Abstract

We present the design, bench-top setup, and experimental results of a compact heterodyne interferometer that achieves picometer-level displacement sensitivities in air over frequencies above 100 MHz. The optical configuration with spatially separated beams prevents frequency and polarization mixing, and therefore eliminates periodic errors. The interferometer is designed to maximize common-mode optical laser beam paths to obtain high rejection of environmental disturbances, such as temperature fluctuations and acoustics. The results of our experiments demonstrate the short- and long-term stabilities of the system during stationary and dynamic measurements. In addition, we provide measurements that compare our interferometer prototype with a commercial system, verifying our higher sensitivity of 3 pm, higher thermal stability by a factor of two, and periodic-error-free performance.

© 2020 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Harding, Handbook of Optical Dimensional Metrology (CRC Press, 2013), chap. 4.
  2. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [Crossref]
  3. R. Pierce, J. Leitch, M. Stephens, P. Bender, and R. Nerem, “Intersatellite range monitoring using optical interferometry,” Appl. Opt. 47, 5007–5019 (2008).
    [Crossref]
  4. F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
    [Crossref]
  5. S. Lu, C. Chiueh, and C. Lee, “Differential wavelength-scanning heterodyne interferometer for measuring large step height,” Appl. Opt. 41, 5866–5871 (2002).
    [Crossref]
  6. G. E. Sommargren, “Differential plane mirror interferometer,” Patent4,693,605 (September15, 1987).
  7. V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
    [Crossref]
  8. G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
    [Crossref]
  9. N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26, 2676–2682 (1987).
    [Crossref]
  10. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [Crossref]
  11. Y. Xie and Y. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
    [Crossref]
  12. C. Wu and R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37, 6696–6700 (1998).
    [Crossref]
  13. T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
    [Crossref]
  14. Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
    [Crossref]
  15. S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
    [Crossref]
  16. D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
    [Crossref]
  17. H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
    [Crossref]
  18. J. D. Ellis, M. Baas, K. N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36, 261–269 (2012).
    [Crossref]
  19. H. Fu, P. Hu, J. Tan, and Z. Fan, “Simple method for reducing the first-order optical nonlinearity in a heterodyne laser interferometer,” Appl. Opt. 54, 6321–6326 (2015).
    [Crossref]
  20. C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
    [Crossref]
  21. T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
    [Crossref]
  22. M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
    [Crossref]
  23. C. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. 38, 4089–4094 (1999).
    [Crossref]
  24. K. N. Joo, J. D. Ellis, J. W. Spronck, P. J. V. Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. 34, 386–388 (2009).
    [Crossref]
  25. K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express 18, 1159–1165 (2010).
    [Crossref]
  26. P. Hu, P. Chen, X. Ding, and J. Tan, “Balanced plane-mirror heterodyne interferometer with subnanometer periodic nonlinearity,” Appl. Opt. 53, 5448–5452 (2014).
    [Crossref]
  27. M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement 39, 120–129 (2006).
    [Crossref]
  28. M. Tröbs and G. Heinzel, “Corrigendum to ‘Improved spectrum estimation from digitized time series on a logarithmic frequency axis’ (Measurement 39 (2006) 120–129),” Measurement 42, 170 (2009).
    [Crossref]
  29. Zygo, “Differential plane mirror interferometer,” https://www.zygo.com/met/markets/stageposition/zmi/interferometers/differential/Differential_Plane_Mirror_Interferometer.pdf .
  30. J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
    [Crossref]
  31. T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
    [Crossref]

2017 (1)

C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
[Crossref]

2015 (1)

2014 (1)

2012 (1)

J. D. Ellis, M. Baas, K. N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36, 261–269 (2012).
[Crossref]

2010 (1)

2009 (3)

K. N. Joo, J. D. Ellis, J. W. Spronck, P. J. V. Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. 34, 386–388 (2009).
[Crossref]

M. Tröbs and G. Heinzel, “Corrigendum to ‘Improved spectrum estimation from digitized time series on a logarithmic frequency axis’ (Measurement 39 (2006) 120–129),” Measurement 42, 170 (2009).
[Crossref]

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

2008 (2)

R. Pierce, J. Leitch, M. Stephens, P. Bender, and R. Nerem, “Intersatellite range monitoring using optical interferometry,” Appl. Opt. 47, 5007–5019 (2008).
[Crossref]

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

2007 (1)

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

2006 (2)

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement 39, 120–129 (2006).
[Crossref]

2004 (1)

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

2003 (2)

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

2002 (3)

S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
[Crossref]

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

S. Lu, C. Chiueh, and C. Lee, “Differential wavelength-scanning heterodyne interferometer for measuring large step height,” Appl. Opt. 41, 5866–5871 (2002).
[Crossref]

2001 (1)

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[Crossref]

2000 (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

1999 (1)

1998 (1)

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[Crossref]

1992 (2)

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

Y. Xie and Y. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
[Crossref]

1989 (1)

M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[Crossref]

1987 (1)

Baas, M.

J. D. Ellis, M. Baas, K. N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36, 261–269 (2012).
[Crossref]

Bender, P.

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[Crossref]

N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26, 2676–2682 (1987).
[Crossref]

Braxmaier, C.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Buice, E. S.

Burnham-Fay, E. D.

C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
[Crossref]

Cervantes, F. G.

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

Chen, P.

Chiueh, C.

Choi, H.

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

Choi, T.

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

Cosijns, S.

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
[Crossref]

Cosijns, S. J.

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

Danzmann, K.

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

Deng, Y.

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

Deslattes, R. D.

Ding, X.

Ellis, J. D.

C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
[Crossref]

J. D. Ellis, M. Baas, K. N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36, 261–269 (2012).
[Crossref]

K. N. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express 18, 1159–1165 (2010).
[Crossref]

K. N. Joo, J. D. Ellis, J. W. Spronck, P. J. V. Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. 34, 386–388 (2009).
[Crossref]

Eom, T.

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[Crossref]

Fan, Z.

Fu, H.

García, A.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Gohlke, M.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Guzmán, F.

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

Haitjema, H.

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
[Crossref]

Harding, K.

K. Harding, Handbook of Optical Dimensional Metrology (CRC Press, 2013), chap. 4.

Heinzel, G.

M. Tröbs and G. Heinzel, “Corrigendum to ‘Improved spectrum estimation from digitized time series on a logarithmic frequency axis’ (Measurement 39 (2006) 120–129),” Measurement 42, 170 (2009).
[Crossref]

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement 39, 120–129 (2006).
[Crossref]

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Hou, W.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

Hoyland, D.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Hu, J.

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

Hu, P.

Jansen, M. J.

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

Jennrich, O.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Jeong, K.

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[Crossref]

Johann, U.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Joo, K. N.

Kan, P. J. V.

Kessler, E.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

Kim, J.

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[Crossref]

Knarren, B.

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

Lawall, J.

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

C. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. 38, 4089–4094 (1999).
[Crossref]

Lee, C.

Lee, K.

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

Lee, S.

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

Leitch, J.

Li, X.

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

Loner, D.

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

Lu, S.

Middleton, K.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Nakayama, K.

M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[Crossref]

Nerem, R.

Pierce, R.

Ressel, S.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Robertson, D.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Roset, N.

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

Rüdiger, A.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Schallakans, P.

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

Schellekens, P.

S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
[Crossref]

Schilling, R.

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Schmidt, R. H. M.

Schuldt, T.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Sommargren, G. E.

G. E. Sommargren, “Differential plane mirror interferometer,” Patent4,693,605 (September15, 1987).

Spannagel, R.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Spronck, J. W.

Steier, F.

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

Stephens, M.

Tan, J.

Tanaka, M.

M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[Crossref]

Tröbs, M.

M. Tröbs and G. Heinzel, “Corrigendum to ‘Improved spectrum estimation from digitized time series on a logarithmic frequency axis’ (Measurement 39 (2006) 120–129),” Measurement 42, 170 (2009).
[Crossref]

M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement 39, 120–129 (2006).
[Crossref]

Wand, V.

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

Wang, C.

C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
[Crossref]

Wanner, G.

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

Weise, D.

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Wilkening, G.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

Wu, C.

Wu, Y.

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

Y. Xie and Y. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
[Crossref]

Xie, Y.

Yamagami, T.

M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[Crossref]

Yao, J.

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

AIP Conf. Proc. (1)

V. Wand, F. Guzmán, G. Heinzel, and K. Danzmann, “LISA phasemeter development,” AIP Conf. Proc. 873, 689 (2006).
[Crossref]

Appl. Opt. (8)

Appl. Phys. B (1)

F. G. Cervantes, F. Steier, G. Wanner, G. Heinzel, and K. Danzmann, “Subtraction of test mass angular noise in the LISA technology package interferometer,” Appl. Phys. B 90, 395–400 (2008).
[Crossref]

CIRP Ann. (1)

D. Loner, B. Knarren, S. Cosijns, H. Haitjema, and P. Schallakans, “Laser polarization state measurement in heterodyne interferometry,” CIRP Ann. 52, 439–442 (2003).
[Crossref]

Class. Quantum Grav. (1)

G. Heinzel, V. Wand, A. García, O. Jennrich, C. Braxmaier, D. Robertson, K. Middleton, D. Hoyland, A. Rüdiger, and R. Schilling, “The LTP interferometer and phasemeter,” Class. Quantum Grav. 21, S581–S587 (2004).
[Crossref]

IEEE Trans. Instrum. Meas (1)

M. Tanaka, T. Yamagami, and K. Nakayama, “Linear interpolation of periodic error in a heterodyne laser interferometer at subnanometer levels (dimension measurement),” IEEE Trans. Instrum. Meas. 38, 552–554 (1989).
[Crossref]

Int. J. Optomechatron. (1)

T. Schuldt, M. Gohlke, R. Spannagel, S. Ressel, D. Weise, U. Johann, and C. Braxmaier, “Sub-nanometer heterodyne interferometry and its application in dilatometry and industrial metrology,” Int. J. Optomechatron. 3, 187–200 (2009).
[Crossref]

Meas. Sci. Technol. (4)

T. Eom, J. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
[Crossref]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[Crossref]

T. Eom, T. Choi, K. Lee, H. Choi, and S. Lee, “A simple method for the compensation of the nonlinearity in the heterodyne interferometer,” Meas. Sci. Technol. 13, 222–225 (2002).
[Crossref]

Y. Deng, X. Li, Y. Wu, J. Hu, and J. Yao, “Analysis of frequency mixing error on heterodyne interferometric ellipsometry,” Meas. Sci. Technol. 18, 3339–3343 (2007).
[Crossref]

Measurement (2)

M. Tröbs and G. Heinzel, “Improved spectrum estimation from digitized time series on a logarithmic frequency axis,” Measurement 39, 120–129 (2006).
[Crossref]

M. Tröbs and G. Heinzel, “Corrigendum to ‘Improved spectrum estimation from digitized time series on a logarithmic frequency axis’ (Measurement 39 (2006) 120–129),” Measurement 42, 170 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Precis. Eng. (4)

J. D. Ellis, M. Baas, K. N. Joo, and J. W. Spronck, “Theoretical analysis of errors in correction algorithms for periodic nonlinearity in displacement measuring interferometers,” Precis. Eng. 36, 261–269 (2012).
[Crossref]

C. Wang, E. D. Burnham-Fay, and J. D. Ellis, “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017).
[Crossref]

S. Cosijns, H. Haitjema, and P. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
[Crossref]

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
[Crossref]

Proc. SPIE (1)

H. Haitjema, S. J. Cosijns, N. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
[Crossref]

Rev. Sci. Instrum. (1)

J. Lawall and E. Kessler, “Michelson interferometry with 10 pm accuracy,” Rev. Sci. Instrum. 71, 2669–2676 (2000).
[Crossref]

Other (3)

Zygo, “Differential plane mirror interferometer,” https://www.zygo.com/met/markets/stageposition/zmi/interferometers/differential/Differential_Plane_Mirror_Interferometer.pdf .

K. Harding, Handbook of Optical Dimensional Metrology (CRC Press, 2013), chap. 4.

G. E. Sommargren, “Differential plane mirror interferometer,” Patent4,693,605 (September15, 1987).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. Optical configuration of the proposed differential heterodyne laser interferometer with spatial beam separation. ${{\rm BS}_1}$, 45° tilted non-polarizing beam splitter; PBS, polarizing beam splitter; ${{\rm BS}_2}$, 45° tilted and axially 90° rotated non-polarizing beam splitter; QWP, 45° rotated quarter-wave plate; ${{\rm M}_{\rm F}}$, fixed mirror; ${{\rm M}_{\rm R}}$, reference mirror; ${{\rm M}_{\rm M}}$, measurement mirror; ${{\rm PD}_{\rm R}}$, reference photodetector; ${{\rm PD}_{\rm M}}$, measurement photodetector.
Fig. 2.
Fig. 2. Top view of the proposed differential heterodyne laser interferometer with spatial beam separation shown in Fig. 1. The inset indicates the rear view of the interferometer with $x {-} y$ coordinates.
Fig. 3.
Fig. 3. (a) Displacement measurement results over 10 s for the short-term stability test and (b) linear spectral density of (a).
Fig. 4.
Fig. 4. (a) Displacement measurement results over 2 h and (b) linear spectral density (LSD) of (a).
Fig. 5.
Fig. 5. (a) Displacement measurement results and temperature variation over 7 h; (b) Allan deviations of original and compensated for displacements for long-term stability.
Fig. 6.
Fig. 6. Linear spectral density (LSD) of fixed individual mirror motions compared to that of the single mirror motion.
Fig. 7.
Fig. 7. (a) Linear spectral density (LSD) plots of 1 Hz and 5 Hz PZT motions with 10 mV and (b) LSD plots of 1 Hz PZT motions with 10 mV, 5 mV, and 2 mV.
Fig. 8.
Fig. 8. Displacement measurement results of the commercial DMI (${{\rm L}_1}$), the proposed interferometer (${{\rm L}_1}$), and (${{\rm L}_1} + {{\rm L}_2}$) for (a) a triangle and (b) a sinusoidal motion with 0.06 Hz.
Fig. 9.
Fig. 9. (${{\rm L}_1} + {{\rm L}_2}$) for 20 s time-averaged displacements.
Fig. 10.
Fig. 10. (a) Displacement measurement results and (b) periodicity of the residual errors for the commercial DMI and the proposed interferometer.
Fig. 11.
Fig. 11. Phase error calculated by the amplitude method.

Equations (2)

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

E 1 = E 0 exp [ i 2 π ( f 0 + δ f 2 ) t + ϕ 2 + ϕ t ( x 0 + δ x , y 0 + δ y ) + ϕ R , E 2 = E 0 exp [ i 2 π ( f 0 + δ f 1 ) t + ϕ 1 + ϕ t ( x 0 + δ x , y 0 ) , E 3 = E 0 exp [ i 2 π ( f 0 + δ f 2 ) t + ϕ 2 + ϕ t ( x 0 , y 0 + δ y ) + ϕ M ,   a n d E 4 = E 0 exp [ i 2 π ( f 0 + δ f 1 ) t + ϕ 1 + ϕ t ( x 0 , y 0 ) ,
I R = I 0 ( 1 + cos ( 2 π δ f t + ϕ R + Δ ϕ ty ) ) , a n d I M = I 0 ( 1 + cos ( 2 π δ f t + ϕ M + Δ ϕ ty ) ) ,

Metrics