Abstract

The refractive index of a layer is a sensitive function of the preparation conditions. Normal incidence measurement of the optical properties can reveal possible inhomogeneity of index. We propose a method of automatic determination of the complex refractive index and thickness of a layer which includes systematic measurement of the degree of inhomogeneity which is represented by a simple model. The usefulness of the technique is demonstrated by examples that form part of an experimental study of a number of useful optical materials including Y2O3, TiO2, MgF2, HfO2, and SiO2. The dispersions of the refractive index, the extinction coefficient, and of the inhomogeneity are represented by Cauchy formulas with accurately determined coefficients. The results can therefore be readily used in computing the optical properties of thin-film multilayers.

© 1982 Optical Society of America

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References

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  1. H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
  2. H. L. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, London, 1981); A. L. Bloom, Appl. Opt. 20, 66 (1981).
    [CrossRef] [PubMed]
  3. H. A. Macleod, Appl. Opt. 20, 82 (1981); P. Bousquet, E. Pelletier, Thin Solid Films 77, 165 (1981).
    [CrossRef] [PubMed]
  4. E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
    [CrossRef]
  5. M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
    [CrossRef]
  6. H. K. Pulker, Appl. Opt. 18, 1969 (1979).
    [CrossRef] [PubMed]
  7. S. Ogura, PhD Thesis, Newcastle-upon-Tyne (1975).
  8. F. Abeles, Prog. Opt. 2, 251 (1963).
  9. R. Jacobsson, Phys. Thin Films, 8, 51 (1975).
  10. P. Jacquinot, Rev. Opt. Theor. Instrum. 21, 15 (1942).
  11. A. Vasicek, J. Phys. Rad. 11, 342 (1950).
    [CrossRef]
  12. P. H. Berning, Phys. Thin Films 1, 69 (1963).
  13. C. M. Horwitz, Appl. Opt. 17, 1771 (1978).
    [CrossRef] [PubMed]
  14. J. A. Nelder, R. MeadComput. J. 7, 308 (1965).
    [CrossRef]
  15. P. Borgogno, B. Lazarides, P. Roche, Thin Solid Films (1982), to be published.
  16. J. P. Borgogno, P. Bousquet, F. Flory, B. Lazarides, E. Pelletier, P. Roche, Appl. Opt. 20, 90 (1981); D. Smith, P. Baumeister, Appl. Opt. 18, 111 (1979).
    [CrossRef] [PubMed]
  17. H. K. Pulker, G. Paesold, E. Ritter, Appl. Opt. 15, 2986 (1976).
    [CrossRef] [PubMed]

1981 (2)

1979 (2)

H. K. Pulker, Appl. Opt. 18, 1969 (1979).
[CrossRef] [PubMed]

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

1978 (1)

1976 (2)

H. K. Pulker, G. Paesold, E. Ritter, Appl. Opt. 15, 2986 (1976).
[CrossRef] [PubMed]

E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

1975 (1)

R. Jacobsson, Phys. Thin Films, 8, 51 (1975).

1965 (1)

J. A. Nelder, R. MeadComput. J. 7, 308 (1965).
[CrossRef]

1963 (2)

P. H. Berning, Phys. Thin Films 1, 69 (1963).

F. Abeles, Prog. Opt. 2, 251 (1963).

1950 (1)

A. Vasicek, J. Phys. Rad. 11, 342 (1950).
[CrossRef]

1942 (1)

P. Jacquinot, Rev. Opt. Theor. Instrum. 21, 15 (1942).

Abeles, F.

F. Abeles, Prog. Opt. 2, 251 (1963).

Berning, P. H.

P. H. Berning, Phys. Thin Films 1, 69 (1963).

Borgogno, J. P.

Borgogno, P.

P. Borgogno, B. Lazarides, P. Roche, Thin Solid Films (1982), to be published.

Bousquet, P.

Flory, F.

Harris, M.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

Horwitz, C. M.

Jacobsson, R.

R. Jacobsson, Phys. Thin Films, 8, 51 (1975).

Jacquinot, P.

P. Jacquinot, Rev. Opt. Theor. Instrum. 21, 15 (1942).

Lazarides, B.

Liddell, H. L.

H. L. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, London, 1981); A. L. Bloom, Appl. Opt. 20, 66 (1981).
[CrossRef] [PubMed]

Macleod, H. A.

H. A. Macleod, Appl. Opt. 20, 82 (1981); P. Bousquet, E. Pelletier, Thin Solid Films 77, 165 (1981).
[CrossRef] [PubMed]

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).

Mead, R.

J. A. Nelder, R. MeadComput. J. 7, 308 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. MeadComput. J. 7, 308 (1965).
[CrossRef]

Ogura, S.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

S. Ogura, PhD Thesis, Newcastle-upon-Tyne (1975).

Paesold, G.

Pelletier, E.

J. P. Borgogno, P. Bousquet, F. Flory, B. Lazarides, E. Pelletier, P. Roche, Appl. Opt. 20, 90 (1981); D. Smith, P. Baumeister, Appl. Opt. 18, 111 (1979).
[CrossRef] [PubMed]

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Pulker, H. K.

Ritter, E.

Roche, P.

J. P. Borgogno, P. Bousquet, F. Flory, B. Lazarides, E. Pelletier, P. Roche, Appl. Opt. 20, 90 (1981); D. Smith, P. Baumeister, Appl. Opt. 18, 111 (1979).
[CrossRef] [PubMed]

E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

P. Borgogno, B. Lazarides, P. Roche, Thin Solid Films (1982), to be published.

Vasicek, A.

A. Vasicek, J. Phys. Rad. 11, 342 (1950).
[CrossRef]

Vidal, B.

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Appl. Opt. (5)

Comput. J. (1)

J. A. Nelder, R. MeadComput. J. 7, 308 (1965).
[CrossRef]

J. Phys. Rad. (1)

A. Vasicek, J. Phys. Rad. 11, 342 (1950).
[CrossRef]

Nouv. Rev. Opt. (1)

E. Pelletier, P. Roche, B. Vidal, Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Phys. Thin Films (2)

P. H. Berning, Phys. Thin Films 1, 69 (1963).

R. Jacobsson, Phys. Thin Films, 8, 51 (1975).

Prog. Opt. (1)

F. Abeles, Prog. Opt. 2, 251 (1963).

Rev. Opt. Theor. Instrum. (1)

P. Jacquinot, Rev. Opt. Theor. Instrum. 21, 15 (1942).

Thin Solid Films (1)

M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, B. Vidal, Thin Solid Films 57, 173 (1979).
[CrossRef]

Other (4)

S. Ogura, PhD Thesis, Newcastle-upon-Tyne (1975).

H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).

H. L. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, London, 1981); A. L. Bloom, Appl. Opt. 20, 66 (1981).
[CrossRef] [PubMed]

P. Borgogno, B. Lazarides, P. Roche, Thin Solid Films (1982), to be published.

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Figures (22)

Fig. 1
Fig. 1

Values, after Ref. 4, of the refractive index n of a zinc sulfide film. The departures from a smooth curve of dispersion of index are due to structural imperfections.

Fig. 2
Fig. 2

Model chosen to represent the inhomogeneous layer between media of indices ns (substrate) and na (air). The index of the layer varies from ni at the inner surface to no at the outer.

Fig. 3
Fig. 3

Calculated reflectance as a function of λ/λo for layers of thickness λo, of index n ¯ = 2.3, and of inhomogeneity defined by Δn/ n ¯. Δn/ n ¯ takes the values +15%, +7.5%, 0, −7.5%, and −15%. For Δn/ n ¯ = 0 and λ/λo = 1, the layer is both homogeneous and absentee: Rc = Rd.

Fig. 4
Fig. 4

Calculated reflectance as a function of λ/λo for various values of n ¯ for layers of optical thickness λo/2 with inhomogeneity Δn/ n ¯ = −15%. For λ/λo = 1, the reflectance Re is independent of the mean index of the material.

Fig. 5
Fig. 5

Variation of the calculated reflectance Re for λ/λo = 1, as a function of the inhomogeneity Δn/ n ¯. The layer has optical thickness λo, and the indices are na = 1, ns = 1.52, and n ¯ = 2.3. The two curves represent results for extinction coefficients of k = 0 and k = 1 × 10−2.

Fig. 6
Fig. 6

Grid showing values of Re and Te as functions of n ¯ and k, with na = 1 and ns = 1.52: (a) homogeneous layer, (b) inhomogeneous layer with Δn/ n ¯ = −15%.

Fig. 7
Fig. 7

Grid showing values of R0 and To as functions of n ¯ and k. The layer has 3λ0/4 optical thickness, and the media are given by na = 1 and ns = 1.52. The grid is unaffected by the degree of inhomogeneity Δn/ n ¯.

Fig. 8
Fig. 8

Flow chart of the determination of the optical constants of a layer with inhomogeneity represented by the ratio Δn/ n ¯ (see Fig. 2).

Fig. 9
Fig. 9

Yttrium oxide. Curve showing the agreement between the measured reflectance (—), the values calculated (++++) from the values of the indices obtained by iteration using a homogeneous model, and the final values calculated (◇ ◇ ◇ ◇) from the Cauchy expression.

Fig. 10
Fig. 10

Yttrium oxide. Index of refraction against wavelength assuming a homogeneous model for the layer. The uncertainty in the corresponding R and T values does not differ by more than 0.02 from those measured. The continuous curve is the index represented by the Cauchy expression.

Fig. 11
Fig. 11

Yttrium oxide. The curve shows the agreement between measured (—) and calculated reflectances, the indices being obtained iteratively. Unlike Fig. 9, inhomogeneity has been included in the calculations.

Fig. 12
Fig. 12

Yttrium oxide. Values of the refractive index as a function of wavelength for one layer. The inhomogeneity Δn/ n ¯ is given in the text with the coefficients of the Cauchy expression for the index.

Fig. 13
Fig. 13

Titanium oxide. Values of n ¯ and the uncertainty δn in n ¯ calculated on the basis of a homogeneous layer.

Fig. 14
Fig. 14

Titanium oxide. Values of n ¯ and of the uncertainty δn calculated using an inhomogeneous layer model. Figure 15 gives the values of the extinction coefficient.

Fig. 15
Fig. 15

Titanium oxide. Values of the extinction coefficient as a function of wavelength. The values were derived using the expression (1 − RT)/T. Figure 14 gives the correspondiong refractive index.

Fig. 16
Fig. 16

Titanium oxide. Representation of the values n ¯ (λ) of Fig. 14 by a Cauchy expression. The inhomogeneous layer model (Fig. 2) allows us to calculate the inner surface index, ni, and the outer surface index, no.

Fig. 17
Fig. 17

Magnesium fluoride. Values of n ¯ and of the uncertainty δn calculated with an inhomogeneous layer model.

Fig. 18
Fig. 18

Magnesium fluoride. Values of n ¯ and the uncertainty δn in n ¯ calculated on the basis of a homogeneous layer.

Fig. 19
Fig. 19

Hafnium oxide. Values of n ¯ and of the uncertainty δn calculated with an inhomogeneous layer model.

Fig. 20
Fig. 20

Hafnium oxide. Values of n ¯ and the uncertainty δn in n ¯ calculated on the basis of a homogeneous layer.

Fig. 21
Fig. 21

Silicon oxide. Values of n ¯ and of the uncertainty δn calculated with an inhomogeneous layer model.

Fig. 22
Fig. 22

Silicon oxide. Values of n ¯ and the uncertainty δn in n ¯ calculated on the basis of a homogeneous layer.

Equations (21)

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R = ( n a n o n i - n g n i n o ) 2 + ( n a n g / n o n i - n o n i ) 2 tan 2 β ( n o n o n i + n g n i n o ) 2 + ( n o n g / n o n i + n o n i ) 2 tan 2 β ,
β = 2 π λ o e n ( z ) d z ,
R o = ( n a n g - n o n i n a n g + n o n i ) 2 ,
R e = ( n a n i - n o n g n a n i + n o n g ) 2 .
R d = ( n a - n g n a + n g ) 2
n l = n ¯ + 2 l - 11 20 Δ n .
l = 1 10 n l e l = n ¯ e .
f i = ( R c - R i ) 2 + ( R c - R i ) 2 + ( T c - T i ) 2 .
n ¯ ( λ ) = n g ( λ ) 1 + R o ( λ ) 1 - R o ( λ ) .
e = k o λ o 4 n ¯ ( λ o ) = ( k o + 2 ) λ o 4 n ¯ ( λ o ) = .
n ¯ ( λ e ) = ( k o + 1 ) λ e 4 e ,             n ¯ ( λ e ) = ( k o + 3 ) λ e 4 e .
n ¯ ( λ ) = A + B λ 2 + C λ 4 + ,
Δ n n ¯ = A + B λ 2 + C λ 4 + .
A = 1.78024 ,             B = 1.228 × 10 4 nm 2 ,             C = - 7.055 × 10 3 nm 4 .
A = 2.2405 ,             B = 6.312 × 10 3 nm 2 ,             C = 7.244 × 10 9 nm 4 .
A = 2.2254 ,             B = 2.338 × 10 3 nm 2 ,             C = 7.688 × 10 9 nm 4 ,
A = 1.3840 ,             B = - 3.651 × 10 3 nm 2 ,             C = 6.429 × 10 8 nm 4 , A = 1.72 × 10 - 2 ,             B = - 4.65 × 10 + 3 nm 2 ,
A = 1.9165 ,             B = 2.198 × 10 4 nm 2 ,             C = - 3.276 × 10 8 nm 4 ,
A = - 5.39 × 10 - 2 ,             B = - 1.77 × 10 + 3 nm 2 .
A = 1.4625 ,             B = 3.069 × 10 3 nm 2 ,             C = - 2.019 × 10 8 nm 4
A = - 9.98 × 10 - 3 ,             B = 3.12 × 10 + 3 nm 2 , C = - 4.18 × 10 + 8 nm 4

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