Linear Canonical Domains and Degrees of Freedom of Signals and Systems

Abstract

We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, 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 monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.

Effect of spatial distribution of partial information on the accurate recovery of optical wave fields

“…Planes perpendicular to the axis of propagation correspond to fractional Fourier domains (FRFDs) [3]. Furthermore, the same is true for all quadratic-phase systems [4], which are systems including an arbitrary sequence of lenses, sections of free space, and quadratic graded-index media [7][8][9][10][11][12][13][14][15][16]. Working with the FRT will allow us to approach the heart of the problem in its purest form and identify the main trends as clearly as possible, while the resulting important observations remain applicable to a broad class of systems of practical interest.…”

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.…”