DTH2B.8.pdf Imaging and Applied Optics © 2014 OSA Near-field Fresnel Reconstruction of Digital Holograms Logan A. Williams 1 , Georges Nehmetallah 2 , Rola Aylo 3 , and Partha P. Banerjee 1 1 Electro-Optics Program, University of Dayton, Dayton, Ohio 45469, USA 2 Electrical Engineering and Computer Science, Catholic University of America, Washington DC 20064, USA 3 Electrona Engineering LLC., 430 Taylor St. N.E., Washington DC 20017, USA Author e-mail address: logan.williams@gmail.com Abstract: Fresnel transform implementation methods are explored which dramatically reduce the minimum reconstruction distance requirements and allow maximal signal recovery with numerically improved resolution scaling at any distance. Methods are illustrated using experimental results. OCIS codes: (090.1995) Digital Holography, (100.3010) Image reconstruction techniques 1. Introduction The Fresnel transform is one of the most common and computationally efficient forms of image reconstruction in digital holography (DH). However, the Fresnel transform asserts certain limitations upon the recording geometry, including an inability to adequately reconstruct fields recorded below a certain minimum object-to-CCD distance. Additionally, the reconstructed image resolution is pre-determined by the physical recording parameters, which may not necessarily provide the desired resolution for a given application. However, simple numerical pre-processing techniques may be used to effectively eliminate these restrictions from the Fresnel transform. 2. Traditional Limits of the Fresnel Transform It is well known that the usefulness of the Fresnel transform in DH reconstruction is generally limited to distances beyond the near-field, where the recording (and hence, reconstruction) distance d is on the order of, or greater, than      , where l is the “feature size” of the object, and λ is the illumination wavelength. Similar to the Fraunhofer length,   is simply the distance at which the Fresnel number  ⁄ , ensuring the object resides in the mid-to-far field. A geometrical argument to determine the minimum value of d based upon the maximum angular frequency recorded using an in-line geometry is also given by Schnars & Jueptner [1], which is         √   (   ), (1) where   is the maximum path length from the farthest transverse extent of the object of size   to the farthest transverse extent of the CCD array, and   is the maximum diffraction angle captured by the CCD array. However both of these criteria are only heuristic. The actual limitation on d is determined more rigorously by the extent of the spatial bandwidth of the object field that can be effectively captured by the CCD array under the Whittaker-Shannon sampling theorem [2]. If   , alternative reconstruction methods may be employed (e.g. angular spectrum method, etc.) to faithfully reconstruct the image. This fact reveals that the image information has indeed been recorded by the CCD at these distances; it is simply not recoverable via standard implementation of the Fresnel transform. Differences between Fresnel reconstruction and the angular spectrum method (or Rayleigh- Sommerfeld diffraction formulation) for sampled diffraction fields have been analyzed by Onural [3]. 3. Reduction of Reconstruction Distance For Fresnel holograms, Goodman [2] has shown the captured bandwidth of the object field,   , to be         , (2) where   and   are the total array lengths in the (image plane) and x (hologram plane) directions, such that      , and      , where   and   are the number of samples in the  and x directions. Properties of the Discrete Fourier Transform (DFT) dictate that     . The minimum sampling interval for a given bandwidth dictated by the Whittaker-Shannon sampling theorem is      , (3) such that the total number of required samples is given by [2]