Abstract

A method has been developed to improve the accuracy with which the polarization state of light can be characterized by the rotating quarter-wave plate technique. Through detailed analysis, verified by experiment, we determine the positions of the optic axes of the retarder and linear polarizer, and the wave plate retardance, to better than 1° for typical signal-to-noise ratios. Accurate determination of the Stokes parameters can be achieved using a single wave plate for a wide range of optical wavelengths using this technique to determine the precise retardance at each of the wavelengths of interest.

© 2011 Optical Society of America

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References

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  1. G. G. Stokes, Mathematical and Physical Papers (Cambridge University, 1901), Vol.  3.
  2. M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835(2010).
    [CrossRef]
  3. F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
    [CrossRef]
  4. F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. R. L. Brooks and E. H. Pinnington, “Polarization measurements of HeI singlet transitions following beam-tilted-foil excitation,” Phys. Rev. A 18, 1454–1458 (1978).
    [CrossRef]
  16. R. L. Brooks, “Polarization studies using beam foil spectroscopy,” Ph.D. thesis (University of Alberta, 1979).
  17. E. Hecht, “A mathematical description of polarization,” in Optics, 4th ed., A.Black, ed. (Addison Wesley, 2002), pp. 373–379.
  18. V. A. Dlugunovich, V. N. Snopko, and O. V. Tsaryuk, “Analysis of a method for measuring polarization characteristics with a Stokes polarimeter having a rotating phase plate,” J. Opt. Technol. 68, 269–273 (2001).
    [CrossRef]
  19. P. A. Williams, “Rotating-wave-plate Stokes polarimeter for differential group delay measurements of polarization-mode dispersion,” Appl. Opt. 38, 6508–6515 (1999).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  21. M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
    [CrossRef]
  22. M. Fox, Optical Properties of Solids (Oxford Univ. Press, 2010).
  23. M. Bass, Handbook of Optics (McGraw-Hill, 2000).
  24. M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
    [CrossRef]

2010 (3)

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

R. Randhawa and R. S. Kaler, “High-speed transmission limitations due to polarization mode dispersion,” Optik 121, 1450–1454 (2010).
[CrossRef]

M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835(2010).
[CrossRef]

2008 (3)

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

2007 (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

2006 (3)

2001 (1)

1999 (1)

1998 (1)

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

1992 (1)

1985 (1)

1978 (1)

R. L. Brooks and E. H. Pinnington, “Polarization measurements of HeI singlet transitions following beam-tilted-foil excitation,” Phys. Rev. A 18, 1454–1458 (1978).
[CrossRef]

1977 (1)

Atfield, J. E.

M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
[CrossRef]

Azzam, R. M. A.

Barrett, D.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Bass, M.

M. Bass, Handbook of Optics (McGraw-Hill, 2000).

Berry, H. G.

Besse, J.

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

Bigue, L.

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Broch, L.

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

Brooks, R. L.

R. L. Brooks and E. H. Pinnington, “Polarization measurements of HeI singlet transitions following beam-tilted-foil excitation,” Phys. Rev. A 18, 1454–1458 (1978).
[CrossRef]

R. L. Brooks, “Polarization studies using beam foil spectroscopy,” Ph.D. thesis (University of Alberta, 1979).

Chenault, D. B.

Collett, E.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

de Wijn, A. G.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Dlugunovich, V. A.

El-Bahrawi, M. S.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Ferraton, M.

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Fischer, C. E.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Flueraru, C.

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

Foreman, M. R.

M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835(2010).
[CrossRef]

Fox, M.

M. Fox, Optical Properties of Solids (Oxford Univ. Press, 2010).

Fraher, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Gabrielse, G.

Goldstein, D. H.

Goldstein, D. L.

Hecht, E.

E. Hecht, “A mathematical description of polarization,” in Optics, 4th ed., A.Black, ed. (Addison Wesley, 2002), pp. 373–379.

Ichimoto, K.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Johann, L.

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

Kaler, R. S.

R. Randhawa and R. S. Kaler, “High-speed transmission limitations due to polarization mode dispersion,” Optik 121, 1450–1454 (2010).
[CrossRef]

Keller, C. U.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Khodier, S. A.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Latoui, S.

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

Legendre, P.

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

Lites, B. W.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Livingston, A. E.

McCarthy, T. G.

M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
[CrossRef]

Meriaudeau, F.

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Morel, O.

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Nagib, N. N.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Pinnington, E. H.

R. L. Brooks and E. H. Pinnington, “Polarization measurements of HeI singlet transitions following beam-tilted-foil excitation,” Phys. Rev. A 18, 1454–1458 (1978).
[CrossRef]

Randhawa, R.

R. Randhawa and R. S. Kaler, “High-speed transmission limitations due to polarization mode dispersion,” Optik 121, 1450–1454 (2010).
[CrossRef]

Schaefer, B.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Shaw, J. A.

Shiell, R. C.

M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
[CrossRef]

Sidki, H. M.

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Smyth, R.

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Snik, F.

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

Snopko, V. N.

Stokes, G. G.

G. G. Stokes, Mathematical and Physical Papers (Cambridge University, 1901), Vol.  3.

Stolz, C.

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Török, P.

M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835(2010).
[CrossRef]

Tsaryuk, O. V.

Tyo, J. S.

Ugray, M.

M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
[CrossRef]

Wei, H.

Williams, P. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Am. J. Phys. (1)

B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
[CrossRef]

Appl. Opt. (5)

Astron. Astrophys. (1)

F. Snik, A. G. de Wijn, K. Ichimoto, C. E. Fischer, C. U. Keller, and B. W. Lites, “Observations of solar scattering polarization at high spatial resolution,” Astron. Astrophys. 519, A18(2010).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’ polarimeter,” IEEE Trans. Instrum. Meas. 57, 731–735 (2008).
[CrossRef]

J. Opt. Technol. (1)

Opt. Laser Technol. (1)

M. S. El-Bahrawi, N. N. Nagib, S. A. Khodier, and H. M. Sidki, “Birefringence of muskovite mica,” Opt. Laser Technol. 30, 411–415 (1998).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. Randhawa and R. S. Kaler, “High-speed transmission limitations due to polarization mode dispersion,” Optik 121, 1450–1454 (2010).
[CrossRef]

Phys. Rev. A (2)

M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835(2010).
[CrossRef]

R. L. Brooks and E. H. Pinnington, “Polarization measurements of HeI singlet transitions following beam-tilted-foil excitation,” Phys. Rev. A 18, 1454–1458 (1978).
[CrossRef]

Phys. Status Solidi C (1)

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5, 1036–1040 (2008).
[CrossRef]

Proc. SPIE (1)

F. Meriaudeau, M. Ferraton, C. Stolz, O. Morel, and L. Bigue, “Polarization imaging for industrial inspection,” Proc. SPIE 6813, 681308 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Ugray, J. E. Atfield, T. G. McCarthy, and R. C. Shiell, “Microcontroller-based wavemeter using compression locking of an internal mirror reference laser,” Rev. Sci. Instrum. 77, 113109 (2006).
[CrossRef]

Other (7)

M. Fox, Optical Properties of Solids (Oxford Univ. Press, 2010).

M. Bass, Handbook of Optics (McGraw-Hill, 2000).

Comar Optics Inc., http://www.comaroptics.com.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

R. L. Brooks, “Polarization studies using beam foil spectroscopy,” Ph.D. thesis (University of Alberta, 1979).

E. Hecht, “A mathematical description of polarization,” in Optics, 4th ed., A.Black, ed. (Addison Wesley, 2002), pp. 373–379.

G. G. Stokes, Mathematical and Physical Papers (Cambridge University, 1901), Vol.  3.

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Figures (5)

Fig. 1
Fig. 1

(a) Illustration of the relevant angles for the calibrated rotating quarter-wave plate method (angles have been exaggerated for clarity). The angle of the LP transmission axis and the offset of the retarder fast axis in its initial position from x ^ are denoted by γ and β 0 , respectively. The front-to-back rotation axis of the polarizer defines y ^ and the misalignment of the retarder’s rotation axis from this is denoted ϕ. (b) Position of the fast axis is shown for the forward (F) and reversed ( F ) orientations of the retarder. When reversed, the fast axis is offset by 2 ϕ β 0 from x ^ due to the misalignment of the retarder’s front-to-back rotation axis from y ^ .

Fig. 2
Fig. 2

Schematic of the experimental setup. The beam under test originates from a fiber-coupled external cavity diode laser. A BE increases the beam width to 8 mm . A pair of PBSs produce a horizontally polarized beam with which to test the calibration method. The first s-polarized reflection is sent to an optical spectrum analyzer (OSA), while the weak second reflection serves as a reference for power normalization. The polarimeter itself consists of a quarter-wave plate ( λ / 4 ) and LP in rotational mounts. The calibration method employs an additional LP ( LP ) set to make S 1 S 2 and is removed for regular beam analysis. The retarder is rotated via a worm gear by the SM and the transmitted light is measured by a PD after attenuation by a neutral density filter (ND). The μC controls the SM and records the PD voltages for transmission to a computer.

Fig. 3
Fig. 3

Dependence of ξ on β 0 and Δ using the values γ = 43.93 ° and ϕ = 0.9221 ° , which were found after 10 search iterations for data simulated using the values β 0 = 2 ° , Δ = 0.26 × 2 π = 93.6 ° , γ = 44 ° , ϕ = 1 ° , and an incident Stokes vector corresponding to 67.5 ° polarized light and with noise added. The dashed lines indicate the correct values of β 0 and Δ.

Fig. 4
Fig. 4

Dependences of ξ on β 0 and Δ for data generated using ϕ = 1 ° with noise added. The two surfaces were plotted assuming values of ϕ = 0 ° (minimum indicated by dashed lines, red) and ϕ = 0.92 ° (minimum located close to the intersection of the solid lines, blue) with a 5 mrad sampling resolution. The solid lines indicate the chosen values of β 0 and Δ.

Fig. 5
Fig. 5

Analysis of calibration results for sim ulated data with added random noise using parameter values { β 0 , Δ , γ , ϕ } = { 2 ° , 0.26 × 2 π = 93.6 ° , 44 ° , 1 ° } . Each datum is the average over 20 sets of results with the same SNR value. For comparison, data used to construct Table 1 had SNR values in the range 40–61 for the three different retarders.

Tables (2)

Tables Icon

Table 1 Experimental Results for Nine Calibrations Using Three Different Quarter-Wave Plates with Design Wavelengths Denoted by λ π / 2 nom , Each with Three Different Values of ϕ nom

Tables Icon

Table 2 Stokes Vectors Derived from Measurements of Horizontally Polarized Light Using Three Different Retarders and Three Intentional Misalignments of the Retarder Vertical Rotation Axis Using the Nine Calibrations Presented in Table 1

Equations (29)

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

P = S 1 2 + S 2 2 + S 3 2 S 0 .
S = M ̲ ̲ pol M ̲ ̲ ret S .
I ( β i , Δ , γ ) = 1 2 { S 0 + S 1 [ cos 2 β i cos 2 ( γ β i ) sin 2 β i sin 2 ( γ β i ) cos Δ ] + S 2 [ sin 2 β i cos 2 ( γ β i ) + cos 2 β i sin 2 ( γ β i ) cos Δ ] + S 3 [ sin 2 ( γ β i ) sin Δ ] } .
I ( β i ) = 1 2 ( a 0 + a 2 cos 2 β i + b 2 sin 2 β i + a 4 cos 4 β i + b 4 sin 4 β i ) .
a 0 = 2 N i = 1 N I i ,
a 2 = 4 N i = 1 N I i cos 2 β i ,
b 2 = 4 N i = 1 N I i sin 2 β i ,
a 4 = 4 N i = 1 N I i cos 4 β i ,
b 4 = 4 N i = 1 N I i sin 4 β i .
S 0 = a 0 1 + cos Δ 1 cos Δ ( a 4 cos 4 γ + b 4 sin 4 γ ) ,
S 1 = 2 1 cos Δ ( a 4 cos 2 γ + b 4 sin 2 γ ) ,
S 2 = 2 1 cos Δ ( b 4 cos 2 γ a 4 sin 2 γ ) ,
S 3 = a 2 sin Δ sin 2 γ = b 2 sin Δ cos 2 γ .
S 0 ( 1 , 3 ) = a 0 ( 1 + cos Δ ) ( 1 cos Δ ) [ a 4 cos 4 ( γ β 0 ) ± b 4 sin 4 ( γ β 0 ) ] ,
S 1 ( 1 , 3 ) = 2 ( 1 cos Δ ) [ a 4 cos 2 ( γ 2 β 0 ) ± b 4 sin 2 ( γ 2 β 0 ) ] ,
S 2 ( 1 , 3 ) = 2 ( 1 cos Δ ) [ b 4 cos 2 ( γ 2 β 0 ) a 4 sin 2 ( γ 2 β 0 ) ] ,
S 3 ( 1 , 3 ) = ± a 2 sin Δ sin 2 ( γ β 0 ) = b 2 sin Δ cos 2 ( γ β 0 ) ,
S 0 ( 2 , 4 ) = a 0 ( 1 + cos Δ ) ( 1 cos Δ ) [ a 4 cos 4 ( γ ± β 0 2 ϕ ) ± b 4 sin 4 ( γ ± β 0 2 ϕ ) ] ,
S 1 ( 2 , 4 ) = 2 ( 1 cos Δ ) [ a 4 cos 2 ( γ ± 2 β 0 4 ϕ ) ± b 4 sin 2 ( γ ± 2 β 0 4 ϕ ) ] ,
S 2 ( 2 , 4 ) = 2 ( 1 cos Δ ) [ b 4 cos 2 ( γ ± 2 β 0 4 ϕ ) a 4 sin 2 ( γ ± 2 β 0 4 ϕ ) ] ,
S 3 ( 2 , 4 ) = ± a 2 sin Δ sin 2 ( γ ± β 0 2 ϕ ) = b 2 sin Δ cos 2 ( γ ± β 0 2 ϕ ) ,
| S 3 ( 1 , 2 , 3 , 4 ) | = a 2 2 + b 2 2 sin Δ .
ξ = j = 1 4 [ ( S 1 ( j ) S 0 ( j ) S 1 S 0 ) 2 + ( S 2 ( j ) S 0 ( j ) S 2 S 0 ) 2 + ( 1 P ( j ) ) 2 ] .
5 ° β 0 5 ° ,
( λ π / 2 nom λ ) 85 ° Δ ( λ π / 2 nom λ ) 95 ° ,
40 ° γ 50 ° ,
5 ° ϕ 5 ° .
S = ( 1.000 0.995 ± 0.001 0.121 ± 0.002 0.0158 ± 0.0003 ) ,
P = 1.002 ± 0.001 .

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