Abstract

Simple analytic formulas are given for the real and imaginary parts of the dielectric function of an absorbing medium in terms of the TE and TM reflectances Rs and Rp. An analysis of the formulas shows zero/zero instability at 0°, 45°, and 90° angles of incidence. The instability (extreme sensitivity to experimental error) at 45° is related to the result that Rp=Rs2 at 45° incidence, for all absorbing or nonabsorbing media. It is shown that for materials of large refractive index the deduced values of the real and imaginary parts of the dielectric function are very sensitive to experimental error, even at the optimum angle of incidence.

© 1997 Optical Society of America

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References

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  1. L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, 1988), Sec. 2.6.
  2. R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Chap. 13.
  4. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Chap. 8.
  5. J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
    [CrossRef] [PubMed]
  6. J. Lekner, “Determination of complex refractive index and thickness of a homogeneous layer by combined reflection and transmission ellipsometry,” J. Opt. Soc. Am. A 11, 2156–2158 (1994).
    [CrossRef]
  7. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
    [CrossRef]
  8. D. W. Berreman, “Simple relation between reflectances of polarized components of a beam when the angle of incidence is 45°,” J. Opt. Soc. Am. 56, 1784 (1966).
    [CrossRef]
  9. R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta 26, 113–115, 301 (1979).
    [CrossRef]
  10. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  11. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

1994 (2)

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

1983 (1)

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

1979 (1)

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta 26, 113–115, 301 (1979).
[CrossRef]

1966 (1)

1961 (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta 26, 113–115, 301 (1979).
[CrossRef]

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Chap. 13.

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Lekner, J.

Ward, L.

L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, 1988), Sec. 2.6.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Chap. 13.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. Phys., Condens. Matter (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys., Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Opt. Acta (1)

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta 26, 113–115, 301 (1979).
[CrossRef]

Proc. Phys. Soc. London (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Rev. Sci. Instrum. (1)

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Chap. 13.

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Chap. 8.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, 1988), Sec. 2.6.

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Figures (4)

Fig. 1
Fig. 1

Reflectances Rs and Rp for glass (=2.25), silicon ( =15+0.15i), and aluminum (=-56+21i), versus the angle of incidence θ. In all cases the s and p reflectances are equal at normal incidence, and RsRp at all angles incidence.

Fig. 2
Fig. 2

Common denominator D=2C[s(1-sp)-p(1 -s2)C]2 of inversion formulas (15) and (16) as a function of the angle of incidence for Al(-56+21i), Si(15+0.15i), and glass (=2.25). The reciprocal of D is a measure of sensitivity to error; note the zeros at 0°, 45°, and 90°.

Fig. 3
Fig. 3

Plot of the error multiplier Δr, which multiplies experimental uncertainties in C=cos2 θ, p=(1-Rp)/(1+Rp), and s=(1-Rs)/(1+Rs) to estimate the uncertainty in r/1.

Fig. 4
Fig. 4

Plot of the error multiplier ΔI, which gives the uncertainty in i22/12 for glass, silicon, and aluminum.

Equations (34)

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r=nr2-ni2,i=2nrni.
(cq/ω)2=-1 sin2 θ=r-1 sin2 θ+ii
qr2-qi2=ωc2(r-1 sin2 θ),
2qrqi=i(ω/c)2.
qr=ωc 12 [r-1 sin2 θ+(r-1 sin2 θ)2+i2]1/2.
Rs=(q1-qr)2+qi2(q1+qr)2+qi2,
Rp=(Q1-Qr)2+Qi2(Q1+Qr)2+Qi2,
Qr=rqr+iqir2+i2,Qi=rqi-iqrr2+i2.
s=1-Rs1+Rs=2q1q2q12+qr2+qi2,
p=1-Rp1+Rp=2Q1QrQ12+Qr2+Qi2,
Qr2=r2qr2+i2qi2+ri2(ω/c)2(r2+i2)2.
ρ=(r-1 sin2 θ)2+i2.
z1(r, i, s2, p2, θ)=ρs-ρp
z2(r, i, s2, θ)=(r-1 sin2 θ)2+i2-ρs2.
r1=c0+c1C+c2C2+c3C32C[s(1-sp)-p(1-s2)C]2,
c0=s2(p-s)2,
c1=2s(1-s2)(2s-p-sp2),
c2=2(1-s2)[p(p-s)-2s2(1-p2)],
c3=2p(1-s2)[2s-p(1+s2)],
11=s(p-s)|1-2C|[d0+d1C+d2C2+d3C3+d4C4]1/22C[s(1-sp)-p(1-s2)C]2,
d0=-s2(p-s)2,
d1=4s(1-s2)(p-s),
d2=4(1-s2)[s2(1+p2)-p(s+p)],
d3=8p2(1-s2)2,
d4=-4p2(1-s2)2.
Rs(45°)=81i2+2rv2-v381i2+2rv2+v3
v2=21{2r-1+[(2r-1)2+i2]1/2}.
Rs2-Rp=16n1no(meno-mone)(no3ne-n12mome)(none+n1me)2(n1+mo)4,
mo2=2no2-n12me2=2ne2-n12.
dr=rC dC+rp dp+rs ds.
(dr)2=rC2(dC)2+rp2(dp)2+rs2(ds)2.
Δr=rC2+rp2+rs21/2.
ΔI=IC2+Ip2+Is21/2,
112 i2Ci=0=(r-1)(1-r-C)(rC+C-1)C(1-2C)(1-C).

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