Abstract

In the previous two sections of “Flatland optics” [J. Opt. Soc. Am. A 17, 1755 (2000); J. Opt. Soc. Am. A 18, 1056 (2001)] we described the basic principles of two-dimensional (2D) optics and showed that a wavelength λ in three-dimensional (3D) space (x, y, z) may appear in Flatland (x, z) as a wave with another wavelength Λ=λ/cos α. The tilt angle α can be modified by a 3D-Spaceland individual, who then is able to influence the 2D optics in a way that must appear to be magical to 2D-Flatland individuals—in the spirit of E. A. Abbott’s science fiction story of 1884 [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)]. Here we show how the light from a white source can be perceived in Flatland as perfectly monochromatic, so diffraction with white light will be free of color blurring and the contrast of interference fringes can be 100%. The basic considerations for perfectly achromatic diffraction are presented, along with experimental illustration of Talbot self-imaging performed with broadband illumination.

© 2001 Optical Society of America

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References

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  1. E. A. Abbot, Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952).
  2. A. W. Lohmann, A. Pe’er, D. Wang, A. A. Friesem, “Flatland optics. I. Fundamentals,” J. Opt. Soc. Am. A 17, 1755–1762 (2000).
    [Crossref]
  3. A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Flatland optics. II. Basic experiments,” J. Opt. Soc. Am. A 18, 1056–1061 (2000).
    [Crossref]
  4. B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
    [Crossref]
  5. E. E. Sicre, N. Bolognini, M. Garavaglia, “Partial achromatization of the self-imaging phenomenon,” Appl. Opt. 24, 929–930 (1985).
    [Crossref]
  6. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
    [Crossref]
  7. P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
    [Crossref]
  8. E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
    [Crossref]
  9. N. Guerineau, J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
    [Crossref]

2000 (2)

1999 (2)

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

N. Guerineau, J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16, 293–298 (1999).
[Crossref]

1993 (1)

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

1988 (1)

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

1985 (1)

1984 (1)

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[Crossref]

Abbot, E. A.

E. A. Abbot, Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952).

Andres, P.

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

Bolognini, N.

Bonet, E.

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

Bryngdahl, O.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[Crossref]

Climent, V.

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[Crossref]

Fernandez-Alonso, M.

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

Friesem, A. A.

Garavaglia, M.

Guerineau, N.

Indebetouw, G.

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

Lancis, J.

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

Lohmann, A. W.

Packross, B.

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[Crossref]

Pe’er, A.

Primot, J.

Sicre, E. E.

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

E. E. Sicre, N. Bolognini, M. Garavaglia, “Partial achromatization of the self-imaging phenomenon,” Appl. Opt. 24, 929–930 (1985).
[Crossref]

Tajahuerce, E.

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

Wang, D.

Appl. Opt. (1)

J. Mod. Opt. (2)

G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988).
[Crossref]

E. Tajahuerce, P. Andres, M. Fernandez-Alonso, V. Climent, “White-light array generation with a diffractive lenslet array,” J. Mod. Opt. 46, 49–63 (1999).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

B. Packross, R. Eschbach, O. Bryngdahl, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[Crossref]

P. Andres, J. Lancis, E. E. Sicre, E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104, 39–45 (1993).
[Crossref]

Other (1)

E. A. Abbot, Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952).

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Figures (5)

Fig. 1
Fig. 1

Optical setup in which a 1D object at z=0 is illuminated by a plane wave, tilted by an angle α in the y direction.

Fig. 2
Fig. 2

Optical setup for tuning the tilt angle α (λ) by means of a diffraction grating oriented orthogonally to the object.

Fig. 3
Fig. 3

Spectrum of a source in 3D Spaceland and in 2D Flatland obtained with the setup of Fig. 2.

Fig. 4
Fig. 4

Experimental result of an achromatic Talbot self-image in Flatland. The illumination spectrum is spread along the y axis. The image is composed of two CCD captures taken at the same longitudinal distance z, one at the red–yellow spectrum and the other at the yellow–green spectrum.

Fig. 5
Fig. 5

Optical setup for transforming monochromatic light in 3D Spaceland into polychromatic light in 2D Flatland.

Equations (11)

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Λ=λcos α,
|V(x, y, z; n, λ)|2=I(x, y, z),
n(λ)=n1λ.
k(λ)=2πnλ=2πn1.
Λ=λcos α=Λ(λ, α).
cos α(λ)=λ×const.
sin β(λ)=λ/D,
β(λ)+α(λ)=π/2.
cos α(λ)=sin β(λ)=λ/D.
V(z, t)=exp[i(K1z-ωt)]+exp[i(K2z-ωt)]=exp[i(K¯z-ωt)]2 cos(ΔKz),
|V(z, t)|2=2[1+cos(2ΔKz)].

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