Abstract

Binary holograms have become more interesting since spatial light modulators, capable of binary phase modulation, were developed. The primary restriction of these dynamic elements when used with the Fourier transform is the symmetry of the hologram reconstruction. I describe a new type of hologram that uses the Hartley transform instead of the Fourier transform. Despite their binary form, Hartley holograms offer maximal efficiency (theoretically 100%). Thus they can be presented as high diffraction-efficiency programmable elements. For practical reasons, I also propose a modified version for Hartley holograms that is easy to use. A theoretical analysis as well as experimental results are given.

© 1996 Optical Society of America

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References

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  1. S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
    [CrossRef]
  2. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Binary logic based purely on Fresnel diffraction,” Appl. Opt. 34, 5901–5906 (1995).
    [CrossRef] [PubMed]
  3. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Array illuminators using multi-layer binary phase plates at fractional Talbot planes,” Appl. Opt. 35, 1820–1826 (1996).
    [CrossRef] [PubMed]
  4. D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical correlator performance of binary phase-only filters using Fourier and Hartley transforms,” Appl. Opt. 26, 3755–3761 (1987).
    [CrossRef] [PubMed]
  5. R. N. Bracewell, “Discrete Hartley transform,” J. Opt. Soc. Am. 73, 1832–1835 (1983).
    [CrossRef]
  6. R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Steibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
    [CrossRef] [PubMed]
  7. M. O. Freeman, T. A. Brown, D. M. Walba, “Quantized complex ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 31, 3917–3929 (1992).
    [CrossRef] [PubMed]
  8. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Diffraction efficiency of quantized phase elements: practical assessments,” Pure Appl. Opt. (to be published).
  9. F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
    [CrossRef]
  10. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Programmable joint fractional talbot computer generated holograms,” J. Opt. Soc. Am. A 12, 314–324 (1995).
    [CrossRef]
  11. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]

1996

1995

1992

M. O. Freeman, T. A. Brown, D. M. Walba, “Quantized complex ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 31, 3917–3929 (1992).
[CrossRef] [PubMed]

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

1987

1985

1983

1970

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Bartelt, H.

Bracewell, R. N.

Broomfield, S. E.

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Brown, T. A.

Cottrell, D. M.

Davis, J. A.

Day, T.

de Bougrenet de la Tocnaye, J. L.

Freeman, M. O.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Hamam, H.

Lilly, R. A.

Lohmann, A. W.

Neil, M. A.

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Paige, E. S.

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Steibl, N.

Walba, D. M.

Wyrowski, F.

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

Yang, G. G.

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

Appl. Opt.

Electron. Lett.

S. E. Broomfield, M. A. Neil, E. S. Paige, G. G. Yang, “Programmable binary phase only optical device based on FLC SLM,” Electron. Lett. 28, 26–28 (1992).
[CrossRef]

IBM J. Res. Dev.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

Other

H. Hamam, J. L. de Bougrenet de la Tocnaye, “Diffraction efficiency of quantized phase elements: practical assessments,” Pure Appl. Opt. (to be published).

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Figures (6)

Fig. 1
Fig. 1

Projection of the spectrum value onto the closest quantization level for two cases: (a) Fourier and (b) Hartley.

Fig. 2
Fig. 2

Physical encoding of the data distribution (a) original and (b) modified.

Fig. 3
Fig. 3

Two implementations of the mirroring element: (a) in transmission (lenses L1 and L2 have, respectively, the focal lengths f and 2f) and (b) in reflection for which polarization and phase change have to be taken into account.

Fig. 4
Fig. 4

Optical setup to implement the Hartley transform.

Fig. 5
Fig. 5

Experimental result of the reconstruction of a modified Hartley hologram.

Fig. 6
Fig. 6

Reconstruction of the second elementary diffractive element of only the modified Hartley hologram.

Equations (17)

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H H ( u ) = HT { h ( x ) } ( u ) = h ( x ) cas ( 2 π x u ) d x ,
H H ( u ) = 1 / 2 { exp [ i ( π / 4 ) ] H ( u ) + exp [ i ( π / 4 ) ] H ( u ) } ,
H ( u ) = FT { h ( x ) } ( u ) = h ( x ) exp ( i 2 π x u ) d x ,
H ( u ) = H ( u ) * ,
η = { | R ( u ) | cos [ φ ( u ) ] d u Δ U 2 } 2 ,
η = [ | R ( u ) | d u ] Δ U 2 2 = 1 .
FT { h ( x ) } ( u ) = 1 × FT { h ( x ) } ( u ) = 1 × H ( u ) ,
H H ( u ) = exp [ i ( π / 2 ) ] FT { g ( x ) } ( u ) ,
g ( x ) = 1 / 2 { exp [ i ( π / 4 ) ] h ( x ) + exp [ i ( π / 4 ) ] h ( x ) } .
h ( x ) = h ( x d / 2 ) ,
g ( x ) = h ( x , Z T / 4 ) ,
H H ( u ) = exp [ i ( π / 2 ) ] FT { h ( x , Z T / 4 ) } ( u ) ,
h ( x , Z T / 4 ) = 1 / 2 { exp [ i ( π / 4 ) ] h ( x ) + exp [ i ( π / 4 ) ] h ( x d / 2 ) } .
h ˜ ( x ) = h ( x ) , h ˜ ( x d / 2 ) = h ( x ) .
h ˜ ( x , Z T / 4 ) = g ( 3 d / 2 x ) .
h ˜ ( x , Z T / 4 ) = h ˜ ( x d / 2 , Z T / 4 ) * .
h ˜ ( x , Z T / 4 ) = h ˜ ( d x , Z T / 4 ) * .

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