A method is presented, based on recognition of the Cauchy dispersion equation as a series of Stieltjes, for the bounded extrapolation of long-wavelength refractivity data into the far ultraviolet. The extrapolation is performed with the [n, n−1] and [n,n] Padé approximants, which improve the convergence of the Cauchy equation and provide upper and lower bounds to its exact sum. Illustrative applications are given for atomic hydrogen, the inert gases, and molecular hydrogen, oxygen, and nitrogen. Comparison with the exact dispersion curve for atomic hydrogen and with available theoretical and experimental ultraviolet-dispersion data for the inert and diatomic gases indicates that the Padé approximants converge rapidly to accurate refractivity values. In addition, recognition of the Cauchy equation as a series of Stieltjes provides nontrivial constraints on the Cauchy coefficients; these afford a test of their accuracy and allow estimates of higher-order coefficients from measured refractivity data.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Coefficients appearing in the dispersion equation, at NTP (0°C, 1 atm), (n−1)=2πN0(α0+α1ω2+α2ω4+α3ω6), with ω measured in atomic units and 2πN0=0.2502×10−4.
For atomic hydrogen, the Cauchy coefficients are taken directly from the theoretical calculations of Ref. 15.
Table II
Coefficients for the [2,2]f Padé approximants to atomic and molecular refractivities.a
Atom or molecule
ωa
ωb
fa
fb
H
0.3797
0.6166
0.4940
0.4083
He
0.8227
1.485
0.6097
1.065
Ne
0.6401
1.672
0.3755
4.896
Ar
0.4808
0.8899
1.069
5.123
Kr
0.4226
0.7272
1.150
5.448
Xe
0.3265
0.9318
1.098
14.80
H2
0.4736
0.7441
0.8627
0.8822
N2
0.4929
0.8531
1.244
4.820
O2
0.5121
0.6200
2.033
1.095
Coefficients appearing in the approximant, at NTP (0°C, 1 atm), (n−1)≥2πN0[2,2]f=2πN0[2,1]α=2πN0[fa/(ωa2–ω2)+fb/(ωb2−ω2)], with ω measured in atomic units and 2πN0=0.2502×10−4. See Eqs. (12b) and (16a).
Table III
Predicted refractivities [(n − 1) × 104] for the inert gases.a
All refractivity values quoted are at NTP (0°C, 1 atm).
Values shown are from the Padé approximants as discussed in the text.
A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964). See also, G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).
Table IV
Comparison of measured and predicted argon refractivity [(n − 1) × 104] in the ultraviolet.a
All refractivity values quoted are at NTP (0°C, 1 atm).
The uncertainties on the wavelength values correspond to the range used in the measurements.
D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963); P. Gill and D. W. O. Heddle, J. Opt. Soc. Am. 53, 847 (1963).
A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964).
F. Marmo, as listed in Liggett and Levinger, Ref. 4.
G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).
Table V
Comparison of measured and predicted molecular nitrogen refractivity [(n − 1) × 104] in the vuv.a
All refractivity values quoted are at NTP (0°C, 1 atm).
P. G. Wilkinson, J. Opt. Soc. Am. 50, 1002 (1960).
For previous prediction of molecular nitrogen refractivity in the uv using absorption data and the Kramers–Heisenberg dispersion formula, see A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967).
Table VI
Cauchy coefficients for atomic and molecular hydrogen obtained from constrained nonlinear least-squares fit to dispersion data.a
Dispersion data for atomic hydrogen taken from the exact theoretical refractivity curve of Ref. 7 in the frequency interval 0.01 ≤ ω ≤ 0.15 (15 points, accurate to eight significant figures); data for molecular hydrogen taken from the experimental measurements of M. Kirn, Ann. Physik 64, 566 (1921) (15 points, accurate to six significant figures). All values in the table are in atomic units.
Refers to the root-mean-square deviation of the Cauchy equation of N terms from the dispersion data. That the rms deviations decrease relatively slowly with increase in the number of terms N is a direct consequence of the constraints, or restrictions, of Eqs. (5), (28), and (A2)–(A7), as discussed in the Appendix.
Tables (6)
Table I
Cauchy coefficients obtained from experimental refractivity data and Stieltjes constraints.a
Coefficients appearing in the dispersion equation, at NTP (0°C, 1 atm), (n−1)=2πN0(α0+α1ω2+α2ω4+α3ω6), with ω measured in atomic units and 2πN0=0.2502×10−4.
For atomic hydrogen, the Cauchy coefficients are taken directly from the theoretical calculations of Ref. 15.
Table II
Coefficients for the [2,2]f Padé approximants to atomic and molecular refractivities.a
Atom or molecule
ωa
ωb
fa
fb
H
0.3797
0.6166
0.4940
0.4083
He
0.8227
1.485
0.6097
1.065
Ne
0.6401
1.672
0.3755
4.896
Ar
0.4808
0.8899
1.069
5.123
Kr
0.4226
0.7272
1.150
5.448
Xe
0.3265
0.9318
1.098
14.80
H2
0.4736
0.7441
0.8627
0.8822
N2
0.4929
0.8531
1.244
4.820
O2
0.5121
0.6200
2.033
1.095
Coefficients appearing in the approximant, at NTP (0°C, 1 atm), (n−1)≥2πN0[2,2]f=2πN0[2,1]α=2πN0[fa/(ωa2–ω2)+fb/(ωb2−ω2)], with ω measured in atomic units and 2πN0=0.2502×10−4. See Eqs. (12b) and (16a).
Table III
Predicted refractivities [(n − 1) × 104] for the inert gases.a
All refractivity values quoted are at NTP (0°C, 1 atm).
Values shown are from the Padé approximants as discussed in the text.
A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964). See also, G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).
Table IV
Comparison of measured and predicted argon refractivity [(n − 1) × 104] in the ultraviolet.a
All refractivity values quoted are at NTP (0°C, 1 atm).
The uncertainties on the wavelength values correspond to the range used in the measurements.
D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963); P. Gill and D. W. O. Heddle, J. Opt. Soc. Am. 53, 847 (1963).
A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964).
F. Marmo, as listed in Liggett and Levinger, Ref. 4.
G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).
Table V
Comparison of measured and predicted molecular nitrogen refractivity [(n − 1) × 104] in the vuv.a
All refractivity values quoted are at NTP (0°C, 1 atm).
P. G. Wilkinson, J. Opt. Soc. Am. 50, 1002 (1960).
For previous prediction of molecular nitrogen refractivity in the uv using absorption data and the Kramers–Heisenberg dispersion formula, see A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967).
Table VI
Cauchy coefficients for atomic and molecular hydrogen obtained from constrained nonlinear least-squares fit to dispersion data.a
Dispersion data for atomic hydrogen taken from the exact theoretical refractivity curve of Ref. 7 in the frequency interval 0.01 ≤ ω ≤ 0.15 (15 points, accurate to eight significant figures); data for molecular hydrogen taken from the experimental measurements of M. Kirn, Ann. Physik 64, 566 (1921) (15 points, accurate to six significant figures). All values in the table are in atomic units.
Refers to the root-mean-square deviation of the Cauchy equation of N terms from the dispersion data. That the rms deviations decrease relatively slowly with increase in the number of terms N is a direct consequence of the constraints, or restrictions, of Eqs. (5), (28), and (A2)–(A7), as discussed in the Appendix.
Add to BibSonomy