Abstract

The fitting of measured optical index data to the Sellmeier dispersion formula, using the variable projection algorithm, is described. Examples of fits obtained by this method to several Schott optical glasses and non-glass materials are given.

© 1984 Optical Society of America

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References

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  1. O. N. Stavroudis, L. E. Sutton, “Rapid Method for Interpolating Refractive Index Measurements,” J. Opt. Soc. Am. 51, 368 (1961).
    [CrossRef]
  2. L. E. Sutton, O. N. Stavroudis, “Fitting Refractive Index Data by Least Squares,” J. Opt. Soc. Am. 51, 901 (1961).
    [CrossRef]
  3. B. Tatian, “Interpolation of Glass Indices with Applications to First Order Axial Chromatic Aberration,” Itek Corp. Report OR-63-20, Lexington, Mass. (1964).
  4. M. Herzberger, “Colour Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
    [CrossRef]
  5. A recent article, P. N. Robb, R. I. Mercado, “Calculation of Refractive Indices Using Buchdahl’s Chromatic Coordinate,” Appl. Opt. 22, 1198 (1983), makes the statement, “A theoretical model should have an accuracy comparable to that of precision glass melts, which can be supplied to an index accuracy of 0.0002.” Even standard precision melt sheets supplied by Schott Optical Glass, Inc. specify the dispersion values to ±2 × 10−5, and high-precision melt sheets give the dispersion to ±2 × 10−6. (Dispersion ≡ Nλ0 − Nλ, where λ0 is a wavelength near the middle of the wavelength band considered.) It is the departure of the nominal index, Nλ0, of the melt from the catalog value that can be as high as ±2 × 10−4. This is about ±10 waves/in. of glass, and an error this large in the dispersion would be totally unacceptable for higher-order color effects.
    [CrossRef] [PubMed]
  6. G. H. Golub, V. Pereyra, “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Siam J. Numer. Anal. 10, 413 (1973).
    [CrossRef]
  7. F. T. Krogh, “Efficient Implementation of a Variable Projection Algorithm for Nonlinear Least Squares Problems,” Commun. ACM 17, 167 (1974).
    [CrossRef]
  8. R. Penrose, “On Best Approximate Solutions of Linear Matrix Equations,” Proc. Cambridge Philos. Soc. 52, 17 (1956).
    [CrossRef]
  9. Optical Glass Catalog 3111E, Schott Optical Glass, Inc., Duryea, Pa.
  10. Raytran Infrared Materials, Raytheon Co., Research Division, Waltham, Mass.
  11. M. J. Dodge, “Refractive Properties of CVD Zinc Sulfide,” in Proceedings, Symposium on Laser-Induced Damage in Optical Materials (U.S. GPO, Washington, 1977), pp. 83–88.
  12. I. H. Malitson, “Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55, 1205 (1965).
    [CrossRef]
  13. W. L. Wolfe, Ed., Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 7, p. 7–89.
  14. W. L. Wolfe, Ref. 13, p. 7–102.
  15. I. H. Malitson, “A Redetermination of Some Optical Properties of CaF2,” Appl. Opt. 2, 1103 (1963).
    [CrossRef]
  16. I. H. Malitson, “Refractive Properties of BaF2,” J. Opt. Soc. Am. 54, 628 (1964).
    [CrossRef]
  17. I. H. Malitson, “Refraction and Dispersion of Synthetic Sapphire,” J. Opt. Soc. Am. 52, 1377 (1962).
    [CrossRef]
  18. W. S. Rodney, I. H. Malitson, T. A. King, “Refractive Index of Arsenic Trisulfide,” J. Opt. Soc. Am. 48, 633 (1958).
    [CrossRef]
  19. W. S. Rodney, “Optical Properties of Cesium Iodide,” J. Opt. Soc. Am. 45, 987 (1955).
    [CrossRef]
  20. W. S. Rodney, I. H. Malitson, “Refraction and Dispersion of Thallium Bromide-Iodide,” J. Opt. Soc. Am. 46, 956 (1956).
    [CrossRef]
  21. H. H. Li, “Refractive Index of Alkali Halides and Its Wavelength and Temperature Derivatives,” J. Phys. Chem. Ref. Data 5, 329 (1976).
    [CrossRef]
  22. B. J. Pernick, “Nonlinear Regression Analysis for the Sellmeier Dispersion Equation of CdS,” Appl. Opt. 22, 1133 (1983).
    [CrossRef] [PubMed]
  23. F. Vilches, J. M. Guerra, M. S. Gomez, “Nonlinear Regression Analysis of a Sellmeier Equation with Various Resonances: Best Fit of CdS Dispersion,” Appl. Opt. 23, 2044 (1984).
    [CrossRef] [PubMed]
  24. M. Debenham, “Refractive Indices of Zinc Sulfide in the 0.405–13-μm Wavelength Range,” Appl. Opt. 23, 2238 (1984).
    [CrossRef] [PubMed]

1984 (2)

1983 (2)

1976 (1)

H. H. Li, “Refractive Index of Alkali Halides and Its Wavelength and Temperature Derivatives,” J. Phys. Chem. Ref. Data 5, 329 (1976).
[CrossRef]

1974 (1)

F. T. Krogh, “Efficient Implementation of a Variable Projection Algorithm for Nonlinear Least Squares Problems,” Commun. ACM 17, 167 (1974).
[CrossRef]

1973 (1)

G. H. Golub, V. Pereyra, “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Siam J. Numer. Anal. 10, 413 (1973).
[CrossRef]

1965 (1)

1964 (1)

1963 (1)

1962 (1)

1961 (2)

1959 (1)

M. Herzberger, “Colour Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
[CrossRef]

1958 (1)

1956 (2)

W. S. Rodney, I. H. Malitson, “Refraction and Dispersion of Thallium Bromide-Iodide,” J. Opt. Soc. Am. 46, 956 (1956).
[CrossRef]

R. Penrose, “On Best Approximate Solutions of Linear Matrix Equations,” Proc. Cambridge Philos. Soc. 52, 17 (1956).
[CrossRef]

1955 (1)

Debenham, M.

Dodge, M. J.

M. J. Dodge, “Refractive Properties of CVD Zinc Sulfide,” in Proceedings, Symposium on Laser-Induced Damage in Optical Materials (U.S. GPO, Washington, 1977), pp. 83–88.

Golub, G. H.

G. H. Golub, V. Pereyra, “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Siam J. Numer. Anal. 10, 413 (1973).
[CrossRef]

Gomez, M. S.

Guerra, J. M.

Herzberger, M.

M. Herzberger, “Colour Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
[CrossRef]

King, T. A.

Krogh, F. T.

F. T. Krogh, “Efficient Implementation of a Variable Projection Algorithm for Nonlinear Least Squares Problems,” Commun. ACM 17, 167 (1974).
[CrossRef]

Li, H. H.

H. H. Li, “Refractive Index of Alkali Halides and Its Wavelength and Temperature Derivatives,” J. Phys. Chem. Ref. Data 5, 329 (1976).
[CrossRef]

Malitson, I. H.

Mercado, R. I.

Penrose, R.

R. Penrose, “On Best Approximate Solutions of Linear Matrix Equations,” Proc. Cambridge Philos. Soc. 52, 17 (1956).
[CrossRef]

Pereyra, V.

G. H. Golub, V. Pereyra, “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Siam J. Numer. Anal. 10, 413 (1973).
[CrossRef]

Pernick, B. J.

Robb, P. N.

Rodney, W. S.

Stavroudis, O. N.

Sutton, L. E.

Tatian, B.

B. Tatian, “Interpolation of Glass Indices with Applications to First Order Axial Chromatic Aberration,” Itek Corp. Report OR-63-20, Lexington, Mass. (1964).

Vilches, F.

Wolfe, W. L.

W. L. Wolfe, Ref. 13, p. 7–102.

Appl. Opt. (5)

I. H. Malitson, “A Redetermination of Some Optical Properties of CaF2,” Appl. Opt. 2, 1103 (1963).
[CrossRef]

A recent article, P. N. Robb, R. I. Mercado, “Calculation of Refractive Indices Using Buchdahl’s Chromatic Coordinate,” Appl. Opt. 22, 1198 (1983), makes the statement, “A theoretical model should have an accuracy comparable to that of precision glass melts, which can be supplied to an index accuracy of 0.0002.” Even standard precision melt sheets supplied by Schott Optical Glass, Inc. specify the dispersion values to ±2 × 10−5, and high-precision melt sheets give the dispersion to ±2 × 10−6. (Dispersion ≡ Nλ0 − Nλ, where λ0 is a wavelength near the middle of the wavelength band considered.) It is the departure of the nominal index, Nλ0, of the melt from the catalog value that can be as high as ±2 × 10−4. This is about ±10 waves/in. of glass, and an error this large in the dispersion would be totally unacceptable for higher-order color effects.
[CrossRef] [PubMed]

M. Debenham, “Refractive Indices of Zinc Sulfide in the 0.405–13-μm Wavelength Range,” Appl. Opt. 23, 2238 (1984).
[CrossRef] [PubMed]

B. J. Pernick, “Nonlinear Regression Analysis for the Sellmeier Dispersion Equation of CdS,” Appl. Opt. 22, 1133 (1983).
[CrossRef] [PubMed]

F. Vilches, J. M. Guerra, M. S. Gomez, “Nonlinear Regression Analysis of a Sellmeier Equation with Various Resonances: Best Fit of CdS Dispersion,” Appl. Opt. 23, 2044 (1984).
[CrossRef] [PubMed]

Commun. ACM (1)

F. T. Krogh, “Efficient Implementation of a Variable Projection Algorithm for Nonlinear Least Squares Problems,” Commun. ACM 17, 167 (1974).
[CrossRef]

J. Opt. Soc. Am. (8)

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

H. H. Li, “Refractive Index of Alkali Halides and Its Wavelength and Temperature Derivatives,” J. Phys. Chem. Ref. Data 5, 329 (1976).
[CrossRef]

Opt. Acta (1)

M. Herzberger, “Colour Correction in Optical Systems and a New Dispersion Formula,” Opt. Acta 6, 197 (1959).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

R. Penrose, “On Best Approximate Solutions of Linear Matrix Equations,” Proc. Cambridge Philos. Soc. 52, 17 (1956).
[CrossRef]

Siam J. Numer. Anal. (1)

G. H. Golub, V. Pereyra, “The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate,” Siam J. Numer. Anal. 10, 413 (1973).
[CrossRef]

Other (6)

B. Tatian, “Interpolation of Glass Indices with Applications to First Order Axial Chromatic Aberration,” Itek Corp. Report OR-63-20, Lexington, Mass. (1964).

Optical Glass Catalog 3111E, Schott Optical Glass, Inc., Duryea, Pa.

Raytran Infrared Materials, Raytheon Co., Research Division, Waltham, Mass.

M. J. Dodge, “Refractive Properties of CVD Zinc Sulfide,” in Proceedings, Symposium on Laser-Induced Damage in Optical Materials (U.S. GPO, Washington, 1977), pp. 83–88.

W. L. Wolfe, Ed., Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 7, p. 7–89.

W. L. Wolfe, Ref. 13, p. 7–102.

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Figures (19)

Fig. 1
Fig. 1

Fitting error, BK7—fourteen points from 0.37 to 1.01 μm: ▲ = optimum two-term fit; + = three-term fit, B2 = 0.17; Y = three-term fit, B2 = 0.26; □ = optimum three-term fit.

Fig. 2
Fig. 2

Results from infit program for four glasses.

Fig. 3
Fig. 3

Fitting error, zinc sulfide—fifty-nine points from 0.42 to 18.2 μm.

Fig. 4
Fig. 4

Fitting error, zinc selenide—fifty-six points from 0.54 to 18.2 μm.

Fig. 5
Fig. 5

Fitting error, fused silica—sixty points from 0.21 to 3.71 μm.

Fig. 6
Fig. 6

Fitting error, germanium—seventeen points from 2.06 to 13.0 μm.

Fig. 7
Fig. 7

Fitting error, silicon—thirty points from 1.36 to 11.0 μm.

Fig. 8
Fig. 8

Fitting error, fluorite—forty-six points from 0.23 to 9.72 μm.

Fig. 9
Fig. 9

Fitting error, barium fluoride—forty-five points from 0.27 to 10.3 μm.

Fig. 10
Fig. 10

Fitting error, sapphire—forty-six points from 0.27 to 5.58 μm.

Fig. 11
Fig. 11

Fitting error, arsenic trisulfide-twenty-six points from 0.58 to 11.9 μm.

Fig. 12
Fig. 12

Fitting error, cesium iodide—fifty-four points from 0.30 to 53.1 μm.

Fig. 13
Fig. 13

Fitting error, cesium iodide—fifty-four points from 0.30 to 53.1 μm, three-term case vs four-term case.

Fig. 14
Fig. 14

Fitting error, thallium—bromide-iodide-thirty-eight points from 0.58 to 39.4 μm.

Fig. 15
Fig. 15

Fitting error, cesium bromide—thirty-seven points from 0.37 to 39.2 μm.

Fig. 16
Fig. 16

Fitting error, cesium bromide—forty-six points from 0.25 to 40 μm generated by literature formula.

Fig. 17
Fig. 17

Fitting error, cadmium sulfide O—forty-one points from 0.51 to 1.40 μm.

Fig. 18
Fig. 18

Fitting error, cadmium sulfide E—forty-four points from 0.51 to 1.40 μm.

Fig. 19
Fig. 19

Fitting error, zinc sulfide—twenty-nine points from 0.40 to 13.0 μm.

Tables (4)

Tables Icon

Table I Sellmeier Coefficients for Selected Schott Glasses

Tables Icon

Table II Eleven Nonglass Materials Whose Data were Fitted to the Sellmeier Formula

Tables Icon

Table III Sellmeier Coefficients for Eleven Nonglass Materials

Tables Icon

Table IV Sellmeier Coefficients for Fits in Figs. 15 and 1719

Equations (10)

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N 2 = 1 + j = 1 k A j λ 2 λ 2 - B j 2 ;
min a , α r ( a , α ) 2 = N - Φ ( α , λ ) a 2 .
min a r ( a ) 2 = N - Φ ( λ ) a 2 ,
a = Φ + N ,
r ( a + d a , α + d α ) = r ( a , α ) - ( Φ , Φ α ) ( d a , d α ) T .
min d a , d α r ( a , α ) - ( Φ , Φ α ) ( d a , d α ) T 2 ,
( d a , d α ) T = ( Φ , Φ α ) + r ( a , α ) .
min α N - Φ ( α ) Φ + ( α ) N 2 ,
DP = Q + Q T , Q = ( I - P ) ( D Φ ) Φ + .
x = [ log e ( λ - 0.25 ) - 0.3394896 ] / 2.2060417.

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