Abstract

I investigate the dependence of shot-noise-limited uncertainties of the ellipsometric parameters ψ and Δ for the rotating-analyzer ellipsometer (RAE) and the rotating-compensator ellipsometer (RCE) of the polarizer–sample–compensator–analyzer type. The development is general and takes into account correlations among the Fourier coefficients of the transmitted intensity, in particular the average intensity, which is necessarily correlated with all other coefficients through normalization. The results are expressed in terms of the traditional uncertainties δψ and δΔ of the ellipsometric parameters ψ and Δ, respectively, although a more appropriate measure of uncertainty is the differential area 2δψ×sin ψδΔ on the unit-radius Poincaré sphere. Numerical results for broadband operation from 1.5 to 6.0 eV with a Si sample show that the optimum measurement conditions for both configurations occur when the intensity of light reflected from the sample is approximately balanced between the TE and the TM modes, and, for the RCE, when the analyzer azimuth is essentially equal to that of the polarizer. Under typical broadband operating conditions in which components cannot be optimized on a wavelength-by-wavelength basis, the RCE is better at determining Δ, whereas the RAE is better at determining ψ. The approach is easily generalized to other configurations and other types of experimental uncertainty, both random and systematic.

© 2004 Optical Society of America

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References

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  1. P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
    [CrossRef]
  2. D. E. Aspnes, “Fourier transform detection system for rotating-analyzer ellipsometers,” Opt. Commun. 8, 222–225 (1973).
    [CrossRef]
  3. J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
    [CrossRef]
  4. J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
    [CrossRef]
  5. B. Drevillon, “Phase-modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
    [CrossRef]
  6. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  7. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
    [CrossRef]
  8. Z. M. Huang, J. H. Chu, “Optimizing precision of fixed-polarizer, rotating-polarizer, sample, and fixed-analyzer spectroscopic ellipsometry,” Appl. Opt. 39, 6390–6395 (2000).
    [CrossRef]
  9. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  10. C. V. Kent, J. Lawson, “A photoelectric method for the determination of the parameters of elliptically polarized light,” J. Opt. Soc. Am. 27, 117–119 (1937).
    [CrossRef]
  11. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984).
    [CrossRef] [PubMed]
  12. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]

2000

1998

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

1997

1993

B. Drevillon, “Phase-modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
[CrossRef]

1984

1983

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

1974

1973

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

D. E. Aspnes, “Fourier transform detection system for rotating-analyzer ellipsometers,” Opt. Commun. 8, 222–225 (1973).
[CrossRef]

1937

An, I.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Aspnes, D. E.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
[CrossRef]

D. E. Aspnes, “Fourier transform detection system for rotating-analyzer ellipsometers,” Opt. Commun. 8, 222–225 (1973).
[CrossRef]

Azzam, R. M. A.

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

Bashara, N. M.

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

Chen, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Chu, J. H.

Collins, R. W.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Dill, F. H.

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

Dodge, M. J.

Drevillon, B.

B. Drevillon, “Phase-modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
[CrossRef]

Fanton, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Hauge, P. S.

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

Huang, Z. M.

Jellison, G. E.

Kent, C. V.

Lawson, J.

Lee, J.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Leng, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Modine, F. A.

Opsal, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Rovira, P. I.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Senko, M.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Uhrich, C.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Wei, L.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Zaiser, C.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

D. E. Aspnes, “Fourier transform detection system for rotating-analyzer ellipsometers,” Opt. Commun. 8, 222–225 (1973).
[CrossRef]

Phys. Rev. B

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Prog. Cryst. Growth Charact.

B. Drevillon, “Phase-modulated ellipsometry from the ultraviolet to the infrared: in situ application to the growth of semiconductors,” Prog. Cryst. Growth Charact. 27, 1–87 (1993).
[CrossRef]

Rev. Sci. Instrum.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating-compensator multichannel ellipsometry: applications for real-time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Thin Solid Films

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313–314, 58–62 (1998).
[CrossRef]

Other

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

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Figures (4)

Fig. 1
Fig. 1

Dependences of 〈δψ〉 (top) and 〈δΔ〉 (bottom) on analyzer azimuth A for a broadband RCE operating with P=30° and a MgF2 monoplate exhibiting 90° retardation at 4.0 eV. Other parameters are described in the text.

Fig. 2
Fig. 2

As in Fig. 1, but for various values of P=A.

Fig. 3
Fig. 3

As in Fig. 1, but with P=A=30° and for MgF2 wave plates of various retardations expressed as 90° phase shifts at specific energies.

Fig. 4
Fig. 4

Comparison of the performances of an RCE and an RAE for the configuration described in the text. The “RAE, OPTIMAL” curve of the bottom panel shows the dependence of 〈δΔ〉 on E, assuming that the relative phase of the TE and TM modes entering the rotating analyzer is continually adjusted to 90°. The “RAE, ACTUAL” curve corresponds to the operation in which no such adjustment is made.

Equations (36)

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δIj=±ηIj
IRAE=I02[(|r˜p|2+|r˜s|2)+(|r˜p|2-|r˜s|2)cos 2A+2 Re(r˜pr˜s*)sin 2A]
=a0+a2 cos 2A+b2 sin 2A
=a0(1+α2 cos 2A+β2 sin 2A)
=12r2I0(1-cos 2ξ cos 2A+sin 2ξ cos Δ sin 2A),
r˜p=r˜p cos P=r sin ξeiΔ,
r˜s=r˜s sin P=r cos ξ,
tan ψ=tan ξ tan P
IRCE=I02(|r˜p|2+|r˜s|2)+cos2δ2[(|r˜p|2-|r˜s|2)cos 2A+2 Re(r˜pr˜s*)sin 2A]-2 Im(r˜pr˜s*)sin δ sin(2C-2A)+sin2δ2[(|r˜p|2-|r˜s|2)cos(4C-2A)+2 Re(r˜pr˜s*)sin(4C-2A)]
=c0+c2 cos(2C-2A)+s2 sin(2C-2A)+c4 cos(4C-2A)+s4 sin(4C-2A)
=a0+a2 cos(2C)+b2 sin(2C)+a4 cos(4C)+b4 sin(4C)
=a0[1+α2 cos(2C)+β2 sin(2C)+α4 cos(4C)+β4 sin(4C)]
=12r2I01-cos2δ2(cos 2ξ cos 2A-sin 2ξ cos Δ sin 2A)-sin 2ξ sin Δ sin δ sin(2C-2A)-sin2δ2[cos 2ξ cos(4C-2A)-sin 2ξ cos Δ sin(4C-2A)].
a0+δa0=1Nj=1NIj+1Nj=1NδIj
=I02(|rp|2+|rs|2)+0,
a2=2Nj=1NIj cos 2Aj+2Nj=1NδIj cos 2Aj=I02(|rp|2-|rs|2)+0,
δa02=δa0δa0=1N2j,k=1N(δIj)(δIk)
=1N2j=1NδIj2
=1Nη2a0.
δa22=δb22=2Nη2a0,
δa22=2Nη2(a0+a4),
δb22=2Nη2(a0-a4),
δa42=δb42=2Nη2a0.
δψ=κψ0δa0+κψ2δa2+,
δψ2=κψ02δa02+κψ22δa22+,
δψ=rsδrp-rpδrsrp2 sin2 P+rs2 cos2 Psin P cos P.
δrp=δa0+δa22I0rp,
δrs=δa0-δa22I0rs,
δψ=1I0 sin 2P(rp2+rs2)δa0rsrp-rprs+δa2rsrp+rprs,
δΔ=δa0 cos Δrsrp+rprs+δa2 cos Δrsrp-rprs-2δb22I0rprs sin Δ.
δ=j=1Nc[γj-gj(ψ, Δ)-Δψgj(ψ, Δ)/ψ-ΔΔgj(ψ, Δ)/Δ]2,
v1=j=15[γj-gj(ψ, Δ)][gj(ψ, Δ)/rp],
v2=j=15[γj-gj(ψ, Δ)][gj(ψ, Δ)/rs],
v3=j=15[γj-gj(ψ, Δ)][gj(ψ, Δ)/Δ];
m11=j=15[gj(rp,rs, δ)/δrp]2;
ηNI0=ωfI0.

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