Abstract

In order to minimize moiré patterns in autostereoscopic parallax displays, the optical component, which is used for forming viewing zones, is analyzed with varying period and slant angle. First, horizontal-parallax autostereoscopic displays (HPAD) can be approximated as the superposition of three corresponding binary gratings. Referring to the unification of indicial equation method and Fourier analysis, a singular state and two stable moiré-free states are obtained according to the superposition of the equivalent grating of LCD and special radial grating. Two stable moiré-free states are valid for HPAD. For full-parallax autostereoscopic displays (FPAD), a special radial grid grating is introduced to simulate a 2D optical component with progressively varying period and slant angle. Similarly, two corresponding stable moiré-free states for FPAD can be used for eliminating moiré patterns.

© 2013 Optical Society of America

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References

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  1. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier: Amsterdam, 1993).
  2. O. Kafri and I. Glatt, The Physics of Moiré Metrology (John Wiley & Sons, 1989).
  3. A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng.1(2), 61–69 (1979).
    [CrossRef]
  4. H. Takasaki, “Moiré topography,” Appl. Opt.9(6), 1467–1472 (1970).
    [CrossRef] [PubMed]
  5. A. J. Durelli and V. J. Parks, Moiré Analysis of strain (Prentice-Hall, Englewood Cliffs: New Jersey, 1970).
  6. K. Creath and J. C. Wyant, “Moiré and fringe projection techniques,” in Optical Shop Testing, 2nd ed. (John Wiley & Sons: NY, 1995), 653–685.
  7. I. Amidror, The Theory of the Moiré Phenomenon, 2nd ed (Springer, 2009).
  8. J. P. Allebach, “Random nucleated halftone screen,” Photogr. Sci. Eng22, 89–91 (1978).
  9. D. Blattner, C. Chaves, G. Fleishman, and S. Roth, RealWorld Scanning and Halftones (Berkeley, CA: Peachpit, 2004).
  10. K. Mashitani, G. Hamagishi, M. Sakata, A. Yamashita, E. Nakayama,and M. Inoue, “New autostereoscopic (no-glasses) LCD image splitter displays,” in 3 Dimensional Image Conf (1996), 90–95.
  11. J. Hong, Y. Kim, H. J. Choi, J. Hahn, J. H. Park, H. Kim, S. W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt.50(34), H87–H115 (2011).
    [CrossRef] [PubMed]
  12. K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
    [CrossRef]
  13. T. Koike, M. Oikawa, and K. Utsugi, “Moiré reduction for integral photography,” International Display Workshops (Institute of Image Information and Television Engineers of Japan and Society of Information Display, 2007), 1917–1918.
  14. M. Okui, M. Kobayashi, J. Arai, and F. Okano, “Moire fringe reduction by optical filters in integral three-dimensional imaging on a color flat-panel display,” Appl. Opt.44(21), 4475–4483 (2005).
    [CrossRef] [PubMed]
  15. L. Lipton and M. Feldman, “A new autostereoscopic display technology: the SynthaGram,” Proc. SPIE4660, 229–235 (2002).
    [CrossRef]
  16. Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt.48(11), 2178–2187 (2009).
    [CrossRef] [PubMed]
  17. V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
    [CrossRef]
  18. V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
    [CrossRef]
  19. L. S. Kong, G. Jin, T. Wang, S. Cai, X. Zhong, and K. Xu, “Parameter design of a parallax barrier based on the color moiré patterns in autostereoscopic display,” Appl. Opt.50(34), H153–H158 (2011).
    [CrossRef] [PubMed]
  20. J. W. Goodman, Introduction to Fourier Optics (Roberts: 2005).
  21. K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
    [CrossRef]
  22. S. Yokozeki and K. Patorski, “moiré fringe profile prediction method and its application to fringe sharpening,” Appl. Opt.17(16), 2541–2547 (1978).
    [PubMed]
  23. D. H. Shin, E. S. Kim, and B. Lee, “Computational Reconstruction of three-Dimensional Objects in Integral Imaging using Lenslet Array,” Jpn. J. Appl. Phys.44(11), 8016–8018 (2005).
    [CrossRef]
  24. D. H. Shin, B. Lee, and E. S. Kim, “Improved Viewing Quality of 3-D Images in Computational Integral Imaging Reconstruction Based on Lenslet Array Model,” ETRI Journal28(4), 521–524 (2006).
    [CrossRef]
  25. H. Choi, J. Kim, S. W. Cho, Y. Kim, J. B. Park, and B. Lee, “Three-dimensional-two-dimensional mixed display system using integral imaging with an active pinhole array on a liquid crystal panel,” Appl. Opt.47(13), 2207–2214 (2008).
    [CrossRef] [PubMed]
  26. G. Lippmann, “La photographie integrale,” C. R. Acad. Sci146, 446–451 (1908).
  27. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt.36(7), 1598–1603 (1997).
    [CrossRef] [PubMed]
  28. H. Hoshino, F. Okano, H. Isono, and I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A15(8), 2059–2065 (1998).
    [CrossRef]
  29. S. H. Kaplan, “Theory of parallax barrier,” J.SMPTE59(1), 11–21 (1976).
    [CrossRef]
  30. P. S. Theocaris, “radial gratings as moiré gauges,” J. Phys. E1(6), 613–618 (1968).
    [CrossRef]
  31. M. Abolhassani and M. Mirzaei, “Unification of formulation of moiré fringe spacing in parametric equation and Fourier analysis methods,” Appl. Opt.46(32), 7924–7926 (2007).
    [CrossRef] [PubMed]
  32. V. V. Saveljev, “Orientations and branches of moiré waves in three-dimensional displays,” J. Korean Phys. Soc.57(6), 1392–1396 (2010).
    [CrossRef]

2011 (2)

2010 (1)

V. V. Saveljev, “Orientations and branches of moiré waves in three-dimensional displays,” J. Korean Phys. Soc.57(6), 1392–1396 (2010).
[CrossRef]

2009 (1)

2008 (2)

2007 (1)

2006 (1)

D. H. Shin, B. Lee, and E. S. Kim, “Improved Viewing Quality of 3-D Images in Computational Integral Imaging Reconstruction Based on Lenslet Array Model,” ETRI Journal28(4), 521–524 (2006).
[CrossRef]

2005 (4)

D. H. Shin, E. S. Kim, and B. Lee, “Computational Reconstruction of three-Dimensional Objects in Integral Imaging using Lenslet Array,” Jpn. J. Appl. Phys.44(11), 8016–8018 (2005).
[CrossRef]

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

M. Okui, M. Kobayashi, J. Arai, and F. Okano, “Moire fringe reduction by optical filters in integral three-dimensional imaging on a color flat-panel display,” Appl. Opt.44(21), 4475–4483 (2005).
[CrossRef] [PubMed]

2002 (1)

L. Lipton and M. Feldman, “A new autostereoscopic display technology: the SynthaGram,” Proc. SPIE4660, 229–235 (2002).
[CrossRef]

1998 (1)

1997 (1)

1979 (1)

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng.1(2), 61–69 (1979).
[CrossRef]

1978 (2)

1976 (2)

K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
[CrossRef]

S. H. Kaplan, “Theory of parallax barrier,” J.SMPTE59(1), 11–21 (1976).
[CrossRef]

1970 (1)

1968 (1)

P. S. Theocaris, “radial gratings as moiré gauges,” J. Phys. E1(6), 613–618 (1968).
[CrossRef]

1908 (1)

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci146, 446–451 (1908).

Abolhassani, M.

Allebach, J. P.

J. P. Allebach, “Random nucleated halftone screen,” Photogr. Sci. Eng22, 89–91 (1978).

Arai, J.

Cai, S.

Chen, N.

Cho, S. W.

Choi, H.

Choi, H. J.

Chun, J.-H.

V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
[CrossRef]

Feldman, M.

L. Lipton and M. Feldman, “A new autostereoscopic display technology: the SynthaGram,” Proc. SPIE4660, 229–235 (2002).
[CrossRef]

Fukushima, R.

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Hahn, J.

Hirayama, Y.

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Hong, J.

Hoshino, H.

Isono, H.

Javidi, B.

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

Jin, G.

Jung, J.-H.

Kaplan, S. H.

S. H. Kaplan, “Theory of parallax barrier,” J.SMPTE59(1), 11–21 (1976).
[CrossRef]

Kim, D.-S.

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

Kim, E. S.

D. H. Shin, B. Lee, and E. S. Kim, “Improved Viewing Quality of 3-D Images in Computational Integral Imaging Reconstruction Based on Lenslet Array Model,” ETRI Journal28(4), 521–524 (2006).
[CrossRef]

D. H. Shin, E. S. Kim, and B. Lee, “Computational Reconstruction of three-Dimensional Objects in Integral Imaging using Lenslet Array,” Jpn. J. Appl. Phys.44(11), 8016–8018 (2005).
[CrossRef]

Kim, H.

Kim, J.

Kim, S.-K.

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

Kim, Y.

Kobayashi, H.

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Kobayashi, M.

Kong, L. S.

Kwack, K.-D.

V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
[CrossRef]

Lee, B.

Lippmann, G.

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci146, 446–451 (1908).

Lipton, L.

L. Lipton and M. Feldman, “A new autostereoscopic display technology: the SynthaGram,” Proc. SPIE4660, 229–235 (2002).
[CrossRef]

Min, S. W.

Mirzaei, M.

Okano, F.

Okui, M.

Park, G.

Park, J. B.

Park, J. H.

Patorski, K.

S. Yokozeki and K. Patorski, “moiré fringe profile prediction method and its application to fringe sharpening,” Appl. Opt.17(16), 2541–2547 (1978).
[PubMed]

K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
[CrossRef]

Saishu, T.

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Saveljev, V. V.

V. V. Saveljev, “Orientations and branches of moiré waves in three-dimensional displays,” J. Korean Phys. Soc.57(6), 1392–1396 (2010).
[CrossRef]

V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
[CrossRef]

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

Shepherd, A. T.

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng.1(2), 61–69 (1979).
[CrossRef]

Shin, D. H.

D. H. Shin, B. Lee, and E. S. Kim, “Improved Viewing Quality of 3-D Images in Computational Integral Imaging Reconstruction Based on Lenslet Array Model,” ETRI Journal28(4), 521–524 (2006).
[CrossRef]

D. H. Shin, E. S. Kim, and B. Lee, “Computational Reconstruction of three-Dimensional Objects in Integral Imaging using Lenslet Array,” Jpn. J. Appl. Phys.44(11), 8016–8018 (2005).
[CrossRef]

Son, J.-Y.

V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
[CrossRef]

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
[CrossRef]

Taira, K.

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Takasaki, H.

Theocaris, P. S.

P. S. Theocaris, “radial gratings as moiré gauges,” J. Phys. E1(6), 613–618 (1968).
[CrossRef]

Wang, T.

Xu, K.

Yokozeki, S.

S. Yokozeki and K. Patorski, “moiré fringe profile prediction method and its application to fringe sharpening,” Appl. Opt.17(16), 2541–2547 (1978).
[PubMed]

K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
[CrossRef]

Yuyama, I.

Zhong, X.

Appl. Opt. (9)

H. Takasaki, “Moiré topography,” Appl. Opt.9(6), 1467–1472 (1970).
[CrossRef] [PubMed]

S. Yokozeki and K. Patorski, “moiré fringe profile prediction method and its application to fringe sharpening,” Appl. Opt.17(16), 2541–2547 (1978).
[PubMed]

F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt.36(7), 1598–1603 (1997).
[CrossRef] [PubMed]

M. Okui, M. Kobayashi, J. Arai, and F. Okano, “Moire fringe reduction by optical filters in integral three-dimensional imaging on a color flat-panel display,” Appl. Opt.44(21), 4475–4483 (2005).
[CrossRef] [PubMed]

M. Abolhassani and M. Mirzaei, “Unification of formulation of moiré fringe spacing in parametric equation and Fourier analysis methods,” Appl. Opt.46(32), 7924–7926 (2007).
[CrossRef] [PubMed]

H. Choi, J. Kim, S. W. Cho, Y. Kim, J. B. Park, and B. Lee, “Three-dimensional-two-dimensional mixed display system using integral imaging with an active pinhole array on a liquid crystal panel,” Appl. Opt.47(13), 2207–2214 (2008).
[CrossRef] [PubMed]

Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt.48(11), 2178–2187 (2009).
[CrossRef] [PubMed]

J. Hong, Y. Kim, H. J. Choi, J. Hahn, J. H. Park, H. Kim, S. W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt.50(34), H87–H115 (2011).
[CrossRef] [PubMed]

L. S. Kong, G. Jin, T. Wang, S. Cai, X. Zhong, and K. Xu, “Parameter design of a parallax barrier based on the color moiré patterns in autostereoscopic display,” Appl. Opt.50(34), H153–H158 (2011).
[CrossRef] [PubMed]

C. R. Acad. Sci (1)

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci146, 446–451 (1908).

ETRI Journal (1)

D. H. Shin, B. Lee, and E. S. Kim, “Improved Viewing Quality of 3-D Images in Computational Integral Imaging Reconstruction Based on Lenslet Array Model,” ETRI Journal28(4), 521–524 (2006).
[CrossRef]

J. Display Technol (2)

V. V. Saveljev, J.-Y. Son, B. Javidi, S.-K. Kim, and D.-S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Display Technol1(2), 347–353 (2005).
[CrossRef]

V. V. Saveljev, J.-Y. Son, J.-H. Chun, and K.-D. Kwack, “About a Moiré-less condition for non-square grids,” J. Display Technol4(3), 332–339 (2008).
[CrossRef]

J. Korean Phys. Soc. (1)

V. V. Saveljev, “Orientations and branches of moiré waves in three-dimensional displays,” J. Korean Phys. Soc.57(6), 1392–1396 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. E (1)

P. S. Theocaris, “radial gratings as moiré gauges,” J. Phys. E1(6), 613–618 (1968).
[CrossRef]

J.SMPTE (1)

S. H. Kaplan, “Theory of parallax barrier,” J.SMPTE59(1), 11–21 (1976).
[CrossRef]

Jpn. J. Appl. Phys. (2)

K. Patorski, S. Yokozeki, and T. Suzuki, “moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys.15(3), 443–456 (1976).
[CrossRef]

D. H. Shin, E. S. Kim, and B. Lee, “Computational Reconstruction of three-Dimensional Objects in Integral Imaging using Lenslet Array,” Jpn. J. Appl. Phys.44(11), 8016–8018 (2005).
[CrossRef]

Photogr. Sci. Eng (1)

J. P. Allebach, “Random nucleated halftone screen,” Photogr. Sci. Eng22, 89–91 (1978).

Precis. Eng. (1)

A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng.1(2), 61–69 (1979).
[CrossRef]

Proc. SPIE (2)

L. Lipton and M. Feldman, “A new autostereoscopic display technology: the SynthaGram,” Proc. SPIE4660, 229–235 (2002).
[CrossRef]

K. Taira, R. Fukushima, T. Saishu, H. Kobayashi, and Y. Hirayama, “Autostereoscopic liquid crystal display using mosaic color pixel arrangement,” Proc. SPIE5664, 349–359 (2005).
[CrossRef]

Other (9)

T. Koike, M. Oikawa, and K. Utsugi, “Moiré reduction for integral photography,” International Display Workshops (Institute of Image Information and Television Engineers of Japan and Society of Information Display, 2007), 1917–1918.

J. W. Goodman, Introduction to Fourier Optics (Roberts: 2005).

A. J. Durelli and V. J. Parks, Moiré Analysis of strain (Prentice-Hall, Englewood Cliffs: New Jersey, 1970).

K. Creath and J. C. Wyant, “Moiré and fringe projection techniques,” in Optical Shop Testing, 2nd ed. (John Wiley & Sons: NY, 1995), 653–685.

I. Amidror, The Theory of the Moiré Phenomenon, 2nd ed (Springer, 2009).

D. Blattner, C. Chaves, G. Fleishman, and S. Roth, RealWorld Scanning and Halftones (Berkeley, CA: Peachpit, 2004).

K. Mashitani, G. Hamagishi, M. Sakata, A. Yamashita, E. Nakayama,and M. Inoue, “New autostereoscopic (no-glasses) LCD image splitter displays,” in 3 Dimensional Image Conf (1996), 90–95.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier: Amsterdam, 1993).

O. Kafri and I. Glatt, The Physics of Moiré Metrology (John Wiley & Sons, 1989).

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Figures (14)

Fig. 1
Fig. 1

The principle of multiview autostereoscopic displays based on PBs.

Fig. 2
Fig. 2

The interpretation of the relative period of PBs.

Fig. 3
Fig. 3

(a) The equivalent grating of LCD (b) The special radial grating.

Fig. 4
Fig. 4

Superposition of the equivalent grating of LCD and special radial grating.

Fig. 5
Fig. 5

(a) The equivalent grating of LCD (b) Vector sum of Fourier frequencies.

Fig. 6
Fig. 6

Vector sum of Fourier fundamental frequencies of LCD and PB.

Fig. 7
Fig. 7

(a) Frequencies of (1, 0, 1)-moiré and (1, 1, 1)-moiré (b) Frequencies of predominant moiré patterns.

Fig. 8
Fig. 8

(a) θ=2° , P2 = 3P (b) θ=5° , P2 = 3P (c) θ=10° , P2 = 3P (d) θ=2° , P2 = 2.7P (e) θ=2° , P2 = 3.3P (f) θ=2° , P2 = 4.5P (g) θ=5° , P2 = 4.5P (h) θ=10° , P2 = 4.5P (i) θ=26.565° , P2 = 2.4P (j) θ=26.565° , P2 = 3P (k) θ=26.565° , P2 = 3.6P.

Fig. 9
Fig. 9

(a) The equivalent 2D binary grating (b) The special radial grid grating.

Fig. 10
Fig. 10

Superposition of equivalent grating of LCD and special radial grid grating.

Fig. 11
Fig. 11

Radial and circular gratings which are resolved from special radial grid grating.

Fig. 12
Fig. 12

(a) Superposition of equivalent grating of LCD and radial grating (b) Superposition of equivalent grating of LCD and circular grating.

Fig. 13
Fig. 13

Interpretation of the coactions of Figs. 12(a) and 12(b).

Fig. 14
Fig. 14

(a) θ=2° , P3 = 3P (b) θ=5° , P3 = 3P (c) θ=10° , P3 = 3P (d) θ=2° , P3 = 2.7P (e) θ=2° , P3 = 3.3P (f) θ=2° , P3 = 4.5P (g) θ=5° , P3 = 4.5P (h) θ=10° , P3 = 4.5P (i) θ=26.565° , P3 = 2.4P (j) θ=26.565° , P3 = 3P (k) θ=26.565° , P3 = 3.6P.

Tables (2)

Tables Icon

Table 1 Parameters of PB in the singular state

Tables Icon

Table 2 Parameters of PB in the angle stable state

Equations (5)

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

z optimum =g( e+i i )
b=Ni( z optimum g z optimum )
b`= z zg b
| f 1 |sin(α)= 2 | f 1 |sin(45α)
tan(α)=0.5,α= 26.565

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