Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?

TL;DR

This paper investigates the accuracy of Rayleigh-Sommerfeld diffraction as a reference for high-speed diffraction algorithms, introducing a new sampling validity condition and a method to extend RSD applicability beyond its traditional limits.

Contribution

It presents a novel validity condition for sampling in RSD and a unified approach to apply RSD outside its standard validity domain, enhancing the accuracy of diffraction simulations.

Findings

Sampling restrictions are fundamental and can be eliminated below the Abbe limit.

Violating the sampling validity causes anomalies in RSD calculations.

A combined forward-reverse approach extends RSD applicability beyond traditional limits.

Abstract

In several areas of optics and photonics like wave propagation, digital holography, holographic microscopy, diffraction imaging, biomedical imaging and diffractive optics, the behavior of the electromagnetic waves has to be calculated with the scalar theory of diffraction by computational methods. Many of these high speed diffraction algorithms based on a fast Fourier transformation are in principle approximations of the Rayleigh-Sommerfeld Diffraction (RSD) theory. However, to investigate their numerical accuracy, they should be compared with and verified by RSD. All numerical simulations are in principle based on a sampling of the analogue continuous field. In this article we demonstrate a novel validity condition for the well-sampling in RSD, which makes a systematic treatment of sampling in RSD possible. We show the fundamental restrictions due to this condition and the anomalies caused by its violation. We also demonstrate that the restrictions are completely removed by a sampling below the Abbe resolution limit. Furthermore, we present a very general unified approach for applying the RSD outside its validity domain by the combination of a forward and reverse calculation.