Abstract

Focusing diffractive optical elements encoded in liquid-crystal spatial light modulators yields an inherent equivalent apodization of the focused spot as a result of the pixelated nature of these devices and the finite extent of each pixel. We present a theoretical explanation for and experimental evidence of this effect. We demonstrate an experimental procedure for measuring the apodization and a method to compensate for this effect.

© 2000 Optical Society of America

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References

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  1. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. D. M. Cottrell, J. A. Davis, T. R. Hedman, R. A. Lilly, “Multiple imaging phase-encoded optical elements written as programmable spatial light modulators,” Appl. Opt. 29, 2505–2509 (1990).
    [CrossRef] [PubMed]
  4. K. J. Weible, H. P. Herzig, “Optical optimization of binary fan-out elements,” Opt. Commun. 113, 9–14 (1994).
    [CrossRef]
  5. E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. J. A. Davis, J. C. Escalera, J. Campos, A. Marquez, M. J. Yzuel, C. Iemmi, “Programmable axial apodizing and hyperresolving amplitude filters with a liquid-crystal spatial light modulator,” Opt. Lett. 24, 628–630 (1999).
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  9. J. P. Kirk, A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).
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    [CrossRef]
  11. W. J. Dallas, “Computer Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1980), Chap. 6.

1999

1998

1994

K. J. Weible, H. P. Herzig, “Optical optimization of binary fan-out elements,” Opt. Commun. 113, 9–14 (1994).
[CrossRef]

E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
[CrossRef] [PubMed]

1991

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1990

1989

1971

Arrizon, V.

Bosch, S.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Campos, J.

Carcolé, E.

Carreon, E.

Connely, S. W.

Cottrell, D. M.

Dallas, W. J.

W. J. Dallas, “Computer Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1980), Chap. 6.

Davis, J. A.

Escalera, J. C.

Gonzalez, L. A.

Hedman, T. R.

Herzig, H. P.

K. J. Weible, H. P. Herzig, “Optical optimization of binary fan-out elements,” Opt. Commun. 113, 9–14 (1994).
[CrossRef]

Iemmi, C.

Jones, A. L.

Kirk, J. P.

Lilly, R. A.

Marquez, A.

Moreno, I.

Tsai, P.

Weible, K. J.

K. J. Weible, H. P. Herzig, “Optical optimization of binary fan-out elements,” Opt. Commun. 113, 9–14 (1994).
[CrossRef]

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Yzuel, M. J.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

K. J. Weible, H. P. Herzig, “Optical optimization of binary fan-out elements,” Opt. Commun. 113, 9–14 (1994).
[CrossRef]

Opt. Lett.

Rep. Prog. Phys.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other

W. J. Dallas, “Computer Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics Series (Springer-Verlag, Berlin, 1980), Chap. 6.

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Figures (6)

Fig. 1
Fig. 1

Normalized amplitude transmission and phase dephasing for the phase-only configuration of the LCSLM.

Fig. 2
Fig. 2

Optical setup: f, focal length; other abbreviations defined in text.

Fig. 3
Fig. 3

Annular ring with central radius ρ R , where we can see the encoded lens.

Fig. 4
Fig. 4

Apodizing effect of the LCSLM along the radius of the lens. Experimental points measured with annular masks (filled squares), sinc fitting (filled circles), and parabolic fitting (dashed curve).

Fig. 5
Fig. 5

Theoretically calculated intensity for lens behavior (a) in the best image plane and (b) in the axial direction. Lens with no apodizing effect, solid curves; lens with apodizing effect, dashed curves.

Fig. 6
Fig. 6

Experimental measurements of the intensity of the focused spot. (a) Combination of saturated images in the best image plane (upper half, encoded lens; lower half, compensated lens). (b), (c) Planes at different axial positions produced by the encoded lens and the compensated lens: best image plane, (b1), (c1); first minimum planes, (b2), (c2); first maximum planes, (b3), (c3).

Equations (17)

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Z*u=exp-iϕQr=exp-iπu2/λf.
Tu=Z*un δu-nDrectu/Lrectu/W,
Ux= TuZx-udu.
Ux= Z*un δu-nDrect×Zx-udurectxW.
Ux=expi πλf x2sincLxλf n δx-n λfDrectxW,
Ux=sincLx/λfrectx/W.
τu=sincWu/λf.
Tu=n δu-nDrectu/Lrectu/WZ*u.
Ux=expi πλf x2sincLfλf n δx-n λfDsincWxλf,
Ur=0=R τρ, ϕρdρdϕ,
Ur=0=2π R τρρdρ.
Ur=0=AτρR.
τρR=Ir=0/A.
Tρ=τIρexp-iϕQρ.
Tρ=exp-iτIρϕQρ.
Tρ=n=- Tnρexp-inϕQρ,
Tnρ=sinπn-τIρπn-τIρ.

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