Abstract

Mueller matrix imaging polarimetry of liquid-crystal-on-silicon (LCoS) panels provides detailed information useful for the diagnosis of LCoS problems and to understand the interaction of LCoS panels with other projector components. Data reduction methods are presented for the analysis of LCoS Mueller matrix images yielding contrast ratio, efficiency, spatial uniformity, and the calculation of optimum trim retarders. The effects of nonideal retardance, retardance orientation, and depolarization on LCoS system performance are described. The white-state and dark-state Mueller matrix images of an example LCoS panel are analyzed in terms of LCoS performance metrics typical for red-green-blue wavelengths of 470, 550, and 640  nm. Variations of retardance, retardance orientation, and depolarization are shown to have different effects on contrast ratio, efficiency, and brightness. Thus Mueller matrix images can diagnose LCoS problems in a way different from radiometric testing. The calculation of optimum trim retarders in the presence of spatial variations is discussed. The relationship of the LCoS retardance in single-pass (from front to back) to the double-pass retardance (from entrance to exit) is established and used to clarify coordinate system issues related to Mueller matrices for reflection devices.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
    [CrossRef]
  15. J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.
  16. P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
    [CrossRef]
  17. H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
    [CrossRef]
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    [CrossRef]
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  23. D. Goldstein, Polarized Light (Marcel Dekker, 2004).
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  26. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).
  27. J. A. Davis, I. Moreno, and P. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998).

2003 (3)

J. Wolfe and R. A. Chipman, "High-speed imaging polarimeter," in Polarization Science and Remote Sensing, J. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 24-32 (2003).
[CrossRef]

R. A. Chipman and J. Wolfe, "Characterization of polarization aberrations in liquid crystal devices," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 1-5 (2003).
[CrossRef]

J. X. Guo and X. W. Sun, "Retardation-film-compensated reflective bistable twisted nematic liquid-crystal displays," Appl. Opt. 42, 3853-3863 (2003).

2002 (3)

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998).

1996 (1)

1995 (3)

1994 (1)

1993 (2)

1992 (1)

1990 (1)

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

1978 (1)

Albu, R.

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

Armitage, D.

Azzam, R. M. A.

Bodammer, G.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Brennesholtz, M. S.

E. H. Stupp and M. S. Brennesholtz, Projection Displays (Wiley, 1999), p. 14.

Calster, A. V.

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

Calton, D.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Chipman, R.

J. Wolfe and R. Chipman, "Method of time resolved liquid crystal angle of incidence and depolarization characterization," in Proceedings of the 23rd International Display Research Conference (Society for Information Display, 2003), paper 5.5.

Chipman, R. A.

J. Wolfe and R. A. Chipman, "High-speed imaging polarimeter," in Polarization Science and Remote Sensing, J. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 24-32 (2003).
[CrossRef]

R. A. Chipman and J. Wolfe, "Characterization of polarization aberrations in liquid crystal devices," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 1-5 (2003).
[CrossRef]

S. Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

J. L. Pezzaniti and R. A. Chipman, "Phase-only modulation of a twisted nematic liquid-crystal TV by the use of eigen polarizations," Opt. Lett. 18, 1567-1569 (1993).

J. L. Pezzaniti, S. C. McClain, R. A. Chipman, and S. Y. Lu, "Depolarization in liquid crystal televisions," Opt. Lett. 18, 2071-2073 (1993).

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

R. A. Chipman, "Polarimetry," in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.

Choi, S. C.

Collings, N.

Cuypers, D.

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

Davis, J.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

Davis, J. A.

De Vleeschouwer, H.

Dean, J.

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

DeMeyere, A.

Doorselaer, G. V.

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

Fornier, J.

Friends, M.

Gandhi, J.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, 2004).

Goldstein, D. H.

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

Gourlay, J.

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).

Guo, J. X.

Hareb, S.

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

Janssen, P.

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

Ji, Y.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

Johnson, K.

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

Johnson, K. M.

Jones, M. W.

Kazlas, P.

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

Kim, T. H.

Kinell, D. K.

Klouda, R.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

Kowel, S. T.

Kulick, J. H.

Lee, I. W.

Lee, Y.

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

Lee, Y. W.

Lindquist, R. G.

Lu, S. Y.

McClain, S. C.

McKnight, D. J.

Miremont, C.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Moreno, I.

Nasiatka, P. J.

Nordin, G. P.

Parkes, W.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Pezzaniti, J. L.

Proudley, G. M.

Serati, R. A.

Shimizu, J. A.

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

Smet, H. D.

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

Stace, C.

Steen, J. V.

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

Stefanov, M.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

Stupp, E. H.

E. H. Stupp and M. S. Brennesholtz, Projection Displays (Wiley, 1999), p. 14.

Sun, X. W.

Tsai, P.

Underwood, I.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Vass, D. G.

Vermeirsch, K.

White, H. J.

Wolfe, J.

R. A. Chipman and J. Wolfe, "Characterization of polarization aberrations in liquid crystal devices," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 1-5 (2003).
[CrossRef]

J. Wolfe and R. A. Chipman, "High-speed imaging polarimeter," in Polarization Science and Remote Sensing, J. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 24-32 (2003).
[CrossRef]

J. Wolfe and R. Chipman, "Method of time resolved liquid crystal angle of incidence and depolarization characterization," in Proceedings of the 23rd International Display Research Conference (Society for Information Display, 2003), paper 5.5.

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).

Zheng, W.

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

Appl. Opt. (8)

Displays (3)

C. Miremont, G. Bodammer, D. Calton, W. Parkes, W. Zheng, and I. Underwood, "Improving the flatness of silicon backplanes for high quality FLCoS microdisplays," Displays 23, 115-119 (2002).
[CrossRef]

P. Janssen, J. A. Shimizu, J. Dean, and R. Albu, "Design aspects of a scrolling color LCoS display," Displays 23, 99-108 (2002).
[CrossRef]

H. D. Smet, D. Cuypers, A. V. Calster, J. V. Steen, and G. V. Doorselaer, "Design fabrication and evaluation of a high performance XGA VAN-LCOS microdisplay," Displays 23, 89-98 (2002).
[CrossRef]

J. Opt. Soc. Am A (1)

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

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. L. Pezzaniti and R. A. Chipman, "Mueller matrix imaging polarimetry," Opt. Eng. 34, 1558-1568 (1995).
[CrossRef]

Opt. Lett. (3)

Proc. SPIE (3)

P. Kazlas, K. Johnson, Y. Lee, and S. Hareb, "Assembly and packaging of liquid-crystal-on-silicon displays," in Micro-Optics Integration and Assemblies, M. R. Feldman and Y.-C. Lee, eds., Proc. SPIE 3289, 52-59 (1998).
[CrossRef]

J. Wolfe and R. A. Chipman, "High-speed imaging polarimeter," in Polarization Science and Remote Sensing, J. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 24-32 (2003).
[CrossRef]

R. A. Chipman and J. Wolfe, "Characterization of polarization aberrations in liquid crystal devices," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 1-5 (2003).
[CrossRef]

Other (7)

In addition to the references above, Displays, Vol. 23, has a section devoted entirely to LCoS displays.

J. Gandhi, Y. Ji, R. Klouda, J. Davis, and M. Stefanov, "Comparison of process tolerance of various LCoS modes," in Conference Record of the 20th International Display Research Conference(Society for Information Display, 2000), pp. 233-236.

J. Wolfe and R. Chipman, "Method of time resolved liquid crystal angle of incidence and depolarization characterization," in Proceedings of the 23rd International Display Research Conference (Society for Information Display, 2003), paper 5.5.

E. H. Stupp and M. S. Brennesholtz, Projection Displays (Wiley, 1999), p. 14.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).

R. A. Chipman, "Polarimetry," in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.

D. Goldstein, Polarized Light (Marcel Dekker, 2004).

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Figures (25)

Fig. 1
Fig. 1

Simplified side cutaway of a LCoS panel showing a cover glass with an ITO layer, LC, and a mirror over a silicon integrated circuit.

Fig. 2
Fig. 2

Layout of a LCoS projector.

Fig. 3
Fig. 3

Configuration for normal-incidence retroreflection polarization measurements.

Fig. 4
Fig. 4

Measured black-state Mueller matrix image of a LCoS panel illuminated at normal incidence at 550 nm showing spatial variations of retardance, depolarization, and diattenuation. In these normalized Mueller matrices, 1 is white; –1 is black, and 0 is gray. The gray scale is nonlinear to emphasize small variations near zero.

Fig. 5
Fig. 5

Ideal black-state Mueller matrix image. Such a Mueller matrix has no retardance, diattenuation, or depolarization and is uniform.

Fig. 6
Fig. 6

Spatial variation of retardance (Ret) (in degrees) for (a) 470, (b) 550, and (c) 640 nm. The vertical axis is retardance and the horizontal axis is spatial position. The spatial variation of the dark-state single-pass retardance is well correlated between wavelengths as is expected for LC thickness variations.

Fig. 7
Fig. 7

Retardance variation (in degrees) along the central row of pixels for (a) 470, (b) 550, and (c) 640 nm. The vertical axis is retardance and the horizontal axis is spatial position.

Fig. 8
Fig. 8

Spatial variation of the dark-state retardance (Ret) orientation (in degrees) for (a) 470, (b) 550, and (c) 640 nm is more than 10°.

Fig. 9
Fig. 9

Retardance orientation variation (in degrees) along the central row of pixels for (a) 470, (b) 550, and (c) 640 nm. The orientation varies by 8° at 470 nm, 12° at 550 nm, and 15° at 640 nm. These significant orientation variations are associated with very low retardances of less than 1∕20 of a wave. The vertical axis is retardance and the horizontal axis is spatial position.

Fig. 10
Fig. 10

Ideal white-state Mueller matrix image rotates the plane of horizontal or vertical light by 90°.

Fig. 11
Fig. 11

Measured white-state Mueller matrix image at 550 nm. The gray scale is nonlinear to emphasize small variations near zero.

Fig. 12
Fig. 12

Spatial variation of the white-state single-pass retardance (Ret) (in degrees) for (a) 470, (b) 550, and (c) 640 nm.

Fig. 13
Fig. 13

White-state single-pass retardance variations (in degrees) along the central row of pixels for (a), 470, (b) 550, and (c) 640 nm.

Fig. 14
Fig. 14

Spatial variation of the white-state retardance (Ret) orientation (in degrees) for (a) 470, (b) 550, and (c) 640 nm increases with shorter wavelengths.

Fig. 15
Fig. 15

Variation of the dark-state retardance orientation (in degrees) along a central row of pixels for (a) 470, (b) 550, and (c) 640 nm shows a well-correlated variation. The vertical axis is retardance and the horizontal axis is spatial position.

Fig. 16
Fig. 16

Spatial variation of dark-state depolarization for (a) 470, (b) 550, and (c) 640 nm. The depolarization is represented as the output DoP averaged over the Poincaré sphere The depolarization appears uncorrelated between wavelengths. Dark-state depolarization is a critical parameter.

Fig. 17
Fig. 17

Spatial variation of white-state depolarization for (a) 470, (b) 550, and (c) 640 nm. White-state depolarization is much greater than dark-state depolarization and is less critical.

Fig. 18
Fig. 18

Dark-state retardance variation in degrees across a row of pixels where δ0 = 11.8°.

Fig. 19
Fig. 19

Dark-state retardance orientation variation in degrees across a row of pixels where θ0 = 30.9°.

Fig. 20
Fig. 20

Dark-state leakage of system due to second-order retardance variations calculated from a series expansion of approximation (5) for the data of Figs. 18 and 19.

Fig. 21
Fig. 21

Dark-state leakage of system due to third and fourth-order terms in the series expansion of approximation (5) due to retardance variations.

Fig. 22
Fig. 22

Comparison of exact leakage and the leakage from the fourth-order series expansion.

Fig. 23
Fig. 23

Leakage across one row of the LCoS panel due to depolarization.

Fig. 24
Fig. 24

Leakage at 640 nm with optimum trim retarder applied for (a) the entire area and (b) a central row of pixels.

Fig. 25
Fig. 25

Coordinate system for the Mueller matrix polarimeter operating in retroreflection shows a change of coordinate upon reflection. The incident Stokes vector is described in (x′, y′, z′) coordinates and the reflected Stokes vector with (x, y, z) coordinates. The orientation of the linear retardance θ is specified relative to the output coordinates.

Tables (6)

Tables Icon

Table 1 Average Degree of Polarization for the Panel and a Mirror

Tables Icon

Table 2 Optimum Trim Retarder Values

Tables Icon

Table 3 Efficiency, Contrast, and Leakage for Trim Retarder Optimized for Single Wavelengths

Tables Icon

Table 4 Efficiency, Contrast, and Leakage for a Trim Retarder Optimized Only for 550 nm

Tables Icon

Table 5 Performance Specification Issues Related to Polarization Causes and Liquid-Crystal-on-Silicon Defects

Tables Icon

Table 6 Polarization Defects and Resulting On-Screen Effect

Equations (20)

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

ideal   MM = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] .
ideal   MM = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
L V = L P ( 90 ° ) T R ( δ 0 , θ 0 90 ° ) LCoS ( δ , θ ) T R [ δ 0 , ( θ 0 90 ° ) ] L P ( 0 ° ) .
S 1 = ( L V S ) 1 = { cos ( Δ δ + δ 0 ) sin ( δ 0 ) sin ( 2 θ 0 ) sin ( Δ δ + δ 0 ) [ cos ( 2 θ 0 ) sin ( 2 Δθ ) + cos ( 2 Δθ ) cos ( δ 0 ) sin ( 2 θ 0 ) ] } ,
leakage ( δ 0 , Δ δ , θ 0 , Δ θ ) Δ δ 2 [ 1 cos ( 2 θ 0 ) 2 ]
+ Δ θ 2 { 2 cos ( 2 δ 0 ) 2 [ 1 cos ( 4 θ 0 ) ] + 4 cos ( 2 θ 0 ) 2 } + 2 Δ θ Δ δ sin ( δ 0 ) sin ( 4 θ 0 ) + Δ θ 2 Δ δ [ sin ( 2 δ 0 ) + 3 cos ( 4 θ 0 ) sin ( 2 δ 0 ) ] + 2 Δ δ 2 Δ θ cos ( δ 0 ) sin ( 4 θ 0 ) + Δ θ 2 Δ δ 2 [ 2 cos ( δ 0 ) 2 4 cos ( 2 θ 0 ) 2 + 6 cos ( δ 0 ) 2 cos ( 4 θ 0 ) ] .
leakage = 1 DoP 2 .
S exiting = M S incident = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ] [ S 0 S 1 S 2 S 3 ] = [ S 0 S 1 S 2 S 3 ] .
M = M Δ M R M D ,
DoP ( S ) = S 1    2 + S 2    2 + S 3    2 S 0 .
average   DoP ( M )
= 0 π π / 2 π / 2 DoP [ M S ( θ , ϕ ) ] cos ( ϕ ) d ϕ d θ 4 π .
S ( θ , ϕ ) = [ 1 cos ( 2 θ ) cos ( ϕ ) sin ( 2 θ ) cos ( ϕ ) sin ( ϕ ) ] .
M M measured = M M T M M LCoS M M R ,
M M LCoS = ( M M T ) 1 M M measured ( M M R ) 1 .
LCoS ( δ , θ ) = L R ( δ , θ ) M i r LR ( δ , θ ) ,
M i r = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
L R ( δ , θ ) = [ 1 0 0 0 0 cos 2 ( 2 θ ) + cos ( δ ) [ 1 cos 2 ( 2 θ ) ] [ 1 cos ( δ ) ] cos ( 2 θ ) sin ( 2 θ ) sin ( δ ) sin ( 2 θ ) 0 [ 1 cos ( δ ) ] cos ( 2 θ ) sin ( 2 θ ) 1 cos 2 ( 2 θ ) + cos ( δ ) cos 2 ( 2 θ ) cos ( 2 θ ) sin ( δ ) 0 sin ( δ ) sin ( 2 θ ) cos ( 2 θ ) sin ( δ ) cos ( δ ) ] .
LCoS ( δ , θ ) = [ 1 0 0 0 0 cos 2 ( δ ) [ cos ( 4 θ ) 1 ] + cos ( 4 θ ) [ cos 2 ( δ ) 1 ] sin ( 4 θ ) sin ( 2 δ ) sin ( 2 θ ) 0 [ 1 cos 2 ( δ ) ] sin ( 4 θ ) 2 cos 2 ( δ ) cos 2 ( 2 θ ) + cos ( 4 θ ) cos ( 2 θ ) sin ( 2 δ ) 0 sin ( 2 δ ) sin ( 2 θ ) cos ( 2 θ ) sin ( 2 δ ) cos ( 2 δ ) ] .
LCoS ( δ , θ ) M i r = L R ( 2 δ , θ ) = [ 1 0 0 0 0 cos 2 ( δ ) [ 1 + cos ( 4 θ ) ] + cos ( 4 θ ) [ 1 cos 2 ( δ ) ] sin ( 4 θ ) 4 cos ( δ ) cos ( θ ) sin ( δ ) sin ( θ ) 0 [ 1 cos 2 ( δ ) ] sin ( 4 θ ) cos 2 ( δ ) + [ 1 + cos 2 ( δ ) ] cos ( 4 θ ) cos ( 2 θ ) sin ( 2 δ ) 0 sin ( 2 δ ) sin ( 2 θ ) cos ( 2 θ ) sin ( 2 δ ) cos ( 2 δ ) ] .

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