Abstract

A balanced plane-mirror heterodyne interferometer with a polarizing beam splitter used to recombine the reference and measurement beams is proposed to reduce periodic nonlinearity and to eliminate thermal error. Experimental results indicated that the periodic error due to ghost reflection was kept within ±36pm, and the interferometer proposed was immune from thermal error.

© 2014 Optical Society of America

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References

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  1. C. M. Wu and R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37, 6696–6700 (1998).
    [CrossRef]
  2. J. Ahn, J. A. Kim, C. S. Kang, J. W. Kim, and S. Kim, “High resolution interferometer with multiple pass optical configuration,” Opt. Express 17, 21042–21049 (2009).
    [CrossRef]
  3. T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001).
    [CrossRef]
  4. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002).
    [CrossRef]
  5. A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
    [CrossRef]
  6. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30, 337–346 (2006).
    [CrossRef]
  7. C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interferometry,” Opt. Commun. 215, 17–23 (2003).
    [CrossRef]
  8. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992).
    [CrossRef]
  9. H. Haitjema, S. J. A. G. Cosijns, N. J. J. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003).
    [CrossRef]
  10. C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
    [CrossRef]
  11. C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. 38, 4089–4094 (1999).
    [CrossRef]
  12. K. Joo, J. D. Ellis, J. W. Spronck, P. J. M. v. Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with subnanometer periodic errors,” Opt. Lett. 34, 386–388 (2009).
    [CrossRef]
  13. K. 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]
  14. V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000).
    [CrossRef]
  15. T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105–2114 (2002).
    [CrossRef]

2012 (1)

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

2010 (1)

2009 (2)

2006 (1)

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30, 337–346 (2006).
[CrossRef]

2003 (2)

C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interferometry,” Opt. Commun. 215, 17–23 (2003).
[CrossRef]

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

2002 (2)

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

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105–2114 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000).
[CrossRef]

1999 (1)

1998 (1)

1992 (1)

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

1990 (1)

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Ahn, J.

Andreas, B.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Badami, V. G.

V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000).
[CrossRef]

Beckwith, J. F.

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105–2114 (2002).
[CrossRef]

Bobroff, N.

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Buice, E. S.

Cosijns, S. J. A. G.

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

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

Deslattes, R. D.

Ellis, J. D.

Eom, T.

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

Flügge, J.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Haitjema, H.

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

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

Hou, W.

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30, 337–346 (2006).
[CrossRef]

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

Jansen, M. J.

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

Jeong, K.

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

Joo, K.

Kan, P. J. M. v.

Kang, C. S.

Kim, J. A.

Kim, J. W.

Kim, J. Y.

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

Kim, S.

Köchert, P.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Köning, R.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Kuetgens, U.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Lawall, J.

Patterson, S. R.

V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

Roset, N. J. J.

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

Schellekens, P. H. J.

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

Schmidt, R. H. M.

Schmitz, T. L.

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105–2114 (2002).
[CrossRef]

Spronck, J. W.

Weichert, C.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[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. M.

Yacoot, A.

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacement-measuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105–2114 (2002).
[CrossRef]

Meas. Sci. Technol. (2)

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

C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10  pm,” Meas. Sci. Technol. 23, 094005 (2012).
[CrossRef]

Opt. Commun. (1)

C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interferometry,” Opt. Commun. 215, 17–23 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Precis. Eng. (5)

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

V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000).
[CrossRef]

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

A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990).
[CrossRef]

W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30, 337–346 (2006).
[CrossRef]

Proc. SPIE (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

Schematic diagram of balanced plane-mirror heterodyne interferometer. BS, beams splitter; PBS1 and PBS2, polarizing beam splitters; QWP1 and QWP2, quarter-wave plates; RAP, right angle prism; M1, M2, and M3, mirrors; AP1 and AP2, absorber plates; PDR and PDM, reference and measurement photodetectors.

Fig. 2.
Fig. 2.

Optical source with two spatially separated beams driven by two AOFSs at slightly different frequencies from a stabilized source. BS, beam splitter; AOFS1 and AOFS2, acousto-optic frequency shifters; HWP1 and WHP2, half-wave plates; PH, pinhole.

Fig. 3.
Fig. 3.

Periodic errors measured using frequency domain method at first harmonic frequency (500 kHz) and second harmonic frequency (429.70 kHz).

Fig. 4.
Fig. 4.

Periodic errors with 1.4 μm/s from phase quadrature measurement method. (a) Demodulated periodic phase measured with a digital lock-in amplifier; (b) periodic error calculated with (1/4)*(λ0/2π)*(dR/R) (4 is number of optical paths to target M2, λ0=632.8nm is laser wavelength); (c) FFT result of periodic error 4(b). The dominant frequency is approximately 9 Hz, the first harmonic peak amplitude (at 9 Hz) is ±36pm, and the second harmonic peak amplitude (at 18 Hz) is ±10pm.

Fig. 5.
Fig. 5.

Experimental result of thermal error. The error curve (dotted) represents the case with a glass plate 3 mm thick of the same material as PBS1 placed closely between QWP1 and M2; the error curve (solid) represents the other case without the glass plate.

Equations (8)

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

Ircos(2πf1t+θ1)cos(2πf2t+θ2),
Imcos(2πf1t+θ1+δ1+θm)cos(2πf2t+θ2+δ2),
Ircos(2πfst+θ1θ2),
Imcos(2πfst+θ1θ2+δ1δ2+θm).
Im|A1|2+|A2|2+|ε12|2+|ε21|2DCterms+2R{A1*A2ei(2πfst+ϕm)Base signal+(A2ε21*+A1*ε12)ei(2πfst)First harmonic error terms++ε12ε21*ei(2πfstϕm)Second harmonic error term+(A1ε21*+A2*ε12)eiϕmQuasi-DCterms},
|dϕ|=|dRR|.
{R(n)=[C2(n)+S2(n)]1/2,ϕ(n)=tan1[S(n)/C(n)],R¯=1NR(n),dϕ(n)=dR(n)R¯=R(n)R¯R¯,Filter(dϕ(n)),dL(n)=14*λ02π*dϕ(n),
FT(dL)={i=1Δn[FFT[dL(i:Δn:N)·*window()]]}*2N,

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