Abstract

We develop a new approximation for the amplitude reflection coefficients of a slightly inhomogeneous thin film. This approximation incorporates exactly the interference effects at the substrate and the ambient interfaces. Interference effects inside the inhomogeneous film are incorporated in the Born approximation. We also develop a new approach to the reconstruction of the refractive-index profile from ellipsometric spectra. It is based on a physically sound parameterization of the refractive-index profile. The new approach is tested on the model reconstruction problem.

© 1998 Optical Society of America

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References

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  1. J. C. Charmet, P. G. de Gennes, “Ellipsometric formulas for an inhomogeneous layer with arbitrary refractive-index profile,” J. Opt. Soc. Am. 73, 1777–1784 (1983).
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  2. C. K. Carniglia, “Ellipsometric calculations for nonabsorbing thin films with linear refractive-index gradients,” J. Opt. Soc. Am. A 7, 848–856 (1990).
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  3. G. Parjadis de Lariviere, J. M. Frigerio, J. Rivory, F. Abeles, “Estimate of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6056–6061 (1992).
    [CrossRef]
  4. V. A. Shvets, “Profiles of optical constants of inhomogeneous layers determined by ellipsometric measurements in situ,” Optoelectron. Instrum. Data Process. 6, 25–32 (1993).
  5. S. Y. Kim, “Numerical calculation of ellipsometric spectra of layer with arbitrary refractive index profiles,” Opt. Eng. 32, 88–93 (1993).
    [CrossRef]
  6. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
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  7. J. P. Borgogno, F. Flory, P. Roche, B. Schmitt, G. Albrand, E. Pelletier, H. A. Macleod, “Refractive index and inhomogeneity of thin films,” Appl. Opt. 23, 3567–3570 (1984).
    [CrossRef] [PubMed]
  8. H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).
  9. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Sec. 2, 247–286.
    [CrossRef]
  10. R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.
  11. M. Kildemo, O. Hunderi, B. Drevillon, “Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films,” J. Opt. Soc. Am. A 14, 931–939 (1997).
    [CrossRef]
  12. K. Vedam, S. Y. Kim, L. D’Aries, A. H. Guenther, “Nondestructive depth profiling of ZnS and MgO films by spectroscopic ellipsometry,” Opt. Lett. 12, 456–458 (1987).
    [CrossRef] [PubMed]
  13. S. Y. Kim, K. Vedam, “Simultaneous determination of dispersion relation and depth profile of thorium fluoride thin films by spectroscopic ellipsometry,” Thin Solid Films 166, 325–334 (1988).
    [CrossRef]
  14. K. Vedam, S. Y. Kim, “Simultaneous determination of refractive index, its dispersion and depth-profile of magnesium oxide thin film by spectroscopic ellipsometry,” Appl. Opt. 28, 2691–2694 (1989).
    [CrossRef] [PubMed]
  15. S. Y. Kim, “Simultaneous determination of refractive index, extinction coefficient, and void distribution of titanium dioxide thin films by optical methods,” Appl. Opt. 35, 6703–6707 (1996).
    [CrossRef] [PubMed]
  16. J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
    [CrossRef]
  17. B. Dugnoille, O. Virlet, “Optical profile of surface layers on a float glass determined by ellipsometry,” Appl. Opt. 33, 5853–5858 (1994).
    [CrossRef] [PubMed]
  18. D. Tonova, A. Konova, “Damage depth profiles determination by ellipsometry: a new numerical algorithm,” Surf. Sci. 293, 195–201 (1993).
    [CrossRef]
  19. A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).
  20. Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 2.
  21. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
  22. A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).
  23. Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 1.

1997 (1)

1996 (1)

1994 (1)

1993 (4)

D. Tonova, A. Konova, “Damage depth profiles determination by ellipsometry: a new numerical algorithm,” Surf. Sci. 293, 195–201 (1993).
[CrossRef]

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).

V. A. Shvets, “Profiles of optical constants of inhomogeneous layers determined by ellipsometric measurements in situ,” Optoelectron. Instrum. Data Process. 6, 25–32 (1993).

S. Y. Kim, “Numerical calculation of ellipsometric spectra of layer with arbitrary refractive index profiles,” Opt. Eng. 32, 88–93 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (1)

1988 (2)

S. Y. Kim, K. Vedam, “Simultaneous determination of dispersion relation and depth profile of thorium fluoride thin films by spectroscopic ellipsometry,” Thin Solid Films 166, 325–334 (1988).
[CrossRef]

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

1987 (1)

1984 (1)

1983 (1)

1982 (1)

1941 (1)

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Abeles, F.

Albrand, G.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Borgogno, J. P.

Carniglia, C. K.

Charmet, J. C.

D’Aries, L.

de Gennes, P. G.

Drevillon, B.

Dugnoille, B.

Flory, F.

Frigerio, J. M.

Furman, Sh.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 1.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 2.

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Guenther, A. H.

Hunderi, O.

Jacobson, R.

R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Sec. 2, 247–286.
[CrossRef]

Kaiser, J. H.

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

Kildemo, M.

Kim, S. Y.

Konova, A.

D. Tonova, A. Konova, “Damage depth profiles determination by ellipsometry: a new numerical algorithm,” Surf. Sci. 293, 195–201 (1993).
[CrossRef]

Lazarides, B.

Macleod, H. A.

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Parjadis de Lariviere, G.

Pelletier, E.

Rivory, J.

Roche, P.

Schmitt, B.

Schröder, H.

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Shvets, V. A.

V. A. Shvets, “Profiles of optical constants of inhomogeneous layers determined by ellipsometric measurements in situ,” Optoelectron. Instrum. Data Process. 6, 25–32 (1993).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Tikhonravov, A. V.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 1.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 2.

Tonova, D.

D. Tonova, A. Konova, “Damage depth profiles determination by ellipsometry: a new numerical algorithm,” Surf. Sci. 293, 195–201 (1993).
[CrossRef]

Trubetskov, M. K.

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).

Vedam, K.

Virlet, O.

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Ann. Phys. (Leipzig) (1)

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichtige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Appl. Opt. (6)

Appl. Phys. B (1)

J. H. Kaiser, “Regularization in ellipsometry,” Appl. Phys. B 45, 1–5 (1988).
[CrossRef]

Comp. Maths. Math. Phys. (1)

A. N. Tikhonov, A. V. Tikhonravov, M. K. Trubetskov, “Second order optimization methods in the synthesis of multilayer coatings,” Comp. Maths. Math. Phys. 33, 1339–1352 (1993).

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

S. Y. Kim, “Numerical calculation of ellipsometric spectra of layer with arbitrary refractive index profiles,” Opt. Eng. 32, 88–93 (1993).
[CrossRef]

Opt. Lett. (1)

Optoelectron. Instrum. Data Process. (1)

V. A. Shvets, “Profiles of optical constants of inhomogeneous layers determined by ellipsometric measurements in situ,” Optoelectron. Instrum. Data Process. 6, 25–32 (1993).

Surf. Sci. (1)

D. Tonova, A. Konova, “Damage depth profiles determination by ellipsometry: a new numerical algorithm,” Surf. Sci. 293, 195–201 (1993).
[CrossRef]

Thin Solid Films (1)

S. Y. Kim, K. Vedam, “Simultaneous determination of dispersion relation and depth profile of thorium fluoride thin films by spectroscopic ellipsometry,” Thin Solid Films 166, 325–334 (1988).
[CrossRef]

Other (6)

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 2.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, Sec. 2, 247–286.
[CrossRef]

R. Jacobson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.

Sh. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), Chap. 1.

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Figures (8)

Fig. 1
Fig. 1

Schematic of a slightly inhomogeneous film.

Fig. 2
Fig. 2

Subdivision of the film shown in Fig. 1 into three parts: the boundary between media with the refractive indices n s and n, the inhomogeneous film surrounded by the media with the refractive index n, and the boundary between media with the refractive indices n and n a .

Fig. 3
Fig. 3

Explanation of zero variations of the ellipsometric angle Ψ at the QW points in the case of the linear refractive-index profile (see the text for details).

Fig. 4
Fig. 4

Model refractive-index profiles: 1: linear, 2: hyperbolic tangent, 3: convex upward, 4: convex downward.

Fig. 5
Fig. 5

Comparison of the ellipsometric spectra of the model inhomogeneous films with the ellipsometric spectra of the unperturbed homogeneous film: (a) 1: spectrum of the linear profile, 2: spectrum of the hyperbolic profile, hom: spectrum of the homogeneous profile; (b) 3: spectrum of the convex upward profile, 4: spectrum of the convex downward profile, hom: spectrum of the homogeneous profile.

Fig. 6
Fig. 6

Results of the reconstruction of the model refractive-index profile (dashed curve) by using a homogeneous model (plot 1), a one-parameter model of the inhomogeneity factor (plot 2), a three-parameter model of the inhomogeneity factor (plot 3), and a seven-parameter model of the inhomogeneity factor (plot 4).

Fig. 7
Fig. 7

Correspondence between theoretical and experimental angles Ψ at 65°: crosses: simulated experimental data, 1: theoretical data for the homogeneous film (plot 1 in Fig. 6), 2: theoretical data for the film with the linear refractive-index profile (plot 2 in Fig. 6).

Fig. 8
Fig. 8

Correspondence between theoretical and experimental angles Δ at 65°: crosses: simulated experimental data, solid curve: theoretical data for the film with the linear refractive-index profile (plot 2 in Fig. 6).

Equations (52)

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n = n 0 + n i / 2 .
δ = n 0 - n i / n .
C = 1 / t r * / t * r / t 1 / t * .
C 1 = 1 / t 1 r 1 / t 1 r 1 / t 1 1 / t 1 ,     C ˜ = 1 / t ˜ r ˜ * / t ˜ * r ˜ / t ˜ 1 / t ˜ * , C 2 = 1 / t 2 r 2 / t 2 r 2 / t 2 1 / t 2 .
C = C 1 C ˜ C 2 .
r = 1 / t ˜ r 1 + 1 / t ˜ * r 2 + r ˜ * / t ˜ * r 1 r 2 + r ˜ / t ˜ 1 / t ˜ + 1 / t ˜ * r 1 r 2 + r ˜ * / t ˜ * r 2 + r ˜ / t ˜ r 1 .
r p / r s = tan   Ψ   exp i Δ .
1 / t ˜ s , p = exp i φ a ,
φ a = 2 π λ 0 z a n 2 z - α 2 d z .
α = n a sin   Θ a .
r ˜ s , p / t ˜ s , p = - iA s , p exp - i φ a 0 φ a   η φ exp 2 i φ d φ ,
A s = n n 2 - α 2 ,     A p = n 2 - 2 α 2 n n 2 - α 2 .
φ = 2 π λ 0 z n 2 ξ - α 2 d ξ ,
r 1 s , p = n a s , p - n s , p n a s , p + n s , p ,     r 2 s , p = n s , p - n s s , p n s , p + n s s , p ,
n s = n 2 - α 2 ,     n p = n 2 / n 2 - α 2 .
r = n a - n s cos   φ a + n a + n s μ + i n a n s / n - n sin   φ a - n a n s / n + n ν n a + n s cos   φ a + n a - n s μ + i n a n s / n + n sin   φ a - n a n s / n - n ν ,
μ = A   0 φ a   η φ sin 2 φ - φ a d φ ,
ν = A   0 φ a   η φ sin 2 φ - φ a d φ ,
φ a = m π / 2 ,
r = n a + n s μ + i n a n s / n - n - 1 m - 1 / 2 - n a n s / n + n ν n a - n s μ + i n a n s / n + n - 1 m - 1 / 2 - n a n s / n - n ν ,
r = n a - n s - 1 m / 2 + n a + n s μ - i n a n s / n + n ν n a + n s - 1 m / 2 + n a - n s μ - i n a n s / n - n ν
| r | 2 = n a n s / n - n 2 - 2 - 1 m - 1 / 2 n a 2 n s 2 / n 2 - n 2 ν n a n s / n + n 2 - 2 - 1 m - 1 / 2 n a 2 n s 2 / n 2 - n 2 ν
| r | 2 = n a - n s 2 + 2 - 1 m / 2 n a 2 - n s 2 μ n a + n s 2 + 2 - 1 m / 2 n a 2 - n s 2 μ
n z ,   λ = q z n λ .
n λ = n + A λ 2 + B λ 4 .
q z = 1 + a 1 T 1 z + a 2 T 2 z + ,
a z = 1 + a 1 2 z a z - z a 2 .
F = 1 2 J j = 1 J Ψ λ j - Ψ ˜ λ j δ Ψ j 2 + Δ λ j - Δ ˜ λ j δ Δ j 2 .
n z = n 1 - δ 2 tanh   a z - z a / 2 tanh az a / 2
d u d z = i   2 π λ   v ,     d u d z = i   2 π λ n 2 z - α 2 u ,
α = n a sin   Θ a .
u 0 = 1 ,     v 0 = n 2 - α 2 ,
r ˜ = n 2 - α 2 u z a - v z a n 2 - α 2 u z a + v z a , t ˜ = 2 n 2 - α 2 n 2 - α 2 u z a + v z a .
f 1 z = 0.5 u z + v z / n 2 - α 2 .
f 2 z = 0.5 u z - v z / n 2 - α 2 .
f 1 z a = 1 / t ˜ ,     f 2 z a = r ˜ / t ˜ .
d f 1 d z = i   π λ n 2 - α 2 f 1 - f 2 + n 2 z - α 2 n 2 - α 2 f 1 + f 2 ,
d f 2 d z = i   π λ n 2 - α 2 f 1 - f 2 - n 2 z - α 2 n 2 - α 2 f 1 + f 2 ,
f 1 0 = 1 ,
f 2 0 = 0 .
d f 1 d z = i   2 π λ n 2 - α 2 + n η z n 2 - α 2   f 1 ,
d f 2 d z = - i   2 π λ n 2 - α 2 + n η z n 2 - α 2   f 2 - i   2 π λ n η z n 2 - α 2   f 1 .
n 2 - α 2 + n η z n 2 - α 2 = n 2 z - α 2 ,
d f 1 d z = i   2 π λ n 2 z - α 2 f 1 ,
d f 2 d z = - i   2 π λ n 2 z - α 2 f 2 - i   2 π λ n η z n 2 - α 2   f 1 .
f 1 z = exp i   2 π λ 0 z n 2 s - α 2 d s .
f 2 z = - i   2 π λ n n 2 - α 2 exp - i   2 π λ 0 z n 2 s - α 2 d s × 0 z   η s exp - i   2 π λ 0 s n 2 ξ - α 2 d ξ d s .
φ = 2 π λ 0 z n 2 s - α 2 d s .
φ a = 2 π λ 0 z a n 2 z - α 2 d z .
1 / t ˜ = f 1 z a = f 1 φ a ,     r ˜ / t ˜ = f 2 z a = f 2 φ a .
1 / t ˜ = exp i φ a ,
r ˜ / t ˜ = - i   n n 2 - α 2 exp - i φ a 0 φ a   η φ exp 2 i φ d φ .

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