| Phys 198 | April 7, 1997 |
Once holointerferograms (fringe pattern images) have been obtained, the problem arises as to how to analyze them quantitatively. How are the direction and magnitude of the displacements obtained from the fringe patterns?
The first problem is to determine the direction of the object displacement which caused the fringes seen in the image. It turns out that a single holointerferogram will give only a single component of the displacement at any point.This is not as limiting as might be thought since frequently the motions of interest are either in-plane or out-of-plane, so that full three-dimensional studies are not required. For the time being, it will be assumed that the displacements of interest are basically out-of-plane from a relatively flat surface. How can the physical setup for taking the hologram be arranged to be sensitive to motion in this normal-to-the-surface direction?
Vector Direction
This question (as far as direction is concerned) is answered by introducing a quantity known as the sensitivity vector. This vector is aligned along the direction of maximum sensitivity of the holographic setup. How can this vector be controlled? Although the class text (p. 366)gives a derivation in terms of three-dimensional motion and points out some limitations, the result for many practical cases can be simply described by stating that the sensitivity vector bisects the angle between the incoming illumination beam (assumed collimated) which falls on the object and the observation direction. Note that the direction of the reference has nothing to do with direction of the sensitivity vector, although the direction of the reference beam with respect to the illumination beam determines the sensitivity of the resulting fringe display. Thus if the illumination vector is almost parallel to the viewing direction, the direction of the sensitivity vector will be almost parallel to the viewing direction. Mathematically, if e1 is a unit vector in the direction of the illumination beam, and e2 is a unit vector in the viewing direction, then the sensitivity vector is described as g = e2 - e1. Note that although e1 and e2 are unit vectors, g is not, and its magnitude varies from [0, 2] depending on the angle between the two unit vectors.
Vector Magnitude
To get the magnitude of the displacement between two points on the interferogram, the fringe count between them must first be determined. As noted above, the displacement effects shown by the fringe pattern are due to the component of the total displacement vector d in the direction of g; that is, the projection of d onto g is what is measured: g·d = Dn l . For a normal displacement d with g essentially parallel to the displacement and the two unit vectors are symmetrically oriented at a small angle to the normal, the measured value for out-of-plane motion will be d' = (Dn l )/2. Here d' is the displacement in the direction of the sensitivity vector between two points separated by n fringes.
Vector Sign
Finally, the direction of the displacement must be determined. The magnitude and the line-of-motion has been obtained, but the displacement holointerferogram does not give the direction of the displacement, i.e., for normal motion is the surface at the two points moving in or out? The common method of answering this question is to introduce a known varying phase across the surface under study, a varying phase which for simplicity is usually linear. If then a linearly changing phase is impressed across the image, then any deformation-caused phase changing in the same direction as the linear phase will make the phase change more rapidly and so the fringes will appear closer together, and conversely a deformation-caused phase change in the opposite direction to the known phase will cause the phase to change less rapidly, and the fringes will be spaced further apart.
Thus to get quantitative results from a holointerferometric study of a surface, the observer should create two holointerferograms: the first of the fringe pattern obtained when the initial state of the surface has a known linear phase change across it, and the second of the surface in its final state with a fringe pattern due to both the linear phase change and the deformation-caused phase change. Besides these two interferograms, the observer must understand how the linear phase change was created, so as to know the basic direction of the imposed linear phase change.
Remember that a single holographic setup gives only one component of motion, the observed motion which represents the projection of the real displacement vector upon the sensitivity vector.
Summary:
| g = e2 - e1 | where e1 and e2 are unit illumination and viewing direction vectors |
| g·d = Dn l | projection of displacement vector onto sensitivity vector |
| g = [0, 2] | magnitude varies with directions of illumination and viewing vectors |
Send Mail or Comments: matthysd@vms.csd.mu.edu
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