Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space–bandwidth product

Abstract

Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.

Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

“…It is known that the discrete FRT approximately maps the samples of a function to the samples of its FRT in the same sense that the ordinary discrete FT does for the ordinary FT [10][11][12][13]. Therefore, the values of the diffracted field at the natural sampling grid points can be well approximated by the discrete FRT of the samples of the input field.…”

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

“…The discrete FRT does the same for the FRT [38][39][40][41][42][43]. It follows that if we are given the values of the samples of the input field, the discrete FRT can be used to approximately compute the values of the field on the natural sampling grid.…”

Phase-space window and degrees of freedom of optical systems with multiple apertures

“…(For a discussion of the implications of this decomposition on the propagation of light through first-order optical systems, see [19,37]. For a discussion of the implications for sampling optical fields, see [38,39].)…”

“…This implies the assumption of a rectangular space-frequency region. However, the set of input signals may not exhibit a rectangular space-frequency support, and even if they do, this support will not remain rectangular as it propagates through the system [18][19][20]. Likewise, the space-frequency windows of multicomponent optical systems, as we will see in this paper, do not in general exhibit rectangular shapes.…”