Abstract

There are two parameters in the one-term Sellmeier formula for refractive index. Two different ways are proposed to express them in terms of the number of electrons, the dielectric constant, and the first excitation energy of the molecule. In the normal-dispersion region, one construction will make the formula an upper bound and the other, a lower bound. Furthermore, if a measured value at a particular frequency is used for the determination of the parameters, then the Sellmeier formula is an upper bound for frequencies below that reference frequency and a lower bound for frequencies above it. The various bounding properties are illustrated for the case of atomic hydrogen.

© 1972 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.
  2. H. A. Kramers and W. Heisenberg, Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, (McGraw–Hill, New York, 1960), p. 154.
    [Crossref]
  3. W. Sellmeier, Pogg. Ann. Physik 143, 271 (1871); Pogg. Ann. Physik 145, 399, 520 (1872); Pogg. Ann. Physik 147, 386, 525 (1872).
  4. See, for example, S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
    [Crossref]
  5. C. R. Mansfield and E. R. Peck, J. Opt. Soc. Am. 59, 199 (1969).
    [Crossref]
  6. See, for example, J. M. Stone, Radiation and Optics (McGraw–Hill, New York, 1963), p. 377.
  7. A. Maryott and F. Burkley, Tables of Dielectric and Electric Dipole Moments of Substances in the Gaseous State, Natl. Bur. Std. (U. S.) Circ. 537 (U. S. Government Printing Office, Washington, D. C., 1953).
  8. K. T. Tang, Phys. Rev. Letters 23, 1271 (1969).
    [Crossref]
  9. W. Kuhn, Z. Physik 33, 408 (1925); W. Thomas, Naturwiss. 13, 627 (1925); F. Reiche and W. Thomas, Z. Physik 34, 510 (1925).
    [Crossref]
  10. K. T. Tang, Phys. Rev. 177, 108 (1969).
    [Crossref]
  11. M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
    [Crossref]

1969 (3)

K. T. Tang, Phys. Rev. Letters 23, 1271 (1969).
[Crossref]

K. T. Tang, Phys. Rev. 177, 108 (1969).
[Crossref]

C. R. Mansfield and E. R. Peck, J. Opt. Soc. Am. 59, 199 (1969).
[Crossref]

1963 (1)

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
[Crossref]

1932 (1)

See, for example, S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
[Crossref]

1925 (2)

W. Kuhn, Z. Physik 33, 408 (1925); W. Thomas, Naturwiss. 13, 627 (1925); F. Reiche and W. Thomas, Z. Physik 34, 510 (1925).
[Crossref]

H. A. Kramers and W. Heisenberg, Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

1871 (1)

W. Sellmeier, Pogg. Ann. Physik 143, 271 (1871); Pogg. Ann. Physik 145, 399, 520 (1872); Pogg. Ann. Physik 147, 386, 525 (1872).

Breit, G.

See, for example, S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
[Crossref]

Burkley, F.

A. Maryott and F. Burkley, Tables of Dielectric and Electric Dipole Moments of Substances in the Gaseous State, Natl. Bur. Std. (U. S.) Circ. 537 (U. S. Government Printing Office, Washington, D. C., 1953).

Heisenberg, W.

H. A. Kramers and W. Heisenberg, Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

Karplus, M.

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
[Crossref]

Kolker, H. J.

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
[Crossref]

Korff, S. A.

See, for example, S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
[Crossref]

Kramers, H. A.

H. A. Kramers and W. Heisenberg, Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

Kuhn, W.

W. Kuhn, Z. Physik 33, 408 (1925); W. Thomas, Naturwiss. 13, 627 (1925); F. Reiche and W. Thomas, Z. Physik 34, 510 (1925).
[Crossref]

Lorentz, H. A.

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.

Mansfield, C. R.

Maryott, A.

A. Maryott and F. Burkley, Tables of Dielectric and Electric Dipole Moments of Substances in the Gaseous State, Natl. Bur. Std. (U. S.) Circ. 537 (U. S. Government Printing Office, Washington, D. C., 1953).

Peck, E. R.

Sellmeier, W.

W. Sellmeier, Pogg. Ann. Physik 143, 271 (1871); Pogg. Ann. Physik 145, 399, 520 (1872); Pogg. Ann. Physik 147, 386, 525 (1872).

Stone, J. M.

See, for example, J. M. Stone, Radiation and Optics (McGraw–Hill, New York, 1963), p. 377.

Tang, K. T.

K. T. Tang, Phys. Rev. 177, 108 (1969).
[Crossref]

K. T. Tang, Phys. Rev. Letters 23, 1271 (1969).
[Crossref]

J. Chem. Phys. (1)

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

K. T. Tang, Phys. Rev. 177, 108 (1969).
[Crossref]

Phys. Rev. Letters (1)

K. T. Tang, Phys. Rev. Letters 23, 1271 (1969).
[Crossref]

Physik (1)

H. A. Kramers and W. Heisenberg, Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

Pogg. Ann. Physik (1)

W. Sellmeier, Pogg. Ann. Physik 143, 271 (1871); Pogg. Ann. Physik 145, 399, 520 (1872); Pogg. Ann. Physik 147, 386, 525 (1872).

Rev. Mod. Phys. (1)

See, for example, S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
[Crossref]

Z. Physik (1)

W. Kuhn, Z. Physik 33, 408 (1925); W. Thomas, Naturwiss. 13, 627 (1925); F. Reiche and W. Thomas, Z. Physik 34, 510 (1925).
[Crossref]

Other (3)

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.

See, for example, J. M. Stone, Radiation and Optics (McGraw–Hill, New York, 1963), p. 377.

A. Maryott and F. Burkley, Tables of Dielectric and Electric Dipole Moments of Substances in the Gaseous State, Natl. Bur. Std. (U. S.) Circ. 537 (U. S. Government Printing Office, Washington, D. C., 1953).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Refractive index of atomic hydrogen at NTP (0°C, 1 atm). ——, Exact value taken from Ref. 11; — · — · —, upper bound calculated from the single-term Sellmeier formula with parameters chosen according to Eq. (36); — · · —, lower bound calculated with the same formula but with parameters chosen according to Eq. (37); – – – –, average of upper and lower bounds.

Fig. 2
Fig. 2

Upper part: Refractive index of atomic hydrogen at NTP (0°C, 1 atm). – – – –, Approximate value calculated from the single-term Sellmeier formula with parameters chosen according to Eq. (38). The reference frequency is chosen at 0.182 a.u. (λ = 2500 Å). Lower part: The difference between the exact and the approximate value of the refractive index. Below the reference frequency, the approximation is an upper bound; above the reference frequency, the approximation is a lower bound.

Fig. 3
Fig. 3

Refractive index of atomic hydrogen. ——, Upper and lower bounds calculated with the additional information of the first two oscillator strengths; ······, exact value and approximate value with the refractivity at 0.182 a.u. incorporated in the calculation.

Equations (55)

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

( n 2 - 1 ) / ( n 2 + 2 ) = ( 4 π / 3 ) N 0 α ( ω ) ,
n - 1 = 2 π N 0 α ( ω ) .
α ( ω ) = n f n ω n 2 - ω 2 ,
n - 1 = i = 1 m c i η i 2 - ω 2 ,
n - 1 = c / ( η 2 - ω 2 ) .
α a ( ω ) = α 0 η 2 / ( η 2 - ω 2 ) ,
α 0 = ( - 1 ) / 4 π N 0 .
g ( ω ) = α a ( ω ) - α ( ω ) = α 0 η 2 η 2 - ω 2 - n f n ω n 2 - ω 2 .
α 0 = α ( 0 ) = n f n ω n 2
g ( ω ) = n η 2 f n ( η 2 - ω 2 ) ω n 2 - n f n ω n 2 - ω 2 = ω 2 η 2 - ω 2 n ω n 2 - η 2 ω n 2 ( ω n 2 - ω 2 ) f n .
η = ω 1 ,
g ( ω ) = ω 2 ω 1 2 - ω 2 n ω n 2 - ω 1 2 ω n 2 ( ω n 2 - ω 2 ) f n .
ω i ω 1
g ( ω ) > 0.
α a ( ω ) = α 0 ω 1 2 / ( ω 1 2 - ω 2 )
η 2 = N / α 0
N = n f n ,
g ( ω ) = A ( ω ) B ( ω ) ,
A ( ω ) = ω 2 η 2 - ω 2 = ω 2 α 0 [ l f l ( ω l 2 - ω 2 ) ω l 2 ] - 1
B ( ω ) = n = 1 ( ω n 2 - η 2 ) f n ω n 2 ( ω n 2 - ω 2 ) = 1 α 0 n = 1 i = 1 ( ω n 2 - ω i 2 ) f i f n ω n 2 ω i 2 ( ω n 2 - ω 2 ) = 1 α 0 | n > i i ( ω n 2 - ω i 2 ) f i f n ω n 2 ω i 2 ( ω n 2 - ω 2 ) + n > i i ( ω i 2 - ω n 2 ) f i f n ω i 2 ω n 2 ( ω i 2 - ω 2 ) | = - 1 α 0 n > i i ( ω n 2 - ω i 2 ) 2 f i f n ω n 2 ω i 2 ( ω n 2 - ω 2 ) ( ω i 2 - ω 2 ) .
A ( ω ) > 0
B ( ω ) < 0.
g ( ω ) < 0.
α a ( ω ) = N / [ ( N / α 0 ) - ω 2 ]
α ( ω 0 ) = α 0 η 2 / ( η 2 - ω 0 2 )
η 2 = [ α ( ω 0 ) · ω 0 2 ] / [ α ( ω 0 ) - α 0 ] .
α a ( ω 0 ) = α ( ω 0 ) ,
g ( ω 0 ) = ω 0 2 η 2 - ω 0 2 n = 1 ( ω n 2 - η 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) = 0 .
n ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) = 0 .
η - ω k > 0 ;             η - ω k + 1 0.
n = 1 k ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) = m = k + 1 η 2 - ω m 2 f m ω m 2 ( ω m 2 - ω 0 2 ) .
d g ( ω ) d ω | ω = ω 0 = - 2 ω 0 3 η 2 - ω 0 2 [ n = 1 k ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) 2 - m = k + 1 η 2 - ω m 2 f m ω m 2 ( ω m 2 - ω 0 2 ) 2 ] .
n = 1 k ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) 2 > 1 ω k 2 - ω 0 2 n = 1 k ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) .
m = k + 1 η 2 - ω m 2 f m ω m 2 ( ω m 2 - ω 0 2 ) 2 < 1 ω k 2 - ω 0 2 m = k + 1 η 2 - ω m 2 f m ω m 2 ( ω m 2 - ω 0 2 ) .
n = 1 k ( η 2 - ω n 2 ) f n ω n 2 ( ω n 2 - ω 0 2 ) 2 > m = k + 1 η 2 - ω m 2 f m ω m 2 ( ω m 2 - ω 0 2 ) 2 .
ω 0 3 η 2 - ω 0 2 = ω 0 3 [ α ( ω 0 ) ω 0 2 α ( ω 0 ) - α 0 - ω 0 2 ] - 1 = ω 0 3 [ α 0 ω 0 2 α ( ω 0 ) - α 0 ] - 1
α ( ω 0 ) > α ( 0 ) = α 0 ,
ω 0 3 / ( η 2 - ω 0 2 ) > 0.
g ( ω ) > 0 for ω < ω 0 < 0 for ω > ω 0 .
g ( ω ) > 0 for ω < ω 0 < 0 for ω > ω 0 .
α a ( ω ) > α ( ω )             for             0 < ω < ω 0
α a ( ω ) < α ( ω )             for             ω 0 < ω < ω 1 .
c = 2 π N 0 α 0 ω 1 2 ,
η = ω 1 ,
c = 2 π N 0 N ,
η = ( N / α 0 ) 1 2 .
c = 2 π N 0 α 0 { [ α ( ω 0 ) · ω 0 2 ] / [ α ( ω 0 ) - α 0 ] } ,
η = { [ α ( ω 0 ) · ω 0 2 ] / [ α ( ω 0 ) - α 0 ] } 1 2 ,
α ( ω ) = i = 2 f i ω i 2 - ω 2 = α ( ω ) - f 1 ω 1 2 - ω 2
α 0 = α 0 - ( f 1 / ω 1 2 ) ,
N = N - f 1 ;
α a ( ω ) = f 1 / ( ω 1 2 - ω 2 ) + [ α 0 ω 2 2 / ( ω 2 2 - ω 2 ) ]
α a ( ω ) = f ω / ( ω 1 2 - ω 2 ) + { N / [ ( N / α 0 ) - ω 2 ] }
α a ( ω ) = [ f 1 / ( ω 1 2 - ω 2 ) ] + [ α 0 η 2 / ( η 2 - ω 2 ) ]
η 2 = [ α ( ω 0 ) ω 0 2 ] / [ α ( ω 0 ) - α 0 ] ,