Abstract

By numerical analysis of the general formulas for standard ellipsometiric measurement, positions of the compensator, polarizer, and analyzer can be found such that the errors of Δ and ψ are minimized. Such optimal conditions can be found for any value of Δ and ψ from plots that are given. The improvement of precision over that of usual methods can be one order of magnitude or better in some Δ, ψ regions.

© 1970 Optical Society of America

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References

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  1. R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago, 1968).
  2. A. Vašíček, Měření a vytváření tenkých vrstev v optice (Publishing House of CSAV, Prague, 1957).
  3. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
    [Crossref]
  4. A. B. Winterbottom, Optical Studies of Metal Surfaces, Roy. Norwegian Sci. Soc. Rep. 1 (F. Bruns, Trondheim, Norway, 1955).
  5. A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960).
  6. O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.
  7. F. Abelès, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963).

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Abelès, F.

F. Abelès, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963).

Archer, R. J.

R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago, 1968).

Heavens, O. S.

O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Vašícek, A.

A. Vašíček, Měření a vytváření tenkých vrstev v optice (Publishing House of CSAV, Prague, 1957).

A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960).

Winterbottom, A. B.

A. B. Winterbottom, Optical Studies of Metal Surfaces, Roy. Norwegian Sci. Soc. Rep. 1 (F. Bruns, Trondheim, Norway, 1955).

J. Res. Natl. Bur. Std. (U. S.) (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Std. (U. S.) 67A, 363 (1963).
[Crossref]

Other (6)

A. B. Winterbottom, Optical Studies of Metal Surfaces, Roy. Norwegian Sci. Soc. Rep. 1 (F. Bruns, Trondheim, Norway, 1955).

A. Vašíček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam, 1960).

O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

F. Abelès, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963).

R. J. Archer, Manual of Ellipsometry (Gaertner Scientific Corp., Chicago, 1968).

A. Vašíček, Měření a vytváření tenkých vrstev v optice (Publishing House of CSAV, Prague, 1957).

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Figures (7)

Fig. 1
Fig. 1

Plot of errors δQ, δP, and δA vs position of compensator Q0 calculated for Δ = 166.7°, ψ = 12.06°, and θ = 90° from Eq. (5.)

Fig. 2
Fig. 2

Plot of errors δΔP, δΔQ, δψPQ, δψAQ, and δψAP vs Q0 calculated from Eqs. (6) and (7) for the same data as in Fig. 1. Scales for P0 and A0 are also presented.

Fig. 3
Fig. 3

The plot of QoptΔ and δΔQ vs Δ for the value ψ marked at the appropriate curve. The scale on the left side is in degrees (Q) and relative units (δΔ); on the right side, the numbers in brackets indicate the beginning of the scale for the curve for ψ of that value.

Fig. 4
Fig. 4

The plot of Qoptψ and δψmin vs Δ and values of ψ as in Fig. 3. The solid line is for Qoptψ, where δψPQ is minimized. Similarly, δψAP corresponds to the broken line and δψAQ to the dotted line. The breaks between branches of the curve for one value ψ are marked (— · — ·).

Fig. 5
Fig. 5

Plot of I vs P − P0 [Fig. 5 (a)] vs QQ0 [Fig. 5 (b)] vs AA0 [Fig. 5 (c)]. The numbers on the curves are values of Q0. The solid line is calculated from Eq. (1) and the crosses are experimental results (θ = 90°, Δ = 166.7°, and ψ = 12.06°).

Fig. 6
Fig. 6

The plot of the ratio rΔ = δΔ(Q = 45°)/δΔQ (solid line) and rΔ = δΔ(A = 45°)/δΔQ (broken line) vs Δ with ψ marked as on Fig. 3.

Fig. 7
Fig. 7

The plot of ratio rψQ = δψ(Q = 45°)/δψmin (left) and rψA = δψ(A = 45°)/δψmin (right) vs Δ, with ψ marked as on Fig. 3.

Equations (14)

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I = g I 0 { cos 2 ( P - Q ) [ p cos 2 A cos 2 Q + s sin 2 A sin 2 Q ] + sin 2 ( P - Q ) ( p cos 2 A sin 2 Q + s sin 2 A cos 2 Q ) + 0.5 cos θ sin 2 ( P - Q ) sin 2 Q ( s sin 2 A - p cos 2 A ) + 0.5 ( p s ) 1 2 sin 2 A [ cos Δ sin 2 Q cos 2 ( P - Q ) + sin 2 ( P - Q ) ] [ cos 2 Q cos ( θ + Δ ) - sin 2 Q cos ( θ - Δ ) ] } ,
p = c sin 2 ψ ,             s = c cos 2 ψ ,
tan Δ = - sin θ [ cos θ cos 2 Q + sin 2 Q tan - 1 2 ( P - Q ) ] - 1 ,
tan ψ = S tan A ,
S = ( tan 2 Q + tan 2 ( P - Q ) + 2 tan ( P - Q ) tan Q cos θ 1 + tan 2 Q tan 2 ( P - Q ) - 2 tan ( P - Q ) tan Q cos θ ) 1 2 .
I / Q = I / P = I / A = 0.
δ Q = ( 2 I Q 2 ) - 1 δ I Q ,             δ P = ( 2 I P 2 ) - 1 δ I P , δ A = ( 2 I A 2 ) - 1 δ I A ,
δ I Q = δ I P = δ I A = δ D
δ Δ P = ( Δ / P ) δ P ,
δ ψ P Q = { [ ( ψ / P ) δ P ] 2 + [ ( ψ / Q ) δ Q ] 2 } 1 2
f ( Q , P , A , ψ , Δ ) = f ( Q , P , - A , ψ , Δ + 180 ° )
f ( Q , P , A , ψ , Δ ) = f ( 180 ° - Q , 180 ° - P , - A , ψ , Δ )
f ( Q , P , A , ψ , Δ ) = f ( 90 ° - Q , 180 ° - P , A , 180 ° - Δ , ψ )
f ( Q , P , A , ψ , Δ ) = f ( 90 ° - Q , 90 ° - P , A + 90 ° , 90 ° - ψ , 180 ° - Δ )