Abstract

Absolute constraints, namely, the Schwarz inequality and a complementary expression derived by us, are used to obtain corresponding absolute constraints on the Fourier coefficients of the intensity transmitted through rotating-compensator polarimeters and ellipsometers. These expressions allow the investigation of artifacts that result in mixed or apparently mixed polarization states over the cross section of the beam, the averaging time of the detector, or the frequency passband of the dispersing element. Examples include multiple internal reflections or inhomogeneous strain within an element, scattered light, and other types of system and component defects that cannot be accessed by means of polarization-state data alone. We apply these results to our polarizer-sample-compensator-analyzer (PSCA) ellipsometer to illustrate capabilities. A simple analytic model is shown to give a quantitative description of depolarization in systems for which the resolution is finite and the retardation varies with wavelength.

© 2001 Optical Society of America

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References

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  1. See, for example, H. G. Tompkins, W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, New York, 1999).
  2. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).
  4. S. Li, J. Fanton, J. Opsal, “Artifacts of elements used in ellipsometry,” available from J. Fanton, Therma-Wave, Inc., 1250 Reliance Way, Fremont, Calif. 94539.
  5. See, for example, the Proceedings of the Second International Conference on Spectroscopic Ellipsometry, Thin Solid Films, 313–314 (1998).
  6. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
    [CrossRef]
  7. P. R. Halmos, Finite-Dimensional Vector Spaces (Van Nostrand, Princeton, N.J., 1958).
  8. S. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
    [CrossRef]

2000

1998

See, for example, the Proceedings of the Second International Conference on Spectroscopic Ellipsometry, Thin Solid Films, 313–314 (1998).

1997

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).

Born, M.

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

Fanton, J.

S. Li, J. Fanton, J. Opsal, “Artifacts of elements used in ellipsometry,” available from J. Fanton, Therma-Wave, Inc., 1250 Reliance Way, Fremont, Calif. 94539.

Halmos, P. R.

P. R. Halmos, Finite-Dimensional Vector Spaces (Van Nostrand, Princeton, N.J., 1958).

Jellison, G. E.

Li, S.

S. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
[CrossRef]

S. Li, J. Fanton, J. Opsal, “Artifacts of elements used in ellipsometry,” available from J. Fanton, Therma-Wave, Inc., 1250 Reliance Way, Fremont, Calif. 94539.

McGahan, W. A.

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

Modine, F. A.

Opsal, J.

S. Li, J. Fanton, J. Opsal, “Artifacts of elements used in ellipsometry,” available from J. Fanton, Therma-Wave, Inc., 1250 Reliance Way, Fremont, Calif. 94539.

Tompkins, H. G.

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

Wolf, E.

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

Appl. Opt.

J. Opt. Soc. Am. A

Proceedings of the Second International Conference on Spectroscopic Ellipsometry, Thin Solid Films

See, for example, the Proceedings of the Second International Conference on Spectroscopic Ellipsometry, Thin Solid Films, 313–314 (1998).

Other

P. R. Halmos, Finite-Dimensional Vector Spaces (Van Nostrand, Princeton, N.J., 1958).

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

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

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

S. Li, J. Fanton, J. Opsal, “Artifacts of elements used in ellipsometry,” available from J. Fanton, Therma-Wave, Inc., 1250 Reliance Way, Fremont, Calif. 94539.

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Figures (3)

Fig. 1
Fig. 1

P=D versus λ for our RCE polarimeter for PCA operation with a single-plate MgF2 compensator.

Fig. 2
Fig. 2

As in Fig. 1, but for a biplate compensator and a silicon sample covered with 760.8-nm-thick oxide.

Fig. 3
Fig. 3

Comparison between the data of Fig. 2 and the result of a model calculation for depolarization resulting from wavelength-dependent phase and a detector of finite resolution, as described by Eq. (28e). Points connected by curves, data; dashed and heavy solid curves, first- and second-order calculations, respectively; thin solid curve, phase.

Equations (69)

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ρ=ρxxρxyρyxρyy=ExEx*ExEy*Ex*EyEyEy*,
s0=ExEx*+EyEy*,
s1=ExEx*-EyEy*,
s2=ExEy*+Ex*Ey=2 ReExEy*,
s3=-i(ExEy*-Ex*Ey)=2 ImExEy*.
ExEx*=12 (s0+s1), EyEy*=12 (s0-s1),
ExEy*=12 (s2+is3).
S=η[C0+C2cos(2ωt)+S2sin(2ωt)+C4cos(4ωt)+S4sin(4ωt)],
C0=s0+s1cos2 (δ/2),
C2=-s3sin(δ)sin(2θ0),
S2=-s3sin(δ)cos(2θ0),
C4=sin2(δ/2)[-s1cos(4θ0)+s2sin(4θ0)],
S4=sin2(δ/2)[s1sin(4θ0)+s2cos(4θ0)].
s02s12+s22+s32.
P=IpolItot=(s12+s22+s32)1/2s0.
P=1-4 (|Ex|2|Ey|2 -|ExEy*|2)s021/2.
0P1.
s12+s22=(S42+C42)csc2(δ/2).
s32=(S22+C22)csc2(δ),
s0=C0-s1cos2(δ/2).
s0=C0-ϑ cot2(δ/2).
ϑ=[C4(S22-C22)-2S4S2C2]S22+C22,
0P=[4(S42+C42)+(S22+C22)tan2(δ/2)]1/2/{2[C0sin2(δ/2)-ϑ cos2(δ/2)]}1.
0P=[4(S42+C42)+S22tan2(δ/2)]1/2/{2[C0sin2(δ/2)-C4cos2(δ/2)]}1.
0limδ±nπP=limδ±nπ[(C42+S42)csc4(δ/2)+(S22+C22)csc2(δ)]1/2C0
1,
limδ±nπcsc2(δ)limδ±nπC02-(C42+S42)csc4(δ/2)S22+C22.
limδ±nπcsc2(δ),limδ±nπcsc2(δ/2)=1,
0(C42+S42+S22+C22)1/2C01.
0limδ±nπP=[(S22+C22)2(S42+C42)]1/2C0(S22+C22)sin2(δ/2)+2C2S2S4-C4(S22-C22)
1.
0P=(S42+C42)1/2C0sin2(δ/2)-C4cos2(δ/2)1.
0(s12+s22)1/2s01.
Jsys=exp(-iAσy)JAexp(iAσy)exp(-iSσy)JS×exp(iSσy)exp(-iCσy)JCexp(iCσy)JP,
S=η Tr(JsysρinJsys),
S=η Tr(ρGD),
ρG=exp(-iCσy)JCexp(iCσy)JPρinJPexp(-iCσy)JC×exp(iCσy),
D=exp(-iSσy)JSexp[i(S-A)σy]JAJA×exp[i(A-S)σy]JSexp(iSσy).
ρG=s0I+s2σx-s3σy+s1σz,
D=d0I+dxσx+dyσy+dzσz,
S=η Tr(ρGD)=η(ExEy)DExEy,
DExEy=βExEy
β+=d0+(dx2+dy2+dx2)1/2
β-=d0-(dx2+dy2+dx2)1/2.
d0(dx2+dy2+dx2)1/20,
0D=(dx2+dy2+dz2)1/2d01.
D=ςmax-ςminςmax+ςmin=β+-β-β++β-.
ρG=Iin[Σ0-ΣS2sin(2C)+ΣC4cos(4C)+ΣS4sin(4C)],
Iin=04[Tr(ρin)+Tr(ρinσz)].
Σ0=I+cos2(δ/2)σz,
ΣS2=sin(δ)σy,
ΣC4=sin2(δ/2)σz,
ΣS4=cos2(δ/2)σx.
S=C0+C2cos(2ωt)+S2sin(2ωt)+C4cos(4ωt)+S4sin(4ωt),
C0=ηIin[d0+dzcos2(δ/2)],
C2=ηIindysin(2θ0)sin(δ),
S2=ηIindycos(2θ0)sin(δ),
C4=ηIinsin2(δ/2)[dzcos(4θ0)+dxsin(4θ0)],
S4=ηIinsin2(δ/2)[dxcos(4θ0)-dzsin(4θ0)].
0D=[4(S42+C42)+(S22+C22)tan2(δ/2)]1/22[C0sin2(δ/2)-ϑ cos2(δ/2)]1.
ExEy=Ex1tan ψ exp(iΔ)
Δ(λ)=Δ0+Δ(λ-λ0)+Δ(λ-λ0)2.
|Ex|2=|Ex|2-exp-(λ-λ0)2σ2dλ,
|Ey|2=|Ex|2tan2 ψ-exp-(λ-λ0)2σ2dλ,
Ex*Ey=|Ex|2tan ψ exp(iΔ0)-exp-(λ-λ0)2σ2×exp{i[(Δ(λ-λ0)+Δ(λ-λ0)2)]}dλ.
-exp-(λ-λ0)2σ2dλ=πσ,
-exp-(λ-λ0)2σ2exp{i[(Δ(λ-λ0)
+Δ(λ-λ0)2)]}dλ=πσ˜ exp-14 (Δσ˜)2;
P=1-sin2 (2ψ)×1-1(1+Δ2σ4)1/2×exp-Δ2σ22(1+Δ2σ4)1/2.

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