Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwell's equation numerical solutions.

Internal structure of the nanogratings generated inside bulk fused sílica by ultrafast laser direct writing

The aim of the present work was to characterize the internal structure of nanogratings generated inside bulk fused silica by ultrafast laser processing and to study the influence of diluted hydrofluoric acid etching on their structure. The nanogratings were inscribed at a depth of 100 mu m within fused silica wafers by a direct writing method, using 1030 nm radiation wavelength and the following processing parameters: E = 5 mu J, tau = 560 fs, f = 10 kHz, and v = 100 mu m/s. The results achieved show that the laser-affected regions are elongated ellipsoids with a typical major diameter of about 30 mu m and a minor diameter of about 6 mu m. The nanogratings within these regions are composed of alternating nanoplanes of damaged and undamaged material, with an average periodicity of 351 +/- 21 nm. The damaged nanoplanes contain nanopores randomly dispersed in a material containing a large density of defects. These nanopores present a roughly bimodal size distribution with average dimensions for each class of pores 65 +/- 20 x 16 +/- 8 x 69 +/- 16 nm(3) and 367 +/- 239 x 16 +/- 8 x 360 +/- 194 nm(3), respectively. The number and size of the nanopores increases drastically when an hydrofluoric acid treatment is performed, leading to the coalescence of these voids into large planar discontinuities parallel to the nanoplanes. The preferential etching of the damaged material by the hydrofluoric acid solution, which is responsible for the pores growth and coalescence, confirms its high defect density. (C) 2014 AIP Publishing LLC.

Morphology and structure of particles produced by femtosecond laser ablation of fused silica

The aim of the present work was to study the morphology and structure of the nanoparticles produced by femtosecond laser ablation of fused silica. Ultrashort laser pulses of 1030 nm wavelength and 550 fs duration were tightly focused by a high numerical aperture microscope objective at the surface of fused silica samples while scanning the sample in relation to the stationary laser beam. Laser tracks were created with pulse energies in the range 5-100 mu J, resulting in ablation debris of different morphologies. The debris were examined by scanning and transmission electron microscopy for their morphology and crystal structure in relation to the incident laser pulse energy. Ejected particles with sizes ranging from a few nanometers to a few microns were found. Their morphologies can be broadly classified into three categories: very fine round nanoparticles with diameters lower than 20 nm, nanoparticles with intermediate sizes between 50 and 200 nm, and big irregular particles with typical size between 0.5 and 1.5 mu m. The fine nanoparticles of the first category are predominantly observed at higher pulse energies and tend to aggregate to form web-like and arborescent-like structures. The nanoparticles with intermediate sizes are observed for all pulse energies used and may appear isolated or aggregated in clusters. Finally, the larger irregular particles of the third category are observed for all energies and appear normally isolated.