Abstract

Most homodyne interferometers have a quadrature detector system that includes two polarizing beam splitters that cause nonlinearity of the order of a few nanometers by phase mixing. Detectors should have the same gains to reduce nonlinearity under the assumption that there is no loss in optical components. However, optical components exhibit some loss. We show that nonlinearity can be reduced to an order of 0.01 nm when the detector gains are adjusted by simulation to include the optical characteristics. The compensated nonlinearity is 18 times smaller than that when the four detector gains are set to be equal.

© 2004 Optical Society of America

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References

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  1. T. Ito, S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
    [CrossRef] [PubMed]
  2. E. C. Teague, “Generating and measuring displacements up to 0.1 m to an accuracy of 0.1 nm: Is it possible?” in Technology of Proximal Probe Lithography, C. R. K. Marrian, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 322–363.
  3. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981).
    [CrossRef] [PubMed]
  4. K. P. Birch, “Optical fringe subdivision with nanometric accuracy,” Precis. Eng. 12, 195–198 (1990).
    [CrossRef]
  5. V. Greco, G. Molesini, F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. 66, 3729–3734 (1995).
    [CrossRef]
  6. C.-M. Wu, C.-S. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
    [CrossRef]
  7. C.-M. Wu, C.-S. Su, G.-S. Peng, “Correction of nonlinearity in one-frequency optical interferometry,” Meas. Sci. Technol. 7, 520–524 (1996).
    [CrossRef]
  8. F. Petru, O. Cip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
    [CrossRef]
  9. 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]
  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. M. J. Downs, “A proposed design for an optical interferometer with sub-nanometric resolution,” Nanotechnology 1, 27–30 (1990).
    [CrossRef]
  12. M. J. Downs, W. R. C. Rowley, “A proposed design for a polarization-insensitive optical interferometer system with subnanometric capacity,” Precis. Eng. 15, 281–286 (1993).
    [CrossRef]
  13. M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
    [CrossRef]
  14. These specifications are from Spectra-Physics, Inc., http://www.oriel.com/netcat/VolumeIII/Descrippage/v3t3polbs.htm .
  15. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), Chap. 1.
  16. This specification is from ThorLabs Company, http://www.thorlabs.com/ShowProducts.cfm?DID=6&CATID=151&ObjectGroup_ID=152&Queries=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]

2000 (1)

T. Ito, S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef] [PubMed]

1999 (2)

F. Petru, O. Cip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[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]

1996 (2)

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

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

1995 (2)

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

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[CrossRef]

1993 (1)

M. J. Downs, W. R. C. Rowley, “A proposed design for a polarization-insensitive optical interferometer system with subnanometric capacity,” Precis. Eng. 15, 281–286 (1993).
[CrossRef]

1990 (2)

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

M. J. Downs, “A proposed design for an optical interferometer with sub-nanometric resolution,” Nanotechnology 1, 27–30 (1990).
[CrossRef]

1981 (1)

Birch, K. P.

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[CrossRef]

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

Cip, O.

F. Petru, O. Cip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Cox, M. G.

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[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]

Downs, M. J.

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[CrossRef]

M. J. Downs, W. R. C. Rowley, “A proposed design for a polarization-insensitive optical interferometer system with subnanometric capacity,” Precis. Eng. 15, 281–286 (1993).
[CrossRef]

M. J. Downs, “A proposed design for an optical interferometer with sub-nanometric resolution,” Nanotechnology 1, 27–30 (1990).
[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.

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]

Ito, T.

T. Ito, S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef] [PubMed]

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]

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.

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]

Molesini, G.

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

Nunn, J. W.

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[CrossRef]

Okazaki, S.

T. Ito, S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef] [PubMed]

Peng, G.-S.

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

Petru, F.

F. Petru, O. Cip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Quercioli, F.

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

Rowley, W. R. C.

M. J. Downs, W. R. C. Rowley, “A proposed design for a polarization-insensitive optical interferometer system with subnanometric capacity,” Precis. Eng. 15, 281–286 (1993).
[CrossRef]

Su, C.-S.

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

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

Teague, E. C.

E. C. Teague, “Generating and measuring displacements up to 0.1 m to an accuracy of 0.1 nm: Is it possible?” in Technology of Proximal Probe Lithography, C. R. K. Marrian, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 322–363.

Wu, C.-M.

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

C.-M. Wu, C.-S. Su, “Nonlinearity in measurement of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (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]

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), Chap. 1.

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. (1)

Meas. Sci. Technol. (3)

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

C.-M. Wu, C.-S. 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]

Nanotechnology (1)

M. J. Downs, “A proposed design for an optical interferometer with sub-nanometric resolution,” Nanotechnology 1, 27–30 (1990).
[CrossRef]

Nature (1)

T. Ito, S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[CrossRef] [PubMed]

Precis. Eng. (4)

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

M. J. Downs, W. R. C. Rowley, “A proposed design for a polarization-insensitive optical interferometer system with subnanometric capacity,” Precis. Eng. 15, 281–286 (1993).
[CrossRef]

M. J. Downs, K. P. Birch, M. G. Cox, J. W. Nunn, “Verification of a polarization-insensitive optical interferometer system with subnanometric capability,” Precis. Eng. 17, 84–88 (1995).
[CrossRef]

F. Petru, O. Cip, “Problems regarding linearity of data of a laser interferometer with a single-frequency laser,” Precis. Eng. 23, 39–50 (1999).
[CrossRef]

Rev. Sci. Instrum. (2)

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]

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

Other (4)

E. C. Teague, “Generating and measuring displacements up to 0.1 m to an accuracy of 0.1 nm: Is it possible?” in Technology of Proximal Probe Lithography, C. R. K. Marrian, ed. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 322–363.

These specifications are from Spectra-Physics, Inc., http://www.oriel.com/netcat/VolumeIII/Descrippage/v3t3polbs.htm .

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), Chap. 1.

This specification is from ThorLabs Company, http://www.thorlabs.com/ShowProducts.cfm?DID=6&CATID=151&ObjectGroup_ID=152&Queries=1 .

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Figures (4)

Fig. 1
Fig. 1

General homodyne interferometer with a quadrature detector system; it has two parts, an interferometer part and a detector. BS, beam splitter; other abbreviations defined in text.

Fig. 2
Fig. 2

Remaining error of the homodyne interferometer with a quadrature detector system. After gain adjustment, the remaining error is dramatically reduced by approximately 18×: (a) single-pass interferometer, (b) double-pass interferometer, (c) four-pass interferometer.

Fig. 3
Fig. 3

Remaining error of the four-pass homodyne interferometer; all wave plates have retardation error ε in λ/500. When all the gains of the detectors are equal to 10, the PV value is ±0.078 nm, but after gain adjustment the PV value is ±0.013 nm, an ∼6× reduction. The reduced signal is biased.

Fig. 4
Fig. 4

Simulation result of introducing random noise from measuring the intensities; the PV value is ±0.075 nm.

Equations (22)

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E=αβ,
λ2=i2-1-1-11, λ4=121-i-i1.
Tj=Tpj00Tsj, Rj=Rpj00Rsj,
Tr1=R1λ4expiϕλ4T1+T1λ4λ4R1E,Tr2=T1λ4expiϕλ4R1 R1λ4expiϕλ4T1+R1λ4λ4T1×T1λ4λ4R1E,Tr4=T1λ4expiϕλ4R1 R1λ4expiϕλ4T1×T1λ4expiϕλ4R1 R1λ4expiϕλ4T1+R1λ4λ4T1 T1λ4λ4R1 R1λ4λ4T1×T1λ4λ4R1E,
D1=12T2λ4Tri=E1e1,D2=12R2λ4Tri=e2E2,D3=12T3λ2λ4Tri=E3e3,D4=12R3λ2λ4Tri=e4E4.
Ê1=1, 0D1+1, 0D2=E1+e2,Ê2=0, 1D2+0, 1D1=E2+e1,Ê3=1, 0D3+1, 0D4=E3+e4,Ê4=0, 1D4+0, 1D3=E4+e3.
Ii=kiÊiÊi*,
Ix=I3-I4=D+E cosnϕ,
Iy=I1-I2=A+B sinnϕ+δ,
Iy=A+B sinnϕ+δ=A+γ cosnϕ+C sinnϕ,
IyIx=I1-I2I3-I4=sinnϕcosnϕ.
ϕ=1ntan-1EIy-ACIx-D-γC.
A=-α2+β2Tp12nRs12n+Rp12nTs12nk1Tp2+Rp22-k2Ts2+Rs22,γ=-2α2+β2k1Tp2+Rp22-k2Rs2+Ts22Tp1nRp1nTs1nRs1n,C=2αβTp12nRs12n-Rp12nTs12nk1Tp2+Rp22+k2Ts2+Rs22,D=k3Tp3+Rp324αβTp1nRp1nTs1nRs1n+α2+β2×Tp12nRs12n+Rp12nTs12n+k4Ts3+Rs324αβTp1nRp1nTs1nRs1n-α2+β2×Tp12nRs12n+Rp12nTs12n,E=2k3Tp3+Rp32α2+β2Tp1nRp1nTs1nRs1n+αβTp12nRs12n+Rp12nTs12n-k4Ts3+Rs32α2+β2Tp1nRp1nTs1nRs1n-αβTp12nRs12n+Rp12nTs12n.
k1k2=Ts2+Rs22Tp2+Rp22,
k3k4=Ts3+Rs32Tp3+Rp32,
k1Tp2+Rp22+k2Ts2+Rs22k3Tp3+Rp32+k4Ts3+Rs32=1.
k1Tp2+Rp22k3Tp3+Rp32=1 or k2Ts2+Rs22k4Ts3+Rs32=1.
k1k2=k3k4=Ts+Rs2Tp+Rp2, k1=k3, k2=k4.
ΔL=12πλ21ntan-1IyIx - tan-1sinnϕcosnϕ nm.
W=cos θ-sin θsin θcos θ×exp-iΓ/2+00expiΓ/2+×cos θsin θ-sin θcos θ.
λ445°=121-/2-i1+/2-i1+/21-/2,λ222.5°=12-i-/2-i-ii-/2.
Iy=¼α2+β2-k1+k2+α2-β2k1+k2+2αβk1+k2sinnϕ,Ix=¼α2+β2k3-k4+2α2-β2k3+k4+2αβk3+k4cosnϕ.

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