Refraction of nonlinear light beams in nematic liquid crystals

Optical spatial solitons in nematic liquid crystals, termed nematicons, have become an excellent test bed for nonlinear optics, ranging from fundamental effects to potential uses, such as designing and demonstrating all-optical switching and routing circuits in reconfigurable settings and guided-wave formats. Following their demonstration in planar voltage-assisted nematic liquid crystal cells, the spatial routing of nematicons and associated waveguides have been successfully pursued by exploiting birefringent walkoff, interactions between solitons, electro-optic controlling, lensing effects, boundary effects, solitons in twisted arrangements, refraction and total internal reflection and dark solitons. Refraction and total internal reflection, relying on an interface between two dielectric regions in nematic liquid crystals, provides the most striking results in terms of angular steering. In this thesis, the refraction and total internal reflection of self-trapped optical beams in nematic liquid crystals in the case of a planar cell with two separate regions defined by independently applied bias voltages have been investigated with the aim of achieving a broader understanding of the nematicons and their control. The study of the refraction of nematicons is then extended to the equivalent refraction of optical vortices. The equations governing nonlinear optical beam propagation in nematic liquid crystals are a system consisting of a nonlinear Schr¨odinger-type equation for the optical beam and an elliptic Poisson equation for the medium response. This system of equations has no exact solitary wave solution or any other exact solutions. Although numerical solutions of the governing equations can be found, it has been found that modulation theories give insight into the mechanisms behind nonlinear optical beam evolution, while giving approximate solutions in good to excellent agreement with full numerical solutions and experimental results. The modulation theory reduces the infinite-dimensional partial differential equation problem to a finite dynamical system of comparatively simple ordinary differential equations which are, then easily solved numerically. The modulation theory results on the refraction and total internal reflection of nematicons are in excellent agreement with experimental data and numerical simulations, even when accounting for the birefringent walkoff. The modulation theory also gives excellent results for the refraction of optical vortices of +1 topological charge. The modulation theory predicts that the vortices can become unstable on interaction with the nematic interface, which is verified in quantitative detail by full numerical solutions. This prediction of their azimuthal instability and their break-up into bright beams still awaits an experimental demonstration, but the previously obtained agreement of modulation theory models with the behaviour of actual nematicons leads us to expect the forthcoming observation of the predicted effects with vortices as well

Park City Lectures on Mechanics, Dynamics, and Symmetry

In these ve lectures, I cover selected items from the following topics: 1. Reduction theory for mechanical systems with symmetry, 2. Stability, bifurcation and underwater vehicle dynamics, 3. Systems with rolling constraints and locomotion, 4. Optimal control and stabilization of balance systems, 5. Variational integrators. Each topic itself could be expanded into several lectures, but I limited myself to what I could reasonably explain in the allotted time. The hope is that the overview is informative enough so that the reader can understand the fundamental ideas and can intelligently choose from the literature for additional details on topics of interest. Compatible with the theme of the PCI graduate school, I assume that the readers are familiar with the elements of geometric mechanics, including the basics of symplectic and Poisson geometry. The reader can find the needed background in, for example, Marsden and Ratiu [1998]

Nonlinear Evolution Equations: Analysis and Numerics

The workshop was devoted to the analytical and numerical investigation of nonlinear evolution equations. The main aim was to stimulate a closer interaction between experts in analytical and numerical methods for areas such as wave and Schrödinger equations or the Navier–Stokes equations and fluid dynamics

Geometric Numerical Integration (hybrid meeting)

The topics of the workshop included interactions between geometric numerical integration and numerical partial differential equations; geometric aspects of stochastic differential equations; interaction with optimisation and machine learning; new applications of geometric integration in physics; problems of discrete geometry, integrability, and algebraic aspects

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time

Interaction and steering of nematicons

The waveguiding effect of spatial solitary waves in nonlinear optical media has been suggested as a potential basis for future all-optical devices, such as optical interconnects. It has been shown that low power (∼ mW) beams, which can encode information, can be optically steered using external electric fields or through interactions with other beams. This opens up the possibility of creating reconfigurable optical interconnects. Nematic liquid crystals are a potential medium for such future optical interconnects, possessing many advantageous properties, including a “huge” nonlinear response at comparatively low input power levels. Consequently, a thorough understanding of the behaviour of spatial optical solitary waves in nematic liquid crystals, termed nematicons, is needed. The investigation of multiple beam interaction behaviour will form an essential part of this understanding due to the possibility of beam-on-beam control. Here, the interactions of two nematicons of different wavelengths in nematic liquid crystals, and the optical steering of nematicons in dye-doped nematic liquid crystals will be investigated with the aim of achieving a broader understanding of nematicon interaction and steering. The governing equations modelling nematicon interactions are nonintegrable, which means that nematicon collisions are inelastic and radiative losses occur during and after collision. Consequently numerical techniques have been employed to solve these equations. However, to fully understand the physical dynamics of nematicon interactions in a simple manner, an approximate variational method is used here which reduces the infinite-dimensional partial differential equation problem to a finite dynamical system of comparatively simple ordinary differential equations. The resulting ordinary differential equations are modified to include radiative losses due to beam evolution and interaction, and are then quickly solved numerically, in contrast to the original governing partial differential equations. N¨other’s Theorem is applied to find various conservation laws which determine the final steady states, aid in calculating shed radiation and accurately compute the trajectories of nematicons. Solutions of the approximate equations are compared with numerical solutions of the original governing equations to determine the accuracy of the approximation. Excellent agreement is found between full numerical solutions and approximate solutions for each physical situation modelled. Furthermore, the results obtained not only confirm, but explain theoretically, the interaction phenomena observed experimentally. Finally, the relationship between the nature of the nonlinear response of the medium, the trajectories of the beams and radiation shed as the beams evolve is investigated

Complex extreme nonlinear waves: classical and quantum theory for new computing models

The historical role of nonlinear waves in developing the science of complexity, and also their physical feature of being a widespread paradigm in optics, establishes a bridge between two diverse and fundamental fields that can open an immeasurable number of new routes. In what follows, we present our most important results on nonlinear waves in classical and quantum nonlinear optics. About classical phenomenology, we lay the groundwork for establishing one uniform theory of dispersive shock waves, and for controlling complex nonlinear regimes through simple integer topological invariants. The second quantized field theory of optical propagation in nonlinear dispersive media allows us to perform numerical simulations of quantum solitons and the quantum nonlinear box problem. The complexity of light propagation in nonlinear media is here examined from all the main points of view: extreme phenomena, recurrence, control, modulation instability, and so forth. Such an analysis has a major, significant goal: answering the question can nonlinear waves do computation? For this purpose, our study towards the realization of an all-optical computer, able to do computation by implementing machine learning algorithms, is illustrated. The first all-optical realization of the Ising machine and the theoretical foundations of the random optical machine are here reported. We believe that this treatise is a fundamental study for the application of nonlinear waves to new computational techniques, disclosing new procedures to the control of extreme waves, and to the design of new quantum sources and non-classical state generators for future quantum technologies, also giving incredible insights about all-optical reservoir computing. Can nonlinear waves do computation? Our random optical machine draws the route for a positive answer to this question, substituting the randomness either with the uncertainty of quantum noise effects on light propagation or with the arbitrariness of classical, extremely nonlinear regimes, as similarly done by random projection methods and extreme learning machines

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

The geometry of differential constraints for a class of evolution PDEs

The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in order to obtain explicit differential constraints for PDEs belonging to this family. Several examples, with applications in non-linear stochastic filtering theory, stochastic perturbation of soliton equations and non-isospectral integrable systems, are discussed in detail to verify the effectiveness of the method