Abstract

The dielectric constant of an isotropic homogeneous layer on an isotropic substrate is shown to satisfy a quintic equation, with the coefficients determined by |ρ|2 and Re(ρ), where ρ = r p/r s, is found ellipsometrically. This algebraic equation eliminates many (but not all) of the nonphysical roots in the inversion of ellipsometric data. A simple form is obtained if the angle of incidence is equal to the Brewster angle of the substrate. The problem of inversion for thin films is also discussed.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. J. Lekner, “Ellipsometry of anisotropic media,” J. Opt. Soc. Am. A 10, 1579–1581 (1993).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  4. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).
  5. D. Charlot, A. Maruani, “Ellipsometric data processing: an efficient method and an analysis of the relative errors,” Appl. Opt. 24, 3368–3373 (1985).
    [CrossRef] [PubMed]
  6. M. C. Dorf, J. Lekner, “Reflection and transmission ellipsometry of a uniform layer,” J. Opt. Soc. Am. A 4, 2096–2100 (1987).
    [CrossRef]
  7. R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983).
    [CrossRef]
  8. R. M. A. Azzam, “Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air,” Appl. Opt. 30, 2801–2806 (1991).
    [CrossRef] [PubMed]
  9. J. Lekner, “Inversion of transmission ellipsometric data for transparent films,” Appl. Opt. 33, (1994).
    [CrossRef] [PubMed]
  10. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  11. J. Lekner, “Ellipsometry of a thin film between similar media,” J. Opt. Soc. Am. A 5, 1041–1043 (1988).
    [CrossRef]
  12. H. Hilton, Plane Algebraic Curves (Clarendon, Oxford, 1920).

1994 (1)

J. Lekner, “Inversion of transmission ellipsometric data for transparent films,” Appl. Opt. 33, (1994).
[CrossRef] [PubMed]

1993 (1)

1991 (1)

1990 (1)

1988 (1)

1987 (1)

1985 (1)

1983 (1)

Azzam, R. M. A.

Bashara, N. M.

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, Oxford, 1965).

Charlot, D.

Dorf, M. C.

Hilton, H.

H. Hilton, Plane Algebraic Curves (Clarendon, Oxford, 1920).

Lekner, J.

Maruani, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

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

H. Hilton, Plane Algebraic Curves (Clarendon, Oxford, 1920).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Paths of ρ = r p /r s = tan ψ exp iΔ for variable thicknesses of the layer, at fixed angle of incidence θ1 = 60°: (a) ρ = x + iy plane, (b) ψΔ plane. The layer index is 1.33 or 1.36 as marked, n 1 = 1, and the substrate index is 1.50. The layer-thickness values, indicated in increments of 10 nm, correspond to a wavelength of 633 nm.

Fig. 2
Fig. 2

Same as Fig. 1(a), with the refractive index of the layer varying from 1.1 to 1.9 in steps of 0.1. Note that at n = n 2(= 1.5) the curve collapses to the point ρ+ (the layer is optically undetectable). The indices are not shown on the curves inside the n = 1.9 locus, of which the smallest is n = 1.6. Note the directions of increasing thickness, indicated by the arrows.

Fig. 3
Fig. 3

Curves of ρ = x + iy for thin films of water on glass (n 1 = 1, n = 1.33, n 2 = 1.5). As the angle of incidence varies from 0° to 90°, ρ moves from (1, 0) to (−1, 0) in the complex plane. The film thicknesses are such that (ω/cz = 0.2, 0.4, and 0.6; for λ = 633 nm this corresponds to film thicknesses of approximately 20, 40, and 60 nm. The solid curves are calculated from Eq. (1), and the dashed curves are calculated from Eq. (34).

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

ρ = r p r s = p 1 + p 2 Z 1 + p 1 p 2 Z 1 + s 1 s 2 Z s 1 + s 2 Z ,
s 1 = q 1 - q q 1 + q ,             s 2 = q - q 2 q + q 2 , p 1 = Q - Q 1 Q + Q 1 ,             p 2 = Q 2 - Q Q 2 + Q .
q 1 2 = 1 ( ω / c ) 2 - K 2 , q 2 = ( ω / c ) 2 - K 2 , q 2 2 = 2 ( ω / c ) 2 - K 2 .
c K / ω = n 1 sin θ 1 = n sin θ = n 2 sin θ 2 .
Z = exp ( 2 i q Δ z ) .
α Z 2 + β Z + γ = 0 ,
α = p 2 s 2 ( ρ p 1 - s 1 ) , β = ρ ( p 1 s 1 p 2 + s 2 ) - p 2 - p 1 s 1 s 2 , γ = ρ s 1 - p 1 .
( α 2 - γ 2 ) 2 - α * β - β * γ 2 = 0.
ρ 2 = 4 p 1 s 1 p 2 s 2 c 2 + 2 { p 1 p 2 [ 1 + ( s 1 s 2 ) 2 ] + s 1 s 2 ( p 1 2 + p 2 2 ) } c + ( p 1 2 + p 2 2 ) [ 1 + ( s 1 s 2 ) 2 ] 4 p 1 s 1 p 2 s 2 c 2 + 2 { p 1 p 2 ( s 1 2 + s 2 2 ) + s 1 s 2 [ 1 + ( p 1 p 2 ) 2 ] } c + ( s 1 2 + s 2 2 ) [ 1 + ( p 1 p 2 ) 2 ] ,
Re ( ρ ) = 2 ( p 1 2 + s 1 2 ) p 2 s 2 c 2 + [ p 1 s 2 ( 1 + s 1 2 ) ( 1 + p 2 2 ) + s 1 p 2 ( 1 + p 1 2 ) ( 1 + s 2 2 ) ] c + p 1 s 1 ( 1 + p 2 2 ) ( 1 + s 2 2 ) + p 2 s 2 ( 1 - p 1 2 ) ( 1 - s 1 2 ) 4 p 1 s 1 p 2 s 2 c 2 + 2 { p 1 p 2 ( s 1 2 + s 2 2 ) + s 1 s 2 [ 1 + ( p 1 p 2 ) 2 ] } c + ( s 1 2 + s 2 2 ) [ 1 + ( p 1 p 2 ) 2 ] .
A 1 c 2 + B 1 c + C 1 = 0 ,             A 2 c 2 + B 2 c + C 2 = 0.
A 1 ( c - c 0 ) ( c - c 1 ) = 0 ,             A 2 ( c - c 0 ) ( c - c 2 ) = 0.
B 1 / A 1 = - ( c 0 + c 1 ) , C 1 / A 1 = c 0 c 1 , B 2 / A 2 = - ( c 0 + c 2 ) , C 2 / A 2 = c 0 c 2 .
( A 1 B 2 - B 1 A 2 ) ( B 1 C 2 - C 1 B 2 ) - ( C 1 A 2 - A 1 C 2 ) 2 = 0 ,
c 0 = C 1 A 2 - A 1 C 2 A 1 B 2 - B 1 A 2 = B 1 C 2 - C 1 B 2 C 1 A 2 - A 1 C 2 .
( ω / c ) 2 = q 2 + K 2 ,
q 8 ( q - q 1 ) 4 ( q - q 2 ) 4 ( q 2 + K 2 ) 2 ( q q 2 - K 2 ) × ( q 1 - q 2 ) 2 ( q 1 + q 2 ) 2 ( q 1 2 + K 2 ) ,
Δ z = 1 2 q arccos ( c 0 ) .
Q 1 = Q 2 = ω / c ( 1 + 2 ) 1 / 2 ,             K = ω c ( 1 2 1 + 2 ) 1 / 2 , q 1 = ω c 1 ( 1 + 2 ) 1 / 2 ,             q 2 = ω c 2 ( 1 + 2 ) 1 / 2 , q = ω c [ ( 1 + 2 ) - 1 2 1 + 2 ] 1 / 2 .
r 2 ρ 2 ,             x Re ( ρ ) ,
f 1 = ( 1 + 2 ) - 1 2 ,             f 2 = ( 1 + 3 2 ) + 2 ( 2 - 1 ) .
( f 1 - 1 2 ) ( 2 + f 1 ) f 2 2 r 4 + ( 1 + 2 ) ( 2 - ) × f 2 [ ( 3 1 + 5 2 ) 3 + ( 1 2 - 4 1 2 + 3 2 2 ) 2 - 1 2 ( 5 1 + 7 2 ) + ( 2 1 2 ) 2 ] r 2 x + ( 1 + 2 ) 2 ( 2 - ) 2 [ ( 3 1 + 2 ) 2 - ( 1 2 + 4 1 2 - 2 2 ) + 1 2 ( 1 - 2 ) ] r 2 + 2 2 ( 1 + 2 ) 2 ( - 1 ) ( 2 - ) 2 ( 3 2 + f 1 ) x 2 + ( 1 + 2 ) 3 ( - 1 ) ( 2 - ) 3 x = 0.
a r 4 + b r 2 x + c r 2 + d x 2 + e x + f = 0 ,
ρ ± = ρ ( Z = ± 1 ) = p 1 ± p 2 1 ± p 1 p 2 1 ± s 1 s 2 s 1 ± s 2 .
ρ + = q 1 q 2 - K 2 q 1 q 2 + K 2 ,
ρ - = ( q 2 + q 1 q 2 ) ( q 1 q 2 2 - q 2 1 2 ) ( q 2 - q 1 q 2 ) ( q 1 q 2 2 + q 2 1 2 ) = ( q 2 + q 1 q 2 ) ( Q 1 Q 2 - Q 2 ) ( q 2 - q 1 q 2 ) ( Q 1 Q 2 + Q 2 ) .
K 2 = 1 2 - 2 1 + 2 - 2 ( ω c ) 2 ,
tan 2 θ 0 = 1 2 - 2 ( 1 - ) 2 .
= ( K 2 + q 1 q 2 ) ( c ω ) 2 = 1 [ sin 2 θ 1 + cos θ 1 ( 2 / 1 - sin 2 θ 1 ) 1 / 2 ] .
ρ = tan ψ exp ( i Δ ) ,             0 ψ π / 2.
ρ + = ρ ( Z = 1 ) = q 1 q 2 - K 2 q 1 q 2 + K 2 = p 0 s 0 ,
I 1 = ( 1 - ) ( - 2 ) Δ z .
p 1 + p 2 1 + p 1 p 2 1 + s 1 s 2 s 1 + s 2 = p 0 s 0 ,
( p 2 p 1 + p 2 - p 1 p 2 1 + p 1 p 2 + s 1 s 2 1 + s 1 s 2 - s 2 s 1 + s 2 ) q Δ z = ( Q 1 Q 1 2 - Q 2 2 ) K 2 I 1 1 2 ,
r p r s = p 0 s 0 - 1 s 0 2 i Q 1 K 2 I 1 1 2 ( Q 1 + Q 2 ) 2 + .
I 1 = d z ( 1 - ) ( - 2 ) .
a r 4 + b r 2 x + c r 2 + d x 2 + e x + f = 0.
q 2 = q 1 q 2 u ,             K 2 = q 1 q 2 v ,             q 2 = q 1 w .
a = ( u - 1 ) ( v + 1 ) { u [ v + w + v w ( 2 + v + w ) ] + ( u 2 + v 2 ) w } [ u ( v + 2 w + v w ) + v w ( 1 + 2 v + w ) ] 2 , b = 2 [ u ( v + 2 w + v w ) + v w ( 1 + 2 v + w ) ]             ( quartic in u ) , c = quintic in u , d = 4 ( v + w ) ( v 2 w - u ) { u 2 ( v - 2 ) w + u [ v + 2 w + v w ( - 4 + 2 v + w ) ] + v w ( 1 - 2 v ) } { u 2 ( v + 2 ) w + u [ v + 2 w + v w ( 4 + 2 v + w ) ] + v w ( 1 + 2 v ) } , e = - 2 [ u ( v + 2 w - v w ) + v w ( 1 - 2 v - w ) ]             ( quartic in u ) , f = ( u + 1 ) ( v - 1 ) { u [ v + w - v w ( 2 - v - w ) ] - ( u 2 + v 2 ) w } [ u ( v + 2 w - v w ) + v w ( 1 - 2 v - w ) ] 2 .
ρ + = 1 - v 1 + v , ρ - = ( u + 1 ) [ ( u + v ) 2 w - u ( 1 + v w ) ( v + w ) ] ( u - 1 ) [ ( u + v ) 2 w + u ( 1 + v w ) ( v + w ) ] .
x 2 = u ( v + 2 w - v w ) + v w ( 1 - 2 v - w ) u ( v + 2 w + v w ) + v w ( 1 + 2 v + w ) .
a x 4 + b x 3 + ( c + d ) x 2 + e x + f = a ( x - ρ + ) ( x - ρ - ) ( x - x 2 ) 2 .
b = - a ( ρ + + ρ - + 2 x 2 ) , c + d = a [ ρ + ρ - + 2 ( ρ + + ρ - ) x 2 + x 2 2 ] , e = - a x 2 [ 2 ρ + ρ - + ( ρ + + ρ - ) x 2 ] , f = a ρ + ρ - x 2 .
a [ x 2 - ( ρ + + ρ - ) x + ρ + ρ - ] ( x 2 - 2 x 2 x + x 2 2 ) .
a [ r 2 - ( ρ + + ρ - ) x + ρ + ρ - ] ( r 2 - 2 x 2 x + x 2 2 ) + 4 v 2 ( 1 + v w ) ( u - w ) 4 ( u w - v 2 ) y 2 = 0.
( c K ω ) 2 = 2 1 + 2 .
x + = u - v w u + v w ,             x - = [ 1 + w ( v + w - v w ) ] u 2 - 2 w ( 1 + v 2 w ) u + v w ( v + w + v w 2 - 1 ) [ 1 + w ( v + w + v w ) ] u 2 - 2 w ( 1 - v 2 w ) u - v w ( u + w + v w 2 + 1 ) .
n - d = f + ( x - ρ + ) ( x - x + ) = f + [ x 2 - ( ρ + + x + ) x + ρ + x + ] , n + d = f - ( x - ρ - ) ( x - x - ) = f - [ x 2 - ( ρ - + x - ) x + ρ - x - ] ,
f + = 2 u ( u + v ) 2 ( u + v w ) ( 1 + v ) ( 1 + w ) 2 ( 1 - w ) 2 , f - = 2 ( u - 1 ) { w u 2 + [ v + w + v w ( 2 + v + w ) ] u + v 2 w } ( denominator of x - ) .
g + = r 2 - ( ρ + + x + ) x + ρ + x + , g - = r 2 - ( ρ - + x - ) x + ρ - x - .
cos 2 q Δ z = 1 - h 1 + h ,             or tan 2 q Δ z = h .

Metrics