Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem

Abstract

Diffraction calculations are widely used in applications that require numerical simulation of optical wave propagation. Different numerical diffraction calculation methods have their own transform and sampling properties. In this study, we provide a unified analysis where five popular fast diffraction calculation methods are analyzed from the perspective of phase space optics and the sampling theorem: single fast Fourier transform-based Fresnel transform, Fresnel transfer function approach, Fresnel impulse response approach, angular spectrum method, and Rayleigh–Sommerfeld convolution. The evolutions of an input signal’s space-bandwidth product (SBP) during wave propagation are illustrated with the help of a phase space diagram (PSD) and an ABCD matrix. It is demonstrated that all of the above methods cannot make full use of the SBP of the input signal after diffraction; and some transform properties have been ignored. Each method has its own restrictions and applicable range. The reason why different methods have different applicable ranges is explained with physical models. After comprehensively studying and comparing the effect on the SBP and sampling properties of these methods, suggestions are given for choosing the proper method for different applications and overcoming the restrictions of corresponding methods. The PSD and ABCD matrix are used to illustrate the properties of these methods intuitively. Numerical results are presented to verify the analysis, and potential ways to develop new diffraction calculation methods are also discussed.

Diffractive optical system design by cascaded propagation

“…One way to mitigate the sampling constraints is by doing two single-step propagations [42,43]. All those methods and their variations have different sampling requirements and often requiring drastic zero-padding to reduce aliasing [38]. When simulating free space propagation these methods can be viable; However, the Collins approach allows for fewer propagation calculations and choosing planes with minimal field supports to reduce the sampling constraints.…”

“…Efficient methods to evaluate eq. ( 1) have been extensively studied [38]. In this work we focus mainly on an angular-spectrum method termed the Collins integral [26].…”

Diffraction integral computation using sinc approximation

“…Thus, to compute either (1) or (2), the ASM requires that we compute u numerically, multiply it by (10) or (12), and then apply the inverse discrete Fourier transform to the result.…”

“…The most straightforward way to reduce the errors caused by artificial periodicity is to increase the size of the domain, but at the cost of additional computation. There have been attempts to repair FFT-based methods, such as the band-limited angular spectrum method [11]; see also the recent article [12] and the references therein. Unfortunately, all such modifications only mitigate the artefacts of artificial periodicity under certain circumstances; they cannot eliminate the underlying problem entirely.…”

Diffraction integral computation using sinc approximation

“…The most straightforward way to reduce the errors caused by artificial periodicity is to increase the size of the domain, but at the cost of additional computation. There have been attempts to repair FFT-based methods, such as the band-limited angular spectrum method [6]; see also the recent article [7] and the references therein. Unfortunately, all such modifications only mitigate the artefacts of artificial periodicity under certain circumstances; they cannot eliminate the underlying problem entirely.…”