Abstract

The linear errors of Mueller matrix measurements are formulated for misalignment, depolarization, and incorrect retardation of the polarimetric components. The measured errors of a Mueller matrix depend not only on the imperfections of the measuring system but also on the Mueller matrix itself. The error matrices for different polarimetric systems are derived and also evaluated for the straight-through case. The error matrix for a polarizer–sample–analyzer system is much simpler than those for more complicated systems. The general error matrix is applied to null ellipsometry, and the obtained errors in ellipsometric parameters ψ and Δ are identical to the errors specifically derived for null ellipsometry with depolarization.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
    [CrossRef]
  3. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  4. D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
    [CrossRef]
  5. B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
    [CrossRef]
  6. S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
    [CrossRef]
  7. S.-M. F. Nee, “Depolarization and principal Mueller matrix measured by null ellipsometry,” Appl. Opt. 40, 4933–4939 (2001).
    [CrossRef]
  8. K. A. O’Donnell, M. E. Knotts, “Polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  9. T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
    [CrossRef]
  10. R. M. A. Azzam, N. M. Bashara, “Ellipsometry with im-perfect components including incoherent effects,” J. Opt. Soc. Am. 61, 1380–1391 (1977).
    [CrossRef]
  11. S.-M. F. Nee, “Error reduction for a serious compensator imperfection for null ellipsometry,” J. Opt. Soc. Am. A 8, 314–321 (1991).
    [CrossRef]
  12. S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
    [CrossRef]
  13. S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
    [CrossRef]
  14. S.-M. F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000).
    [CrossRef]
  15. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
  16. W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).
  17. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1977).
  18. R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 22.
  19. S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC and IEEE, Boca Raton, 1999), Chap. 60.
  20. S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of infrared polarizer,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 213–224 (1997).

2002

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

2001

2000

1999

1998

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

1996

1994

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

1992

1991

1990

1978

1977

Azzam, R. M. A.

Ballard, S. S.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Bashara, N. M.

Bjork, D. R.

T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
[CrossRef]

Boulbry, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Bousquet, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Burge, D.

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of infrared polarizer,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 213–224 (1997).

Cariou, J.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Chipman, R. A.

D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 22.

Cole, T.

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of infrared polarizer,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 213–224 (1997).

Goldstein, D. H.

Knotts, M. E.

Le Jeunne, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Lotrian, J.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Ludema, K. C.

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Nee, S.-M. F.

O’Donnell, K. A.

Pellen, F.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Ramsey, D. A.

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Schiff, T. F.

T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Stover, J. C.

T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
[CrossRef]

Swimley, B. D.

T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
[CrossRef]

Yoo, C.

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of infrared polarizer,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 213–224 (1997).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, J. Lotrian, “Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source,” Meas. Sci. Technol. 13, 1563–1573 (2002).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

D. A. Ramsey, K. C. Ludema, “The influences of roughness on film thickness measurements by Mueller matrix ellipsometry,” Rev. Sci. Instrum. 65, 2874–2881 (1994).
[CrossRef]

Thin Solid Films

S.-M. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Other

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

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

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), Chap. 22.

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC and IEEE, Boca Raton, 1999), Chap. 60.

S.-M. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of infrared polarizer,” in Polarization: Measurement, Analysis and Remote Sensing, D. H. Goldstein, R. A. Chipman, eds., Proc. SPIE3121, 213–224 (1997).

T. F. Schiff, J. C. Stover, B. D. Swimley, D. R. Bjork, “Mueller matrix measurements of scattered light,” in Stray Radiation in Optical Systems II, R. P. Breult, ed., Proc. SPIE1753, 269–277 (1992).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (1)

Tables Icon

Table 1 Ideal Null Positions of Polarizer Angle P and Analyzer Angle A for the Four Zones in Null Ellipsometry

Equations (65)

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

I=D  M  X,
I=(Do+δD)  M  (Xo+δX)=Do  M*  Xo,
M*=(1+δmd)M(1+δmx)=M+δMx+δMd.
δM=δMx+δMd=Mδmx+δmdM.
δMo=δmx+δmd.
M=T1+D-cos 2ψ00-cos 2ψ1+D-2Dv0000sin 2ψ cos Δsin 2ψ sin Δ00-sin 2ψ sin Δsin 2ψ cos Δ.
Po=T1100110000000000.
Xo=Po(P)  So,
So=(1, 0, 0, 0)T.
X=P(P+δP)  So=R(-δP)Po(P)R(δP)  So=R(-δP)  Xo,
δR(δP)=0000002δP00-2δP000000.
P(0)=1+Dp10011+Dp-2Dvp0000000000.
X=P(P+δP)  So=R(-δP)  (1+Dp, cos 2P, sin 2P, 0)T=R(-δP)1+Dp000010000100001  1cos 2Psin 2P0.
δDp=Dp000000000000000.
δmx=δR(-δP)+δDp=Dp00000-2δP002δP000000.
δmd=δR(δA)+δDa=Da000002δA00-2δA000000.
δMx=Mδmx=DpM002δPM02-2δPM010DpM102δPM12-2δPM110DpM202δPM22-2δPM210DpM302δPM32-2δPM310,
δMd=δmdM=DaM00DaM01DaM02DaM032δAM202δAM212δAM222δAM23-2δAM10-2δAM11-2δAM12-2δAM130000.
δMo=Da+Dp000002δA-2δP002δP-2δA000000.
C(0, τ)
=Tc1+Dc00001+Dc-2Dvc0000cos τsin τ00-sin τcos τ.
C(0, τ+δτ)=C(0, δτ)Co(0, τ)=[1+δc(δτ)]Co(0, τ),
δc(δτ)=Dc0000Dc-2Dvc00000δτ00-δτ0.
X=C(C+δC, τ+δτ)P(P+δP)  So=(1+δmx)Co(C, τ)Po(P)  So,
δmx=δR(-δC)+R(-C)δc(δτ)R(C)+R(-C)CoδR(δC-δP)Co-1R(C)+δDp.
δR(-δC)=000000-2δC002δC000000,
R(-C)δc(δτ)R(C)+δDp=Dc+Dp0000(Dc-2Dvc)cos2 2C½(Dc-2Dvc)sin 4C-δτ sin 2C0½(Dc-2Dvc)sin 4C(Dc-2Dvc)sin2 2Cδτ cos 2C0δτ sin 2C-δτ cos 2C0,
R(-C)CoδR(δC-δP)Co-1R(C)=2(δC-δP)000000cos τ-cos 2C sin τ0-cos τ0-sin 2C sin τ0cos 2C sin τsin 2C sin τ0.
δmx(C=0|90°)
=Dc+Dp0000Dc-2Dvc-2δC2(δC-δP)02δC0±δτc0±2(δC-δP)δτc0,
δmx(C=±45°)
=Dc+Dp00000-2δCδτc02δCDc-2Dvc2(δC-δP)0±δτc±2(δC-δP)0.
δmd(Q=0|90°)
=Dq+Da0000Dq-2Dvq2δQ2(δA-δQ)0-2δQ0±δτq0±2(δA-δQ)±δτq0,
δmd(Q=±45°)
=Dq+Da000002δQ±δτq0-2δQDq-2Dvq2(δA-δQ)0δτq±2(δA-δQ)0,
 
δMo=Dc+Dp+Da00(Dc-2Dvc)cos2 2C0-2(δA-δC)+(Dc-2Dvc)sin 2C cos 2C0δτ sin 2C+2(δC-δP)cos 2C
002(δA-δC)+(Dc-2Dvc)sin 2C cos 2C-δτ sin 2C-2(δC-δP)cos 2C(Dc-2Dvc)sin2 2Cδτ cos 2C-2(δC-δP)sin 2C-δτ cos 2C+2(δC-δP)sin 2C0.
δMo(0|90°)=Dp+Da+Dc0000Dc-2Dvc2(δA-δC)2(δC-δP)0-2(δA-δC)0±δτc0±2(δC-δP)δτc0,
δMo(±45°)=Dp+Da+Dc000002(δA-δC)δτc0-2(δA-δC)Dc-2Dvc2(δC-δP)0±δτc±2(δC-δP)0.
δM00=Dp+Da+Dc+Dq,
δM01=δM02=δM03=δM10=δM20=δM30=δM33=0,
δM11=(Dc-2Dvc)cos2 2C+(Dq-2Dvq)cos2 2Q,
δM12=2δQ-2δC+(Dc-2Dvc)sin 2C cos 2C+2(δC-δP)cos τc+(Dq-2Dvq)sin 2Q cos 2Q+2(δA-δQ)cos τq,
δM13=-δM31=-δτcsin 2C-2(δC-δP)cos 2C sin τc+δτqsin 2Q-2(δA-δQ)cos 2Q sin τq,
δM21=2δC-2δQ+(Dc-2Dvc)sin 2C cos 2C-2(δC-δP)cos τc+(Dq-2Dvq)sin 2Q cos 2Q-2(δA-δQ)cos τq,
δM22=(Dc-2Dvc)sin2 2C+(Dq-2Dvq)sin2 2Q,
δM23=-δM32=δτccos 2C-2(δC-δP)sin 2C sin τc-δτqcos 2Q-2(δA-δQ)sin 2Q sin τq.
M=T1Px00Px1-2Dv0000PyPz00-PzPy,
Px=-Pcos 2ψ,
Py=Psin 2ψ cos Δ,
Pz=Psin 2ψ sin Δ,
D=1-P=Du+Dv.
δM=Dc+Dp+DaDaPxNANAPx(Dc+Dp)0NANANANA(Dc-2Dvc)Py±2(δC-δP)Pz2(δC-δP)PyNANA±2(δC-δP)Py-(Dc-2Dvc)Pz±2(δC-δP)Pz.
δPx=Px(Dc+Dp),
δPy=(Dc-2Dvc)Py±2(δC-δP)Pz,
δPz=2(δC-δP)Py.
δPx=Px(Dc+Dp),
δPy=(Duc-Dvc)Py,
δPz=0.
tan Δ=Pz/Py,
tan 2ψ=(Py2+Pz2)1/2/Px.
δΔ=sin Δ cos Δ(δPz/Pz-δPy/Py)=-(Duc-Dvc)sin Δ cos Δ.
δψ=½ sin 2ψ cos 2ψPzδPz+PyδPyPy2+Pz2-δPxPx=¼ sin 4ψ[(Duc-Dvc)cos2 Δ-(Dc+Dp)].

Metrics