Abstract

A three-spherical-mirror test method that uses a Fabry-Pérot (FP) cavity is proposed for a radius of curvature measurement system, especially for radii larger than 10 m. By using the three-spherical-mirror test with mode spacing measurement in an FP cavity, the local value of the radius of curvature of a mirror can be determined in situ and this mirror can then be used as the reference spherical mirror in a radius of curvature measurement system. We demonstrated determinations of radii of curvature of around 10 m using the three-spherical-mirror test with uncertainties of around 1.5 × 10−4 and then measured the radius of curvature of around 20 m with uncertainties of around 3.1 × 10−4 by using the spherical mirror, of which the radius of curvature was determined by the three-spherical-mirror test, as the reference sphere. The proposed system has high practical applicability because measurements can be conducted under usual air conditions and the measurement results are directly traceable to the time standard because beat frequency measurement is used.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. N. Uehara and K. Ueda, “Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry-Perot interferometer,” Appl. Opt. 34(25), 5611–5619 (1995).
    [Crossref] [PubMed]
  2. Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 808–825.
  3. M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).
  4. W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
    [Crossref]
  5. B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).
  6. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal radius measurement,” Opt. Express 18(3), 2345–2360 (2010).
    [Crossref] [PubMed]
  7. J. Yang, L. Qiu, W. Zhao, X. Zhang, and X. Wang, “Radius measurement by laser confocal technology,” Appl. Opt. 53(13), 2860–2865 (2014).
    [Crossref] [PubMed]
  8. M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
    [Crossref]
  9. W. Zhao, X. Zhang, Y. Wang, and L. Qiu, “Laser reflection differential confocal large-radius measurement,” Appl. Opt. 54(31), 9308–9314 (2015).
    [Crossref] [PubMed]
  10. Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
    [Crossref] [PubMed]
  11. Q. Wang, U. Griesmann, and J. A. Soons, “Holographic radius test plates for spherical surfaces with large radius of curvature,” Appl. Opt. 53(20), 4532–4538 (2014).
    [Crossref] [PubMed]
  12. H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
    [Crossref]
  13. C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
    [Crossref]
  14. J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
    [Crossref]
  15. Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
    [Crossref]
  16. Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
    [Crossref] [PubMed]
  17. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985).
    [Crossref]
  18. Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
    [Crossref] [PubMed]
  19. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996).
    [Crossref] [PubMed]
  20. JCGM 2008, Evaluation of Measurement Data - Guide to the Expression of Uncertainty in Measurement, (BIPM, 2008).

2018 (1)

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

2016 (1)

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (3)

2013 (1)

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

2010 (1)

2009 (2)

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
[Crossref] [PubMed]

2005 (2)

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

2001 (1)

1996 (1)

1995 (1)

1985 (1)

1983 (1)

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

1980 (1)

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Bitou, Y.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
[Crossref] [PubMed]

Burge, J. H.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Ciddor, P. E.

Gao, D.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Gerchman, M. C.

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Griesmann, U.

Guo, J.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Hall, J. L.

Hunter, G. C.

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Lawall, J. R.

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Meng, J.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Murtyand, M. V. R. K.

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

Pratt, J. R.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Qiu, L.

Sha, D.

Shukla, R. P.

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

Soons, J. A.

Sun, R.

Takei, Y.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Takeuchi, H.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Telada, S.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Truax, B.

B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).

Tsutsumi, H.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Ueda, K.

Uehara, N.

Wang, Q.

Wang, X.

Wang, Y.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

W. Zhao, X. Zhang, Y. Wang, and L. Qiu, “Laser reflection differential confocal large-radius measurement,” Appl. Opt. 54(31), 9308–9314 (2015).
[Crossref] [PubMed]

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Wong, N. C.

Xiang, Y.

Yang, J.

Yoshizumi, K.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Yu, Y.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Zehnder, R.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Zhang, X.

Zhao, C.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Zhao, W.

Appl. Opt. (6)

J. Opt. Soc. Am. B (1)

Laser Focus World (1)

B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).

Meas. Sci. Technol. (1)

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Opt. Commun. (1)

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Opt. Eng. (3)

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

Opt. Express (1)

Opt. Lett. (1)

Precis. Eng. (1)

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Proc. SPIE (1)

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Rev. Sci. Instrum. (1)

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Other (2)

JCGM 2008, Evaluation of Measurement Data - Guide to the Expression of Uncertainty in Measurement, (BIPM, 2008).

Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 808–825.

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Figures (7)

Fig. 1
Fig. 1 Basic configuration of the Fabry–Pérot cavity used in the proposed system.
Fig. 2
Fig. 2 Transmission spectrum lines of the FP cavity with transverse modes.
Fig. 3
Fig. 3 Experimental setup of the proposed system. ECLD: external-cavity laser diode, EOM: electro-optic modulator, AFC: adjustable fiber collimator, PBS: polarizing beam splitter, DBM: double-balanced mixer, PM: polarization-maintaining.
Fig. 4
Fig. 4 Photograph of the FP cavity part.
Fig. 5
Fig. 5 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were both 10 m
Fig. 6
Fig. 6 Temporal variation in the measured beat frequency fbeat3.
Fig. 7
Fig. 7 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were 10 and 20 m, respectively.

Tables (1)

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Table 1 Measurement results of three-spherical-mirror test

Equations (12)

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ν m,j+k = c 2n L ab ( m+(j+k+1)γ ),
γ= 1 π cos 1 ( 1 L ab R a )( 1 L ab R b ) ,
ν FSRab = v m+1,0 v m,0 = c 2nL ab .
ν TRab = v m,1 v m,0 = 1 i ( v m,i v m,0 ),
R= L sin 2 ( π ν TR ν FSR ) = c 2n ν FSR sin 2 ( π ν TR ν FSR ) .
δR R = RL R e ,
( 1 L ab R a )( 1 L ab R b ) =cos( π v TRab v FSRab ).
( 1 L ac R a )( 1 L ac R c ) =cos( π v TRac v FSRac ),
( 1 L bc R b )( 1 L bc R c ) =cos( π v TRbc v FSRbc ).
f beat1 = f 2 f 1 = v m,0 v m1,0 , f beat2 = v m,i v m1,0 , f beat3 = v m+1,0 v m1,0 .
ν FSRab = f beat3 f beat1 = v m+1,0 v m,0 , ν TRab = 1 i ( f beat2 f beat1 )= 1 i ( v m,i v m,0 ).
R d = L cd 1 cos 2 ( π ν TRcd ν FSRcd ) 1 L cd R c .

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