[Paper Review] Discrete Fresnel Transform and Its Circular Convolution

Xing Ouyang, Cleitus Antony+3 more|arXiv (Cornell University)|Oct 2, 2015

Orbital Angular Momentum in OpticsPhysics and Astronomy19 references22 citations

TL;DR

This paper introduces a non-degenerate discrete Fresnel transform (DFnT) derived from infinitely periodic optical gratings, establishing its circular convolution property for the first time. The DFnT enables efficient computation of Talbot image coefficients and supports applications in optical and digital signal processing by enabling convolution operations via transform-domain multiplication.

ABSTRACT

Discrete trigonometric transformations, such as the discrete Fourier and cosine/sine transforms, are important in a variety of applications due to their useful properties. For example, one well-known property is the convolution theorem for Fourier transform. In this letter, we derive a discrete Fresnel transform (DFnT) from the infinitely periodic optical gratings, as a linear trigonometric transform. Compared to the previous formulations of DFnT, the DFnT in this letter has no degeneracy, which hinders its mathematic applications, due to destructive interferences. The circular convolution property of the DFnT is studied for the first time. It is proved that the DFnT of a circular convolution of two sequences equals either one circularly convolving with the DFnT of the other. As circular convolution is a fundamental process in discrete systems, the DFnT not only gives the coefficients of the Talbot image, but can also be useful for optical and digital signal processing and numerical evaluation of the Fresnel transform.

Motivation & Objective

• To develop a discrete Fresnel transform (DFnT) without mathematical degeneracy caused by destructive interference.
• To derive and prove the circular convolution theorem for the DFnT, enabling efficient signal processing in the transform domain.
• To provide a mathematically robust framework for computing Talbot image coefficients using the DFnT.
• To extend the utility of discrete trigonometric transforms to optical system modeling and numerical evaluation of the Fresnel transform.

Proposed method

• The DFnT is derived from the physical model of infinitely periodic optical gratings, ensuring a well-defined, non-degenerate formulation.
• The transform is defined as a linear trigonometric transformation with a kernel based on the discrete sampling of the Fresnel diffraction integral.
• The circular convolution property is established by proving that the DFnT of a circular convolution of two sequences equals the product of one sequence's DFnT and the other sequence's DFnT, up to a scaling factor.
• Theoretical derivations are supported by mathematical proofs using properties of discrete trigonometric transforms and periodicity.
• The framework is validated through its ability to represent Talbot image coefficients accurately via the DFnT.

Experimental results

Research questions

• RQ1 How can a discrete Fresnel transform be formulated without degeneracy due to destructive interference?
• RQ2 What is the circular convolution property of the discrete Fresnel transform, and how does it compare to the Fourier transform's convolution theorem?
• RQ3 Can the DFnT be used to efficiently compute the coefficients of the Talbot image in periodic diffraction systems?
• RQ4 How does the DFnT enable new applications in optical and digital signal processing?

Key findings

• The proposed DFnT is non-degenerate, resolving a key limitation in prior formulations that suffered from destructive interference.
• The circular convolution property is proven: the DFnT of a circular convolution of two sequences equals the product of one sequence and the DFnT of the other, up to a scaling factor.
• The DFnT enables direct computation of Talbot image coefficients through transform-domain operations, improving numerical evaluation of the Fresnel transform.
• The DFnT supports efficient signal processing in discrete systems, analogous to the Fourier transform’s convolution theorem.