Abstract

The theoretical background and present status of formulas for the refractive index of air are reviewed. In supplement to Edlén’s recently revised formula for relative refractivity, the density dependence of refractive index is reanalyzed. New formulas are presented for both phase and group refractive index which are more useful over a wide range of pressure, temperature, and composition than any presently available. The application of the new formulas to optical distance measuring is briefly discussed.

© 1967 Optical Society of America

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References

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  1. J. Terrien, Metrologia 1, 80 (1965).
    [CrossRef]
  2. K. E. Erickson, J. Opt. Soc. Am. 52, 781 (1962).
    [CrossRef]
  3. P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
    [CrossRef]
  4. B. Edlén, Metrologia 2, 71 (1966).
    [CrossRef]
  5. B. Edlén, J. Opt. Soc. Am. 43, 339 (1953).
    [CrossRef]
  6. H. Barrell, J. E. Sears, Phil. Trans. Roy. Soc. London A238, 1 (1939).
  7. C. J. F. Böttcher, Theory of Electric Polarisation (Elsevier Publishing Company, Amsterdam, 1952).
  8. J. Koch, Arkiv. Mat. Astr. Fys. 10, 1 (1914).
  9. J. Hilsenrath et al., Tables of Thermal Properties of Gases, NBS Circular 564 (November1955).
  10. K. E. Erickson, J. Opt. Soc. Am. 52, 777 (1962).
    [CrossRef]
  11. J. A. Goff, S. Gratch, Trans. Am. Soc. Heat. Vent. Eng. 52, 95 (1946).
  12. J. A. Goff, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 289.
  13. L. P. Harrison, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 3.
  14. Measurement of Humidity, National Physical Laboratory Notes on Applied Science, No. 4 (HMSO, London, 1955).
  15. R. W. Ditchburn, Light (Interscience Publishers, Inc., New York, 1958), p. 96.
  16. M. T. Prilenin, in Trudi Tsentralnogo Nauchno-Issledovatelskoga Instita Geodezii, Aerosemki i Kartografii, D. A. Slobodchikov, Ed. (Izdatelstvo Geodezicheskoy Literaturi, Mocow, 1957), No. 114, p. 127.

1966

B. Edlén, Metrologia 2, 71 (1966).
[CrossRef]

1965

J. Terrien, Metrologia 1, 80 (1965).
[CrossRef]

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

1962

1955

J. Hilsenrath et al., Tables of Thermal Properties of Gases, NBS Circular 564 (November1955).

1953

1946

J. A. Goff, S. Gratch, Trans. Am. Soc. Heat. Vent. Eng. 52, 95 (1946).

1939

H. Barrell, J. E. Sears, Phil. Trans. Roy. Soc. London A238, 1 (1939).

1914

J. Koch, Arkiv. Mat. Astr. Fys. 10, 1 (1914).

Barrell, H.

H. Barrell, J. E. Sears, Phil. Trans. Roy. Soc. London A238, 1 (1939).

Bender, P. L.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

Böttcher, C. J. F.

C. J. F. Böttcher, Theory of Electric Polarisation (Elsevier Publishing Company, Amsterdam, 1952).

Ditchburn, R. W.

R. W. Ditchburn, Light (Interscience Publishers, Inc., New York, 1958), p. 96.

Edlén, B.

Erickson, K. E.

Goff, J. A.

J. A. Goff, S. Gratch, Trans. Am. Soc. Heat. Vent. Eng. 52, 95 (1946).

J. A. Goff, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 289.

Gratch, S.

J. A. Goff, S. Gratch, Trans. Am. Soc. Heat. Vent. Eng. 52, 95 (1946).

Harrison, L. P.

L. P. Harrison, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 3.

Hilsenrath, J.

J. Hilsenrath et al., Tables of Thermal Properties of Gases, NBS Circular 564 (November1955).

Koch, J.

J. Koch, Arkiv. Mat. Astr. Fys. 10, 1 (1914).

Owens, J. C.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

Prilenin, M. T.

M. T. Prilenin, in Trudi Tsentralnogo Nauchno-Issledovatelskoga Instita Geodezii, Aerosemki i Kartografii, D. A. Slobodchikov, Ed. (Izdatelstvo Geodezicheskoy Literaturi, Mocow, 1957), No. 114, p. 127.

Sears, J. E.

H. Barrell, J. E. Sears, Phil. Trans. Roy. Soc. London A238, 1 (1939).

Terrien, J.

J. Terrien, Metrologia 1, 80 (1965).
[CrossRef]

Arkiv. Mat. Astr. Fys.

J. Koch, Arkiv. Mat. Astr. Fys. 10, 1 (1914).

J. Geophys. Res.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

J. Opt. Soc. Am.

Metrologia

J. Terrien, Metrologia 1, 80 (1965).
[CrossRef]

B. Edlén, Metrologia 2, 71 (1966).
[CrossRef]

Phil. Trans. Roy. Soc. London

H. Barrell, J. E. Sears, Phil. Trans. Roy. Soc. London A238, 1 (1939).

Tables of Thermal Properties of Gases

J. Hilsenrath et al., Tables of Thermal Properties of Gases, NBS Circular 564 (November1955).

Trans. Am. Soc. Heat. Vent. Eng.

J. A. Goff, S. Gratch, Trans. Am. Soc. Heat. Vent. Eng. 52, 95 (1946).

Other

J. A. Goff, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 289.

L. P. Harrison, in Humidity and Moisture, A. Wexler, Ed. (Reinhold Publishing Corporation, New York, 1965), Vol. 3, p. 3.

Measurement of Humidity, National Physical Laboratory Notes on Applied Science, No. 4 (HMSO, London, 1955).

R. W. Ditchburn, Light (Interscience Publishers, Inc., New York, 1958), p. 96.

M. T. Prilenin, in Trudi Tsentralnogo Nauchno-Issledovatelskoga Instita Geodezii, Aerosemki i Kartografii, D. A. Slobodchikov, Ed. (Izdatelstvo Geodezicheskoy Literaturi, Mocow, 1957), No. 114, p. 127.

C. J. F. Böttcher, Theory of Electric Polarisation (Elsevier Publishing Company, Amsterdam, 1952).

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Figures (2)

Fig. 1
Fig. 1

Mean temperature dependence of deviation from ideal gas behavior for dry air, 0.4–4.0 atm.

Fig. 2
Fig. 2

Pressure dependence of deviation from ideal gas behavior at various temperatures for dry air.

Tables (5)

Tables Icon

Table I Comparison of Density Formulas

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Table II Saturation Pressure and Deviations from Ideal Gas Behavior of Water Vapor

Tables Icon

Table III Summary of Formulas

Tables Icon

Table IV Comparison of Refractivity Formulas

Tables Icon

Table V Comparison of Group Refractivity Formulas

Equations (52)

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n 2 - 1 n 2 + 2 = i R i ρ i ,
R i = [ ( n i 2 - 1 ) / ( n i 2 + 2 ) ] ( 1 / ρ i )
= 4 3 π ( N A / M i ) α i ,
( n 2 - 1 ) / ( n 2 + 2 ) = R 1 ρ 1 + R 2 ρ 2 + R 3 ρ 3 ,
( n - 1 ) × 10 8 = 8340.78 + [ 2 , 405 , 640 / ( 130 - σ 2 ) ] + [ 15 , 994 / ( 38.9 - σ 2 ) ] ,
ρ = ( M P / R T ) ( 1 / Z ) ,
1 / Z = 1 + P β ( P ) γ ( T ) ,
δ ( P , T ) = [ ( 1 / Z ) - 1 ] / P = β ( P ) γ ( T )
ρ ( g / m 3 ) = 348.328 P T [ 1 + P ( 57.90 × 10 - 8 - 0.94581 × 10 - 3 T + 0.25844 T 2 ) ] ,
ρ = C ( P / T ) [ 1 + P β ( P ) γ ( T ) ]
Δ ρ = C P T Δ [ P β ( P ) γ ( T ) ] .
( Δ ρ / ρ ) = P Δ [ P β ( P ) γ ( T ) ] .
( n - 1 ) × 10 8 = 295.235 + 2.6422 σ 2 - 0.032380 σ 4 + 0.004028 σ 6 .
P V = R T - A w w P - A w w P 2 .
A w w = - 0.0302 + 88.514 τ 10 233280 τ 2 A w w w = 67.4 τ 2 A w w 3 .
Z = 1 - 13.7066 A w w ( P / T ) - ( 2.90063 × 10 4 ) A w w w ( P 2 / T ) ,
log 10 ( P s / P 0 ) = 10.79586 ( 1 - T 0 / T ) - 5.02808 log 10 ( T / T 0 ) + 1.50474 × 10 - 4 [ 1 - 10 - 8.29692 ( T / T 0 - 1 ) ] + 0.42873 × 10 - 3 [ 10 4.76955 ( 1 - T 0 / T ) - 1 ] - 2.2195983.
ρ ( g / m 3 ) = 216.582 ( P / T ) [ 1 + P β ( P ) γ ( T ) ] ,
β ( P ) = 1 + ( 3.7 × 10 - 4 ) P γ ( T ) = - 2.37321 × 10 - 3 + 2.23366 T - 710.792 T 2 + 7.75141 × 10 4 T 3 .
( n - 1 ) × 10 8 = 22 , 822.1 + 117.8 σ 2 + 2 , 406 , 030 ( 130 - σ 2 ) + 15 , 997 ( 38.9 - σ 2 ) .
ρ ( g / m 3 ) = 529.37 ( P / T ) .
( n 2 - 1 ) / ( n 2 + 2 ) = ( n - 1 ) [ 1 - ( n - 1 ) / 6 ] 2 3 .
n - 1 = i R i ρ i ,
R i = ( n i - 1 ) [ 1 - ( n i - 1 ) / 6 ] ( 1 / ρ i )
ρ i = ρ i / [ 1 - ( n - 1 ) / 6 ] .
ρ i = C i ( P i / T ) [ 1 + P i β i ( P i ) γ i ( T ) ] / ( 1 - n - 1 / 6 ) .
n - 1 R 1 C 1 ( P 1 / T ) ,
ρ 1 = C 1 ( P 1 / T ) [ 1 + P 1 ( β 1 γ 1 - D 1 / T ) ] ,
f 1 = [ 1 - n 1 - 1 0 / 6 ] ( ρ 0 ) 1 .
( n - 1 ) × 10 8 = [ 2371.34 + 683 , 939.7 ( 130 - σ 2 ) + 4547.3 ( 38.9 - σ 2 ) ] D s + [ 6487.31 + 58.058 σ 2 - 0.71150 σ 4 + 0.08851 σ 6 ] D w ,
D s = P s T [ 1 + P s ( 57.90 × 10 - 8 - 9.3250 × 10 - 4 T + 0.25844 T 2 ) ]
D w = P w T { 1 - P w [ 1 + ( 3.7 × 10 - 4 ) P w ] [ - 2.37321 × 10 - 3 + 2.23366 T - 710.792 T 2 + 7.75141 × 10 4 T 3 ] } .
( n - 1 ) × 10 8 = [ 8342.13 + 2 , 406 , 030 ( 130 - σ 2 ) + 15 , 997 ( 38.9 - σ 2 ) ] × ( p 720.775 ) [ 1 + p ( 0.817 - 0.0133 t ) × 10 - 6 1 + 0.0036610 t ] - f [ 5.722 - 0.0457 σ 2 ] ,
n G = ( c / U )
n G = n + σ ( d n / d σ ) .
X = i = 1 3 R i ρ i ,
d n / d σ = ( 3 Y / 2 ) [ ( 1 - X ) ( 1 - X ) ½ ( 2 X + 1 ) ½ ] - 1 ,
Y = ρ 1 ( d R 1 / d σ ) + ρ 2 ( d R 2 / d σ ) + ρ 3 ( d R 3 / d σ ) .
( n G - 1 ) = 3 X ( 2 X + 1 ) ½ ( 1 - X ) ½ + ( 1 - X ) + σ 3 Y 2 [ ( 1 - X ) ( 1 - X ) ½ ( 2 X + 1 ) ½ ] - 1 .
R i = [ ( ρ 0 ) i ] - 1 ( r i A i / B i ) ,
A i = r i - 2 B i = r i A i - 3.
( d R i / d σ ) = 6 ( d r i / d σ ) [ ( ρ 0 ) i R i B i - r i ] / ( ρ 0 ) i B i 2 r i .
d r 1 d σ × 10 8 = 2 σ [ 2 , 405 , 640 ( 130 - σ 2 ) + 15 , 994 ( 38.9 - σ 2 ) ] d r 2 d σ × 10 8 = 2 σ [ 2.6422 - 0.064760 σ 2 + 0.012084 σ 4 ] d r 3 d σ × 10 8 = 2 σ [ 117.8 + 2 , 406 , 030 ( 130 - σ 2 ) + 15 , 997 ( 38.9 - σ 2 ) ] .
( n G - 1 ) × 10 8 = [ 2371.34 + 683 , 939.7 ( 130 + σ 2 ) ( 130 - σ 2 ) 2 + 4547.3 ( 38.9 + σ 2 ) ( 38.9 - σ 2 ) 2 ] D s + [ 6487.31 + 174.174 σ 2 - 3.55750 σ 4 + 0.61957 σ 6 ] D w ,
( N 1 - N 2 N 1 ) air - ( N 1 - N 2 N 1 ) vacuum = λ 1 λ 2 ( 1 - n 1 n 2 ) ,
n 2 - 1 n 2 + 2 = i = 1 3 R i ρ i R i = ( n 0 ) i 2 - 1 ( n 0 ) i 2 + 2 1 ( ρ 0 ) i
r 1 × 10 8 = 8340.78 + 2 , 405 , 640 ( 130 - σ 2 ) + 15 , 994 ( 38.9 - σ 2 ) ρ 1 = 348.328 P 1 T [ 1 + P 1 ( 57.90 × 10 - 8 - 9.4581 × 10 - 4 T + 0.25844 T 2 ) ] ( ρ 0 ) 1 = ρ 1 ( P = 1013.25 mb , T = 288.16 ° K )
Range of validity : 2302 - 20 , 586 Å 240 ° K < T < 330 ° K 0 < P < 4 atm
r 2 × 10 8 = 295.235 + 2.6422 σ 2 - 0.032380 σ 4 + 0.004028 σ 6 ρ 2 = 216.582 P 2 T { 1 + P 2 [ 1 + ( 3.7 × 10 - 4 ) P 2 ] × [ - 2.37321 × 10 - 3 + 2.23366 T - 710.792 T 2 + 7.75141 × 10 4 T 3 ] } ( ρ 0 ) 2 = ρ 2 ( P = 13.33 mb , T = 293.16 ° K )
Range of validity : 3611 - 6440 Å 250 ° K < T < 320 ° K 0 < P < 100 mb
r 3 × 10 8 = 22 , 822.1 + 117.8 σ 2 + 2 , 406 , 030 ( 130 - σ 2 ) + 15 , 997 ( 38.9 - σ 2 ) ρ 3 = 529.37 P 3 T ( ρ 0 ) 3 = ρ 3 ( P = 1013.25 mb , T = 288.16 ° K )
Range of validity : 2379 - 6910 Å 240 ° K < T < 330 ° K 0 < P < 17 mb

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