Abstract

The technique of generalized ellipsometry is briefly reviewed. An improved criterion for computing the normalized 2 × 2 complex reflection matrix of an anisotropic surface from multiple-null ellipsometer measurements (in excess of three) is given. Generalized ellipsometry, together with the recently developed 4 × 4-matrix methods for the study of the reflection and transmission of polarized light by stratified anisotropic media, provide the basic tools to carry out and to interpret ellipsometric measurements on anisotropic structures. As an example, the case of uniaxial (absorbing) crystals, with the optic axis parallel to the surface, is considered. From three or more null measurements at a single unknown orientation of the optic axis, the five parameters (the ordinary and extraordinary complex indices of refraction and the angle of inclination of the optic axis from the plane of incidence) that characterize a uniaxial crystal of calcite are all determined.

© 1974 Optical Society of America

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References

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  1. Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Stand. (U. S.) Misc. Publ. 256 (U. S. Government Printing Office, Washington, D. C., 1964).
  2. Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).
  3. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), pp. 38 and 615.
  4. Reference 1, p. 9.
  5. Reference 3, p. 51.
  6. A. B. Winterbottom, Optical Studies of Metal Surfaces (F. Burns, Trondheim, 1955).
  7. R. A. W. Graves, J. Opt. Soc. Am. 59, 1225 (1969).
    [Crossref]
  8. A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
    [Crossref]
  9. D. den Engelsen, J. Opt. Soc. Am. 61, 1460 (1971).
    [Crossref]
  10. D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
    [Crossref]
  11. F. Meyer, E. E. de Kluizenaar, and D. den Engelsen, J. Opt. Soc. Am. 63, 529 (1973).
    [Crossref]
  12. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
    [Crossref]
  13. R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
    [Crossref]
  14. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1375A (1972); J. Opt. Soc. Am. 62, 1521 (1972).
  15. S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830 (1970).
    [Crossref]
  16. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
    [Crossref]
  17. D. J. De Smet, J. Opt. Soc. Am. 63, 958 (1973).
    [Crossref]
  18. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [Crossref]
  19. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 63, 508A (1973); Appl. Phys. 2, 59 (1973).
  20. Many other possible applications of generalized ellipsometry are mentioned in Ref. 14.
  21. This method of obtaining and spacing multiple nulls is explained in Ref. 13. Other nulling schemes are detailed in Refs. 12 and 22.
  22. R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
    [Crossref]
  23. This assumes that the principal axes of both the absorption and the refraction indicatrices are coincident.
  24. T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
    [Crossref]
  25. American Institute of Physics Handbook, 2nd ed. (McGraw–Hill, New York, 1963), Ch. 6, p. 18. These values were obtained by interpolating between values given in Table 6b-5a at wavelengths immediately above and below 6328 Å. If we use our values for N¯o and N¯e [Eq. (15)], instead, the computed curves in Figs. 2 and 3 stay very much the same.
  26. Note that three ellipsometer null measurements made on a single surface of a uniaxial absorping crystal positioned in one orientation are just enough to determine all of the refractive properties of such a crystal, even when the optic axis makes arbitrary unknown angles θ and ω(less than 90°) with the surface and the plane of incidence, respectively. In this more-general case, we have six known quantities (the magnitude and angles, or the real and imaginary parts of Rpp/Rss, Rps/Rss, and Rsp/Rss) and six unknown parameters (no, ko; ne, ke; ω and θ). However, in this case, the reflection coefficients are no longer given by Eqs. (11)–(13). They may be derived using the methods of Refs. 15 and 16.

1973 (4)

F. Meyer, E. E. de Kluizenaar, and D. den Engelsen, J. Opt. Soc. Am. 63, 529 (1973).
[Crossref]

D. J. De Smet, J. Opt. Soc. Am. 63, 958 (1973).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 63, 508A (1973); Appl. Phys. 2, 59 (1973).

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[Crossref]

1972 (7)

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[Crossref]

D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1375A (1972); J. Opt. Soc. Am. 62, 1521 (1972).

D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[Crossref]

1971 (1)

1970 (2)

1969 (1)

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 63, 508A (1973); Appl. Phys. 2, 59 (1973).

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1375A (1972); J. Opt. Soc. Am. 62, 1521 (1972).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 63, 508A (1973); Appl. Phys. 2, 59 (1973).

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1375A (1972); J. Opt. Soc. Am. 62, 1521 (1972).

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[Crossref]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[Crossref]

Berreman, D. W.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), pp. 38 and 615.

Bundy, T. L.

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[Crossref]

de Kluizenaar, E. E.

De Smet, D. J.

den Engelsen, D.

Graves, R. A. W.

Henvis, B. W.

Meyer, F.

Sosnowski, T. P.

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

Teitler, S.

Winterbottom, A. B.

A. B. Winterbottom, Optical Studies of Metal Surfaces (F. Burns, Trondheim, 1955).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), pp. 38 and 615.

Wünsche, A.

A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

Ann. Phys. (Leipz.) (1)

A. Wünsche, Ann. Phys. (Leipz.) 25, 201 (1970).
[Crossref]

J. Opt. Soc. Am. (10)

J. Phys. Chem. (1)

D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[Crossref]

Opt. Commun. (3)

R. M. A. Azzam and N. M. Bashara, Opt. Commun. 5, 5 (1972).
[Crossref]

R. M. A. Azzam, T. L. Bundy, and N. M. Bashara, Opt. Commun. 7, 110 (1973).
[Crossref]

T. P. Sosnowski, Opt. Commun. 4, 408 (1972).
[Crossref]

Other (11)

American Institute of Physics Handbook, 2nd ed. (McGraw–Hill, New York, 1963), Ch. 6, p. 18. These values were obtained by interpolating between values given in Table 6b-5a at wavelengths immediately above and below 6328 Å. If we use our values for N¯o and N¯e [Eq. (15)], instead, the computed curves in Figs. 2 and 3 stay very much the same.

Note that three ellipsometer null measurements made on a single surface of a uniaxial absorping crystal positioned in one orientation are just enough to determine all of the refractive properties of such a crystal, even when the optic axis makes arbitrary unknown angles θ and ω(less than 90°) with the surface and the plane of incidence, respectively. In this more-general case, we have six known quantities (the magnitude and angles, or the real and imaginary parts of Rpp/Rss, Rps/Rss, and Rsp/Rss) and six unknown parameters (no, ko; ne, ke; ω and θ). However, in this case, the reflection coefficients are no longer given by Eqs. (11)–(13). They may be derived using the methods of Refs. 15 and 16.

This assumes that the principal axes of both the absorption and the refraction indicatrices are coincident.

Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. R. Stromberg, and J. Kruger, Natl. Bur. Stand. (U. S.) Misc. Publ. 256 (U. S. Government Printing Office, Washington, D. C., 1964).

Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North–Holland, Amsterdam, 1969).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), pp. 38 and 615.

Reference 1, p. 9.

Reference 3, p. 51.

A. B. Winterbottom, Optical Studies of Metal Surfaces (F. Burns, Trondheim, 1955).

Many other possible applications of generalized ellipsometry are mentioned in Ref. 14.

This method of obtaining and spacing multiple nulls is explained in Ref. 13. Other nulling schemes are detailed in Refs. 12 and 22.

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Figures (3)

Fig. 1
Fig. 1

The ellipsometer arrangement used. P, C, S, and A represent the polarizer, compensator, surface of the uniaxial crystal under measurement, and analyzer, respectively. ϕ is the angle of incidence and ω is the angle of inclination of the optic axis of the crystal (which is parallel to the surface) from the plane of incidence.

Fig. 2
Fig. 2

The magnitude of the ratio of the diagonal, complex reflection coefficients, Rpp/Rss, as a function of the angle of inclination ω of the optic axis from the plane of incidence. The continuous line (——) represents theoretical values computed from Eqs. (11)(13) using ϕ = 70°, λ = 6328 Å, n = 1, N ¯ o = 1.6556 + j0, N ¯ e = 1.4852 + j0 with ω variable from 0° − 180°. The experimental points (○) have been shifted horizontally (parallel to the ω axis) by the same amount (25.85°) so as to be as near as possible to the theoretical curve. This horizontal shift is equal to the reading of the divided scale used to define the orientation of the optic axis of the crystal, when the latter is parallel to the plane of incidence.

Fig. 3
Fig. 3

Same as in Fig. 2, for the magnitude of the ratio of complex reflection coefficients Rps/Rss.

Equations (21)

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E r = R E i ,
R = [ R p p R p s R s p R s s ] ,
[ E r p E r s ] = [ R p p R p s R s p R s s ] [ E i p E i s ] .
χ i = E i s / E i p ,             χ r = E r s / E r p ,
χ r = R s s χ i + R s p R p s χ i + R p p .
R p p / R s s = χ i 2 - χ i 1 H - χ r 1 + χ r 2 H ,
R p s / R s s = H - 1 - χ r 1 + χ r 2 H ,
R s p / R s s = χ i 2 χ r 1 - χ i 1 χ r 2 H - χ r 1 + χ r 2 H ,
H = ( χ r 3 - χ r 1 ) ( χ i 3 - χ i 2 ) ( χ r 3 - χ r 2 ) ( χ i 3 - χ i 1 ) .
χ i = tan C + ρ c tan ( P - C ) 1 - ρ c tan C tan ( P - C ) ,             χ r = - cot A ,
χ i = tan P ,             χ r = - 1 - ρ c tan C tan ( A - C ) tan C + ρ c tan ( A - C ) .
N ( χ 1 , χ 2 ) = χ 1 χ 1 * χ 2 χ 2 * + χ 1 χ 2 * + χ 1 * χ 2 + 1 χ 1 χ 1 * χ 2 χ 2 * + χ 1 χ 1 * + χ 2 χ 2 * + 1 ,
S = k = 1 N N ( χ r k , χ r k ) ,
R p p = ( A 1 B 4 + A 2 B 3 ) / ( A 1 + A 2 ) ,
R p s = A 1 A 2 ( B 2 - B 1 ) / ( A 1 + A 2 ) ,
R s p = ( B 4 - B 3 ) / ( A 1 + A 2 ) ,
R s s = ( A 1 B 1 + A 2 B 2 ) / ( A 1 + A 2 ) ,
A 1 = J / ( n sin 2 ϕ + J cos ϕ ) tan ω , A 2 = N ¯ o tan ω ( I + n N ¯ o cos ϕ ) / ( I N ¯ o cos ϕ + n J 2 ) , B 1 = ( n cos ϕ - J ) / ( n cos ϕ + J ) , B 2 = ( n N ¯ o cos ϕ - I ) / ( n N ¯ o cos ϕ + I ) , B 3 = ( N ¯ o 2 cos ϕ - n J ) / ( N ¯ o 2 cos ϕ + n J ) , B 4 = ( I 2 N ¯ o cos ϕ - n J 2 I ) / ( I 2 N ¯ o cos ϕ + n J 2 I ) ,
I 2 = N ¯ o 2 N ¯ e 2 - n 2 sin 2 ϕ ( N ¯ o 2 sin 2 ω + N ¯ e 2 cos 2 ω ) , J 2 = N ¯ o 2 - n 2 sin 2 ϕ .
R p p / R s s = 0.3722 + j 0.0050 , R p s / R s s = - 0.0468 + j 0.0067 , R s p / R s s = 0.0473 - j 0.0055
N ¯ o = 1.6448 - j 0.0309 , N ¯ e = 1.4802 - j 0.0042 , ω = 47.7265° .