Abstract

We report on a linearly graded phase model for the design of binary diffraction gratings of liquid crystals (LCs) associated with the periodic interfacial effect. The binary nature of the LC grating is produced by use of periodic striped domains in an alternating homeotropic and hybrid geometry. In our graded phase model the diffraction patterns and the diffracted intensities of the LC binary grating is primarily governed by three length scales: the cell thickness and two distortion parameters scaled by the grating period at two domain boundaries. The experimental data agree well with theoretical predictions made in our linearly graded phase model as well as the elastic continuum theory.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. K. Herl, J. Bischoff, U. Mohaupt, M. Palme, B. Schnabel, L. Wenke, R. Bödefeld, W. Theobald, E. Welsch, R. Sauerbrey, H. Heyer, “High-efficiency dielectric reflection gratings: design, fabrication, and analysis,” Appl. Opt. 38, 6257–6271 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
    [CrossRef]
  17. I. Fujieda, “Liquid-crystal phase grating based on in-plane switching,” Appl. Opt. 40, 6252–6259 (2001).
    [CrossRef]
  18. G. A. Lester, A. M. Strudwick, “A liquid crystal phase grating for instrumentation applications,” J. Mod. Opt. 47, 1959–1967 (2000).
  19. P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 3rd ed. (Oxford U. Press, New York, 1993).
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    [CrossRef]

2003

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

2001

2000

G. A. Lester, A. M. Strudwick, “A liquid crystal phase grating for instrumentation applications,” J. Mod. Opt. 47, 1959–1967 (2000).

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

E. Sokolova, V. Sokolov, A. Nunes, “Two-channel tunable monochromatic illuminator with a concave diffraction grating and an original scanning system,” Appl. Opt. 39, 3854–3863 (2000).
[CrossRef]

1999

1998

1997

1996

1995

1993

1989

1988

1980

Ailawadi, N. K.

Arimoto, Y.

W. Klaus, M. Ide, Y. Hayano, Y. Arimoto, “Efficient liquid crystal wavefront modulator,” in Liquid Crystal Materials, Devices, and Applications V, R. Shashidhar, ed., Proc. SPIE3015, 84–92 (1997).
[CrossRef]

Baker, K. M.

Bischoff, J.

Bödefeld, R.

Bos, P. J.

C. M. Titus, 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, D. L. Johnson, “An electro-optically controlled liquid crystal diffraction grating,” Appl. Phys. Lett. 67, 2588–2590 (1995).
[CrossRef]

Bouvier, M.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Boyd, R. D.

Britten, J. A.

Cambril, E.

Chavel, P.

Chen, J.

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

Chung, S.-Y.

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

de Beaucoudrey, N.

de Gennes, P. G.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 3rd ed. (Oxford U. Press, New York, 1993).

Decker, D.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Friends, M. W.

Fujieda, I.

George, N.

Gerritsen, H. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

Harvey, P.

Hayano, Y.

W. Klaus, M. Ide, Y. Hayano, Y. Arimoto, “Efficient liquid crystal wavefront modulator,” in Liquid Crystal Materials, Devices, and Applications V, R. Shashidhar, ed., Proc. SPIE3015, 84–92 (1997).
[CrossRef]

Herl, K.

Heyer, H.

Ide, M.

W. Klaus, M. Ide, Y. Hayano, Y. Arimoto, “Efficient liquid crystal wavefront modulator,” in Liquid Crystal Materials, Devices, and Applications V, R. Shashidhar, ed., Proc. SPIE3015, 84–92 (1997).
[CrossRef]

Jahns, J.

Jannson, T.

Jarem, J. M.

Jepsen, M. L.

Jiang, W.

Johnson, D. L.

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

Kim, J.

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

Klaus, W.

W. Klaus, M. Ide, Y. Hayano, Y. Arimoto, “Efficient liquid crystal wavefront modulator,” in Liquid Crystal Materials, Devices, and Applications V, R. Shashidhar, ed., Proc. SPIE3015, 84–92 (1997).
[CrossRef]

Kowel, S. T.

Kulick, J. H.

Lee, S.-D.

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

Leslie, T. M.

Lester, G. A.

G. A. Lester, A. M. Strudwick, “A liquid crystal phase grating for instrumentation applications,” J. Mod. Opt. 47, 1959–1967 (2000).

Li, L.

Lindquist, R. G.

Mansfield, W. M.

Miller, J. M.

Mohaupt, U.

Moslehi, B.

Mulgrew, P.

Ng, J.

Nunes, A.

Palme, M.

Park, J.-H.

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

Perry, M. D.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Prost, J.

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 3rd ed. (Oxford U. Press, New York, 1993).

Roberts, C. W.

Sauerbrey, R.

Scharf, T.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Schnabel, B.

Shannon, C.

Shealy, D. L.

Shore, B. W.

Shults, E.

Sokolov, V.

Sokolova, E.

Stone, T.

Strudwick, A. M.

G. A. Lester, A. M. Strudwick, “A liquid crystal phase grating for instrumentation applications,” J. Mod. Opt. 47, 1959–1967 (2000).

Tai, A.

Tennant, D. M.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Theobald, W.

Titus, C. M.

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

Turunen, J.

Unno, Y.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

Vithana, H.

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

Walker, S. J.

Welsch, E.

Wenke, L.

West, L. C.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, Chichester, U.K., 1988).

Yu, C.-J.

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

Appl. Opt.

E. Sokolova, V. Sokolov, A. Nunes, “Two-channel tunable monochromatic illuminator with a concave diffraction grating and an original scanning system,” Appl. Opt. 39, 3854–3863 (2000).
[CrossRef]

J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, M. W. Friends, T. M. Leslie, “Electrostatic and diffraction analysis of a liquid-crystal device utilizing fringing fields: applications to three-dimensional displays,” Appl. Opt. 34, 1901–1922 (1995).
[CrossRef] [PubMed]

W. Jiang, D. L. Shealy, K. M. Baker, “Development and testing of a holographic projection system,” Appl. Opt. 35, 5994–5998 (1996).
[CrossRef] [PubMed]

T. Stone, N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
[CrossRef] [PubMed]

Y. Unno, “Fabrication of N-level binary optical elements by use of M mask patterns with N in the range of 2M-1 + 1 ≤ N ≤ 2M,” Appl. Opt. 37, 8012–8020 (1998).
[CrossRef]

K. Herl, J. Bischoff, U. Mohaupt, M. Palme, B. Schnabel, L. Wenke, R. Bödefeld, W. Theobald, E. Welsch, R. Sauerbrey, H. Heyer, “High-efficiency dielectric reflection gratings: design, fabrication, and analysis,” Appl. Opt. 38, 6257–6271 (1999).
[CrossRef]

L. Li, S. J. Walker, J. Jahns, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, W. M. Mansfield, “Design and fabrication of high-efficiency beam splitters and beam deflectors for integrated planar micro-optic systems,” Appl. Opt. 32, 2494–2501 (1993).
[CrossRef] [PubMed]

J. M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, E. Cambril, “Design and fabrication of binary slanted surface-relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
[CrossRef] [PubMed]

I. Fujieda, “Liquid-crystal phase grating based on in-plane switching,” Appl. Opt. 40, 6252–6259 (2001).
[CrossRef]

Appl. Phys. Lett.

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

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

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

J. Mod. Opt.

G. A. Lester, A. M. Strudwick, “A liquid crystal phase grating for instrumentation applications,” J. Mod. Opt. 47, 1959–1967 (2000).

Opt. Eng.

M. Bouvier, T. Scharf, “Analysis of nematic-liquid-crystal binary gratings with high spatial frequency,” Opt. Eng. 39, 2129–2137 (2000).
[CrossRef]

Opt. Lett.

Other

W. Klaus, M. Ide, Y. Hayano, Y. Arimoto, “Efficient liquid crystal wavefront modulator,” in Liquid Crystal Materials, Devices, and Applications V, R. Shashidhar, ed., Proc. SPIE3015, 84–92 (1997).
[CrossRef]

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 3rd ed. (Oxford U. Press, New York, 1993).

P. Yeh, Optical Waves in Layered Media (Wiley, Chichester, U.K., 1988).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

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Figures (8)

Fig. 1
Fig. 1

LC binary grating structure consisting of periodic striped domains in an alternating homeotropic and hybrid geometry.

Fig. 2
Fig. 2

Gray-scale representation of the distribution of the molecular tilt in our binary grating obtained in the elastic continuum theory. The hybrid alignment is in the region of 0.25Ly ≤ 0.75L, and the homeotropic alignment is in the other region.

Fig. 3
Fig. 3

(a) Phase profiles of the LC binary grating evaluated in the elastic continuum theory and in our linearly graded phase model, and (b) the intensity at each diffraction order calculated by the discrete Fourier transformation of the corresponding phase profile.

Fig. 4
Fig. 4

Phase profile of the LC binary grating as a function of the distance scaled by the grating period along the grating vector (the y axis) in our linearly graded phase model.

Fig. 5
Fig. 5

Experimental setup to measure the diffraction patterns of the LC grating. The polarization state of the incident light is perpendicular to the direction of the grating vector.

Fig. 6
Fig. 6

Diffraction patterns of the LC binary grating as a function of the normalized distance with the grating periods of (a) L = 38 μm and (b) L = 150 μm for the cell thickness d = 4.28 μm. Open circles and solid curves represent the measured intensities and the calculated patterns, respectively.

Fig. 7
Fig. 7

Distorted lengths, scaled by a grating period L, into (a) homeotropic (ξ1) and (b) hybrid (ξ2) regions at the interface of the two domains as a function of the inverse grating period (1/L). Open symbols were determined from Eqs. (1) and (3) for various grating periods. Solid lines represent the least-squares fits of the data to straight lines.

Fig. 8
Fig. 8

Peak intensities of the diffractions at (a) the zeroth order, (b) the first order, and (c) the second order as a function of the distorted parameters of ξ1 and ξ2 for the cell thickness d = 4.28 μm.

Equations (7)

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

η0=4ξ-2sin2Δϕ/2+1-2ξ+cosΔϕ/2+2ξ+sincΔϕ/22
sincu=sinuu,
ηk=cos2πk2+πkξ-ξ+Φ+-1πk Ψ-2+sin2πk2+πkξ-ξ+Φ--1πk Ψ+2,
Φ±=sincΔϕ2+πkξ+±sincΔϕ2-πkξ+,
Ψ±=sinΔϕ2+πkξ+±sinΔϕ2-πkξ+.
η0*=1+cosΔϕ2.
ηk*=1-cosΔϕ2sinc2πk2.

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