Abstract

It is possible to reduce the diffraction peaks of a Spatial Light Modulator (SLM) by breaking the periodicity of the pixels shape. We propose a theoretical investigation of a SLM that would be based on a Voronoi diagram, obtained by deforming a regular grid, and show that for a specific deformation parameter the diffraction peaks disappear and are replaced with a speckle-like diffraction halo. We also develop a simple model to determine the shape and the level of this halo.

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References

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  1. S.-W. Chung and Y.-K. Kim, “Design and fabrication of 10×10 micro-spatial light modulator array for phase and amplitude modulation,” Sens. Actuators A 78(1), 63–70 (1999).
    [Crossref]
  2. J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
    [Crossref]
  3. J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” Proc. SPIE 2641, 64–75 (1995).
    [Crossref]
  4. L. J. Hornbeck, “Deformable mirror spatial light modulator,” Spat. Light Modulators Appl. 3 SPIE Crit. Rev. 1150, 86–102 (1990).
  5. R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).
  6. S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
    [Crossref]
  7. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36(7), 1517–1524 (1997).
    [Crossref] [PubMed]
  8. D. M. Cottrell, J. A. Davis, T. R. Hedman, and R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29(17), 2505–2509 (1990).
    [Crossref] [PubMed]
  9. W-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119–232 (1978).
    [Crossref]
  10. G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 134, 198–287 (1908).
    [Crossref]

1999 (1)

S.-W. Chung and Y.-K. Kim, “Design and fabrication of 10×10 micro-spatial light modulator array for phase and amplitude modulation,” Sens. Actuators A 78(1), 63–70 (1999).
[Crossref]

1998 (1)

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
[Crossref]

1997 (1)

1995 (1)

J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” Proc. SPIE 2641, 64–75 (1995).
[Crossref]

1993 (1)

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[Crossref]

1990 (2)

1978 (1)

W-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119–232 (1978).
[Crossref]

1908 (1)

G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 134, 198–287 (1908).
[Crossref]

Apte, R. B.

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).

Banyai, W. C.

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).

Bloom, D. M.

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).

Chung, S.-W.

S.-W. Chung and Y.-K. Kim, “Design and fabrication of 10×10 micro-spatial light modulator array for phase and amplitude modulation,” Sens. Actuators A 78(1), 63–70 (1999).
[Crossref]

Cottrell, D. M.

Davis, J. A.

Hedman, T. R.

Hornbeck, L. J.

L. J. Hornbeck, “Deformable mirror spatial light modulator,” Spat. Light Modulators Appl. 3 SPIE Crit. Rev. 1150, 86–102 (1990).

Kim, H. Y.

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
[Crossref]

Kim, Y.-K.

S.-W. Chung and Y.-K. Kim, “Design and fabrication of 10×10 micro-spatial light modulator array for phase and amplitude modulation,” Sens. Actuators A 78(1), 63–70 (1999).
[Crossref]

Lee, S. H.

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
[Crossref]

Lee, S. L.

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
[Crossref]

Lee, W-H.

W-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119–232 (1978).
[Crossref]

Lilly, R. A.

Love, G. D.

Sandejas, F. S. A.

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).

Voronoi, G. F.

G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 134, 198–287 (1908).
[Crossref]

Younse, J. M.

J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” Proc. SPIE 2641, 64–75 (1995).
[Crossref]

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. H. Lee, S. L. Lee, and H. Y. Kim, “Electro-optic characteristics and switching principle of a nematic liquid crystal cell controlled by fringe-field switching,” Appl. Phys. Lett. 73, 2881–2883 (1998).
[Crossref]

IEEE Spectrum (1)

J. M. Younse, “Mirrors on a chip,” IEEE Spectrum 30(11), 27–31 (1993).
[Crossref]

J. Reine Angew. Math. (1)

G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 134, 198–287 (1908).
[Crossref]

Proc. SPIE (1)

J. M. Younse, “Projection display systems based on the digital micromirror device (DMD),” Proc. SPIE 2641, 64–75 (1995).
[Crossref]

Prog. Opt. (1)

W-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119–232 (1978).
[Crossref]

Sens. Actuators A (1)

S.-W. Chung and Y.-K. Kim, “Design and fabrication of 10×10 micro-spatial light modulator array for phase and amplitude modulation,” Sens. Actuators A 78(1), 63–70 (1999).
[Crossref]

Spat. Light Modulators Appl. 3 SPIE Crit. Rev. (1)

L. J. Hornbeck, “Deformable mirror spatial light modulator,” Spat. Light Modulators Appl. 3 SPIE Crit. Rev. 1150, 86–102 (1990).

Other (1)

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” Solid-State Sensor and Actuator Workshop, Hilton-Head, SC, 1–6 (1994).

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Figures (8)

Fig. 1.
Fig. 1.

Voronoi diagram

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Fig. 2.
Fig. 2.

Effect of an increasing randomness in the distribution of the cell centres. The initial square grid (a) has a pitch equal to d = 100µm. Pupil radius R=2mm. (c, e, g) : Associated Voronoi diagrams. The centres are moved according to a uniform distribution on a square of side α = ad with a = 0.5, a = 1.27 and a = 1.5. (b, d, f, h) : Corresponding histograms of wall orientations.

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Fig. 3.
Fig. 3.

Fraunhofer diffraction patterns of the components of Fig.2 (a, c, e, g) and their horizontal cross sections through the center (b, d, f, h) for λ = 0.5µm, d = 100µm, l = 5µm and R = 2mm. The images as well as the cross-sections are shown in a logarithmic scale.

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Fig. 4.
Fig. 4.

Diffraction by a rectangular slit of width l, length L and orientation α

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Fig. 5.
Fig. 5.

Comparison, at λ = 0.5µm, of the diffracted intensity computed by the slits cloud simulation (dotted line) and the theoretical model given by Eq. (3) (solid line) for N = 2000 slits with random orientations and positions according to a uniform distribution over [0,π] and [ − C 2 , C 2 ] . The length of all the slits is L = 100µm and the width : (a) l = 5µm, (b) l = 10µm, (c) l = 15µm and (d) l = 20µm.

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Fig. 6.
Fig. 6.

Horizontal section of diffracted intensity pattern at λ = 0.5µm. Comparison between Voronoi simulations and the theoretical model given by Eq. (3). For all slits, l = 5µm. For the Voronoi simulations, the initial grid pitch is d = 100µm, a = √2 and R = 2mm. For Eq. (3), L = 100µm for all the slits.

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Fig. 7.
Fig. 7.

1D Voronoi component.

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Fig. 8.
Fig. 8.

Average of the relative intensities on 20 realizations of 2D Voronoi components at λ = 0.5µm. The initial grid pitch is d = 100µm, the length of the wall is e = 5µm, and R = 2mm.

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Equations (27)

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a = α d
f θ k ( x , y ) = rect [ x k ˜ sin α k − y k ˜ cos α k l ] rect [ x k ˜ cos α k + y k ˜ sin α k L ]
〈 I ( θ diff ) 〉 ≃ Nl 2 L 2 π λ 2 f 2 ∫ 0 π sin 2 ( π l λ tan θ diff sin φ ) sin 2 ( π L λ tan θ diff cos φ ) π 4 l 2 L 2 4 λ 4 tan 4 θ diff sin 2 ( 2 φ ) d φ
〈 I ( 0 ) 〉 = ∣ ∑ k f ˜ θ k ( 0 , 0 ) ∣ 2 = 1 λ 2 f 2 ∣ N ε ∣ 2
D ( x ) = e ∑ k = − ∞ ∞ δ ( x − kd − α k ′ )
〈 U ( v ) 〉 = e 〈 exp ( − 2 ι π ν α k ′ ) 〉 ∑ k = − ∞ ∞ exp ( − 2 ι π ν k d )
∑ k = − ∞ ∞ exp ( − 2 ι π ν k d ) = 1 d ∑ k = − ∞ ∞ δ ( ν − k d )
〈 U ( v ) 〉 = e d ∑ k = − ∞ ∞ P ˜ α ′ ( k d ) δ ( v − k d )
P ˜ α ′ ( ν ) = 1 2 exp ( − ι π ν d ) [ P ˜ α ( ν 2 ) ] 2 = 1 2 exp ( − ι π ν d ) [ sinc ( ν α 2 ) ] 2
〈 U ( v ) 〉 = e 2 d ∑ k = − ∞ ∞ exp ( − ι k π ) [ sinc ( k a 2 ) ] 2 δ ( ν − k d )
O k = ∣ 〈 U ( k d ) 〉 ∣ 2 = e 2 4 d 2 [ sinc ( ka 2 ) ] 4
A ( x , y ) = ∑ k = 1 N f θ k ( x , y )
I ( ξ , η ) = 1 λ 2 f 2 ∑ k ∣ f ˜ θ k ( ξ , η ) ∣ 2 + 1 λ 2 f 2 ∑ k ∑ l , k ≠ l f ˜ θ k ( ξ , η ) f ˜ θ l * ( ξ , η )
f ˜ θ k ( ξ , η ) = lL sinc [ l λ f ( ξ sin α k − η cos α k ) ] sinc [ L λ f ( ξ cos α k + η sin α k ) ]
exp [ − 2 ι π λ f ( ξ x k + η y k ) ]
〈 1 λ 2 f 2 ∑ k ∣ f ˜ θ k ( ξ , η ) ∣ 2 〉 = N λ 2 f 2 ∫ 0 π ∣ f ˜ θ k ( ξ , η ) ∣ 2 P α ( α k ) d α k
∣ f ˜ θ k ( ξ , η ) ∣ 2 = l 2 L 2 sinc 2 [ l ( ξ sin α k − η cos α k ) λ f ] sinc 2 [ L ( ξ cos α k + η sin α k ) λ f ]
∣ f ˜ θ k ( ρ , ψ ) ∣ 2 = l 2 L 2 sinc 2 [ l λ f ( ρ sin ( α k − ψ ) ) ] sinc 2 [ L λ f ( ρ cos ( α k − ψ ) ) ]
∣ f ˜ θ k ( θ diff , φ ) ∣ 2 = l 2 L 2 sinc 2 [ l λ ( tan θ diff sin φ ) ] sinc 2 [ L λ ( tan θ diff cos φ ) ]
〈 1 λ 2 f 2 ∑ k ∣ f ˜ θ k ( ξ , η ) ∣ 2 〉 ≃ N λ 2 f 2 π ∫ 0 π ∣ f ˜ θ k ( θ diff , φ ) ∣ 2 d φ
≃ Nl 2 L 2 π λ 2 f 2 ∫ 0 π sin 2 ( π l λ tan θ diff sin φ ) sin 2 ( π L λ tan θ diff cos φ ) π 4 l 2 L 2 4 λ 4 tan 4 θ diff sin 2 ( 2 φ ) d φ
〈 ∑ k ∑ l , k ≠ l f ˜ θ k ( ξ , η ) f ˜ θ l * ( ξ , η ) 〉 = N ( N − 1 ) ∫ ∫ ∫ ∫ ∫ ∫ f ˜ θ k ( ξ , η ) f ˜ θ l * ( ξ , η )
P α ( α k ) P x ( x k ) P y ( y k ) P α ( α l ) P x ( x l ) P y ( y l ) d α k dx k dy k d α l dx l dy l
〈 ∑ k ∑ l , k ≠ l f ˜ θ k ( ξ , η ) f ˜ θ l * ( ξ , η ) 〉 = N ( N − 1 ) L 2 l 2 I a 2 ( ξ , η ) sinc 2 ( ξ C λ f ) sinc 2 ( η C λ f )
I a ( ξ , η ) = ∫ 0 π sinc [ l λ f ( ξ sin α k − η cos α k ) ]
sinc [ L λ f ( ξ cos α k + η sin α k ) ] P α ( α k ) d α k
〈 I ( 0 ) 〉 = ∣ ∑ k f ˜ θ k ( 0 , 0 ) ∣ 2 = 1 λ 2 f 2 ∣ N ε ∣ 2

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