EXAMPLE OF IN-PLANE HOLOINTERFEROMETRY

As an example of a relatively simple holointerferometric measurement of in-plane measurement, consider the case of a thin glass disk, 88 mm in diameter, 1.6 mm in thickness, mounted with its circular boundary held rigid. A small sphere attached to the end of a micrometer is used to push out the center of the disk, creating a bulge in the center. How can the normal displacement over the surface be measured? If you did not already know the direction of displacement, how could you determine the direction of the displacement from examination of the holointerferometric images?

Recall from the lecture of April 7 that two images are required: an image of the fringes produced on the disk when a uniform phase shift is introduced,producing a linear fringe pattern; an image of the fringe pattern obtained when both the uniform phase shift and the phase changes due to the displacement are present. Also, you have to know the direction of the linear phase shift. In this example, the phase shift was introduced by rotating the bottom of the disk a few milliradians towards the viewer about a horizontal axis through the center of the disk. This means that the phase shift will be linear and increasing from bottom to top.

Figure 1 shows the optical setup for obtaining the fringe patterns. It is a typical two beam off-axis setup for taking transmission holograms.

To obtain the required interferograms, first the center of the disk is deformed by pressing against its center with a ball-loaded micrometer tip. The fringe pattern due to this deformation is shown in Fig. 2.














The carrier fringe pattern must be considerably higher than the frequencies of the deformation fringe pattern. To introduce a suitable controlled phase shift, the disk deformation is removed and then the bottom of the disk is rotated around a horizontal in-plane axis through the center of the disk. This produces the fringe pattern shown in Figure 3, where each fringe represents a phase change of 2p or a projected displacement of a wavelength (HeNe laser gives light of wavelength 0.633 microns). Since the center of the disk does not move, the central fringe in numbered as the reference zero fringe, and the other fringes are numbered, from bottom to top, from +10 to -10.

When the holographic image of the original flat disk is superposed with the tilted and deformed disk, the fringe pattern shown in Fig.4 is obtained.


















If now the fringe order versus distance from the bottom row up along column 128 is plotted for each of Figures 3 and 4 is plotted, the curves shown in Fig. 5 are obtained. The curve labeled a is for the carrier wave, while b is for the deformation plus tilt pattern.



















Since what is wanted is just the fringe count due to the deformation, the curves in Fig. 5 are subtracted from each other, giving the result shown in Fig. 6. This gives the contour of the vertical midline of the deformed disk. With proper scaling, the vertical axis can be expressed in microns. Of course, to get the full profile of the disk, a similar operation must be performed for all the columns across the entire disk. This would produce a roughly cone-shaped mountain display for the disk surface. Notice that the scale of the vertical axis is orders-of-magnitude different from disk size. The disk is 88 mm in diameter and 1.6 mm thick, while the deformation is only a few microns.

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