Abstract

The dielectric constant of a uniform unsupported or embedded layer is shown to satisfy a cubic equation with coefficients determined by the angle of incidence and the measured complex ellipsometric ratio ρ = rp/rs. Analytic inversion is thus possible. The consequence of measurement errors on the deduced dielectric constant and layer thickness is explored.

© 1990 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. F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
    [Crossref]
  3. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Sect. 2–4.
  4. M. C. Dorf, J. Lekner, “Reflection and transmission ellipsometry of a uniform layer,” J. Opt. Soc. Am. A 4, 2096–2100 (1987).
    [Crossref]
  5. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions, No. 55 of the U.S. National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), Sec. 3.8.2.
  6. D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), Chap. 15.
  7. D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980).
  8. R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).
  9. A. R. Reinberg, “Ellipsometer data analysis with a small programmable desk calculator,” Appl. Opt. 11, 1273–1274 (1972).
    [Crossref] [PubMed]

1987 (1)

1983 (1)

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

1980 (1)

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

1972 (1)

1963 (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), Chap. 15.

Azzam, R. M. A.

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

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).

Dorf, M. C.

Lekner, J.

M. C. Dorf, J. Lekner, “Reflection and transmission ellipsometry of a uniform layer,” J. Opt. Soc. Am. A 4, 2096–2100 (1987).
[Crossref]

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Sect. 2–4.

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Reinberg, A. R.

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Stromberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Appl. Opt. (1)

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

J. Phys. (Paris) (1)

R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (Paris) C10, 67–70 (1983).

J. Res. Natl. Bur. Stand. Sect. A (1)

F. L. McCrackin, E. Passaglia, R. R. Stromberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67A, 363–377 (1963).
[Crossref]

Physica (1)

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

Other (4)

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

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Sect. 2–4.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions, No. 55 of the U.S. National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1964), Sec. 3.8.2.

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), Chap. 15.

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Figures (3)

Fig. 1
Fig. 1

Inversion of computer-generated values of ρ = x + iy for a uniform layer of dielectric constant 2.25 and thickness parameter ωΔz/c = 0.3. In this case there are three real roots of Eq. (10) for ɛ/ɛ0; one less than sin2θ0 has been discarded. Of the two remaining, one is nonphysical since it leads to variable ɛ and Δz (dotted curves). The input values of ɛ and Δz are reproduced by the other solution (solid lines).

Fig. 2
Fig. 2

Scatter in ɛ/ɛ0 and ωΔz/c values produced by solving Eqs. (10) and (16) if random errors are introduced into the ellipsometric data. Here uniformly distributed errors of up to 0.1° in Ψ and 0.2° in Δ were put in at the start of the inversion process. Note the large scatter near normal incidence.

Fig. 3
Fig. 3

As for Fig. 2 but with random errors δx and δy (instead of δΨ and δΔ), uniformly distributed up to as 0.002.

Equations (19)

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ρ = x + i y = tan Ψ exp ( i Δ ) .
ρ = p s 1 s 2 Z 1 p 2 Z , Z exp ( 2 i q Δ z ) ,
s = q 0 q q 0 + q , p = Q Q 0 Q + Q 0 .
Z = ρ s p p s ( ρ p s ) .
( ρ s p ) ( ρ * s p ) = p 2 s 2 ( ρ p s ) ( ρ * p s ) .
| ρ | 2 ( ρ + ρ * ) p s 1 ( p s ) 2 1 p 4 + ( p s ) 2 1 s 4 1 p 4 = 0.
p s 1 s 2 1 p 2 = 1 ( 1 + f ) sin 2 θ 0 F ,
p s 1 + s 2 1 + p 2 = F 1 + f 2 f sin 2 θ 0 1 + f ( 1 + f 2 ) sin 2 θ 0 ,
2 p s 1 ( p s ) 2 1 p 4 = F ( 1 + f ) [ 2 ( 1 + f ) sin 2 θ 0 ] 1 + f ( 1 + f 2 ) sin 2 θ 0 ,
a 0 + a 1 g + a 2 g 2 + a 3 g 3 = 0.
ρ = x + i y , sin 2 θ 0 = σ .
a 0 = σ 2 ( 1 2 σ x ) , a 1 = σ [ 2 + 7 σ 4 σ 2 + 3 x ( 1 σ ) x 2 y 2 ] , a 2 = ( 1 σ ) ( 1 5 σ + 2 σ 2 ) x ( 2 6 σ + 3 σ 2 ) + x 2 + y 2 , a 3 = ( 1 σ ) [ 1 σ x ( 2 σ ) + x 2 + y 2 ] .
u = ( 3 a 1 a 3 a 2 2 ) / 9 a 3 2 , υ = ( 9 a 1 a 2 a 3 27 a 0 a 3 2 2 a 2 3 ) / 54 a 3 3
t 1 = ( υ + w ) 1 / 3 , t 2 = ( υ w ) 1 / 3 ,
s a 2 / 3 a 3 , s / 2 a 2 / 3 a 3 ± i ( 3 / 2 ) d ,
ε ε 0 = 2 ( u ) 1 / 2 cos ( ϕ + 2 π m 3 ) a 2 3 a 3 ,
Δ z = ( 2 q ) 1 arctan y ( p 2 s 2 ) p s ( x 2 + y 2 ) ( p 2 + s 2 ) x + p s .
δ x = 1 + x 2 + y 2 ( x 2 + y 2 ) 1 / 2 x δ Ψ y δ Δ , δ y = 1 + x 2 + y 2 ( x 2 + y 2 ) 1 / 2 y δ Ψ + x δ Δ .
y x s 2 sin 2 q Δ z 1 s 2 cos 2 q Δ z , s 2 ( ε ε 0 ε + ε 0 ) 2 ,

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