Abstract

The linear errors of Mueller matrix measurements, using a partially polarized light source, have been formulated for imperfections of misalignment, depolarization, and nonideal ellipsometric parameters of the polarimetric components. The error matrices for a source-polarizer system and a source–polarizer–compensator system are derived. A polarized light source, when used with an imperfect polarizer, generates extra errors in addition to those for an unpolarized source. The compensator redistributes these errors to different elements of the error matrix. The errors of the Mueller matrices for the polarizer–sample–analyzer and the polarizer–compensator–sample–analyzer systems are evaluated for a straight through case. This error analysis is applied to a Stokes method and an experiment was performed to show the errors by a polarized light source. This general analysis can be used to evaluate errors for ellipsometry and polarimetry.

© 2006 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, "Ellipsometry with imperfect components including incoherent effects," J. Opt. Soc. Am. 61, 1380-1391 (1971).
    [CrossRef]
  2. R. M. A. Azzam, "Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148-150 (1978).
    [CrossRef] [PubMed]
  3. D. H. Goldstein and R. A. Chipman, "Error analysis of a Mueller matrix polarimeter," J. Opt. Soc. Am. A 7, 693-700 (1990).
    [CrossRef]
  4. G. E. Jellison, Jr., and F. A. Modine, "Two modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
    [CrossRef]
  5. G. E. Jellison, Jr., and F. A. Modine, "Two modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
    [CrossRef]
  6. B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
    [CrossRef]
  7. S.-M. F. Nee, "Error reduction for a serious compensator imperfection for null ellipsometry," J. Opt. Soc. Am. A 8, 314-321 (1991).
    [CrossRef]
  8. S.-M. F. Nee, "Error analysis of null ellipsometry with depolarization," Appl. Opt. 38, 5388-5398 (1999).
    [CrossRef]
  9. S.-M. F. Nee, "Depolarization and principal Mueller matrix measured by null ellipsometry," Appl. Opt. 40, 4933-4939 (2001).
    [CrossRef]
  10. S. F. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003).
    [CrossRef]
  11. S.-M. F. Nee, C. Yoo, T. Cole, and D. Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998).
    [CrossRef]
  12. S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
    [CrossRef]
  13. K. A. O'Donnell and E. R. Mendez, "Experimental study of scattering from characterized random surfaces," J. Opt. Soc. Am. A 4, 1194-1205 (1987).
    [CrossRef]
  14. V. J. Iafelice and W. S. Bickel, "Polarized light-scattering matrix elements for select perfect and perturbed optical surfaces," Appl. Opt. 26, 2410-2415 (1987).
    [CrossRef]
  15. M. E. Knotts and K. A. O'Donnell, "Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness," J. Opt. Soc. Am. A 11, 697-710 (1994).
    [CrossRef]
  16. E. R. Mendez, A. G. Navarrete, and R. E. Luna, "Statistics of the polarization properties of one-dimensional randomly rough surfaces," J. Opt. Soc. Am. A 12, 2507-2516 (1995).
    [CrossRef]
  17. G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555- 6566 (2002).
    [CrossRef] [PubMed]
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  19. S.-M. F. Nee, "Polarization measurement," in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC Press, 1999), Chap. 60.
  20. R. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 22.
  21. S.-M. F. Nee, "Polarization of specular reflection and near-specular scattering by a rough surface," Appl. Opt. 35, 3570-3582 (1996).
    [CrossRef]
  22. S.-M. F. Nee and T. Cole, "Effects of depolarization of optical components on null ellipsometry," Thin Solid Films 313-314, 90-96 (1998).
    [CrossRef]
  23. S.-M. F. Nee, "Depolarization and retardation of a birefringent slab," J. Opt. Soc. Am. A 17, 2067-2073 (2000).
    [CrossRef]

2003 (1)

2002 (2)

G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, "Transmission two-modulator generalized ellipsometry measurements," Appl. Opt. 41, 6555- 6566 (2002).
[CrossRef] [PubMed]

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (2)

S.-M. F. Nee, C. Yoo, T. Cole, and D. Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998).
[CrossRef]

S.-M. F. Nee and T. Cole, "Effects of depolarization of optical components on null ellipsometry," Thin Solid Films 313-314, 90-96 (1998).
[CrossRef]

1997 (3)

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
[CrossRef]

G. E. Jellison, Jr., and F. A. Modine, "Two modulator generalized ellipsometry: experiment and calibration," Appl. Opt. 36, 8184-8189 (1997).
[CrossRef]

G. E. Jellison, Jr., and F. A. Modine, "Two modulator generalized ellipsometry: theory," Appl. Opt. 36, 8190-8198 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1991 (1)

1990 (1)

1987 (2)

1978 (1)

1971 (1)

Azzam, R. M. A.

Bashara, N. M.

Bickel, W. S.

Boulbry, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Bousquet, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Burge, D.

S.-M. F. Nee, C. Yoo, T. Cole, and D. Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
[CrossRef]

Cariou, J.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Chipman, R. A.

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. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 22.

Cole, T.

S.-M. F. Nee and T. Cole, "Effects of depolarization of optical components on null ellipsometry," Thin Solid Films 313-314, 90-96 (1998).
[CrossRef]

S.-M. F. Nee, C. Yoo, T. Cole, and D. Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
[CrossRef]

Goldstein, D. H.

Griffiths, C. O.

Holcomb, D. E.

Iafelice, V. J.

Jellison, G. E.

Knotts, M. E.

Le Jeunne, B.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Lotrian, J.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Luna, R. E.

Mendez, E. R.

Modine, F. A.

Navarrete, A. G.

Nee, S. F.

Nee, S.-M. F.

O'Donnell, K. A.

Pellen, F.

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Rouleau, C. M.

Yoo, C.

S.-M. F. Nee, C. Yoo, T. Cole, and D. Burge, "Characterization for imperfect polarizers under imperfect conditions," Appl. Opt. 37, 54-64 (1998).
[CrossRef]

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Meas. Sci. Technol. (1)

B. Boulbry, B. Le Jeunne, B. Bousquet, F. Pellen, J. Cariou, and J. Lotrian, "Error analysis and calibration of a spectroscopic Mueller matrix polarimeter using a short-pulse laser source," Meas. Sci. Technol. 13, 1563-1573 (2002).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, "Characterization of infrared polarizers," in Polarization: Measurement, Analysis, and Remote Sensing, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 3121, 213-224 (1997).
[CrossRef]

Thin Solid Films (1)

S.-M. F. Nee and T. Cole, "Effects of depolarization of optical components on null ellipsometry," Thin Solid Films 313-314, 90-96 (1998).
[CrossRef]

Other (3)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

S.-M. F. Nee, "Polarization measurement," in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC Press, 1999), Chap. 60.

R. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds. (McGraw-Hill, 1995), Vol. II, Chap. 22.

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Figures (2)

Fig. 1
Fig. 1

(Color online) Difference of the measured polarizations for the straight through case by using a Glan polarizer and a wire-grid polarizer as the polarizer of the polarimeter.

Fig. 2
Fig. 2

(Color online) Measured P x * ( Grid ) - P x * ( Glan ) versus α P y of the grid and P z * ( Grid ) - P z * ( Glan ) versus α P z of the grid for the straight through case. The parentheses (Grid) or (Glan) represent that a wire-grid polarizer or a Glan polarizer is used as the polarizer of the polarimeter.

Tables (2)

Tables Icon

Table 1 Characteristics of the Light Source and the Wire-Grid Polarizer Used in the Polarimeter

Tables Icon

Table 2 Measured Polarizations for the Straight Through Case Using the Polarimeter with Its Polarizer Being a Glan–Thompson Polarizer or a Wire-Grid Polarizer

Equations (58)

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I = D · M · X = D o · M * · X o .
δ X = δ m x · X o , δ D = D o · δ m d .
δ M = M δ m x + δ m d M .
S = ( 1 , α , β , γ ) T .
M = T ( 1 P x 0 0 P x 1 2 D v 0 0 0 0 P y P z 0 0 P z P y ) ,
P x = P cos 2 ψ ,
P y = P sin 2 ψ cos Δ ,
P z = P sin 2 ψ sin Δ ,
D = 1 P = D u + D v .
P ( 0 ) = T p ( 1 + D p 1 ξ p     2 / 2 0 0 1 ξ p     2 / 2 1 + D p 2 D v p 0 0 0 0 ε p η p 0 0 η p ε p ) .
X o ( P ) = P o ( P ) · S o = ( 1 , cos 2 P , sin 2 P , 0 ) T .
X ( P ) = P ( P ) · S = ( 1 + E ) X o ( P ) + δ X ( P ) ,
E = α   cos   2 P + β   sin   2 P ,
δ x ( P ) = δ X ( P ) / ( 1 + E ) = δ m x · X o .
P ( P + δ P ) = R ( δ P ) P o ( P ) R ( δ P )
= P o ( P ) + δ P m ( P ) ,
δ P m ( P ) = δ R ( δ P ) P o ( P ) + P o ( P ) δ R ( δ P ) .
δ X m ( P ) = δ P m ( P ) · ( 1 , α , β , γ ) T = [ δ R ( δ P ) P o ( P ) + P o ( P ) δ R ( δ P ) ] · S .
δ m x m = 2 δ P ( 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ) + 2 B δ P 1 + E ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ) ,
B = β   cos   2 P α   sin   2 P .
δ X i ( P ) = R ( P ) δ P i ( 0 ) R ( P ) · ( 1 , α , β , γ ) T .
δ P i ( 0 ) = ( D p ξ p     2 / 2 0 0 ξ p     2 / 2 D p 2 D v p 0 0 0 0 ε p η p 0 0 η p ε p ) .
δ m x i = 1 1 + E ( D p E ξ p     2 / 2 0 0 0 0 ξ p     2 / 2 + ( D p 2 D v p ) E ε p B γ η p 0 0 ε p B + γ η p ξ p     2 / 2 + ( D p 2 D v p ) E 0 ε p γ β η p α η p 0 ) .
δ m x = ( D p 0 0 0 0 ξ p     2 / 2 2 δ P 0 0 2 δ P ξ p     2 / 2 0 ε p γ 0 0 0 ) + 1 1 + E ( 2 B δ P ( D p + ξ p     2 / 2 ) E 0 0 0 0 G ε p B γ η p 0 0 ε p B + γ η p G 0 ε p γ E β η p α η p 0 ) ,
G = 2 δ P B + E ( D p 2 D v p + ξ p     2 / 2 ) .
δ m x = ( D p 0 0 0 0 ξ p     2 / 2 2 δ P 0 0 2 δ P - ξ p     2 / 2 0 0 0 0 0 ) + 1 1 E 2 × ( ( D p + ξ p     2 / 2 ) E 2 2 B E δ P α ( D p + ξ p     2 / 2 ) + 2 β δ P β ( D p + ξ p     2 / 2 ) 2 α δ P 0 G   cos   2 P + ( γ η p E ε p B ) sin   2 P G E ε p B E γ η p 0 G   sin   2 P + ( ε p B γ η p E ) cos   2 P ε p B E + γ η p G E 0 ε p γ η p B E ε p α γ β η p α η p ε p β γ 0 ) .
X = C ( C , τ ) P ( P ) · S = ( 1 + E ) ( 1 + δ m x ) · X o c ,
X o c = C o ( C ) P o ( P ) · S o ,
C o ( C ) P o ( P ) · S = ( 1 + E ) X o c .
C o ( C , τ ) = ( 1 0 0 0 0 cos 2 2 C + sin 2 2 C   cos   τ sin   2 C   cos   2 C ( 1 cos   τ ) sin   2 C   sin   τ 0 sin   2 C   cos   2 C ( 1 cos   τ ) sin 2 2 C + cos 2 2 C   cos   τ cos   2 C   sin   τ 0 sin   2 C   sin   τ cos   2 C   sin   τ cos   τ ) ,
X o c ( P , C , τ ) = ( 1 cos   2 C   cos   2 ( C P ) + cos τ   sin   2 C   sin   2 ( C P ) sin   2 C   cos   2 ( C P ) cos τ   cos   2 C   sin   2 ( C P ) sin τ sin 2 ( C P ) ) .
δ m x = δ m x c + δ m x p .
C ( 0 , τ ) = ( 1 + δ C o ) C o ( 0 , τ ) ,
δ C o = ( D c χ 0 0 χ D c 2 D v c 2 δ C ( 1 cos   τ ) 2 δ C   sin   τ 0 2 δ C ( 1 cos   τ ) 0 δ τ 0 2 δ C   sin   τ δ τ 0 ) .
δ m x c = R ( C ) δ C o R ( C ) ,
δ m x c = ( D c χ cos 2 C χ   sin   2 C 0 χ cos 2 C 0.5 ( D c 2 D v c ) ( 1 + cos   4 C ) 0.5 ( D c 2 D v c ) sin   4 C + 2 δ C ( 1 cos   τ ) δ τ sin 2 C + 2 δ C   sin   τ   cos   2 C χ   sin   2 C 0 0.5 ( D c 2 D v c ) sin   4 C 2 δ C ( 1 cos   τ ) δ τ sin 2 C 2 δ C   sin   τ   cos   2 C 0.5 ( D c 2 D v c ) ( 1 cos   4 C ) δ τ   cos   2 C 2 δ C   sin   τ sin 2 C δ τ   cos   2 C + 2 δ C   sin   τ sin 2 C 0 ) .
δ m x p = C o ( C , τ ) δ m p C o     1 ( C , τ ) .
δ m x p = ( D p 0 ε p γ   sin   τ   sin   2 C 0.5 ξ p     2 ( 1 sin 2 τ sin 2 2 C ) ε p γ   sin   τ   cos   2 C 0.25 ξ p     2 sin 2 τ   sin   4 C + 2 δ P   cos   τ ε p γ   cos   τ 0.25 ξ p     2   sin   2 τ   sin   2 C 2 δ P   sin   τ   cos   2 C 0 0 0.25 ξ p 2 sin 2 τ sin 4 C - 2 δ P cos τ - 0.25 ξ p 2 sin 2 τ sin 2 C + 2 δP sin τ cos 2 C - 0.5 ξ p 2 ( 1 - sin 2 τ cos 2 2 C ) 0. 25 ξ p 2 sin 2 τ cos 2 C + 2 δP sin τ sin 2 C 0.25 ξ p 2 sin 2 τ cos 2 C - 2 δP sin τ sin 2 C 0.5 ξ p 2 sin 2 τ ) .
δ m x = ( D c + D p 0 ± χ 0 ε p γ 0 2 δ C δ τ ± χ 2 δ C D c 2 D v c ξ p     2 / 2 ± 2 ( δ P δ C ) 0 ± δ τ 2 ( δ P δ C ) ξ p     2 / 2 ) + 1 1 E 2 × ( ( D p + ξ p     2 / 2 ) E 2 2 B E δ P 0 β ( D p + ξ p     2 / 2 ) 2 α δ P α ( D p + ξ p     2 / 2 ) ± 2 β δ P ( ε p γ E η p B ) E 0 ± ( ε p β γ α η p ) ε p αγ + β η p G   sin   2 P + ( ε p B γ η p E ) cos   2 P 0 G E ( ε p B E γ η p ) ± G   cos   2 P ( ε p B γ η p E ) sin   2 P 0 ± ( ε p B E γ η p ) G E ) .
δ m x = ( D c + D p ± χ 0 0 ± χ D c 2 D v c ξ p     2 / 2 2 δ C ± 2 ( δ P δ C ) 0 2 δ C 0 ± δ τ 0 ± 2 ( δ C δ P ) δ τ ξ p     2 / 2 ) + 1 1 E 2 × ( ( D p + ξ p     2 / 2 ) E 2 2 B E δ P α ( D p + ξ p     2 / 2 ) + 2 β δ P 0 ± β ( D p + ξ p     2 / 2 ) ± 2 α δ P G   cos   2 P ( ε p B γ η p E ) sin   2 P G E 0 ( ε p B E γ η p ) ± ( ε p γ η p B E ) ( ε p α γ + β η p ) 0 ε p β γ α η p G   sin   2 P ( ε p B γ η p E ) cos   2 P ± ( ε p B E γ η p ) 0 G E ) .
δ m o = δ m x + δ m d .
δ m o = ( D a + D p 0 0 0 0 ( ξ p     2 + ξ a     2 ) / 2 2 δ A 2 δ P 0 0 2 δ P 2 δ A ( ξ p     2 + ξ a     2 ) / 2 0 0 0 0 0 ) + 1 1 E 2 × ( ( D p + ξ p     2 / 2 ) E 2 2 B E δ P α ( D p + ξ p     2 / 2 ) + 2 β δ P β ( D p + ξ p     2 / 2 ) 2 α δ P 0 G   cos   2 P + ( γ η p E ε p B ) sin   2 P G E ε p B E γ η p 0 G   sin   2 P + ( ε p B γ η p E ) cos   2 P ε p B E + γ η p G E 0 ε p γ η p B E ε p α γ β η p α η p ε p β γ 0 ) .
δ m o = ( D p + D a + D c 0 ± χ 0 0 ξ a     2 / 2 2 ( δ A δ C ) δ τ c ± χ 2 ( δ A δ C ) D c 2 D v c ( ξ p     2 + ξ a     2 ) / 2 2 ( δ C δ P ) 0 ± δ τ c ± 2 ( δ C δ P ) ξ p     2 / 2 ) + 1 1 E 2 × ( ( D p + ξ p     2 / 2 ) E 2 2 B E δ P α ( D p + ξ p     2 / 2 ) + 2 β δ P 0 ± β ( D p + ξ p     2 / 2 ) ± 2 α δ P G   cos   2 P ( ε p B γ η p E ) sin   2 P G E 0 ( ε p B E γ η p ) ± ( ε p γ η p B E ) ( ε p α γ + β η p ) 0 ε p βγ α η p G   sin   2 P ( ε p B γ η p E ) cos   2 P ± ( ε p B E γ η p ) 0 G E ) .
I ( A , P ) = 1 + δ M 00 + ( P x + δ M 01 ) cos   2 P + δ M 02   sin   2 P + cos   2 A [ P x + δ M 10 + ( 1 2 D v + δ M 11 ) cos   2 P + δ M 12   sin   2 P ] + sin 2 A [ δ M 20 + δ M 21 cos 2 P + ( P y + δ M 22 ) sin   2 P ] .
δ M i j = k = 0 3 ( δ m d i k M k j + M i k δ m x k j ) .
P x * = I ( 0 , ± p 45 ° ) I ( 90 , ± p 45 ° ) I ( 0 , ± p 45 ° ) + I ( 90 , ± p 45 ° ) = P x + δ M 10 ± p δ M 12 1 + δ M 00 ± p δ M 02 ,
P y * = ± p [ I ( 45 , ± p 45 ° ) I ( 45 , ± p 45 ° ) ] I ( 45 , ± p 45 ° ) + I ( 45 , ± p 45 ° ) = P y + δ M 22 ± p δ M 20 1 + δ M 00 ± p δ M 02 .
δ P x = δ M 10 P x δ M 00 , δ P y = δ M 22 P y δ M 00 .
δ M 00 = D a + D p + [ ( D p + ξ p     2 / 2 ) β 2 + 2 α β δ P + P x ( γ η p β + ε p α ) ] / ( 1 β 2 ) ,
δ M 10 = P x ( D a + D p ) + { γ η p β + ε p α + P x [ ( D p + ξ p     2 / 2 ) × β 2 + 2 α β δ P ] } / ( 1 β 2 ) ,
δ M 22 = P y ( ξ a     2 + ξ p     2 ) / 2 + { P y [ 2 α β δ P β 2 ( D p 2 D v p + ξ p     2 / 2 ) ] + P z ( α η p ε p β γ ) } / ( 1 β 2 ) .
P z * = ± p [ I ( 45 , 0 , ± p 45 ° ) I ( 45 , 0 , ± p 45 ° ) ] I ( 45 , 0 , ± p 45 ° ) + I ( 45 , 0 , ± p 45 ° ) = P z + δ M 23 p δ M 20 1 + δ M 00 p δ M 03 ,
δ P z = δ M 23 P z δ M 00 .
δ M 00 = D a + D c + D p + χ P x + [ ( D p + ξ p     2 / 2 ) ( β 2 α P x ) + 2 β δ P ( α + P x ) ] / ( 1 β 2 ) ,
δ M 23 = P z ( ξ a     2 + ξ p     2 ) / 2 + P y δ τ + { P y ( ε p βγ - α η p ) + P z [ 2 δP αβ - β 2 ( D p - 2 D vp + ξ p     2 / 2 ) ] } ( 1 - β 2 ) .
δ P x = P x * = ( η p β γ + ε p α ) / ( 1 β 2 ) ,
δ P y = ( D a + D p ) ( ξ a     2 + ξ p     2 ) / 2 2 β 2 × ( D p D v p + ξ p     2 / 2 ) / ( 1 β 2 ) ,
δ P z = P z * = δ τ + ( ε p β γ - η p α ) / ( 1 β 2 ) .

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