Abstract

The author’s recent studies of the refractive index of air are extended, and several assumptions made therein are further examined. It is shown that the alternative dispersion equations for CO2, which are due to Edlen [Metrologia 2, 71 (1966)] and Old et al. [J. Opt. Soc. Am. 61, 89 (1971)] result in differences of less than 2 × 10-9 in the phase refractive index and less than 3 × 10-9 in the group refractive index for current and predicted concentrations of CO2. However, because the dispersion equation given by Old et al. is consistent with experimental data in the near infrared, it is preferable to the equation used by Edlen, which is valid only in the ultraviolet and the visible. The classical measurement by Barrell and Sears [Philos. Trans. R. Soc. London Ser. A 238, 1 (1939)] on the refractivity of moist air is shown to have some procedural errors in addition to the one discussed by Birch and Downs [Metrologia 30, 155 (1993)]. It is shown that for normal atmospheric conditions the higher refractivity virial coefficients related to the Lorentz-Lorenz relation are adequately incorporated into the empirically determined first refractivity virial. As a guide to users the practical limits to the calculation of the refractive index of the atmosphere that result from the uncertainties in the measurement of the various atmospheric parameters are summarized.

© 2002 Optical Society of America

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References

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  1. P. E. Ciddor, “Refractive index of air: new equations for the visible and the near infrared,” Appl. Opt. 35, 1566–1573 (1996).
    [CrossRef] [PubMed]
  2. P. E. Ciddor, R. J. Hill, “Refractive index of air: 2. Group index,” Appl. Opt. 38, 1663–1667 (1999).
    [CrossRef]
  3. B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  4. C. Cuthbertson, M. Cuthbertson, “On the refraction and dispersion of carbon dioxide, carbon monoxide, and methane,” Proc. R. Soc. London A 97, 152–159 (1920).
    [CrossRef]
  5. J. G. Old, K. L. Gentili, E. R. Peck, “Dispersion of carbon dioxide,” J. Opt. Soc. Am. 61, 89–90 (1971).
    [CrossRef]
  6. A. C. Simmons, “The refractive index and Lorentz-Lorenz functions of propane, nitrogen and carbon-dioxide in the spectral range 15803–22002 cm-1 and at 944 cm-1,” Opt. Commun. 25, 211–214 (1978).
    [CrossRef]
  7. P. Giacomo, “Equation for the determination of the density of moist air (1981),” Metrologia 18, 33–40 (1982).
    [CrossRef]
  8. R. S. Davis, “Equation for the determination of the density of moist air (1981/1991),” Metrologia 29, 67–70 (1992).
    [CrossRef]
  9. H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
    [CrossRef]
  10. J. C. Owens, “Optical refractive index of air,” Appl. Opt. 6, 51–59 (1967).
    [CrossRef] [PubMed]
  11. K. P. Birch, M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
    [CrossRef]
  12. K. P. Birch, M. J. Downs, “Correction to the updated Edlen equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
    [CrossRef]
  13. R. J. Hill, “Refractive index of atmospheric gases,” in The Upper Atmosphere, W. Diemenger, G. Hartmann, R. Leitinger, eds. (Springer, Berlin, 1995), pp. 261–270.
  14. K. P. Birch, M. J. Downs, “The precise determination of the refractive index of air,” Rep. MOM90, (National Physical Laboratory, Teddington, Middlesex, TW11 0LW, U.K., 1988), pp. 1–35.
  15. K. P. Birch, 36 Fircroft Road, Chessington, Surrey, KT9 1RW, U.K. (personal communication, 2000).
  16. H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
    [CrossRef]
  17. G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
    [CrossRef]
  18. R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
    [CrossRef]
  19. See, for example, R. Carmichael, “A table of the standard atmosphere to 65,000 feet,” (Public Domain Aeronautical Software, Santa Cruz, Calif., 12June2001), http://www.pdas.com/m1.htm .
  20. I. G. Enting, T. M. L. Wigley, M. Heimann, “Future emissions and concentrations of carbon dioxide: key ocean/atmosphere/land analyses,” Tech. Paper 31 (CSIRO Division of Atmospheric Research, Melbourne, Australia, 1994), p. 120. On-line edition available at http://www.dar.csiro.au , (search for publications\enting_2001a0.pdf ).
  21. R. J. Hill, “Infrared refractive index software, IR_N, 2000,” Environmental Research Laboratories, NOAA, 325 Broadway, Boulder, Colorado 80303-3328 (personal communication, May2000). (A limited number of copies of this software are available on compact disk from P. E. Ciddor or from J.M.Rueger@unsw.edu.au).
  22. Association Internationale de Géodesie, Handbook of Geodesy (on-line edition, 2000),” http://www.gfy.ku.dk/∼iag/HB2000/part2/iag_res.htm .
  23. J. A. Stone, J. H. Zimmerman, “Index of refraction of air,” (National Institute of Science and Technology, Gaithersburg, Maryland, 12June2001), http://patapsco.nist.gov/mel/div821 .
  24. M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

1999 (1)

1996 (1)

1994 (1)

K. P. Birch, M. J. Downs, “Correction to the updated Edlen equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

1993 (1)

K. P. Birch, M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

1992 (1)

R. S. Davis, “Equation for the determination of the density of moist air (1981/1991),” Metrologia 29, 67–70 (1992).
[CrossRef]

1991 (1)

H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
[CrossRef]

1986 (1)

R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
[CrossRef]

1983 (1)

G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
[CrossRef]

1982 (1)

P. Giacomo, “Equation for the determination of the density of moist air (1981),” Metrologia 18, 33–40 (1982).
[CrossRef]

1978 (1)

A. C. Simmons, “The refractive index and Lorentz-Lorenz functions of propane, nitrogen and carbon-dioxide in the spectral range 15803–22002 cm-1 and at 944 cm-1,” Opt. Commun. 25, 211–214 (1978).
[CrossRef]

1971 (1)

1967 (1)

1966 (1)

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

1939 (1)

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

1920 (1)

C. Cuthbertson, M. Cuthbertson, “On the refraction and dispersion of carbon dioxide, carbon monoxide, and methane,” Proc. R. Soc. London A 97, 152–159 (1920).
[CrossRef]

Achtermann, H. J.

H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
[CrossRef]

Barrell, H.

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Birch, K. P.

K. P. Birch, M. J. Downs, “Correction to the updated Edlen equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

K. P. Birch, M. J. Downs, “The precise determination of the refractive index of air,” Rep. MOM90, (National Physical Laboratory, Teddington, Middlesex, TW11 0LW, U.K., 1988), pp. 1–35.

K. P. Birch, 36 Fircroft Road, Chessington, Surrey, KT9 1RW, U.K. (personal communication, 2000).

Bose, T. K.

H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
[CrossRef]

Burns, R. C.

R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
[CrossRef]

Ciddor, P. E.

P. E. Ciddor, R. J. Hill, “Refractive index of air: 2. Group index,” Appl. Opt. 38, 1663–1667 (1999).
[CrossRef]

P. E. Ciddor, “Refractive index of air: new equations for the visible and the near infrared,” Appl. Opt. 35, 1566–1573 (1996).
[CrossRef] [PubMed]

R. J. Hill, “Infrared refractive index software, IR_N, 2000,” Environmental Research Laboratories, NOAA, 325 Broadway, Boulder, Colorado 80303-3328 (personal communication, May2000). (A limited number of copies of this software are available on compact disk from P. E. Ciddor or from J.M.Rueger@unsw.edu.au).

Coulon, R.

G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
[CrossRef]

Cuthbertson, C.

C. Cuthbertson, M. Cuthbertson, “On the refraction and dispersion of carbon dioxide, carbon monoxide, and methane,” Proc. R. Soc. London A 97, 152–159 (1920).
[CrossRef]

Cuthbertson, M.

C. Cuthbertson, M. Cuthbertson, “On the refraction and dispersion of carbon dioxide, carbon monoxide, and methane,” Proc. R. Soc. London A 97, 152–159 (1920).
[CrossRef]

Davis, R. S.

R. S. Davis, “Equation for the determination of the density of moist air (1981/1991),” Metrologia 29, 67–70 (1992).
[CrossRef]

Downs, M. J.

K. P. Birch, M. J. Downs, “Correction to the updated Edlen equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

K. P. Birch, M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

K. P. Birch, M. J. Downs, “The precise determination of the refractive index of air,” Rep. MOM90, (National Physical Laboratory, Teddington, Middlesex, TW11 0LW, U.K., 1988), pp. 1–35.

Edlen, B.

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

Fen, K.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Gentili, K. L.

Giacomo, P.

P. Giacomo, “Equation for the determination of the density of moist air (1981),” Metrologia 18, 33–40 (1982).
[CrossRef]

Giardini, W. J.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Graham, C.

R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
[CrossRef]

Hill, R. J.

P. E. Ciddor, R. J. Hill, “Refractive index of air: 2. Group index,” Appl. Opt. 38, 1663–1667 (1999).
[CrossRef]

R. J. Hill, “Refractive index of atmospheric gases,” in The Upper Atmosphere, W. Diemenger, G. Hartmann, R. Leitinger, eds. (Springer, Berlin, 1995), pp. 261–270.

R. J. Hill, “Infrared refractive index software, IR_N, 2000,” Environmental Research Laboratories, NOAA, 325 Broadway, Boulder, Colorado 80303-3328 (personal communication, May2000). (A limited number of copies of this software are available on compact disk from P. E. Ciddor or from J.M.Rueger@unsw.edu.au).

Kenny, M. J.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Leistner, A. J.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Magnus, G.

H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
[CrossRef]

Montixi, G.

G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
[CrossRef]

Occelli, R.

G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
[CrossRef]

Old, J. G.

Owens, J. C.

Peck, E. R.

Sears, J. E.

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Simmons, A. C.

A. C. Simmons, “The refractive index and Lorentz-Lorenz functions of propane, nitrogen and carbon-dioxide in the spectral range 15803–22002 cm-1 and at 944 cm-1,” Opt. Commun. 25, 211–214 (1978).
[CrossRef]

Walsh, C. J.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Ward, B. R.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Weeler, A. R. M.

R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
[CrossRef]

Wielunski, L. S.

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

Appl. Opt. (3)

Can. J. Phys. (1)

G. Montixi, R. Coulon, R. Occelli, “Coefficients du viriel de la réfractivité de l’azote à 25 °C,” Can. J. Phys. 61, 473–479 (1983).
[CrossRef]

J. Chem. Phys. (1)

H. J. Achtermann, G. Magnus, T. K. Bose, “Refractivity virial coefficients of gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar,” J. Chem. Phys. 94, 5669–5684 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Metrologia (5)

P. Giacomo, “Equation for the determination of the density of moist air (1981),” Metrologia 18, 33–40 (1982).
[CrossRef]

R. S. Davis, “Equation for the determination of the density of moist air (1981/1991),” Metrologia 29, 67–70 (1992).
[CrossRef]

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

K. P. Birch, M. J. Downs, “An updated Edlen equation for the refractive index of air,” Metrologia 30, 155–162 (1993).
[CrossRef]

K. P. Birch, M. J. Downs, “Correction to the updated Edlen equation for the refractive index of air,” Metrologia 31, 315–316 (1994).
[CrossRef]

Mol. Phys. (1)

R. C. Burns, C. Graham, A. R. M. Weeler, “Direct measurement and calculation of the second refractivity virial coefficients of gases,” Mol. Phys. 59, 41–64 (1986).
[CrossRef]

Opt. Commun. (1)

A. C. Simmons, “The refractive index and Lorentz-Lorenz functions of propane, nitrogen and carbon-dioxide in the spectral range 15803–22002 cm-1 and at 944 cm-1,” Opt. Commun. 25, 211–214 (1978).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

H. Barrell, J. E. Sears, “The refraction and dispersion of air for the visible spectrum,” Philos. Trans. R. Soc. London Ser. A 238, 1–64 (1939).
[CrossRef]

Proc. R. Soc. London A (1)

C. Cuthbertson, M. Cuthbertson, “On the refraction and dispersion of carbon dioxide, carbon monoxide, and methane,” Proc. R. Soc. London A 97, 152–159 (1920).
[CrossRef]

Other (9)

See, for example, R. Carmichael, “A table of the standard atmosphere to 65,000 feet,” (Public Domain Aeronautical Software, Santa Cruz, Calif., 12June2001), http://www.pdas.com/m1.htm .

I. G. Enting, T. M. L. Wigley, M. Heimann, “Future emissions and concentrations of carbon dioxide: key ocean/atmosphere/land analyses,” Tech. Paper 31 (CSIRO Division of Atmospheric Research, Melbourne, Australia, 1994), p. 120. On-line edition available at http://www.dar.csiro.au , (search for publications\enting_2001a0.pdf ).

R. J. Hill, “Infrared refractive index software, IR_N, 2000,” Environmental Research Laboratories, NOAA, 325 Broadway, Boulder, Colorado 80303-3328 (personal communication, May2000). (A limited number of copies of this software are available on compact disk from P. E. Ciddor or from J.M.Rueger@unsw.edu.au).

Association Internationale de Géodesie, Handbook of Geodesy (on-line edition, 2000),” http://www.gfy.ku.dk/∼iag/HB2000/part2/iag_res.htm .

J. A. Stone, J. H. Zimmerman, “Index of refraction of air,” (National Institute of Science and Technology, Gaithersburg, Maryland, 12June2001), http://patapsco.nist.gov/mel/div821 .

M. J. Kenny, C. J. Walsh, A. J. Leistner, K. Fen, W. J. Giardini, L. S. Wielunski, B. R. Ward, “Determination of the Avogadro constant from precision density measurements on a silicon sphere,” in Proceedings of the Conference on Precision Electromagnetic Measurements, J. Hunter, L. Johnson, eds. (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 2000), pp. 184–185.

R. J. Hill, “Refractive index of atmospheric gases,” in The Upper Atmosphere, W. Diemenger, G. Hartmann, R. Leitinger, eds. (Springer, Berlin, 1995), pp. 261–270.

K. P. Birch, M. J. Downs, “The precise determination of the refractive index of air,” Rep. MOM90, (National Physical Laboratory, Teddington, Middlesex, TW11 0LW, U.K., 1988), pp. 1–35.

K. P. Birch, 36 Fircroft Road, Chessington, Surrey, KT9 1RW, U.K. (personal communication, 2000).

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Figures (2)

Fig. 1
Fig. 1

Dispersion curves for CO2 at a pressure of 101,325 Pa and a temperature of 0 °C according to Old et al.5 (solid curve) and Edlen3 (broken curve). The quantity plotted is 108(n - 1) versus wavelength. Both equations have been extrapolated to cover the same wavelength range.

Fig. 2
Fig. 2

Difference (δn) between the equations of Old et al. and Edlen for the refractivity of CO2 at a pressure of 45 Pa and a temperature of 0 °C. The quantity plotted is 108δn versus wavelength, where δn = n Edlen - n Old. Both equations have been extrapolated to cover the same wavelength range.

Tables (2)

Tables Icon

Table 1 Air Density and Phase and Group Refractivities at 450 ppm of CO2 a

Tables Icon

Table 2 Experimental Limits to Determination of Phase Refractive Indexa

Equations (12)

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

ρ=pM/ZRT,
ρ=p/ZRTxdMd+xwMw+xcMc.
ρi=p/ZRTxiMi,
108nEDLEN-1=22822.1+117.8σ2+2406030130-σ2+1599738.9-σ2,
108nOLD-1=154.4890.0584738-σ2+8309192.7210.92417-σ2+287641.9060.122959-σ2,
L=i DiLi,
Li=ni2-1ni2+22/3ni-11-ni-1/6.
n=1+2L/1-L1/21+3/2L+3/8L2.
ng=n+σn2+22/nini/ni2+22×Didni/dσ n+σ/4-ni4-ni×Didni/dσ,
Ln2-1/n2+2=ARρ+BRρ2+Oρ3ARρ1+BRρ/AR
L*AR*ρ,
AR*L0/ρ0AR1+BRρ0/AR.

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