Dimensionless formulation of the convolution and angular spectrum reconstruction methods in digital holography

Abstract

The evaluation of the Rayleigh-Sommerfeld diffraction formula by means of numerical convolution and angular spectrum filtering are two of the most usual reconstruction methods in digital holography. Both of them are normally implemented by using a discrete Fourier transform and a sample of, respectively, the free space impulse response function and the corresponding transfer function. In this communication we propose a modified formulation of the sampled free space impulse response and transfer functions in terms of five dimensionless parameters: the wavelength to horizontal pixel size ratio, the reconstruction distance to horizontal field size ratio, the field and pixel aspect ratios and the number of pixels in the horizontal direction. This formulation simplifies the task of comparing and finding equivalences between holographic reconstruction situations with different distance, wavelength, field and pixel sizes. The reconstruction range for each of the methods is expressed in terms of the aforementioned dimensionless parameters by analyzing the resolution limits for the impulse response and the transfer function, respectively. This notation makes very simple to decide which of the two methods should be used for given conditions as well as to tailor range extension strategies based on the effects of hologram manipulations such as zero padding or pixel splitting. The details of the implementation of the convolution and angular spectrum algorithms with the proposed formulation are disclosed paying particular attention to the consequences of the sacrificial zero-padding required to avoid aliasing in Fourier-transform based cyclic convolution.