Numerical sampling rules for paraxial regime pulse diffraction calculations

Abstract

Sampling rules for numerically calculating ultrashort pulse fields are discussed. Such pulses are not monochromatic but rather have a finite spectral distribution about some central (temporal) frequency. Accordingly, the diffraction pattern for many spectral components must be considered. From a numerical implementation viewpoint, one may ask how many of these spectral components are needed to accurately calculate the pulse field. Using an analytical expression for the Fresnel diffraction from a 1-D slit, we examine this question by varying the number of contributing spectral components. We show how undersampling the spectral profile produces erroneous numerical artifacts (aliasing) in the spatial-temporal domain. A guideline, based on graphical considerations, is proposed that determines appropriate sampling conditions. We show that there is a relationship between this sampling rule and a diffraction wave that emerges from the aperture edge; comparisons are drawn with boundary diffraction waves. Numerical results for 2-D square and circular apertures are presented and discussed, and a potentially time-saving calculation technique that relates pulse distributions in different z planes is described.

Diffraction of an optical pulse as an expansion in ultrashort orthogonal Gaussian beam modes

“…While the study of ultrashort pulse dynamics is a comparatively recent addition to the field of optics, massive experimental advancements have been made in the last decade or so, with electromagnetic fields now being generated with durations on attosecond timescales [1,2]. In applied optics, the numerical techniques used to predict the temporal behavior of ultrashort signals are naturally derived from the models of continuous wave (CW) propagation, and methods vary from computationally intensive full-wave solutions of Maxwell's equations such as the finite-difference time-domain (FDTD) algorithm [3][4][5][6] to, with more relevance to large optical systems, the diffraction integral approach [7][8][9][10][11][12][13]. To introduce in this paper the dynamics involved in pulse diffraction, a generic illustration of the near-field propagation of an ultrashort plane wave pulse truncated at a narrow aperture is shown in Fig.…”