Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures
JOSA A, Vol. 15, Issue 5, pp. 1100-1113 (1998)
http://dx.doi.org/10.1364/JOSAA.15.001100
Acrobat PDF (1959 KB)
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
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
Sort: Year | Journal | Reset
References
- K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
- O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
- A. T. Shepherd, “25 years of moiré fringe measurement,” Precis. Eng. 1, 61–69 (1979).
- H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
- P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, UK, 1969).
- Y. Nishijima and G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements,” J. Opt. Soc. Am. 54, 1–5 (1964).
- G. Oster, M. Wasserman, and C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
- O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. 64, 1287–1294 (1974).
- O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. 65, 685–694 (1975).
- I. Amidror and 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).
- See, for example, K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 16–21.
- 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.
- The equivalence between a 2D mapping from R^{2} onto itself and a coordinate change in R^{2} is discussed and illustrated in R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 133–140.
- If g_{1}(x, y) and g_{2}(x, y) are dependent, for instance, g_{2}(x, y)=g_{1}(x, y)^{2}, then the 2D transformation g(x, y) is degenerate, and it maps R^{2} into a 1D curve in R^{2}; see R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, p. 155.
- See, for example, R. Courant, Differential and Integral Calculus (Wiley-Interscience, New York, 1988), Vol. II, pp. 154–155. If g_{1}(x, y) and g_{2}(x, y) satisfy also the Cauchy-Riemann conditions (a) ∂g_{1}/∂x=∂g_{2}/∂y, ∂g_{1}/∂y= −∂g_{2}/∂x or (b) ∂g_{1}/∂x=−∂g_{2}/∂y, ∂g_{1}/∂y= −∂g_{2}/∂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 g_{1}(x, y) and g_{2}(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.
- A. W. Lohmann and D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
- A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, UK, 1968), Vol. 1, p. 36.
- Note that this particular case has already been proposed in Ref. 16.
Cited By |
Alert me when this paper is cited |
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
« Previous Article | Next Article »
OSA is a member of CrossRef.