Coherent gradient sensing (CGS) is a diffraction-based, noncontact optical technique for measuring the slight curvature of a nearly flat thin film or specimen surface. CGS is especially useful for measuring curvatures of micromechanical structures and of thin films in electronic devices, to enable the determination of stresses in, and mechanical properties of, such structures and films. CGS is a full-field, real-time technique: Unlike in some other techniques, it is not necessary to acquire images of the specimen surface at different times under different conditions of curvature, nor is it necessary to scan a narrow beam of light over the surface; instead, CGS yields data on curvature over the entire surface area of interest, in as little time as it takes to acquire, digitize, and process a video image. Moreover, CGS is insensitive to rotation or displacement of the specimen.

Figure 1. A Typical CGS Apparatus is set up to measure the curvature of a specularly reflective specimen.

Figure 1 schematically illustrates a reflection-mode CGS optical setup. A coherent, collimated laser beam is directed onto a specularly reflective specimen via a beam splitter. The reflected beam passes through the beam splitter, then through two identical high-density (40 lines/mm) Ronchi gratings separated by a distance Δ. A lens spatially filters the portions of light diffracted to various orders to form distinct diffraction spots on a filter plane. An aperture is placed in the filter plane to select a diffraction order of interest and reject other orders. The order of interest is then imaged on the focal plane of a video camera. The video image is digitized and processed to extract information on the curvature of the specimen surface.

Figure 2. Diffraction, Interference, and Spatial Filtering are utilized in CGS to obtain an image containing interference fringes that are indicative of the curvature of the specimen.

Suppose that the grating lines are oriented along the x1axis. Figure 2 illustrates the formation of the first few diffraction orders by each Ronchi grating, and the formation of some of the resulting images on the filter plane. The reflected wavefront incident on Ronchi grating G1is diffracted into wavefronts E1, E0, and E -1, among others, these orders corresponding to diffraction orders 1,0, and -1, respectively. (For the sake of simplicity, only these orders are shown, though many others could be present.) Each of these wavefronts is diffracted by Ronchi grating G2, yielding wavefronts E1,1 through E -1, -1, among others. Various sets of parallel diffracted beams are combined by the filtering lens to form diffraction spots D1, D0, and D-1, among others.

The net effect of the gratings is a lateral (in this case, along x2) shift or "shearing" of the incident wavefront, leading to the formation of interference fringes. The fringe pattern has been analyzed theoretically, using the simplifying approximations that the wavefront (and thus the specimen) is nearly flat and that diffraction angles are small. The analysis reveals that the curvature of the specimen surface can be obtained from the CGS interference-fringe pattern via the equation

where a = 1 or 2, b = 1 or 2, Καβ is the curvature tensor, p is the grating pitch, and n(α) denotes the cardinal number of a fringe observed in shearing along the xα direction.

This work was done by Ares J. Rosakis, Raman P. Singh, Elzbieta Kolawa, and Nicholas R. Moore, Jr., of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at under the Physical Sciences category.

In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to

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Refer to NPO-20189, volume and number of this NASA Tech Briefs issue, and the page number.

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Coherent gradient sensing for measuring curvature

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This article first appeared in the August, 1998 issue of Photonics Tech Briefs Magazine.

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