Abstract

A method of designing achromatic elliptical polarizers using a combination of multiple birefringent waveplates is demonstrated. This approach has a simple geometric interpretation and simplifies the problem of designing an achromatic elliptical polarizer to find overlapping arcs on the Poincaré sphere. The technique is applied to the design of achromatic elliptical polarizers for a broadband division-of-focal-plane full-Stokes imaging polarimeter for visible wavelength band (λ = 450nm to 650nm). An achromatic elliptical polarizer sample with a two-layer retarder is fabricated using liquid crystal polymer. The performance of the polarizer sample is measured and compared with the theoretical calculation. For comparison, a superachromatic polarizer design (λ = 400nm to 1μm) is also presented by using three-layer and four-layer retarder configurations.

© 2017 Optical Society of America

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References

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  1. P. Hariharan and P. E. Ciddor, “Broad-band superachromatic retarders and circular polarizers for the UV, visible and near infrared,” J. Mod. Opt. 51(15), 2315–2322 (2004).
    [Crossref]
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    [Crossref]
  4. J. M. Herrera-Fernandez, J. L. Vilas, L. M. Sanchez-Brea, and E. Bernabeu, “Design of superachromatic quarter-wave retarders in a broad spectral range,” Appl. Opt. 54(33), 9758–9762 (2015).
    [Crossref] [PubMed]
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    [Crossref]
  7. A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51(12), 1976–1980 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  18. L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  26. X. Tu and S. Pau, “Optimized design of N optical filters for color and polarization imaging,” Opt. Express 24(3), 3011–3024 (2016).
    [Crossref] [PubMed]
  27. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  29. S. T. Wu, “Birefringence dispersions of liquid crystals,” Phys. Rev. A Gen. Phys. 33(2), 1270–1274 (1986).
    [Crossref] [PubMed]

2016 (5)

2015 (2)

2014 (2)

2013 (1)

2012 (4)

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51(12), 1976–1980 (2012).
[Crossref] [PubMed]

G. Myhre, W. L. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-stokes division of focal plane polarimeter,” Opt. Express 20(25), 27393–27409 (2012).
[Crossref] [PubMed]

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “New achromatic quarter-wave combination of birefringent plates,” Opt. Eng. 51(1), 013001 (2012).
[Crossref]

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

2010 (3)

2008 (1)

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

2005 (2)

2004 (1)

P. Hariharan and P. E. Ciddor, “Broad-band superachromatic retarders and circular polarizers for the UV, visible and near infrared,” J. Mod. Opt. 51(15), 2315–2322 (2004).
[Crossref]

2000 (1)

1997 (1)

1995 (1)

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

1986 (1)

S. T. Wu, “Birefringence dispersions of liquid crystals,” Phys. Rev. A Gen. Phys. 33(2), 1270–1274 (1986).
[Crossref] [PubMed]

1955 (1)

S. Pancharatnam, “Achromatic combination of birefringent plates,” Proc. Indiana Acad. Sci. XLI(4), 137 (1955).

Azzam, R. M. A.

Balakrishnan, K.

Bermak, A.

Bernabeu, E.

Bhattacharya, K.

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51(12), 1976–1980 (2012).
[Crossref] [PubMed]

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “New achromatic quarter-wave combination of birefringent plates,” Opt. Eng. 51(1), 013001 (2012).
[Crossref]

Boussaid, F.

Brock, N.

Broer, D. J.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Chakraborty, A. K.

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “New achromatic quarter-wave combination of birefringent plates,” Opt. Eng. 51(1), 013001 (2012).
[Crossref]

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51(12), 1976–1980 (2012).
[Crossref] [PubMed]

Chang, Z.

Chen, L.

Chen, Z.

Chigrinov, V. G.

Chipman, R. A.

Choi, H. C.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Ciddor, P. E.

P. Hariharan and P. E. Ciddor, “Broad-band superachromatic retarders and circular polarizers for the UV, visible and near infrared,” J. Mod. Opt. 51(15), 2315–2322 (2004).
[Crossref]

Davis, J.

Deng, X.

Denker, C.

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

Dereniak, E. L.

Descour, M. R.

Fan, X.

Gruev, V.

Hariharan, P.

P. Hariharan and P. E. Ciddor, “Broad-band superachromatic retarders and circular polarizers for the UV, visible and near infrared,” J. Mod. Opt. 51(15), 2315–2322 (2004).
[Crossref]

Herrera-Fernandez, J. M.

Hsu, W. L.

Hui, B.

Ibn-Elhaj, M.

Iwata, K.

Jin, G.

Kang, G.

Kang, W. S.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Kemme, S. A.

Kikuta, H.

Kim, B. K.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Kroto, S.

LaCasse, C.

Lee, G. D.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Lee, J. H.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Lee, S. H.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Liang, R.

Lim, Y. J.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Liu, F.

Liu, X.

Lub, J.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Luo, H.

Ma, J.

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

Mol, G. N.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Mu, T.

Mun, B. J.

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

Myhre, G.

Ohira, Y.

Pan, X.

Pancharatnam, S.

S. Pancharatnam, “Achromatic combination of birefringent plates,” Proc. Indiana Acad. Sci. XLI(4), 137 (1955).

Pau, S.

Peinado, A.

Phipps, G. S.

Powell, S. B.

Sabatke, D. S.

Saha, A.

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “New achromatic quarter-wave combination of birefringent plates,” Opt. Eng. 51(1), 013001 (2012).
[Crossref]

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “Achromatic quarter-wave plate using crystalline quartz,” Appl. Opt. 51(12), 1976–1980 (2012).
[Crossref] [PubMed]

Sanchez-Brea, L. M.

Sayyad, A.

Sciortino, P. F.

She, J.

Shen, S.

Sweatt, W. C.

Tan, Q.

Tao, T.

Tu, X.

Vilas, J. L.

Wang, H. M.

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

Wang, J. J.

Wang, J. S.

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

Wang, L.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Wang, M.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Wang, X.

Wu, S. T.

S. T. Wu, “Birefringence dispersions of liquid crystals,” Phys. Rev. A Gen. Phys. 33(2), 1270–1274 (1986).
[Crossref] [PubMed]

Xu, P.

Yang, D.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Yang, H.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Yu, H.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Zhang, C.

Zhang, J.

Zhang, L.

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Zhao, X.

Appl. Opt. (3)

Chin. J. Astron. Astrophys. (1)

J. Ma, J. S. Wang, C. Denker, and H. M. Wang, “Optical design of multilayer achromatic waveplate by simulated annealing algorithm,” Chin. J. Astron. Astrophys. 8(3), 349–361 (2008).
[Crossref]

J. Appl. Phys. (1)

W. S. Kang, B. J. Mun, G. D. Lee, J. H. Lee, B. K. Kim, H. C. Choi, Y. J. Lim, and S. H. Lee, “Optimal design of quarter-wave plate with wideband and wide viewing angle for three-dimensional liquid crystal display,” J. Appl. Phys. 111(10), 103119 (2012).
[Crossref]

J. Mod. Opt. (1)

P. Hariharan and P. E. Ciddor, “Broad-band superachromatic retarders and circular polarizers for the UV, visible and near infrared,” J. Mod. Opt. 51(15), 2315–2322 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Liq. Cryst. (1)

L. Zhang, M. Wang, L. Wang, D. Yang, H. Yu, and H. Yang, “Polymeric infrared reflective thin films with ultra-broad bandwidth,” Liq. Cryst. 43(6), 750–757 (2016).
[Crossref]

Nature (1)

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Opt. Eng. (1)

A. Saha, K. Bhattacharya, and A. K. Chakraborty, “New achromatic quarter-wave combination of birefringent plates,” Opt. Eng. 51(1), 013001 (2012).
[Crossref]

Opt. Express (11)

T. Mu, Z. Chen, C. Zhang, and R. Liang, “Optimal design and performance metric of broadband full-Stokes polarimeters with immunity to Poisson and Gaussian noise,” Opt. Express 24(26), 29691–29704 (2016).
[Crossref] [PubMed]

J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24(18), 20799–20807 (2016).
[Crossref] [PubMed]

X. Zhao, X. Pan, X. Fan, P. Xu, A. Bermak, and V. G. Chigrinov, “Patterned dual-layer achromatic micro-quarter-wave-retarder array for active polarization imaging,” Opt. Express 22(7), 8024–8034 (2014).
[Crossref] [PubMed]

S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21(18), 21039–21055 (2013).
[Crossref] [PubMed]

G. Kang, Q. Tan, X. Wang, and G. Jin, “Achromatic phase retarder applied to MWIR & LWIR dual-band,” Opt. Express 18(2), 1695–1703 (2010).
[Crossref] [PubMed]

G. Myhre, W. L. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-stokes division of focal plane polarimeter,” Opt. Express 20(25), 27393–27409 (2012).
[Crossref] [PubMed]

W. L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014).
[Crossref] [PubMed]

W. L. Hsu, J. Davis, K. Balakrishnan, M. Ibn-Elhaj, S. Kroto, N. Brock, and S. Pau, “Polarization microscope using a near infrared full-Stokes imaging polarimeter,” Opt. Express 23(4), 4357–4368 (2015).
[Crossref] [PubMed]

X. Zhao, A. Bermak, F. Boussaid, and V. G. Chigrinov, “Liquid-crystal micropolarimeter array for full Stokes polarization imaging in visible spectrum,” Opt. Express 18(17), 17776–17787 (2010).
[Crossref] [PubMed]

X. Tu and S. Pau, “Optimized design of N optical filters for color and polarization imaging,” Opt. Express 24(3), 3011–3024 (2016).
[Crossref] [PubMed]

G. Myhre, A. Sayyad, and S. Pau, “Patterned color liquid crystal polymer polarizers,” Opt. Express 18(26), 27777–27786 (2010).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. A Gen. Phys. (1)

S. T. Wu, “Birefringence dispersions of liquid crystals,” Phys. Rev. A Gen. Phys. 33(2), 1270–1274 (1986).
[Crossref] [PubMed]

Proc. Indiana Acad. Sci. (1)

S. Pancharatnam, “Achromatic combination of birefringent plates,” Proc. Indiana Acad. Sci. XLI(4), 137 (1955).

Other (3)

R. Chipman, “Polarimetry,” in OSA Handbook of Optics (McGraw-Hill, 2010).

K. Adlem, O. L. Parri, D. Wilkes, and C. Topping, “Polymer comprising cyclohexylene groups and its use in films with negative optical dispersion,” United States patent US8802813 B2.

M. Garcia, S. Gao, C. Edmiston, T. York, and V. Gruev, “A 1300 × 800, 700 mW, 30 fps spectral polarization imager,” in Circuits and Systems (ISCAS)2015IEEE International Symposium (IEEE 2015), 1106–1109.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 The schematics of an achromatic elliptical polarizer are shown with (a) a linear polarizer and with (b) a circular polarizer.
Fig. 2
Fig. 2 The schematics show the trajectory of the output state on the Poincaré sphere, when the input state passes through a linear retarder with fixed fast-axis orientation but varying retardances from 0 to 360 degrees. Each trajectory represents a different fast-axis orientation T. The input state is (a) s = [1, 0, 0] and (b) s = [ 2/3 , 0, 1/3 ].
Fig. 3
Fig. 3 The geometry of overlapping arcs on a Poincaré sphere is shown for (a) an input state of linear polarized light and (b) an input state of circular polarized light
Fig. 4
Fig. 4 (a) The regular tetrahedron adopted for a broadband full-Stokes camera is shown along with the Poincaré sphere. The coordinates of vertices 1, 2, 3, 4 are [ 2/3 , 0, 1/3 ], [ 2/3 , 0, 1/3 ], [0, 2/3 , 1/3 ], [0, 2/3 , 1/3 ] respectively. (b) A design of a DoFP broadband full-Stokes camera shows the array of four elliptical polarizers. The macro pixel of each layer is displayed on right.
Fig. 5
Fig. 5 (a) The retardance curve of liquid crystal polymer RMM141C is plotted as a function of wavelength. (b) The plot shows the deviation curve of the optimized achromatic elliptical polarizer made of RMM141C. (c) The curve of the optimized achromatic elliptical polarizer output state on the Poincaré sphere is shown for wavelength range from 450nm to 650nm; (d) The magnified view of (c) is shown. The green dot is the target state.
Fig. 6
Fig. 6 The schematic shows the fabrication process of an achromatic elliptical polarizer.
Fig. 7
Fig. 7 (a) A comparison of the deviation curve of the fabricated sample and theoretical prediction is shown. The bar represents the measurement error of the Axometrics polarimeter. (b) The trajectory of the sample’s output state is shown on the Poincaré sphere for wavelength range from 450nm to 650nm. (c) The magnified view of the trajectory in (b). The blue curve is the trajectory of theoretical prediction. The green dot is the target state.
Fig. 8
Fig. 8 The deviation curve of optimized 2, 3, 4 retarder layer configurations for wavelength range of (a) λ = 450nm to 650nm and (b) λ = 400nm to 1μm.

Equations (25)

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s(λ)- s target m =max( ( s 1 (λ)-cos(2ε)) 2 + s 2 (λ) 2 + ( s 3 (λ)-sin(2ε)) 2 ) 450nmλ650nm
P (t)=rcos(t)* u +rsin(t)*( n × u )+ C
n A =[ cos(2 θ 1 ),sin(2 θ 1 ),0 ], u A =[ sin(2 θ 1 ),cos(2 θ 1 ),0 ] C C =[ m 1 *cos(2 θ 1 ), m 1 *sin(2 θ 1 ),0 ], r C = 1 m 1 2 , 180-δ< t A <180+δ n B =[ cos(2 θ 2 ),sin(2 θ 2 ),0 ], u B =[ sin(2 θ 2 ),cos(2 θ 2 ),0 ] C B =[ m 2 *cos(2 θ 2 ), m 2 *sin(2 θ 2 ),0 ], r B = 1 m 2 2 , 180- t 2 δ< t B <180+ t 2 δ
θ LP = θ 1 + cos 1 ( m 1 )/2
P A (180)=[ 1- m 1 2 sin(2 θ 1 )+ m 1 cos(2 θ 1 ),- 1- m 1 2 cos(2 θ 1 )+ m 1 sin(2 θ 1 ),0 ] P B (180)=[ 1- m 2 2 sin(2 θ 2 )+ m 2 cos(2 θ 2 ),- 1- m 2 2 cos(2 θ 2 )+ m 2 sin(2 θ 2 ),0 ]
P A (180-δ)=[ 1- m 1 2 cos(δ)sin(2 θ 1 )+ m 1 cos(2 θ 1 ), - 1- m 1 2 cos(δ)cos(2 θ 1 )+ m 1 sin(2 θ 1 ), 1- m 1 2 *sin(δ)] P B (180- t 2 δ)= 1- m 2 2 cos( t 2 δ)sin(2 θ 2 )+ m 2 cos(2 θ 2 ), - 1- m 2 2 cos( t 2 δ)cos(2 θ 2 )+ m 2 sin(2 θ 2 ), 1- m 2 2 sin( t 2 δ)]
P A (180) P B (180)=( 1- m 1 2 1- m 2 2 + m 1 m 2 )*cos[2( θ 1 - θ 2 )] +( 1- m 1 2 m 2 - 1- m 2 2 m 1 )*sin[2( θ 1 - θ 2 )] P A (180-δ) P B (180- t 2 δ)=( 1- m 1 2 1- m 2 2 cos(δ)cos( t 2 δ)+ m 1 m 2 )* cos[2( θ 1 - θ 2 )]+ 1- m 1 2 1- m 2 2 sin(δ)sin( t 2 δ) +( 1- m 1 2 m 2 cos(δ)- 1- m 2 2 m 1 cos( t 2 δ))* sin[2( θ 1 - θ 2 )]
P A (180-δ) P B (180- t 2 δ)= P A (180) P B (180)
1cos(δ)cos( t 2 δ) tan[2( θ 1 θ 2 )] sin(δ)sin( t 2 δ) sin[2( θ 1 θ 2 )] = m 1 (1cos( t 2 δ)) 1 m 1 2 m 2 (1cos(δ)) 1 m 2 2
m 2 =cos(2ε)*cos(2 θ 2 ) t 2 = cot 1 (cot(2ε)*sin(2 θ 2 ))/180
P A (180) P B (180)=G( θ 1 , θ 2 )
n C =[ cos(2 θ 1 ),sin(2 θ 1 ),0 ], u C =[ 0,0,1 ] C C =[ 0,0,0 ], r C =1, 90-δ/2< t C <90+δ/2
P C (90)=[ sin(2 θ 1 ),-cos(2 θ 1 ),0 ]
P C (90-δ/2)=[cos(δ/2)sin(2 θ 1 ),-cos(δ/2)cos(2 θ 1 ),sin(δ/2)]
P C (90) P B (180)= 1- m 2 2 cos[2( θ 1 - θ 2 )]+ m 2 sin[2( θ 1 - θ 2 )] P C (90-δ/2) P B (180- t 2 δ)= 1- m 2 2 cos(δ/2)cos( t 2 δ)cos[2( θ 1 - θ 2 )] + 1- m 2 2 sin(δ/2)sin( t 2 δ) + m 2 cos(δ/2)sin[2( θ 1 - θ 2 )]
m 2 1 m 2 2 = (1cos δ 2 cos( t 2 δ))cos[2( θ 1 θ 2 )]sin δ 2 sin( t 2 δ) (cos δ 2 1)sin[2( θ 1 θ 2 )]
a( θ 2 )cos[2Δθ]+b( θ 2 )sin[2Δθ]=c( θ 2 )
a( θ 2 )=1cos δ 2 cos( t 2 δ) b( θ 2 )= m 2 1 m 2 2 (1cos δ 2 ) c( θ 2 )=sin δ 2 sin( t 2 δ)
Δθ= sin 1 ( c a 2 + b 2 )ϕ 2 [sinϕ,cosϕ]=[ a a 2 + b 2 , b a 2 + b 2 ]
P A (180) P B (180)=H( θ 2 )
[ I 0 I 1 I 2 I 3 ]=W[ S 0 S 1 S 2 S 3 ]
M analyzer = M LP ( θ LP ) M LR ( δ N , θ N ) M LR ( δ 2 , θ 2 ) M LR ( δ 1 , θ 1 )
M LR ( δ, θ ) = [ 1 0 0 0 0 cos 2 (2θ)+ cos(δ) sin 2 (2θ) sin(2θ)cos(2θ) (1-cos(δ)) sin(2θ)sin(δ) 0 sin(2θ)cos(2θ) (1-cos(δ)) cos(δ) cos 2 (2θ)+ sin 2 (2θ) -cos(2θ)sin(δ) 0 -sin(2θ)sin(δ) cos(2θ)sin(δ) cos(δ) ] M LP ( θ ) = 1 2 [ 1 cos(2θ) sin(2θ) 0 cos(2θ) cos 2 (2θ) sin(2θ)cos(2θ) 0 sin(2θ) sin(2θ)cos(2θ) sin 2 (2θ) 0 0 0 0 0 ]
M analyzer T = M LR T ( δ 1 , θ 1 ) M LR T ( δ 2 , θ 2 ) M LR T ( δ N , θ N ) M LP T ( θ LP ) = M LR ( δ 1 , θ 1 +π/2) M LR ( δ 2 , θ 2 +π/2) M LR ( δ N , θ N +π/2) M LP ( θ LP )
Δϕ(t,λ)= 2πt λ Δn=Ct λ λ * 2 λ 2 λ * 2

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