Numerical calculation of the Fresnel transform

Abstract

In this paper, we address the problem of calculating Fresnel diffraction integrals using a finite number of uniformly spaced samples. General and simple sampling rules of thumb are derived that allow the user to calculate the distribution for any propagation distance. It is shown how these rules can be extended to fast-Fourier-transform-based algorithms to increase calculation efficiency. A comparison with other theoretical approaches is made.

Simulation of Coherent X-Ray Imaging of Tilted Objects in the Fourier Space

“…The Fourier space is often used in optical computing [12,18,19]. In this approach, the free-space propagation from the shadow plane of the field Fourier transform u(p, z) is given by a simple formula [14], which is equivalent to (3):…”

“…(3) For objects located normal to the wave vector k, the PWE is equivalent to the Fresnel integral (FI) approximation, which is widely used in imaging and optical sciences, has been confirmed in numerous experiments, and is well elaborated for simulations and computing [12].…”

Numerical solution of nonparaxial scalar diffraction integrals for focused fields

“…In general, these effects cannot be completely avoided and, hence, their influence on the calculation results has to be kept reasonably small. Various authors have derived sampling criteria for the ASM and the RSC, highlighting suitable combinations of sampling intervals, field sizes, and propagation distances [14][15][16][17][18][19][20][21][22][23][24]. The Nyquist-Shannon sampling theorem is commonly used as a starting point for sampling considerations (e.g., [14][15][16]).…”

“…To extend the range of propagation distances allowed by the Nyquist-Shannon sampling theorem, modified versions of the ASM and the RSC were introduced [16][17][18]. Alternative sampling criteria are based on the Wigner distribution and the space-bandwidth product [19][20][21][22][23]. Several authors noted that, for focal field calculations, the phase distribution of the input field should be taken into account, and will result in different regions within which the ASM and the RSC provide accurate results (e.g., [16,24]).…”

Measuring optical phase digitally in coherent metrology systems

“…17 We won't need a reference wave however and the optical setup is much simpler. In many situations these are significant advantages.…”