Abstract

The coherency matrix formalism based on Pauli matrices is applied to analyze a general ellipsometer that is described by Jones matrices. Here the Jones matrices are represented as sums of appropriate coefficients times the Pauli matrices and the identity matrix, and intensities are represented as traces of coherency matrices. This approach allows us not only to treat partial polarizations explicitly but also to take advantage of various identities to reduce to algebra the operations necessary for system analysis. A general expression is obtained for the intensity transmitted through a polarizer–sample–compensator–analyzer (PSCA) ellipsometer. This general expression is applied to an ideal PSCA ellipsometer, and then the results are reduced to describe several simpler but commonly used configurations. The results provide insight regarding general capabilities and limitations and allow us to compare different ellipsometer systems directly. Finally, this expression is extended to include artifacts, the explicit representation of which allows a complete determination of their defects.

© 2000 Optical Society of America

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References

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  1. H. G. Tompkins, W. A. McGahan, Spectroscopic Ellipsometry and Reflectrometry: A User’s Guide (Wiley, New York, 1999).
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
    [CrossRef]
  4. G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento (Ser. 10) 15, 370–388 (1960).
    [CrossRef]
  5. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  6. See, for example, D. Bohm, Quantum Theory (Prentice-Hall, New York, 1954).
  7. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).
  8. W. Swindell, ed., Polarized Light (Dowden Hutchinson & Ross, Stroudsburg, Pa., 1975).
  9. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento (Ser. 10) 13, 1165–1181 (1959).
    [CrossRef]
  10. D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [CrossRef] [PubMed]
  11. C. V. Kent, J. Lawson, “A photometric method for the determination of the parameters of elliptically polarized light,” J. Opt. Soc. Am. 27, 117–119 (1937).
    [CrossRef]
  12. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
    [CrossRef]

1975

1974

1960

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento (Ser. 10) 15, 370–388 (1960).
[CrossRef]

1959

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento (Ser. 10) 13, 1165–1181 (1959).
[CrossRef]

1954

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
[CrossRef]

1937

Aspnes, D. E.

Azzam, R. M. A.

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

Bashara, N. M.

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

Bohm, D.

See, for example, D. Bohm, Quantum Theory (Prentice-Hall, New York, 1954).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

Fano, U.

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
[CrossRef]

Kent, C. V.

Lawson, J.

McGahan, W. A.

H. G. Tompkins, W. A. McGahan, Spectroscopic Ellipsometry and Reflectrometry: A User’s Guide (Wiley, New York, 1999).

Parrent, G. B.

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento (Ser. 10) 15, 370–388 (1960).
[CrossRef]

Roman, P.

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento (Ser. 10) 15, 370–388 (1960).
[CrossRef]

Studna, A. A.

Tompkins, H. G.

H. G. Tompkins, W. A. McGahan, Spectroscopic Ellipsometry and Reflectrometry: A User’s Guide (Wiley, New York, 1999).

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento (Ser. 10) 13, 1165–1181 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Appl. Opt.

J. Opt. Soc. Am.

Nuovo Cimento (Ser. 10)

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento (Ser. 10) 15, 370–388 (1960).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento (Ser. 10) 13, 1165–1181 (1959).
[CrossRef]

Phys. Rev.

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1954).
[CrossRef]

Other

H. G. Tompkins, W. A. McGahan, Spectroscopic Ellipsometry and Reflectrometry: A User’s Guide (Wiley, New York, 1999).

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

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

See, for example, D. Bohm, Quantum Theory (Prentice-Hall, New York, 1954).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

W. Swindell, ed., Polarized Light (Dowden Hutchinson & Ross, Stroudsburg, Pa., 1975).

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Equations (77)

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σx=0110,σy=0-ii0,σz=100-1.
σi=σi,
[σi, σj]=σiσj-σjσi=i2ϵijkσk,
σiσj+σjσi=2δijI,
I=1001.
σiσj=δijI+iϵijkσk.
(a0I+A·σ)(b0I+B·σ)
=(a0b0+A·B)I+(b0A+a0B)·σ+i(A×B)·σ,
J·σ=Jxσx+Jyσy+Jzσz=JzJx-iJyJx+iJy-Jx.
JJ11J12J21J22=J0I+J·σ,
J0=J11+J222,Jx=J12+J212,
Jy=i J12-J212,Jz=J11-J222.
Ji=12Tr(Jσi),
JP=1000=12(I+σz).
JP=1iαpiαp11-iχp-iχpγ1-iαp-iαp1
=12[(1+γ)I+(1-γ)σz-2iχpσx+2ασy],
Js=rp00rs=r+I+r-σz,r±=12(rp±rs),
JC=100exp(iδc)=exp(iδc/2)cosδc2I-i sinδc2σz.
JC=1iαciαc1100ξ1-iαc-iαc1=12[(1+ξ)I+(1-ξ)σz+2αc(1-ξ)σy],
Je=[(cos θ)I-i(sin θ)σy]Je[(cos θ)I+i(sin θ)σy]
=exp(-iθσy)Je exp(iθσy),
ExEy=[(cos θ)I-i(sin θ)σy]ExEy
=exp(-iθσy)ExEy.
ρ=12ϵ0|Ex|2ExEy*EyEx*|Ey|2,
I=12ϵ0(|Ex|2+|Ey|2)=Tr(ρ).
ρ=14ϵ0(S0I+S·σ),
S0=|Ex|2+|Ey|2,
S1=|Ex|2-|Ey|2,
S2=EyEx*+ExEy*=2 ReExEy*,
S3=iEyEx*-ExEy*=2 ImExEy*.
ExEy=Jsys(λ)ExEyin.
ρout(λ)=Jsys(λ)ρin(λ)Jsys(λ).
Iout=Tr(JsysρinJsys)=Tr(JsysJsysρin),
Iout=Δλdλ Tr(ρout)=Δλdλ Tr[Jsys(λ)Jsys(λ)ρin(λ)],
Iout=Tr[Jsys(λ0)Jsys(λ0)Δλdλ ρin(λ)],
Jsys(λ0)=exp(-iAσy)JA×exp(iAσy)exp(-iCσy)JC exp(iCσy)×JS exp(-iPσy)JP exp(iPσy),
ρin=Δλdλ ρin(λ)=12Iin[I+αyσy+αz exp(-iSσy)σz exp(-iSσy)],
Iout=Tr(TAJCTPJC),
TP=exp(iCσy)JS exp(-iPσy)JP exp(iPσy)ρin×exp(-iPσy)JP exp(iPσy)JS exp(-iCσy)=Iin2(p0I+p·σ),
TA=exp[i(C-A)σy]JAJA exp[-i(C-A)σy]=12[a0I+a·σ],
JC=12(c0I+c·σ).
TAJCTP=Iin8{(a0p0c0+a0c·p+a·m)I+[a0m+(c0p0+c·p)a+i(a×m)]·σ},
Iout=Iin8{(a0p0c0+a0c·p+a·m)c0*+[a0m+(c0p0+c·p)a+i(a×m)]·c*}.
(c*×a)·(c×p)=(c*·c)(a·p)-(c*·p)(a·c),
Iout=Iin8(Ξ1+Ξ2),
Ξ1=[|c0|2(a0p0+a·p)+|c|2(a0p0-a·p)]+p·(c*c+cc*)·a,
Ξ2=(c0*c+c0c*)·(a0p+p0a)+i(c*×c)·(a0p-p0a)+i(c0*c-c0c*)·(p×a).
a0=1,c0=1+ξ,p0=It[1-Θz cos(2P)],
a=(sin[2(A-C)], 0, cos[2(A-C)]),
c=(0, 0, 1-ξ),
px=It{Θx sin(2P)cos(2C)+[Θz-cos(2P)]sin(2C)},
py=ItΘy sin(2P),
pz=It{Θx sin(2P)sin(2C)+[cos(2P)-Θz]cos(2C)},
It=12(|rp|2+|rs|2){1+αz cos[2(S-P)]}Iin
Θ=(sin(2ψ)cos Δ,sin(2ψ)sin Δ,cos(2ψ)),
ρ=rprs=(tan ψ)exp(iΔ).
Iout=It8(Σ1-Σ2-Σ3+Σ4),
Σ1=2(1+|ξ|2)[1-Θz cos(2P)]-|1+ξ|2×{[Θz-cos(2P)]cos(2A)-Θx sin(2P)sin(2A)},
Σ2=2i(ξ*-ξ)Θy sin(2P)sin[2(A-C)],
Σ3=|1-ξ|2{[Θz-cos(2P)]cos(2A-4C)+Θx sin(2P)sin(2A-4C)},
Σ4=2(|ξ|2-1){[Θz-cos(2P)]cos(2C)-Θx sin(2P)sin(2C)+[Θz cos(2P)-1]cos[2(A-C)]}.
TP=exp[-i(P-C)σy]JP exp(iPσy)ρin exp(-iPσy)×JP exp[i(P-C)σy]=Iin2(p0I+p·σ),
TA=exp(iCσy)JS exp(-iAσy)JAJA exp(iAσy)×JS exp(-iCσy)=12(a0I+a·σ).
CA=C-A,CP=C-P,
ua+iva=χa,up+ivp=χp-iγp×αz sin[2(P-S)]1+αz cos[2(P-S)].
a0=1,c0=1+ξ,p0=1,
a=(-sin(2CA)+2va cos(2CA), 2(αa+ua),cos(2CA)+2v0 sin(2CA)),
p=(-sin(2CP)+2vp cos(2CP), 2(αp-up),cos(2CP)+2vp sin(2CP)),
c=(1-ξ)(0, 2αc, 1).
Iout=I¯t8{dc+C4 cos[2(CA+CP)]+S4 sin[2(CA+CP)]+C2 cos(CA+CP)+S2 sin(CA+CP)},
I¯t={1+αz cos[2(S-P)]+2 Im(χp)αz sin[2(S-P)]}Iin
dc=2|1+ξ|2 cos2(A-P)+|1-ξ|2+[8 Im(ξ)αc+2|1+ξ|2(vp-va)]×sin[2(A-P)],
C4=|1-ξ|2,S4=2|1-ξ|2(vp+va),
C2=8 Im(ξ)[(up-ua)-(αa+αp)]sin(A-P)+4(1-|ξ|2)cos(A-P),
S2=-8 Im(ξ)[(up+ua)+(αa-αp)]cos(A-P).
[8 Im(ξ)αc+2|1+ξ|2(vp-va)]fromdcandC4,|1-ξ|2(vp+va)fromdcandS4,
Im(ξ)(up-αp),Im(ξ)(ua+αa),

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