Abstract

We propose a liquid crystal (LC) polarization grating that conserves the polarization state of incident light, wherein the variation range of the twist angle is 2π. The design scheme for theoretically 100% diffraction efficiency of the first-diffraction order is derived, and a prototype LC grating is evaluated. Under zero voltage, the fabricated LC grating exhibits high efficiency of the first-order diffraction, validating the proposed design scheme. The high efficiency of the second-order diffraction can also be achieved under a high voltage so that the LC director in the midplane is vertical to the substrate plane. The circular polarization sense of the second-order diffraction is identical to that of the incident light as in the case of the first-order diffraction. This grating functions as a beam deflector, steering the input beam in three different directions (zeroth-, first-, and second-order diffractions) by adjusting the applied voltage.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2012

2009

2008

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett.33(20), 2287–2289 (2008).
[CrossRef] [PubMed]

L. Shi, P. F. McManamon, and P. J. Bos, “Liquid crystal optical phase plate with a variable in-plane gradient,” J. Appl. Phys.104(3), 033109 (2008).
[CrossRef]

2007

J.-C. Chao, W.-Y. Wu, and A. Y.-G. Fuh, “Diffraction characteristics of a liquid crystal polarization grating analyzed using the finite-difference time-domain method,” Opt. Express15(25), 16702–16711 (2007).
[CrossRef] [PubMed]

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(2), 021701 (2007).
[CrossRef] [PubMed]

2006

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

V. Presnyakov, K. Asatryan, T. Galstian, and V. Chigrinov, “Optical polarization grating induced liquid crystal micro-structure using azo-dye command layer,” Opt. Express14(22), 10558–10564 (2006).
[CrossRef] [PubMed]

C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett.89(12), 121105 (2006).
[CrossRef]

H. Sarkissian, S. V. Serak, N. V. Tabiryan, L. B. Glebov, V. Rotar, and B. Y. Zeldovich, “Polarization-controlled switching between diffraction orders in transverse-periodically aligned nematic liquid crystals,” Opt. Lett.31(15), 2248–2250 (2006).
[CrossRef] [PubMed]

S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: potential application for projection displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)451(1), 1–19 (2006).
[CrossRef]

2005

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

2004

M. Honma and T. Nose, “Liquid-crystal blazed grating with azimuthally distributed liquid-crystal directors,” Appl. Opt.43(27), 5193–5197 (2004).
[CrossRef] [PubMed]

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

2003

M. Honma and T. Nose, “Polarization-independent liquid crystal grating fabricated by microrubbing process,” Jpn. J. Appl. Phys.42(Part 1, No. 11), 6992–6997 (2003).
[CrossRef]

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys.94(3), 1298–1303 (2003).
[CrossRef]

2002

1999

1998

1990

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys.67(6), 2853–2856 (1990).
[CrossRef]

Arbuzov, A. V.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

Asatryan, K.

Barnik, M. I.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

Bastiaansen, C.

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

Bastiaansen, C. W. M.

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

Blinov, L. M.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

G. Cipparrone, A. Mazzulla, and L. M. Blinov, “Permanent polarization gratings in photosensitive Langmuir-Blodgett films for polarimetric applications,” J. Opt. Soc. Am. B19(5), 1157–1161 (2002).
[CrossRef]

Bos, P. J.

L. Shi, P. F. McManamon, and P. J. Bos, “Liquid crystal optical phase plate with a variable in-plane gradient,” J. Appl. Phys.104(3), 033109 (2008).
[CrossRef]

Broer, D. J.

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

Broer, D.J.

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

Callan-Jones, A.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

Chao, J.-C.

Chen, Y. S.

S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

Chigrinov, V.

Cipparrone, G.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett.89(12), 121105 (2006).
[CrossRef]

G. Cipparrone, A. Mazzulla, and L. M. Blinov, “Permanent polarization gratings in photosensitive Langmuir-Blodgett films for polarimetric applications,” J. Opt. Soc. Am. B19(5), 1157–1161 (2002).
[CrossRef]

Crawford, G. P.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

Davis, J. A.

de Boer, D. K. G.

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

Eakin, J. N.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

Emoto, A.

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys.94(3), 1298–1303 (2003).
[CrossRef]

Escuti, M. J.

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett.33(20), 2287–2289 (2008).
[CrossRef] [PubMed]

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(2), 021701 (2007).
[CrossRef] [PubMed]

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

Fuh, A. Y.-G.

J.-C. Chao, W.-Y. Wu, and A. Y.-G. Fuh, “Diffraction characteristics of a liquid crystal polarization grating analyzed using the finite-difference time-domain method,” Opt. Express15(25), 16702–16711 (2007).
[CrossRef] [PubMed]

S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

Galstian, T.

Glebov, L. B.

Gori, F.

Guo, J. H.

S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

Hasegawa, T.

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys.94(3), 1298–1303 (2003).
[CrossRef]

Hoke, L.

Honma, M.

M. Honma and T. Nose, “Temperature-independent achromatic liquid-crystal grating with spatially distributed twisted-nematic orientation,” Appl. Phys. Express5(6), 062501 (2012).
[CrossRef]

M. Honma and T. Nose, “Liquid-crystal blazed grating with azimuthally distributed liquid-crystal directors,” Appl. Opt.43(27), 5193–5197 (2004).
[CrossRef] [PubMed]

M. Honma and T. Nose, “Polarization-independent liquid crystal grating fabricated by microrubbing process,” Jpn. J. Appl. Phys.42(Part 1, No. 11), 6992–6997 (2003).
[CrossRef]

Kawatsuki, N.

M. Kuzuwata, T. Sasaki, N. Kawatsuki, and H. Ono, “Fabrication of twisted nematic structure and vector grating cells by one-step exposure on photocrosslinkable polymer liquid crystals,” Opt. Lett.37(6), 1115–1117 (2012).
[CrossRef] [PubMed]

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys.94(3), 1298–1303 (2003).
[CrossRef]

Kim, J.

Kimball, B. R.

Komanduri, R. K.

R. K. Komanduri and M. J. Escuti, “Elastic continuum analysis of the liquid crystal polarization grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.76(2), 021701 (2007).
[CrossRef] [PubMed]

Kuzuwata, M.

Lee, S.-D.

Lien, A.

A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys.67(6), 2853–2856 (1990).
[CrossRef]

Mazzulla, A.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

G. Cipparrone, A. Mazzulla, and L. M. Blinov, “Permanent polarization gratings in photosensitive Langmuir-Blodgett films for polarimetric applications,” J. Opt. Soc. Am. B19(5), 1157–1161 (2002).
[CrossRef]

McManamon, P. F.

L. Shi, P. F. McManamon, and P. J. Bos, “Liquid crystal optical phase plate with a variable in-plane gradient,” J. Appl. Phys.104(3), 033109 (2008).
[CrossRef]

Moreno, I.

Na, J. H.

Nersisyan, S. R.

Nose, T.

M. Honma and T. Nose, “Temperature-independent achromatic liquid-crystal grating with spatially distributed twisted-nematic orientation,” Appl. Phys. Express5(6), 062501 (2012).
[CrossRef]

M. Honma and T. Nose, “Liquid-crystal blazed grating with azimuthally distributed liquid-crystal directors,” Appl. Opt.43(27), 5193–5197 (2004).
[CrossRef] [PubMed]

M. Honma and T. Nose, “Polarization-independent liquid crystal grating fabricated by microrubbing process,” Jpn. J. Appl. Phys.42(Part 1, No. 11), 6992–6997 (2003).
[CrossRef]

Oh, C.

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett.33(20), 2287–2289 (2008).
[CrossRef] [PubMed]

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

Ono, H.

M. Kuzuwata, T. Sasaki, N. Kawatsuki, and H. Ono, “Fabrication of twisted nematic structure and vector grating cells by one-step exposure on photocrosslinkable polymer liquid crystals,” Opt. Lett.37(6), 1115–1117 (2012).
[CrossRef] [PubMed]

H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa, “Highly stable polarization gratings in photocrosslinkable polymer liquid crystals,” J. Appl. Phys.94(3), 1298–1303 (2003).
[CrossRef]

Pagliusi, P.

C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett.89(12), 121105 (2006).
[CrossRef]

Palto, S. P.

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

Park, B.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: potential application for projection displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)451(1), 1–19 (2006).
[CrossRef]

Pelcovits, R. A.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

Petschek, R. G.

Presnyakov, V.

Provenzano, C.

C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett.89(12), 121105 (2006).
[CrossRef]

L. M. Blinov, G. Cipparrone, A. Mazzulla, C. Provenzano, S. P. Palto, M. I. Barnik, A. V. Arbuzov, and B. A. Umanskii, “A nematic liquid crystal as an amplifying replica of a holographic polarization grating,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)449(1), 147–160 (2006).
[CrossRef]

Radcliffe, M. D.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys.98(12), 123102 (2005).
[CrossRef]

Rosenblatt, C.

Rotar, V.

Sanchez, C.

M. J. Escuti, C. Oh, C. Sanchez, C. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE6302, 630207, 630207-11 (2006).
[CrossRef]

Sarkissian, H.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: potential application for projection displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)451(1), 1–19 (2006).
[CrossRef]

H. Sarkissian, S. V. Serak, N. V. Tabiryan, L. B. Glebov, V. Rotar, and B. Y. Zeldovich, “Polarization-controlled switching between diffraction orders in transverse-periodically aligned nematic liquid crystals,” Opt. Lett.31(15), 2248–2250 (2006).
[CrossRef] [PubMed]

Sasaki, T.

Serak, S. V.

Shi, L.

L. Shi, P. F. McManamon, and P. J. Bos, “Liquid crystal optical phase plate with a variable in-plane gradient,” J. Appl. Phys.104(3), 033109 (2008).
[CrossRef]

Steeves, D. M.

Tabirian, N.

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[CrossRef]

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[CrossRef]

Tsai, P.

Umanskii, B. A.

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[CrossRef]

Varghese, S.

S. Varghese, G. P. Crawford, C. W. M. Bastiaansen, D. K. G. de Boer, and D.J. Broer, “Microrubbing technique to produce high pretilt multidomain liquid crystal alignment,” Appl. Phys. Lett.85(2), 230–232 (2004).
[CrossRef]

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Wu, S.-T.

S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

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Zeldovich, B.

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: potential application for projection displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)451(1), 1–19 (2006).
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Appl. Opt.

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[CrossRef]

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[CrossRef]

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S.-T. Wu, Y. S. Chen, J. H. Guo, and A. Y.-G. Fuh, “Fabrication of twisted nematic gratings using polarization hologram based on azo-dye-doped liquid crystals,” Jpn. J. Appl. Phys.45(12), 9146–9151 (2006).
[CrossRef]

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[CrossRef]

H. Sarkissian, B. Park, N. Tabirian, and B. Zeldovich, “Periodically aligned liquid crystal: potential application for projection displays,” Mol. Cryst. Liq. Cryst. (Phila. Pa.)451(1), 1–19 (2006).
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Figures (6)

Fig. 1
Fig. 1

Definitions of twist angle (Φ) and azimuthal angle of surface LC directors on input and output substrates (Ψi and Ψo). Twist angle is defined to be Φ = Ψo − Ψi.

Fig. 2
Fig. 2

(a) Microrubbing pattern and (b) surface LC molecular orientation of the LC polarization grating, where the grating period is Λ = 80 µm.

Fig. 3
Fig. 3

(a) Microscope image of the LC polarization grating under crossed polarizers (Λ = 80 µm). (b) LC molecular orientation model. (c) Spatial distribution profile of twist angle Φ(x), where profile is approximated by eight steps.

Fig. 4
Fig. 4

(a) Relationship between normalized retardation R/λ and applied voltage, where λ = 633 nm and T = 24°C. Inset shows temperature dependence of normalized retardation. Temperature dependence of diffraction efficiency when (b) right- and (c) left-handed circularly polarized light beams are incident.

Fig. 5
Fig. 5

Relationship between diffraction efficiency and applied voltage when incident light is (a) right- and (b) left-handed circularly polarized, where λ = 633 nm and T = 24°C.

Fig. 6
Fig. 6

LC molecular orientation models in regimes (a) I and (b) II. Rubbing directions on the upper and lower polyimide surfaces are the same (parallel). Light propagation behavior for RCP and LCP incidences is also illustrated.

Equations (15)

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W TN E in =a E in ,
a j =p+i ( 1 ) j 1 p 2 ,
p= 1 1+ u 2 sinΘsinΦ+cosΘcosΦ
u= πΔnd Φλ
Θ=Φ 1+ u 2 .
a j =exp{ i ( 1 ) j ϕ },
cosϕ= 1 1+ u 2 sinΘsinΦ+cosΘcosΦ
Θ=Nπ
E out = ( 1 ) N exp{ i ( 1 ) j Φ } E in .
Δnd=Nλ.
E in ( j ) = 1 2 ( 1 ( 1 ) j+1 i ),
E out ( j ) = 1 2 ( 1 ) N exp{ ( 1 ) j iΦ }( 1 ( 1 ) j+1 i ).
η m = | 1 Λ 0 Λ Λ d E ox exp( i 2πm Λ x )dx | 2 + | 1 Λ 0 Λ Λ d E oy exp( i 2πm Λ x )dx | 2 ,
η 1 = ( Λ Λ d Λ ) 2 ,
η= sin 2 ( π/L ) ( π/L ) 2 ,

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