Abstract

An ultra-sensitive angle sensor employing single-cell photodiodes, which allows tighter focusing leading to a higher angular resolution better than 0.001 arc-second, has been designed based on laser autocollimation. Aiming to investigate the influences of spherical aberrations in the optical system on the sensor sensitivity, an optical model has been established based on wave optics. Computer simulation has been carried out by using the model, and its feasibility has been verified in experiments. In addition, a prototype optical angle sensor has been designed in a compact size of 100 mm × 150 mm, and its measurement resolution has been verified in experiments.

© 2016 Optical Society of America

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References

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  1. E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
    [Crossref]
  2. W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
    [Crossref]
  3. M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
    [Crossref]
  4. Keysight data sheet, “55280B linear Measurement Kit,” http://literature.cdn.keysight.com (accessed 15 December 2015).
  5. Renishaw plc. data sheet, “RLE System Performance: L-9904-2391-03-B,” http://resources.renishaw.com (accessed 15th December 2015).
  6. J. B. Bryan, “The Abbe principle revisited: an updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
    [Crossref]
  7. H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
    [Crossref]
  8. R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
    [Crossref]
  9. E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
    [Crossref]
  10. T. Ruijl, “Ultra precision coordinate measuring machine—design, calibration and error compensation,” Technische Universiteit Delft, Eindhoven. (ISBN 90–6464–287–7, Ph.D. thesis). (2001)
  11. 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,” Ann. CIRP 60(1), 515–518 (2011).
    [Crossref]
  12. A. H. Slocum, Precision Machine Design (Society of Manufacturing Engineers, 1992).
  13. A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4(1), 5–8 (1982).
    [Crossref]
  14. Möller-Werdel optical GmbH data sheet, “Electric Autocollimators,” http://www.moeller-wedel-optical.com (accessed 15 December 2015).
  15. Newport data sheet, “CONEX-LDS Electronic Autocollimator for Metrology and Industrial Applications,” http://assets.newport.com/webDocuments-EN/images/CONEX-LDS_Data_Sheet.pdf (accessed 22nd January 2016).
  16. Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
    [Crossref]
  17. W. Gao, Precision Nanometrology - Sensors and Measuring Systems for Nanomanufacturing (Springer, 2010).
  18. S. J. Bennett and J. W. C. Gates, “The design of detector arrays for laser alignment systems,” J. Phys. E Sci. Instrum. 3(1), 65–68 (1970).
    [Crossref]
  19. E. Hecht, Optics (Addison-Wesley, 2002).
  20. W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw Hill, 2007).
  21. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).
  22. V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (Society of Photo Optical, 1998).
  23. Hamamatsu Photonics technical paper, “Si photodiodes: Chapter 2,” http://www.hamamatsu.com/jp/en/index.html (accessed 11th December 2015).

2012 (2)

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

2010 (1)

Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
[Crossref]

2009 (1)

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

2007 (1)

W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
[Crossref]

2000 (1)

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
[Crossref]

1993 (1)

H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
[Crossref]

1982 (1)

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4(1), 5–8 (1982).
[Crossref]

1979 (1)

J. B. Bryan, “The Abbe principle revisited: an updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

1970 (1)

S. J. Bennett and J. W. C. Gates, “The design of detector arrays for laser alignment systems,” J. Phys. E Sci. Instrum. 3(1), 65–68 (1970).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
[Crossref]

Bennett, S. J.

S. J. Bennett and J. W. C. Gates, “The design of detector arrays for laser alignment systems,” J. Phys. E Sci. Instrum. 3(1), 65–68 (1970).
[Crossref]

Brien, J.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Bryan, J. B.

J. B. Bryan, “The Abbe principle revisited: an updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

Buice, E.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Caskey, G.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Ennos, A. E.

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4(1), 5–8 (1982).
[Crossref]

Fesperman, R.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Flugge, J.

H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
[Crossref]

Füßl, R.

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
[Crossref]

Gates, J. W. C.

S. J. Bennett and J. W. C. Gates, “The design of detector arrays for laser alignment systems,” J. Phys. E Sci. Instrum. 3(1), 65–68 (1970).
[Crossref]

Hausotte, T.

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Hocken, R.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
[Crossref]

Hocken, R. J.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Holmes, M.

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
[Crossref]

Jäger, G.

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Kim, W. J.

W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
[Crossref]

Kunzmann, H.

H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
[Crossref]

Lemmons, T.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Manske, E.

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

Otten, D.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Ozturk, O.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Pfeifer, T.

H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
[Crossref]

Phipps, J.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Ruben, S.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
[Crossref]

Shakir, H.

W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

Smith, S. T.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Trumper, D.

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
[Crossref]

Trumper, D. L.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Tsao, T. C.

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

Verma, S.

W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
[Crossref]

Virdee, M. S.

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4(1), 5–8 (1982).
[Crossref]

Yang, R. H.

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

Ann. CIRP (2)

H. Kunzmann, T. Pfeifer, and J. Flugge, “Scales vs laser interferometers, performance and comparison of two measuring systems,” Ann. CIRP 42(2), 753–767 (1993).
[Crossref]

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,” Ann. CIRP 60(1), 515–518 (2011).
[Crossref]

J. Phys. E Sci. Instrum. (1)

S. J. Bennett and J. W. C. Gates, “The design of detector arrays for laser alignment systems,” J. Phys. E Sci. Instrum. 3(1), 65–68 (1970).
[Crossref]

Meas. Sci. Technol. (2)

E. Manske, G. Jäger, T. Hausotte, and R. Füßl, “Recent developments and challenges of nanopositioning and nanomeasuring technology,” Meas. Sci. Technol. 23(7), 074001 (2012).
[Crossref]

Y. Saito, Y. Arai, and W. Gao, “Investigation of an optical sensor for small tilt angle detection of a precision linear stage,” Meas. Sci. Technol. 21(5), 054006 (2010).
[Crossref]

Precis. Eng. (6)

J. B. Bryan, “The Abbe principle revisited: an updated interpretation,” Precis. Eng. 1(3), 129–132 (1979).
[Crossref]

R. Fesperman, O. Ozturk, R. Hocken, S. Ruben, T. C. Tsao, J. Phipps, T. Lemmons, J. Brien, and G. Caskey, “Multi-scale alignment and positioning system – MAPS,” Precis. Eng. 36(4), 517–537 (2012).
[Crossref]

E. Buice, D. Otten, R. H. Yang, S. T. Smith, R. J. Hocken, and D. L. Trumper, “Design evaluation of a single-axis precision controlled positioning stage,” Precis. Eng. 33(4), 418–424 (2009).
[Crossref]

W. J. Kim, S. Verma, and H. Shakir, “Design and precision construction of novel magnetic-levitation-based multi-axis nanoscale positioning systems,” Precis. Eng. 31(4), 337–350 (2007).
[Crossref]

M. Holmes, R. Hocken, and D. Trumper, “The long-range scanning stage: a novel platform for scanned-probe microscopy,” Precis. Eng. 24(3), 191–209 (2000).
[Crossref]

A. E. Ennos and M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4(1), 5–8 (1982).
[Crossref]

Other (12)

Möller-Werdel optical GmbH data sheet, “Electric Autocollimators,” http://www.moeller-wedel-optical.com (accessed 15 December 2015).

Newport data sheet, “CONEX-LDS Electronic Autocollimator for Metrology and Industrial Applications,” http://assets.newport.com/webDocuments-EN/images/CONEX-LDS_Data_Sheet.pdf (accessed 22nd January 2016).

A. H. Slocum, Precision Machine Design (Society of Manufacturing Engineers, 1992).

E. Hecht, Optics (Addison-Wesley, 2002).

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw Hill, 2007).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).

V. N. Mahajan, Optical Imaging and Aberrations: Ray Geometrical Optics (Society of Photo Optical, 1998).

Hamamatsu Photonics technical paper, “Si photodiodes: Chapter 2,” http://www.hamamatsu.com/jp/en/index.html (accessed 11th December 2015).

Keysight data sheet, “55280B linear Measurement Kit,” http://literature.cdn.keysight.com (accessed 15 December 2015).

Renishaw plc. data sheet, “RLE System Performance: L-9904-2391-03-B,” http://resources.renishaw.com (accessed 15th December 2015).

T. Ruijl, “Ultra precision coordinate measuring machine—design, calibration and error compensation,” Technische Universiteit Delft, Eindhoven. (ISBN 90–6464–287–7, Ph.D. thesis). (2001)

W. Gao, Precision Nanometrology - Sensors and Measuring Systems for Nanomanufacturing (Springer, 2010).

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Figures (17)

Fig. 1
Fig. 1 A schematic of motion error components of a precision linear stage.
Fig. 2
Fig. 2 A schematic of the optical configuration for the ultra-sensitive optical angle sensor employing single-cell photodiodes (SPDs) (A laser beam with a high truncation ratio is used as the measurement laser beam).
Fig. 3
Fig. 3 An influence of the spherical aberration of the collimator objective on the focused light spot. (a) without the influence of the spherical aberration; (b) with the influence of the spherical aberration.
Fig. 4
Fig. 4 An optical configuration for computer simulation based on wave optics.
Fig. 5
Fig. 5 Wavefront errors due to the spherical aberration of the collimator objective applied to the simulation model (results in the case of D = 14 mm). (a) without the spherical aberration; (b) with the spherical aberration, at paraxial focal plane; (c) with the spherical aberration, at best focal plane.
Fig. 6
Fig. 6 Calculated intensity distribution of the focused light spot on the SPD plane. (a) intensity distributions in 3-D plots (D = 2 mm); (b) intensity distributions in 3-D plots (D = 14 mm); (c) cross sections of the intensity distributions.
Fig. 7
Fig. 7 Variation of the focused light spot diameter on the SPD plane.
Fig. 8
Fig. 8 Calculation procedure for the simulation of the optical angle sensor output.
Fig. 9
Fig. 9 Simulated sensor outputs. (a) uSPD_out-Δθ curves (D = 2 mm); (b) uSPD_out-Δθ curves (D = 14 mm); (c) variation of the sensitivity as a function of D.
Fig. 10
Fig. 10 Optical setup for measurement of the focused light spot.
Fig. 11
Fig. 11 Measured focused light spot diameter.
Fig. 12
Fig. 12 Setup for sensitivity evaluation. (a) a schematic of the optical setup; (b) a photograph of the photosensitive plane of the photodiode.
Fig. 13
Fig. 13 Variation of the sensor output with the tilt angle applied to the reflective mirror. (a) uSPD_out-Δθ curves acquired in the experiments; (b) sensor sensitivity.
Fig. 14
Fig. 14 The optical sensor head designed in a compact size of 100 mm × 150 mm for the optical angle sensor. (a) a schematic of the optical configuration; (b) a photograph of the optical setup.
Fig. 15
Fig. 15 A typical waveform of the SPD output through the trans-impedance amplifier.
Fig. 16
Fig. 16 Variation of the sensor sensitivity with respect to the measurement laser beam diameter D. (a) uSPD_out-Δθ curves acquired by the computer simulation; (b) uSPD_out-Δθ curves acquired by the experiments; (c) sensor sensitivity.
Fig. 17
Fig. 17 Measurement of an angular motion with the amplitude of 0.001 arc-second (measurement laser beam diameter D was set to be 5 mm).

Tables (1)

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Table 1 Parameters used in the computer simulation.

Equations (12)

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Δu(v)=ftan2 θ Y(Z) 2fΔ θ Y(Z)
u SPD_out = S1 I 1 (u,v)dudv S1_total I 1 (u,v)dudv ×100
v SPD_out = S2 I 2 (u,v)dudv S2_total I 2 (u,v)dudv ×100
u SPD_out = 1 2 π ( d 1 /2 ) 2 + d 1 Δu π ( d 1 /2 ) 2 ×100=50+ 8fΔ θ Y π d 1 ×100
v SPD_out = 1 2 π ( d 2 /2 ) 2 + d 2 Δv π ( d 2 /2 ) 2 ×100=50+ 8fΔ θ Z π d 2 ×100
d diff = 2.44fλ D
d TA = D 3 32 f 2 ( n1 ) 2 [ n 2 ( 2n+1 ) R 2 R 2 R 1 + n+2 n ( R 2 R 2 R 1 ) 2 ]
I 1 ( u,v )= 1 λ 2 f 2 | U l ( x,y ) P(x+u,y+v)exp[ j 2π λf ( xu+yv ) ]dxdy | 2
P( x,y )={ 1(whenρ(x,y)<1) 0(when1<ρ(x,y))
ρ( x,y )= x 2 + y 2 D/2
U l ( x,y )={ A(withoutaberration) Aexp[ j 2π λ S ρ 4 ( x,y ) ](withaberration,atparaxialfocus) Aexp[ j 2π λ S( ρ 4 ( x,y ) ρ 2 ( x,y ) ) ](withaberration,atbestfocus)
S= [ n 3 +( n+2 )+( 3n+2 ) ( n1 ) 2 4( n 2 1 ) ] D 4 32n ( n1 ) 2 f 3

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