Abstract

An experimental test system capable of measuring the refractive index variation of a salt-water solution at two wavelengths, with temporally varying temperature and concentration, is constructed by using a Mach–Zehnder interferometer. An experimental data-reduction method is developed to take the data generated in the experimental test program and to determine the variation of the refractive index with temperature and concentration. The experimental data obtained from the test system are reduced and the results are fitted by using a nonlinear regression algorithm to determine the functional dependence of the refractive index with temperature and concentration.

© 1993 Optical Society of America

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  1. A. Ecker, “Two-wavelength holographic measurement of temperature and concentration during alloy solidifications,” J. Thermophys. Heat Transfer 2, 193–196 (1988).
    [CrossRef]
  2. D. Beysens, P. Calmettes, “Temperature dependence of the refractive indices of liquids: deviation from the Lorentz–Lorenz formula,” J. Chem. Phys. 66, 766–771 (1977).
    [CrossRef]
  3. C. Böttcher, Theory of Electric Polarization (Elsevier, New York, 1952).
  4. C. G. Murphy, S. S. Alpert, “Dependence of refraction index temperature coefficient on the thermal expansivity of liquids,” Am. J. Phys. 39, 834–836 (1971).
    [CrossRef]
  5. L. W. Tilton, J. K. Taylor, “Refractive index and dispersion of distilled water for visible radiation, and temperatures of 0 to 60 °C,” J. Res. Natl. Bur. Stand. 20, 419–477 (1938).
    [CrossRef]
  6. H. M. Dobbins, E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. 63, 318–320 (1973).
    [CrossRef]
  7. D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
    [CrossRef]
  8. W. Hauf, U. Grigull, Optical Methods in Heat Transfer, Vol. 6 of Advances in Heat Transfer (Academic, New York, 1970).
  9. A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
    [CrossRef]
  10. V. P. Semenova, “Character of the temperature variations of the refractive index of benzene and seignette salt,” Opt. Spectrosc. (USSR) 41, 299–300 (1976).
  11. G. Schödel, “Kombinierte Wärmelestrahlung in Konvektions-freien Flussigheitsschichten,” Ph.D. dissertation (Institut A für Thermodynamik, Technische Hochschule München, München, Germany, 1969).
  12. J. A. Wasasjerna, “Lösnongars optiska egenskaper,” Acta. Soc. Sci. Fenn. Ser. A. 50, 1–63 (1920).
  13. J. M. Mehta, W. M. Worck, “Analysis of refractive errors for interferometric measurements in multicomponent systems,” Appl. Opt. 23, 928–933 (1984).
    [CrossRef] [PubMed]
  14. B. W. Grange, W. H. Stevenson, R. Viskanta, “Refractive index of liquid solutions at low temperatures: an accurate measurement,” Appl. Opt. 15, 858–859 (1976).
    [CrossRef] [PubMed]
  15. W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
    [CrossRef]
  16. W. T. Lewis, “Interferometric study of buoyancy-induced mixing in salt stratified fluid layers,” M.S. thesis (Purdue University, Lafayette, Ind., 1980).
  17. R. L. Boxman, D. J. Shlien, “Interferometric technique for measuring the refractive index variation of a liquid with temperature,” Rev. Sci. Instrum. 49, 861–863 (1978).
    [CrossRef] [PubMed]
  18. R. C. Weast, ed., Handbook of Chemistry and Physics, 56th ed. (CRC, Cleveland, Ohio, 1975).
  19. R. W. Stineman, “A consistently well-balanced method of interpolation,” Creat. Comp. (July1980), pp. 54–57.
  20. A. Savitzky, M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
    [CrossRef]
  21. J. E. Dennis, R. B. Schabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice Hall, Englewood Cliffs, N. J., 1983).

1988 (1)

A. Ecker, “Two-wavelength holographic measurement of temperature and concentration during alloy solidifications,” J. Thermophys. Heat Transfer 2, 193–196 (1988).
[CrossRef]

1984 (1)

1982 (1)

W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
[CrossRef]

1980 (1)

R. W. Stineman, “A consistently well-balanced method of interpolation,” Creat. Comp. (July1980), pp. 54–57.

1978 (1)

R. L. Boxman, D. J. Shlien, “Interferometric technique for measuring the refractive index variation of a liquid with temperature,” Rev. Sci. Instrum. 49, 861–863 (1978).
[CrossRef] [PubMed]

1977 (1)

D. Beysens, P. Calmettes, “Temperature dependence of the refractive indices of liquids: deviation from the Lorentz–Lorenz formula,” J. Chem. Phys. 66, 766–771 (1977).
[CrossRef]

1976 (2)

V. P. Semenova, “Character of the temperature variations of the refractive index of benzene and seignette salt,” Opt. Spectrosc. (USSR) 41, 299–300 (1976).

B. W. Grange, W. H. Stevenson, R. Viskanta, “Refractive index of liquid solutions at low temperatures: an accurate measurement,” Appl. Opt. 15, 858–859 (1976).
[CrossRef] [PubMed]

1973 (1)

1971 (1)

C. G. Murphy, S. S. Alpert, “Dependence of refraction index temperature coefficient on the thermal expansivity of liquids,” Am. J. Phys. 39, 834–836 (1971).
[CrossRef]

1964 (2)

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
[CrossRef]

A. Savitzky, M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[CrossRef]

1942 (1)

A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
[CrossRef]

1938 (1)

L. W. Tilton, J. K. Taylor, “Refractive index and dispersion of distilled water for visible radiation, and temperatures of 0 to 60 °C,” J. Res. Natl. Bur. Stand. 20, 419–477 (1938).
[CrossRef]

1920 (1)

J. A. Wasasjerna, “Lösnongars optiska egenskaper,” Acta. Soc. Sci. Fenn. Ser. A. 50, 1–63 (1920).

Alpert, S. S.

C. G. Murphy, S. S. Alpert, “Dependence of refraction index temperature coefficient on the thermal expansivity of liquids,” Am. J. Phys. 39, 834–836 (1971).
[CrossRef]

Beysens, D.

D. Beysens, P. Calmettes, “Temperature dependence of the refractive indices of liquids: deviation from the Lorentz–Lorenz formula,” J. Chem. Phys. 66, 766–771 (1977).
[CrossRef]

Böttcher, C.

C. Böttcher, Theory of Electric Polarization (Elsevier, New York, 1952).

Boxman, R. L.

R. L. Boxman, D. J. Shlien, “Interferometric technique for measuring the refractive index variation of a liquid with temperature,” Rev. Sci. Instrum. 49, 861–863 (1978).
[CrossRef] [PubMed]

Calmettes, P.

D. Beysens, P. Calmettes, “Temperature dependence of the refractive indices of liquids: deviation from the Lorentz–Lorenz formula,” J. Chem. Phys. 66, 766–771 (1977).
[CrossRef]

Coumou, D. J.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
[CrossRef]

Dennis, J. E.

J. E. Dennis, R. B. Schabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice Hall, Englewood Cliffs, N. J., 1983).

Dobbins, H. M.

Dorinson, A.

A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
[CrossRef]

Ecker, A.

A. Ecker, “Two-wavelength holographic measurement of temperature and concentration during alloy solidifications,” J. Thermophys. Heat Transfer 2, 193–196 (1988).
[CrossRef]

Golay, M. J. E.

A. Savitzky, M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[CrossRef]

Grange, B. W.

Grigull, U.

W. Hauf, U. Grigull, Optical Methods in Heat Transfer, Vol. 6 of Advances in Heat Transfer (Academic, New York, 1970).

Hauf, W.

W. Hauf, U. Grigull, Optical Methods in Heat Transfer, Vol. 6 of Advances in Heat Transfer (Academic, New York, 1970).

Hijmans, J.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
[CrossRef]

Incropera, F. P.

W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
[CrossRef]

Lewis, W. T.

W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
[CrossRef]

W. T. Lewis, “Interferometric study of buoyancy-induced mixing in salt stratified fluid layers,” M.S. thesis (Purdue University, Lafayette, Ind., 1980).

Mackor, E. L.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
[CrossRef]

McCorkle, M. R.

A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
[CrossRef]

Mehta, J. M.

Murphy, C. G.

C. G. Murphy, S. S. Alpert, “Dependence of refraction index temperature coefficient on the thermal expansivity of liquids,” Am. J. Phys. 39, 834–836 (1971).
[CrossRef]

Peck, E. R.

Ralston, A. W.

A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
[CrossRef]

Savitzky, A.

A. Savitzky, M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[CrossRef]

Schabel, R. B.

J. E. Dennis, R. B. Schabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice Hall, Englewood Cliffs, N. J., 1983).

Schödel, G.

G. Schödel, “Kombinierte Wärmelestrahlung in Konvektions-freien Flussigheitsschichten,” Ph.D. dissertation (Institut A für Thermodynamik, Technische Hochschule München, München, Germany, 1969).

Semenova, V. P.

V. P. Semenova, “Character of the temperature variations of the refractive index of benzene and seignette salt,” Opt. Spectrosc. (USSR) 41, 299–300 (1976).

Shlien, D. J.

R. L. Boxman, D. J. Shlien, “Interferometric technique for measuring the refractive index variation of a liquid with temperature,” Rev. Sci. Instrum. 49, 861–863 (1978).
[CrossRef] [PubMed]

Stevenson, W. H.

Stineman, R. W.

R. W. Stineman, “A consistently well-balanced method of interpolation,” Creat. Comp. (July1980), pp. 54–57.

Taylor, J. K.

L. W. Tilton, J. K. Taylor, “Refractive index and dispersion of distilled water for visible radiation, and temperatures of 0 to 60 °C,” J. Res. Natl. Bur. Stand. 20, 419–477 (1938).
[CrossRef]

Tilton, L. W.

L. W. Tilton, J. K. Taylor, “Refractive index and dispersion of distilled water for visible radiation, and temperatures of 0 to 60 °C,” J. Res. Natl. Bur. Stand. 20, 419–477 (1938).
[CrossRef]

Viskanta, R.

W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
[CrossRef]

B. W. Grange, W. H. Stevenson, R. Viskanta, “Refractive index of liquid solutions at low temperatures: an accurate measurement,” Appl. Opt. 15, 858–859 (1976).
[CrossRef] [PubMed]

Wasasjerna, J. A.

J. A. Wasasjerna, “Lösnongars optiska egenskaper,” Acta. Soc. Sci. Fenn. Ser. A. 50, 1–63 (1920).

Worck, W. M.

Acta. Soc. Sci. Fenn. Ser. A. (1)

J. A. Wasasjerna, “Lösnongars optiska egenskaper,” Acta. Soc. Sci. Fenn. Ser. A. 50, 1–63 (1920).

Am. J. Phys. (1)

C. G. Murphy, S. S. Alpert, “Dependence of refraction index temperature coefficient on the thermal expansivity of liquids,” Am. J. Phys. 39, 834–836 (1971).
[CrossRef]

Anal. Chem. (1)

A. Savitzky, M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[CrossRef]

Appl. Opt. (2)

Creat. Comp. (1)

R. W. Stineman, “A consistently well-balanced method of interpolation,” Creat. Comp. (July1980), pp. 54–57.

J. Chem. Phys. (1)

D. Beysens, P. Calmettes, “Temperature dependence of the refractive indices of liquids: deviation from the Lorentz–Lorenz formula,” J. Chem. Phys. 66, 766–771 (1977).
[CrossRef]

J. Chem. Soc. (1)

A. Dorinson, M. R. McCorkle, A. W. Ralston, “Refractive indices and densities of normal saturated fatty acids in the liquid state,” J. Chem. Soc. 64, 2739–2741 (1942).
[CrossRef]

J. Fluid Mech. (1)

W. T. Lewis, F. P. Incropera, R. Viskanta, “Interferometric study of stable salinity gradients heated from below or cooled from above,” J. Fluid Mech. 116, 411–430 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Res. Natl. Bur. Stand. (1)

L. W. Tilton, J. K. Taylor, “Refractive index and dispersion of distilled water for visible radiation, and temperatures of 0 to 60 °C,” J. Res. Natl. Bur. Stand. 20, 419–477 (1938).
[CrossRef]

J. Thermophys. Heat Transfer (1)

A. Ecker, “Two-wavelength holographic measurement of temperature and concentration during alloy solidifications,” J. Thermophys. Heat Transfer 2, 193–196 (1988).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

V. P. Semenova, “Character of the temperature variations of the refractive index of benzene and seignette salt,” Opt. Spectrosc. (USSR) 41, 299–300 (1976).

Rev. Sci. Instrum. (1)

R. L. Boxman, D. J. Shlien, “Interferometric technique for measuring the refractive index variation of a liquid with temperature,” Rev. Sci. Instrum. 49, 861–863 (1978).
[CrossRef] [PubMed]

Trans. Faraday Soc. (1)

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquids,” Trans. Faraday Soc. 60, 1539–1547 (1964).
[CrossRef]

Other (6)

W. Hauf, U. Grigull, Optical Methods in Heat Transfer, Vol. 6 of Advances in Heat Transfer (Academic, New York, 1970).

G. Schödel, “Kombinierte Wärmelestrahlung in Konvektions-freien Flussigheitsschichten,” Ph.D. dissertation (Institut A für Thermodynamik, Technische Hochschule München, München, Germany, 1969).

C. Böttcher, Theory of Electric Polarization (Elsevier, New York, 1952).

R. C. Weast, ed., Handbook of Chemistry and Physics, 56th ed. (CRC, Cleveland, Ohio, 1975).

W. T. Lewis, “Interferometric study of buoyancy-induced mixing in salt stratified fluid layers,” M.S. thesis (Purdue University, Lafayette, Ind., 1980).

J. E. Dennis, R. B. Schabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice Hall, Englewood Cliffs, N. J., 1983).

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Figures (8)

Fig 1
Fig 1

Two-wavelength Mach–Zehnder interferometer.

Fig 2
Fig 2

Experimental test system for refractive index measurement.

Fig 3
Fig 3

Systems for (a) the test cell, (b) cooling, and (c) salt-water injection.

Fig 4
Fig 4

Data-acquisition system.

Fig 5
Fig 5

Comparison of the current results with those of Tilton and Taylor5 for pure water at (a) λair = 632.8 nm and (b) λair = 457.9 nm.

Fig 6
Fig 6

Typical intensity signal from a photodetector and the raw data points.

Fig 7
Fig 7

(a) Definition of the acceptance band used to detect false peaks. (b) False-peak detection method. (c) Curve-fitting technique used to minimize signal irregularity.

Fig 8
Fig 8

Variation of the refractive index with temperature at different concentrations for (a) λair = 632.8 nm and (b) λair = 457.9 nm.

Tables (1)

Tables Icon

Table 1 Measurement Results of the Experimental Samples

Equations (22)

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C = 1158.05518 2454.72729 d + 1296.91602 d 2 .
N = S 1 S 2 ,
S = Δ λ ,
Δ = test leg n d l reference leg n d l ,
λ S 1 = Δ 1 = test leg , 1 n d l reference leg , 1 n d l ,
λ S 2 = Δ 2 = test leg , 2 n d l reference leg , 2 n d l ,
λ N = Δ 1 Δ 2 = test leg , 1 n d l reference leg , 1 n d l test leg , 2 n d l + reference leg , 2 n d l .
λ N = cell , 1 n d l cell , 2 n d l .
cell n d l = optical glass n d l + cell frame n d l air n d l .
Δ L α L g Δ T = 558.8 nm ,
n Δ L 1.47 × 558.8 = 821.4 nm .
λ S 1 = Δ 1 = n g L g , T 1 + n fluid , T 1 L T 1 n air ( L T 1 + L g , T 1 ) ,
λ S 2 = Δ 2 = n g L g , T 2 + n fluid , T 2 L T 2 n air ( L T 2 + L g , T 2 ) ,
L g , T 2 = L g , T 1 [ 1 + α g ( T 2 T 1 ) ] ,
L T 2 = L T 1 [ 1 + α cell ( T 2 T 1 ) ] .
λ N = Δ 1 Δ 2 = n g L g , T 1 + n fluid , T 1 L T 1 n air ( L T 1 + L g , T 1 ) n g L g , T 2 n fluid , T 2 L T 2 + n air ( L T 2 + L g , T 2 ) .
n fluid , T 2 = { L T 1 [ 1 + α ( T 2 T 1 ) ] } 1 { λ N α ( T 1 T 2 ) × [ n air ( L T 1 + L g , T 1 ) n g L g , T 1 ] + n fluid , T 1 } .
n fluid , T 2 = ( n fluid , T 1 λ N ) L T 1 α Δ T [ n fluid , T 1 L T 1 λ N + n g L g , T 1 n air ( L T 1 + L g , T 1 ) ] L T 1 ( α Δ T ) 2 [ n air ( L T 1 + L g , T 1 ) n g L g , T 1 ] L T 1 ,
n fluid , C 2 = n fluid , C 1 λ N L cell frame , T room .
n ( T , C ) = A 1 + A 2 T + A 3 T 2 + A 4 C + A 5 C 2 + A 6 C T + A 7 T 2 C 2 .
A 1 = 1.332876681 , A 2 = 2.637899 × 10 5 , A 3 = 1.513549 × 10 6 , A 4 = 1.477013 × 10 3 , A 5 = 2.12523 × 10 4 , A 6 = 1.137696 × 10 5 , A 7 = 2.192279 × 10 9 .
A 1 = 1.339675476 , A 2 = 1.909458 × 10 5 , A 3 = 1.674018 × 10 6 , A 4 = 1.781316 × 10 3 , A 5 = 2.31062 × 10 4 , A 6 = 2.191775 × 10 5 , A 7 = 5.880504 × 10 10 .

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