Abstract

A measurement method based on interferometry with two different reference cavity lengths is presented and applied in air refractive index measurement in which the two cavity lengths and a laser wavelength are combined to generate two wavelength equivalents of cavity. Corresponding calculation equations are derived, and the optical path configuration is designed, which is inspired by the traditional synthetic wavelength method. Theoretical analyses indicate that the measurement uncertainty of the determined index of refraction is about 2.3×108, which is mainly affected by the length precision of the long vacuum cavity and the ellipticity of polarization components of the dual-frequency laser, and the range of nonambiguity is 3.0×105, which is decided by the length difference of the two cavities. Experiment results show that the accuracy of air refractive index measurement is better than 5.0×108 when the laboratory conditions changes slowly. The merit of the presented method is that the classical refractometry can be also used without evacuation of the gas cavity during the experiment. Furthermore, the application of the traditional synthetic wavelength method may be extended by using the wavelength equivalents of cavity, any value of which can be easily acquired by changing cavity length rather than using actual wavelengths whose number is limited.

© 2012 Optical Society of America

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References

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  1. M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).
  2. H. Tuomas, M. Mikko, V. Markku, S. Jeremias, and L. Antti, “High-precision diode-laser-based temperature measurement for air refractive index compensation,” Appl. Opt. 50, 5990–5998 (2011).
    [CrossRef]
  3. B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  4. C. R. Tilford, “Analytical procedure for determining length from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
    [CrossRef]
  5. M. R. Benoit, “Application des phénomènes d’interférence á des déterminations méteorologiques,” J. Phys. Theor. Appl. 7, 57–68 (1898).
    [CrossRef]
  6. G. L. Bourdet and A. G. Orszag, “Absolute distance measurements by CO2 laser multiwavelength interferometry,” Appl. Opt. 18, 225–227 (1979).
    [CrossRef]
  7. K. Alzahrani, D. Burton, F. Lilley, M. Gdeisat, F. Bezombes, and M. Qudeisat, “Absolute distance measurement with micrometer accuracy using a Michelson interferometer and the iterative synthetic wavelength principle,” Opt. Express 20, 5658–5682 (2012).
    [CrossRef]
  8. J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
    [CrossRef]
  9. L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
    [CrossRef]
  10. M. H. Karl and A. Z. Ahmed, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19, (2008).
    [CrossRef]
  11. I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
    [CrossRef]
  12. C. R. Tilford, “Analytical procedure for determining length from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
    [CrossRef]
  13. Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
    [CrossRef]
  14. P. de Groot and H. A. Hill, “Superheterodyne interferometer and method for compensating the refractive index of air using electronic frequency multiplication,” U.S. patent 5,838,485(17November1998).
  15. A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys. 28, L473–L475(1989).
    [CrossRef]
  16. H. A. Hill, “Apparatus and methods for measuring intrinsic optical properties of a gas,” U.S. patent 6,124,931(26September2000).

2012

2011

H. Tuomas, M. Mikko, V. Markku, S. Jeremias, and L. Antti, “High-precision diode-laser-based temperature measurement for air refractive index compensation,” Appl. Opt. 50, 5990–5998 (2011).
[CrossRef]

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

2008

M. H. Karl and A. Z. Ahmed, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19, (2008).
[CrossRef]

2005

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

1999

L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
[CrossRef]

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

1989

A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys. 28, L473–L475(1989).
[CrossRef]

1979

1977

1966

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

1898

M. R. Benoit, “Application des phénomènes d’interférence á des déterminations méteorologiques,” J. Phys. Theor. Appl. 7, 57–68 (1898).
[CrossRef]

Ahmed, A. Z.

M. H. Karl and A. Z. Ahmed, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19, (2008).
[CrossRef]

Alzahrani, K.

Antti, L.

Benoit, M. R.

M. R. Benoit, “Application des phénomènes d’interférence á des déterminations méteorologiques,” J. Phys. Theor. Appl. 7, 57–68 (1898).
[CrossRef]

Bezombes, F.

Bourdet, G. L.

Burton, D.

Chen, Q. H.

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

de Groot, P.

P. de Groot and H. A. Hill, “Superheterodyne interferometer and method for compensating the refractive index of air using electronic frequency multiplication,” U.S. patent 5,838,485(17November1998).

Diao, X. F.

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

Edlen, B.

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Fujima, I.

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

Gdeisat, M.

Hidekazu, O.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

Hideyuki, N.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

Hill, H. A.

P. de Groot and H. A. Hill, “Superheterodyne interferometer and method for compensating the refractive index of air using electronic frequency multiplication,” U.S. patent 5,838,485(17November1998).

H. A. Hill, “Apparatus and methods for measuring intrinsic optical properties of a gas,” U.S. patent 6,124,931(26September2000).

Hiroki, M.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

Hisayoshi, S.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

Hu, P. C.

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

Ishida, A.

A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys. 28, L473–L475(1989).
[CrossRef]

Iwasaki, S.

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

Jeremias, S.

Kaoru, M.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

Karl, M. H.

M. H. Karl and A. Z. Ahmed, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19, (2008).
[CrossRef]

Lilley, F.

Lin, D. J.

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

Markku, V.

Matsumoto, H. A.

L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
[CrossRef]

Mikko, M.

Orszag, A. G.

Qudeisat, M.

Seta, K.

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
[CrossRef]

Tan, J. B.

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

Tilford, C. R.

Tuomas, H.

Wu, J.

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

Xie, G. P.

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

Yan, J. Q.

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

Yang, H. X.

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

Yin, C. Y.

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

Zeng, L. J.

L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
[CrossRef]

Appl. Opt.

Int. J. Autom. Technol.

M. Kaoru, O. Hidekazu, N. Hideyuki, M. Hiroki, and S. Hisayoshi, “Two-wavelength laser interferometer system which reduces the uncertainty caused by the fluctuation of the refractive index of air,” Int. J. Autom. Technol. 5, 126–131 (2011).

J. Phys. Theor. Appl.

M. R. Benoit, “Application des phénomènes d’interférence á des déterminations méteorologiques,” J. Phys. Theor. Appl. 7, 57–68 (1898).
[CrossRef]

Jpn. J. Appl. Phys.

A. Ishida, “Two-wavelength displacement-measuring interferometer using second-harmonic light to eliminate air-turbulence-induced errors,” Jpn. J. Appl. Phys. 28, L473–L475(1989).
[CrossRef]

Meas. Sci. Technol.

J. B. Tan, H. X. Yang, P. C. Hu, and X. F. Diao, “Identification and elimination of half-synthetic wavelength error for multi-wavelength long absolute distance measurement,” Meas. Sci. Technol. 22, 115301 (2011).
[CrossRef]

M. H. Karl and A. Z. Ahmed, “Refractive index determination in length measurement by two-colour interferometry,” Meas. Sci. Technol. 19, (2008).
[CrossRef]

Q. H. Chen, D. J. Lin, J. Wu, J. Q. Yan, and C. Y. Yin, “Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser,” Meas. Sci. Technol. 16, 2030–2037 (2005).
[CrossRef]

Metrologia

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Opt. Express

Proc. SPIE

I. Fujima, S. Iwasaki, G. P. Xie, and K. Seta, “Precise measurement of the difference of the air refractive indices between visible and near-infrared wavelengths using two-color interferometer,” Proc. SPIE 3897, 767–776 (1999).
[CrossRef]

Rev. Sci. Instrum.

L. J. Zeng, H. A. Matsumoto, and K. Seta, “Group-phase refractive index method for improving the accuracy in two-color interferometric length measurements,” Rev. Sci. Instrum. 70, 2917–2920 (1999).
[CrossRef]

Other

P. de Groot and H. A. Hill, “Superheterodyne interferometer and method for compensating the refractive index of air using electronic frequency multiplication,” U.S. patent 5,838,485(17November1998).

H. A. Hill, “Apparatus and methods for measuring intrinsic optical properties of a gas,” U.S. patent 6,124,931(26September2000).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of air refractive index measurement setup based on presented method using dual vacuum cavities. 1, Beam splitter plate; 2, annular optical compensator; 3, quarter-wave plate; 4, vacuum cavity; 5, pyramid prism; 6, PBS; 7, total reflection prism; 8, detector; 9, phase meter; 10, precision translation stage.

Fig. 2.
Fig. 2.

Error difference result from the quarter-wave plate.

Fig. 3.
Fig. 3.

Error difference result from the light source.

Fig. 4.
Fig. 4.

Stability curve of air refractive index measurement system.

Fig. 5.
Fig. 5.

Experiment comparison of air refractive index measurement between two methods. 1. Presented method; 2. method of Edlen equations.

Equations (9)

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

(n1)L=(N+ε)λ02,
{I1=cos[2π(f1f2)t+(φ01φ02)+Δφ],I2=cos[2π(f1f2)t+(φ01φ02)Δφ],
(n1)=(Ni+εi)Ei,i=1,2,
(n1)=(Ns+εs)Es,
(Ns+εs)=2(n1)L1λ02(n1)L1λ0=2Δφ12Δφ22π.
δn(n1)=δL1L1+δ(ε1+N1)(ε1+N1)+δλ0λ0,
Rn=λ04(L1L2),
{δns<Rn14δns=(ns1)(δLsLs+δ(εs+Ns)(εs+Ns)+δλ0λ0),Rn1=λ04L1,
Δ=Δ12+Δ22+2Δ32+Δ42+Δ52+Δ62=2.3×108.

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