Abstract

The ratio ρ = Rp/Rs of the complex amplitude-reflection coefficients Rp and Rs 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 d. Both constant-angle-of-incidence contours (CAIC) and constant-thickness contours (CTC) of the ellipsometric function ρ(ϕ,d) in the complex ρ plane are examined. For transparent films, ρ(ϕ,d) is a periodic function of d with period Dϕ that is a function of ϕ. For a given angle of incidence ϕ and film thickness d (0 ≤ ϕ ≤ 90, 0 ≤ d < Dϕ), the equispaced linear array of points (ϕ,d + mDϕ) (m = 0, 1, 2,…) in the real (ϕ,d) plane is mapped by the periodic function ρ(ϕ,d) 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 d, d + Dϕ, d + 2Dϕ, …. 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.

© 1975 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. Stromberg, and J. Kruger, Natl. 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–SiO2System 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. Winterbofctom, Optical Studies of Metal Surfaces, Kgl. Norske Vidensk. Selsk. Skrift., Vol. 1 (F. Bruns, Trondheim, 1955).
  5. Such contours are 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/r01sr12s, 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 p-suppressing reflection polarizer, ρ is set equal to zero.
  9. M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 62, 1188 (1972).
    [Crossref]
  10. M. Ruiz-Urbieta and E. M. Sparrow, J. Opt. Soc. Am. 63, 194 (1973).
    [Crossref]
  11. M. Ruiz-Urbieta, E. M. Sparrow, and G. W. Goldman, Appl. Opt. 12, 590 (1973).
    [Crossref] [PubMed]
  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.
    [Crossref] [PubMed]

1973 (2)

1972 (2)

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Other (10)

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.

Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Natl. 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–SiO2System 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. Winterbofctom, Optical Studies of Metal Surfaces, Kgl. Norske Vidensk. Selsk. Skrift., Vol. 1 (F. Bruns, Trondheim, 1955).

Such contours are 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/r01sr12s, 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 p-suppressing reflection polarizer, ρ is set equal to zero.

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Figures (14)

FIG. 1
FIG. 1

Film–substrate system. Medium 0 is the ambient, 1 is the film, and 2 is the substrate.

FIG. 2
FIG. 2

Left: The unit circle in the complex X plane. Right: Constant-angle-of-incidence contour (CAIC) of the ellipsometric function ρ at an angle of incidence ϕ = 60° for Si–SiO2 system at a wavelength λ = 6328 Å.

FIG. 3
FIG. 3

Thickness period Dϕ angstroms for SiO2 film (N1 = 1.46) at λ = 6328 Å as a function of the angle of incidence ϕ, where ϕ is in degrees. Point A ≡ (ϕ, d) can be brought down vertically, at the same ϕ, into point A1 ≡ (ϕ,dDϕ) that realizes the same ellipsometric function ρ as pointA, at a different film thickness (dDϕ).

FIG. 4
FIG. 4

Constant-angle-of-incidence contours (CAIC’s) in the complex ρ plane for Si–SiO2 system at λ = 6328 Å starting near ρ = −1 at an angle of incidence ϕ = 30° with a step of 5°. On each contour, the arrow indicates the direction in which the film thickness increases.

FIG. 5
FIG. 5

Zero-thickness contour (ZTC) in the complex ρ plane for Si–SiO2 system at λ = 6328 Å.

FIG. 6
FIG. 6

A constant-thickness contour (CTC) for a film thickness d = 0.24λ Å superimposed on the constant-angle-of-incidence contours of ϕ = 55° to ϕ = 85°, with a step of 5°, in the complex ρ plane for Si–SiO2 system at λ = 6328 Å.

FIG. 7
FIG. 7

Constant-thickness contour (CTC) in the complex ρ plane for Si-SiO2 system at λ = 6328 Å and film thickness d = 1.5 μm.

FIG. 8
FIG. 8

The angle ψ as a function of the angle of incidence ϕ for Si–SiO2 system at λ = 6328 Å and film thickness d = 1.5 μm. Both ψ and ϕ are in degrees.

FIG. 9
FIG. 9

The angle Δ as a function of the angle of incidence ϕ for Si–SiO2 system at λ = 6328 Å and film thickness d = 1.5 μm. Both Δ and ϕ are in degrees.

FIG. 10
FIG. 10

A sequence of constant-thickness contours (CTC’s) in the complex ρ plane for the Si–SiO2 system at λ = 6328 Å for different film thicknesses. Upper left: d = 0.21λ; upper right: d = 0.22λ; lower left: d = 0.23λ; and lower right: d = 0.24λ.

FIG. 11
FIG. 11

A sequence of constant-thickness contours (CTC’s) in the complex ρ plane for the Si–SiO2 system at λ = 6328 Å for different film thicknesses. Upper left: d = λ; upper right: d = 2λ; lower left: d = 3λ; and lower right: d = 4λ.

FIG. 12
FIG. 12

This group of curves represent the design and performance of the p-suppressing polarizer for the Si-SiO2 system at λ = 6328 Å. Upper left: The quantity 1 − |X|, where X is the thickness complex-exponential function, plotted against the angle of incidence ϕ, where ϕ is in degrees. Upper right: The constant-thickness contour (CTC) for film thickness d = 2891.73 Å. Lower left: The angle ψ as a function of the angle of incidence ϕ for the same film thickness. Both ψ and ϕ are in degrees. Lower right: The angle Δ as a function of the angle of incidence ϕ for the same film thickness. Both Δ and ϕ are in degrees.

FIG. 13
FIG. 13

This group of curves represent the design and performance of the s-suppressing polarizer for the Si–SiO2 system at λ = 6328 Å. Upper left: The quantity 1 − |X|, where X is the thickness complex-exponential function plotted against the angle of incidence ϕ, where ϕ is in degrees. Upper right: The constant-thickness contour (CTC) for film thickness ds = 1417.69 Å. The part of the curve around zero is shown in the box on a different scale. Lower left: The angle ψ as a function of the angle of incidence ϕ for the same film thickness. Both ψ and ϕ are in degrees. Lower right: The angle Δ as a function of the angle of incidence ϕ for the same film thickness. Both Δ and ϕ are in degrees.

FIG. 14
FIG. 14

This group of curves represent the design and performance of the quarter-wave-retarder for the Si–SiO2 system at λ = 6328 Å. Upper left: The quantity 1 − |X|, where X is the thickness complex-exponential function, plotted against the angle of incidence ϕ, where ϕ is in degrees. Upper right: The constant-thickness contour (CTC) at film thickness dQWR = 1056.29 Å. Lower left: The angle ψ as a function of the angle of incidence ϕ for the same film thickness. Both ψ and ϕ are in degrees. Lower right: The angle Δ as a function of the angle of incidence ϕ for the same film thickness. Both Δ and ϕ are in degrees.

Equations (34)

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R ν = r 01 ν + r 12 ν e j 2 β 1 + r 01 ν r 12 ν e j 2 β , ν = p , s
β = 2 π ( d / λ ) ( N 1 2 N 0 2 sin 2 ϕ 0 ) 1 / 2 ,
ρ = R p / R s ,
ρ = tan ψ e j Δ ,
ρ = r 01 p + r 12 p e j 2 β 1 + r 01 p r 12 p e j 2 β × 1 + r 01 s r 12 s e j 2 β r 01 s + r 12 s e j 2 β .
R p = a + b X 1 + a b X , R s = c + d X 1 + c d X ,
ρ = ( a + b X ) ( 1 + c d X ) ( 1 + a b X ) ( c + d X ) ,
ρ = A + B X + C X 2 D + E X + F X 2 .
X = e j 2 β ,
( a , b ) = ( r 01 p , r 12 p ) , ( c , d ) = ( r 01 s , r 12 s ) ,
A = a , B = ( b + a c d ) , C = b c d , D = c , E = ( d + a b c ) , F = a b d .
X = exp [ j 4 π ( d / λ ) ( N 1 2 sin 2 ϕ ) 1 / 2 ] ,
X = exp [ j 2 π ( d / D ϕ ) ] ,
D ϕ = λ 2 ( N 1 2 sin 2 ϕ ) 1 / 2 .
0 d < D ϕ .
0 ϕ 90 ° ,
0 < ϕ < ϕ s ,
ϕ s < ϕ < 90 ° ,
ρ = ρ 0 = r 02 p / r 02 s ,
R p = 0 ,
R s = 0 .
r 01 p + r 12 p X = 0 , X = ( r 01 p / r 12 p ) .
| X | = 1 ,
r 01 p = | r 01 p | e j δ 01 p , r 12 p = | r 12 p | e j δ 12 p ,
e j 2 π ( d / D ϕ ) = ( | r 01 p | / | r 12 p | ) e j ( δ 01 p δ 12 p ) .
e j 2 π ( d p / D ϕ p ) = e j ( δ 01 p δ 12 p ) ϕ p .
2 π ( d p / D ϕ p ) + 2 π ( m + 1 2 ) = ( δ 01 p δ 12 p ) ϕ p , ( d p / D ϕ p ) = 1 2 π ( δ 12 p δ 01 p ) ϕ p + ( m + 1 2 ) ,
d p = 1 2 π ( δ 12 p δ 01 p ) ϕ p D ϕ p + ( m + 1 2 ) D ϕ p .
X = ( B ρ E ) ± [ ( B ρ E ) 2 4 ( C ρ F ) ( A ρ D ) ] 1 / 2 2 ( C ρ F ) ,
2 π ( d / D ϕ ) + m 2 π = α ,
d = α 2 π D ϕ + m D ϕ ,
ρ = e j Δ ,
ln ( X ) = j 2 π ( d / D ϕ ) ,
d = j ( λ / 4 π ) ( N 1 2 sin 2 ϕ ) 1 / 2 ln ( X ) + 1 2 m λ ( N 1 2 sin 2 ϕ ) 1 / 2 ,