Abstract

For the first time, to our knowledge, we report ray-tracing simulations of an advanced liquid-crystal gradient-index lens structure for application in switchable two-dimensional/three-dimensional (3D) autostereoscopic displays. We present ray-tracing simulations of the angular-dependent lens action. From the results we conclude that the lens action of the advanced optical design corresponds to the desired performance for small viewing angles. For oblique viewing angles of approximately 30° and higher, the lens action becomes significantly weaker compromising the 3D performance of an autostereoscopic display. The general approach and the advanced ray-optics analysis procedures presented form a useful tool in the search for improvements for high viewing angles and enable a better understanding of the liquid-crystal technology discussed.

© 2009 Optical Society of America

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References

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  1. D. F. McAllister, Stereoscopic Computer Graphics and Other True 3D Technologies (Princeton U. Press, 1993).
  2. S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.
  3. W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).
  4. D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
    [CrossRef]
  5. M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
    [CrossRef]
  6. H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
    [CrossRef]
  7. SHINTECH, Inc., http://www.shintech.jp.
  8. AUTRONIC MELCHERS GmbH, http://www.autronic-melchers.com.
  9. S. Sato, “LC-lens cell with variable focal length,” Jpn. J. Appl. Phys. 18, 1679-1684 (1979).
    [CrossRef]
  10. D. W. Berreman, “Variable-focus LC-lens system,” U.S. patent 4,190,330 (February 26, 1980).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. J. S. Patel and K. Rastani, “Electrically controlled polarization-independent liquid-crystal Fresnel lens arrays,” Opt. Lett. 16, 532-534 (1991).
    [CrossRef] [PubMed]
  14. A. F. Naumov, M. Y. Loktev, I. R. Guralnik, and G. Vdovin, “Liquid-crystal adaptive lenses with modal control,” Opt. Lett. 23, 992-994 (1998).
    [CrossRef]
  15. H. Ren and S. T. Wu, “Adaptive liquid crystal lens with focal length tunability,” Opt. Express 14, 11292-11298 (2006).
    [CrossRef] [PubMed]
  16. G. E. Nevskaya and M. G. Tomilin, “Adaptive lenses based on liquid crystals,” J. Opt. Technol. 75, 563-573 (2008).
    [CrossRef]
  17. C. Jenkins, R. Bingham, K. Moore, and G. D. Love, “Ray equation for a spatially variable uniaxial crystal and its use in the optical design of liquid-crystal lenses,” J. Opt. Soc. Am. A 24, 2089-2096 (2007).
    [CrossRef]
  18. M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).
  19. M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260-1273 (2008).
    [CrossRef]
  20. T. C. Kraan, T. van Bommel, and R. A. M. Hikmet, “Modeling liquid-crystal gradient-index lenses,” J. Opt. Soc. Am. A 24, 3467-3477 (2007).
    [CrossRef]
  21. R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).
  23. The Merck Group, Chemicals, www.merck.de.

2009

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

2008

2007

2006

2005

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).

1998

1991

1988

1984

1979

S. Sato, “LC-lens cell with variable focal length,” Jpn. J. Appl. Phys. 18, 1679-1684 (1979).
[CrossRef]

Berreman, D. W.

D. W. Berreman, “Variable-focus LC-lens system,” U.S. patent 4,190,330 (February 26, 1980).

Bingham, R.

Braat, J. J. M.

Brinkley, P. F.

Chu, C.

Clevery, D. S.

de Boer, D. K. G.

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260-1273 (2008).
[CrossRef]

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

de Zwart, S. T.

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).

S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Dekker, T.

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).

S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Guralnik, I. R.

Hiddink, M. G. H.

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

Hikmet, R. A. M.

T. C. Kraan, T. van Bommel, and R. A. M. Hikmet, “Modeling liquid-crystal gradient-index lenses,” J. Opt. Soc. Am. A 24, 3467-3477 (2007).
[CrossRef]

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Hong, H.

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

IJzerman, W. L.

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).

S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.

Jenkins, C.

Jung, S.

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

Kornreich, P. G.

Kowel, S. T.

Kraan, T. C.

T. C. Kraan, T. van Bommel, and R. A. M. Hikmet, “Modeling liquid-crystal gradient-index lenses,” J. Opt. Soc. Am. A 24, 3467-3477 (2007).
[CrossRef]

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Krijn, M. P. C. M.

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Kusters, L. H. C.

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Lee, B.

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

Loktev, M. Y.

Love, G. D.

McAllister, D. F.

D. F. McAllister, Stereoscopic Computer Graphics and Other True 3D Technologies (Princeton U. Press, 1993).

Moore, K.

Naumov, A. F.

Nevskaya, G. E.

Patel, J. S.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Rastani, K.

Ren, H.

Sato, S.

S. Sato, “LC-lens cell with variable focal length,” Jpn. J. Appl. Phys. 18, 1679-1684 (1979).
[CrossRef]

Shin, H.

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

Sluijter, M.

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260-1273 (2008).
[CrossRef]

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Tomilin, M. G.

van Bommel, T.

T. C. Kraan, T. van Bommel, and R. A. M. Hikmet, “Modeling liquid-crystal gradient-index lenses,” J. Opt. Soc. Am. A 24, 3467-3477 (2007).
[CrossRef]

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Vdovin, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Willemsen, O. H.

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

Wolter, W. A. M.

S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.

Wu, S. T.

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Technol.

J. Soc. Inf. Disp.

W. L. IJzerman, S. T. de Zwart, and T. Dekker, “Design of 2D/3D switchable displays,” J. Soc. Inf. Disp. 36, 98-101 (2005).

M. P. C. M. Krijn, S. T. de Zwart, D. K. G. de Boer, O. H. Willemsen, and M. Sluijter, “2D/3D displays based on switchable lenticulars,” J. Soc. Inf. Disp. 16, 847-855 (2008).
[CrossRef]

H. Hong, S. Jung, B. Lee, and H. Shin, “Electric-field-driven LC lens for 3-D/2-D autostereoscopic display,” J. Soc. Inf. Disp. 17, 399-406 (2009).
[CrossRef]

Jpn. J. Appl. Phys.

S. Sato, “LC-lens cell with variable focal length,” Jpn. J. Appl. Phys. 18, 1679-1684 (1979).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

D. K. G. de Boer, M. G. H. Hiddink, M. Sluijter, O. H. Willemsen, and S. T. de Zwart, “Switchable lenticular-based 2D/3D displays,” Proc. SPIE 6490, 64900R (2007).
[CrossRef]

Other

D. F. McAllister, Stereoscopic Computer Graphics and Other True 3D Technologies (Princeton U. Press, 1993).

S. T. de Zwart, W. L. IJzerman, T. Dekker, and W. A. M. Wolter, “A 20″switchable auto-stereoscopic 2D/3D display,” in 11th International Display Workshop (2004), pp. 1459-1460.

SHINTECH, Inc., http://www.shintech.jp.

AUTRONIC MELCHERS GmbH, http://www.autronic-melchers.com.

D. W. Berreman, “Variable-focus LC-lens system,” U.S. patent 4,190,330 (February 26, 1980).

M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, 1965).

R. A. M. Hikmet, T. van Bommel, T. C. Kraan, L. H. C. Kusters, S. T. de Zwart, O. H. Willemsen, and M. P. C. M. Krijn, “Beam-shaping device,” U.S. patent pending WO/2008/126049A1 (2008).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

The Merck Group, Chemicals, www.merck.de.

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Figures (13)

Fig. 1
Fig. 1

Schematic principle of an autostereoscopic lenticular-based 3D imaging display. In (a), multiple images are projected to multiple viewing directions. The neighboring images form stereo pairs, thus enabling the binocular disparity and motion parallax depth cues. The lenticular is placed in front of a display as depicted in (b). The light from subpixels of the display is collimated by the lenticular and directed toward different viewing directions. The contribution of all pixels of a display produces the individual views.

Fig. 2
Fig. 2

Schematic working principle of a liquid-crystal GRIN lens integrated in a 3D display. (a) shows a liquid-crystal layer between two transparent substrates. The figure shows one unit cell, which is repeated in the y direction (see also Fig. 1). An ITO electrode structure induces an electric field along which the liquid-crystal molecules align (indicated by the black stripes). In (b), we show how the effective index of refraction typically varies with position between n o and n e , with n e > n o . The propagating wavefront of an incident plane wave (polarized in the y z plane) is depicted in (c). The ray paths that correspond to these wavefronts are depicted in (d), but this time for a plane wave with an angle of incidence ϑ i n . Ideally, the ray paths focus at a distance f in the pixel plane of the display.

Fig. 3
Fig. 3

Schematic cross section of the advanced optical design of a liquid-crystal GRIN lens structure [21]. The liquid-crystal mixture that is used is TL213, for which n o = 1.5271 and n e = 1.7659 .

Fig. 4
Fig. 4

Measured angle of refraction ϑ out ϑ in (in air) as a function of the position y for the GRIN lens defined in Fig. 3 for ϑ in = 0 ° 40 ° . For ϑ in = 0 ° in the region where | y | 45 μ m , ϑ out is a linear function of y: ϑ out = 0.85   mrad / μ m . In the region where | y | > 45 μ m , ϑ o u t is not linear.

Fig. 5
Fig. 5

Evolution of the Huygens spheres in a liquid-crystal layer. The Huygens spheres evolve differently at the positions y and y + d y since at these positions the averaged effective index of refraction n ¯ eff ( y ) varies over the distance d y . The figure shows an incident plane wave with an angle of incidence ϑ in and the corresponding emerging plane wave with an angle of refraction ϑ out . Here it is assumed that n ¯ eff ( y ) < n ¯ eff ( y + d y ) . The liquid-crystal layer has a thickness h and the indices of refraction of the two glass substrates are indicated by n glass .

Fig. 6
Fig. 6

(a) n ¯ eff as a function of y calculated from the experimentally obtained result for ϑ in = 0 ° depicted in Fig. 4. At the position y = 0 , the value for n ¯ eff is the extraordinary index of refraction n e = 1.7659 . (b) The angle α (in degrees) between the director d ̂ and the vertical z direction (i.e., the direction of propagation s ̂ ) as a function of the position y. Then the director is given by d ̂ = ( 0 , sin   α , cos   α ) .

Fig. 7
Fig. 7

Ray-tracing results of the Huygens method for ϑ out ( y ) ϑ in for ϑ in = 0 ° , , 60 ° with steps of 10°. Clearly, the linearity of ϑ out decreases with increasing ϑ in .

Fig. 8
Fig. 8

(a) Simulated director profile d ̂ ( y , z ) of the liquid-crystal layer defined by the optical configuration depicted in Fig. 3. The result is obtained using the simulation program LCD Master [7]. The director profile is inhomogeneous in both the y and z directions and can be used for the Hamiltonian principle. (b) The resulting director profile d ̂ ( y ) when averaged over the vertical z direction.

Fig. 9
Fig. 9

Ray-tracing results of the Huygens method (dashed curves) for ϑ in = 0 ° , , 50 ° , with steps of 10°. The experimental result for ϑ in = 0 ° is also depicted. The figure also shows the ray-tracing results of the Hamiltonian method applied to the averaged director profile depicted in Fig. 8b (solid curves).

Fig. 10
Fig. 10

Averaged effective index of refraction n ¯ eff as a function of y for ϑ in = 0 ° , 20°, and 40°.

Fig. 11
Fig. 11

Ray-tracing results of the Hamiltonian method applied to the director profile d ̂ ( y , z ) depicted in Fig. 8a for ϑ in = 0 ° , , 50 ° , with steps of 10°. The results are presented together with the Huygens results of Fig. 9. The experimental result for ϑ in = 0 ° is also depicted. Clearly, the Hamiltonian method predicts a stronger lens action than the Huygens method does.

Fig. 12
Fig. 12

In (a), ray-tracing results of the Hamiltonian method and the Huygens method are compared with the experimental result for ϑ in = 20 ° . In (b), ϑ in = 40 ° .

Fig. 13
Fig. 13

Ray paths of light rays in the glass cover plate for incident plane waves with ϑ in = 0 ° , , 50 ° with steps of 10°. The light rays for ϑ in = 0 ° are focused in the pixel plane at approximately z = 1800 μ m . The lens action for oblique angles of incidence decreases with ϑ i n .

Tables (1)

Tables Icon

Table 1 Liquid-Crystal Properties of TL213 Mixture [23]

Equations (10)

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tan   ϑ out = y f ,
n eff ( r , s ) = n o 2 n e 2 n o 2 [ 1 ( s ̂ , d ̂ ) 2 ] + n e 2 ( s ̂ , d ̂ ) 2 ,
n ¯ eff ( y ) = 1 h 0 h n eff ( r ) d z ,
n glass   sin   ϑ out = n glass   sin   ϑ in + h d n ¯ eff ( y ) d y ,
sin   ϑ out ( y ) h = d n ¯ eff ( y ) d y .
n ¯ eff ( y ) = 1 h 0 y sin   ϑ out ( y ) d y + n e ,
d r ( τ ) d τ = s H ( d ̂ ) ,
d s ( τ ) d τ = r H ( d ̂ ) ,
H i = 2 ( n e 2 n o 2 ) ( s d ̂ ) ( s x d ̂ x i + s y d ̂ y i + s z d ̂ z i ) ,
H s i = 2 n o 2 s i + 2 ( n e 2 n o 2 ) ( s d ̂ ) d ̂ i ,     i = x , y , z ,

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