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.
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Abbreviations
- \(\bar A\) :
-
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
-
: -
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 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
:References
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Livnat, A., Post, D. The governing equations for moiré interferometry and their identity to equations of geometrical moiré. Experimental Mechanics 25, 360–366 (1985). https://doi.org/10.1007/BF02321334
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DOI: https://doi.org/10.1007/BF02321334