Abstract

Errors and error sources occurring in rotating-analyzer ellipsometry are discussed. From general considerations it is shown that a rotating-analyzer ellipsometer is inaccurate if applied at P = 0° and in cases when ψ = 0° or where Δ is near 0° or 180°. Window errors, component imperfections, azimuth errors and all other errors may, to first order, be treated independently and can subsequently be added. Explicit first-order expressions for the errors δΔ and δψ caused by windows, component imperfections, and azimuth errors are derived, showing that all of them, except the window errors, are eliminated in a two-zone measurement. Higher-order errors that are due to azimuth errors are studied numerically, revealing that they are in general less than 0.1°. Statistical errors are also discussed. Errors caused by noise and by correlated perturbations, i.e., periodic fluctuations of the light source, are also considered. Such periodic perturbations do cause random errors, especially when they have frequencies near 2ωA and 4ωA.

© 1988 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1971).
    [Crossref]
  2. R. M. A. Azzam, N. M. Bashara, “Unified analysis of ellipsometry errors due to imperfect components, cell-window birefringence, and incorrect azimuth angles,” J. Opt. Soc. Am. 61, 600–607 (1971).
    [Crossref]
  3. R. M. A. Azzam, N. M. Bashara, “General treatment of the effect of cell windows in ellipsometry,” J. Opt. Soc. Am. 61, 773–776 (1971).
    [Crossref]
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  5. D. E. Aspnes, “Measurement and correction of first-order errors in ellipsometry,” J. Opt. Soc. Am. 61, 1077–1085 (1971).
    [Crossref]
  6. A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
    [Crossref]
  7. F. L. McCrackin, “Analyses and corrections of instrumental errors in ellipsometry,” J. Opt. Soc. Am. 60, 57–63 (1970).
    [Crossref]
  8. R. H. Muller, “Present status of automatic ellipsometers,” Surf. Sci. 56, 19–36 (1976).
    [Crossref]
  9. P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
    [Crossref]
  10. D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [PubMed]
  11. Y. I. van der Meulen, N. C. Hien, “Design and operation of an automated, high-temperature ellipsometer,” J. Opt. Soc. Am. 64, 804–811 (1974).
    [Crossref]
  12. R. M. A. Azzam, N. M. Bashara, “Analysis of systematic errors in rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 1459–1469 (1974).
    [Crossref]
  13. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
    [Crossref]
  14. D. E. Aspnes, “Effects of component optical activity in data reduction and calibration of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 812–819 (1974).
    [Crossref]
  15. G. A. Korr, T. M. Kom, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

1980 (1)

A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
[Crossref]

1976 (2)

R. H. Muller, “Present status of automatic ellipsometers,” Surf. Sci. 56, 19–36 (1976).
[Crossref]

P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[Crossref]

1975 (1)

1974 (4)

1971 (4)

1970 (1)

Aspnes, D. E.

Azzam, R. M. A.

Bashara, N. M.

Bootsma, G. A.

A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
[Crossref]

Hanekamp, L. J.

A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
[Crossref]

Hauge, P. S.

P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[Crossref]

Hien, N. C.

Kom, T. M.

G. A. Korr, T. M. Kom, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

Korr, G. A.

G. A. Korr, T. M. Kom, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

McCrackin, F. L.

Muller, R. H.

R. H. Muller, “Present status of automatic ellipsometers,” Surf. Sci. 56, 19–36 (1976).
[Crossref]

Straaijer, A.

A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
[Crossref]

Studna, A. A.

van der Meulen, Y. I.

Appl. Opt. (1)

J. Opt. Soc. Am. (9)

Surf. Sci. (3)

R. H. Muller, “Present status of automatic ellipsometers,” Surf. Sci. 56, 19–36 (1976).
[Crossref]

P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[Crossref]

A. Straaijer, L. J. Hanekamp, G. A. Bootsma, “The influence of cell window imperfections on the calibration and measured data of two types of rotating-analyzer ellipsometers,” Surf. Sci. 96, 217–231 (1980).
[Crossref]

Other (2)

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

G. A. Korr, T. M. Kom, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of an ideal ellipsometer. The optical part of the system consists of a polarizer, an optional compensator, the sample, and an analyzer. In the case of a RAE the analyzer rotates with constant angular velocity. The angles P, C, and A are the angles of the characteristic axes of the components measured relative to the plane of incidence.

Fig. 2
Fig. 2

Errors caused by a polarizer azimuth error δP = 1°. If a two-zone measurement is performed, errors δΔ and δψ caused by an azimuth error of the polarizer δP are eliminated to first order. This plot shows the remaining higher-order error [expressed as (δη)2 = (δΔ)2 + (δψ)2] in the ψP plane; Δ has no influence.

Fig. 3
Fig. 3

The error δη caused by a badly aligned analyzer as a function of Δ and for some arbitrary values for ψ and P. All curves clearly exhibit minima for Δ = 0°, 90° and maxima for Δ = 45°, 135°.

Fig. 4
Fig. 4

Second-order errors δη caused by bad alignment of the analyzer (δA = 1°). The picture presents the lower and upper bounds for the error δη in the ψP plane, denoted by the dashed and solid lines, respectively.

Fig. 5
Fig. 5

Second-order errors δη caused by a misaligned compensator (δC = 1°). δη is independent of Δ and only slightly dependent on ψ and consequently is depicted in the QP plane. Solid lines denote the results for ψ = 45°, and dashed lines correspond to ψ = 30°.

Fig. 6
Fig. 6

The generalized sinc(x; ϕ) function as defined by Eq. (34) for ϕ = 0 and ϕ = π/2 rad. The figure clearly shows the localized and bounded character of the function.

Tables (1)

Tables Icon

Table 1 First-Order Errors δΔ and δψ Caused by Polarizer Imperfections, Azimuth Errors, and Windows

Equations (55)

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I = I 0 [ 1 C 2 P C 2 ψ + ( C 2 P C 2 ψ ) cos 2 ω A t + S 2 P S 2 ψ C Δ sin 2 ω A t ] ,
I det ( t ) = α 0 + α c cos 2 ω A t + α s sin 2 ω A t .
α 0 = I det ,
α c + i α s = 2 τ 0 τ I det ( t ) exp ( i 2 ω A t ) d t , τ = 4 π ω A N ,
C 2 P C 2 ψ 1 C 2 P C 2 ψ = α c α 0 ,
S 2 P S 2 ψ C Δ 1 C 2 P C 2 ψ = α s α 0 .
[ δ Δ δ ψ ] = 1 C 2 P C 2 ψ S 2 P 2 S 2 ψ 2 S Δ [ ( C 2 ψ C 2 P ) C Δ S 2 P S 2 ψ ½ ( 1 C 2 P C 2 ψ ) S 2 ψ S Δ 0 ] [ δ α ̂ c δ α ̂ s ] .
[ δ Δ δ ψ ] = 1 S 2 P 2 S 2 ψ 2 S Δ [ ( 1 C 2 P C 2 ψ ) C Δ ( C 2 ψ C 2 P ) C Δ S 2 P S 2 ψ ½ ( C 2 ψ C 2 P ) S 2 ψ S Δ ½ ( 1 C 2 P C 2 ψ ) S 2 ψ S Δ 0 ] [ δ α 0 δ α c δ α s ] ,
S f ° = T A ° · ( A ° ) T S ° T C ° 1 ( P ° ) T P ° S i ° ,
S f = T A ( A ) T w T S T w 1 ( C ) T C ( C ) 1 ( P ) T P S i .
( θ ) = ( θ ° ) + d d θ δ θ + 1 2 d 2 d θ 2 δ θ 2 + .
T ( ξ ) = T ° + d T d ξ δ ξ + 1 2 d 2 T d ξ 2 δ ξ 2 + .
S f = S f ° + k δ S f k + k , l k l δ S f k , l + k δ S f k , k + ,
S f S f ° + k δ S f k = S f ° + [ T A ° ( A ° ) T S ° T C ° 1 ( P ° ) d T P d γ P δ γ P + T A ° ( A ° ) T S ° T C ° d 1 d P δ P T P ° + . + d T A d γ A δ γ A ( A ° ) T S ° T C ° 1 ( P ° ) T P ° ] S i ° ,
δ 1 · S f = ( S f ) 0 = I det = δ 1 T A ° · ( A ° ) S w = ( 1 , C 2 A , S 2 A , 0 ) S w ,
S w = [ α 0 α c α s ] , δ S w = [ δ α 0 δ α c δ α s ] .
1 1 + γ 2 [ 1 i γ + i γ γ 2 ] ,
T P = ½ [ 1 + 2 γ 2 1 0 + 2 γ 1 1 2 γ 2 0 + 2 γ 0 0 4 γ 2 0 + 2 γ + 2 γ 0 0 ] .
δ T p = γ [ 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 ] .
S P = ( 1 , 1 , 0 , 0 ) + 2 γ p ( 0 , 0 , 0 , 1 ) + 0 ( γ 2 ) .
δ S f p = ( 1 , C 2 A , S 2 A , 0 ) T S ° T C ° 1 ( P ° ) δ T P S i ° .
[ δ α 0 δ α c δ α s ] = + 2 γ P S 2 ψ [ 0 0 S Δ + Q C Δ + Q ] ,
[ δ Δ δ ψ ] = 2 γ P [ 1 / S 2 P 0 ]
[ δ α 0 δ α c δ α s ] = 2 γ A [ S 2 P S 2 ψ S Δ + Q 0 0 ] ,
[ δ Δ δ ψ ] = γ A S 2 P S 2 ψ [ 2 ( 1 C 2 P C 2 ψ ) C Δ + Q ( C 2 ψ C 2 P ) S 2 ψ S Δ + Q ] .
δ ( θ ) = [ 0 0 0 0 0 sin θ cos θ 0 0 cos θ sin θ 0 0 0 0 0 ] .
[ δ α 0 δ α c δ α s ] = T S ° T C ° δ ( P ) T P ° S i ° = 2 δ P [ C 2 ψ S 2 P S 2 P S 2 ψ C Δ + Q C 2 P S 2 ψ S Δ + Q C 2 P ] ,
[ δ Δ δ ψ ] = δ P [ 0 S 2 ψ / S 2 P ] .
δ Δ = δ Δ ( P , ψ ) , δ ψ = δ ψ ( P , ψ ) .
δ I det = δ 1 T A d d A δ A T S ° T C ° 1 ( P ° ) T P ° S i ° = 2 δ A ( 0 , S 2 A , C 2 A , 0 ) T S ° T C ° 1 ( P ° ) T P ° S i ° .
[ α 0 α c α s ] = 2 δ A [ 0 S 2 ψ C Δ + Q S 2 P C 2 ψ C 2 P ] .
[ δ Δ δ ψ ] = δ A S 2 P S 2 ψ [ 2 ( C 2 P C 2 ψ ) S Δ + Q ( 1 C 2 P C 2 ψ ) S 2 ψ C Δ + Q ] ,
[ δ α 0 δ α c δ α s ] = T s ° ( d 1 d c T C + T C d d C ) δ C ( P ° ) T P ° S i ° = [ ( 1 C Q ) C 2 ψ S 2 P ( 1 C Q ) S 2 P ( C Δ C Δ + Q ) C 2 P ] δ C .
[ δ Δ δ ψ ] = δ C S 2 P [ 2 S Q C 2 P ( 1 C Q ) S 2 ψ ] .
T w = 1 + [ 0 0 0 0 0 0 0 S 2 ϕ w 0 0 0 C 2 ϕ w 0 S 2 ϕ w C 2 ϕ w 0 0 ] δ w ,
[ δ α 0 δ α c δ α s ] = [ S Q C 2 ψ S 2 P S 2 ϕ w S Q S 2 P S 2 ϕ w { S Δ S 2 ψ C 2 P S 2 ϕ w S Δ + Q S 2 ψ S 2 P C 2 ϕ w } ] δ w .
[ δ Δ δ ψ ] = δ w S 2 P [ S 2 P C 2 ϕ w C Q C 2 P S 2 ϕ w ½ S Q S 2 ψ S 2 ϕ w ] .
[ δ α 0 δ α c δ α s ] = S Δ + Q S 2 P S 2 ψ [ 0 S 2 ϕ C 2 ϕ w ] δ w ,
[ δ Δ δ ψ ] = δ w S 2 ψ S 2 P [ C Δ + Q ( C 2 ψ C 2 P ) S 2 ϕ w + S 2 ψ S 2 P C 2 ϕ w ½ S Δ + Q S 2 ψ ( 1 C 2 P C 2 ψ ) S 2 ϕ w ] .
σ c = σ s = ( 2 σ 0 ) 1 / 2 .
σ 0 ( τ I 0 ) 1 / 2 .
σ f 2 = ( δ f δ x ) 2 σ x 2 + ( δ f δ y ) 2 σ y 2 + ,
[ σ Δ 2 σ ψ 2 ] = 1 S 2 P 4 S 2 ψ 4 S Δ 2 [ ( C 2 ψ C 2 P ) 4 + S 2 P 4 S 2 ψ 4 ( 1 C 2 P C 2 ψ ) 2 C Δ 2 ( C 2 ψ C 2 P ) 2 C Δ 2 S 2 ψ 2 S 2 P 2 ¼ ( C 2 ψ C 2 P ) 2 S 2 ψ 2 S Δ 2 ¼ ( 1 C 2 P C 2 ψ ) 2 S 2 ψ 2 S Δ 2 0 ] [ σ 0 2 σ c 2 σ s 2 ] .
I ( t ) = I ° + δ I ( ω ) cos ( ω t + ϕ ) ,
I det ( t ) = [ I ° + δ I ( ω ) cos ( ω t + ϕ ) ] [ 1 C 2 P C 2 ψ + ( C 2 P C 2 ψ ) cos 2 ω A t + S 2 P S 2 ψ C Δ sin 2 ω A t ] .
α 0 = ( 1 C 2 P C 2 ψ ) I ° + ( 1 C 2 P C 2 ψ ) δ I ( ω ) sinc ( ω τ ; ϕ ) × ( C 2 P C 2 ψ ) δ I ( ω ) sinc [ ( 2 ω A ω ) τ ; ϕ ] × S 2 P S 2 ψ C Δ δ I ( ω ) sinc [ ( 2 ω A ω ) τ ; ϕ + π 2 ] ,
α c = ( C 2 P C 2 ψ ) I ° + ( 1 C 2 P C 2 ψ ) δ I ( ω ) sinc [ ( 2 ω A ω ) τ ; ϕ ] × ( C 2 P C 2 ψ ) δ I ( ω ) { sinc ( ω τ ; ϕ ) sinc [ ( 4 ω A ω ) τ ; ϕ ] } × S 2 P S 2 ψ C Δ δ I ( ω ) sinc [ ( 4 ω A ω ) τ ; ϕ + π 2 ] ,
α s = S 2 P S 2 ψ C Δ I ° + ( 1 C 2 P C 2 ψ ) δ I ( ω ) sinc [ ( 2 ω A ω ) τ ; ϕ + π 2 ] × ( C 2 P C 2 ψ ) δ I ( ω ) sinc [ ( 4 ω A ω ) τ ; ϕ + π 2 ] × S 2 P S 2 ψ C Δ δ I ( ω ) { sinc [ ( 4 ω A ω ) τ ; ϕ ] + sinc ( ω τ ; ϕ ) } .
sinc ( x ; ϕ ) = sin ( x + ϕ ) sin ϕ x .
sinc ( x , ϕ ) 0 if | x | > 8 π .
0 < ω < 2 ω A N , 2 ω A 2 ω A N < ω < 2 ω A + 2 ω A N , 4 ω A 2 ω A N < ω < 4 ω A + 2 ω A N ,
S 2 P C 2 ϕ w C Q C 2 P S 2 ϕ w S 2 P δ w
S Q S 2 ψ S 2 ϕ w 2 S 2 P δ w
C Δ + Q ( C 2 ψ C 2 P ) S 2 ϕ w + S 2 ψ S 2 P C 2 ϕ w S 2 ψ S 2 P δ w
S Δ + Q S 2 ϕ w 2 S 2 P ( 1 C 2 P C 2 ψ ) δ w

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