T Tatiana Alieva

On fractional Fourier transform moments

Based on the relation between the ambiguity function represented in a quasipolar coordinate system and the fractional power spectra, the fractional Fourier transform (FT) moments are introduced. Important equalities for the global second order fractional FT moments are derived, and their applications for signal analysis are discussed. The connection between the local moments and the angle derivative of the fractional power spectra is established. This permits us to solve the phase retrieval problem if only two close fractional power spectra are known.

Classification of lossless first-order optical systems and the linear canonical transformation

Based on the eigenvalues of the ray transformation matrix, a classification of ABCD systems is proposed and some nuclei (i.e., elementary members) in each class are described. In the one-dimensional case, possible nuclei are the magnifier, the lens, and the fractional Fourier transformer. In the two-dimensional case we have-in addition to the obvious concatenations of one-dimensional nuclei-the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently two-dimensional. Any ABCD system belongs to one of the classes described in this paper and is similar (in the sense of matrix similarity of the ray transformation matrices) to the corresponding nucleus. Knowledge of a nucleus may be helpful in finding eigenfunctions of the corresponding class of first-order optical systems: one only has to find eigenfunctions of the nucleus and to determine how these functions propagate through a first-order optical system.

Powers of transfer matrices determined by means of eigenfunctions

The parameters of the transfer matrix describing a first-order optical system that is a cascade of k identical subsystems defined by the transfer matrix M are determined from consideration of the subsystem’s eigenfunctions. A condition for the cascade to be cyclic is derived. Particular examples of cyclic first-order optical systems are presented. Structure and properties of eigenfunctions of cyclic transforms are considered. A method of optical signal encryption by use of cyclic first-order systems is proposed.

Fractionalization of the linear cyclic transforms

In this study the general algorithm for the fractionalization of the linear cyclic integral transforms is established. It is shown that there are an infinite number of continuous fractional transforms related to a given cyclic integral transform. The main properties of the fractional transforms used in optics are considered. As an example, two different types of fractional Hartley transform are introduced, and the experimental setups for their optical implementation are proposed.

Phase-space distributions in quasi-polar coordinates and the fractional Fourier transform

The ambiguity function and Cohens class of bilinear phase-space distributions are represented in a quasipolar coordinate system instead of in a Cartesian system. Relationships between these distributions and the fractional Fourier transform are derived; in particular, derivatives of the ambiguity function are related to moments of the fractional power spectra. A simplification is achieved for the description of underspread signals, for optical beam characterization, and for the generation of signal-adaptive phase-space distributions.

Wigner distribution moments in fractional Fourier transform systems

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. This general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

Generating function for Hermite–Gaussian modes propagating through first-order optical systems

We consider the field that appears at the output of a first-order optical system when its input field is a Hermite-Gaussian mode. The generating function for the output modes is determined. The orthonormality property of the output modes is confirmed, and derivative and recurrence relations for these modes are derived. Mode converters that generate Laguerre-Gaussian and Hermite-Laguerre-Gaussian modes are considered as examples.

Partially coherent stable and spiral beams

Stable and spiral coherent beams, which do not change the form of their intensity distribution apart from possible scaling and rotation during propagation and therefore possess self-healing properties, are widely applied in science and technology. On the other hand, it has been found that partially coherent light often provides better output than coherent light. Here we consider two methods for the design and experimental generation of partially coherent stable and spiral beams.

Fractional-Fourier-domain weighted Wigner distribution

A fractional-Fourier-domain realization of the weighted Wigner distribution (or S-method), producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the short-time Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the analysis of the second-order fractional Fourier transform moments. Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain.

Wigner distribution and fractional Fourier transform for two-dimensional symmetric optical beams

A useful relationship between the fractional Fourier transform power spectra of a two-dimensional symmetric optical beam, on the one hand, and its Wigner distribution, on the other, is established. This relationship allows a significant simplification of the standard procedure for the reconstruction of the Wigner distribution from the field intensity distributions in the fractional Fourier domains. The Wigner distribution of a symmetric optical beam is analyzed, both in the coherent and in the partially coherent case.