 ( β 23 ) RR2 6 3 ( β 21 ) RR2 ) h 2 ( 2 3 3 ( β 13 ) RR2 + 2 3 3 ( β 23 ) RR2 2 3 3 ( β 21 ) RR2 ) ( HC ) 2 +n( 2 6 3 ( β 13 ) RR2 6 3 ( β 23 ) RR2 + 6 3 ( β 21 ) RR2 ) ( Ah ) 2 +( σ H + φ H )A( σ H + φ H )h( 1 1 n ) δ zx2 = z QD2 2 +D ε xx + ε yx h( 1 1 n ) ( 2 ( β 23 ) RR2 + 2 ( β 21 ) RR2 ) h 2 ( 2 3 3 ( β 13 ) RR2 2 3 3 ( β 23 ) RR2 + 2 3 3 ( β 21 ) RR2 ) ( DB ) 2 n( 2 ( β 23 ) RR2 + 2 ( β 21 ) RR2 ) ( Ah ) 2 +( σ V + φ V )A( σ V + φ V )h( 1 1 n ) .
ε xx = ( z QD1 z QD2 ) φ V 2A+ φ V 2h( 1 1 n ) +( ( ϕ V ) RR1 ( ϕ V ) RR2 )( Ah )+( ( ϕ V ) RR1 n ( ϕ V ) RR2 n )h ( 2 3 3 ( β 13 ) RR1 2 3 3 ( β 23 ) RR1 + 2 3 3 ( β 21 ) RR1 )( DB ) +( 2 3 3 ( β 13 ) RR2 2 3 3 ( β 23 ) RR2 + 2 3 3 ( β 21 ) RR2 )( DB ) 4D ,
n r = R ABC n beamb = [ 1 2 ε zx + σ H + φ H 2 ε yx + σ V + φ V 0 ] T ,
{ ε yx = z PSD1 2 f 1 + σ V + φ V 2 ε zx = y PSD1 2 f 1 σ H + φ H 2 .
{ σ H = y PSD2 f 2 ( ϕ H ) RR1 σ V = z PSD2 f 2 ( ϕ V ) RR1 .
δ zx = z QD1 2 + ε yx h(1 1 n ).

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