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

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## 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 of

*m*th 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

- β
_{m} emergence angle of the ray of

*m*th 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 of

*m*th 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

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