Numerical analysis for finite Fresnel transform

• Published in: Optical review (Tokyo, Japan) Vol. 23; no. 5; pp. 865 - 869
• Main Authors: Aoyagi, Tomohiro, Ohtsubo, Kouichi, Aoyagi, Nobuo
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-10-2016
• Subjects: Atomic; Germany; Lasers; Microwaves; Molecular; Optical and Plasma Physics; Optical Devices; Optics; Photonics; Physics; Physics and Astronomy; Quantum Optics; RF and Optical Engineering; Special Section: Regular Paper; The 10th International Conference on Optics-Photonics Design & Fabrication (ODF'16); Weingarten; Bessel function; Sampling theorem; Fresnel transform; Approximation
• ISSN: 1340-6000; 1349-9432
• DOI: 10.1007/s10043-016-0258-y

The Fresnel transform is a bounded, linear, additive, and unitary operator in Hilbert space and is applied to many applications. In this study, a sampling theorem for a Fresnel transform pair in polar coordinate systems is derived. According to the sampling theorem, any function in the complex plane can be expressed by taking the products of the values of a function and sampling function systems. Sampling function systems are constituted by Bessel functions and their zeros. By computer simulations, we consider the application of the sampling theorem to the problem of approximating a function to demonstrate its validity. Our approximating function is a circularly symmetric function which is defined in the complex plane. Counting the number of sampling points requires the calculation of the zeros of Bessel functions, which are calculated by an approximation formula and numerical tables. Therefore, our sampling points are nonuniform. The number of sampling points, the normalized mean square error between the original function and its approximation function and phases are calculated and the relationship between them is revealed.