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Optical Engineering

Meaning of integrability and image aberration for the analysis of fringe curvature with two separate holograms in relatively large strains at the surface of an opaque body
Author(s): Walter Schumann
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Paper Abstract

Beginning with the linear equation of holographic interferometry for small displacements, a technique with a recording of the undeformed and the deformed states of an opaque body on two separate holograms and a modification at the reconstruction is investigated in the general nonlinear case of relatively large strain and large rotation. First, the problem of recovering previously invisible fringes and the following fringe analysis for the strain determination is briefly discussed. Necessary equations from two first derivatives of the optical path difference should ensure, by a careful choice of the modification parameters, at least locally proper spacing and sufficient contrast of the fringes. To enlarge the domain of visibility, the second derivative must also be considered; the remaining parameters should be determined such that the path difference becomes quasi stationary in an extended domain. Here the principal subject is a detailed analysis of this second derivative of the optical path difference, which reveals peculiar properties as a by-product of the mentioned purpose. It contains out-of-plane terms with changes of the surface curvature and in-plane terms with the derivative of the surface dilatation. The latter terms are obtained by the basic integrability equation for curved surfaces. This equation enables, on one hand, the elimination of the variable rotation part of the deformation and shows, on the other hand, a relation to the dislocation theory of continuum mechanics. Three quadratic forms appear besides terms with the geodesic curvature and the first derivative in the final result of the second derivative along any curve. One of the forms contains the deformation of the object surface, the two others imply the virtual deformation of the images due to the modification. The duality property in holographic imaging enables the elimination of a bilinear form. The expression also enables us to analyze the fringe curvature.

Paper Details

Date Published: 1 August 1996
PDF: 11 pages
Opt. Eng. 35(8) doi: 10.1117/1.600820
Published in: Optical Engineering Volume 35, Issue 8
Show Author Affiliations
Walter Schumann, Swiss Federal Institute of Technology (Switzerland)

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