Abstract

An analytical technique based on the Mueller matrix method and the Stokes parameters is proposed for extracting five effective parameters on the principal axis angle, phase retardance, diattenuation axis angle, diattenuation and optical rotation angle of anisotropic optical materials. The linear birefringence (LB) / circular birefringence (CB) properties and linear diattenuation (LD) properties are decoupled within the analytical model. The analytical method is then integrated with a genetic algorithm to extract the optical properties of samples with linear birefringence property using a fiber-based polarimeter. The result demonstrates the feasibility of analytical model in characterizing five effective parameters of anisotropic optical material. Also, it confirms that the proposed fiber-based polarimeter provides a simple alternative to existing fiber-based probes for parameter measurement in the near field or the remote environment. A low birefringence fiber-based polarimeter based on effective parameters and genetic algorithm without using a fiber polarization controller is first proposed confirmatively.

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References

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  1. G. F. Smith, Constitutive equations for anisotropic and isotropic materials, (North-Holland, 1994).
  2. D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardation spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32(19), 3513–3519 (1993).
    [CrossRef] [PubMed]
  3. D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. 17, 4223–4227 (1992).
  4. M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
    [CrossRef] [PubMed]
  5. L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
    [CrossRef] [PubMed]
  6. L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
    [CrossRef]
  7. A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002).
    [CrossRef]
  8. D. B. Chenault, R. A. Chipman, and S. Y. Lu, “Electro-optic coefficient spectrum of cadmium telluride,” Appl. Opt. 33(31), 7382–7389 (1994).
    [CrossRef] [PubMed]
  9. E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997).
    [CrossRef]
  10. P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express 17(18), 15860–15884 (2009).
    [CrossRef] [PubMed]
  11. I. C. Khoo, and F. Simoni, Physics of Liquid Crystalline Materials, (Gorden and Breach Science Publishers, 1991), Chap. 13.
  12. H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).
  13. T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Light. Tech. 25, (2007).
    [CrossRef]
  14. W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).
  15. Z. Michalewicz, Genetic Algorithm+ Data Structure = Evolution Programs, (Springer-Verlag, New York, 1994).

2009 (2)

P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express 17(18), 15860–15884 (2009).
[CrossRef] [PubMed]

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

2007 (1)

T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Light. Tech. 25, (2007).
[CrossRef]

2005 (2)

H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).

L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
[CrossRef]

2003 (2)

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
[CrossRef] [PubMed]

2002 (1)

A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002).
[CrossRef]

1997 (1)

E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. 17, 4223–4227 (1992).

Beers, K. L.

Bryant, G. W.

Campillo, A. L.

A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002).
[CrossRef]

Chen, P. C.

Chenault, D. B.

Cheng, H.C.

H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).

Chipman, R. A.

E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997).
[CrossRef]

D. B. Chenault, R. A. Chipman, and S. Y. Lu, “Electro-optic coefficient spectrum of cadmium telluride,” Appl. Opt. 33(31), 7382–7389 (1994).
[CrossRef] [PubMed]

D. B. Chenault and R. A. Chipman, “Measurements of linear diattenuation and linear retardation spectra with a rotating sample spectropolarimeter,” Appl. Opt. 32(19), 3513–3519 (1993).
[CrossRef] [PubMed]

D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. 17, 4223–4227 (1992).

DeRege, P.

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

Fasolka, M. J.

L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
[CrossRef]

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
[CrossRef] [PubMed]

Goldie, S. N.

L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
[CrossRef]

Goldner, L. S.

L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
[CrossRef]

L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
[CrossRef] [PubMed]

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

Hsu, J. W. P.

A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002).
[CrossRef]

Hwang, J.

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
[CrossRef] [PubMed]

Lin, J. F.

P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express 17(18), 15860–15884 (2009).
[CrossRef] [PubMed]

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

Lin, W. L.

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

Lo, Y. L.

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express 17(18), 15860–15884 (2009).
[CrossRef] [PubMed]

T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Light. Tech. 25, (2007).
[CrossRef]

H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).

Lu, S. Y.

Nguyen, H. P.

Nougier, S.

Sornsin, E. A.

E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997).
[CrossRef]

Swager, T.

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

Thomas, E. L.

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

L. S. Goldner, M. J. Fasolka, S. Nougier, H. P. Nguyen, G. W. Bryant, J. Hwang, K. D. Weston, K. L. Beers, A. Urbas, and E. L. Thomas, “Fourier analysis near-field polarimetry for measurement of local optical properties of thin films,” Appl. Opt. 42(19), 3864–3881 (2003).
[CrossRef] [PubMed]

Urbas, A.

Urbas, A. M.

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

Weston, K. D.

Yang, T. T.

Yu, T. C.

P. C. Chen, Y. L. Lo, T. C. Yu, J. F. Lin, and T. T. Yang, “Measurement of linear birefringence and diattenuation properties of optical samples using polarimeter and Stokes parameters,” Opt. Express 17(18), 15860–15884 (2009).
[CrossRef] [PubMed]

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Light. Tech. 25, (2007).
[CrossRef]

Appl. Opt. (3)

Appl. Scanned Probe Microscopy Polymers (1)

L. S. Goldner, M. J. Fasolka, and S. N. Goldie, “Measurement of the local diattenuation and retardance of thin polymer films using near-field polarimetry,” Appl. Scanned Probe Microscopy Polymers 897, 65–84 (2005).
[CrossRef]

J. Appl. Phys. (1)

A. L. Campillo and J. W. P. Hsu, “Near-field scanning optical microscope studies of the anisotropic stress variations in patterned SiN membranes,” J. Appl. Phys. 91(2), 646–651 (2002).
[CrossRef]

J. Light. Tech. (3)

H.C. Cheng and Y. L. Lo, “The synthesis of multiple parameters of arbitrary FBGs via a genetic algorithm and two thermally modulated intensity spectra,” J. Light. Tech. 23, (2005).

T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Light. Tech. 25, (2007).
[CrossRef]

W. L. Lin, T. C. Yu, Y. L. Lo, and J. F. Lin, “A hybrid approach for measuring the parameters of twisted-nematic liquid crystal cells utilizing the stokes parameter method and a genetic algorithm,” J. Light. Tech. 27, (2009).

Opt. Express (1)

Opt. Lett. (1)

D. B. Chenault and R. A. Chipman, “Infrared birefringence spectra for cadmium-sulfide and cadmium selenide,” Opt. Lett. 17, 4223–4227 (1992).

Phys. Rev. Lett. (1)

M. J. Fasolka, L. S. Goldner, J. Hwang, A. M. Urbas, P. DeRege, T. Swager, and E. L. Thomas, “Measuring local optical properties: near-field polarimetry of photonic block copolymer morphology,” Phys. Rev. Lett. 90(1), 016107 (2003).
[CrossRef] [PubMed]

SPIE (1)

E. A. Sornsin and R. A. Chipman, “Visible Mueller matrix spectropolarimetry,” SPIE 3121, 156–160 (1997).
[CrossRef]

Other (3)

I. C. Khoo, and F. Simoni, Physics of Liquid Crystalline Materials, (Gorden and Breach Science Publishers, 1991), Chap. 13.

Z. Michalewicz, Genetic Algorithm+ Data Structure = Evolution Programs, (Springer-Verlag, New York, 1994).

G. F. Smith, Constitutive equations for anisotropic and isotropic materials, (North-Holland, 1994).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of model used to characterize anisotropic material.

Fig. 2
Fig. 2

Correlation between input value of principal axis angle (α) and extracted value of principal axis angle (α’).

Fig. 3
Fig. 3

Correlation between input value of phase retardance (β) and extracted value of phase retardance (β’).

Fig. 4
Fig. 4

Correlation between input value diattenuation axis angle (θd) and extracted value of diattenuation axis angle (θd’).

Fig. 5
Fig. 5

Correlation between input value of diattenuation (D) and extracted value of diattenuation (D’).

Fig. 6
Fig. 6

Correlation between input value of optical rotation angle (γ) and extracted value of optical rotation angle (γ’).

Fig. 7
Fig. 7

Correlation between (a) α and α’, (b) β and β’, (c) θd and θd’, (d) D and D’, and (e) γ and γ’, respectively.

Fig. 8
Fig. 8

Schematic illustration of measurement system used to characterize an optical fiber.

Fig. 9
Fig. 9

Schematic diagram of fiber-type polarimeter used to extract LB sample parameters using genetic algorithm.

Fig. 10
Fig. 10

Flowchart of GA optimization procedure used to extract LB sample parameters.

Fig. 11
Fig. 11

Experimental results for birefringence of quarter-wave plate (without considering the optical rotation angle).

Fig. 12
Fig. 12

Experimental results for birefringence of quarter-wave plate (with considering the optical rotation angle).

Tables (2)

Tables Icon

Table 2 Calculated values of five effective optical parameters of optical fiber

Tables Icon

Table 1 Calculated values of four effective optical parameters of optical fiber

Equations (42)

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M l b = ( 1 0 0 0 0 cos ( 4 α ) sin 2 ( β / 2 ) + cos 2 ( β / 2 ) sin ( 4 α ) sin 2 ( β / 2 ) sin ( 2 α ) sin ( β ) 0 sin ( 4 α ) sin 2 ( β / 2 ) cos ( 4 α ) sin 2 ( β / 2 ) + cos 2 ( β / 2 ) cos ( 2 α ) sin ( β ) 0 sin ( 2 α ) sin ( β ) cos ( 2 α ) sin ( β ) cos ( β ) )
M l d = ( 1 2 ( 1 + 1 D 1 + D ) 1 2 cos ( 2 θ d ) ( 1 1 D 1 + D ) 1 2 sin ( 2 θ d ) ( 1 1 D 1 + D ) 0 1 2 cos ( 2 θ d ) ( 1 1 D 1 + D ) 1 4 ( ( 1 + 1 D 1 + D ) 2 + cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) 1 4 sin ( 4 θ d ) ( 1 1 D 1 + D ) 2 0 1 2 sin ( 2 θ d ) ( 1 1 D 1 + D ) 1 4 sin ( 4 θ d ) ( 1 1 D 1 + D ) 2 1 4 ( ( 1 + 1 D 1 + D ) 2 cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) 0 0 0 0 1 D 1 + D )
M c b = ( 1 0 0 0 0 cos ( 2 γ ) sin ( 2 γ ) 0 0 sin ( 2 γ ) cos ( 2 γ ) 0 0 0 0 1 )
S c = [ S 0 S 1 S 2 S 3 ] c = [ M l d ] [ M l b ] [ M c b ] S ^ c = ( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 0 m 42 m 43 m 44 ) ( S ^ 0 S ^ 1 S ^ 2 S ^ 3 ) c
m 11 = 1 2 ( 1 + 1 D 1 + D )
m 14 = 1 2 ( cos ( 2 θ d ) ( 1 1 D 1 + D ) ) sin ( 2 α ) sin ( β ) 1 2 ( cos ( 2 θ d ) ( 1 1 D 1 + D ) ) cos ( 2 α ) sin ( β )
m 21 = 1 2 cos ( 2 θ d ) ( 1 1 D 1 + D )
m 24 = 1 4 ( ( 1 + 1 D 1 + D ) 2 + cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) sin ( 2 α ) sin ( β ) 1 4 ( sin ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) cos ( 2 α ) sin ( β )
m 31 = 1 2 sin ( 2 θ d ) ( 1 1 D 1 + D )
m 34 = 1 4 ( sin ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) sin ( 2 α ) sin ( β ) 1 4 ( ( 1 + 1 D 1 + D ) 2 cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) cos ( 2 α ) sin ( β )
m 42 = 1 D 1 + D sin ( 2 α + 2 γ ) sin ( β )
m 43 = 1 D 1 + D cos ( 2 α + 2 γ ) sin ( β )
m 44 = 1 D 1 + D cos ( β )
S 0 0 = [ m 11 + m 12 , m 21 + m 22 , m 31 + m 32 , m 42 ] T
S 45 0 = [ m 11 + m 13 , m 21 + m 23 , m 31 + m 33 , m 43 ] T
S 90 0 = [ m 11 m 12 , m 21 m 22 , m 31 m 32 , m 42 ] T
S 135 0 = [ m 11 m 13 , m 21 m 23 , m 31 m 33 , m 43 ] T
S R H C = [ m 11 + m 14 , m 21 + m 24 , m 31 + m 34 , m 44 ] T
S L H C = [ m 11 m 14 , m 21 m 24 , m 31 m 34 , m 44 ] T
S 0 ( S 3 ) S 45 ( S 3 ) = m 42 m 43 = 1 D 1 + D sin ( 2 α + 2 γ ) sin ( β ) 1 D 1 + D cos ( 2 α + 2 γ ) sin ( β )
2 α + 2 γ = tan 1 ( S 0 ( S 3 ) S 45 ( S 3 ) )
S 45 0 ( S 3 ) S R H C ( S 3 ) = m 43 m 44 = 1 D 1 + D cos ( 2 α + 2 γ ) sin ( β ) 1 D 1 + D sin ( β )
β = tan 1 ( S 45 ( S 3 ) cos ( 2 α + 2 γ ) S R H ( S 3 ) )
S 0 ( S 0 ) + S 90 ( S 0 ) = 2m 11 = ( 1 + 1 D 1 + D )
S 0 ( S 1 ) + S 90 ( S 1 ) = 2m 21 = cos ( 2 θ d ) ( 1 1 D 1 + D )
S 45 ( S 2 ) + S 135 ( S 2 ) = 2m 31 = sin ( 2 θ d ) ( 1 1 D 1 + D )
2 θ d = tan 1 ( S 45 ( S 2 ) + S 135 ( S 2 ) S 0 ( S 1 ) + S 90 ( S 1 ) )
D = S 0 ( S 1 ) + S 90 ( S 1 ) cos ( 2 θ d ) [ S 0 ( S 0 ) + S 90 ( S 0 ) ]
D = S 45 ( S 2 ) + S 135 ( S 2 ) sin ( 2 θ d ) [ S 0 ( S 0 ) + S 90 ( S 0 ) ]
S R H C ( S 2 ) S L H C ( S 2 ) = 2m 24 = 2 [ C 1 sin ( 2 α ) C 2 cos ( 2 α ) ]
S R H C ( S 3 ) S L H C ( S 3 ) = 2m 34 = 2 [ C 2 sin ( 2 α ) C 3 cos ( 2 α ) ]
C 1 = 1 4 ( ( 1 + 1 D 1 + D ) 2 + cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) sin β
C 2 = 1 4 ( sin ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) sin β
C 3 = 1 4 ( ( 1 + 1 D 1 + D ) 2 cos ( 4 θ d ) ( 1 1 D 1 + D ) 2 ) sin β
2 α = tan 1 ( C 3 [ S R H C ( S 2 ) S L H C ( S 2 ) ] C 2 [ S R H C ( S 3 ) S L H C ( S 3 ) ] C 2 [ S R H C ( S 2 ) S L H C ( S 2 ) ] C 1 [ S R H C ( S 3 ) S L H C ( S 3 ) ] )
2 γ = tan 1 ( S 0 ( S 3 ) S 45 ( S 3 ) ) 2 α
Q c = [ S 0 S 1 S 2 S 3 ] = [ M l b S ] [ M l d ] [ M l b ] [ M c b ] S ^ c = [ M l b S ] ( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 0 m 42 m 43 m 44 ) ( S ^ 0 S ^ 1 S ^ 2 S ^ 3 ) c
M l b S = ( 1 0 0 0 0 cos ( 4 α S ) sin 2 ( β S / 2 ) + cos 2 ( β S / 2 ) sin ( 4 α S ) sin 2 ( β S / 2 ) sin ( 2 α S ) sin ( β S ) 0 sin ( 4 α S ) sin 2 ( β S / 2 ) cos ( 4 α S ) sin 2 ( β S / 2 ) + cos 2 ( β S / 2 ) cos ( 2 α S ) sin ( β S ) 0 sin ( 2 α S ) sin ( β S ) cos ( 2 α S ) sin ( β S ) cos ( β S ) )
Get closer: { P 1 = P 1 + δ ( P 1 P 2 ) P 2 = P 1 δ ( P 1 P 2 )
Pull away: { P 1 = P 1 + δ ( P 2 P 1 ) P 2 = P 1 δ ( P 2 P 1 )
P " = r a n d o m _ v a l u e + P '
Error = E ϕ = i = 1 9 ( ϕ i , Experiment ϕ i , Compute ) 2

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