Abstract

The ratio ρ = <i>R</i><sub>p</sub>/<i>R</i><sub>s</sub> of the complex amplitude-reflection coefficients <i>R</i><sub>p</sub> and <i>R</i><sub>s</sub> for light polarized parallel (p) and perpendicular (s) to the plane of incidence, reflected from an optically isotropic film-substrate system, is investigated as a function of the angle of incidence ϕ and the film thickness <i>d</i>. Both constant-angle-of-incidence contours (CAIC) and constant-thickness contours (CTC) of the ellipsometric function ρ(ϕ,<i>d</i>) in the complex ρ plane are examined. For transparent films, ρ(ϕ,<i>d</i>) is a periodic function of <i>d</i> with period <i>D</i><sub>ϕ</sub> that is a function of ϕ. For a given angle of incidence ϕ and film thickness <i>d</i> (0 ≤ ϕ ≤ 90, 0 ≤ <i>d</i> ≤ <i>D</i><sub>ϕ</sub>), the equispaced linear array of points (ϕ,<i>d</i> + <i>m</i><i>D</i><sub>ϕ</sub>) (<i>m</i> = 0, 1, 2,…) in the real (ϕ,<i>d</i>) plane is mapped by the periodic function ρ(ϕ,<i>d</i>) into one distinct point in the complex ρ plane. Conversely, for a given film-substrate system, any value of the ellipsometric function ρ can be realized at one particular angle of incidence ϕ and an associated infinite series of film thicknesses <i>d</i>, <i>d</i> + <i>D</i><sub>ϕ</sub>, <i>d</i> + 2<i>D</i><sub>ϕ</sub>,.... This analysis leads to (1) a unified scheme for the design of all reflection-type optical devices, such as polarizers and retarders, (2) a novel null ellipsometer without a compensator, for the measurement of films whose thicknesses are within certain permissible ranges, and (3) an inversion procedure for the nonlinear equations of reflection ellipsometry that separates the determination of the optical constants (refractive indices and extinction coefficients) of the film and substrate from the film thickness. Extension of the work to absorbing films is discussed.

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  1. Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Natd. Bur. Std. (U.S.) Misc. Pub. 1, No. 256 (U. S. Government Printing Office, Washington, D. C., 1961).
  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, 1968).
  3. Ellipsometric Tables of the Si-SiO2 System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).
  4. The significance of this was appreciated by A. B. Winterbottom, Optical Studies of Metal Surfaces, Kgl. Norske Vidensk. Selsk. Skrift., Vol. 1 (F. Bruns, Trondheim, 1955).
  5. Such contours arle also called the polar curves. See the paper by P. C. S. Hayfield, Ref. 1, pp. 157ff.
  6. The principal angle is defined for a bare substrate as that angle at which δ=π/2.
  7. These thicknesses are spaced by an integral multiple of the thickness period Dϕ evaluated at the common angle of incidence ϕ.
  8. Two branches are shown in Fig. 12 (upper left). The lower branch is generated from the condition that Rs = ∞ which has no physical existence but leads to ρ=0. In this case, x=-1/γ01sγ12s, from which the lower branch is calculated. Both the upper and lower branches are predicted by the general design equation, Eq. (25), in which, for the present case of a ρ-suppressing reflection polarizer, ρ is set equal to zero.
  9. M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).
  10. M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 63, 194 (1973).
  11. M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).
  12. So also is the design of reflection polarizers. For the p-suppressing and s-suppressing reflection polarizers, the values of ρ to be substituted in the right-hand side of Eq. (25) are 0 and ∞, respectively.
  13. This film thickness is the least one; to get the correct film thickness the appropriate multiple of Dϕ should be added.
  14. A. R. Reinberg, Appl. Opt. 11, 1273 (1972). We want to thank a reviewer for calling this reference to our attention.

Goldman, G. W.

M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).

Hayfield, P. C. S.

Such contours arle also called the polar curves. See the paper by P. C. S. Hayfield, Ref. 1, pp. 157ff.

Reinberg, A. R.

A. R. Reinberg, Appl. Opt. 11, 1273 (1972). We want to thank a reviewer for calling this reference to our attention.

Ruiz-Urbieta, M.

M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 63, 194 (1973).

Sparrow, E. M.

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 63, 194 (1973).

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).

M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).

Winterbottom, A. B.

The significance of this was appreciated by A. B. Winterbottom, Optical Studies of Metal Surfaces, Kgl. Norske Vidensk. Selsk. Skrift., Vol. 1 (F. Bruns, Trondheim, 1955).

Other (14)

Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Natd. Bur. Std. (U.S.) Misc. Pub. 1, No. 256 (U. S. Government Printing Office, Washington, D. C., 1961).

Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1968).

Ellipsometric Tables of the Si-SiO2 System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).

The significance of this was appreciated by A. B. Winterbottom, Optical Studies of Metal Surfaces, Kgl. Norske Vidensk. Selsk. Skrift., Vol. 1 (F. Bruns, Trondheim, 1955).

Such contours arle also called the polar curves. See the paper by P. C. S. Hayfield, Ref. 1, pp. 157ff.

The principal angle is defined for a bare substrate as that angle at which δ=π/2.

These thicknesses are spaced by an integral multiple of the thickness period Dϕ evaluated at the common angle of incidence ϕ.

Two branches are shown in Fig. 12 (upper left). The lower branch is generated from the condition that Rs = ∞ which has no physical existence but leads to ρ=0. In this case, x=-1/γ01sγ12s, from which the lower branch is calculated. Both the upper and lower branches are predicted by the general design equation, Eq. (25), in which, for the present case of a ρ-suppressing reflection polarizer, ρ is set equal to zero.

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).

M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 63, 194 (1973).

M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).

So also is the design of reflection polarizers. For the p-suppressing and s-suppressing reflection polarizers, the values of ρ to be substituted in the right-hand side of Eq. (25) are 0 and ∞, respectively.

This film thickness is the least one; to get the correct film thickness the appropriate multiple of Dϕ should be added.

A. R. Reinberg, Appl. Opt. 11, 1273 (1972). We want to thank a reviewer for calling this reference to our attention.

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