Abstract

Computation of a binary spatial light modulator (SLM) pattern that generates a desired light field is a challenging quantization problem for which several algorithms have been proposed, mainly for far-field or Fourier plane reconstructions. We study this problem assuming that the desired light field is synthesized within a volumetric region in the non-far-field range after free space propagation from the SLM plane. We use Fresnel and Rayleigh–Sommerfeld scalar diffraction theories for propagation of light. We show that, when the desired field is confined to a sufficiently narrow region of space, the ideal gray-level complex-valued SLM pattern generating it becomes sufficiently low pass (oversampled) so it can be successfully halftoned into a binary SLM pattern by solving two decoupled real-valued constrained halftoning problems. Our simulation results indicate that, when the synthesis region is considered, the binary SLM is indistinguishable from a lower resolution full complex gray-level SLM. In our approach, free space propagation related computations are done only once at the beginning, and the rest of the computation time is spent on carrying out standard image halftoning.

© 2011 Optical Society of America

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2011

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

2010

2008

2007

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
[CrossRef]

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Signal Process. Image Commun. 22, 169–177 (2007).
[CrossRef]

2006

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Y. Chang, P. Zhou, and J. H. Burge, “Analysis of phase sensitivity for binary computer-generated holograms,” Appl. Opt. 45, 4223–4234 (2006).
[CrossRef] [PubMed]

2005

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

2004

2003

2001

L. Bigue and P. Ambs, “Optimal multicriteria approach to the iterative Fourier transform algorithm,” Appl. Opt. 40, 5886–5893(2001).
[CrossRef]

T. Kreis, P. Aswendt, and R. Hofling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. 40, 926–933 (2001).
[CrossRef]

2000

1999

1998

1997

M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” ACM SIGGRAPH Comput. Graph. 31, 63–67 (1997).
[CrossRef]

R. Piestun, B. Spektor, and J. Shamir, “On-axis binary-amplitude computer-generated holograms,” Opt. Commun. 136, 85–92(1997).
[CrossRef]

L. Legeard, P. Refregier, and P. Ambs, “Multicriteria optimality for iterative encoding of computer-generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
[CrossRef]

1996

1995

1994

1993

1992

A. Kirk, K. Powell, and T. Hall, “A generalization of the error diffusion method for binary computer-generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

O. K. Ersoy, J. Zhuang, and J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt. 31, 6894–6901 (1992).
[CrossRef] [PubMed]

1991

1990

1989

1987

1984

1981

1979

1975

1974

1971

1970

1968

1967

1966

Abookasis, D.

Allebach, J. P.

Ambs, P.

Aswendt, P.

T. Kreis, P. Aswendt, and R. Hofling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. 40, 926–933 (2001).
[CrossRef]

Athale, R. A.

J. A. Neff, R. A. Athale, and S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Bernet, S.

Bigue, L.

Brede, J.

Brown, B. R.

Bryngdahl, O.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

P. Thorston, F. Wyrowski, and O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

R. Hauck and O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10(1984).
[CrossRef]

O. Bryngdahl and F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E.Wolf, ed. (Elsevier, 1990), Vol.  28, pp. 1–86.
[CrossRef]

Burge, J. H.

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Castaneda, J. O.

Cathey, W. T.

Chang, Y.

Chavel, P.

Chen, C.

Chevallier, R.

Chhetri, B. B.

Cottrell, D. M.

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in Digital Holography and Three-Dimensional Display, T.C.Poon, ed. (Springer, 2006), pp. 1–49.
[CrossRef]

Davis, J. A.

Dietrich, C. H.

Dudgeon, D. E.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, 1990).

Dudley, D.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” White Paper (Texas Instruments, 2003).

Duncan, W.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” White Paper (Texas Instruments, 2003).

Eriksson, N.

Ersoy, O. K.

Eschbach, R.

Fan, Z.

Fetthauer, F.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
[CrossRef]

Fiddy, M. A.

Frauel, Y.

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Gabel, R. A.

Galyean, T. A.

M. Lucente and T. A. Galyean, “Rendering interactive holographic images,” in Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1995), pp. 387–394.

Goodman, J. W.

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

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Gotchev, A.

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Hagberg, M.

Hall, T.

A. Kirk, K. Powell, and T. Hall, “A generalization of the error diffusion method for binary computer-generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Hanak, I.

Hauck, R.

Heggarty, K.

Henton, R. F.

Hofling, R.

T. Kreis, P. Aswendt, and R. Hofling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. 40, 926–933 (2001).
[CrossRef]

Janda, M.

Javidi, B.

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Jennison, B. K.

Jesacher, A.

Kang, H.

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

F. Yaras, H. Kang, and L. Onural, “State of the art in holographic displays: A survey,” J. Display Technol. 6, 443–454 (2010).
[CrossRef]

Kirk, A.

A. Kirk, K. Powell, and T. Hall, “A generalization of the error diffusion method for binary computer-generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Kreis, T.

T. Kreis, P. Aswendt, and R. Hofling, “Hologram reconstruction using a digital micromirror device,” Opt. Eng. 40, 926–933 (2001).
[CrossRef]

Lalor, E.

Larsson, A.

Lee, S. H.

J. A. Neff, R. A. Athale, and S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Lee, W.

Legeard, L.

Li, M.

Liu, B.

Lohmann, A. W.

Lucente, M.

M. Lucente, “Interactive three-dimensional holographic displays: seeing the future in depth,” ACM SIGGRAPH Comput. Graph. 31, 63–67 (1997).
[CrossRef]

M. Lucente and T. A. Galyean, “Rendering interactive holographic images,” in Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1995), pp. 387–394.

Mait, J. N.

Manner, R.

Matoba, O.

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Maurer, C.

Mersereau, R. M.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, 1990).

Naughton, T. J.

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Neff, J. A.

J. A. Neff, R. A. Athale, and S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Noehte, S.

Onural, L.

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

F. Yaras, H. Kang, and L. Onural, “State of the art in holographic displays: A survey,” J. Display Technol. 6, 443–454 (2010).
[CrossRef]

M. Janda, I. Hanak, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096(2008).
[CrossRef]

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
[CrossRef]

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Signal Process. Image Commun. 22, 169–177 (2007).
[CrossRef]

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
[CrossRef]

Ozaktas, H. M.

L. Onural and H. M. Ozaktas, “Signal processing issues in diffraction and holographic 3DTV,” Signal Process. Image Commun. 22, 169–177 (2007).
[CrossRef]

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Paris, D. P.

Piestun, R.

R. Piestun, B. Spektor, and J. Shamir, “On-axis binary-amplitude computer-generated holograms,” Opt. Commun. 136, 85–92(1997).
[CrossRef]

Powell, K.

A. Kirk, K. Powell, and T. Hall, “A generalization of the error diffusion method for binary computer-generated hologram design,” Opt. Commun. 92, 12–18 (1992).
[CrossRef]

Ransom, P. L.

Refregier, P.

Ritsch-Marte, M.

Rosen, J.

Sass, A. R.

Schwaighofer, A.

Seldowitz, M. A.

Shamir, J.

R. Piestun, B. Spektor, and J. Shamir, “On-axis binary-amplitude computer-generated holograms,” Opt. Commun. 136, 85–92(1997).
[CrossRef]

Sherman, G. C.

Shimomura, T.

Slaughter, J.

D. Dudley, W. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” White Paper (Texas Instruments, 2003).

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Spektor, B.

R. Piestun, B. Spektor, and J. Shamir, “On-axis binary-amplitude computer-generated holograms,” Opt. Commun. 136, 85–92(1997).
[CrossRef]

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Stoykova, E.

L. Onural, A. Gotchev, H. M. Ozaktas, and E. Stoykova, “A survey of signal processing problems and tools in holographic three-dimensional television,” IEEE Trans. Circuits Syst. Video Technol. 17, 1631–1646 (2007).
[CrossRef]

Suh, H. H.

Sweeney, D. W.

Tajahuerce, E.

Y. Frauel, T. J. Naughton, O. Matoba, E. Tajahuerce, and B. Javidi, “Three-dimensional imaging and processing using computational holographic imaging,” Proc. IEEE 94, 636–653 (2006).
[CrossRef]

Tao, S. H.

Thorston, P.

P. Thorston, F. Wyrowski, and O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. 40, 591–600 (1993).
[CrossRef]

Tricoles, G.

Tucker, S. B.

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, 1987).

Valadez, K. O.

Waters, J. P.

Weissbach, S.

F. Fetthauer, S. Weissbach, and O. Bryngdahl, “Equivalence of error diffusion and minimal average error algorithms,” Opt. Commun. 113, 365–370 (1995).
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Figures (15)

Fig. 1
Fig. 1

Real part of u z ( x , y ) .

Fig. 2
Fig. 2

Actual SLM pattern s ¯ [ m , n ] .

Fig. 3
Fig. 3

Output produced when s ¯ [ m , n ] is written on the SLM (only the central diffraction order is displayed).

Fig. 4
Fig. 4

Low-pass filtered SLM pattern s ¯ L [ m , n ] .

Fig. 5
Fig. 5

Output produced when s ¯ L [ m , n ] is written on the SLM. Also shown are the borders of the synthesis region given in Eq. (22).

Fig. 6
Fig. 6

Desired field. Entire central diffraction order is displayed.

Fig. 7
Fig. 7

Real part of ideal gray-level SLM pattern.

Fig. 8
Fig. 8

Three-level SLM pattern for real part obtained by solving the first constrained halftoning problem in Eq. (30). Even pixels are ± 1 , odd pixels are 0.

Fig. 9
Fig. 9

Binary SLM pattern obtained by adding the three-level SLM patterns for real and imaginary parts.

Fig. 10
Fig. 10

Output produced by the binary SLM pattern in Fig. 9.

Fig. 11
Fig. 11

Updated version of the binary SLM pattern in Fig. 9 to be used when the mask is removed and oblique illumination is used.

Fig. 12
Fig. 12

Output produced by the binary SLM pattern in Fig. 11 when normally incident illumination is used instead of oblique illumination.

Fig. 13
Fig. 13

Binary SLM pattern.

Fig. 14
Fig. 14

Output produced by the binary SLM pattern in Fig. 13 at z = 0.8 m .

Fig. 15
Fig. 15

Output produced by the binary SLM pattern in Fig. 13 at z = 1.0 m .

Equations (34)

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u z ( x , y ) = u 0 ( x , y ) * * h z ( x , y ) = u 0 ( x , y ) h z ( x x , y y ) d x d y ,
h z ( x , y ) = z j λ e j k R R 2 ,
H z ( f x , f y ) = F { h z ( x , y ) } = h z ( x , y ) exp { j 2 π ( x f x + y f y ) } d x d y = exp { j k z 1 ( λ f x ) 2 ( λ f y ) 2 } rect ( ( λ f x ) 2 + ( λ f y ) 2 2 ) ,
h z ( x , y ) = e j k z j λ z e j π λ z ( x 2 + y 2 ) .
H z ( f x , f y ) = e j k z exp { j π λ z ( f x 2 + f y 2 ) } ,
h z ( x , y ) * * B x B y sinc ( x B x ) sinc ( y B y ) h z ( x , y ) rect ( x λ z B x ) rect ( y λ z B y ) .
s a ( x , y ) = m = M M n = N N s ¯ [ m , n ] a ( x m Δ x , y n Δ y ) .
s ( x , y ) = m = M M n = N N s ¯ [ m , n ] sinc ( x m Δ x Δ x ) sinc ( y n Δ y Δ y ) .
s a ( x , y ) = a ( x , y ) * * { s ( x , y ) m = n = δ ( x m Δ x , y n Δ y ) } = 1 Δ x Δ y a ( x , y ) * * s ( x , y ) + 1 Δ x Δ y p = ( p , q ) q = ( 0 , 0 ) a ( x , y ) * * [ s ( x , y ) exp { j 2 π ( p x Δ x + q y Δ y ) } ] ,
1 Δ x Δ y e j π λ z ( p 2 Δ x 2 + q 2 Δ y 2 ) u z ( x p λ z Δ x , y q λ z Δ y ) exp { j 2 π ( p x Δ x + q y Δ y ) } ,
| x | < λ z 2 Δ x , | y | < λ z 2 Δ y .
u z ( x , y ) Δ x Δ y rect ( x Δ x λ z ) rect ( y Δ y λ z ) m = M M n = N N s ¯ [ m , n ] h z ( x m Δ x , y n Δ y ) .
u z a ( x , y ) 1 Δ x Δ y u z ( x , y ) for     | x | < λ z 2 Δ x , | y | < λ z 2 Δ y .
u z a ( x , y ) = e j k z j λ z e j π λ z ( x 2 + y 2 ) m = M M n = N N s ¯ [ m , n ] e j π λ z ( m 2 Δ x 2 + n 2 Δ y 2 ) e j 2 π λ z ( x m Δ x + y n Δ y )
u ¯ z a [ m , n ] = u z a ( m λ z M s Δ x , n λ z N s Δ y ) = e j k z j λ z e j π λ z ( m 2 M s 2 Δ x 2 + n 2 N s 2 Δ y 2 ) m = M M n = N N s ¯ [ m , n ] e j π λ z ( m 2 Δ x 2 + n 2 Δ y 2 ) e j 2 π ( m m M s + n n N s )
s ¯ [ m , n ] = j λ z e j k z M s N s e j π λ z ( m 2 Δ x 2 + n 2 Δ y 2 ) m = M M n = N N u ¯ z a [ m , n ] e j π λ z ( m 2 M s 2 Δ x 2 + n 2 N s 2 Δ y 2 ) e j 2 π ( m m M s + n n N s )
s ¯ L [ m , n ] = g ¯ [ m , n ] s ¯ [ m , n ] = m = n = g ¯ [ m , n ] s ¯ [ m m , n n ] .
g ( x , y ) = B x B y sinc ( x B x ) sinc ( y B y ) ,
G ( f x , f y ) = rect ( f x B x ) rect ( f y B y ) .
B x < 1 Δ x , B y < 1 Δ y ,
s L ( x , y ) = Δ x Δ y m = M M n = N N s ¯ [ m , n ] g ( x m Δ x , y n Δ y ) .
| x | < λ z B x 2 , | y | < λ z B y 2 .
u z L ( x , y ) Δ x Δ y m = M M n = N N s ¯ [ m , n ] h z ( x m Δ x , y n Δ y ) u z ( x , y ) ,
u z L a ( x , y ) u z a ( x , y ) rect ( x λ z B x ) rect ( y λ z B y ) .
u ¯ z a [ m , n ] = { d ¯ [ m , n ] for samples within the synthesis region 0 for samples within the do not care region ,
s ¯ b [ m , n ] g ¯ [ m , n ] s ¯ i [ m , n ] ,
T ( x , y ) = m = M M n = N N T ¯ [ m , n ] a ( x m Δ x , y n Δ y ) ,
T ¯ [ m , n ] = { 1 when   m + n   is even j when   m + n   is odd .
s ¯ T [ m , n ] = { ± 1 when   m + n   is even ± j when   m + n   is odd .
s ¯ T R [ m , n ] = { ± 1 when   m + n   is even 0 when   m + n   is odd , s ¯ T I [ m , n ] = { 0 when   m + n   is even ± 1 when   m + n   is odd .
s ¯ T R [ m , n ] g ¯ [ m , n ] R { s ¯ i [ m , n ] } , s ¯ T I [ m , n ] g ¯ [ m , n ] I { s ¯ i [ m , n ] } ,
I ( x , y ) = exp { j π 2 ( x Δ x + y Δ y ) } .
I ¯ [ m , n ] = exp { j π 2 ( m + n ) } ,
s ¯ b 1 [ m , n ] = { s ¯ b [ m , n ] when   T ¯ [ m , n ] I ¯ [ m , n ] = 1 s ¯ b [ m , n ] when   T ¯ [ m , n ] I ¯ [ m , n ] = 1 ,

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