Fractional Fourier optics

Abstract

There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.

The fractional Fourier transform as a simulation tool for lens-based X-ray microscopy

“…The advantages of using FrFT for cascades of axially centered lenses are well known in the context of visible-light optics (Ozaktas et al, 2001;Ozaktas & Mendlovic, 1995). More recently, its use for simulating free space propagation of X-rays has been presented by Le Bolloc'h et al (Le Bolloc'h et al, 2012;Mas et al, 1999).…”

Optimal representation and processing of optical signals in quadratic-phase systems

“…One of the possible decompositions involves three stages. The first is a FRT operation, the second is a magnification operation, and the final stage is a chirp multiplication operation [28][29][30][31][32]:…”

Optimal filtering in fractional Fourier domains

“…Several applications of the fractional Fourier transform have been suggested or explored to varying degrees. These include optical diffraction and beam propagation, and optical signal processing [2], [7]- [12], [22]- [25], quantum optics [26]- [30], phase retrieval, signal detection, pattern recognition, noise representation, time-variant filtering and multiplexing, data compression, study of space/TF distributions, and sweptfrequency filters [2], [4], [31], [32]. The fractional Fourier transform is also related to a number of recently developed signal processing tools [33]- [35].…”

“…Since fractional Fourier transform is unitary and it preserves norms, is also equal to the MSE in the th domain (12) where from (6) varies with the choice of since varies. This functional is to be minimized with respect to Let us substitute , where complex scalar parameter, optimum filter, arbitrary perturbation term.…”

“…A related concept ("swept-frequency filters") has been discussed by Almeida [4]. Filtering in a fractional Fourier domain can be implemented as efficiently as filtering in the conventional Fourier domain since the fractional Fourier transformation has a fast digital algorithm [5], [6], and can also be optically realized much like the usual Fourier transform [7]- [12]. The problem considered in this paper is to minimize the MSE for arbitrary degradation models and nonstationary processes by filtering in fractional Fourier domains.…”