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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 15, Iss. 5 — May. 1, 1998
  • pp: 1100–1113

Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures

Isaac Amidror and Roger D. Hersch  »View Author Affiliations


JOSA A, Vol. 15, Issue 5, pp. 1100-1113 (1998)
http://dx.doi.org/10.1364/JOSAA.15.001100


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Abstract

The best method for investigating moiré phenomena in the superposition of periodic layers is based on the Fourier approach. However, superposition moiré effects are not limited to periodic layers, and they also occur between repetitive structures that are obtained by geometric transformations of periodic layers. We present in this paper the basic rules based on the Fourier approach that govern the moiré effects between such repetitive structures. We show how these rules can be used in the analysis of the obtained moirés as well as in the synthesis of moirés with any required intensity profile and geometric layout. In particular, we obtain the interesting result that the geometric layout and the periodic profile of the moiré are completely independent of each other; the geometric layout of the moiré is determined by the geometric layouts of the superposed layers, and the periodic profile of the moiré is determined by the periodic profiles of the superposed layers. The moiré in the superposition of two geometrically transformed periodic layers is a geometric transformation of the moiré formed between the original layers, the geometric transformation being a weighted sum of the geometric transformations of the individual layers. We illustrate our results with several examples, and in particular we show how one may obtain a fully periodic moiré even when the original layers are not necessarily periodic.

© 1998 Optical Society of America

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.0080) Geometric optics : Geometric optics
(120.4120) Instrumentation, measurement, and metrology : Moire' techniques

History
Original Manuscript: August 1, 1997
Revised Manuscript: November 25, 1997
Manuscript Accepted: December 8, 1997
Published: May 1, 1998

Citation
Isaac Amidror and Roger D. Hersch, "Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures," J. Opt. Soc. Am. A 15, 1100-1113 (1998)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-5-1100


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References

  1. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  2. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
  3. A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979). [CrossRef]
  4. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970). [CrossRef] [PubMed]
  5. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, UK, 1969).
  6. Y. Nishijima, G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements,” J. Opt. Soc. Am. 54, 1–5 (1964). [CrossRef]
  7. G. Oster, M. Wasserman, C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964). [CrossRef]
  8. O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. 64, 1287–1294 (1974). [CrossRef]
  9. O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. 65, 685–694 (1975). [CrossRef]
  10. I. Amidror, R. D. Hersch, “Fourier-based analysis of phase shifts in the superposition of periodic layers and their moiré effects,” J. Opt. Soc. Am. A 13, 974–987 (1996). [CrossRef]
  11. See, for example, K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 16–21.
  12. Note that the term curvature is defined in mathematics in a different way; see, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 86.
  13. The equivalence between a 2D mapping from ℝ2 onto itself and a coordinate change in ℝ2 is discussed and illustrated in R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 133–140.
  14. If g1(x, y) and g2(x, y) are dependent, for instance, g2(x, y)=g1(x, y)2, then the 2D transformation g(x, y) is degenerate, and it maps ℝ2 into a 1D curve in ℝ2; see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 155.
  15. See, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 154–155. If g1(x, y) and g2(x, y) satisfy also the Cauchy-Riemann conditions (a) ∂g1/∂x=∂g2/∂y,∂g1/∂y=-∂g2/∂x or (b) ∂g1/∂x=-∂g2/∂y,∂g1/∂y=-∂g2/∂x, then the transformation g(x) is conformal [see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 166–167], and it maps the straight lines x=constant,y=constant into curve families x′=constant and y′=constant, which intersect at right angles. This orthogonality is clearly stronger than the mere independence of g1(x, y) and g2(x, y); and indeed, condition (a) implies J(x, y)>0, and condition (b) implies J(x, y)<0. Such an orthogonality is not required for our needs [see for instance Fig. 2(b)], but it is advantageous; for example, it guarantees that the two curvilinear gratings that together form our curved grid r(x, y) do not generate moirés between themselves within the curved grid itself.
  16. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967). [CrossRef] [PubMed]
  17. A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, UK, 1968), Vol. 1, p. 36.
  18. Note that this particular case has already been proposed in Ref. 16.

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