Abstract

We consider the dynamics of a variable optical prism based on Pancharatnam phase. The device basics, using liquid crystals (LCs) as the electro-optical material, have been previously proposed. In this paper, we study the dynamics of discrete changes in the phase profile, and also continuous changes in the phase profile through acquired data and numerical modeling. We show that a design based on LCs whose dielectric anisotropy can change sign (as a function of frequency) allows continuous tuning with reasonable response times.

© 2010 Optical Society of America

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References

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  1. K. Hirabayashi, T. Yamamoto, and M. Yamaguchi, “Free-space optical interconnections using liquid crystal microprism arrays,” Appl. Opt. 34, 2571-2580 (1995).
    [CrossRef] [PubMed]
  2. E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
    [CrossRef]
  3. B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
    [CrossRef]
  4. P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
    [CrossRef]
  5. J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
    [CrossRef]
  6. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
    [CrossRef]
  7. D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
    [CrossRef] [PubMed]
  8. S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. 41, 137-144 (1955).
  9. M. Honma and T. Nose, “Liquid-crystal Fresnel zone plate fabricated by microrubbing,” Jpn. J. Appl. Phys. 44, 287-290(2005).
    [CrossRef]
  10. M. Honma and T. Nose, “Liquid-crystal blazed grating with azimuthally distributed liquid-crystal directors,” Appl. Opt. 43, 5193-5197 (2004).
    [CrossRef] [PubMed]
  11. G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
    [CrossRef]
  12. M. J. Escuti and W. M. Jones, “Polarization-independent switching with high contrast from a liquid-crystal polarization gratings,” in SID Symposium Digest (Society for Information Display, 2006), Vol. 37, pp. 1443-1446
    [CrossRef]
  13. J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
    [CrossRef]
  14. L. Shi, P. F. McManamon, and P. J. Bos, “Liquid crystal optical phase plate with a variable in-plane gradient,” J. Appl. Phys. 104, 033109 (2008).
    [CrossRef]
  15. W. H. de Jeu, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, 1980).

2008 (1)

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

2005 (2)

M. Honma and T. Nose, “Liquid-crystal Fresnel zone plate fabricated by microrubbing,” Jpn. J. Appl. Phys. 44, 287-290(2005).
[CrossRef]

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

2004 (1)

2003 (1)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
[CrossRef]

1999 (1)

E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
[CrossRef]

1997 (2)

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

1996 (2)

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

1995 (1)

1993 (1)

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[CrossRef]

1955 (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. 41, 137-144 (1955).

Barnes, L. J.

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[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, 033109 (2008).
[CrossRef]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
[CrossRef]

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Crawford, G. P.

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

de Jeu, W. H.

W. H. de Jeu, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, 1980).

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[CrossRef]

Duncan, B. D.

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
[CrossRef]

Eakin, J. N.

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

Escuti, M. J.

M. J. Escuti and W. M. Jones, “Polarization-independent switching with high contrast from a liquid-crystal polarization gratings,” in SID Symposium Digest (Society for Information Display, 2006), Vol. 37, pp. 1443-1446
[CrossRef]

Fainman, Y.

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

Friedman, L. J.

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Hirabayashi, K.

Hobbs, D. S.

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Holz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Honma, M.

M. Honma and T. Nose, “Liquid-crystal Fresnel zone plate fabricated by microrubbing,” Jpn. J. Appl. Phys. 44, 287-290(2005).
[CrossRef]

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

Johnson, K.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Jones, A. C.

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

Jones, W. M.

M. J. Escuti and W. M. Jones, “Polarization-independent switching with high contrast from a liquid-crystal polarization gratings,” in SID Symposium Digest (Society for Information Display, 2006), Vol. 37, pp. 1443-1446
[CrossRef]

Lasher, M.

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[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, 033109 (2008).
[CrossRef]

E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[CrossRef]

Miller, D. T.

E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
[CrossRef]

Nguyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Nose, T.

M. Honma and T. Nose, “Liquid-crystal Fresnel zone plate fabricated by microrubbing,” Jpn. J. Appl. Phys. 44, 287-290(2005).
[CrossRef]

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

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. 41, 137-144 (1955).

Pelcovits, R. A.

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

Radcliffe, M. D.

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

Resler, D. P.

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Serati, S.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Sergan, V.

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
[CrossRef]

Sharp, G.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Sharp, R. C.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

D. P. Resler, D. S. Hobbs, R. C. Sharp, L. J. Friedman, and T. A. Dorschner, “High efficiency liquid-crystal optical phased array beam steering,” Opt. Lett. 21, 689-691 (1996).
[CrossRef] [PubMed]

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, 033109 (2008).
[CrossRef]

Soltan, P.

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

Stockley, J.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Thomas, J.

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

Walsh, K.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Wang, P.

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

Watson, E. A.

E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[CrossRef]

Yamaguchi, M.

Yamamoto, T.

Appl. Opt. (2)

J. Appl. Phys. (2)

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

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

Jpn. J. Appl. Phys. (1)

M. Honma and T. Nose, “Liquid-crystal Fresnel zone plate fabricated by microrubbing,” Jpn. J. Appl. Phys. 44, 287-290(2005).
[CrossRef]

Opt. Eng. (2)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide angle achromatic prism beam steering for infrared countermeasures applications,” Opt. Eng. 42, 1038-1047 (2003).
[CrossRef]

P. F. McManamon, E. A. Watson, T. A. Dorschner, and L. J. Barnes, “Applications look at the use of liquid crystal writable gratings for steering passive radiation,” Opt. Eng. 32, 2657-2664 (1993).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84, 268-298 (1996).
[CrossRef]

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates,” Proc. Indian Acad. Sci. 41, 137-144 (1955).

Proc. SPIE (3)

J. Stockley, S. Serati, G. Sharp, P. Wang, K. Walsh, and K. Johnson, “Broad-band beam steering,” Proc. SPIE 3131, 111-123 (1997).
[CrossRef]

E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and requirements for nonmechanical beam steering in active electro-optic sensors,” Proc. SPIE 3633, 216-255 (1999).
[CrossRef]

J. Thomas, M. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speed random-access beam steering,” Proc. SPIE 3131, 124-132 (1997).
[CrossRef]

Other (2)

W. H. de Jeu, Physical Properties of Liquid Crystalline Materials (Gordon and Breach, 1980).

M. J. Escuti and W. M. Jones, “Polarization-independent switching with high contrast from a liquid-crystal polarization gratings,” in SID Symposium Digest (Society for Information Display, 2006), Vol. 37, pp. 1443-1446
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

In-plane spiral configuration of the optic axis of a half-wave retarder with one full period. Circularly polarized light is considered to be normally incident onto this plane.

Fig. 2
Fig. 2

Surface alignment and electrode voltages to form the desired in-plane spiral director orientation. The arrows represent the preferred director orientation at the surfaces of the substrates. The shading of the electrodes corresponds to the voltage applied to them. Going from the lightest to darkest, the voltages are 0, 2, 10, and 12 V .

Fig. 3
Fig. 3

Views of in-plane spiral director configuration for four consecutive time steps as the period of the spiral is changed.

Fig. 4
Fig. 4

Surface alignment of a large angle digital beam deflector. The arrows represent the preferred orientation of the surface directors that are only slightly tilted from the vertical direction.

Fig. 5
Fig. 5

Side view of LC cell where the high-frequency voltage between the substrates is fixed at 80 V , but where the potential of offset by 40 V for the middle top and bottom electrodes. The lines are the equal potential lines, and the arrows show the local director orientation. Where the arrows appear to be short, they are pointing into the plane of the drawing.

Fig. 6
Fig. 6

Side view of a LC cell where the low-frequency voltage applied to the electrodes is shown. The lines are the equal potential lines, and the arrows show the local director orientation.

Fig. 7
Fig. 7

Polarization microscopy of a V-COPA device with crossed polarizer and analyzer, with analyzer aligned at: (a)  0 ° , (b)  22.5 ° , and (c)  45 ° to the electrode axis,. The two horizontal lines are at a fixed position in the aperture to show the offsetting of the extinction lines.

Fig. 8
Fig. 8

Tuning of a V-COPA device in polarization microscopy. (a) Initial half-period of 23 μm , (b) half-period changed to 46 μm , (c) half-period changed to 69 μm , and (d) half-period changed to 92 μm .

Fig. 9
Fig. 9

Deflection of transmitted light beam for different periods of the V-COPA device. Shown is a spot profile for the case of no voltage applied to the device (left-most peak), and for four different spiral half-periods. Going from right to left, the peaks correspond to spiral half-periods of 23, 46, 69, and 92 μm .

Fig. 10
Fig. 10

Modeling of the resetting method for changing the pitch of the spiral director configuration. The graph’s view is from a normal to the plane of the LC cell looking at the director configuration midway between the two substrates. The arrows indicate the director orientation. The length of the arrows is proportional to their projection onto the plane of the cell. Figures (a)–(d) show the director configuration with the voltages applied to the electrodes as given in Table 1.

Fig. 11
Fig. 11

Polarized optical microscopy images with the resetting method. Picture is taken of a cell between crossed polarizers. Figures (a)–(d) correspond to the director configuration and voltages given in Fig. 10 and Table 1.

Fig. 12
Fig. 12

(a) Side and (b) top view of a V-COPA device with a short pitch. The modeled device is 3 μm thick and the width of the simulation area shown is 21 μm (one z   grid unit = 0.25 μm , one x   grid unit = 0.25 μm ). The voltage applied to the electrodes is related to the darkness of the bars that represent them. The lightest shades have 0 V , and the darkest shades have 12 V , with the intermediate levels being 2 and 10 V . The arrows represent the director orientation, and the lines are the equal potential lines.

Fig. 13
Fig. 13

Initialization of a two-frequency V-COPA device in the negative mode. The electrodes are shown “end on” as bars. The voltages applied to the top electrodes (left to right) are 12, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 10 V ; and for the bottom substrate are 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0 V . The figure shows (a) side and (b) top views of the device. One z   grid unit = 0.25 μm and one x   grid unit = 0.25 μm .

Fig. 14
Fig. 14

Period tuning of two-frequency V-COPA device (director configuration in the x y plane of middle of the cell) with a 15 ms time period between each step. The voltages applied to the electrodes are given in Tables 2, 3. The open rectangles show the location of electrodes on the top substrate that have no voltage applied, while a high-frequency (negative mode) voltage of 80 V is applied on the bottom. The rectangles with a lighter shade show where a pair of equal low-frequency (positive mode) voltages is applied on both substrates, and those with the darker shade are where a pair of high-frequency (negative mode) voltages is applied on both substrates. The dashed rectangles in step ( j , k , l ) is denoted with no high-frequency voltage applied on both substrates.

Fig. 15
Fig. 15

(a) Side and (b) top view of the V-COPA device described in the text after the initial configuration shown in Fig. 13, followed by the 12-step process shown in Fig. 14, followed by the same initialization voltages used to create Fig. 13 applied to the left half only, to form a new spiral.

Tables (3)

Tables Icon

Table 1 Voltage Profile of Each Step for the Resetting Method a

Tables Icon

Table 2 Driving Scheme for Changing the Spiral Pitch a

Tables Icon

Table 3 Driving Scheme for Changing the Spiral Pitch a

Equations (1)

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d f E d n i = 1 2 ε 0 · [ d V + d i · j ( 1 2 · d V + d j · n j ) · Δ ε + + d V d i · j ( 1 2 · d V d j · n j ) · Δ ε ] , i , j = x , y , z ,

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