Abstract

A polarimetry based method able to characterize optical properties of linear Liquid Crystal Displays (LCDs), even in presence of time-fluctuations of the phase, is proposed in this work. In particular, mean linear retardance, Liquid Crystal (LC) fast axis orientation and phase fluctuation amplitude of LCDs can be obtained with the proposed alternative technique. This technique enables to achieve these important features of LCDs with a set-up significantly less complicated to build up and with faster measurements than previously proposed techniques, which are based on diffraction or interferometry experiments. The validity of the technique is tested by measuring two different LCDs: one monopixel PA-LC panel working in transmission and a reflective PA-LCoS display. The technique provides similar results than those obtained by using previously proposed methods, confirming the validity of our alternative technique.

© 2013 OSA

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References

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  1. P. Ambs and L. Bigué, “Characterization of an analog ferroelectric spatial light modulator – Application to dynamic diffractive Optical elements and Optical information processing,” in Opt. Inf. Proc.: Opt. Inf. Syst. CR81, 365–393 (2001).
  2. A. Márquez, C. Iemmi, J. Campos, J. C. Escalera, and M. J. Yzuel, “Programmable apodizer to compensate chromatic aberration effects using a liquid Crystal spatial light modulator,” Opt. Express13(3), 716–730 (2005).
    [CrossRef] [PubMed]
  3. C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
    [CrossRef]
  4. Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
    [CrossRef]
  5. A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
    [CrossRef]
  6. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley & Sons Inc., Chichester, 2005).
  7. N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
    [CrossRef]
  8. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt.43(34), 6278–6284 (2004).
    [CrossRef] [PubMed]
  9. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt.45(8), 1688–1703 (2006).
    [CrossRef] [PubMed]
  10. A. Lizana, I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Time-resolved Mueller matrix analysis of a liquid crystal on silicon display,” Appl. Opt.47(23), 4267–4274 (2008).
    [CrossRef] [PubMed]
  11. A. Lizana, I. Moreno, C. Iemmi, A. Márquez, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
    [CrossRef]
  12. A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express18(10), 10581–10593 (2010).
    [CrossRef] [PubMed]
  13. J. García-Márquez, V. López, A. González-Vega, and E. Noé, “Flicker minimization in an LCoS spatial light modulator,” Opt. Express20(8), 8431–8441 (2012).
    [CrossRef] [PubMed]
  14. F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).
  15. M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot Mueller matrix polarimeter by wavelength polarization coding,” Opt. Express15(21), 13660–13668 (2007).
    [CrossRef] [PubMed]
  16. M. Dubreuil, S. Rivet, B. Le Jeune, and L. Dupont, “Time-resolved switching analysis of a ferroelectric liquid crystal by snapshot Mueller matrix polarimetry,” Opt. Lett.35(7), 1019–1021 (2010).
    [CrossRef] [PubMed]
  17. D. Goldstein, Polarized Light (Mercel Dekker, New York, 2003).
  18. A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
    [CrossRef]
  19. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express17(10), 8491–8505 (2009).
    [CrossRef] [PubMed]

2012

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

J. García-Márquez, V. López, A. González-Vega, and E. Noé, “Flicker minimization in an LCoS spatial light modulator,” Opt. Express20(8), 8431–8441 (2012).
[CrossRef] [PubMed]

2011

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

2010

2009

2008

2007

M. Dubreuil, S. Rivet, B. Le Jeune, and J. Cariou, “Snapshot Mueller matrix polarimeter by wavelength polarization coding,” Opt. Express15(21), 13660–13668 (2007).
[CrossRef] [PubMed]

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

2006

J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt.45(8), 1688–1703 (2006).
[CrossRef] [PubMed]

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

2005

2004

1994

C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
[CrossRef]

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
[CrossRef]

Álvarez, M. L.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

Beaudoin, N.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

Beléndez, A.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

Blümel, T.

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

Campos, J.

Campos, J.

Cariou, J.

Chipman, R. A.

Christmas, J.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

Chu, D.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

Collings, N.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

Crossland, B.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

Davey, T.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

Dubreuil, M.

Dupont, L.

Escalera, J. C.

Estapé, M.

Fernández, E.

Gallego, S.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

García-Márquez, J.

González-Vega, A.

Hermerschmidt, A.

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

Iemmi, C.

A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express18(10), 10581–10593 (2010).
[CrossRef] [PubMed]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express17(10), 8491–8505 (2009).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Time-resolved Mueller matrix analysis of a liquid crystal on silicon display,” Appl. Opt.47(23), 4267–4274 (2008).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef]

A. Márquez, C. Iemmi, J. Campos, J. C. Escalera, and M. J. Yzuel, “Programmable apodizer to compensate chromatic aberration effects using a liquid Crystal spatial light modulator,” Opt. Express13(3), 716–730 (2005).
[CrossRef] [PubMed]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt.43(34), 6278–6284 (2004).
[CrossRef] [PubMed]

Knopp, J.

C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
[CrossRef]

Krüger, S.

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

Le Jeune, B.

Lizana, A.

Lobato, L.

López, V.

Lu, G.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
[CrossRef]

Márquez, A.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express18(10), 10581–10593 (2010).
[CrossRef] [PubMed]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express17(10), 8491–8505 (2009).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Time-resolved Mueller matrix analysis of a liquid crystal on silicon display,” Appl. Opt.47(23), 4267–4274 (2008).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef]

A. Márquez, C. Iemmi, J. Campos, J. C. Escalera, and M. J. Yzuel, “Programmable apodizer to compensate chromatic aberration effects using a liquid Crystal spatial light modulator,” Opt. Express13(3), 716–730 (2005).
[CrossRef] [PubMed]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt.43(34), 6278–6284 (2004).
[CrossRef] [PubMed]

Martín, N.

Martínez, A.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

Martínez, F. J.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

Monroe, S. E.

C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
[CrossRef]

Moreno, I.

A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express18(10), 10581–10593 (2010).
[CrossRef] [PubMed]

A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express17(10), 8491–8505 (2009).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Time-resolved Mueller matrix analysis of a liquid crystal on silicon display,” Appl. Opt.47(23), 4267–4274 (2008).
[CrossRef] [PubMed]

A. Lizana, I. Moreno, C. Iemmi, A. Márquez, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008).
[CrossRef]

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt.43(34), 6278–6284 (2004).
[CrossRef] [PubMed]

Noé, E.

Ortuño, M.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

Osten, S.

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

Rivet, S.

Rodange, Y.

Sánchez-López, M. M.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

Soutar, C.

C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
[CrossRef]

Velásquez, P.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

Wolfe, J. E.

Yu, F. T. S.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
[CrossRef]

Yzuel, M. J.

Zhang, Z.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
[CrossRef]

Appl. Opt.

J. Disp. Tech.

N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of Phase-Only Liquid Crystal on Silicon Devices,” J. Disp. Tech.7(3), 112–119 (2011).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

A. Martínez, N. Beaudoin, I. Moreno, M. M. Sánchez-López, and P. Velásquez, “Optimization of the contrast ratio of a ferroelectric liquid crystal optical modulator,” J. Opt. A, Pure Appl. Opt.8(11), 1013–1018 (2006).
[CrossRef]

Opt. Eng.

C. Soutar, S. E. Monroe, and J. Knopp, “Measurement of the complex transmittance of the Epson liquid crystal television,” Opt. Eng.33(4), 1061–1069 (1994).
[CrossRef]

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal television,” Opt. Eng.33(9), 3018–3022 (1994).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

F. J. Martínez, S. Gallego, A. Márquez, M. Ortuño, M. L. Álvarez, and A. Beléndez, “Analysis of the linear polarimeter to measure the retardance in the presence of phase flicker: application to LCoS devices,” Opt. Lasers Eng. (2012).

Opt. Lett.

Proc. SPIE

A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating liquid-crystal based micro-display with HDTV resolution,” Proc. SPIE6584, 65840E, 65840E-10 (2007).
[CrossRef]

Other

D. Goldstein, Polarized Light (Mercel Dekker, New York, 2003).

P. Ambs and L. Bigué, “Characterization of an analog ferroelectric spatial light modulator – Application to dynamic diffractive Optical elements and Optical information processing,” in Opt. Inf. Proc.: Opt. Inf. Syst. CR81, 365–393 (2001).

S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley & Sons Inc., Chichester, 2005).

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Figures (7)

Fig. 1
Fig. 1

Triangle function based model for the phase variation in a period of time.

Fig. 2
Fig. 2

Set-up used for the monopixel parallel aligned liquid crystal panel measurement: (a) parallel and crossed polarizers case; (b) right-handed and left-handed case.

Fig. 3
Fig. 3

Phase modulation as a function of the voltage addressed to the liquid crystal.

Fig. 4
Fig. 4

Contrast ratio β as a function of the voltage.

Fig. 5
Fig. 5

Set-up used for the parallel aligned LCoS display measurements.

Fig. 6
Fig. 6

Phase modulation as a function of the gray level for the sequence “5-5 2π linear 633 nm.”

Fig. 7
Fig. 7

Contrast ratio β as a function of the GLs for the sequence “5-5 2π linear 633 nm”.

Equations (25)

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

S exit = M LP2 ( 2θ ) M WP ( Δ,α )( 1 cos( 2θ ) sin( 2θ ) 0 ),
I || =A{ 1+ cos 2 [ 2( θα ) ]+cosΔ sin 2 [ 2( θα ) ] }.
I =A{ 1 cos 2 [ 2( θα ) ]cosΔ sin 2 [ 2( θα ) ] }.
S exit = M LP2 ( 2θ ) M WP ( Δ,α )( 1 0 0 S 3 ).
I R =B{ 1+sinΔsin[ 2( θα ) ] },
I L =B{ 1sinΔsin[ 2( θα ) ] }.
I || =A{ 1+ cos 2 [ 2( θα ) ]+ cosδ sin 2 [ 2( θα ) ] },
I =A{ 1 cos 2 [ 2( θα ) ] cosδ sin 2 [ 2( θα ) ] },
I R =B{ 1+ sinδ sin[ 2( θα ) ] },
I L =B{ 1 sinδ sin[ 2( θα ) ] },
cosδ = 1 T 0 T 2 cos ( δ 1 ( t ) )dt+ 1 T T 2 T cos ( δ 2 ( t ) )dt= sin( a ) a cosΔ=βcosΔ,
{ δ 1 = 4a T t+Δa δ 2 = 4a T t+Δ+3a t[ 0, T 2 ] t[ T 2 ,T ] ,
sinδ =βsinΔ.
I || =A{ 1+ cos 2 [ 2( θα ) ]+βcosΔ sin 2 [ 2( θα ) ] },
I =A{ 1 cos 2 [ 2( θα ) ]βcosΔ sin 2 [ 2( θα ) ] },
I R =B{ 1+βsinΔsin[ 2( θα ) ] },
I L =B{ 1βsinΔsin[ 2( θα ) ] }.
I || Norm = I || I || + I = 1+ cos 2 [ 2( θα ) ]+βcosΔ sin 2 [ 2( θα ) ] 2 ,
I R Norm = I R I R + I L = 1+βsinΔsin[ 2( θα ) ] 2 .
erro r L = n=0 m [ 1+ cos 2 [ 2( θ n α ) ]+βcosΔ sin 2 [ 2( θ n α ) ] 2 I n, || Norm / ] 2 ,
erro r C = n=0 m [ 1+βsinΔsin[ 2( θ n α ) ] 2 I n, R Norm / ] 2 ,
I n, || Norm / = 1+βcosΔ 2 ,
I n, R Norm / = 1+βsinΔ 2 .
Δ=arctg( 2 I R Norm / 1 2 I || Norm / 1 ),
β=2 I || Norm / 2 + I R Norm / 2 I || Norm / I R Norm / + 1 2 .

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