No abstract

We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are combined into a fractional Hilbert transform, which is characterized by two parameters. Computer simulations are presented.

An optical setup to achieve superresolution in microscopy using holographic recording is presented. The technique is based on off-axis illumination of the object and a simple optical image processing stage after the imaging system for the interferometric recording process. The superresolution effect can be obtained either in one step by combining a spatial multiplexing process and an incoherent addition of different holograms or it can be implemented sequentially. Each hologram holds the information of each different frequency bandpass of the object spectrum. We have optically implemented the approach for a low-numerical-aperture commercial microscope objective. The system is simple and robust because the holographic interferometric recording setup is done after the imaging lens.

The ability of dynamic extraction of remote sounds is very appealing. In this manuscript we propose an optical approach allowing the extraction and the separation of remote sound sources. The approach is very modular and it does not apply any constraints regarding the relative position of the sound sources and the detection device. The optical setup doing the detection is very simple and versatile. The principle is to observe the movement of the secondary speckle patterns that are generated on top of the target when it is illuminated by a spot of laser beam. Proper adaption of the imaging optics allows following the temporal trajectories of those speckles and extracting the sound signals out of the processed trajectory. Various sound sources are imaged in different spatial pixels and thus blind source separation becomes a very simple task.

The Gerchberg-Saxton (G-S) algorithm is a well-known procedure used in various optical implementations. One of its most common applications is beam shaping of an input plane. In that application the desired beam shape is obtained in the Fourier plane. We propose an algorithm in which the desired shape is obtained in the fractional Fourier or the Fresnel domain. Computer simulations illustrate that the newly defined algorithm can reduce the error between the theoretically desired shape and the actual output by 30 times over the conventional G-S algorithm.