Third-order nonlinearity of Er(3+)-doped lead phosphate glass

The third-order optical susceptibility and dispersion of the linear refractive index of Er(3+)-doped lead phosphate glass were measured in the wavelength range between 400 and 1940 nm by using the spectrally resolved femtosecond Maker fringes technique. The nonlinear refractive index obtained from the third-order susceptibility was found to be five times higher than that of silica, indicating that Er(3+)-doped lead phosphate glass is a potential candidate to be used as the base component for the fabrication of photonic devices. For comparison purposes, the Z-scan technique was also employed to obtain the values of the nonlinear refractive index of Er(3+)-doped lead phosphate glass at several wavelengths, and the values obtained using the two techniques agree to within 15%.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic gamma, we prove the existence of a locally constant integer valued map Lambda(gamma) on the unit circle with the property that the Morse index of the iterated gamma(N) is equal, up to a correction term epsilon(gamma) is an element of {0,1}, to the sum of the values of Lambda(gamma) at the N-th roots of unity. The discontinuities of Lambda(gamma) occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincare map of gamma. We discuss some applications of the theory.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

Microwave-Assisted Derivatization of Cellulose in an Ionic Liquid: An Efficient, Expedient Synthesis of Simple and Mixed Carboxylic Esters

Microwave (MW)-assisted cellulose dissolution in ionic liquids (ILs) has routinely led either to incomplete biopolymer solubilization, or its degradation. We show that these problems can be avoided by use of low-energy MW heating, coupled with efficient stirring. Dissolution of microcrystalline cellulose in the IL 1-allyl-3-methylimidazolium chloride has been achieved without changing its degree of polymerization; regenerated cellulose showed pronounced changes in its index of crystallinity, surface area, and morphology. MW-assisted functionalization of MCC by ethanoic, propanoic, butanoic, pentanoic, and hexanoic anhydrides has been studied. Compared with conventional heating, MW irradiation has resulted in considerable decrease in dissolution and reaction times. The value of the degree of substitution (DS) was found to be DS(ethanoate) > DS(propanoate) > DS(butanoate). The values of DS(pentanoate) and DS(hexanoate) were found to be slightly higher than DS(ethanoate). This surprising dependence on the chain length of the acylating agent has been reported before, but not rationalized. On the basis of the rate constants and activation parameters of the hydrolysis of ethanoic, butanoic, and hexanoic anhydrides in aqueous acetonitrile (a model acyl transfer reaction), we suggest that this result may be attributed to the balance between two opposing effects, namely, steric crowding and (cooperative) hydrophobic interactions between the anhydride and the cellulosic surface, whose lipophilicity has increased, due to its partial acylation. Four ethanoate-based mixed esters were synthesized by the reaction with a mixture of the two anhydrides; the ethanoate moiety predominated in all products. The DS is reproducible and the IL is easily recycled. (C) 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 134-143, 2010

Application of 1-Allyl-3-(1-butyl)imidazolium Chloride in the Synthesis of Cellulose Esters: Properties of the Ionic Liquid, and Comparison with Other Solvents

The ionic liquid (IL), 1-allyl-3-(1-butyl)imidazolium chloride (AlBuImCl), has been synthesized and its properties determined. Increase in the temperature increased its conductivity and decreased its density, polarity, and viscosity. Microcrystalline cellulose (MCC), dissolves in thisIL by heating at 80 degrees C; this did not affect its degree of polymerization, decreased its index of crystallinity (Ic), and changed in morphology after regeneration. Convenient acylation of MCC was achieved by using 50% excess anhydride at 80 degrees C, for 24 or 48 h for acetic and butyric anhydride, respectively. The composition of the mixed esters depended on the initial ratio of the anhydrides, and their order of addition.