Abstract

A least squares method for the precise nulling of an arbitrary ellipsometer arrangement is presented, and the standard errors of the null settings are calculated. The general expressions obtained are applied to the most common null ellipsometers, i.e., the PCSA and PSCA arrangements. For these ellipsometers we arrive at simple closed formulas that show the influence of the noise of the detected light, the extinction ratio of the ellipsometer, and the reflectivity and ellipsometric angles of the sample. In addition, coupling is discussed. It is shown, especially for the PSCA ellipsometer, that coupling has little effect on precision if the least squares method is used.

© 1981 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. G. Wedler, Chemisorption—An Experimental Approach (Butterworths, London, 1976), p. 89.
  3. U. Merkt, P. Wissmann, Thin Solid Films 57, 65 (1979); Z. Phys. Chem. Frankfurt am Main 115, 55 (1979); Surf. Sci. 96, 529 (1980).
    [CrossRef]
  4. E. Schmidt, J. Opt. Soc. Am. 60, 490 (1970). The last three lines of Eq. (1) in this reference should read 0.5RsRp sin2A{cosΔ sin2Q cos2(P-Q)+sin2(P-Q) [cos2Q cos(θ+Δ)-sin2Q cos(θ-Δ)]}.
    [CrossRef]
  5. D. E. As’pnes, Appl. Opt. 14, 1131 (1975).
    [CrossRef] [PubMed]
  6. D. L. Confer, R. M. A. Azzam, N. M. Bashara, Appl. Opt. 15, 2568 (1976).
    [CrossRef] [PubMed]
  7. M. P. Kothiyal, Appl. Opt. 18, 1019 (1979).
    [CrossRef] [PubMed]
  8. As unit for plane angle degree used in the text but radian in the equations.
  9. E. V. Haynsworth, K. Goldberg, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), p. 804.
  10. U. Merkt, Dissertation, U. Erlangen-Nürnberg, 1978.
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 43.
  12. In the PCSA arrangement ψi is given by cos2ψi = −cos2C cos2(P − C).
  13. Ref. 5, Eq. (13).
  14. Ref. 1, p. 385.
  15. The coefficients for the PCSA (Cfixed = arbitrary) ellipsometer can be derived from Eq. (1) in Ref. 4. The results, omitting a common constant 2rI0, are a = sin2ψ cos2A0 + cos2ψ sin2A0, b = sin2ψ sin2C + cos2ψ cos2C − cos2 (P0 − C) cos2C cos2ψ, and c = sin2(P0 − C) sin2A0 cos2C − cos2A0 sin2ψ cos2(P0 − C) sinΔ cos22C. The null settings of the polarizer P0 and analyzer A0 can be calculated using Eqs. (2) and (3) in Ref. 4. With the given coefficients the standard errors can be determined according to Eq. (7) of this paper.
  16. A. Rothen, General Theory and Operating Instructions for the Ellipsometer (Rudolph Research, Fairfield, Conn.1958).
  17. H. G. Jerrard, J. Opt. Soc. Am. 42, 159 (1952).
    [CrossRef]
  18. G. Szivessy, in Handbuch der Physik, H. Geiger, K. Scheel, Eds. (Springer, Berlin, 1928), Vol. 19, Chap. 28, p. 924, Eq. (30).
  19. For an arbitrary ellipsometer the number of steps necessary to null it is n = 1 + ln(∑/σ)/ln(4ab/c2) for each adjustable component.

1979

U. Merkt, P. Wissmann, Thin Solid Films 57, 65 (1979); Z. Phys. Chem. Frankfurt am Main 115, 55 (1979); Surf. Sci. 96, 529 (1980).
[CrossRef]

M. P. Kothiyal, Appl. Opt. 18, 1019 (1979).
[CrossRef] [PubMed]

1976

1975

1970

1952

As’pnes, D. E.

Azzam, R. M. A.

D. L. Confer, R. M. A. Azzam, N. M. Bashara, Appl. Opt. 15, 2568 (1976).
[CrossRef] [PubMed]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

D. L. Confer, R. M. A. Azzam, N. M. Bashara, Appl. Opt. 15, 2568 (1976).
[CrossRef] [PubMed]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 43.

Confer, D. L.

Goldberg, K.

E. V. Haynsworth, K. Goldberg, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), p. 804.

Haynsworth, E. V.

E. V. Haynsworth, K. Goldberg, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), p. 804.

Jerrard, H. G.

Kothiyal, M. P.

Merkt, U.

U. Merkt, P. Wissmann, Thin Solid Films 57, 65 (1979); Z. Phys. Chem. Frankfurt am Main 115, 55 (1979); Surf. Sci. 96, 529 (1980).
[CrossRef]

U. Merkt, Dissertation, U. Erlangen-Nürnberg, 1978.

Rothen, A.

A. Rothen, General Theory and Operating Instructions for the Ellipsometer (Rudolph Research, Fairfield, Conn.1958).

Schmidt, E.

Szivessy, G.

G. Szivessy, in Handbuch der Physik, H. Geiger, K. Scheel, Eds. (Springer, Berlin, 1928), Vol. 19, Chap. 28, p. 924, Eq. (30).

Wedler, G.

G. Wedler, Chemisorption—An Experimental Approach (Butterworths, London, 1976), p. 89.

Wissmann, P.

U. Merkt, P. Wissmann, Thin Solid Films 57, 65 (1979); Z. Phys. Chem. Frankfurt am Main 115, 55 (1979); Surf. Sci. 96, 529 (1980).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 43.

Appl. Opt.

J. Opt. Soc. Am.

Thin Solid Films

U. Merkt, P. Wissmann, Thin Solid Films 57, 65 (1979); Z. Phys. Chem. Frankfurt am Main 115, 55 (1979); Surf. Sci. 96, 529 (1980).
[CrossRef]

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

G. Wedler, Chemisorption—An Experimental Approach (Butterworths, London, 1976), p. 89.

G. Szivessy, in Handbuch der Physik, H. Geiger, K. Scheel, Eds. (Springer, Berlin, 1928), Vol. 19, Chap. 28, p. 924, Eq. (30).

For an arbitrary ellipsometer the number of steps necessary to null it is n = 1 + ln(∑/σ)/ln(4ab/c2) for each adjustable component.

As unit for plane angle degree used in the text but radian in the equations.

E. V. Haynsworth, K. Goldberg, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), p. 804.

U. Merkt, Dissertation, U. Erlangen-Nürnberg, 1978.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 43.

In the PCSA arrangement ψi is given by cos2ψi = −cos2C cos2(P − C).

Ref. 5, Eq. (13).

Ref. 1, p. 385.

The coefficients for the PCSA (Cfixed = arbitrary) ellipsometer can be derived from Eq. (1) in Ref. 4. The results, omitting a common constant 2rI0, are a = sin2ψ cos2A0 + cos2ψ sin2A0, b = sin2ψ sin2C + cos2ψ cos2C − cos2 (P0 − C) cos2C cos2ψ, and c = sin2(P0 − C) sin2A0 cos2C − cos2A0 sin2ψ cos2(P0 − C) sinΔ cos22C. The null settings of the polarizer P0 and analyzer A0 can be calculated using Eqs. (2) and (3) in Ref. 4. With the given coefficients the standard errors can be determined according to Eq. (7) of this paper.

A. Rothen, General Theory and Operating Instructions for the Ellipsometer (Rudolph Research, Fairfield, Conn.1958).

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Figures (4)

Fig. 1
Fig. 1

Intensity transmitted through an ellipsometer near a null (x0,y0). The determination of the square root of the extinction ratio with the analyzer azimuth y is indicated by a double-headed arrow. The inset shows the quadratic calculation mesh (N = 3).

Fig. 2
Fig. 2

Influence of the step width s on the standard errors σx and σy of the null settings. The optimal step width sopt and the corresponding errors are given in Eqs. (8) and (9), respectively.

Fig. 3
Fig. 3

Precision functions of the (a) PCA and (b) PSCA ellipsometers.

Fig. 4
Fig. 4

Geometrical illustration of the successive minima leading to a null of the PSCA ellipsometer: ∑ denotes the initial azimuth CC0 of the compensator, σ the ultimate precision attainable.

Tables (1)

Tables Icon

Table I Expansion Coefficients and Precision Functions for the PCSA and PSCA Ellipsometers

Equations (18)

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I ( x , y ) = a x 2 + b y 2 + c x y + d x + e y + f .
σ I 2 = 1 N 2 - 6 i , j = 1 N [ I ( x i , y j ) - I i j ] 2 .
x 0 = e c - 2 b d 4 a b - c 2 ,             y 0 = d c - 2 a e 4 a b - c 2 ,
σ x = σ I 4 a b - c 2 [ 16 b 2 x 0 2 + 4 c 2 y 0 s 0 ( s 0 s 4 - s 2 2 ) + ( e + 2 c x 0 ) 2 s 2 2 + 4 b 2 + c 2 s 0 2 s 2 ] 1 / 2 , s n = i = 1 N x i n = i = 1 N y i n .
c I = σ I / I av .
I a v = ( a + b ) s 2 N 2 / 12 + f
σ x = c I N 3 / 2 f ( a + b ) ( 4 b 2 + c 2 ) 4 a b - c 2 ( s s opt + s opt s ) ,
s opt = 1 N 12 f a + b .
σ x , y = 2 c I N - 3 / 2 ( f / b ) 1 / 2 F x , y ( a / b , c / b ) .
γ = f 2 r I 0 ( sin 2 ψ sin 2 ψ i + cos 2 ψ cos 2 ψ i ) - 1 .
α = - arctan 4 a b - c 2 2 c ( a + b )
I = r I 0 2 [ 1 - cos 2 ψ cos 2 A - sin 2 ψ sin 2 A sin ( Δ - 2 P ) ] .
σ P , A = 2 c I N - 3 / 2 γ 1 / 2 F P , A ( ψ )
σ A = c I γ 1 / 2 ,             σ p = σ A sin - 1 2 ψ
tan Δ = tan 2 sin - 2 2 β ,             cos 2 ψ = cos 2 cos 2 β .
σ C , A = 2 c I N - 3 / 2 γ 1 / 2 F C , A ( ) ,
σ A = c I γ 1 / 2 ,             σ C = σ A ( 1 + cos 2 2 ) - 1 / 2 .
n = 1 + ln ( Σ / σ ) ln ( 1 + cos 2 2 ) .

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