# Basic optical law in the interpretation of moiré patterns applied to the analysis of strains—part 1

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## Abstract

The objective of this paper is to generalize the optical law that relates the displacement field to the fringes of moiré patterns. Until now, a discontinuous relationship has been applied. Displacements that are equal to an integral number of times the master grid pitch or half the master grid pitch are related to the points of maximum and minimum light intensity, respectively.

The generalized optical law gives a continuous relation-ship between displacements and light intensities. By applying this law it is possible to increase the precision of moiré far beyond the actual limits.

## Keywords

Mechanical Engineer Light Intensity Fluid Dynamics Displacement Field Actual Limit## List of symbols

*a*radius of the pupil of the optical system equal to

*r*/λ*R**A*area of the exit pupil of the optical system

*b*width of the light-intensity rectangular pulse

- \(\bar b\)
average value of

*b**c*_{n}coefficient of the Fourier expansion of the grid transmission function n=0,1, 2, 3...

*E*amplitude of the light-intensity rectangular pulse

*f(x, y)*grid transmission function

*f*_{T}(*x*,*y*)transmission function resulting from the superposition of two grids

- \(\bar fr(x'y')\)
cross-correlation function of

*f*_{ A }(*x*,*y*) and*f*_{ B }(*x*,*y*)*F(g, h)*Fourier transform of

*f(x, y)*- \(\bar F_r (g, h)\)
Fourier transform of\(\bar f_r (x, y)\)

*g*spacial frequency

*G(g, h)*pupil function of the image-forming system

*h*spacial frequency

*i*imaginary unit

*I*_{0}image average background light intensity

*I*_{1}light-intensity amplitude of the image first harmonic

*I(x)*light intensity at a point of coordinate

*x*in the image plane- \(\Im (g, h)\)
Fourier transform of the light-intensity distribution function in the image plane

*k*integer 1, 2, 3...

*K*coefficient smaller than one

*l*integer 1, 2, 3....

*L*range of integration for the computation of the Fourier transform

- \(\mathfrak{L}(g, h)\)
frequency-response function of the image-forming system

*n*integer 1, 2, 3....

*m*magnification of the optical system

*p*grid pitch

*p*_{1},*p*_{2}pitches of the grids

*A*and*B*, respectively- \(\bar p\)
average pitch

*p*_{x}grid pitch in the

*x*-axis direction*p(x)*local pitch

- \(\bar p_x\)
average value of

*p*_{ x }*p*_{m}grid pitch of the image of the grid

*R*radius of the Gaussian sphere

*r*radius of the pupil of the optical system

*S*grid area

*s*integer 1, 2, 3...

*u*displacement in the direction of the

*x*axis*U*_{o}^{*}(*x*,*y*)*U*_{0}(*x*,*y*)disturbances arriving at and leaving the grid, respectively

*x*coordinate

*x′*relative displacement in the

*x*-axis direction independent of*x**y*coordinate

*y′*relative displacement in the

*y*-axis direction independent of*y*- β
angle whose cosine is

*h*/2*a*- δ
fringe spacing in the

*x*-axis direction- \(\delta (\xi - \xi _1 )\)
Dirac delta function

- Δ
*p(x)* local change of pitch

- Δθ(
*x*) local change of the angle θ

- Δπ(
*x*) local change of π

- θ
rotation of grid

*B*with respect to the*x*axis

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## References

- 1.
*Aciammarella, C. A.*, and*Durelli, A. J.*, “*Moiré Fringes as a Means of Analyzing Strains*,”*Transactions of the ASCE Paper No. 3329, 127, pt I, 582–603*(*1962*).Google Scholar - 2.
*Guild, G., “The Interference System of Crossed Diffraction Gratings. Theory of Moiré Fringes,” Oxford at the Clarendon Press (1956)*.Google Scholar - 3.
*Born, A., and Wolf, E., “Principle of Optics,” Pergamon Press, 479 (1959)*.Google Scholar - 4.
*Encyclopedia of Physics. Edited by S. Flugge, XXIV, Fundamentals of Optics. “Interferences, diffraction et polarization,” 331 (1956)*.Google Scholar