A Lower Bound for the Differences of Powers of Linear Operators

Let T be a bounded linear operator in a Banach space, with σ（T）={1}. In 1983, Esterle-Berkani＇ s conjecture was proposed for the decay of differences （I - T） T^n as follows： Eitheror lim inf （n→∞（n＋1）｜｜（I-T）T^n｜｜≥1/e or T = I. We prove this claim and discuss some of its consequences.     

Elementary-field modeling of surface-plasmon excitation with partially coherent light

We present a mathematical model and its numerical implementation for the analysis of the interaction of spatially partially coherent electromagnetic fields with micro- and nanostructured objects. The model is based on the decomposition of the incident field into a set of fully coherent but mutually uncorrelated elementary field modes, and the use of the Fourier Modal Method (FMM) with the S-matrix propagation algorithm. We apply the model to studies of the excitation of surface plasmons in thin metallic slabs, nanowires, and resonant structures. We demonstrate, e.g., that the plasmon excitation efficiency is not essentially affected by the degree of spatial coherence. However, certain plasmon interference effects can be efficiently smoothed out by using illumination with reduced coherence.     

Transverse laser-mode structure determination from spatial coherence measurements: Experimental results

We demonstrate experimentally the feasibility of a certain coherence-theoretic algorithm for determining the relative modal weights of a laser beam that consists of multiple Hermite-Gaussian transverse modes. A computer-controlled Young's two-pinhole interferometer with a linear CCD array is used for the coherence measurements. The required scaling parameter (beam width) is obtained iteratively by library routines. Numerical simulations on the stability of the algorithm in the presence of experimental noise are also presented.     

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

In this paper we give several global characterisations of the Hörmander class \(\Psi ^m(G)\) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.     

Propagation-invariant spot arrays

Optical fields with scale- and shape-invariant transverse intensity profiles consisting of a prescribed distribution of bright spots are introduced. Approximations of such fields are demonstrated by means of binary annular-aperture diffractive elements.     

Novel use of mesoporous aluminas as supports for Cp2ZrCl2 and Cp*ZrMe3: Ethylene polymerization and formation of polyethylene nanofibers

Mesoporous aluminas with average pore sizes of 4.3–7.8 nm were prepared by anodization of an aluminum film (AAO), and by a sol–gel templating method (TPL). In addition to a commercial alumina and sulfated TPL, the aluminas were used as supports for cyclopentadienyl zirconocene dichloride (Cp₂ZrCl₂) and trimethyl(η5‐pentamethylcyclopentadienyl)zirconium (Cp*ZrMe₃) and tested in the polymerization of ethylene. The metallocenes supported on the alumina prepared with the templating method and its sulfated modification exhibited polymerization activities of 440 and 350 kg_PE/(mol_Zr × h × bar), respectively, comparable to that obtained with silica‐supported metallocenes (390 kg_PE/(mol_Zr × h × bar)). The acid site distribution of the aluminas was studied with FTIR and temperature programmed desorption (TPD) of pyridine, and also the amount of medium and strong acid sites was determined gravimetrically from the adsorption of pyridine. Relative to the surface area, AAO with the highest amount of acid sites (2.10 μmol_py/m) adsorbed Cp₂ZrCl₂ and Cp*ZrMe₃ the most. Study of the polymers' morphology with a scanning electron microscope revealed polyethylene nanofibers in all the polymer samples, also in those obtained from the reference polymerizations with homogeneous Cp₂ZrCl₂ and Cp*ZrMe₃. This finding suggests that a catalyst support with a tubular pore structure is not a prerequisite for the formation of polymer nanofibers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4002–4012, 2007     

Probing surface plasmons by bare V-shaped tips: modeling by geometrical optics and rigorous diffraction theory

Optical Review - We consider probing inhomogeneous waves in the near fields of metallic nanostructures with the aid of a dielectric V-shaped wedge connected to a waveguide. A geometrical model...     

Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser

Recently we observed a strong reduction of spatial coherence of the emission of large-aperture vertical-cavity surface-emitting lasers when they are driven by microsecond electrical pulses [Opt. Express 13, 9337 (2005)]. We study the influence of this partial spatial coherence on the propagation characteristics. The spatial decoherence manifests itself in the formation of a Gaussian far-field intensity distribution. We measure the transverse pulse profile from near to far field and see that the shape-invariant far-field regime starts after 140 microm in the pulsed regime as opposed to several centimeters in continuous wave operation. This value matches quantitatively calculations made with a novel partially coherent propagation model without any free fitting parameters.