Exact solution of the Bragg-diffraction problem in sillenites

A method for the exact solution of the Bragg-difrraction problem for a photorefractive grating in sillenite crystals based on Pauli matrices is proposed. For the two main optical configurations explicit analytical expressions are found for the diffraction efficiency and the polarization of the scattered wave. The exact solution is applied to a detailed analysis of a number of particular cases. For the known limiting cases there is agreement with the published results.

Exact solution of the Bragg-diffraction problem in sillenites

A method for the exact solution of the Bragg-difrraction problem for a photorefractive grating in sillenite crystals based on Pauli matrices is proposed. For the two main optical configurations explicit analytical expressions are found for the diffraction efficiency and the polarization of the scattered wave. The exact solution is applied to a detailed analysis of a number of particular cases. For the known limiting cases there is agreement with the published results.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Using interior point algorithms for the solution of linear programs with special structural features

Linear Programming (LP) is a powerful decision making tool extensively used in various economic and engineering activities. In the early stages the success of LP was mainly due to the efficiency of the simplex method. After the appearance of Karmarkar's paper, the focus of most research was shifted to the field of interior point methods. The present work is concerned with investigating and efficiently implementing the latest techniques in this field taking sparsity into account. The performance of these implementations on different classes of LP problems is reported here. The preconditional conjugate gradient method is one of the most powerful tools for the solution of the least square problem, present in every iteration of all interior point methods. The effect of using different preconditioners on a range of problems with various condition numbers is presented. Decomposition algorithms has been one of the main fields of research in linear programming over the last few years. After reviewing the latest decomposition techniques, three promising methods were chosen the implemented. Sparsity is again a consideration and suggestions have been included to allow improvements when solving problems with these methods. Finally, experimental results on randomly generated data are reported and compared with an interior point method. The efficient implementation of the decomposition methods considered in this study requires the solution of quadratic subproblems. A review of recent work on algorithms for convex quadratic was performed. The most promising algorithms are discussed and implemented taking sparsity into account. The related performance of these algorithms on randomly generated separable and non-separable problems is also reported.

Factors associated with degeneration of the thallus centre in foliose lichens

Degeneration of the older parts of foliose lichen thalli often lead to the formation of a space or 'window' in the centre of the colonies. The percentage of thalli of different size which exhibited 'windows' was studied in twenty saxicolous lichen populations in south Gwynedd, Wales. The proportion of thalli with 'windows' increased with thallus size. The size class at which 50% and 100% of thalli exhibited 'windows' varied between populations. Differences between populations were not correlated with distance from the sea, aspect, slope or porosity of the substrate or the total number of lichen species present. However, a higher percentage of smaller thalli had 'windows' on rock surfaces with a greater lichen cover. There were no significant differences in the levels of Ca, Mg, Cu or Zn in large (>4 cm) and small (<2 cm) Parmelia conspersa (Ehrh. ex Ach.) Ach. thalli or in the centres and marginal lobes of these thalli. The concentration of ribitol, arabitol and mannitol was significantly reduced in the centre of large thalli compared with the margin of large thalli and the centre of small thalli. However, carbohydrate levels were similar in the centre of large thalli and the margin of small thalli. The data suggest that loss of the thallus centre is a degenerative process related to thallus size. In the field, the formation of 'windows' may be related to the intensity of competition on a substrate. Central degeneration was not associated with a deficiency or an accumulation of Ca, Mg, Cu and Zn in the thallus centre. However, degeneration may be associated with a reduction in carbohydrates in the centre compared with the marginal lobes.

3D reconstruction of complex dielectric function of the glass during the femtosecond micro-fabrication

We measure complex amplitude of scattered wave in the far field, and justify theoretically and numerically solution of the inverse scattering problem. This allows single-shot reconstructing of dielectric function distribution during direct femtosecond laser micro-fabrication.

Theoretical studies of the mid-latitude ionosphere

Following a brief description of the atmosphere and ionosphere in Chapter I we describe how the equations of continuity and momentum for 0+, H+, He+, 0++ are derived from the formulations of St-Maurice and Schunk(1977) and Quegan et al.(1981) in Chapter II. In Chapter III we investigate the nature of the downward flow of protons in a collapsing post-sunset ionosphere. We derive an analytical form for the limiting temperature, we also note the importance of the polarization field term and concluded that the flow will remain subsonic for realistic conditions. The time-dependent behaviour of He+ under sunspot minimum conditions is investigated in Chapter IV. This is achieved by numerical solution of the 0+, H+ and,He+ continuity and momentum equations, treating He+ as a minor ion with 0+ , H+ as major ions. We found that He+ flows upwards during the day-time and downwards during the nighttime. He+ flux tube content reached a maximum on the 8th day of the integration period and started to decreasing. This is due to the large amount of H+ present at the late stages of the integration period which makes He+ unable to diffuse through the H+ layer away from the loss region. In Chapter V we investigate the behaviour of 0++ using sunspot maximum parameters. Although our results support the findings of Geis and Young (1981) that the large amounts of 0++ at the equator are caused mainly by thermal diffusion, the model used by Geis and Young overemphesizes the effect of thermal diffusion. The importance of 0++ - 0+ collision frequency is also noted. In Chapter VI we extend the work of Chapter IV, presenting a comparative study of H and He at sunspot minimum and sunspot maximum.In this last Chapter all three ions, O+ ,H+ and He+ , are treated theoretically as major ions and we concentrate mainly on light ion contents and fluxes. The results of this Chapter indicate that by assuming He+ as a minor ion we under-estimate He+ and over-estimate. H+. Some interesting features concerning the day to day behaviour of the light ion fluxes arise. In particular the day-time H+ fluxes decrease from day to day in contrast to the work of Murphy et al.(1976). In appendix.A we derive some analytical forms for the optical depth so that the models can include a realistic description of photoionization.

Parabolic optical pulses under the action of the third-order dispersion

Recent developments in nonlinear optics reveal an interesting class of pulses with a parabolic intensity profile in the energy-containing core and a linear frequency chirp that can propagate in a fiber with normal group-velocity dispersion. Parabolic pulses propagate in a stable selfsimilar manner, holding certain relations (scaling) between pulse power, width, and chirp parameter. In the additional presence of linear amplification, they enjoy the remarkable property of representing a common asymptotic state (or attractor) for arbitrary initial conditions. Analytically, self-similar (SS) parabolic pulses can be found as asymptotic, approximate solutions of the nonlinear Schr¨odinger equation (NLSE) with gain in the semi-classical (largeamplitude/small-dispersion) limit. By analogy with the well-known stable dynamics of solitary waves - solitons, these SS parabolic pulses have come to be known as similaritons. In practical fiber systems, inherent third-order dispersion (TOD) in the fiber always introduces a certain degree of asymmetry in the structure of the propagating pulse, eventually leading to pulse break-up. To date, there is no analytic theory of parabolic pulses under the action of TOD. Here, we develop aWKB perturbation analysis that describes the effect of weak TOD on the parabolic pulse solution of the NLSE in a fiber gain medium. The induced perturbation in phase and amplitude can be found to any order. The theoretical model predicts with sufficient accuracy the pulse structural changes induced by TOD, which are observed through direct numerical NLSE simulations.

Parabolic optical pulses under the action of the third-order dispersion

Recent developments in nonlinear optics reveal an interesting class of pulses with a parabolic intensity profile in the energy-containing core and a linear frequency chirp that can propagate in a fiber with normal group-velocity dispersion. Parabolic pulses propagate in a stable selfsimilar manner, holding certain relations (scaling) between pulse power, width, and chirp parameter. In the additional presence of linear amplification, they enjoy the remarkable property of representing a common asymptotic state (or attractor) for arbitrary initial conditions. Analytically, self-similar (SS) parabolic pulses can be found as asymptotic, approximate solutions of the nonlinear Schr¨odinger equation (NLSE) with gain in the semi-classical (largeamplitude/small-dispersion) limit. By analogy with the well-known stable dynamics of solitary waves - solitons, these SS parabolic pulses have come to be known as similaritons. In practical fiber systems, inherent third-order dispersion (TOD) in the fiber always introduces a certain degree of asymmetry in the structure of the propagating pulse, eventually leading to pulse break-up. To date, there is no analytic theory of parabolic pulses under the action of TOD. Here, we develop aWKB perturbation analysis that describes the effect of weak TOD on the parabolic pulse solution of the NLSE in a fiber gain medium. The induced perturbation in phase and amplitude can be found to any order. The theoretical model predicts with sufficient accuracy the pulse structural changes induced by TOD, which are observed through direct numerical NLSE simulations.

3D reconstruction of complex dielectric function of the glass during the femtosecond micro-fabrication

We measure complex amplitude of scattered wave in the far field, and justify theoretically and numerically solution of the inverse scattering problem. This allows single-shot reconstructing of dielectric function distribution during direct femtosecond laser micro-fabrication.