Abstract

An orthogonal form of a general three-term dispersion equation is presented which is useful for determination of the refractive index of transparent optical materials from measured spectral data. The orthogonal basis functions are dependent on the spectral range of the data and whether the spacing of the data is uniform with respect to wavelength or wavenumber. The application of the technique to the reduction of ellipsometric measurements of a single layer thin film of zirconia on a fused silica substrate is presented. The measured data were fit to a Conrady dispersion equation and to the corresponding orthogonal basis set. The convergence to the solution was an order of magnitude faster for the orthogonal form of the dispersion equation.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. P. Arndt et al., “Multiple Determination of the Optical Constants of Thin Film Coating Materials,” Appl. Opt. 23, 3571–3596 (1984).
    [CrossRef] [PubMed]
  2. G. Arfken, Mathematical Methods for Physicists (Academic, London, 1970), p. 441.
  3. B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).
  4. R. W. Stobie, B. Rao, M. J. Dignam, “Analysis of a Novel Ellipsometric Technique with Special Advantages for Infrared Spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
    [CrossRef]
  5. R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

1986 (1)

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

1984 (1)

1975 (2)

R. W. Stobie, B. Rao, M. J. Dignam, “Analysis of a Novel Ellipsometric Technique with Special Advantages for Infrared Spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
[CrossRef]

R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, London, 1970), p. 441.

Arndt, D. P.

Carniglia, C. K.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

Dignam, M. J.

R. W. Stobie, B. Rao, M. J. Dignam, “Analysis of a Novel Ellipsometric Technique with Special Advantages for Infrared Spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
[CrossRef]

R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

Pond, B.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

Raj, T.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

Rao, B.

R. W. Stobie, B. Rao, M. J. Dignam, “Analysis of a Novel Ellipsometric Technique with Special Advantages for Infrared Spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
[CrossRef]

R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

Schmell, R. A.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

Stobie, R. W.

R. W. Stobie, B. Rao, M. J. Dignam, “Analysis of a Novel Ellipsometric Technique with Special Advantages for Infrared Spectroscopy,” J. Opt. Soc. Am. 65, 25–28 (1975).
[CrossRef]

R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

Appl. Opt. (2)

D. P. Arndt et al., “Multiple Determination of the Optical Constants of Thin Film Coating Materials,” Appl. Opt. 23, 3571–3596 (1984).
[CrossRef] [PubMed]

R. W. Stobie, B. Rao, M. J. Dignam, “Automatic Ellipsometer with High Sensitivitiy and Special Advantages for Infrared Spectroscopy of Adsorbed Species,” Appl. Opt. 14, 99–1003 (1975).

J. Opt. Soc. Am. (1)

NIST Spec. Publ. (1)

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the Optical Properties of Some High-Index Oxide Films Prepared by Ion Beam Sputter Deposition with Those of Electron Beam Evaporated Films,” NIST Spec. Publ. 752, 410–417 (1986).

Other (1)

G. Arfken, Mathematical Methods for Physicists (Academic, London, 1970), p. 441.

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 (6)

Fig. 1
Fig. 1

Comparison of the terms for the Cauchy dispersion equation for two similar dispersion curves. The solid curves represent A (lower curve), A + Bλ−2 (middle curve), and A + Bλ−2 + Cλ−4 (upper curve), where A = 1.632, B = 0.133, and C = 0.02. The dashed curves represent similar terms with A = 1.664, B = 0.066, and C = 0.04.

Fig. 2
Fig. 2

Functions Z m ( λ 1 , λ 2 ) / λ 1 m λ 2 m plotted vs λ12 for several values of m. The values of m are indicated for selected curves.

Fig. 3
Fig. 3

Orthogonal basis functions χi for the Cauchy dispersion equation expressed in terms of wavelength for the case where λ1 = 0.5 and λ2 = 2.0. The curves are labeled with the corresponding value of i.

Fig. 4
Fig. 4

Same as Fig. 3 for the Conrady dispersion equation expressed in terms of wavenumber.

Fig. 5
Fig. 5

Merit function vs time illustrating the convergence for the case of the Conrady dispersion equation (upper curve) and the orthogonal basis functions (lower curve) for a single layer zirconia thin film.

Fig. 6
Fig. 6

Dispersion curves for a a single-layer zirconia thin film at various stages of the fitting process. The solid curve corresponds to the fit to the Conrady equation after one iteration cycle; the dashed curve is the fit after four cycles. The other curves, which are indistinguishable, represent the fit to the Conrady equation after the filial iteration cycle as well as the fits to the orthogonal basis functions after the first, fourth, and final iteration cycles.

Tables (1)

Tables Icon

Table I Evaluation of Zm12) for Integer Values of m

Equations (34)

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

n ( λ ) = A + B λ 2 + C λ 4 ,
MF = i [ n i n ( λ i ) ] 2
MF = i [ Δ i Δ ( λ i ) ] 2 + [ Ψ i Ψ ( λ i ) ] 2 ,
n ( λ ) = A + B λ p + C λ q .
p q 0 .
χ 1 = 1 ,
χ 2 = λ p b ,
χ 3 = λ q c λ p + d .
0 = λ 1 λ 2 χ i ( λ ) χ j ( λ ) d λ .
0 < λ 1 < λ 2 < .
Z m ( λ 1 , λ 2 ) 1 λ 2 λ 1 λ 1 λ 2 λ m d λ
Z m ( λ 1 , λ 2 ) = 1 ( m + 1 ) ( λ 2 m + 1 λ 1 m + 1 λ 2 λ 1 ) for m 1
Z m ( λ 1 2 ) = ln ( λ 2 / λ 1 ) / ( λ 2 λ 1 ) for m = 1 .
b = Z p ,
c = ( Z q Z p Z p + q ) / D .
d = ( Z 2 p Z q Z p Z p + q ) / D .
D = Z p 2 Z 2 p .
n ( λ ) = A χ 1 + B χ 2 + C χ 3 ,
A = A B b + C d ,
B = B C c ,
C = C .
n ( σ ) = A + B σ p + C σ q ,
χ 1 = 1 ,
χ 2 = σ p b ,
χ 3 = σ q c σ p + d .
0 = σ 1 σ 2 χ i ( σ ) χ j ( σ ) d σ ,
σ 1 = 1 / λ 2 ,
σ 2 = 1 / λ 1 .
b = 1 / λ 1 λ 2 ,
c = ( 3 a 2 ) / ( 5 λ 1 λ 2 ) ,
d = ( 4 a 1 ) / ( 15 λ 1 2 λ 2 2 ) ,
a = ( λ 1 + λ 2 ) 2 / λ 1 λ 2 .
n ( σ ) = A + B σ + C σ 3 . 5 .
b = ( σ 1 + σ 2 ) / 2 .

Metrics