Abstract

The group refractive index of air in laboratory conditions is measured directly between adjacent femtosecond laser pulses by a new interferometry technique. Measurement of the repetition rate of the mode-locked pulse train that gives the maximum amplitude of the interference-signal envelope enables us to determine the group refractive index of air within a standard deviation of 2 × 10-7. This simple method without vacuum reference is attractive for measuring the group refractive index needed for precise distance measurements in open fields.

© 2002 Optical Society of America

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References

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  1. K. Minoshima, H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39, 5512–5517 (2000).
    [CrossRef]
  2. I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
    [CrossRef]
  3. P. E. Ciddor, R. J. Hill, “Refractive index of air: 2. Group index,” Appl. Opt. 38, 1663–1667 (1999).
    [CrossRef]
  4. Y. S. Galkin, R. A. Tatevian, “The problem of obtaining formulae for the refractive index of air for high-precision EDM,” J. Geodesy 71, 483–485 (1997).
    [CrossRef]
  5. Y. S. Galkin, R. A. Tatevian, “Influence of resonances on the phase and the group refractive indices of air,” J. Geodesy 71, 680–684 (1997).
    [CrossRef]
  6. M. J. Downs, K. P. Birch, “Bidirectional fringe counting interference refractometer,” Precis. Eng. 5, 105–110 (1983).
    [CrossRef]
  7. H. Matsumoto, “Measurement of the group refractive index of air by two-wavelength interferometry,” Opt. Commun. 44, 5–7 (1982).
    [CrossRef]
  8. Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
    [CrossRef]
  9. B. Edlén, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  10. R. Muijlwijk, “Update of the Edlén formulae for the refractive index of air,” Metrologia 25, 189 (1988).
    [CrossRef]
  11. T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
    [CrossRef]
  12. Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
    [CrossRef]
  13. ISO/DIS 3650, “Geometrical product specification (GPS)—length standards—gauge blocks,” (International Organization for Standardization, Geneva, Switzerland, 1995).

2001 (1)

T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
[CrossRef]

1997 (2)

Y. S. Galkin, R. A. Tatevian, “The problem of obtaining formulae for the refractive index of air for high-precision EDM,” J. Geodesy 71, 483–485 (1997).
[CrossRef]

Y. S. Galkin, R. A. Tatevian, “Influence of resonances on the phase and the group refractive indices of air,” J. Geodesy 71, 680–684 (1997).
[CrossRef]

1990 (1)

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

1988 (1)

R. Muijlwijk, “Update of the Edlén formulae for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

1983 (1)

M. J. Downs, K. P. Birch, “Bidirectional fringe counting interference refractometer,” Precis. Eng. 5, 105–110 (1983).
[CrossRef]

1982 (1)

H. Matsumoto, “Measurement of the group refractive index of air by two-wavelength interferometry,” Opt. Commun. 44, 5–7 (1982).
[CrossRef]

1966 (1)

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

Birch, K. P.

M. J. Downs, K. P. Birch, “Bidirectional fringe counting interference refractometer,” Precis. Eng. 5, 105–110 (1983).
[CrossRef]

Bitou, Y.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Bor, Z.

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Ciddor, P. E.

Downs, M. J.

M. J. Downs, K. P. Birch, “Bidirectional fringe counting interference refractometer,” Precis. Eng. 5, 105–110 (1983).
[CrossRef]

Edlén, B.

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

Fujima, I.

I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
[CrossRef]

Galkin, Y. S.

Y. S. Galkin, R. A. Tatevian, “Influence of resonances on the phase and the group refractive indices of air,” J. Geodesy 71, 680–684 (1997).
[CrossRef]

Y. S. Galkin, R. A. Tatevian, “The problem of obtaining formulae for the refractive index of air for high-precision EDM,” J. Geodesy 71, 483–485 (1997).
[CrossRef]

Hill, R. J.

Hirai, A.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Hong, F.-L.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Iwasaki, S.

I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
[CrossRef]

Matsumoto, H.

T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
[CrossRef]

K. Minoshima, H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39, 5512–5517 (2000).
[CrossRef]

H. Matsumoto, “Measurement of the group refractive index of air by two-wavelength interferometry,” Opt. Commun. 44, 5–7 (1982).
[CrossRef]

Minoshima, K.

T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
[CrossRef]

K. Minoshima, H. Matsumoto, “High-accuracy measurement of 240-m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39, 5512–5517 (2000).
[CrossRef]

Muijlwijk, R.

R. Muijlwijk, “Update of the Edlén formulae for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

Onae, A.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Osvay, K.

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Rá;cz, B.

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Seta, K.

I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
[CrossRef]

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Szabó;, G.

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Tatevian, R. A.

Y. S. Galkin, R. A. Tatevian, “The problem of obtaining formulae for the refractive index of air for high-precision EDM,” J. Geodesy 71, 483–485 (1997).
[CrossRef]

Y. S. Galkin, R. A. Tatevian, “Influence of resonances on the phase and the group refractive indices of air,” J. Geodesy 71, 680–684 (1997).
[CrossRef]

Yasui, T.

T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
[CrossRef]

Yoshimori, H.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Zhang, Y.

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum. Electron. (1)

T. Yasui, K. Minoshima, H. Matsumoto, “Stabilization of femtosecond mode-locked Ti:sapphire laser for high-accuracy pulse interferometry,” IEEE J. Quantum. Electron. 37, 12–19 (2001).
[CrossRef]

J. Geodesy (2)

Y. S. Galkin, R. A. Tatevian, “The problem of obtaining formulae for the refractive index of air for high-precision EDM,” J. Geodesy 71, 483–485 (1997).
[CrossRef]

Y. S. Galkin, R. A. Tatevian, “Influence of resonances on the phase and the group refractive indices of air,” J. Geodesy 71, 680–684 (1997).
[CrossRef]

Meas. Sci. Technol. (1)

I. Fujima, S. Iwasaki, K. Seta, “High-resolution distance meter using optical intensity modulation at 28 GHz,” Meas. Sci. Technol. 9, 1049–1052 (1998).
[CrossRef]

Metrologia (2)

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

R. Muijlwijk, “Update of the Edlén formulae for the refractive index of air,” Metrologia 25, 189 (1988).
[CrossRef]

Opt. Commun. (2)

H. Matsumoto, “Measurement of the group refractive index of air by two-wavelength interferometry,” Opt. Commun. 44, 5–7 (1982).
[CrossRef]

Z. Bor, K. Osvay, B. Rá;cz, G. Szabó;, “Group refractive-index measurement by Michelson interferometer,” Opt. Commun. 78, 109–112 (1990).
[CrossRef]

Precis. Eng. (1)

M. J. Downs, K. P. Birch, “Bidirectional fringe counting interference refractometer,” Precis. Eng. 5, 105–110 (1983).
[CrossRef]

Other (2)

Y. Bitou, A. Hirai, H. Yoshimori, F.-L. Hong, Y. Zhang, A. Onae, K. Seta, “Gauge block interferometer using three frequency-stabilized lasers,” in Recent Developments in Traceable Dimensional Measurements, J. E. Decker, ed., Proc. SPIE4401, 288–297 (2001).
[CrossRef]

ISO/DIS 3650, “Geometrical product specification (GPS)—length standards—gauge blocks,” (International Organization for Standardization, Geneva, Switzerland, 1995).

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Figures (6)

Fig. 1
Fig. 1

(A) Rough sketch of experimental setup I for group-refractive-index measurement in which an unbalanced Michelson interferometer was used with the stabilized repetition rate of a femtosecond pulse laser: M, mirror; BS, beam splitter; PD, photodiode; OF, optical flat; GB, 1-cm gauge block; LGB, 1-m gauge block. (B) Interference-signal envelopes [P, interference-signal envelope between pulse 1 (BS1-GB-OF-GB-BS1) and the next pulse 2 (BS1-M1-BS1); Q, interference-signal envelope between the pulse (BS1-OF-BS1) and the same pulse (BS1-M1-BS1)].

Fig. 2
Fig. 2

(A) Rough sketch of experimental setup II for group-refractive-index measurement with a Fabry-Pérot-like interferometer with a scanned repetition rate of a femtosecond pulse laser. (B) Interference-signal envelope between pulse 1 (BS1-GB-OF-GB-BS1) and the next pulse 2 (BS1-OF-BS1).

Fig. 3
Fig. 3

Actual experimental setup for direct measurement of the group refractive index of air in which experimental setup II is used: GB, 1-cm gauge block; LGB, 1-m gauge block; M, mirror; BS, beam splitter; OF, optical flat; PD, photodiode; FG, function generator; Counter, frequency counter. Pulse 1, which travels twice between OF and GB, is overlapped by pulse 2 that is reflected by OF.

Fig. 4
Fig. 4

Interference fringe-pattern signal (open squares) between the two adjacent pulses in a mode-locked train. Solid curve, envelope of the squared interference signal. Although open squares look asymmetric because of the small amount of data, the asymmetry disappears in the solid curve when the square of data is taken.

Fig. 5
Fig. 5

(A) Variation of environmental parameters (solid curve, air pressure, dashed curve, air temperature). (B) Closed dots, measured group refractive index of air depending on environmental parameters; open dots, solid line, group refractive index at 293.15 K, 1013.25 hPa, and ∼50% humidity calibrated by the density factor D t,p ; dashed line, value of the group refractive index at 293.15 K, 1013.25 hPa, and ∼50% humidity calculated by Edlén’s equation.

Fig. 6
Fig. 6

Wavelength dependence of the group refractive index of air: solid dots, average group refractive index at each wavelength; solid line, linear least-squares fitting for the group refractive indices that were obtained; dashed line, group refractive index derived from Edlén’s equation.

Equations (8)

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ng=cTl,
ng=c2fs2L-ap,
ng=c4fpL.
L=Ls1+αt-tsLs1+αΔt,
δng2=ngfp2δfp2+ngL2δL2δfp2fp2+δL2L2.
δL2=LLs2δLs2+Lα2δα2+LΔt2δΔt2=1+αΔt2δLs2+Ls2Δt2δα2+Ls2α2δΔt2.
δng2=δfp2fp2+1L21+αΔt2δLs2+Ls2Δt2δα2+Ls2α2δΔt2δfp2fp2+δLs2L2+Δt2δα2+α2δΔt2.
d2kdω2=-λ22πc2ngλ.

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