Abstract

We investigate the technical feasibility of the optical implementation of a four-phase-level diffractive element with two {0, π}-phase spatial light modulators in a fractional Talbot configuration. The space–bandwidth product of the spatial light modulators is seen as the main theoretical limitation of the proposed approach. We investigate the robustness of technological and geometrical parameters on the diffraction efficiency of the whole system. Ferroelectric liquid-crystal silicon backplane spatial light modulators are chosen because of their high reconfiguration rates and good electro-optics interface. Similarly, we assess the influence of liquid-crystal technical parameters on system performance.

© 1998 Optical Society of America

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References

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  1. I. Underwood, D. G. Vass, A. O’Hara, D. C. Burns, P. W. McOwan, J. Gourlay, “Improving the performance of liquid-crystal-over-silicon spatial light modulators: issues and achievements,” Appl. Opt. 33, 2768–2774 (1994).
    [CrossRef] [PubMed]
  2. D. J. McKnight, K. M. Johnson, R. A. Serati, “256 × 256 liquid-crystal-on-silicon spatial light modulator,” Appl. Opt. 33, 2775–2784 (1994).
    [CrossRef] [PubMed]
  3. D. C. Burns, I. Underwood, J. Gourlay, A. O’Hara, D. G. Vass, “A 256 × 256 SRAM-XOR pixel ferroelectric liquid crystal over silicon spatial light modulators,” Opt. Commun. 119, 623–632 (1995).
    [CrossRef]
  4. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 12, 478–484 (December1969).
  5. T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Dynamic asymmetric binary holograms using FLC SLM’s,” Opt. Commun. 109, 222–226 (1994).
    [CrossRef]
  6. 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]
  7. S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “Programmable multiple-level phase modulation using ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 34, 6652–6665 (1995).
    [CrossRef] [PubMed]
  8. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Fractional Talbot four-level hologram using ferroelectric liquid-crystal spatial light modulators,” Opt. Lett. 19, 1654–1656 (1994).
    [CrossRef] [PubMed]
  9. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).
  10. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  11. J. L. de Bougrenet de la Tocnaye, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
    [CrossRef]
  12. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Programmable joint fractional Talbot computer-generated holography,” J. Opt. Soc. Am. A 12, 314–324 (1995).
    [CrossRef]
  13. R. Moignard, J. L. de Bougrenet de la Tocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
    [CrossRef]
  14. A. O’Hara, J. R. Hannah, I. Underwood, D. G. Vass, R. J. Holwill, “Mirror quality and efficiency improvements of reflective spatial light modulators by the use of dielectric coatings and chemical–mechanical polishing,” Appl. Opt. 32, 5549–5556 (1993).
    [CrossRef]
  15. J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
    [CrossRef] [PubMed]
  16. P. Berthelé, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Efficient beam steering in 1.55 micron window using large tilt FLC 1d array,” Ferroelectrics214 (to be published).
  17. M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
    [CrossRef]

1998

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

1997

1996

R. Moignard, J. L. de Bougrenet de la Tocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

1995

J. L. de Bougrenet de la Tocnaye, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
[CrossRef]

D. C. Burns, I. Underwood, J. Gourlay, A. O’Hara, D. G. Vass, “A 256 × 256 SRAM-XOR pixel ferroelectric liquid crystal over silicon spatial light modulators,” Opt. Commun. 119, 623–632 (1995).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “Programmable multiple-level phase modulation using ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 34, 6652–6665 (1995).
[CrossRef] [PubMed]

H. Hamam, J. L. de Bougrenet de la Tocnaye, “Programmable joint fractional Talbot computer-generated holography,” J. Opt. Soc. Am. A 12, 314–324 (1995).
[CrossRef]

1994

1993

1992

1969

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 12, 478–484 (December1969).

1965

1836

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Berthelé, P.

P. Berthelé, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Efficient beam steering in 1.55 micron window using large tilt FLC 1d array,” Ferroelectrics214 (to be published).

Broomfield, S. E.

Brown, T. A.

Burns, D. C.

D. C. Burns, I. Underwood, J. Gourlay, A. O’Hara, D. G. Vass, “A 256 × 256 SRAM-XOR pixel ferroelectric liquid crystal over silicon spatial light modulators,” Opt. Commun. 119, 623–632 (1995).
[CrossRef]

I. Underwood, D. G. Vass, A. O’Hara, D. C. Burns, P. W. McOwan, J. Gourlay, “Improving the performance of liquid-crystal-over-silicon spatial light modulators: issues and achievements,” Appl. Opt. 33, 2768–2774 (1994).
[CrossRef] [PubMed]

de Bougrenet de la Tocnaye, J. L.

J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
[CrossRef] [PubMed]

R. Moignard, J. L. de Bougrenet de la Tocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

J. L. de Bougrenet de la Tocnaye, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
[CrossRef]

H. Hamam, J. L. de Bougrenet de la Tocnaye, “Programmable joint fractional Talbot computer-generated holography,” J. Opt. Soc. Am. A 12, 314–324 (1995).
[CrossRef]

H. Hamam, J. L. de Bougrenet de la Tocnaye, “Fractional Talbot four-level hologram using ferroelectric liquid-crystal spatial light modulators,” Opt. Lett. 19, 1654–1656 (1994).
[CrossRef] [PubMed]

P. Berthelé, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Efficient beam steering in 1.55 micron window using large tilt FLC 1d array,” Ferroelectrics214 (to be published).

Dupont, L.

Fracasso, B.

P. Berthelé, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Efficient beam steering in 1.55 micron window using large tilt FLC 1d array,” Ferroelectrics214 (to be published).

Freeman, M. O.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 12, 478–484 (December1969).

Gourlay, J.

D. C. Burns, I. Underwood, J. Gourlay, A. O’Hara, D. G. Vass, “A 256 × 256 SRAM-XOR pixel ferroelectric liquid crystal over silicon spatial light modulators,” Opt. Commun. 119, 623–632 (1995).
[CrossRef]

I. Underwood, D. G. Vass, A. O’Hara, D. C. Burns, P. W. McOwan, J. Gourlay, “Improving the performance of liquid-crystal-over-silicon spatial light modulators: issues and achievements,” Appl. Opt. 33, 2768–2774 (1994).
[CrossRef] [PubMed]

Hamam, H.

Hannah, J. R.

Holwill, R. J.

Johnson, K. M.

McKnight, D. J.

McOwan, P. W.

Mears, R. J.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Dynamic asymmetric binary holograms using FLC SLM’s,” Opt. Commun. 109, 222–226 (1994).
[CrossRef]

Moignard, R.

R. Moignard, J. L. de Bougrenet de la Tocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

Mori, K. M.

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

Neil, M. A. A.

O’Brien, D. C.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Dynamic asymmetric binary holograms using FLC SLM’s,” Opt. Commun. 109, 222–226 (1994).
[CrossRef]

O’Hara, A.

Paige, E. G. S.

Serati, R. A.

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 12, 478–484 (December1969).

Talbot, F.

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Thurnes, W. N.

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

Underwood, I.

Vass, D. G.

Vohra, R. T.

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

Walba, D. M.

Wand, M. D.

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

Wilkinson, T. D.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Dynamic asymmetric binary holograms using FLC SLM’s,” Opt. Commun. 109, 222–226 (1994).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Appl. Opt.

Ferroelectrics

M. D. Wand, R. T. Vohra, W. N. Thurnes, K. M. Mori, “High resolution color FLC miniature display and FLC materials optimized for thin operation,” Ferroelectrics 213, 187–194 (1998).
[CrossRef]

IBM J. Res. Dev.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 12, 478–484 (December1969).

J. Opt. (Paris)

J. L. de Bougrenet de la Tocnaye, H. Hamam, “La transformée de Fresnel fractionnaire,” J. Opt. (Paris) 26, 49–56 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

R. Moignard, J. L. de Bougrenet de la Tocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996).
[CrossRef]

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Dynamic asymmetric binary holograms using FLC SLM’s,” Opt. Commun. 109, 222–226 (1994).
[CrossRef]

D. C. Burns, I. Underwood, J. Gourlay, A. O’Hara, D. G. Vass, “A 256 × 256 SRAM-XOR pixel ferroelectric liquid crystal over silicon spatial light modulators,” Opt. Commun. 119, 623–632 (1995).
[CrossRef]

Opt. Lett.

Philos. Mag.

F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Other

P. Berthelé, B. Fracasso, J. L. de Bougrenet de la Tocnaye, “Efficient beam steering in 1.55 micron window using large tilt FLC 1d array,” Ferroelectrics214 (to be published).

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Figures (15)

Fig. 1
Fig. 1

Principle of the realization of a four-phase-level hologram by use of the fractional Talbot effect.

Fig. 2
Fig. 2

(a) First-plane basic pattern. (b) Second-plane basic pattern. (c) Reconstruction plane.

Fig. 3
Fig. 3

Single-wafer SBEASLM: transmission and reflection configurations.

Fig. 4
Fig. 4

Influence of the distance d between the two planes on the diffraction efficiency for 0 cm ≤ d ≤ 64 cm.

Fig. 5
Fig. 5

Influence of the distance between the two planes on the diffraction efficiency near the optimal position.

Fig. 6
Fig. 6

Influence of the pitch and the yaw angles of the two planes with respect to the propagation direction on the diffraction efficiency.

Fig. 7
Fig. 7

Optimal configuration for {0, π} modulation with FLC cells in the real case.

Fig. 8
Fig. 8

Diffraction efficiency with respect to the technological parameter: thickness-constant error and FLC tilt-angle error.

Fig. 9
Fig. 9

Diffraction efficiency with respect to the linear thickness error.

Fig. 10
Fig. 10

Diffraction efficiency with respect to the quadratic thickness error.

Fig. 11
Fig. 11

Influence of the number of repetitions of the basic pattern on the diffraction efficiency.

Fig. 12
Fig. 12

Picture of the diffractive elements on the glass substrate. One can observe the effect of overengraving, which makes the corners of the pixels more rounded.

Fig. 13
Fig. 13

Profile of the diffractive element shown in Fig. 12. The optimal thickness for performing {0, π} modulation with this photoresist at λ = 0.633 μm is 0.494 μm. The measured thickness is 0.499 μm.

Fig. 14
Fig. 14

Measured diffraction efficiency for 20 basic patterns with respect to the distance near the optimal position.

Fig. 15
Fig. 15

Measured diffraction efficiency at the optimal distance with respect to the number of repetitions of the basic pattern.

Tables (1)

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Table 1 Talbot Distances as a Function of λ and Pixel Size

Equations (28)

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H d x , d y x ,   y ,   z = 1 2 exp - i   π 4 H d x , d y x ,   y ,   0 + exp + i   π 4 H d x , d y x ,   y - d y 2 ,   0 ,
z = 1 4   Z T d y = 1 4 2 d y 2 λ = Z T d x = 2 d x 2 λ ,
SBWP SLM s     1 2 SBWP Basic   Pattern 2 ,
ϕ = 2 π λ   Δ ne ,
ϕ x ,   y = π + ϕ 0 ,
ϕ x ,   y = π + α x + β y ,
ϕ x ,   y = π + γ x 2 - T x 2 3 + η y 2 - T y 2 3 ,
e = a c r + b c l ,
e S = a   cos ϕ 2 - jb   sin ϕ 2 exp + 2 js θ t c r + - ja   sin ϕ 2 exp - 2 js θ t + b   cos ϕ 2 c l ,
s = + 1 ,     if   S = + , s = - 1 ,     if   S = - .
e S = s b c r - a c l .
ϕ = ϕ x ,   y ,
θ t = π 4 - c ,
e 1 = b c r - a c l , e 2 = a ¯ c r + b ¯ c l ,
c r = b ¯ e 1 + a e 2 , c l = - a ¯ e 1 + b e 2 .
e S = a b ¯ - a ¯ b cos ϕ 2 + s   sin ϕ 2 | b | 2 exp - 2 jsc + | a | 2 exp + 2 jsc e 1 + a 2 + b 2 cos ϕ 2 + 2 jab   sin ϕ 2 sin 2 c e 2 .
a b ¯ - a ¯ b = 0 , | b | 2 exp - 2 jsc + | a | 2 exp + 2 jsc = 1 ,
| b | 2 exp - 2 jsc + | a | 2 exp + 2 jsc = cos 2 c + js | a | 2 - | b | 2 sin 2 c .
| b | 2 - | a | 2 = 0 , a b ¯ - a ¯ b = 0 .
e S = s   sin ϕ 2 cos 2 c e 1 + cos ϕ 2 + j   sin ϕ 2 sin 2 c e 2 .
p 1 + = 1 2 c r + exp - 2 j θ t c l , p 2 + = 1 2 c r - exp - 2 j θ t c l .
p 1 - = 1 2 c r + exp + 2 j θ t c l , p 2 - = 1 2 c r - exp + 2 j θ t c l .
c r = 1 2 p 1 + + p 2 + = 1 2 p 1 - + p 2 - , c l = 1 2 exp + 2 j θ t p 1 + - p 2 + = 1 2 exp - 2 j θ t p 1 - - p 2 - .
e = 1 2 a + b   exp + 2 j θ t p 1 + + 1 2 a - b   exp + 2 j θ t p 2 + ,
e = 1 2 a + b   exp - 2 j θ t p 1 - + 1 2 a - b   exp - 2 j θ t p 2 - .
e + = 1 2 a + b   exp + 2 j θ t exp - j   ϕ 2 p 1 + + 1 2 a - b   exp + 2 j θ t exp + j   ϕ 2 p 2 + , e - = 1 2 a + b   exp - 2 j θ t exp - j   ϕ 2 p 1 - + 1 2 a - b   exp - 2 j θ t exp + j   ϕ 2 p 2 - .
e S = a   cos ϕ 2 - jb   sin ϕ 2 exp + 2 js θ t c r + - ja   sin ϕ 2 exp - 2 js θ t + b   cos ϕ 2 c l ,
s = + 1 ,     if   S = + , s = - 1 ,     if   S = - .

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