Abstract

We show that a pixelated lens with appropriate parameters exhibits an apodized point-spread function that originates in the finite size of the pixel’s pupil. We evaluate numerically the degree of apodization and the enlargement associated with the point-spread function in terms of the parameters that characterize the pixelated lens.

© 1999 Optical Society of America

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References

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  1. 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]
  2. J. A. Davis, W. V. Brandt, D. M. Cottrell, R. M. Bunch, “Spatial image differentiation using programmable binary optical elements,” Appl. Opt. 30, 4610–4614 (1991).
    [CrossRef] [PubMed]
  3. E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [CrossRef] [PubMed]
  4. E. Carcolé, J. Campos, I. Juvells, S. Bosch, “Diffraction efficiency of low-resolution Fresnel encoded lenses,” Appl. Opt. 33, 6741–6746 (1994).
    [CrossRef] [PubMed]
  5. E. Carcolé, J. Campos, I. Juvells, J. R. de F. Moneo, “Diffraction theory of optimized low-resolution Fresnel encoded lenses,” Appl. Opt. 34, 5952–5960 (1995).
    [CrossRef] [PubMed]
  6. J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
    [CrossRef]
  7. J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
    [CrossRef]
  8. Y. Takaki, “Electro-optical implementation of learning architecture to control point spread function of liquid crystal active lens,” in Optical Implementation of Image Processing, B. Javidi, J. L. Horner, eds., Proc. SPIE2565, 205–214 (1995).
    [CrossRef]
  9. Y. Takaki, H. Ohzu, “Reconfigurable lens with an electro-optical learning system,” Appl. Opt. 35, 6896–6908 (1996).
    [CrossRef] [PubMed]
  10. Y. Takaki, “Electro-optical feedback system for controlling a reconfigurable lens,” in Optical Information Science and Technology (OIST 97): Computer and Holographic Optics and Image Processing, A. L. Mikaelian, ed., Proc. SPIE3348, 178–185 (1998).
  11. V. Arrizón, S. Kinne, S. Sinzinger, “Efficient detour-phase encoding of multilevel phase elements,” Appl. Opt. 37, 5454–5460 (1998).
    [CrossRef]
  12. P. Jacquinot, B. Rozien-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 240.

1998 (1)

1996 (1)

1995 (2)

E. Carcolé, J. Campos, I. Juvells, J. R. de F. Moneo, “Diffraction theory of optimized low-resolution Fresnel encoded lenses,” Appl. Opt. 34, 5952–5960 (1995).
[CrossRef] [PubMed]

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

1994 (2)

1991 (1)

1990 (1)

Arrizón, V.

Bosch, S.

Brandt, W. V.

Bunch, R. M.

Campos, J.

Carcolé, E.

Cottrell, D. M.

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

J. A. Davis, W. V. Brandt, D. M. Cottrell, R. M. Bunch, “Spatial image differentiation using programmable binary optical elements,” Appl. Opt. 30, 4610–4614 (1991).
[CrossRef] [PubMed]

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]

J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
[CrossRef]

Davis, J. A.

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

J. A. Davis, W. V. Brandt, D. M. Cottrell, R. M. Bunch, “Spatial image differentiation using programmable binary optical elements,” Appl. Opt. 30, 4610–4614 (1991).
[CrossRef] [PubMed]

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]

J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
[CrossRef]

de F. Moneo, J. R.

Field, A. M.

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
[CrossRef]

Hedman, T. R.

Heiskala, J. J.

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Rozien-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 240.

Juvells, I.

Kinne, S.

Lilly, R. A.

Ohzu, H.

Rozien-Dossier, B.

P. Jacquinot, B. Rozien-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 240.

Sinzinger, S.

Takaki, Y.

Y. Takaki, H. Ohzu, “Reconfigurable lens with an electro-optical learning system,” Appl. Opt. 35, 6896–6908 (1996).
[CrossRef] [PubMed]

Y. Takaki, “Electro-optical implementation of learning architecture to control point spread function of liquid crystal active lens,” in Optical Implementation of Image Processing, B. Javidi, J. L. Horner, eds., Proc. SPIE2565, 205–214 (1995).
[CrossRef]

Y. Takaki, “Electro-optical feedback system for controlling a reconfigurable lens,” in Optical Information Science and Technology (OIST 97): Computer and Holographic Optics and Image Processing, A. L. Mikaelian, ed., Proc. SPIE3348, 178–185 (1998).

Appl. Opt. (7)

Opt. Eng. (1)

J. A. Davis, A. M. Field, J. J. Heiskala, D. M. Cottrell, “Phase analysis of diffracted beams using multiplexed Fourier transform lenses,” Opt. Eng. 34, 50–55 (1995).
[CrossRef]

Other (4)

Y. Takaki, “Electro-optical implementation of learning architecture to control point spread function of liquid crystal active lens,” in Optical Implementation of Image Processing, B. Javidi, J. L. Horner, eds., Proc. SPIE2565, 205–214 (1995).
[CrossRef]

Y. Takaki, “Electro-optical feedback system for controlling a reconfigurable lens,” in Optical Information Science and Technology (OIST 97): Computer and Holographic Optics and Image Processing, A. L. Mikaelian, ed., Proc. SPIE3348, 178–185 (1998).

P. Jacquinot, B. Rozien-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), p. 240.

J. A. Davis, J. J. Heiskala, A. M. Field, D. M. Cottrell, “Programmable Dammann grating interconnections using Fourier transform lenses written onto spatial light modulators,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 237–243 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Phase values (modulus 2π) of a 1-D PL with 51 pixels and an f-number of f N = 10.

Fig. 2
Fig. 2

Apodized PSF intensity of a PL with r = 1.5 (solid curve), and the PSF intensity of the corresponding continuous lens (dashed curve).

Fig. 3
Fig. 3

Apodization degree as a function of the ratio r = af N (lower curve) and the corresponding PSF’s enlargement En (upper curve).

Fig. 4
Fig. 4

Enlargement of a CGH that encodes a PL with N = 131 pixels and has parameters that yield a ratio of r = 1.37 associated with the maximum axial intensity.

Fig. 5
Fig. 5

Experimental PSF intensity for the holographically encoded PL shown in Fig. 4.

Equations (11)

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Lpx=n=-QQ exp-iπn2p2λfrectx-npa,
Lcx=exp-iπx2λfrectxNp,
Lsx=Lcxn=- δx-np,
Lpx=Lsxrectx/a.
Ucx=Npλf expiπx2λfsincNpxλf,
Usx=Nλf expiπx2λfn=- sincNpλfx-nλfp.
Upx=Nλfn=-expiπx2λfsincNpλfx-nλfprectxa.
ψ0xNλfexpiπx2λfsincNpxλfrectxa.
r=a/λfN,
A=I10/0.0472.
f0.73apN/λ.

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