Abstract

We modeled the index of refraction of water at a temperature of 25 °C, employing a Lorentz model for wavelengths ranging from 200 nm to 200 μm. We determined model parameters by minimizing discrepancies between calculated and experimental data, using an elite genetic algorithm with adaptive mutations. We found that a Lorentz model with six oscillators fits the available data well in the whole range of interest.

© 1998 Optical Society of America

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References

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  1. C. Wohlfarth, B. Wohlfarth, in Optical Constants, Vol. 38, Subvol. A of Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Condensed Matter, M. D. Lechner , ed. (Springer-Verlag, Berlin, 1996), pp. 17–22.
  2. C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
    [CrossRef]
  3. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  4. X. Quan, E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. 34, 3477–3480 (1995).
    [CrossRef] [PubMed]
  5. P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
    [CrossRef]
  6. P. D. T. Huibers, “Models for wavelength dependence of the index of refraction of water,” Appl. Opt. 36, 3785–3787 (1997).
    [CrossRef] [PubMed]
  7. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).
  8. J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
    [CrossRef]
  9. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  10. K. P. Wong, Y. W. Wong, “Genetic and genetic/simulated annealing approaches to economic dispatch,” Proc. Gener. Trans. Distrib. 141(5), 507–513 (1994).
    [CrossRef]
  11. R. Vemuri, R. Vemuri, “Genetic algorithm for MCM partitioning,” Electron. Lett. 30, 1270–1272 (1994).
    [CrossRef]
  12. T. Bäck, H.-P. Schwefel, Genetic Algorithms in Engineering and Computer Science, G. Winter, J. Periaux, M. Galan, P. Cuesta, eds. (Wiley, New York, 1995), pp. 111–140.
  13. A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
    [CrossRef]

1997 (2)

P. D. T. Huibers, “Models for wavelength dependence of the index of refraction of water,” Appl. Opt. 36, 3785–3787 (1997).
[CrossRef] [PubMed]

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

1995 (1)

1994 (2)

K. P. Wong, Y. W. Wong, “Genetic and genetic/simulated annealing approaches to economic dispatch,” Proc. Gener. Trans. Distrib. 141(5), 507–513 (1994).
[CrossRef]

R. Vemuri, R. Vemuri, “Genetic algorithm for MCM partitioning,” Electron. Lett. 30, 1270–1272 (1994).
[CrossRef]

1990 (1)

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

1974 (1)

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

1973 (2)

C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
[CrossRef]

G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

Bäck, T.

T. Bäck, H.-P. Schwefel, Genetic Algorithms in Engineering and Computer Science, G. Winter, J. Periaux, M. Galan, P. Cuesta, eds. (Wiley, New York, 1995), pp. 111–140.

Birkhoff, R. D.

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

Curnette, B.

C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
[CrossRef]

Djurišic, A. B.

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

Elazar, J. M.

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

Fry, E. S.

Gallagher, Q. J. S.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Hale, G. M.

Hamm, R. N.

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

Heller, J. M.

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

Huibers, P. D. T.

Levelt Sengers, J. M. H.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Painter, L. R.

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

Quan, X.

Querry, M. R.

Rakic, A. D.

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

Robertson, C. W.

C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
[CrossRef]

Schiebener, P.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Schwefel, H.-P.

T. Bäck, H.-P. Schwefel, Genetic Algorithms in Engineering and Computer Science, G. Winter, J. Periaux, M. Galan, P. Cuesta, eds. (Wiley, New York, 1995), pp. 111–140.

Straub, J.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Vemuri, R.

R. Vemuri, R. Vemuri, “Genetic algorithm for MCM partitioning,” Electron. Lett. 30, 1270–1272 (1994).
[CrossRef]

R. Vemuri, R. Vemuri, “Genetic algorithm for MCM partitioning,” Electron. Lett. 30, 1270–1272 (1994).
[CrossRef]

Williams, D.

C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
[CrossRef]

Wohlfarth, B.

C. Wohlfarth, B. Wohlfarth, in Optical Constants, Vol. 38, Subvol. A of Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Condensed Matter, M. D. Lechner , ed. (Springer-Verlag, Berlin, 1996), pp. 17–22.

Wohlfarth, C.

C. Wohlfarth, B. Wohlfarth, in Optical Constants, Vol. 38, Subvol. A of Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Condensed Matter, M. D. Lechner , ed. (Springer-Verlag, Berlin, 1996), pp. 17–22.

Wong, K. P.

K. P. Wong, Y. W. Wong, “Genetic and genetic/simulated annealing approaches to economic dispatch,” Proc. Gener. Trans. Distrib. 141(5), 507–513 (1994).
[CrossRef]

Wong, Y. W.

K. P. Wong, Y. W. Wong, “Genetic and genetic/simulated annealing approaches to economic dispatch,” Proc. Gener. Trans. Distrib. 141(5), 507–513 (1994).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (1)

R. Vemuri, R. Vemuri, “Genetic algorithm for MCM partitioning,” Electron. Lett. 30, 1270–1272 (1994).
[CrossRef]

J. Chem. Phys. (1)

J. M. Heller, R. N. Hamm, R. D. Birkhoff, L. R. Painter, “Collective oscillation in liquid water,” J. Chem. Phys. 60, 3483–3486 (1974).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, Q. J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Mol. Phys. (1)

C. W. Robertson, B. Curnette, D. Williams, “The infra-red spectrum of water,” Mol. Phys. 26, 183–191 (1973).
[CrossRef]

Opt. Commun. (1)

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Modeling the optical constants of solids using genetic algorithms with parameter space size adjustment,” Opt. Commun. 134, 407–414 (1997).
[CrossRef]

Proc. Gener. Trans. Distrib. (1)

K. P. Wong, Y. W. Wong, “Genetic and genetic/simulated annealing approaches to economic dispatch,” Proc. Gener. Trans. Distrib. 141(5), 507–513 (1994).
[CrossRef]

Other (4)

T. Bäck, H.-P. Schwefel, Genetic Algorithms in Engineering and Computer Science, G. Winter, J. Periaux, M. Galan, P. Cuesta, eds. (Wiley, New York, 1995), pp. 111–140.

C. Wohlfarth, B. Wohlfarth, in Optical Constants, Vol. 38, Subvol. A of Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Group III: Condensed Matter, M. D. Lechner , ed. (Springer-Verlag, Berlin, 1996), pp. 17–22.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

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Figures (3)

Fig. 1
Fig. 1

Variation of the index of refraction of water around λ = 2 μm (h ν ≈ 0.4 eV).

Fig. 2
Fig. 2

Index of refraction of water as a function of wavelength at t = 25 °C. The results of the Lorentz model were obtained with six oscillators.

Fig. 3
Fig. 3

Difference between calculated and experimental data of Hale and Querry.3

Tables (1)

Tables Icon

Table 1 Values of the Model Parameters

Equations (6)

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r ω = 1 - j = 1 k f j ω p 2 ω 2 - ω j 2 + i ω Γ j = r 1 + i r 2 ,
n = 1 2 r 1 2 + r 2 2 + r 1 1 / 2 .
p k = p 1 k + p u k - p 1 k r ,
F i = f i i = 1 N   f i ,
p new - u k = p old - u k - c p old - u k - μ ˆ k ,
p new - 1 k = p old - 1 k + c μ ˆ k - p old - 1 k ,

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