Abstract

Analytical expressions describing the moire fringes formed by two periodic structures consisting of the evolutes of a circle are derived. The circular and radial moire patterns can be formed by contact and noncontact superimposition of the structures. In the latter case the self-imaging phenomenon is used. Experimental verification and proposals for use are given.

© 1989 Optical Society of America

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References

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  1. See, for example, W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  2. See, for example, P. Theocaris, Moire Fringes in Strain Analysis (Pergamon, Oxford, 1969).
  3. O. Bryngdahl, W. H. Lee, “Shearing Interferometry in Polar Coordinates,” J. Opt. Soc. Am. 64, 1606–1615 (1974).
    [CrossRef]
  4. B. Sandler, E. Keren, A. Livnat, O. Kafri, “Moire Patterns of Skewed Radial Gratings,” Appl. Opt. 26, 772–773 (1987).
    [CrossRef] [PubMed]
  5. P. Szwaykowski, “Self-Imaging in Polar Coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
    [CrossRef]

1988 (1)

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Figures (5)

Fig. 1
Fig. 1

Computer-generated plot of a diffraction grating consisting of ten evolutes of a circle.

Fig. 2
Fig. 2

Geometry of the system of two evolutes of a circle running in opposite directions. Both evolutes are separated by the angle (nk)α.

Fig. 3
Fig. 3

Geometry of the system of two evolutes (running in the same direction) of two slightly different radii of inner circles. Both evolutes are separated by the angle ().

Fig. 4
Fig. 4

Radial moire fringes obtained with two evolute gratings with lines running in opposite directions (separated by the self-image distance).

Fig. 5
Fig. 5

Circular moire fringes obtained with two evolute gratings with lines running in the same direction illuminated by a spherical wavefront beam.

Equations (10)

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x = r 0 cos ( t k φ ) + r 0 t sin ( t k φ ) , y = r sin ( t k φ ) rt cos ( t k φ ) ,
r 0 cos ( t k φ ) + r 0 t sin ( t k φ ) = r 0 cos ( z n φ ) + r 0 z sin ( z + n φ ) , r 0 sin ( t k φ ) r 0 t cos ( t k φ ) = r 0 sin ( z n φ ) + r 0 z cos ( z n φ ) ,
( r 0 cos γ + r 0 t sin γ ) ( 1 cos l φ ) + ( r 0 sin γ r 0 t cos γ ) sin l φ = 0 , ( r 0 sin γ r 0 t cos γ ) ( 1 + cos l φ ) + ( r 0 cos γ + r 0 t sin γ ) sin l φ = 0 ,
= x ˚ 1 cos l φ sin l φ = x ˚ tan ( l φ / 2 ) , = x ˚ sin l φ 1 + cos l φ = x ˚ tan ( 1 φ / 2 ) .
R cos ( t k φ ) + Rt sin ( t k φ ) = r cos ( z n φ ) + rz sin ( z n φ ) , R sin ( t k φ ) Rt cos ( t k φ ) = r sin ( z n φ ) rz cos ( z n φ ) .
R 2 ( 1 + t ˚ 2 ) = r 2 ( 1 + z ˚ 2 ) .
z = t + α .
( cos γ + t sin γ ) ( R r cos β ) = ( r sin γ rt cos γ ) sin β + r α sin ( γ + β ) , ( sin γ t cos γ ) ( R r cos β ) = ( r cos γ + rt sin γ ) sin β r α cos ( γ + β ) ,
x ˚ ( R r cos β ) + r sin β = Rr α sin ( γ + β ) , ( R r cos β ) x ˚ r sin β = Rr α cos ( γ + β ) .
x ˚ 2 + 2 = ( rR α ) 2 ( R r ) 2 + 2 rR sin 2 β .

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