Tatiana Alieva

Self-fractional fourier images

Abstract Self-fractional Fourier images (SFFIs), which are invariant under the two-dimensional fractional Fourier transform, are considered. Necessary and sufficient conditions for an image to be a SFFI for given angles are formulated. An optical procedure for synthesizing SFFIs from a generator image is developed. The physical meaning of SFFI synthesis is discussed. An image decomposition on the set of orthogonal SFFIs, which simplifies image processing in fractional FT optical systems, is proposed.

Fractional Fourier and Radon-Wigner transforms of periodic signals

Abstract In this paper we consider specific features of the fractional Fourier transform (FT) for periodic signals. It is shown that the fractional FT of a periodic signal at some angles is the superposition of its scaled weighted and shifted replicas with additional quadratic phase factor. The modulus of the fractional FT (Radon–Wigner transform) at an angle α ≠π/2+π n of a periodic signal is also periodic. For a certain sequence of angles, the Radon–Wigner transforms are affine to each other. This feature simplifies the tomographic reconstruction procedure of periodic signals.

Self-imaging in fractional Fourier transform systems

The self-imaging phenomenon in a fractional Fourier transform optical system is described in the framework of self-fractional Fourier functions. The main properties of these functions are investigated. The possible application of fractional Fourier transform optical systems in optical signal processing is discussed.

Fractional Fourier analysis of objects with scaling symmetry

The fractional Fourier analysis is used for investigation of fractal structures. As is shown, it reveals the scaling features of deterministic as well as of the random fractals, and allows to determine their main characteristics. As an example we present the fractional Fourier analysis of some self-affine functions. The map of the Radon-Wigner transform of the triadic Cantor set is considered numerically. The application of the fractional Fourier analysis in optical engineering is discussed.

Radon-Wigner transform for optical field analysis

The Radon-Wigner transform, associated with the intensity distribution in the fractional Fourier transform system, is used for the analysis of complex structures of coherent as well as partially coherent optical fields. The application of the Radon-Wigner transform to the analysis of fractal fields is presented.

Macroscopic 1/f behavior of fractal signals generated by additive substitution

We analyze the structure of the discrete Fourier transform of a particular class of self-similar discrete signals which are iteratively generated by an additive substitution procedure, starting from a finite generator sequence. The amplitude spectra of such signals obtained from different generators have a similar form. The amplitude spectrum contains several affine branches, the number of which depends on the iteration level, on the size of the generator, and on the spectrums symmetry. It has a macroscopic 1/f-characteristic.

Self-imaging in first-order optical systems

The structure and main properties of coherent and partially coherent optical fields that are self-reproducible under propagation through a first-order optical system are investigated. A phase space description of self-imaging in first-order optical systems is presented.

About quantum state characterization

Examples of different pure quantum states which are not distinguishable by the finite set of their marginal distributions are presented.