Abstract

Spatial light modulators are often used to implement phase modulation. Since they are pixelated, the phase function is usually approximated by a regularly sampled piecewise constant function, and the periodicity of the pixel sampling generates annoying diffraction peaks. We theoretically investigate two pixelation techniques: the isophase method and a new nonperiodic method derived from the Voronoi tessellation technique. We show that, for a suitable choice of parameters, the diffraction peaks disappear and are replaced by a smoothly varying halo. We illustrate the potential of these two techniques for implementing a lens function and wavefront correction.

© 2011 Optical Society of America

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References

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  1. 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, 2505–2509 (1990).
    [CrossRef] [PubMed]
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  3. S.Esener, J.L.Horner, and K.M.Johnson, eds., feature on Spatial Light Modulators, Appl. Opt. 31 (20) (1992).
  4. A.L.Lentine, J.N.Lee, S.H.Lee, and U.Efron, eds., feature on Spatial Light Modulators, Appl. Opt. 33 (14) (1994).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
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2010 (2)

2009 (1)

2004 (1)

1995 (1)

1994 (2)

1990 (1)

1944 (1)

A. Maréchal, “Etude de l’éclairement au centre de la tache de diffraction dans le cas de diverses aberrations géométriques,” C. R. Hebd. Séances Acad. Sci. 218, 395–397 (1944).

1908 (1)

G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 1908, 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,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head, S.C., USA, 1994.

Ballet, J.

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,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head, S.C., USA, 1994.

Benoit-Pasanau, C.

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,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head, S.C., USA, 1994.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge University, 1999).
[PubMed]

Bosch, S.

Campos, J.

Cano, J. P.

Carcolé, E.

Cathey, T.

Chavel, P.

Cottrell, D. M.

Davis, J. A.

de F. Moneo, J. R.

Diaz, F.

Dowski, E.

Goudail, F.

Hedman, T. R.

Huignard, J.-P.

Juvells, I.

Lilly, R. A.

Loiseaux, B.

Maréchal, A.

A. Maréchal, “Etude de l’éclairement au centre de la tache de diffraction dans le cas de diverses aberrations géométriques,” C. R. Hebd. Séances Acad. Sci. 218, 395–397 (1944).

Peloux, M.

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,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head, S.C., USA, 1994.

Sherif, S.

Taboury, J.

Voronoi, G. F.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge University, 1999).
[PubMed]

Appl. Opt. (6)

C. R. Hebd. Séances Acad. Sci. (1)

A. Maréchal, “Etude de l’éclairement au centre de la tache de diffraction dans le cas de diverses aberrations géométriques,” C. R. Hebd. Séances Acad. Sci. 218, 395–397 (1944).

J. Reine Angew. Math. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Other (4)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge University, 1999).
[PubMed]

R. B. Apte, F. S. A. Sandejas, W. C. Banyai, and D. M. Bloom, “Deformable grating light valves for high resolution displays,” presented at the Solid-State Sensor and Actuator Workshop, Hilton Head, S.C., USA, 1994.

S.Esener, J.L.Horner, and K.M.Johnson, eds., feature on Spatial Light Modulators, Appl. Opt. 31 (20) (1992).

A.L.Lentine, J.N.Lee, S.H.Lee, and U.Efron, eds., feature on Spatial Light Modulators, Appl. Opt. 33 (14) (1994).

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Figures (16)

Fig. 1
Fig. 1

SLM with square-shaped pixel, pixel pitch d = 100 μm . Pupil diameter of 4 mm . (a) Walls only. (b) Implementation of a pixelated lens of focal length f = 2 m . (c) Diffracted intensity by the pixelated SLM. (d) Corresponding cross section of (c) in the horizontal direction.

Fig. 2
Fig. 2

Principle of the pixelation by isophases. The phase function is sliced by equally spaced horizontal planes z = 2 k π γ (k, integer, and γ, a real number).

Fig. 3
Fig. 3

Lens of focal length f = 2 m encoded in an isophase modulator. (a) Isophases only. (b) Phase in the modulator. Pupil of 4 mm diameter.

Fig. 4
Fig. 4

Different phase profiles of the pixelated lens of focal length f = 2 m encoded in different isophase modulators. (a)  γ = 0.05 , (b)  γ = 0.09 , and (c)  γ = 0.15 .

Fig. 5
Fig. 5

Lens of focal length f = 2 m . Comparison of the horizontal sections of the diffracted intensity by a regular square grid ( d = 100 μm ), an optimal Voronoi grid ( a = 1.27 ) and by isophases.

Fig. 6
Fig. 6

Illustration of a Voronoi diagram composed of Voronoi cells separated by walls. A Voronoi cell contains exactly one generating point p i , and every point is closer to its generating point p i than to any other p j ( j i ).

Fig. 7
Fig. 7

Increase of the randomness of the periodic distribution of centers, spaced every d = 100 μm (Fig. 1). (a), (c), and (e) Associated Voronoi diagrams with walls only. Pupil di ameter of 4 mm . The centers are moved according to a uniform distribution on a square of side α so that a = 0.5 , a = 1.27 , and a = 1.5 . (b), (d) and (f) Corresponding Voronoi diagram correcting a f = 2 m lens.

Fig. 8
Fig. 8

Diffracted intensity by the pixelated Voronoi components implementing a lens of focal length f = 2 in Fig. 7 and the corresponding cross section in the horizontal direction. (a), (b)  a = 0.5 . (c), (d)  a = 1.27 . (e), (f)  a = 1.5 . Pupil of 4 mm diameter.

Fig. 9
Fig. 9

Evolution of the relative intensities of the order 0 and of the order 1 as a function of the parameter a for a pixelated lens of focal length f = 2 m encoded on a Voronoi SLM. Average of the efficiencies on 20 realizations of two-dimensional Voronoi components.

Fig. 10
Fig. 10

Night scene observed by a periodic pixelated prototype.

Fig. 11
Fig. 11

Principle of wavefront correction.

Fig. 12
Fig. 12

Structure of the pixelated modulators (first column) and the astigmatism wavefront encoded on them (second column). (a), (b) Regular square grid. (c), (d) Isophases. (e), (f) Optimal Voronoi grid.

Fig. 13
Fig. 13

Correction of astigmatism. Comparison of the horizontal sections of the diffracted intensity by a regular square grid ( d = 100 μm ), by an optimal Voronoi grid ( a = 1.27 ), and by isophases.

Fig. 14
Fig. 14

Structure of different pixelated modulators (first column) and the coma wavefront encoded on them (second column). (a), (b) Regular square grid. (c), (d) Isophases. (e), (f) Optimal Voronoi grid.

Fig. 15
Fig. 15

Correction of coma. Comparison of the vertical sections of the diffracted intensity by a regular square grid ( d = 100 μm ), by an optimal Voronoi grid ( a = 1.27 ) and by isophases.

Fig. 16
Fig. 16

Comparison of the continuous and isophase profiles of the coma.

Tables (3)

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Table 1 Lens of Focal Length f = 2 m a

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Table 2 Correction of Astigmatism a

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Table 3 Correction of Coma a

Equations (9)

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φ d ( x , y ) = π λ f ( x 2 + y 2 ) .
σ Φ 2 = D ( φ ideal ( x , y ) φ p ) 2 d x d y .
φ p D ( φ ideal ( x , y ) φ p ) 2 d x d y = 0.
φ p = 1 s D φ ideal ( x , y ) d x d y .
E ( ξ , η ) = I ( ξ , η ) I airy ,
S = max ( E ( ξ , η ) ) .
a = α d .
φ ast ( X , Y ) = 2 A ast X Y .
φ coma ( X , Y ) = A coma ( 2 Y + 3 Y 3 + 3 X 2 Y ) .

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