Abstract

The dual-rotating-retarder configuration is one of the most common forms of the Mueller matrix polarimeter. I perform an optimization of this polarimeter configuration by minimizing the condition number of the system data reduction matrix. I find the optimum retardance for the rotating retarders to be 127°. If exactly 16 intensity measurements are used for a Mueller matrix calculation, a complex relationship exists between the condition number and the sizes of the angular increments of the two retarders. If many intensity measurements are made, thus overspecifying the calculation, I find broad optimal ranges of angular increments of the two retarders that yield essentially equal performance. Experimental results are given.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. S. N. Jasperson, S. E. Schatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
    [CrossRef]
  6. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometer: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. A. Ambirajan, D. C. Look, “Optimum angles for a Mueller matrix polarimeter,” in Polarization Analysis and Measurement II, D. Goldstein, D. Chenault, eds., Proc. SPIE2265, 314–326 (1994).
    [CrossRef]
  16. J. E. Drewes, R. A. Chipman, M. H. Smith, “Characterizing polarization controllers with Mueller matrix polarimetry,” in Active and Passive Optical Components for WDM Communication, A. Dutta, A. Awwal, N. K. Dutta, K. Okamoto, eds., Proc. SPIE4532, 462–466 (2001).
    [CrossRef]
  17. S. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106–1113 (1996).
    [CrossRef]

2000

1997

1996

S. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106–1113 (1996).
[CrossRef]

1994

1990

1986

1980

1978

1977

1969

S. N. Jasperson, S. E. Schatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Ambirajan, A.

A. Ambirajan, D. C. Look, “Optimum angles for a Mueller matrix polarimeter,” in Polarization Analysis and Measurement II, D. Goldstein, D. Chenault, eds., Proc. SPIE2265, 314–326 (1994).
[CrossRef]

Azzam, R. M. A.

Bottinger, J. R.

Chenault, D. B.

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. Goldstein, R. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

Chipman, R. A.

S. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106–1113 (1996).
[CrossRef]

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

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. Goldstein, R. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

J. E. Drewes, R. A. Chipman, M. H. Smith, “Characterizing polarization controllers with Mueller matrix polarimetry,” in Active and Passive Optical Components for WDM Communication, A. Dutta, A. Awwal, N. K. Dutta, K. Okamoto, eds., Proc. SPIE4532, 462–466 (2001).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 2.

Delplancke, F.

Dereniak, E. L.

Descour, M. R.

Drewes, J. E.

J. E. Drewes, R. A. Chipman, M. H. Smith, “Characterizing polarization controllers with Mueller matrix polarimetry,” in Active and Passive Optical Components for WDM Communication, A. Dutta, A. Awwal, N. K. Dutta, K. Okamoto, eds., Proc. SPIE4532, 462–466 (2001).
[CrossRef]

Fry, E. S.

Goldstein, D. H.

Hauge, P. S.

Jasperson, S. N.

S. N. Jasperson, S. E. Schatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Jellison, G. E.

Keiper, J. B.

See, for example, R. D. Skeel, J. B. Keiper, Elementary Numerical Computing with Mathematica (McGraw-Hill, New York, 1993), Chap. 4.

Kemme, S. A.

Krishnan, S.

Look, D. C.

A. Ambirajan, D. C. Look, “Optimum angles for a Mueller matrix polarimeter,” in Polarization Analysis and Measurement II, D. Goldstein, D. Chenault, eds., Proc. SPIE2265, 314–326 (1994).
[CrossRef]

Lu, S.

S. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106–1113 (1996).
[CrossRef]

Modine, F. A.

Nordine, P. C.

Pezzaniti, J. L.

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. Goldstein, R. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

Phipps, G. S.

Sabatke, D. S.

Schatterly, S. E.

S. N. Jasperson, S. E. Schatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Skeel, R. D.

See, for example, R. D. Skeel, J. B. Keiper, Elementary Numerical Computing with Mathematica (McGraw-Hill, New York, 1993), Chap. 4.

Smith, M. H.

J. E. Drewes, R. A. Chipman, M. H. Smith, “Characterizing polarization controllers with Mueller matrix polarimetry,” in Active and Passive Optical Components for WDM Communication, A. Dutta, A. Awwal, N. K. Dutta, K. Okamoto, eds., Proc. SPIE4532, 462–466 (2001).
[CrossRef]

Sweatt, W. C.

Thompson, R. C.

Tyo, J. S.

Appl. Opt.

J. Opt. Soc. A.

S. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. A. 13, 1106–1113 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Rev. Sci. Instrum.

S. N. Jasperson, S. E. Schatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Other

D. B. Chenault, J. L. Pezzaniti, R. A. Chipman, “Mueller matrix algorithms,” in Polarization Analysis and Measurement, D. Goldstein, R. Chipman, eds., Proc. SPIE1746, 231–246 (1992).
[CrossRef]

A. Ambirajan, D. C. Look, “Optimum angles for a Mueller matrix polarimeter,” in Polarization Analysis and Measurement II, D. Goldstein, D. Chenault, eds., Proc. SPIE2265, 314–326 (1994).
[CrossRef]

J. E. Drewes, R. A. Chipman, M. H. Smith, “Characterizing polarization controllers with Mueller matrix polarimetry,” in Active and Passive Optical Components for WDM Communication, A. Dutta, A. Awwal, N. K. Dutta, K. Okamoto, eds., Proc. SPIE4532, 462–466 (2001).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 2.

See, for example, R. D. Skeel, J. B. Keiper, Elementary Numerical Computing with Mathematica (McGraw-Hill, New York, 1993), Chap. 4.

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Figures (9)

Fig. 1
Fig. 1

Schematic of a DRR Mueller matrix polarimeter.

Fig. 2
Fig. 2

Condition number of the data reduction matrix for a DRR Mueller matrix polarimeter as a function of the retardance value of the linear retarders used in the instrument. The minimum of this curve occurs at 127° (with a mirror image of 233°).

Fig. 3
Fig. 3

Condition number of the data reduction matrix for a DRR Mueller matrix polarimeter operated with 16 evenly spaced angular increments. White areas have a high condition number unacceptable for polarimetry. The regions indicated with checkerboards are the global minima in this space.

Fig. 4
Fig. 4

Condition number of the data reduction matrix for a DRR Mueller matrix polarimeter operated with 30 evenly spaced angular increments. White areas have a high condition number unacceptable for polarimetry. The dark regions have essentially equal condition numbers and are all acceptable for Mueller matrix measurements.

Fig. 5
Fig. 5

Condition number of the data reduction matrix for a DRR Mueller matrix polarimeter operated with 120 evenly spaced angular increments during one revolution of the slowest spinning retarder. The condition number is calculated for different ratios of rotation speeds between the generator and the analyzer retarders.

Fig. 6
Fig. 6

At each different rotation ratio between the generator and the analyzer retarders, 33 Mueller matrices were measured of an empty sample compartment. The rms errors of the measured Mueller matrix elements are plotted.

Fig. 7
Fig. 7

At each different rotation ratio between the generator and the analyzer retarders, 33 Mueller matrices were measured of a 63° linear retarder. The mean retardance δ is plotted. Error bars indicate the standard deviation of the measurements.

Fig. 8
Fig. 8

Mueller matrices were measured of a 63° linear retarder. The standard deviation of the calculated retardance σδ varies with rotation ratio in a manner predicted by the condition number of the data reduction matrix (see Fig. 5).

Fig. 9
Fig. 9

Mueller matrices were measured of a 63° linear retarder. The measured diattenuation D varies from the expected value of zero in a manner predicted by the condition number of the data reduction matrix (see Fig. 5).

Equations (6)

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Pq=Wq·M,
M=WTW-1WTP=WP-1P.
condA=A A-1.
A=max1inj=1n |αij|,
Mair=1.0000-0.0013-0.0015-0.0010-0.00000.9963-0.0083-0.0005-0.00070.0068-0.00490.00290.00130.0033-0.0013-0.0046.
D=Tmax-TminTmax+Tmin=m012+m022+m0321/2m00,

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