The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.

Fractional Fourier transform for partially coherent Gaussian-Schell model beams.

Via the route of developing Dirac's symbolic method by virtue of the technique of integration within an ordered product (IWOP) of operators we derive various entangled state representations, squeezed states, and operator ordering formulae. We show how the entangled states can be applied to many aspects of quantum optics theory. The applications of the entangled states in the interdisciplinary study of quantum optics and Fourier optics are also introduced. All these discussions exhibit the power of Dirac's symbolic method with the aid of the IWOP technique.

Entangled states, squeezed states gained via the route of developing dirac's symbolic method and their applications

Chapter 7 The fractional fourier transform and some of its applications to optics

The fractional Fourier transform (FRT) is applied to study the transformation properties of flattened Gaussian beams (FGBs). An analytical formula is derived for the FRT of FGBs based on the definition of FRT. By using the derived formula, we study the intensity distribution properties of FGB in the FRT plane. The influences of the fractional order on the intensity distribution properties of FGB with different order in the fractional Fourier plane are studied in detail. The evolution of the kurtosis parameter of FGB in the FRT plane is also studied.

Properties of a flattened Gaussian beam in the fractional Fourier transform plane

In recent years, there has been an enormous eort put in the denition and analysis of fractional or fractal operators. Fractional calculus is for example a ourishing eld of active research. In this paper we restrict ourselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. The book by H.M. Ozaktas, Z. Zalevsky, and M.A. Kutay, The fractional Fourier transform, John Wiley, 2001 gives a state of the art of 2001. Because this eld is still in full expansion, we want to summarize in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored

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Recent developments in the theory of the fractional Fourier and linear canonical transforms

We derive the integral representation of a fractional Hankel transform (FRHT) from a fractional Fourier transform. Some basic properties of the FRHT such as Parseval's theorem and its optical implementation are discussed qualitatively.

Yingyang Lu

Papers

Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform.

Fractional Fourier transform and the elliptic gradient-index medium