Abstract

The cyclic error of a homodyne interferometer is caused mainly by phase mixing due to the imperfection of polarizing optical components such as polarizing beam splitters. In Appl. Opt. 43, 2443 ( 2004), we concentrated on the relationship between these imperfect optical characteristics and the cyclic error and found the preamplifier-gains condition for removing the cyclic error. Here we demonstrate the cyclic error correction method experimentally and show that the method can be applied in real time. We obtained 0.04-nm cyclic errors, with a standard deviation above 5 µm.

© 2005 Optical Society of America

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References

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  1. C. Wu, C. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
    [CrossRef]
  2. T. Keem, S. Gonda, I. Misumi, Q. Huang, T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004).
    [CrossRef] [PubMed]
  3. W. Augustyn, “An analysis polarization mixing errors in distance measuring interferometer,” J. Vac. Sci. Technol. B 8, 2032–2036 (1990).
    [CrossRef]
  4. S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
    [CrossRef]
  5. I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
    [CrossRef]
  6. V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
    [CrossRef]
  7. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981).
    [CrossRef] [PubMed]
  8. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
    [CrossRef]
  9. C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
    [CrossRef]
  10. T. B. Eom, J. Y. Kim, K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
    [CrossRef]
  11. P. L. Oliver, D. Serge, “Polarization compensation: a passive approach to reducing heterodyne interferometer nonlinearity,” Opt. Lett. 27, 797–799 (2002).
    [CrossRef]
  12. J. Stone, S. D. Phillips, “Corrections for wavelength variations in precision interferometric displacement measurements,” J. Res. Natl. Inst. Stand. Technol. 101, 671–674 (1996).
    [CrossRef]
  13. Spectra-Physics Inc., Spectra-Physics Model 117A Stabilized Helium-Neon Laser Instruction Manual (Spectra-Physics, 1997), http://www.newport.com/store/product.aspx?lone=&ltwo=&id=5220&lang=1&section=Specs .

2004 (1)

2003 (1)

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

2002 (1)

2001 (1)

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

1999 (1)

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

1996 (3)

C. Wu, C. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

J. Stone, S. D. Phillips, “Corrections for wavelength variations in precision interferometric displacement measurements,” J. Res. Natl. Inst. Stand. Technol. 101, 671–674 (1996).
[CrossRef]

C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

1995 (1)

V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
[CrossRef]

1990 (2)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

W. Augustyn, “An analysis polarization mixing errors in distance measuring interferometer,” J. Vac. Sci. Technol. B 8, 2032–2036 (1990).
[CrossRef]

1981 (1)

Augustyn, W.

W. Augustyn, “An analysis polarization mixing errors in distance measuring interferometer,” J. Vac. Sci. Technol. B 8, 2032–2036 (1990).
[CrossRef]

Birch, K. P.

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

Doi, T.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Eom, T. B.

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

Fujimoto, H.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Gonda, S.

T. Keem, S. Gonda, I. Misumi, Q. Huang, T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Greco, V.

V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
[CrossRef]

Heydemann, P. L. M.

Hisata, N.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Huang, Q.

Jeong, K.

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

Keem, T.

Kim, J. Y.

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

Kurosawa, T.

T. Keem, S. Gonda, I. Misumi, Q. Huang, T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Misumi, I.

T. Keem, S. Gonda, I. Misumi, Q. Huang, T. Kurosawa, “Removing nonlinearity of a homodyne interferometer by adjusting the gains of its quadrature detector systems,” Appl. Opt. 43, 2443–2448 (2004).
[CrossRef] [PubMed]

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

Molesini, G.

V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
[CrossRef]

Oliver, P. L.

Peng, G. S.

C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Phillips, S. D.

J. Stone, S. D. Phillips, “Corrections for wavelength variations in precision interferometric displacement measurements,” J. Res. Natl. Inst. Stand. Technol. 101, 671–674 (1996).
[CrossRef]

Quercioli, F.

V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
[CrossRef]

Serge, D.

Stone, J.

J. Stone, S. D. Phillips, “Corrections for wavelength variations in precision interferometric displacement measurements,” J. Res. Natl. Inst. Stand. Technol. 101, 671–674 (1996).
[CrossRef]

Su, C.

C. Wu, C. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Takamasu, K.

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

Tanimura, Y.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Wu, C.

C. Wu, C. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

Yamagishi, T.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Yukawa, H.

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Appl. Opt. (2)

J. Res. Natl. Inst. Stand. Technol. (1)

J. Stone, S. D. Phillips, “Corrections for wavelength variations in precision interferometric displacement measurements,” J. Res. Natl. Inst. Stand. Technol. 101, 671–674 (1996).
[CrossRef]

J. Vac. Sci. Technol. B (1)

W. Augustyn, “An analysis polarization mixing errors in distance measuring interferometer,” J. Vac. Sci. Technol. B 8, 2032–2036 (1990).
[CrossRef]

Meas. Sci. Technol. (4)

I. Misumi, S. Gonda, T. Kurosawa, K. Takamasu, “Uncertainty in pitch measurement of one-dimensional grating standards using a nanometrological atomic force microscope,” Meas. Sci. Technol. 14, 463–471 (2003).
[CrossRef]

C. Wu, C. Su, G. S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
[CrossRef]

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

C. Wu, C. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

Opt. Lett. (1)

Precis. Eng. (1)

K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
[CrossRef]

Rev. Sci. Instrum. (2)

V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
[CrossRef]

S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata, T. Yamagishi, H. Fujimoto, H. Yukawa, “Real-time, interferometrically measuring atomic force microscope for direct calibration of standards,” Rev. Sci. Instrum. 70, 3362–3368 (1999).
[CrossRef]

Other (1)

Spectra-Physics Inc., Spectra-Physics Model 117A Stabilized Helium-Neon Laser Instruction Manual (Spectra-Physics, 1997), http://www.newport.com/store/product.aspx?lone=&ltwo=&id=5220&lang=1&section=Specs .

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Figures (7)

Fig. 1
Fig. 1

Typical single-pass homodyne interferometer. D’s, detectors; BS, beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate.

Fig. 2
Fig. 2

Phase boundary and two sinusoidal signals.

Fig. 3
Fig. 3

Photograph of the implemented single-pass homodyne interferometer. PD, photodetector; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarizing beam splitter; Ref., reference.

Fig. 4
Fig. 4

Schematic diagram of the experiments: FG, function generator; HV, high-voltage amplifier; ST, piezodriven stage; ADC, analog-to-digital converter; LPF, digital low-pass filter; CA, amplitude and offset calculation; CP, compensation; DAC, digital-to-analog converter; OSC, oscilloscope; DSP, digital signal processor.

Fig. 5
Fig. 5

Comparison of raw and compensated cyclic errors.

Fig. 6
Fig. 6

Compensated cyclic errors.

Fig. 7
Fig. 7

Real-time versus non-real-time compensation. PV, peak-to-valley value.

Equations (18)

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Tr = [ R 0 ( λ 4 ) exp ( i ϕ ) ( λ 4 ) T 0 + T 0 ( λ 4 ) ( λ 4 ) R 0 ] E .
D 1 = 1 2 T 1 ( λ 4 ) Tr = ( E 1 e 1 ) , D 2 = 1 2 R 1 ( λ 4 ) Tr = ( e 2 E 2 ) , D 3 = 1 2 T 2 ( λ 2 ) ( λ 4 ) Tr = ( E 3 e 3 ) , D 4 = 1 2 R 2 ( λ 2 ) ( λ 4 ) Tr = ( e 4 E 4 ) .
Ê 1 = ( 1 , 0 ) D 1 + ( 1 , 0 ) D 2 = E 1 + e 2 , Ê 2 = ( 0 , 1 ) D 2 + ( 0 , 1 ) D 1 = E 2 + e 1 , Ê 3 = ( 1 , 0 ) D 3 + ( 1 , 0 ) D 4 = E 3 + e 4 , Ê 4 = ( 0 , 1 ) D 4 + ( 0 , 1 ) D 3 = E 4 + e 3 ,
I i = k i ( a i Ê i Ê i * + o i ) .
I x = I 3 I 4 = D + E cos ( n ϕ ) ,
I y = I 1 I 2 = A + B sin ( n ϕ + δ ) ,
I y = A + B sin ( n ϕ + δ ) = A + γ cos ( n ϕ ) + C sin ( n ϕ ) .
I 1 = k 1 { a e 1 a 1 [ o s + a s c cos ( n ϕ ) + a s sin ( n ϕ ) ] + o e 1 } , I 2 = k 2 { a e 2 a 2 [ o s + a s c cos ( n ϕ ) + a s sin ( n ϕ ) ] + o e 2 } , I 3 = k 3 { a e 3 a 3 [ o c + a c cos ( n ϕ ) ] + o e 3 } , I 4 = k 4 { a e 4 a 4 [ o c a c cos ( n ϕ ) ] + o e 4 } .
a 1 = 1 / 4 T BS 2 T QD 2 ( R p 1 + T p 1 ) 2 , a 2 = 1 / 4 T BS 2 T QD 2 ( R s 1 + T s 1 ) 2 , a 3 = 1 / 4 R BS 2 T HD 2 T QD 2 ( R p 2 + T p 2 ) 2 ( R s n T p n + R p n T s n ) 2 , a 4 = 1 / 4 R BS 2 T HD 2 T QD 2 ( R s 2 + T s 2 ) 2 ( R s n T p n R p n T s n ) 2 , o s = ( T QR 4 n + T QT 4 n ) ( R s 2 n T p 2 n + R p 2 n T s 2 n ) , a s c = 4 T QR 2 n T QT 2 n R p n R s n T p n T s n , a s = 2 T QR 2 n T QT 2 n ( R s 2 n T p 2 n R p 2 n T s 2 n ) , o c = T QR 4 n + T QT n 4 , a c = 2 T QR 2 n T QT 2 n .
k 2 k 1 = a e 1 a e 2 ( R p 1 + T p 1 ) 2 ( R s 1 + T s 1 ) 2 , k 4 k 3 = a e 3 a e 4 ( R s n T p n + R p n T s n ) 2 ( R s n T p R p n T s n ) 2 ( R p 2 + T p 2 ) 2 ( R s 2 + T s 2 ) 2 a e 3 a e 4 ( R p 2 + T p 2 ) 2 ( R s 2 + T s 2 ) 2 , k 3 k 1 = a e 1 a e 3 T BS 2 R BS 2 T HD 2 R s 2 n T p 2 n R p 2 n T s 2 n ( R s n T p n + R p n T s n ) 2 ( R p 1 + T p 1 ) 2 ( R p 2 + T p 2 ) 2 a e 1 a e 3 T BS 2 R BS 2 T HD 2 ( R p 1 + T p 1 ) 2 ( R p 2 + T p 2 ) 2 .
I 1 = k 1 { a e 1 a 1 [ o s + a s c cos ( n ϕ ) + a s sin ( n ϕ ) ] } = k 1 { a e 1 a 1 [ o s + ( a s c 2 + a s 2 ) 1 / 2 sin ( n ϕ + δ ] } , I 2 = k 1 { a e 1 a 1 [ o s + a s c cos ( n ϕ ) + a s sin ( n ϕ ) ] } = k 1 { a e 1 a 1 [ o s ( a sc 2 + a s 2 ) 1 / 2 sin ( n ϕ + δ ) ] } , I 3 = k 1 { a e 1 a 1 ( R s 2 n T p 2 n R p 2 n T s 2 n ) [ o s + a c cos ( n ϕ ) ] } , I 4 = k 1 { a e 1 a 1 ( R s 2 n T p 2 n R p 2 n T s 2 n ) [ o s + a c cos ( n ϕ ) ] } .
a sc a s = 2 R p n R s n T p n T s n R s 2 n T p 2 n R p 2 n T s 2 n .
I 1 I 2 = 2 k 1 a e 1 a 1 ( a sc 2 + a s 2 ) 1 / 2 cos δ sin ( n ϕ ) , I 3 I 4 = 2 k 1 a e 1 a c ( R s 2 n T p 2 n R p 2 n T s 2 n ) cos ( n ϕ ) .
I 1 I 2 = 2 k 1 a e 1 a 1 a s sin ( n ϕ ) , I 3 I 4 = 2 k 1 a e 1 a 1 a s cos ( n ϕ ) .
I 1 I 2 I 3 I 4 = tan ( n ϕ ) .
I i , j ( t ) = o i , j + a i , j S ( t ) ,
c h i , j + 1 = 1 a i , j [ I i , j + 1 ( t ) o i , j ] .
D = Δ L D Δ λ λ = Δ L D Δ f f ,

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