Abstract

Calibration of polarization-state generators (PSG’s), polarimeters, and Mueller-matrix ellipsometers (MME’s) is an important factor in the practical use of these instruments. A new general procedure, the eigenvalue calibration method (ECM), is presented. It can calibrate any complete MME consisting of a PSG and a polarimeter that generate and measure, respectively, all the states of polarization of light. In the ECM, the PSG and the polarimeter are described by two 4 × 4 matrices W and A, and their 32 coefficients are determined from three or four measurements performed on reference samples. Those references are smooth isotropic samples and perfect linear polarizers. Their optical characteristics are unambiguously determined during the calibration from the eigenvalues of the measured matrices. The ECM does not require accurate alignment of the various optical elements and does not involve any first-order approximation. The ECM also displays an efficient error control capability that can be used to improve the MME behavior. The ECM is illustrated by an experimental calibration, at two wavelengths (458 and 633 nm), of a MME consisting of a coupled phase modulator associated with a prism division-of-amplitude polarimeter.

© 1999 Optical Society of America

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References

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  1. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980), and references therein.
    [CrossRef]
  2. B. Drevillon, Progress in Crystal Growth and Characterization of Materials (Pergamon, Oxford, 1993), and references therein.
  3. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  4. R. M. A. Azzam, A. G. Lopez, “Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements,” J. Opt. Soc. Am. A 6, 1513–1521 (1989).
    [CrossRef]
  5. R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
    [CrossRef] [PubMed]
  6. E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
    [CrossRef]
  7. E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
    [CrossRef]
  8. S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).
  9. E. Collett, Polarized Light—Fundamentals and Application (Marcel Dekker, New York, 1993).
  10. J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Press, Bellingham, Wash., 1995).
  11. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
    [CrossRef]
  12. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  13. R. M. A. Azzam, “Simulation of mechanical rotation by optical rotation: application to the design of a new Fourier polarimeter,” J. Opt. Soc. Am. 68, 518–521 (1978).
    [CrossRef]
  14. E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
    [CrossRef]
  15. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
    [CrossRef] [PubMed]
  16. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
    [CrossRef]
  17. K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
    [CrossRef]
  18. S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A 9, 1615–1622 (1992).
    [CrossRef]
  19. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
    [CrossRef]
  20. R. M. A. Azzam, “Mueller-matrix measurement using the four-detector photopolarimeter,” Opt. Lett. 11, 270–272 (1986).
    [CrossRef] [PubMed]
  21. S. Krishnan, “Calibration, properties, and applications of the division-of-amplitude photopolarimeter at 632.8 and 1523 nm,” J. Opt. Soc. Am. A 9, 1615–1622 (1992).
    [CrossRef]
  22. Y. Cui, R. M. A. Azzam, “Sixteen-beam grating-based division-of-amplitude photopolarimeter,” Opt. Lett. 21, 89–91 (1996).
    [CrossRef] [PubMed]
  23. M. C. Pease, Methods of Matrix Algebra (Academic, London, 1965).
  24. A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.
  25. Dielectric functions are the library of D. E. Aspnes, (Department of Physics, University of North Carolina, Raleigh, N.C. 27645), except for that of Cr, which was measured in the authors’ laboratory with a spectroscopic phase-modulated ellipsometer.
  26. P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
    [CrossRef]
  27. W. Kern, D. A. Puotinen, “Cleaning solutions based on hydrogen peroxyde for use in silicon semiconductor technology,” RCA Rev. 31, 187–206 (1970).

1998 (2)

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

1997 (2)

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
[CrossRef]

1996 (1)

1992 (2)

1991 (1)

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

1989 (1)

1986 (1)

1985 (1)

1984 (2)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

1982 (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

1980 (2)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980), and references therein.
[CrossRef]

R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
[CrossRef] [PubMed]

1978 (3)

1970 (1)

W. Kern, D. A. Puotinen, “Cleaning solutions based on hydrogen peroxyde for use in silicon semiconductor technology,” RCA Rev. 31, 187–206 (1970).

Azzam, R. M. A.

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Bertrand, N.

P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
[CrossRef]

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Bottiger, J. R.

Brudzewski, K.

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

Bulkin, P.

P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
[CrossRef]

Collett, E.

E. Collett, Polarized Light—Fundamentals and Application (Marcel Dekker, New York, 1993).

Compain, E.

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Cui, Y.

Drevillon, B.

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

B. Drevillon, Progress in Crystal Growth and Characterization of Materials (Pergamon, Oxford, 1993), and references therein.

Flannery, B. P.

A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.

Fry, E. S.

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980), and references therein.
[CrossRef]

P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
[CrossRef]

Huard, S.

S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

Kern, W.

W. Kern, D. A. Puotinen, “Cleaning solutions based on hydrogen peroxyde for use in silicon semiconductor technology,” RCA Rev. 31, 187–206 (1970).

Krishnan, S.

Lopez, A. G.

Pease, M. C.

M. C. Pease, Methods of Matrix Algebra (Academic, London, 1965).

Press, W. H.

A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.

Puotinen, D. A.

W. Kern, D. A. Puotinen, “Cleaning solutions based on hydrogen peroxyde for use in silicon semiconductor technology,” RCA Rev. 31, 187–206 (1970).

Stover, J. C.

J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Press, Bellingham, Wash., 1995).

Teukolsky, S. A.

A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.

Thompson, R. C.

Vetterling, W. T.

A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.

Am. J. Phys. (2)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1984).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

K. Brudzewski, “Static Stokes ellipsometer: general analysis and optimization,” J. Mod. Opt. 38, 889–896 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

R. M. A. Azzam, “Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” Opt. Acta 29, 685–689 (1982).
[CrossRef]

Opt. Lett. (4)

RCA Rev. (1)

W. Kern, D. A. Puotinen, “Cleaning solutions based on hydrogen peroxyde for use in silicon semiconductor technology,” RCA Rev. 31, 187–206 (1970).

Rev. Sci. Instrum. (2)

E. Compain, B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Surf. Sci. (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980), and references therein.
[CrossRef]

Thin Solid Films (1)

P. Bulkin, N. Bertrand, B. Drevillon, “Deposition of SiO2 in an integrated distributed electron cyclotron resonance microwave reactor,” Thin Solid Films 296, 66–68 (1997).
[CrossRef]

Other (7)

B. Drevillon, Progress in Crystal Growth and Characterization of Materials (Pergamon, Oxford, 1993), and references therein.

S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

E. Collett, Polarized Light—Fundamentals and Application (Marcel Dekker, New York, 1993).

J. C. Stover, Optical Scattering—Measurement and Analysis (SPIE, Press, Bellingham, Wash., 1995).

M. C. Pease, Methods of Matrix Algebra (Academic, London, 1965).

A useful book on the computation of the ECM algorithms is W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989); books for other computer languages are available as parts of a series by the same publisher.

Dielectric functions are the library of D. E. Aspnes, (Department of Physics, University of North Carolina, Raleigh, N.C. 27645), except for that of Cr, which was measured in the authors’ laboratory with a spectroscopic phase-modulated ellipsometer.

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Figures (9)

Fig. 1
Fig. 1

Measurement configurations involved in the ECM. (a) Direct transmission measurement without a sample: The corresponding measurement is aw. (b) The reflection configuration: the corresponding measurement is amw.

Fig. 2
Fig. 2

Fifty numerical simulations of the calibration accuracy. The measurement errors are simulated by addition of random matrices to the theoretical measurements. Relative errors in the measurements are taken to be equal to 0.5% and result, with the ECM, in average relative error 〈ε A 〉 = 0.25% in A and εW = 0.29% in W.

Fig. 3
Fig. 3

Fifty numerical simulations of the ratio between the actual calibration errors (ε A , ε W ) in (A, W) and the error estimators εECM and εECM.

Fig. 4
Fig. 4

ECM algorithm. Numbers in parentheses refer to the corresponding equations in text.

Fig. 5
Fig. 5

Equivalent polarimetric properties of cSi–Cr samples. The values are plotted as a function of the wavelength for several angles of incidence.

Fig. 6
Fig. 6

Equivalent polarimetric properties of cSi–Pd samples. The values are plotted as a function of the wavelength for several angles of incidence.

Fig. 7
Fig. 7

Determination of the orientations of the two polarizers. The correct orientations θ b and θ a minimize the ratios λ1615 and λ16/λ15.

Fig. 8
Fig. 8

K and K′ eigenvalues at 633 nm. They reflect the behavior of the calibration procedure ruled by the choice of the reference samples and the accuracy of the basic measurements.

Fig. 9
Fig. 9

Measured Mueller matrices of a SiO2-coated c-Si substrate as a function of angle of incidence (ϕ i ) for two wavelengths (458 and 633 nm). The coefficient M11 is the intensity reflectance, and the other coefficients, normalized by M11, display the complete polarimetric response. The error between experimental and theoretical values (solid curves) is less than 0.5%.

Tables (2)

Tables Icon

Table 1 Composition of the Harmonic Vector for Four Modulation Techniques

Tables Icon

Table 2 Measured and Theoretical Values of the Characteristics of the Reference Samples at λ = 458 nm and λ = 633 nm

Equations (53)

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

M=MnMn-1M2M1.
Pτ, 0=τ21100110000000000, Pτ, π/4=τ21010000010100000.
Pτ, θ=UθPτ, 0U-θ,
Uθ=10000cos 2θ-sin 2θ00sin 2θcos 2θ00001.
Rτ, Ψ, Δ=τ ×1-cos 2Ψ00-cos 2Ψ10000sin 2Ψ cos Δsin 2Ψ sin Δ00-sin 2Ψ sin Δsin 2Ψ cos Δ.
Si=Wei.
W=12τ1τ1τ1τ2τ10-τ100τ100000τ2, e1=1000,,  e4=0001.
St=Wet.
Ii=TfiAS  i=1,4,
It=TftAS.
I1I2I3I4=AS.
AMW=Iiji,j=14,
Iij=TfiAMWej.
It=TftAMWet.
I1I2I3I4t=AMWet
I1I2I3I4i=AMWei.
M:M4M4, XMX-Xaw-1amw.
MX=0,  MM1,, Mn.
A=awW-1.
HMX16=0,  MM1,, Mn.
KX16=0,
K=THM1HMnHM1HMn=THM1HM1++THMnHMn.
K=TOλ1000λi000λ16O, λ1 >  > λ15  λ16  0,
W16=TO001.
W is the unique solution of Mx=0, MM when M=RτR, Ψ, Δ, PτP, θ.
the accuracy is maximum when λ15/λ1is maximum.
M=RτR, Ψ, Δ, PτP, θ is optimum when τR = 0.71, Ψ = π/6 or 3π/6, θ =±π/4, Δ=±π/2.
τp=trace aw-1apw,  τr=0.5λr1+λr2, Δ=0.5 argλc1/λc2,  Ψ=arctanλr1/λr2,
Kθ=THRHPθHRHPθ=TOθλ1θ00λ16θOθ.
θ verifies thatλ16θλ15θ 0.
Pb:XPbX-Xar1w-1ar1pbw, Req:XReqX-Xar1w-1ar2w,
Reqτeq, Ψeq, Δeq=R1-1τ1, Ψ1, Δ1R2τ2, Ψ2, Δ2,
τeq=τ2τ11-cos 2Ψ1 cos 2Ψ2sin2 2Ψ1, tan Ψeq=cot Ψ1 tan Ψ2, Δeq=Δ2-Δ1.
Pa:XXPa-apar1war1w-1X, Req:XXReq-ar2war1w-1X.
A16=TO001,
K=THPaHPa+THReqHReq=TOλ100λ16Oλ1>>λ15  λ160.
εW=|W-WECM||W|,  εA=|A-AECM||A|,
|M|=i,j Mij21/2
εECM=λ16/λ15,  εECM=λ16/λ15.
εECM=0.34%, εECM=0.98%.
MX=0  M, X=0,
M, :M4M4,  XMX-XM.
M, X=0, MM
Px=detxI4-M=i=1mx-λiαi,
i=1m αi2.
τp multiplicity, 1,  0 multiplicity, 3
2τR sin2Ψ,2τR cos2Ψ;τR sin2ΨexpiΔ,τR sin2Ψexp-iΔ.
λ16/λ15  1.
λ15/λ1  error level.
|ΔK|/|K|ε,
λ1|K|4λ1.
K+ΔK=K+ε4 λ1I16=tOλ1+ε4 λ1λ15+ε4 λ1ε4 λ1O,
λ15/λ1  ε/4.

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