Fast computation algorithm for the Rayleigh-Sommerfeld diffraction formula using a type of scaled convolution

Abstract

We describe a fast computational algorithm able to evaluate the Rayleigh-Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform. What is new in our approach compared to other algorithms is the use of a more general type of convolution with a scale parameter, which allows for independent sampling intervals in the input and output computation windows. Comparison between the calculations made using our algorithm and direct numeric integration show a very good agreement, while the computation speed is increased by orders of magnitude.

Rendering volumetric haptic shapes in mid-air using ultrasound

“…which can be accelerated using the fast Fourier transform (FFT) algorithm [Nascov and Logofȃtu 2009]. With this, we can generate a look up table that describes how the amplitude and relative phase of the sound wave from one transducer changes spatially.…”

A novel method for the design of diffractive optical elements based on the Rayleigh–Sommerfeld integral

“…This diffraction integral can also be seen as a convolution of a harmonic field and a spherical phase function; thus, it is possible to solve this convolution with the FFT algorithm but will be limited to cases with identical sampling intervals in both the DOE and output plane. Though several fast algorithms have been proposed to address this problem [16][17][18], they all seem relatively complex and more time consuming for DOE design because several times of FFT and IFFT are used.…”

Numerical solution of nonparaxial scalar diffraction integrals for focused fields

“…(27) and (28a), are valid for arbitrary fields. However, we have not specified how to define or to determine the lateral extents and the sampling intervals of the input and output fields.…”

“…Instead of increasing the number of sampling points in the input plane, CZTs may be used to realize independent sampling intervals in the input and output plane [16,[25][26][27][28][29][30][31][32]. In this section, we limit our discussion to one specific realization, which can be combined both with the ASM and with the RSC, and replaces the last calculation step F −1 fŨ z g by a CZT.…”

“…• We extend our sampling conditions to chirp z-transforms (CZTs) [16,[25][26][27][28][29][30][31][32] that enable different sampling intervals at the input and output planes.…”