Abstract

We demonstrate an automated polarimeter based on a rotating polarizer for the measurement of linear retardance independent of laser power and detector gain. The retardance is found when a curve is fitted to a unique normalization of the intensity response of the polarimeter over a range of input polarizer orientations. The performance of this polarimeter is optimal for measurements of quarter-wave retardance and minimal for half-wave retardance. Uncertainties are demonstrated by measurements on six stable double Fresnel rhombs of nominal quarter-wave retardance, yielding expanded uncertainties between0.031° and 0.067°. The accuracy has also been verified by blind comparisons with interferometric and modified null retardance measurement techniques.

© 1997 Optical Society of America

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  1. The term expanded uncertainty refers to our multiplication of the measured uncertainty by a coverage factor of 2 to give an approximate 95% confidence interval. For more details, see B. N. Taylor, C. E. Kuyatt, eds. “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” Technical Note 1297, 1994 (National Institute of Standards and Technology, Gaithersburg, Md.)
  2. K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
    [CrossRef]
  3. J. M. Bennett, “A critical evaluation of rhomb-type quarterwave retarders,” Appl. Opt. 9, 2123–2129 (1970).
    [CrossRef] [PubMed]
  4. A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
    [CrossRef]
  5. K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
    [CrossRef]
  6. K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” , 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).
  7. K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
    [CrossRef]
  8. H. G. Jerrard, “Optical compensators for measurement of elliptical polarization,” J. Opt. Soc. Am. 38, 35–59 (1948).
    [CrossRef]
  9. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 100–103.
  10. Y. Lin, Z. Zhou, R. Wang, “Optical heterodyne measurement of the phase retardation of a quarter-wave plate,” Opt. Lett. 13, 553–555 (1988).
    [CrossRef]
  11. J. E. Hayden, S. D. Jacobs, “Automated spatially scanning ellipsometer for retardation measurements of transparent materials,” Appl. Opt. 32, 6256–6263 (1993).
    [CrossRef] [PubMed]
  12. L.-H. Shyu, C.-L. Chen, D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).
    [CrossRef] [PubMed]
  13. H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).
  14. C. D. Caldwell, “Digital lock-in technique for measurement of polarization of radiation,” Opt. Lett. 1, 101–103 (1977).
    [CrossRef] [PubMed]
  15. D. B. Chenault, R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
    [CrossRef] [PubMed]
  16. B. R. Grunstra, H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966).
    [CrossRef] [PubMed]
  17. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  18. E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
    [CrossRef]
  19. R. C. Plumb, “Analysis of elliptically polarized light,” J. Opt. Soc. Am. 50, 892–894 (1960).
    [CrossRef]
  20. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  21. J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
    [CrossRef]
  22. F. A. Graybill, Theory and Application of the Linear Model (Duxbury, North Scituate, Mass., 1976), pp. 608–615.

1997 (2)

1993 (3)

1992 (1)

1991 (1)

E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
[CrossRef]

1988 (1)

1981 (1)

J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
[CrossRef]

1977 (1)

1970 (1)

1966 (1)

1964 (1)

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

1960 (1)

1948 (1)

1941 (1)

1911 (1)

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[CrossRef]

Agatsuma, T.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Bennett, J. M.

Caldwell, C. D.

Chen, C.-L.

Chenault, D. B.

Chipman, R. A.

Clarke, I. G.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 100–103.

Day, G. W.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Dennis, J. E.

J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
[CrossRef]

Dijkstra, E.

E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
[CrossRef]

Gay, D. M.

J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
[CrossRef]

Goldstein, D. H.

Graybill, F. A.

F. A. Graybill, Theory and Application of the Linear Model (Duxbury, North Scituate, Mass., 1976), pp. 608–615.

Grunstra, B. R.

Hale, P. D.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Hayden, J. E.

Isobe, M.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Jacobs, S. D.

Jerrard, H. G.

Jones, R. C.

Konda, A.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Kremers, M.

E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
[CrossRef]

Lin, Y.

Masaki, T.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Meekes, H.

E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
[CrossRef]

Oxley, A. E.

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[CrossRef]

Perkins, H. B.

Plumb, R. C.

Rochford, K. B.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
[CrossRef]

K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” , 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Rose, A. H.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Shyu, L.-H.

Su, D.-C.

Takasaki, H.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Wang, C. M.

K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” , 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Wang, R.

Watanabe, Y.

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

Welsh, R. E.

J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
[CrossRef]

Williams, P. A.

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

Zhou, Z.

ACM Trans. Math. Software (1)

J. E. Dennis, D. M. Gay, R. E. Welsh, “An adaptive nonlinear least-squares algorithm,” ACM Trans. Math. Software 7, 348–368 (1981).
[CrossRef]

Appl. Opt. (9)

H. Takasaki, M. Isobe, T. Masaki, A. Konda, T. Agatsuma, Y. Watanabe, “An automated retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 343–350 (1964).

B. R. Grunstra, H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966).
[CrossRef] [PubMed]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
[CrossRef] [PubMed]

D. B. Chenault, R. A. Chipman, “Measurements of linear diattenuation and linear retardance spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32, 3513–3519 (1993).
[CrossRef] [PubMed]

L.-H. Shyu, C.-L. Chen, D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).
[CrossRef] [PubMed]

J. E. Hayden, S. D. Jacobs, “Automated spatially scanning ellipsometer for retardation measurements of transparent materials,” Appl. Opt. 32, 6256–6263 (1993).
[CrossRef] [PubMed]

K. B. Rochford, C. M. Wang, “Accurate interferometric retardance measurements,” Appl. Opt. 36, 6473–6479 (1997).
[CrossRef]

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Design and performance of a stable linear retarder,” Appl. Opt. 36, 6458–6465 (1997).
[CrossRef]

J. M. Bennett, “A critical evaluation of rhomb-type quarterwave retarders,” Appl. Opt. 9, 2123–2129 (1970).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Phys. D (1)

E. Dijkstra, H. Meekes, M. Kremers, “The high-accuracy universal polarimeter,” J. Phys. D 24, 1861–1868 (1991).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

A. E. Oxley, “On apparatus for the production of circularly polarized light,” Philos. Mag. 21, 517–532 (1911).
[CrossRef]

Other (5)

K. B. Rochford, C. M. Wang, “Uncertainty in null polarimeter measurements,” , 1996 (National Institute of Standards and Technology, Gaithersburg, Md.).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp. 100–103.

The term expanded uncertainty refers to our multiplication of the measured uncertainty by a coverage factor of 2 to give an approximate 95% confidence interval. For more details, see B. N. Taylor, C. E. Kuyatt, eds. “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” Technical Note 1297, 1994 (National Institute of Standards and Technology, Gaithersburg, Md.)

K. B. Rochford, A. H. Rose, P. A. Williams, C. M. Wang, I. G. Clarke, P. D. Hale, G. W. Day, “Standard polarization components: progress toward an optical retardance standard,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 2–8 (1994).
[CrossRef]

F. A. Graybill, Theory and Application of the Linear Model (Duxbury, North Scituate, Mass., 1976), pp. 608–615.

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Figures (2)

Fig. 1
Fig. 1

Design of polarimeter with (a) detection system used for actual retardance measurement (P2 can be in or out of beam) and (b) detection system for orientation of polarization axes with respect to test retarder’s axes.

Fig. 2
Fig. 2

Typical measurement results of normalized intensity versus polarizer angle. Solid circles represent measured data, solid curve is nonlinear least-squares fit.

Tables (2)

Tables Icon

Table 1 Itemized Uncertainty Estimates for Polarimeter

Tables Icon

Table 2 Summary of Measurement Results and Uncertainties on Six Double-Rhomb Retarders

Equations (6)

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

T + δ ,   β ,   θ = 1 2   Γ + cos 2   β sin   β   cos   β sin   β   cos   β sin 2   β × cos δ / 2 i   sin δ / 2 i   sin δ / 2 cos δ / 2 × cos 2   θ sin   θ   cos   θ sin   θ   cos   θ sin 2   θ ,
I + δ ,   β ,   θ = I 0 Γ + 2 sin 2 δ / 2   cos 2 θ + β + cos 2 δ / 2   sin 2 θ - β ,
I - δ ,   β ,   θ = I 0 Γ - 2 sin 2 δ / 2   sin 2 θ + β + cos 2 δ / 2   cos 2 θ - β ,
I N δ ,   β ,   θ = I + δ ,   β ,   θ I - δ ,   β ,   θ + π / 2 I - δ ,   β ,   θ I + δ ,   β ,   θ + π / 2 1 / 2 ,
I N δ ,   β ,   θ = 1 cos 2 β - θ   cos 2 δ / 2 + sin 2 β + θ   sin 2 δ / 2 - 1 ,
χ 2 = i   I N , Exp θ i - I N , Fit θ i 2 ,

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