Experimental Mechanics

, Volume 25, Issue 4, pp 360–366 | Cite as

The governing equations for moiré interferometry and their identity to equations of geometrical moiré

  • A. Livnat
  • D. Post
Article

Abstract

The equations prescribing the gradient and inclination of fringes in moiré interferometry are derived from the basic laws of diffraction and interference. A vectorial representation of three-dimensional diffraction employs incidence and emergence vectors in the plane of the grating; the representation is especially well suited for this type of analysis. The corresponding equations for geometrical moiré are derived by a remarkably direct vectorial method. The analyses prove that the patterns of moiré interferometry and geometrical moiré are governed by identical relationships.

Keywords

Mechanical Engineer Fluid Dynamics Vectorial Representation Vectorial Method Identical Relationship 

List of Symbols

\(\bar A\)

unit vector defining the direction of an incident ray

\(\bar D_m \)

unit vector defining the direction of an emergent ray ofmth diffraction order

\(\bar f\)

grating vector representing the frequency of the reference (or virtual reference) grating-lines/mm

\(\bar F\)

fringe vector representing the spatial frequency or gradient of fringe order in a fringe pattern-fringes/mm

\(\bar F'\)

fringe vector in the image plane

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grating vector representing the frequency of the specimen grating-lines/mm

m

diffraction order

M

magnification ratio

N

fringe order

U

x component of displacement of a point on the specimen surface

x, y, z

rectilinear coordinates

\(\bar \alpha ,\bar \alpha _x ,\bar \alpha _y \)

incidence vector and its x and y components

α*

angle of incidence with respect to the grating normal

β

difference of emergence angles of rays A and B

βm

emergence angle of the ray ofmth diffraction order

γ

angle of emergence vector\(\bar \theta _1 \) with respect to the (rotated) x axis

γ+π

angle of\(\bar \theta \) and\(\bar F\) with respect to the (rotated) x axis

γ′

angle of the fringes with respect to the (rotated) x axis

ɛ

extension or strain of the specimen grating

\(\bar \theta \)

difference of emergence vectors for rays A and B

\(\bar \theta _{m,} \bar \theta _{mx,} \bar \theta _{my} \)

emergence vector for the ray ofmth diffraction order and its x and y components

\(\bar \theta _m (A),\bar \theta _m (B)\)

emergence vectors for rays A and B, respectively

λ

wavelength of light

ϕ

angle of plane of incidence with respect to the x axis; angle of incidence vector\(\bar \alpha \) with respect to the x axis

ψ

angle of in-plane rotation of the specimen grating

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References

  1. 1.
    Czarnek, R. andPost, D., “Moiré Interferometry with ±45° Gratings,”Experimental Mechanics,24 (1),68–74 (March 1984).CrossRefGoogle Scholar
  2. 2.
    Rogers, G.L., “A Geometrical Approach to Moiré Pattern Calculations,”Optica Acta,24 (1),1–13 (1977).Google Scholar
  3. 3.
    Stetson, K.A., “Homogeneous Deformations: Determination by Fringe Vectors in Holographic Interferometry,”Appl. Opt.,14 (9),2256–2259 (Sept. 1975).Google Scholar
  4. 4.
    Spencer, G.H. andMurty, M.V.R.K., “General Ray-Tracing Procudure,”J. Opt. Soc. Amer.,52 (6),672–678 (June 1962).Google Scholar
  5. 5.
    Post, D., “Moiré Interferometry at VPI & SU,”Experimental Mechanics,23 (2),203–210 (June 1983).CrossRefGoogle Scholar
  6. 6.
    Durelli, A.J. andParks, V.J., “Moiré Fringes as Parametric Curves,”Experimental Mechanics,7 (3),97–104 (March 1967).CrossRefGoogle Scholar
  7. 7.
    Guild, J., The Interference System of Crossed Diffraction Gratings—Theory of Moiré Fringes, Oxford University Press, Oxford (1956).Google Scholar
  8. 8.
    Post, D., “Optical Interference for Deformation Measurements—Classical Holographic and Moiré Interferometry,”Mechanics of Nondestructive Testing, ed. W.W. Stinchcomb, Plenum Press, NY, 1–53 (1980).Google Scholar
  9. 9.
    Durelli, A.J. andParks, V.J., Moiré Analysis of Strain, Prentice-Hall, Inc., Englewood Cliffs, NJ (1970).Google Scholar
  10. 10.
    Theocaris, P.S., Moiré Fringes in Strain Analysis, Pergamon Press, Ltd, Oxford (1969).Google Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 1985

Authors and Affiliations

  • A. Livnat
    • 1
  • D. Post
    • 1
  1. 1.Engineering Science and Mechanics DepartmentVirginia Polytechnic Institute and State UniversityBlacksburg

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