Abstract

“Flatland” is the title of a 120-year-old science fiction story. It describes the life of creatures living in a two-dimensional (2D) Flatland. A superior creature living in the three-dimensional (3D) spaceland, as we do, can easily inspect, for example, the inside of a Flatland house, as well as the content of a flat man’s stomach without leaving any trace. Furthermore, the 3D person has supernatural powers that enable him to change the laws of physics in Flatland. We present here the concept of a 2D Flatland optics with one transversal coordinate x and one longitudinal coordinate z. The other transversal coordinate y allows total inspection of Flatland optics, and the freedom to change the wavelength, without using something like nonlinear optics or a Doppler shift. Monochromatic 3D light can be converted reversibly into polychromatic 2D light. A large variety of 2D systems and 2D effects will be presented here and in follow-up contributions. An epilogue faces the question, how “real” is Flatland optics?

© 2000 Optical Society of America

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References

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  1. E. A. Abbott, Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952).
  2. Y. N. Denisyuk, “Three-dimensional and pseudodeep holograms,” J. Opt. Soc. Am. A 9, 1141–1147 (1992).
    [CrossRef]
  3. A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
    [CrossRef]
  4. W. T. Rhodes, ed., Transformations in Optical Signal Processing, Proc. SPIE373 (1981).
  5. H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
    [CrossRef]
  6. A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.
  7. A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.
  8. H. E. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).
  9. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  10. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]

1998 (1)

1993 (1)

A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
[CrossRef]

1992 (1)

1979 (1)

H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
[CrossRef]

1967 (1)

1836 (1)

H. E. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Abbott, E. A.

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

Arimoto, Y.

Bartelt, H. O.

H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
[CrossRef]

Case, S. K.

H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
[CrossRef]

Denisyuk, Y. N.

Friesem, A. A.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.

Klaus, W.

Kodate, K.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
[CrossRef]

H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
[CrossRef]

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.

Montgomery, W. D.

Ojeda-Castañeda, J.

A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
[CrossRef]

Pe’er, A.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.

Serrano-Heredia, A.

A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
[CrossRef]

Talbot, H. E.

H. E. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Wang, Dayong

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

A. W. Lohmann, J. Ojeda-Castañeda, A. Serrano-Heredia, “Synthesis of 1D complex amplitudes using Young’s experiment,” Opt. Commun. 101, 17–20 (1993).
[CrossRef]

H. O. Bartelt, S. K. Case, A. W. Lohmann, “Visualization of light propagation,” Opt. Commun. 30, 13–19 (1979).
[CrossRef]

Philos. Mag. (1)

H. E. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Other (4)

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

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: basic experiments,” available from the authors.

A. W. Lohmann, A. Pe’er, Dayong Wang, A. A. Friesem, “Flatland optics: achromatic diffraction,” available from the authors.

W. T. Rhodes, ed., Transformations in Optical Signal Processing, Proc. SPIE373 (1981).

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Figures (12)

Fig. 1
Fig. 1

Identification of a creature in Flatland by observation of its angular size (α) from all directions (ϑ). The parameter R is the standard distance from the center of the Flatland creature while it is tested. (a) Measuring α(ϑ) of a circle, (b) three Flatland individuals, (c) angular signatures of the three Flatland individuals.

Fig. 2
Fig. 2

Generic optical configuration of coherent Flatland optics.

Fig. 3
Fig. 3

Size of the Flatland experiment.

Fig. 4
Fig. 4

Dependence of the effective wavelength Λ on the y coordinate yS.

Fig. 5
Fig. 5

Increasing the illumination angle by use of a prism. With the help of the prism, the practical point source at any yS1 is equivalent to a virtual source at yS2.

Fig. 6
Fig. 6

Prism arrangement for obtaining different wavelengths Λ within the different ranges z<zP and z>zP.

Fig. 7
Fig. 7

Deflection of an incoming plane wave by a grating.

Fig. 8
Fig. 8

Source distributions for achromatization: (a) cosine of the deflecting angle as a function of wavelength, (b) spectral distribution.

Fig. 9
Fig. 9

Combination of diffractive and refractive dispersions for approximating the condition for Flatland achromatization.

Fig. 10
Fig. 10

Possible configuration for an exact Flatland achromatization.

Fig. 11
Fig. 11

Conversion of polychromatic 3D light into monochromatic Flatland light.

Fig. 12
Fig. 12

Conversion of monochromatic 3D light into polychromatic Flatland light.

Equations (67)

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V(y, z)=exp2πiλ (y sin α+z cos α).
V(x, y, z=0)=u0(x)exp2πiλ y sin α.
u0(x)=u˜0(ν)exp(2πiνx)dν.
V(x, y, 0)=u˜0(ν)exp2πiλ (xλν+y sin α)dν.
Δ3V(x, y, z)+k2V(x, y, z)=0,
Δ3=2x2+2y2+2z2,k=2π/λ.
V(x, y, z)=u˜0(ν) exp2πiλ {xλν+y sin α+z[1-(λν)2-sin2 α]1/2}dν.
V(x, y, z)=exp2πiy sin αλu˜0(ν)×exp2πixν+z cos αλ×1-λνcos α21/2dν.
u(x, z)=u˜0(ν)×exp2πiΛ {xΛν+z[1-(Λν)2]1/2}dν=V(x, 0, z),
Λ=λ/cos α.
Δ2u(x, z)+k22u(x, z)=0,
Δ2=2x2+2z2,k2=2πΛ=2π cos αλ.
u(x, z)= u˜˜(ν, ρ)exp[2πi(νx+ρz)]dνdρ.
ν2+ρ2=(1/Λ)2=cos2 α/λ2.
νL=1/Λ=cos α/λ1/λ.
Λ=λcos α;1cos α=(1+tan2 α)1/2;tan α=ysf.
Λ/λ=[1+(yS/f )2]1/21+yS2/(2 f2).
sin αm=sin α0+m λD.
Λm=λ(1-sin2 αm)1/2λ01+m2λ22D2,
1Λm=1λ (1-sin2 αm)1/2=1λcos αm.
Δ2u(x, z)+k22u(x, z)=0,
Δ2=2x2+2z2,k2=2πΛ=k cos α.
exp{ik2[xΛν+z1-(Λν)2]},
υ(x, z)=u(x, z)exp(-ik2z).
Δ2υ(x, z)+2ik2υ(x, z)z=0.
2υ(x, z)z22υ(x, z)x2.
2υ(x, z)x2+2ik2υ(x, z)z=0;k2=2πΛ.
u(x, z)exp(-2πiνx)dx
=u˜(ν, z)=u˜0(ν)exp[ik2z1-(Λν)2].
υ˜(ν, z)=υ˜0(ν)exp{ik2z[1-(Λν)2-1]}.
υ˜(ν, z)=υ˜0(ν)exp(-iπΛzν2).
υ(x, z)=υ0(x)*p(x; z).
υ˜0(ν)=u˜0(ν)=υ0(x)exp(-2πiνx)dx
υ(x, z)=υ˜0(ν)exp(ik2{xΛν+z[1-(Λν)2-1]})dν.
p(x; z)=exp(ik2{xΛν+z[1-(Λν)2-1]})dν.
1-(Λν)2-1-(Λν)2/2,
p(x; z)exp-i π4expiπ x2Λz/Λz.
υ(x, z)=υ˜0(ν)expik2xΛν-z(Λν)22dν.
υ0(x)=(m)Amexp(2πimν0x);ν0=1D,
υ˜0(ν)=mAmδ(ν-mν0).
υ(x, z)=(m)Amexp(2πimν0x)exp[-iπz(Λmν0)2].
exp(-2πim2z/zP).
zP=2Λν02=2D2Λ=2D2λcos α=zTcos α,
υ(x, z)=(n)Bn(x)exp2πin zzP.
(n)exp2πin zzPd2Bn(x)dx2-Bn(x) 4πnk2zP=0.
Bn(x)=Bn(0)exp[2πixν0|n|sgn(n)].
υ(x, 0)=(n)Bn(x)=υ0(x).
Λ=λcos α=Λ(λ, α).
Λ(λ, α)=Λ¯.
S(λ, α)=S0(λ)δ[λ-Λ¯cos α(λ)].
|υ(x, z; λ)|2=|υ(x, z)|2.
I(x, z)=|υ(x, z)|2S0(λ)dλ.
Δ2υ(x, z)+2ik¯2υ(x, z)z=0,
sin β=λ/D,
α+β=π/2,
cos α=cos(π/2-β)=sin β=λ/D.
V(x, y, z)=u(x, z)w(y),
w(y)=exp2πiλ y sin α
d2wdy2=-2π sin αλ2w(y).
Δ2u(x, z)+k22u(x, z)=0,
Δ2=2x2+2z2,k2=2πΛ=k cos α.
Δ3V(x, y, z)+k2V(x, y, z)
=w(y)[Δ2u(x, z)+k22u(x, z)]=0.
Λ=2πk2=λcos α.
D sin β=Λ.
2F sin γ=Λ.
2πk2=Λ.

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