Abstract

Our investigation uses the film-interference transmittance spectrum in the computer calculation of thin-film thickness t and complex refractive index n*=n-ik. Titanium oxide films of different thicknesses are studied, and a new approach to determine the film thickness and optical constants is proposed. This new approach is based on the use of numerical optimization methods in transmittance-spectra fitting. Various dispersion equations of the finite-power-series type are used to obtain the best fit between transmittance measurements and calculations. The best-fit results for n(λ) and k(λ) are found to agree with dispersion relations that follow from a quantum theory of light absorption.

© 1998 Optical Society of America

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References

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  1. H. A. Macleod, Thin Film Optical Filters (Macmillan, New York, 1986).
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    [CrossRef] [PubMed]
  3. M. Nenkov, T. Pencheva, “Symmetrical five-layer periods from nonabsorbing high- and low-refractive-index materials,” J. Opt. Soc. Am. A 14, 686–692 (1997).
    [CrossRef]
  4. J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
    [CrossRef]
  5. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
    [CrossRef]
  6. R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
    [CrossRef]
  7. R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. A 8, 1339–1343 (1985).
    [CrossRef]
  8. C. Corrales, J. B. Ramı́rez-Malo, J. Fernández-Peña, P. Villares, R. Swanepoel, E. Márquez, “Determining the refractive index and average thickness of AsSe semiconducting glass films from wavelength measurements only,” Appl. Opt. 34, 7907–7913 (1995).
    [CrossRef] [PubMed]
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  11. Rusli, G. A. J. Amaratunga, “Determination of the optical constants and thickness of thin films on slightly absorbing substrates,” Appl. Opt. 34, 7914–7924 (1995).
  12. T. Pencheva, M. Nenkov, “Some possible applications of chromatic parameters to thin films optical constants determination,” J. Mod. Opt. 43, 2449–2462 (1996).
    [CrossRef]
  13. D. Smith, Ph. Baumeister, “Refractive index of some oxide and fluoride coating materials,” Appl. Opt. 18, 111–115 (1979).
    [CrossRef] [PubMed]
  14. A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7025 (1986).
    [CrossRef]
  15. K. A. Stroud, Engineering Mathematics (Macmillan, London, 1992).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1983).
  17. M. B. Bazaraa, C. M. Shetty, Non-Linear Programming Theory and Algorithms (Wiley, New York, 1979).

1997 (1)

1996 (1)

T. Pencheva, M. Nenkov, “Some possible applications of chromatic parameters to thin films optical constants determination,” J. Mod. Opt. 43, 2449–2462 (1996).
[CrossRef]

1995 (3)

1990 (1)

1986 (1)

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7025 (1986).
[CrossRef]

1985 (1)

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. A 8, 1339–1343 (1985).
[CrossRef]

1984 (1)

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

1983 (2)

1979 (1)

1976 (1)

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Amaratunga, G. A. J.

Baumeister, Ph.

Bazaraa, M. B.

M. B. Bazaraa, C. M. Shetty, Non-Linear Programming Theory and Algorithms (Wiley, New York, 1979).

Bloomer, I.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7025 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1983).

Bovard, B. G.

Chiao, Sh. C.

Corrales, C.

Dobrowolski, J. A.

Fernández-Peña, J.

Fillard, J. P.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Forouhi, A. R.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7025 (1986).
[CrossRef]

Gasiot, J.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Ho, F. C.

Leveque, G.

Macleod, H. A.

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Márquez, E.

Nenkov, M.

M. Nenkov, T. Pencheva, “Symmetrical five-layer periods from nonabsorbing high- and low-refractive-index materials,” J. Opt. Soc. Am. A 14, 686–692 (1997).
[CrossRef]

T. Pencheva, M. Nenkov, “Some possible applications of chromatic parameters to thin films optical constants determination,” J. Mod. Opt. 43, 2449–2462 (1996).
[CrossRef]

Pencheva, T.

M. Nenkov, T. Pencheva, “Symmetrical five-layer periods from nonabsorbing high- and low-refractive-index materials,” J. Opt. Soc. Am. A 14, 686–692 (1997).
[CrossRef]

T. Pencheva, M. Nenkov, “Some possible applications of chromatic parameters to thin films optical constants determination,” J. Mod. Opt. 43, 2449–2462 (1996).
[CrossRef]

Rami´rez-Malo, J. B.

Rusli,

Shetty, C. M.

M. B. Bazaraa, C. M. Shetty, Non-Linear Programming Theory and Algorithms (Wiley, New York, 1979).

Smith, D.

Stroud, K. A.

K. A. Stroud, Engineering Mathematics (Macmillan, London, 1992).

Swanepoel, R.

C. Corrales, J. B. Ramı́rez-Malo, J. Fernández-Peña, P. Villares, R. Swanepoel, E. Márquez, “Determining the refractive index and average thickness of AsSe semiconducting glass films from wavelength measurements only,” Appl. Opt. 34, 7907–7913 (1995).
[CrossRef] [PubMed]

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. A 8, 1339–1343 (1985).
[CrossRef]

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

Villachon-Renard, Y.

Villares, P.

Waldorf, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1983).

Appl. Opt. (6)

J. Mod. Opt. (1)

T. Pencheva, M. Nenkov, “Some possible applications of chromatic parameters to thin films optical constants determination,” J. Mod. Opt. 43, 2449–2462 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. A 8, 1339–1343 (1985).
[CrossRef]

M. Nenkov, T. Pencheva, “Symmetrical five-layer periods from nonabsorbing high- and low-refractive-index materials,” J. Opt. Soc. Am. A 14, 686–692 (1997).
[CrossRef]

J. Phys. E (3)

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n,k and the thickness of a weakly absorbing film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
[CrossRef]

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Phys. Rev. B (1)

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7025 (1986).
[CrossRef]

Other (4)

K. A. Stroud, Engineering Mathematics (Macmillan, London, 1992).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1983).

M. B. Bazaraa, C. M. Shetty, Non-Linear Programming Theory and Algorithms (Wiley, New York, 1979).

H. A. Macleod, Thin Film Optical Filters (Macmillan, New York, 1986).

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Figures (4)

Fig. 1
Fig. 1

Measured transmittance spectra of the film–substrate system T(λ) and the uncoated substrate TS(λ).

Fig. 2
Fig. 2

Numerical values of n(λ) and k(λ) corresponding to the best, fit of the transmittance spectra for samples I and II: results from step 6.

Fig. 3
Fig. 3

Comparison between measured and calculated transmittance by use of the dispersion equations from the best fits (Section 5) and equations from Forouhi-model fits (Section 6): plus signs show measurements, solid curves show the best fits, and the dashed curves show the fits obtained after Forouhi-model application.

Fig. 4
Fig. 4

Final results for n(λ) and k(λ) from Sections 5 and 6 (sample I): The solid curves show results from the best fit (Section 5), and the dotted curves show results from the Forouhi-model fit (Section 6).

Tables (4)

Tables Icon

Table 1 Sample I: Initial Dataa and Final Resultsb for Optical Constants

Tables Icon

Table 2 Sample II: Initial Dataa and Final Resultsb for Optical Constants

Tables Icon

Table 3 Final Numerical Results for Constants in Eqs. (4) for Sample I

Tables Icon

Table 4 Initial and Final Results for Parameters in Forouhi Dispersion Equations, Film Thickness t, and Function δT for Sample I

Equations (28)

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

n2(λ)=a+k2(λ)+bλ2(λ2-c2)(λ2-c2)2+d2λ2,
k2(λ)=12n(λ)bdλ3(λ2-c2)2+d2λ2.
n2(λ)=a+bλ2λ2-c2.
n(E)=n()+B0E+C0E2-BE+C,
k(E)=A(E-Eg)2E2-BE+C,
B0=AQ-B22+EgB-Eg2+C
C0=(A/Q)[(Eg2+C)(B/2)-2EgC]
Q=1/2(4C-B2)1/2,
n(λ)=an+bn/λ+cn/λ2+,
k(λ)=ak+bk/λ+ck/λ2+,
n(λ)=an+bn/λ2+cn/λ4+,
k(λ)=ak+bk/λ2+ck/λ4+,
δT=180i=180[T(λi)calc-T(λi)meas]21/2,
nS=1TS+1TS2-11/2.
n(λ)=an+bn/λ,k(λ)=ak+bk/λ;
n(λ)=an+bn/λ2,k(λ)=ak+bk/λ2.
A(E1-Eg)2+Bk1E1-Ck1=k1E12,
A(E2-Eg)2+Bk2E2-Ck2=k2E22,
A(E3-Eg)2+Bk3E3-Ck3=k3E32,
A(E4-Eg)2+Bk4E4-Ck4=k4E42.
αEg2+2βEg+γ=0,
α=u+v+1+w,
β=-uE1-vE2-E3-wE4,
γ=uE12+vE22+E32+wE42,
u=k3k1(E3-E2)(E3-E4)(E2-E1)(E1-E4),
v=k3k2(E3-E1)(E3-E4)(E1-E2)(E2-E4),
w=k3k4(E3-E1)(E3-E2)(E1-E4)(E4-E2).
Eg=-βα+β2-αγα.

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