Abstract

A system of transparent layers on top of an arbitrary underlying subsystem is considered. It is shown that the layer refractive index and the subsystem generalized Fresnel coefficients can be expressed in a simple and explicit way by the Fourier coefficients of the ellipsometric function ρ with the layer thickness as an expansion parameter. Thus the inverse ellipsometric problem is reduced to a much simpler and well-defined problem of finding these Fourier coefficients. Analysis shows that the ellipsometric inverse task must be considered separately, depending on whether the modulus of the s-polarization Fresnel coefficient for the ambient–layer interface is smaller than, equal to, or greater than that for the layer–substrate system.

© 1999 Optical Society of America

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References

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  1. R. M. A. Azzam, N. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
    [CrossRef]
  3. D. E. Aspnes, “New developments in spectroellipsometry: the challenge of surfaces,” in Proceedings of the 1st International Conference on Spectroscopic Ellipsometry (Elsevier, New York, 1993), pp. 1–8.
  4. T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
    [CrossRef]
  5. A. A. Antippa, R. M. Leblanc, D. Ducharme, “Multiple-wavelength ellipsometry in thin uniaxial nonabsorbing films,” J. Opt. Soc. Am. A 3, 1794–1802 (1986).
    [CrossRef]
  6. R. M. A. Azzam, A.-R. M. Zaghloul, N. Bashara, “Ellipsometric function of a film–substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
    [CrossRef]
  7. R. M. A. Azzam, A.-R. M. Zaghloul, N. Bashara, “Polarizer-surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
    [CrossRef]
  8. F. Scandonne, L. Ballerini, “Théorie de la transmission et de la réflexion dans les systèmes de couches minces multiples,” Nuovo Cimento 5, 81–91 (1946).
  9. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  10. H. Jeffreys, B. Swirles, Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1966).
  11. R. E. Edwards, Fourier Series: A Modern Introduction (Springer-Verlag, Berlin, 1979).
  12. R. M. A. Azzam, “Ellipsometry of unsupported and embedded thin films,” J. Phys. (France) C10, 67–70 (1983).
  13. J. Lekner, “Ellipsometry of a thin film between similar media,” J. Opt. Soc. Am. A 5, 1041–1043 (1988).
    [CrossRef]
  14. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]

1995 (1)

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

1993 (1)

1990 (1)

1988 (1)

1986 (1)

1983 (1)

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

1975 (2)

1946 (1)

F. Scandonne, L. Ballerini, “Théorie de la transmission et de la réflexion dans les systèmes de couches minces multiples,” Nuovo Cimento 5, 81–91 (1946).

Antippa, A. A.

Aspnes, D. E.

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[CrossRef]

D. E. Aspnes, “New developments in spectroellipsometry: the challenge of surfaces,” in Proceedings of the 1st International Conference on Spectroscopic Ellipsometry (Elsevier, New York, 1993), pp. 1–8.

Azzam, R. M. A.

Ballerini, L.

F. Scandonne, L. Ballerini, “Théorie de la transmission et de la réflexion dans les systèmes de couches minces multiples,” Nuovo Cimento 5, 81–91 (1946).

Bashara, N.

Ducharme, D.

Easwarakhanthan, T.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Edwards, R. E.

R. E. Edwards, Fourier Series: A Modern Introduction (Springer-Verlag, Berlin, 1979).

Jeffreys, H.

H. Jeffreys, B. Swirles, Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1966).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Leblanc, R. M.

Lekner, J.

Ravelet, S.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Renard, P.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Scandonne, F.

F. Scandonne, L. Ballerini, “Théorie de la transmission et de la réflexion dans les systèmes de couches minces multiples,” Nuovo Cimento 5, 81–91 (1946).

Swirles, B.

H. Jeffreys, B. Swirles, Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1966).

Zaghloul, A.-R. M.

Appl. Surf. Sci. (1)

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. (France) (1)

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

Nuovo Cimento (1)

F. Scandonne, L. Ballerini, “Théorie de la transmission et de la réflexion dans les systèmes de couches minces multiples,” Nuovo Cimento 5, 81–91 (1946).

Other (5)

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

H. Jeffreys, B. Swirles, Methods of Mathematical Physics (Cambridge U. Press, Cambridge, UK, 1966).

R. E. Edwards, Fourier Series: A Modern Introduction (Springer-Verlag, Berlin, 1979).

D. E. Aspnes, “New developments in spectroellipsometry: the challenge of surfaces,” in Proceedings of the 1st International Conference on Spectroscopic Ellipsometry (Elsevier, New York, 1993), pp. 1–8.

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

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Figures (4)

Fig. 1
Fig. 1

System under consideration consists of an ambient with a real refractive index n0; a homogeneous layer with a real refractive index n1 and thickness d1; and an arbitrary underlying system Sx, characterized by the generalized Fresnel coefficients R1xp and R1xs. The ellipsometric ratio ρ, which corresponds to the different layer thicknesses d1, is measured at a fixed angle of incidence φ0.

Fig. 2
Fig. 2

The unit circle in the complex plane X(d1) and the position of Xs0 point (left-hand side) and the corresponding CAIC ρ(X) (right-hand side) at an angle of incidence φ0=60°, wavelength λ=632.8 nm for the system air(n0=1)SiO2(n1=1.46, d1)Si3N4(n2=2, d2=10/100 nm)Si(n3=3.88-0.018i). (a) d2=10 nm, |r01s|<|R1xs|; (b) d2=100 nm, |r01s|>|R1xs|.

Fig. 3
Fig. 3

Fourier coefficients of the function ρ(X) (CAIC), shown in Fig. 2(a) (case |r01s|<|R1xs|), from which layer and system parameters are computed.

Fig. 4
Fig. 4

Fourier coefficients of the function ρ(X) (CAIC), shown in Fig. 2(b) (case |r01s|>|R1xs|), from which layer and system parameters are computed.

Equations (72)

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ρ=tan Ψ exp(iΔ)=Rp/Rs,
ρ=r01p+R1xp X1+r01p R1xp X1+r01s R1xsXr01s+R1xs X,
X=exp(-i2β),
β=2π(d1/λ)[n12-n02 sin2(φ0)]1/2,
Dφ=(λ/2)[n12-n02 sin2(φ0)]1/2.
I=Lρ(X)dX.
I=0
ρ(X)=f(X)X-X0,
I=Lρ(X)dX=L f(X)X-X0dX=2πif(X0),=2πi Res(ρ)|X0.
1+r01pR1xp X=0,
r01s+R1xsX=0.
Xp0=-1r01pR1xp,
Xs0=-r01sR1xs.
|r01p||R1xp|>1.
|r01s|<|R1xs|.
|r01s|>|R1xs|,
Lρ(X) dX=2πi Res(ρ)|Xs0.
Lρ(X)dX=0.
air(n0=1)SiO2(n1=1.46,d1)
Si3N4(n2=2,d2=10/100nm)
Si(n3=3.88-0.018i).
Res(ρ)|Xs0=r01p+R1xp X1+r01p R1xp X1+r01s R1xs XR1xsX=Xs0.
Lρ(X)dX=-2πi r01p R1xs-r01s R1xpR1xs-r01pr01s R1xp1-r01s2R1xs=-2πi r01p-r01sρ1x1-r01pr01sρ1x1-r01s2R1xs,
ρ1x=R1xp/R1xs
Lρ(X)XdX=2πi r01p-r01s ρ1x1-r01p r01sρ1x1-r01s2R1xs2r01s.
R1xs=-r01s Lρ(X)dXLρ(X)XdX.
Lρ(X)dX=-2π i 1Dφ0Dϕρ(d1)exp(-2πid1/Dφ)dd1=-2π iρ˜1
Lρ(X)XdX=-2π i 1Dϕ0Dφρ(d1)exp(-4πid1/Dφ)dd1=-2π iρ˜2
ρ(d1)=k=-k=+ρ˜k exp2πik d1Dϕ,
ρ˜k=1Dϕ0Dϕρ(d1)exp(-2πikd1/Dϕ)dd1=01ρ(x)exp(-2πikx)dx
d1m=d10+DϕNm,m=0N-1.
ρ˜k=1Nm=0N-1ρm exp(-2πikd1m/Dϕ)=1Nm=0N-1ρmXmk,
R1xs=-r01s ρ˜1ρ˜2=-r01s ρmXmρmXm2,
X=1/X*=1/Z
ρ=r01p Z+R1xpZ+r01p R1xpZ+r01s R1xsr01s Z+R1xs,
Zp0=-r01p R1xp,
Zs0=-R1xsr01s.
Lρ(Z)dZ=2πiR1xp×(1-r01p2)(r01s-r01p ρ1x)1-r01p r01sρ1x,
Lρ(Z)ZdZ=-2πir01p R1xp2×(1-r01p2)(r01s-r01p ρ1x)1-r01p r01sρ1x.
R1p=-1r01pLρ(Z)ZdZLρ(Z)dZ.
R1xp=-1r01pρ˜-2ρ˜-1=-1r01pρmXm*2ρmXm*=-1r01pρm/Xm2ρm/Xm.
AmX m2+Bm Xm+Cm=0,
Am=R1xpR1xs(r01s-r01pρm),
Bm=r01p r01sR1xs+R1xp-(R1xs+r01p r01sR1xp)ρm,
Cm=r01p-r01sρm,
1Nm=0N-1Xm=0,
-r01pρ1xR1xs2ρ˜2-(1+r01p r01sρ1x)R1xsρ˜1
+r01sρ˜0-r01p=0.
ρ1x=r01p r01s-ρ˜0r01p2+r01s2-1-r01p r01sρ˜0.
ρ01=r01pr01s=ρ˜0-ρ˜12ρ˜2.
R1xs=r01p-r01sρ˜0ρ˜1,
R1xp=ρ1x r01p-r01sρ˜0ρ˜1.
n1=n0 sin φ01+(1-ρ01)2(1+ρ01)2tan2 φ01/2.
r01 p=ρ01 ρ01+cos 2φ01+ρ01 cos 2φ0,
r01s=ρ01+cos 2φ01+ρ01 cos 2φ0.
ρ01=(r01p/r01s)=ρ˜0
ρ˜-1=-R1xsr01s(r01p2-1)ρ1x-r01pr01s(r01s2-1),
ρ˜-2=-R1xs2 r01pr01s(1-r01p2)ρ1x2+1-r01s2r01p r01sρ1x-1r01s21-r01s21-r01p2.
Cm/Xm2+Bm/Xm+Am=CmZm2+BmZm+Am=0
1Nm=0N-1Zm=0,
-r01sρ˜-2-(1+r01p r01sρ1x)R1xsρ˜-1
+(r01sρ1x-r01pρ˜0)R1xs2=0.
aρ1x2+bρ1x+c=0,
a=1-r01p2+ρ˜-12ρ˜-2r01p r01s,
b=(1-r01s2)ρ˜-12ρ˜-2-2 r01pr01s,
c=-r01pr01s1-r01s21-r01p2ρ˜-12ρ˜-2-r01pr01s+r01p r01s.
R1xs=r01s2 ρ˜-1r01s(1-r01p2)ρ1x-r01p(1-r01s2),
R1xs=r10s=-r01s,
R1xp=r10p=-r01p,
ρ(X)=ρ01 1-r01s2X1-r01p2X.
r01s=[ρ˜-1/ρ˜0(ρ˜0-1)]1/2,
φ0=12arccosr01s-ρ˜01-r01sρ˜0,

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