Reconstructing Surfaces from Optical Fringe Data

Tom Witelski, Duke University, Durham, NC

Experimental techniques for measuring properties of physical systems have been carefully developed so they do not disturb the systems being measured; these are called nondestructive measurement techniques. The data collected in the measurements are generally indirect, i.e. they must be processed or interpreted in some way to yield the desired measurement information. We will carefully consider the mathematical structure and limitations of the data reconstruction process for optical interferograms.

Interferograms are images composed of alternating bright and dark bands (or ‘’fringes’’) generated by constructive and destructive interference of coherent light being reflected from the surfaces of a thin layer of a material, see Figure 1. The presence of fringes implies a change in the thickness of the material layer and hence interferograms resemble topographical maps of surfaces [1] (the key difficulty is that none of the height contours are labeled). The particular application we are concerned with is determining a height surface for a thin layer of a fluid spreading on a flat solid surface from interferometric data.

Figure 1. Sample interferogram image.

The background of this problem is basic optics from physics and geometry from mathematics, but there are many questions about the reconstruction process that require further analysis:

1. Can surfaces be determined just from information on the positions of the dark/bright bands? What additional assumptions are needed?

2. Would more detailed phase (gray scale) information on the fringes help? Are there surfaces that are ambiguous even with this level of data?

3. Are there difficulties present in two dimensions that do not occur for one-dimensional problems?

4. If there is a small level of uncertainty in the fringe data, what are the bounds on the uncertainty in the reconstructed surface?

5. Caustics are bright bands generated by the interaction of light rays reflected from different portions of the surface [2]. What can be done with interferograms containing caustics?

6. Optical heating can affect the surface tension of fluids and consequently modify the film thickness [2,3]. Can these effects be accounted for?

References:

[1] H. E. Cline, A. S. Holik and W. E. Lorensen, Computer-aided surface reconstruction of interference contours, Applied Optics, 21(24): 4481-4488, 1982.

[2] G. Da Costa and R. Escalona, Time evolution of the caustics of a laser heated liquid film, Applied Optics, 29(7): 1023-1033, 1990.

[3] R. Escalona and C. Rosi, Space and time characterization of a thermal lens using an interferometric technique, Optical Engineering, 38(9): 1591-1595, 1999.