Abstract

The conventional formula for the ellipsometric ratio ρ = rp/rs diverges in the limit when the dielectric constants on either side of an inhomogeneous layer become equal, 1 = 2. The general case, including 1 = 2, necessitates going to second order in the layer thickness. A formula is derived that includes the 1 = 2 case without divergence; the predicted maximum in the imaginary part of ρ when 12 indicates that index matching (of the bounding media) can significantly increase the ellipsometric signal.

© 1988 Optical Society of America

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References

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  1. D. Beaglehole, “Ellipsometry of thin substrates and intruding layers,” submitted by J. Opt. Soc. Am. A.
  2. J. Lekner, “Invariant formulation of the reflection of long waves by interfaces,” Physica 128A, 229–252 (1984).
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  4. D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980).
  5. J. Lekner, “Reflection of long waves by interfaces,” Physica 112A, 544–556 (1982).
  6. J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).
  7. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, 1987).
    [CrossRef]
  8. J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–266 (1912).
    [CrossRef]
  9. J. Lekner, “Reflection of light by a nonuniform film between like media,” J. Opt. Soc. Am. A 3, 9–15 (1986).
    [CrossRef]

1986 (1)

1984 (1)

J. Lekner, “Invariant formulation of the reflection of long waves by interfaces,” Physica 128A, 229–252 (1984).

1982 (2)

J. Lekner, “Reflection of long waves by interfaces,” Physica 112A, 544–556 (1982).

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

1980 (1)

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980).

1912 (1)

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–266 (1912).
[CrossRef]

Azzam, R. M. A.

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

Bashara, N. M.

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

Beaglehole, D.

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980).

D. Beaglehole, “Ellipsometry of thin substrates and intruding layers,” submitted by J. Opt. Soc. Am. A.

Lekner, J.

J. Lekner, “Reflection of light by a nonuniform film between like media,” J. Opt. Soc. Am. A 3, 9–15 (1986).
[CrossRef]

J. Lekner, “Invariant formulation of the reflection of long waves by interfaces,” Physica 128A, 229–252 (1984).

J. Lekner, “Reflection of long waves by interfaces,” Physica 112A, 544–556 (1982).

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, 1987).
[CrossRef]

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–266 (1912).
[CrossRef]

J. Opt. Soc. Am. A (1)

Physica (4)

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980).

J. Lekner, “Reflection of long waves by interfaces,” Physica 112A, 544–556 (1982).

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

J. Lekner, “Invariant formulation of the reflection of long waves by interfaces,” Physica 128A, 229–252 (1984).

Proc. R. Soc. London Ser. A (1)

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–266 (1912).
[CrossRef]

Other (3)

D. Beaglehole, “Ellipsometry of thin substrates and intruding layers,” submitted by J. Opt. Soc. Am. A.

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

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, 1987).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

The ellipsometric parameter ρ ¯ for a thin uniform film (ωΔz/c = 0.05, = 2) between media with dielectric constants 1 = 1 and variable 2. The dashed curve is from Eq. (17), the conventional first-order theory, and diverges at 2 = 1. The solid curve is from expression (25), which is an approximate version of the second-order theory. The points are calculated from the exact reflection amplitudes. The ρ ¯ deduced from the second-order expression. [Eq. (24)] is indistinguishable from the exact ρ ¯ for the small film thickness used here.

Equations (30)

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r s 0 = q 1 q 2 q 1 + q 2 , r p 0 = Q 2 Q 1 Q 2 + Q 1 ,
q 1 2 = 1 ω 2 c 2 K 2 , q 2 2 = 2 ω 2 c 2 K 2 .
( c K / ω ) 2 = 1 sin 2 θ 1 = 2 sin 2 θ 2 ,
r s = r s 0 + r s 1 + r s 2 + , r p = r p 0 + r p 1 + r p 2 + ,
r s 0 = q 1 2 q 2 2 ( q 1 + q 2 ) 2 = Δ ω 2 / c 2 ( q 1 + q 2 ) 2 ,
r p r s = r p 0 r s 0 + r p 1 r s 0 r p 0 r s 1 r s 0 2 + .
r s 1 = 2 i q 1 ω 2 / c 2 ( q 1 + q 2 ) 2 λ 1 ,
r p 1 = 2 i Q 1 ( Q 1 + Q 2 ) 2 ( Q 2 2 λ 1 K 2 1 2 Λ 1 ) ,
λ n = d z ( 0 ) z n 1 ,
Λ n = 1 2 d z ( 1 0 1 ) z n 1 .
r s 0 r p r s = r p 0 2 i Q 1 K 2 1 2 ( Q 1 + Q 2 ) 2 I 1 + ,
I 1 = Λ 1 λ 1 = d z ( 1 ) ( 2 ) .
r p 0 = Δ ( ω 2 / c ) 1 2 ( Q 1 + Q 2 ) 2 [ 1 ( c K ω ) 2 ( 1 1 + 1 2 ) ] .
r p r s = ( q 1 + q 2 ) 2 1 2 ( Q 1 + Q 2 ) 2 { [ 1 ( c K ω ) 2 ( 1 1 + 1 2 ) ] 2 i Q 1 ( c K ω ) 2 I 1 / Δ + } .
( c Q 1 ω ) 2 = ( c Q 2 ω ) 2 = 1 1 + 2 , ( c K ω ) 2 = 1 2 1 + 2 ,
( c q 1 ω ) 2 = 1 2 1 + 2 , ( c q 2 ω ) 2 = 2 2 1 + 2 .
ρ ¯ = ( 1 + 2 ) 1 / 2 2 Δ ω c I 1 + .
r p r s = r p 1 r s 1 + = cos 2 θ 0 Λ 1 λ 1 sin 2 θ 0 + ,
r p r s = r p 0 + r p 1 + r p 2 + r s 0 + r s 1 + r s 2 + .
r s 2 = 2 q 1 ω 2 / c 2 ( q 1 + q 2 ) 2 ( 2 q 2 λ 2 + ω 2 / c 2 q 1 + q 2 λ 1 2 ) ,
r p 2 = 2 Q 1 Q 2 ( Q 1 + Q 2 ) 3 { K 4 Q 2 ( Λ 1 1 2 ) 2 + K 2 [ ( Q 1 Q 2 ) λ 1 Λ 1 1 2 + ( Q 1 + Q 2 ) J ] Q 1 Q 2 2 λ 1 2 2 ( Q 1 Q 2 ) ω 2 c 2 λ 2 } ,
Δ J = j 2 + ( 1 1 + 1 2 ) λ 1 Λ 1 .
i 2 = 2 Δ λ 2 λ 1 2 .
r p r s = ( q 1 + q 2 ) 2 1 2 ( Q 1 + Q 2 ) 2 × ( Δ ) 2 [ 1 ( c K ω ) 2 ( 1 1 + 1 2 ) ] Δ 2 i Q 1 ( c K ω ) 2 I 1 + Δ 2 Q 1 K 2 ( c K / ω ) 2 I 1 2 1 2 ( Q 1 + Q 2 ) 4 q 1 q 2 [ i 2 1 2 ( c K ω ) 2 j 2 + ( 1 1 + 1 2 ) i 2 ] ( Δ ) 2 4 q 1 q 2 i 2 .
r s var 1 1 + 2 i q 0 λ 1 ω 2 / c 2 , r p var 1 1 + 2 i q 0 Λ 1 ω 2 / c 2 .
ρ ¯ 1 2 Δ ( 1 + 2 ) 1 / 2 ω c I 1 ( Δ ) 2 4 1 2 1 + 2 ω 2 c 2 i 2 .
I 1 = ( 1 ) ( 2 ) Δ z , i 2 = ( 1 ) ( 2 ) ( Δ z ) 2 , j 2 = 2 ( 1 ) ( 2 ) ( Δ z ) 2 .
2 1 ( 2 1 ) 1 / 2 ( 1 ) ω c Δ z
2 2 ( 2 1 ) 1 / 2 ( 1 ) ω c Δ z
ρ ¯ m 1 ( 1 ) 4 .

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