Holography at the U.S. Army Research Laboratory: Development of Hologram Transform Equations using the Fresnel Approximation

Report No. ARL-TR-6189
Authors: Karl K. Klett, Jr., Neal Bambha, and Justin Bickford
Date/Pages: September 2012; 18 pages
Abstract: This report describes the development and use of the Fresnel approximation, which is necessary to transform a digital hologram to an image. The result is that a Fourier transform, common to many image processing applications, may be used to produce the image. The U.S. Army Research Laboratory's (ARL) goal is to investigate applications of holographic interferometry and transform the holographic interference pattern to three-dimensional images. The mathematics described in this report are a critical step in this process. Stokes's theorem is used as a starting point, where the spherical wave complex amplitude is used in the evaluation. The final result of this first step is the Helmholtz-Kirchhoff equation. Assumptions are made, recognizing that only the illuminated information in the aperture is important to the problem. These assumptions change the Helmholtz-Kirchhoff equation to the Kirchhoff diffraction integral. Finally, the Fresnel and Fraunhofer approximations, which deal with aperture coordinates, and aperture size and distance, respectively, are used to obtain a practical solution. The result of these calculations is that the Fourier transform may be used to form an image, instead of more cumbersome integral equations. This fact makes the transformation from a hologram interference pattern to an image much easier to calculate.
Distribution: Approved for public release
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Last Update / Reviewed: September 1, 2012