Systematic approach for the design of reconstruction algorithm in digital holography

In digital holography, holograms are recorded by a CCD-array, and the complex amplitude of the object wave is numerically reconstructed via computer. For different recording conditions and different properties of objects, different reconstruction algorithms are required. The conventional reconstruction algorithms were conceived directly by replacing the diffraction integral with summation. Each method has its limitation in the valid range for correctly calculating the diffraction integral. The Single Fourier Transform method is valid for far Fresnel zone hologram, whereas the convolution method is appropriate for near Fresnel holograms. Here, we present a general reconstruction model from the perspective of "Generalized sampling theory". Given that the function space in which the unknown complex amplitude lies, an approximation of the continuous complex amplitude at the CCD can be synthesized from a set of basis functions with the recorded samples as weights. Back-propagation of the approximated complex amplitude to the original object plane yields an expression relating the continuous complex amplitude of the object with the recorded samples. By adopting different basis functions and different formulas for describing the diffraction process, an optimal reconstruction algorithm can be developed for various recording conditions and different diffraction characteristics of the object. Contrary to the conventional algorithms where values are available only at specific grid, complex amplitude at any position of the object can be obtained using this model. In addition, the effect due to the non-zero fill factor of the CCD can also be incorporated into the reconstruction algorithm to be further compensated by over-weighting the high frequency components. Two basis functions: Dirac delta- and Sine-, are studied in detail.