Abstract

Data analysis techniques are reviewed and extended for the measurement of the Stokes vector of partially or completely polarized radiation by the rotating quarter-wave method. It is shown that the conventional technique, based on the Fourier analysis of the recorded signal, can be efficiently replaced by a weighted least-squares best fit, so that the different accuracy of the measured data can be taken into account to calculate the measurement errors of the Stokes vector elements. Measurement errors for the polarization index P and for the azimuth and ellipticity angles ψ and χ of the radiation are also calculated by propagation error theory. For those cases in which the above technique gives a nonphysical Stokes vector (i.e., with a polarization degree of P>1) a constrained least-squares best fit is introduced, and it is shown that in this way a Stokes vector with P = 1 (rather than P1) is always obtained. In addition an analysis technique useful to remove from the measured data systematic errors due to initial misalignment of the rotating quarter-wave axis is described. Examples of experimental Stokes vectors obtained by the above techniques during the characterization of components for a far-infrared polarimeter at λ=118.8μm for applications in plasma diagnostics are presented and discussed. Finally the problem of the experimental determination of physically consistent Mueller matrices (i.e., of Mueller matrices for which the transformed Stokes vector has always P1) is discussed, and it is shown that for simple Mueller matrices of the ABCD type, whose elements can be determined by the measurement of a single Stokes vector, the imposed P1 constraint gives a sufficient condition for physical consistency. On the other hand, the same constraint, when imposed to the set of four basic Stokes vectors conventionally measured for the determination of a full 16-element Mueller matrix, gives only a necessary but not a sufficient condition.

© 2007 Optical Society of America

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References

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  2. P. S. Hauge, "Recent developments in instrumentation in ellipsometry," Surf. Sci. 96, 108-140 (1980).
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  3. R. W. Collins, "Automatic rotating element ellipsometers: calibration, operation, and real-time applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
    [CrossRef]
  4. S. E. Segre, "A review of plasma polarimetry--theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
    [CrossRef]
  5. S. E. Segre, "Determination of both the electron density and the poloidal magnetic field in a tokamak plasma from polarimetric measurements of phase only," Plasma Phys. Control. Fusion 38, 883-888 (1996).
    [CrossRef]
  6. L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).
  7. L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
    [CrossRef]
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    [CrossRef]
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  27. This matrix is not experimental; it has been obtained by perturbating with random noise the Mueller matrix of a combination of half-wave and quarter-wave plates.

2006 (1)

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

2005 (1)

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

2004 (2)

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

2000 (1)

1999 (1)

S. E. Segre, "A review of plasma polarimetry--theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

1996 (1)

S. E. Segre, "Determination of both the electron density and the poloidal magnetic field in a tokamak plasma from polarimetric measurements of phase only," Plasma Phys. Control. Fusion 38, 883-888 (1996).
[CrossRef]

1993 (3)

1990 (2)

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

R. W. Collins, "Automatic rotating element ellipsometers: calibration, operation, and real-time applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
[CrossRef]

1989 (1)

S. R. Cloude, "Conditions for the physical realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

1987 (1)

R. Bakarat, "Conditions for the physical realizability of polarization matrices characterizing passive systems," J. Mod. Opt. 34, 1535-1544 (1987).
[CrossRef]

1986 (1)

S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).

1981 (1)

1980 (1)

P. S. Hauge, "Recent developments in instrumentation in ellipsometry," Surf. Sci. 96, 108-140 (1980).
[CrossRef]

1979 (1)

1978 (1)

1952 (1)

Bakarat, R.

R. Bakarat, "Conditions for the physical realizability of polarization matrices characterizing passive systems," J. Mod. Opt. 34, 1535-1544 (1987).
[CrossRef]

Bardamid, A. F.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Beightler, C. S.

C. S. Beightler, D. T. Phillips, and D. J. Wilde, Foundations of Optimization, 2nd ed. (Prentice-Hall, 1979).

Belyaeva, A. I.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Berezhnyj, V. L.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Brandt, S.

S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, 1976).

Brombin, M.

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

Brosseuau, C.

Cavinato, M.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Chipman, R. A

Cloude, S. R.

S. R. Cloude, "Conditions for the physical realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

S. R. Cloude, "Group theory and polarisation algebra," Optik (Stuttgart) 75, 26-36 (1986).

Collins, R. W.

R. W. Collins, "Automatic rotating element ellipsometers: calibration, operation, and real-time applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
[CrossRef]

De Pasqual, L.

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

Donné, A. J. H.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Fry, E. S.

Galuza, A. A.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Gil, C.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Gil, J. J.

Giudicotti, L.

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Givens, C. R.

Goldstein, D.

D. Goldstein, Polarized Light (Dekker, 2003).
[CrossRef]

Goldstein, D. H.

Graswinckel, M. F.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Hauge, P. S.

P. S. Hauge, "Recent developments in instrumentation in ellipsometry," Surf. Sci. 96, 108-140 (1980).
[CrossRef]

P. S. Hauge, "Mueller matrix ellipsometry with imperfect compensators," J. Opt. Soc. Am. 68, 1519-1528 (1978).
[CrossRef]

Howell, B. J.

Jerrard, H. R.

Kattawar, G. W.

Konovalov, V. G.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Koslowski, H. R.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Kostinski, A. B.

Kwiatkowski, J. M.

Lipa, M.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Malaquais, A.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

McCarthy, P.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Naidenkova, D. I.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Nyhan, C.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Phillips, D. T.

C. S. Beightler, D. T. Phillips, and D. J. Wilde, Foundations of Optimization, 2nd ed. (Prentice-Hall, 1979).

Prunty, S.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Prunty, S. L.

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Ryzhkov, V. I.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Schunke, B.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Segre, S. E.

S. E. Segre, "A review of plasma polarimetry--theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

S. E. Segre, "Determination of both the electron density and the poloidal magnetic field in a tokamak plasma from polarimetric measurements of phase only," Plasma Phys. Control. Fusion 38, 883-888 (1996).
[CrossRef]

Shevchenko, T. I.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Solodovchenko, S. I.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Spillane, M.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Topkov, A. N.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Voitsenya, V. S.

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

Walker, C.

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

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Zilli, E.

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Optik (1)

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Plasma Phys. Control. Fusion (1)

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[CrossRef]

Plasma Phys. Controlled Fusion (1)

S. E. Segre, "A review of plasma polarimetry--theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

Rev. Sci. Instrum. (4)

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[CrossRef]

L. Giudicotti, M. Brombin, S. L. Prunty, L. De Pasqual, and E. Zilli, "Far-infrared polarimetric characterization of metallic mirrors exposed to a tokamak plasma," Rev. Sci. Instrum. 77, 123504 (2006).
[CrossRef]

V. S. Voitsenya, A. J. H. Donné, A. F. Bardamid, T. Shevchenko, A. I. Belyaeva, V. L. Berezhnyj, A. A. Galuza, C. Gil, V. G. Konovalov, M. Lipa, A. Malaquais, D. I. Naidenkova, V. I. Ryzhkov, B. Schunke, S. I. Solodovchenko, and A. N. Topkov, "Simulation of environment effects on retroreflectors in ITER," Rev. Sci. Instrum. 76, 083502 (2004).
[CrossRef]

A. J. H. Donné, M. F. Graswinckel, M. Cavinato, L. Giudicotti, E. Zilli, C. Gil, H. R. Koslowski, P. McCarthy, C. Nyhan, S. Prunty, M. Spillane, and C. Walker, "Poloidal polarimeter for current density measurements in ITER," Rev. Sci. Instrum. 75, 4694-4701 (2004).
[CrossRef]

Surf. Sci. (1)

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[CrossRef]

Other (9)

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[CrossRef]

S. Brandt, Statistical and Computational Methods in Data Analysis (North-Holland, 1976).

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All data analyses have been performed by the mathematical package MathCad (www.mathcad.com). MathCad files for all procedures in this paper are available from the authors.

C. S. Beightler, D. T. Phillips, and D. J. Wilde, Foundations of Optimization, 2nd ed. (Prentice-Hall, 1979).

L. Giudicotti, M. Brombin, S. L. Prunty, and L. De Pasqual, "Experimental investigation of polarization effects in the FIR range of deposited layers on mirrors," Final Report of contract TW3-TPDS-DIADEV D2, EFDA (European Fusion Development Agreement), 2005 (unpublished).

Kapton is the commercial name of a polyimide film produced by DuPont. TPX (polymethylpentene) is a FIR transmitting plastic material.

S. R. Cloude, "Conditions for the physical realisability of matrix operators in polarimetry," Proc. SPIE 1166, 177-185 (1989).

This matrix is not experimental; it has been obtained by perturbating with random noise the Mueller matrix of a combination of half-wave and quarter-wave plates.

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Figures (7)

Fig. 1
Fig. 1

Rotating quarter-wave method for the measurement of the Stokes vector.

Fig. 2
Fig. 2

Schematic of the Stokes polarimeter for the polarimetric characterization of plasma-exposed mirrors. LP, linear polarizer; M, mirror; BS, beam splitter; HW, half-wave plate; QW, quarter-wave plate; RQW, rotating quarter-wave plate.

Fig. 3
Fig. 3

(Color online) Top view of the Stokes polarimeter for the polarimetric characterization of plasma-exposed mirrors.

Fig. 4
Fig. 4

(Color online) Example of data analysis for a Stokes vector measurement. The two waveforms show the reconstruction of the detector signal obtained with the Fourier technique (FT) and with a four-parameter, unconstrained best fit (BF4P).

Fig. 5
Fig. 5

(Color online) Another example of data analysis for a Stokes vector measurement. Waveforms show the reconstruction of the detector signal obtained with the Fourier technique (FT); with a four-parameter, unconstrained best fit (BF4P); and finally by a three- parameter, constrained best fit (BF3P).

Fig. 6
Fig. 6

(Color online) Example of data analysis with subtraction of a systematic error. Upper panel: measured data and fitted waveforms. The significant value of the cos 2 θ component suggests that there is a systematic error δ in the measured data angular positions θ i . The analysis of Section 5 yields δ = 6.26 ° . Lower panel: the same data with new, corrected angular positions θ i = θ i + δ and the fitted waveforms.

Fig. 7
Fig. 7

Contour plot of the degree of polarization P of the M-transformed Stokes vector as a function of the ψ and χ angles of the input Stokes vector. Input polarization states span the entire Poincaré sphere.

Tables (3)

Tables Icon

Table 1 Polarimetric Parameters Obtained from the Data of Fig. 4 by Using Different Analysis Methods

Tables Icon

Table 2 Polarimetric Parameters Obtained from the Data of Fig. 5 by Different Analysis Methods

Tables Icon

Table 3 Polarimetric Parameters Obtained from the Data of Fig. 6 Before and After the Systematic Error Subtraction

Equations (64)

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S D ( θ ) = LP · QW ( θ ) · S ,
LP = 1 2 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] ,
Q W ( θ ) = [ 1 0 0 0 0 cos 2 2 θ sin 2 θ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ sin 2 2 θ cos 2 θ 0 sin 2 θ cos 2 θ 0 ]
I D ( θ ) = ( S D ) 0 = 1 2 ( S 0 + S 1 cos 2 2 θ + S 2 sin 2 θ cos 2 θ S 3 sin 2 θ ) = 1 2 S 0 + 1 4 S 1 S 3 sin 2 θ + 1 4 S 1 cos 4 θ + 1 4 S 2 sin 4 θ .
I D ( θ ) = a 0 2 + n = 1 2 ( a 2 n cos 2 n θ + b 2 n sin 2 n θ ) ,
a 0 = 1 π θ 0 θ 0 + 2 π I D ( θ ) d θ ,
a 2 n = 1 π θ 0 θ 0 + 2 π I D ( θ ) cos 2 n θ d θ n = 1 , 2 ,
b 2 n = 1 π θ 0 θ 0 + 2 π I D ( θ ) sin 2 n θ d θ n = 1 , 2 ,
a 0 = S 0 + 1 2 S 1 ,
a 2 = 0 ,
b 2 = 1 2 S 3 ,
a 4 = 1 4 S 1 ,
b 4 = 1 4 S 2 ,
S 0 = 2 π θ 0 θ 0 + 2 π I D ( θ ) ( 1 2 cos 4 θ ) d θ ,
S 1 = 4 π θ 0 θ 0 + 2 π I D ( θ ) cos 4 θ d θ ,
S 2 = 4 π θ 0 θ 0 + 2 π I D ( θ ) sin 4 θ d θ ,
S 3 = 2 π θ 0 θ 0 + 2 π I D ( θ ) sin 2 θ d θ .
S 0 = 4 N [ i = 0 N I i ( 1 2 cos 4 θ i ) 1 2 ( I 0 + I N ) ( 1 2 cos 4 θ 0 ) ] ,
S 1 = 8 N [ i = 0 N I i cos 4 θ i 1 2 ( I 0 + I N ) cos 4 θ 0 ] ,
S 2 = 8 N [ i = 0 N I i sin 4 θ i 1 2 ( I 0 + I N ) sin 4 θ 0 ] ,
S 3 = 4 N [ i = 0 N I i sin 2 θ i 1 2 ( I 0 + I N ) sin 2 θ 0 ] .
P = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 S 0 1 ,
χ 2 = i = 0 N 1 σ i 2 [ I i I D ( θ i , S ) ] 2 ,
χ 2 S k = 0 , k = 0 , 3 .
i = 0 N w i R i = 0 ,
i = 0 N w i R i cos 2 2 θ i = 0 ,
i = 0 N w i R i sin 2 θ i cos 2 θ i = 0 ,
i = 0 N w i R i sin 2 θ i = 0 .
S = S 0 ( 1 P cos 2 χ cos 2 ψ P cos 2 χ sin 2 ψ P sin 2 χ ) ,
P = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 S 0 ,
ψ = { 1 2 arctan ( S 2 / S 1 ) for   S 1 > 0   and   S 2 0 1 2 arctan ( S 2 / S 1 ) + π for   S 1 > 0   and   S 2 < 0 1 2 arctan ( S 2 / S 1 ) + π 2 for   S 1 < 0 π 2 for   S 1 = 0   and   S 2 0
χ = 1 2   arcsin [ S 3 ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 ] .
C = α 1 ,
α k , l = 1 2 ( χ 2 S k S l ) S .
α = 1 4 [ i = 0 N w i i = 0 N w i cos 2 2 θ i i = 0 N w i sin 2 θ i cos 2 θ i i = 0 N w i sin 2 θ i … … i = 0 N w i cos 4 2 θ i i = 0 N w i sin 2 θ i cos 3 2 θ i i = 0 N w i sin 2 θ i cos 2 2 θ i … … … … i = 0 N w i sin 2 2 θ i cos 2 2 θ i i = 0 N w i sin 2 2 θ i cos 2 θ i … … … … … … i = 0 N w i sin 2 2 θ i ] .
C p = D · C · D T ,
D k , n = ( p n S k ) S ,
D = [ S 1 2 + S 2 2 + S 3 2 S 0 2 S 1 S 0 S 1 2 + S 2 2 + S 3 2 S 2 S 0 S 1 2 + S 2 2 + S 3 2 S 3 S 0 S 1 2 + S 2 2 + S 3 2 0 1 2 S 2 S 1 2 + S 2 2 1 2 S 1 S 1 2 + S 2 2 0 0 1 2 S 1 S 3 S 1 2 + S 2 2 1 2 S 2 S 3 S 1 2 + S 2 2 1 2 S 1 2 + S 2 2 S 1 2 + S 2 2 + S 3 2 ] .
S 0 = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 ,
I D ( θ , S k ) = 1 2 [ ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 + S 1 cos 2 2 θ + S 2 sin 2 θ cos 2 θ S 3 sin 2 θ ] , k = 1 , 3 .
i = 0 N w i R i [ cos 2 2 θ i + S 1 ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 ] = 0 ,
i = 0 N w i R i [ sin 2 θ i cos 2 θ i + S 2 ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 ] = 0 ,
i = 0 N w i R i [ sin 2 θ i + S 3 ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 ] = 0 .
α 11 = 1 2 i = 0 N w i [ 1 2 ( cos 2 2 θ i + S 1 S 1 2 + S 2 2 + S 3 2 ) 2 R i S 2 2 + S 3 2 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
α 22 = 1 2 i = 0 N w i [ 1 2 ( sin 2 θ i cos 2 θ i + S 2 S 1 2 + S 2 2 + S 3 2 ) 2 R i S 1 2 + S 3 2 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
α 33 = 1 2 i = 0 N w i [ 1 2 ( sin 2 θ i + S 3 S 1 2 + S 2 2 + S 3 2 ) 2 R i S 1 2 + S 2 2 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
α 12 = α 21 = 1 2 i = 0 N w i [ 1 2 ( cos 2 2 θ i + S 1 S 1 2 + S 2 2 + S 3 2 ) × ( sin 2 θ i cos 2 θ i + S 2 S 1 2 + S 2 2 + S 3 2 ) + R i S 1 S 2 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
α 23 = α 32 = 1 2 i = 0 N w i [ 1 2 ( sin 2 θ i cos 2 θ i + S 2 S 1 2 + S 2 2 + S 3 2 ) × ( sin 2 θ i + S 3 S 1 2 + S 2 2 + S 3 2 ) + R i S 2 S 3 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
α 13 = α 31 = 1 2 i = 0 N w i [ 1 2 ( cos 2 2 θ i + S 1 S 1 2 + S 2 2 + S 3 2 ) × ( sin 2 θ i + S 3 S 1 2 + S 2 2 + S 3 2 ) + R i S 1 S 3 ( S 1 2 + S 2 2 + S 3 2 ) 3 / 2 ] ,
D = [ 1 2 S 2 S 1 2 + S 2 2 1 2 S 1 S 1 2 + S 2 2 0 1 2 S 1 S 3 S 1 2 + S 2 2 1 2 S 2 S 3 S 1 2 + S 2 2 1 2 S 1 2 + S 2 2 S 1 2 + S 2 2 + S 3 2 ] .
I D ( θ ) = 1 2 S 0 + 1 4 S 1 S 3 sin [ 2 ( θ + δ ) ] + 1 4 S 1 cos [ 4 ( θ + δ ) ] + 1 4 S 2 sin [ 4 ( θ + δ ) ] .
I D ( θ ) = 1 2 S 0 + 1 4 S 1 1 2 S 3 sin 2 δ cos 2 θ 1 2 S 3 cos 2 δ sin 2 θ + 1 4 ( S 1 cos 4 δ + S 2 sin 4 δ ) cos 4 θ + 1 4 ( S 2 cos 4 δ S 1 sin 4 δ ) sin 4 θ .
a 0 = S 0 + 1 2 S 1 ,
a 2 = 1 2 S 3 sin 2 δ ,
b 2 = 1 2 S 3 cos 2 δ ,
a 4 = 1 4 ( S 1 cos 4 δ + S 2 sin 4 δ ) ,
b 4 = 1 4 ( S 2 cos 4 δ + S 1 sin 4 δ ) .
δ = 1 2 arctan ( a 2 b 2 ) .
M = [ A B 0 0 B A 0 0 0 0 C D 0 0 D C ] .
A 2 B 2 + C 2 + D 2 .
S i n = ( 1 0 1 0 ) .
S o u t = M S i n = ( A B C D ) .
S i n 1 = ( 1 1 0 0 ) , S i n 2 = ( 1 1 0 0 ) , S i n 3 = ( 1 0 1 0 ) , S i n 4 = ( 1 0 0 1 ) .
M = [ 0.946 0.019 0.048 0.016 0.024 0.848 0.322 0.314 0.003 0.261 0.087 0.885 0.037 0.293 0.981 0.071 ] .

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