Abstract

An optical Fourier processor that allows the use of broadband light sources and colored inputs is designed, fabricated, and tested. We develop a design technique based on phase manipulation in the Fourier plane to construct an image processor that provides a chromatically corrected image making use of the good aberrations behavior of symmetrical optical systems. Only a small number of diffractive lenses and one achromatic refractive lens are required to obtain a real image. We verify our design experimentally using holographic lenses, which are presented, owing to their versatility, as a good alternative to expensive blazed diffractive elements.

© 2001 Optical Society of America

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References

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  1. R. H. Katyl, “Compensating optical systems. Part 3. Achromatic Fourier transform,” Appl. Opt. 11, 1255–1260 (1972).
    [CrossRef] [PubMed]
  2. G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), Chap. 1.2.
  3. E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
    [CrossRef]
  4. E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
    [CrossRef]
  5. J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier-transform setup,” Opt. Lett. 19, 402–404 (1994).
    [PubMed]
  6. P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, UK, 1987).
  7. M. Quintanilla, I. Arias, “Holographic imaging lenses. Composite lens with high efficiency,” J. Opt. 21, 67–72 (1990).
    [CrossRef]
  8. J. Atencia, I. Arias, M. Quintanilla, A. Garcia, A. M. Lopez, “Field improvement in a uniaxial centered lens composed of two stacked-volume holographic elements,” Appl. Opt. 38, 4011–4018 (1999).
    [CrossRef]
  9. C. Neipp, I. Pascual, A. Beléndez, “Silver halide sensitized gelatin derived from BB-640 holographic emulsion,” Appl. Opt. 38, 1348–1356 (1999).
    [CrossRef]

1999

1998

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

1994

1990

M. Quintanilla, I. Arias, “Holographic imaging lenses. Composite lens with high efficiency,” J. Opt. 21, 67–72 (1990).
[CrossRef]

1972

Andrés, P.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

J. Lancis, P. Andrés, W. D. Furlan, A. Pons, “All-diffractive achromatic Fourier-transform setup,” Opt. Lett. 19, 402–404 (1994).
[PubMed]

Arias, I.

Atencia, J.

Beléndez, A.

Climent, V.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

Fernández-Alonso, M.

Furlan, W. D.

Garcia, A.

Hariharan, P.

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, UK, 1987).

Katyl, R. H.

Lancis, J.

Lopez, A. M.

Morris, G. M.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), Chap. 1.2.

Neipp, C.

Pascual, I.

Pons, A.

Quintanilla, M.

Tajahuerce, E.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998).
[CrossRef]

Zweig, D. A.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), Chap. 1.2.

Appl. Opt.

J. Opt.

M. Quintanilla, I. Arias, “Holographic imaging lenses. Composite lens with high efficiency,” J. Opt. 21, 67–72 (1990).
[CrossRef]

Opt. Commun.

E. Tajahuerce, J. Lancis, V. Climent, P. Andrés, P. Andrés, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Opt. Lett.

Other

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, UK, 1987).

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, San Diego, Calif., 1987), Chap. 1.2.

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Figures (8)

Fig. 1
Fig. 1

Diffractive lens doublet arrangement that produces an achromatic Fourier transformation. DL1 provides a chromatically dispersed Fraunhofer diffraction pattern of the transparency, and DL2 images, to first order, each monochromatic diffraction pattern at the same plane with the same size.

Fig. 2
Fig. 2

Achromatic processor arrangement (a) in the most general case and (b) in a particularly interesting configuration that allows the removal of the last lens if the object is placed close to the first lens and one is interested in intensity measurements.

Fig. 3
Fig. 3

Phase factor Φ(σ) in the Fourier plane versus the wave number (solid curve) and its linear approximation (dashed curve) under white-light illumination. The separation d between DL1 and DL2 was chosen as 3f o (1/2 to cancel the square term in the phase factor dependence on σ, which is plotted with f o (1 = 111 mm and σ o = 1.83 µm-1, the values used in the experimental setup.

Fig. 4
Fig. 4

(a) Plot of the geometric residual chromatic aberration in the Fourier plane for the optical processor in Fig. 2(b), with f o (1 = 111 mm, σ o = 1.83 µm-1, and f o (2/f o (1= 2.25, corresponding to the values used in the experimental setup. (b) Longitudinal chromatic aberration (dashed curve) and transversal chromatic aberration (solid curve) in the image plane for the same processor. White-light illumination is assumed.

Fig. 5
Fig. 5

Schematic diagrams of (a) the recording of two off-axis holograms HI and HII and (b) the reconstruction of holograms HI and HII in which the HI–HII sandwich behaves as a uniaxial centered lens. When the recording is made, the glass plate that supports the holographic emulsion is appropriately placed to guarantee that the aberrations introduced will be canceled in the reconstruction step and also that both emulsions will be protected.

Fig. 6
Fig. 6

(a) Achromatic gray-tone irradiance distribution in the Fourier plane of the setup in Fig. 2(b) when the input is a 6 lines/mm two-dimensional grating placed in the object plane. (b) RGB irradiance profile plot along the horizontal line in (a). (c) Irradiance gray-tone representation of the same input spectrum provided by a conventional achromatic refractive lens. (d) RGB irradiance profile plot along the horizontal line in (c).

Fig. 7
Fig. 7

(a) Irradiance distribution in the achromatic image plane of the setup in Fig. 2(b). (b) Profile plot of the RGB irradiance profile plot along the horizontal line in (a).

Fig. 8
Fig. 8

(a) Achromatic gray-tone irradiance distribution in the image plane of the setup in Fig. 2(b) when a two-dot object is used as input transparency and a one-dimensional grating is placed in the Fourier plane. (b) RGB irradiance profile plot along the horizontal line in (a). (c) Irradiance gray-tone distribution of the image provided by a conventional 4f processor with the same grating as in the Fourier plane. (d) RGB irradiance profile plot along the horizontal line in (a). White-light illumination is used in both cases.

Tables (1)

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Table 1 Z Coordinate and Focal Length for Each Constructed Lens

Equations (12)

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UDx, y, D=expiπ σD1+σBDx2+y2×-- tξ, η, 0expiπ σd-zdz-σAd-z+σ3BA2d2d-zξ2+η2×exp-i2π σ3BADdd-zxξ+yηdξdη,
A=σd-z-σd-σofo(1,  B=σd+σ2Ad2+σofo(2-σD,
Dσ=1d+σofo(2σ-fo(1σσod2-1.
fo(2=-d2fo(1,
0<fo(2fo(1<4.
Φσ=π σD1+σBDx2+y2.
d=3fo(1/2,
Φσ=-πσ9/2fo(1-σo9fo(1x2+y2.
fR=-94 fo(1,  f3σo=92 fo(1.
LCATP=100 Dσo-DσDσo,TCATP=100 yσo-yσyσo,
LCAIP=100 dσo-dσdσo,TCAIP=100 Yσo-YσYσo,
f=σσr-1z1+1z2-1,

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