Random projections and the optimization of an algorithm for phase retrieval

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.Comment: 15 page

Dense packing crystal structures of physical tetrahedra

We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of the parameter, providing an atlas of crystal structures which might be observed in systems of nano-particles with tetrahedral symmetry

Stability of the hard-sphere icosahedral quasilattice

The stability of the hard-sphere icosahedral quasilattice is analyzed using the differential formulation of the generalized effective liquid approximation. We find that the icosahedral quasilattice is metastable with respect to the hard-sphere crystal structures. Our results agree with recent findings by McCarley and Ashcroft [Phys. Rev. B {\bf 49}, 15600 (1994)] carried out using the modified weighted density approximation.Comment: 15 pages, 2 figures available from authors upon request, (revtex), submitted to Phys. Rev.

A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

Global convergence of a non-convex Douglas-Rachford iteration

We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration [2] was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given

Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields

Quasicrystals provide a fascinating class of materials with intriguing properties. Despite a strong potential for numerous technical applications, the conditions under which quasicrystals form are still poorly understood. Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry but no single example with 7 or 9-fold symmetry has ever been observed. Here we report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. Experimentally, colloidal quasicrystals are created by subjecting micron-sized particles to two-dimensional quasiperiodic potential landscapes created by n=5 or seven laser beams. Our results clearly demonstrate that quasicrystalline order is much easier established for n = 5 compared to n = 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with n and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of materials with e.g. 7-fold symmetry

Naturally-phasematched second harmonic generation in a whispering gallery mode resonator

We demonstrate for the first time natural phase matching for optical frequency doubling in a high-Q whispering gallery mode resonator made of Lithium Niobate. A conversion efficiency of 9% is achieved at 30 micro Watt in-coupled continuous wave pump power. The observed saturation pump power of 3.2 mW is almost two orders of magnitude lower than the state-of-the-art. This suggests an application of our frequency doubler as a source of non-classical light requiring only a low-power pump, which easily can be quantum noise limited. Our theoretical analysis of the three-wave mixing in a whispering gallery mode resonator provides the relative conversion efficiencies for frequency doubling in various modes

Chemical composition and radiative properties of nascent particulate matter emitted by an aircraft turbofan burning conventional and alternative fuels

Aircraft engines are a unique source of carbonaceous aerosols in the upper troposphere. There, these particles can more efficiently interact with solar radiation than at ground. Due to the lack of measurement data, the radiative forcing from aircraft exhaust aerosol remains uncertain. To better estimate the global radiative effects of aircraft exhaust aerosol, its optical properties need to be comprehensively characterized. In this work we present the link between the chemical composition and the optical properties of the particulate matter (PM) measured at the engine exit plane of a CFM56-7B turbofan. The measurements covered a wide range of power settings (thrust), ranging from ground idle to take-off, using four different fuel blends of conventional Jet A-1 and hydro-processed ester and fatty acids (HEFA) biofuel. At the two measurement wavelengths (532 and 870&thinsp;nm) and for all tested fuels, the absorption and scattering coefficients increased with thrust, as did the PM mass. The analysis of elemental carbon (EC) and organic carbon (OC) revealed a significant mass fraction of OC (up to 90&thinsp;%) at low thrust levels, while EC mass dominated at medium and high thrust. The use of HEFA blends induced a significant decrease in the PM mass and the optical coefficients at all thrust levels. The HEFA effect was highest at low thrust levels, where the EC mass was reduced by up to 50&thinsp;%–60&thinsp;%. The variability in the chemical composition of the particles was the main reason for the strong thrust dependency of the single scattering albedo (SSA), which followed the same trend as the fraction of OC to total carbon (TC). Mass absorption coefficients (MACs) were determined from the correlations between aerosol light absorption and EC mass concentration. The obtained MAC values (MAC532=7.5±0.3&thinsp;m2&thinsp;g−1 and MAC870=5.2±0.9&thinsp;m2&thinsp;g−1) are in excellent agreement with previous literature values of absorption cross section for freshly generated soot. While the MAC values were found to be independent of the thrust level and fuel type, the mass scattering coefficients (MSCs) significantly varied with thrust. For cruise conditions we obtained MSC532=4.5±0.4&thinsp;m2&thinsp;g−1 and MSC870=0.54±0.04&thinsp;m2&thinsp;g−1, which fall within the higher end of MSCs measured for fresh biomass smoke. However, the latter comparison is limited by the strong dependency of MSC on the particles' size, morphology and chemical composition. The use of the HEFA fuel blends significantly decreased PM emissions, but no changes were observed in terms of EC∕OC composition and radiative properties.</p