Abstract

We present a dynamic polarization grating based on a dye-doped liquid crystal cell that is controllable by a single pump beam in a binary homeotropic-planar configuration produced through selective rubbing. Upon single pump beam irradiation, the azo dyes in the liquid-crystal (LC) layer diffuse and adsorb onto the planar LC-anchoring surface due to trans-cis photo-isomerization. It is found that the dynamic polarization grating effect results mainly from the photo-induced easy axis reorientation by the amount of dye molecules adsorbed on the planar LC-alignment surface in a single-beam control scheme. The initial LC-anchoring conditions and the dynamic behavior of the dye adsorption strongly influence the repetitive writing-erasing processes by the single pump beam.

© 2012 Optical Society of America

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    [CrossRef]

2012

2011

A. G. Iljin, “Transient modulation of order parameter and optical nonlinearity in a chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 543, 143–150 (2011).
[CrossRef]

2009

2008

2007

E. Jang, H.-R. Kim, Y.-J. Na, and S.-D. Lee, “Multistage optical memory of a liquid crystal diffraction grating in a single beam rewriting scheme,” Appl. Phys. Lett. 91, 071109 (2007).
[CrossRef]

2006

V. Presnyakov, K. Asatryan, T. Galstian, and V. Chigrinov, “Optical polarization grating induced liquid crystal micro-structure using azo-dye command layer,” Opt. Express 14, 10558–10564 (2006).
[CrossRef]

E. Jang, H. Baac, Y.-T. Kim, and S.-D. Lee, “Electrically and optically controllable liquid crystal grating with a patterned surface-command layer,” Mol. Cryst. Liq. Cryst. 453, 293–300 (2006).
[CrossRef]

2005

2004

2003

J.-H. Park, C.-J. Yu, J. Kim, S.-Y. Chung, and S.-D. Lee, “Concept of a liquid-crystal polarization beamsplitter based on binary phase gratings,” Appl. Phys. Lett. 83, 1918–1920 (2003).
[CrossRef]

2002

C. V. Brown, E. E. Kriezis, and S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

B. J. Kim, S.-D. Lee, S. Y. Park, and D. H. Choi, “Unusual characteristics of diffraction gratings in a liquid crystal cell,” Adv. Mater. 14, 983–988 (2002).

2001

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

P. Pagliusi, R. Macdonald, S. Busch, G. Cipparrone, and M. Kreuzer, “Nonlocal dynamic gratings and energy transfer by optical two-beam coupling in a nematic liquid crystal owing to highly sensitive photoelectric reorientation,” J. Opt. Soc. Am. B 18, 1632–1638 (2001).
[CrossRef]

2000

G. Barbero and V. Popa-Nita, “Model for the planar-homeotropic anchoring transition induced by trans-cis isomerization,” Phys. Rev. E 61, 6696–6698 (2000).
[CrossRef]

1999

O. Francescangeli, S. Slussarenko, F. Simoni, D. Andrienko, V. Reshetnyak, and Y. Reznikov, “Light-induced surface sliding of the nematic director in liquid crystals,” Phys. Rev. Lett. 82, 1855–1858 (1999).
[CrossRef]

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
[CrossRef]

1998

M. L. Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321–330 (1998).
[CrossRef]

T. V. Galstian, B. Saad, and M.-M. Denariez-Roberge, “Asymmetric resonant optical torque in azo-dye-doped liquid crystals,” IEEE J. Quantum Electron. 34, 790–794 (1998).
[CrossRef]

1997

C. M. Titus and P. J. Bos, “Efficient, polarization-independent, reflective liquid crystal phase grating,” Appl. Phys. Lett. 71, 2239–2241 (1997).
[CrossRef]

S. Slussarenko, O. Francescangeli, F. Simoni, and Y. Reznikov, “High resolution polarization gratings in liquid crystals,” Appl. Phys. Lett. 71, 3613–3615 (1997).
[CrossRef]

1995

J. Chen, P. J. Bos, H. Vithana, and D. L. Johnson, “An electrooptically controlled liquid crystal diffraction grating,” Appl. Phys. Lett. 67, 2588–2590 (1995).
[CrossRef]

S. Masuda, T. Nose, R. Yamaguchi, and S. Sato, “A radial molecular orientation using a flow-induced aligning method in a nematic liquid crystal cell,” Jpn. J. Appl. Phys. 34, 4129–4132 (1995).
[CrossRef]

1992

Y. Ouchi, M. B. Feller, T. Moses, and Y. R. Shen, “Surface memory effect at the liquid-crystal-polymer interface,” Phys. Rev. Lett. 68, 3040–3043 (1992).
[CrossRef]

A. G.-S. Chen and D. J. Brady, “Surface-stabilized holography in an azo-dye-doped liquid crystal,” Opt. Lett. 17, 1231–1233 (1992).
[CrossRef]

Andrienko, D.

O. Francescangeli, S. Slussarenko, F. Simoni, D. Andrienko, V. Reshetnyak, and Y. Reznikov, “Light-induced surface sliding of the nematic director in liquid crystals,” Phys. Rev. Lett. 82, 1855–1858 (1999).
[CrossRef]

Asatryan, K.

Baac, H.

E. Jang, H. Baac, Y.-T. Kim, and S.-D. Lee, “Electrically and optically controllable liquid crystal grating with a patterned surface-command layer,” Mol. Cryst. Liq. Cryst. 453, 293–300 (2006).
[CrossRef]

Barbero, G.

G. Barbero and V. Popa-Nita, “Model for the planar-homeotropic anchoring transition induced by trans-cis isomerization,” Phys. Rev. E 61, 6696–6698 (2000).
[CrossRef]

Bortolozzo, U.

Bos, P. J.

C. M. Titus and P. J. Bos, “Efficient, polarization-independent, reflective liquid crystal phase grating,” Appl. Phys. Lett. 71, 2239–2241 (1997).
[CrossRef]

J. Chen, P. J. Bos, H. Vithana, and D. L. Johnson, “An electrooptically controlled liquid crystal diffraction grating,” Appl. Phys. Lett. 67, 2588–2590 (1995).
[CrossRef]

Brady, D. J.

Brown, C. V.

C. V. Brown, E. E. Kriezis, and S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Busch, S.

Cai, Z.

Caputo, R.

Chang, Y.-M.

Chen, A. G.-S.

Chen, H.-Y.

Chen, J.

J. Chen, P. J. Bos, H. Vithana, and D. L. Johnson, “An electrooptically controlled liquid crystal diffraction grating,” Appl. Phys. Lett. 67, 2588–2590 (1995).
[CrossRef]

Cheng, K.-T.

C.-R. Lee, T.-L. Fu, K.-T. Cheng, T.-S. Mo, and A. Y.-G. Fuh, “Surface-assisted photoalignment in dye-doped liquid-crystal films,” Phys. Rev. E 69, 031704 (2004).
[CrossRef]

Chigrinov, V.

Chin, W.-K.

Choi, D. H.

B. J. Kim, S.-D. Lee, S. Y. Park, and D. H. Choi, “Unusual characteristics of diffraction gratings in a liquid crystal cell,” Adv. Mater. 14, 983–988 (2002).

Chung, S.-Y.

J.-H. Park, C.-J. Yu, J. Kim, S.-Y. Chung, and S.-D. Lee, “Concept of a liquid-crystal polarization beamsplitter based on binary phase gratings,” Appl. Phys. Lett. 83, 1918–1920 (2003).
[CrossRef]

Cipparrone, G.

Davis, J. A.

Denariez-Roberge, M.-M.

T. V. Galstian, B. Saad, and M.-M. Denariez-Roberge, “Asymmetric resonant optical torque in azo-dye-doped liquid crystals,” IEEE J. Quantum Electron. 34, 790–794 (1998).
[CrossRef]

Doucen, M. L.

M. L. Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321–330 (1998).
[CrossRef]

Elston, S. J.

C. V. Brown, E. E. Kriezis, and S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Evans, G. H.

Fedorenko, D.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

Feller, M. B.

Y. Ouchi, M. B. Feller, T. Moses, and Y. R. Shen, “Surface memory effect at the liquid-crystal-polymer interface,” Phys. Rev. Lett. 68, 3040–3043 (1992).
[CrossRef]

Francescangeli, O.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

O. Francescangeli, S. Slussarenko, F. Simoni, D. Andrienko, V. Reshetnyak, and Y. Reznikov, “Light-induced surface sliding of the nematic director in liquid crystals,” Phys. Rev. Lett. 82, 1855–1858 (1999).
[CrossRef]

S. Slussarenko, O. Francescangeli, F. Simoni, and Y. Reznikov, “High resolution polarization gratings in liquid crystals,” Appl. Phys. Lett. 71, 3613–3615 (1997).
[CrossRef]

Fu, T.-L.

C.-R. Lee, T.-L. Fu, K.-T. Cheng, T.-S. Mo, and A. Y.-G. Fuh, “Surface-assisted photoalignment in dye-doped liquid-crystal films,” Phys. Rev. E 69, 031704 (2004).
[CrossRef]

Fuh, A. Y.-G.

J.-R. Wang, C.-R. Lee, M.-R. Lee, and A. Y.-G. Fuh, “Photorefractive effect induced by polarization gratings in dye-doped liquid crystals,” Opt. Lett. 29, 110–112 (2004).
[CrossRef]

C.-R. Lee, T.-L. Fu, K.-T. Cheng, T.-S. Mo, and A. Y.-G. Fuh, “Surface-assisted photoalignment in dye-doped liquid-crystal films,” Phys. Rev. E 69, 031704 (2004).
[CrossRef]

Galstian, T.

Galstian, T. V.

T. V. Galstian, B. Saad, and M.-M. Denariez-Roberge, “Asymmetric resonant optical torque in azo-dye-doped liquid crystals,” IEEE J. Quantum Electron. 34, 790–794 (1998).
[CrossRef]

Gori, F.

Huang, Y.-P.

Iljin, A.

Iljin, A. G.

A. G. Iljin, “Transient modulation of order parameter and optical nonlinearity in a chiral nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 543, 143–150 (2011).
[CrossRef]

Israelachvili, J. N.

J. N. Israelachvili, Intermolecular and Surface Forces(Academic, 1997).

Jang, E.

E. Jang, H.-R. Kim, Y.-J. Na, and S.-D. Lee, “Multistage optical memory of a liquid crystal diffraction grating in a single beam rewriting scheme,” Appl. Phys. Lett. 91, 071109 (2007).
[CrossRef]

E. Jang, H. Baac, Y.-T. Kim, and S.-D. Lee, “Electrically and optically controllable liquid crystal grating with a patterned surface-command layer,” Mol. Cryst. Liq. Cryst. 453, 293–300 (2006).
[CrossRef]

Johnson, D. L.

J. Chen, P. J. Bos, H. Vithana, and D. L. Johnson, “An electrooptically controlled liquid crystal diffraction grating,” Appl. Phys. Lett. 67, 2588–2590 (1995).
[CrossRef]

Jung, M.-S.

Khoo, I.-C.

Kim, B. J.

B. J. Kim, S.-D. Lee, S. Y. Park, and D. H. Choi, “Unusual characteristics of diffraction gratings in a liquid crystal cell,” Adv. Mater. 14, 983–988 (2002).

Kim, H.-R.

E. Jang, H.-R. Kim, Y.-J. Na, and S.-D. Lee, “Multistage optical memory of a liquid crystal diffraction grating in a single beam rewriting scheme,” Appl. Phys. Lett. 91, 071109 (2007).
[CrossRef]

Kim, J.

C.-J. Yu, J.-H. Park, J. Kim, M.-S. Jung, and S.-D. Lee, “Design of binary diffraction gratings of liquid crystals in a linearly graded phase model,” Appl. Opt. 43, 1783–1788 (2004).
[CrossRef]

J.-H. Park, C.-J. Yu, J. Kim, S.-Y. Chung, and S.-D. Lee, “Concept of a liquid-crystal polarization beamsplitter based on binary phase gratings,” Appl. Phys. Lett. 83, 1918–1920 (2003).
[CrossRef]

Kim, Y.-T.

E. Jang, H. Baac, Y.-T. Kim, and S.-D. Lee, “Electrically and optically controllable liquid crystal grating with a patterned surface-command layer,” Mol. Cryst. Liq. Cryst. 453, 293–300 (2006).
[CrossRef]

Kreuzer, M.

Kriezis, E. E.

C. V. Brown, E. E. Kriezis, and S. J. Elston, “Optical diffraction from a liquid crystal phase grating,” J. Appl. Phys. 91, 3495–3500 (2002).
[CrossRef]

Kuksenok, O. V.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

Lee, C.-R.

C.-R. Lee, T.-L. Fu, K.-T. Cheng, T.-S. Mo, and A. Y.-G. Fuh, “Surface-assisted photoalignment in dye-doped liquid-crystal films,” Phys. Rev. E 69, 031704 (2004).
[CrossRef]

J.-R. Wang, C.-R. Lee, M.-R. Lee, and A. Y.-G. Fuh, “Photorefractive effect induced by polarization gratings in dye-doped liquid crystals,” Opt. Lett. 29, 110–112 (2004).
[CrossRef]

Lee, M.-R.

Lee, S.-D.

E. Jang, H.-R. Kim, Y.-J. Na, and S.-D. Lee, “Multistage optical memory of a liquid crystal diffraction grating in a single beam rewriting scheme,” Appl. Phys. Lett. 91, 071109 (2007).
[CrossRef]

E. Jang, H. Baac, Y.-T. Kim, and S.-D. Lee, “Electrically and optically controllable liquid crystal grating with a patterned surface-command layer,” Mol. Cryst. Liq. Cryst. 453, 293–300 (2006).
[CrossRef]

C.-J. Yu, J.-H. Park, J. Kim, M.-S. Jung, and S.-D. Lee, “Design of binary diffraction gratings of liquid crystals in a linearly graded phase model,” Appl. Opt. 43, 1783–1788 (2004).
[CrossRef]

J.-H. Park, C.-J. Yu, J. Kim, S.-Y. Chung, and S.-D. Lee, “Concept of a liquid-crystal polarization beamsplitter based on binary phase gratings,” Appl. Phys. Lett. 83, 1918–1920 (2003).
[CrossRef]

B. J. Kim, S.-D. Lee, S. Y. Park, and D. H. Choi, “Unusual characteristics of diffraction gratings in a liquid crystal cell,” Adv. Mater. 14, 983–988 (2002).

Lee, W.

Liou, J. D.

Macdonald, R.

Masuda, S.

S. Masuda, T. Nose, R. Yamaguchi, and S. Sato, “A radial molecular orientation using a flow-induced aligning method in a nematic liquid crystal cell,” Jpn. J. Appl. Phys. 34, 4129–4132 (1995).
[CrossRef]

Mo, T.-S.

C.-R. Lee, T.-L. Fu, K.-T. Cheng, T.-S. Mo, and A. Y.-G. Fuh, “Surface-assisted photoalignment in dye-doped liquid-crystal films,” Phys. Rev. E 69, 031704 (2004).
[CrossRef]

Moses, T.

Y. Ouchi, M. B. Feller, T. Moses, and Y. R. Shen, “Surface memory effect at the liquid-crystal-polymer interface,” Phys. Rev. Lett. 68, 3040–3043 (1992).
[CrossRef]

Na, Y.-J.

E. Jang, H.-R. Kim, Y.-J. Na, and S.-D. Lee, “Multistage optical memory of a liquid crystal diffraction grating in a single beam rewriting scheme,” Appl. Phys. Lett. 91, 071109 (2007).
[CrossRef]

Nose, T.

S. Masuda, T. Nose, R. Yamaguchi, and S. Sato, “A radial molecular orientation using a flow-induced aligning method in a nematic liquid crystal cell,” Jpn. J. Appl. Phys. 34, 4129–4132 (1995).
[CrossRef]

Ouchi, Y.

Y. Ouchi, M. B. Feller, T. Moses, and Y. R. Shen, “Surface memory effect at the liquid-crystal-polymer interface,” Phys. Rev. Lett. 68, 3040–3043 (1992).
[CrossRef]

Ouskova, E.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

Pagliusi, P.

Park, J.-H.

Park, S. Y.

B. J. Kim, S.-D. Lee, S. Y. Park, and D. H. Choi, “Unusual characteristics of diffraction gratings in a liquid crystal cell,” Adv. Mater. 14, 983–988 (2002).

Pei, Y.

Pellat-Finet, P.

M. L. Doucen and P. Pellat-Finet, “Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals,” Opt. Commun. 151, 321–330 (1998).
[CrossRef]

Popa-Nita, V.

G. Barbero and V. Popa-Nita, “Model for the planar-homeotropic anchoring transition induced by trans-cis isomerization,” Phys. Rev. E 61, 6696–6698 (2000).
[CrossRef]

Presnyakov, V.

Reshetnyak, V.

O. Francescangeli, S. Slussarenko, F. Simoni, D. Andrienko, V. Reshetnyak, and Y. Reznikov, “Light-induced surface sliding of the nematic director in liquid crystals,” Phys. Rev. Lett. 82, 1855–1858 (1999).
[CrossRef]

Residori, S.

Reznikov, Y.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

O. Francescangeli, S. Slussarenko, F. Simoni, D. Andrienko, V. Reshetnyak, and Y. Reznikov, “Light-induced surface sliding of the nematic director in liquid crystals,” Phys. Rev. Lett. 82, 1855–1858 (1999).
[CrossRef]

S. Slussarenko, O. Francescangeli, F. Simoni, and Y. Reznikov, “High resolution polarization gratings in liquid crystals,” Appl. Phys. Lett. 71, 3613–3615 (1997).
[CrossRef]

Saad, B.

T. V. Galstian, B. Saad, and M.-M. Denariez-Roberge, “Asymmetric resonant optical torque in azo-dye-doped liquid crystals,” IEEE J. Quantum Electron. 34, 790–794 (1998).
[CrossRef]

Sato, S.

S. Masuda, T. Nose, R. Yamaguchi, and S. Sato, “A radial molecular orientation using a flow-induced aligning method in a nematic liquid crystal cell,” Jpn. J. Appl. Phys. 34, 4129–4132 (1995).
[CrossRef]

Shen, Y. R.

Y. Ouchi, M. B. Feller, T. Moses, and Y. R. Shen, “Surface memory effect at the liquid-crystal-polymer interface,” Phys. Rev. Lett. 68, 3040–3043 (1992).
[CrossRef]

Shiyanovskii, S. V.

E. Ouskova, D. Fedorenko, Y. Reznikov, S. V. Shiyanovskii, L. Su, J. L. West, O. V. Kuksenok, O. Francescangeli, and F. Simoni, “Hidden photoalignment of liquid crystals in the isotropic phase,” Phys. Rev. E 63, 021701 (2001).
[CrossRef]

Simoni, F.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Cell structure of the DDLC and the experimental conditions used for generating and measuring the dynamic polarization grating effect with a single pump beam. (b) Schematic diagram illustrating the selective rubbing on the planar LC-anchoring surface with a shadow mask and a microscopic image of the shadow mask used in our experiment. (c) Schematic diagram illustrating the LC-axis conditions of the DDLC grating and the polarization-state conditions used in our experiments : γI and γII are the LC-axis angles rotated after the pump beam irradiation in the nonrubbed region (I) and the rubbed region (II), respectively. L is the grating period and α is the length ratio of the nonrubbed region to one period of the grating. R⃗ denotes the rubbing direction. θp, θin, and θ±1 are the polarization states of the pump beam, the probe beam, and the diffracted beam, respectively.

Fig. 2.
Fig. 2.

Polarizing optical microscopy images of the periodically reoriented DDLC cells observed after irradiation with the 45° [for (a) and (b)], 60° [for (c) and (d)], and 90° [for (e) and (f)] polarized single pump beam: (a) and (c) are obtained when the rubbing direction in region II is parallel to the polarizer, and (b), (d)–(f) are obtained by rotating the DDLC cells by an angle of 19°, 40°, 13°, and 62°, respectively, to the polarizer.

Fig. 3.
Fig. 3.

Polarization characteristics of the first-order diffracted beam from the DDLC cells measured after pump beam irradiation with θp=45°, 60°, and 90° conditions: (a) transmitted intensity of the first-order diffracted beam after a linear polarizer measured by varying the transmission axis of the polarizer at the fixed probe beam incident polarization condition (θin=90°) and (b) polarization (θ±1=π/2+Sθin) of the first-order diffracted beam measured under different θin conditions.

Fig. 4.
Fig. 4.

First-order diffraction efficiency measured by varying the applied voltage under different θin conditions. Three polarization gratings were prepared by irradiating the pump beam in three different polarization conditions: θp=45°, 60°, and 90°.

Fig. 5.
Fig. 5.

Polarization states of the first-order diffraction from our polarization gratings measured at different applied voltage conditions. Three polarization gratings were prepared by irradiating the pump beam in three different polarization conditions: θp=45°, 60°, and 90°.

Fig. 6.
Fig. 6.

First-order diffraction efficiency observed by varying the polarization state of the pump beam (θp) under the fixed polarization condition of the probe beam (θin=90°). The pump beam irradiation and relaxation times were both 100 s.

Equations (14)

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E±n=sin(nαπ)nπexp(jnαπ)(e^Ie^II),
e^I=R(γI)[exp(+iψ/2)00exp(iψ/2)]R(γI)[ab]
e^II=R(γII)[exp(+iψ/2)00exp(iψ/2)]R(γII)[ab],
R(ϕ)=[cosϕsinϕsinϕcosϕ].
e^Ie^II=isinψ2[a[cos(2γI)cos(2γII)]+b[sin(2γI)sin(2γII)]b[cos(2γI)cos(2γII)]+a[sin(2γI)sin(2γII)]].
e^Ie^II=2isinψ2sinD[sinScosScosSsinS][ab],
|asinS+bcosS|2+|acosS+bsinS|2=1.
E±n=2isin(nαπ)nπexp(jnαπ)sinψ2sinDe^±n,
e^±n=[sinScosScosSsinS][ab].
η±n=4sin2(nαπ)n2π2sin2ψ2sin2D.
Ei=R(S2)Ei=[acos(S/2)+bsin(S/2)asin(S/2)+bcos(S/2)],
e^±n=R(S2)e^±n=[asin(S/2)+bcos(S/2)acos(S/2)+bsin(S/2)].
θin+θ±n2=π4+S2,
θ±n=π2+Sθin.

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