Abstract

In “Flatland optics: fundamentals” [J. Opt. Soc. Am. A 17, 1755 (2000)] 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/cosα. 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 [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)] of 1884. We now want to establish the reality or objectivity of the 2D wavelength Λ by some basic experiments similar to those that demonstrated roughly 200 years ago the wave nature of light. Specifically, we describe how to measure the 2D wavelength Λ by mean of five different arrangements that involve Young’s biprism configuration, Talbot’s self-imaging effect, measuring the focal length of a Fresnel zone plate, and letting light be diffracted by a double slit and by a grating. We also performed experiments with most of these arrangements. The results reveal that the theoretical wavelength, as predicted by our Flatland optics theory, does indeed coincide with the wavelength Λ as measured by Flatland experiments. Finally, we present an alternative way to understand Flatland optics in the spatial frequency domains of Flatland and Spaceland.

© 2001 Optical Society of America

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References

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  1. A. W. Lohmann, A. Pe’er, D. Wang, A. A. Friesem, “Flatland optics: fundamentals,” J. Opt. Soc. Am. A 17, 1755–1762 (1999).
    [CrossRef]
  2. H. E. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).
  3. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  4. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  5. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  7. D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, “Homogeneous layer models for high-spatial-frequency dielectric surface-relief gratings: conical diffraction and antireflection designs,” Appl. Opt. 33, 2695–2706 (1994).
    [CrossRef] [PubMed]

Arimoto, Y.

Arrizón, V.

Brundrett, D. L.

Friesem, A. A.

Gaylord, T. K.

Glytsis, E. N.

Klaus, W.

Kodate, K.

Leger, J. R.

Lohmann, A. W.

Moharam, M. G.

Ojeda-Castañeda, J.

Pe’er, A.

Swanson, G. J.

Talbot, H. E.

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

Wang, D.

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Figures (11)

Fig. 1
Fig. 1

Generic setup for Flatland optics. A tilted plane wave illuminates the object u0(x). The wave field u(x, z) may be observed at a plane y=constant, at z0.

Fig. 2
Fig. 2

Young’s biprism experiment. Observed, p, β; deduced, 2p sinβ=Λ.

Fig. 3
Fig. 3

Reflection-version Young’s interference fringes at illumination angles (a) α=50° and (b) α=26.5°.

Fig. 4
Fig. 4

Talbot effect. Known, grating period p; observed, longitudinal period zp; deduced (based on Talbot’s formula); 2p2/zp=Λ.

Fig. 5
Fig. 5

Talbot’s self-imaging effect in the Flatland, where angle α of the illumination plane wave is 80°.

Fig. 6
Fig. 6

One-dimensional Fresnel zone plate.

Fig. 7
Fig. 7

Focal power of a Fresnel zone plate. Known, first radius R1; observed, focal power 1/fα; deduced, R12/2fα=Λ.

Fig. 8
Fig. 8

Double-slit diffraction. A double slit at the front focal plane of the Fresenl zone-plate lens. Known, double-slit separation p and focal length fα; observed, fringe period D; deduced pD/fα=Λ.

Fig. 9
Fig. 9

Grating diffraction. Known, grating period p and focal length fα; observed, β from S=fα tan β; deduced, p sin β=Λ.

Fig. 10
Fig. 10

Ewald rings in Flatland.

Fig. 11
Fig. 11

Plano–convex refractive cylindrical lens used in Flatland optics. (a) 3D view of the lens, (b) one slice of the lens.

Tables (1)

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Table 1 Measured Results of the Double-Slit Interference Experimenta

Equations (24)

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2p sin β=Λ.
Λ=λ/cos α.
p1p2=Λ1Λ2=cos α2cos α1,
2p2/zp=Λ.
Λ=λ/cos α.
R12/2fα=Λ.
F(x2)=F(x2+mR12),m=0, 1, 2,,
F(x2)=(n) An exp[2πn(x/R1)2].
R12/2n=Λfα.
λ/Λ=cos α.
pD/fα=Λ.
Λ=λcos α DD0 f0f(α).
p sin β=Λ.
S/fα=tan ββsin β=Λ/p.
Λ=pSfα=pSf 1cos α=λcos α,
Δ3V(x)+(2π/λ)2V(x)=0
vx2+vy2+vz2=1/λ2.
ΔxzV(x)+2π cos αλ2V(x)=0,
vx2+vz2cos αλ2.
Δϕ=2πλ[(nAC+CE)-AD]
=2πλ(n2-sin2 α-cos α)H(x),
H(x)=Δ0+R12-R12-x2,
t(x)=exp(iΔϕ)=exp[-πiλR1(n2-sin2 α-cos α)x2].
1f(α)=(n2-sin2 α-cos α)1R1.

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