Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

Conceptualizing leadership perceptions as attitudes:using attitude theory to further the understanding of the relation between leadership and outcomes

Leadership is one of the most examined factors in relation to understanding employee wellbeing and performance. While there are disparate approaches to studying leadership, they share a common assumption that perceptions of a leader's behavior determine reactions to the leader. The concept of leadership perception is poorly understood in most theoretical approaches. To address this, we propose that there are many benefits from examining leadership perceptions as an attitude towards the leader. In this review, we show how research examining a number of aspects of attitudes (content, structure and function) can advance understanding of leadership perceptions and how these affect work-related outcomes. Such a perspective provides a more multi-faceted understanding of leadership perceptions than previously envisaged and this can provide a more detailed understanding of how such perceptions affect outcomes. In addition, we examine some of the main theoretical and methodological implications of viewing leadership perceptions as attitudes to the wider leadership area. The cross-fertilization of research from the attitudes literature to understanding leadership perceptions provides new insights into leadership processes and potential avenues for further research. (C) 2015 Elsevier Inc. All rights reserved

Implementing an agent-based model with a spatial visual display in discrete-event simulation software

There has been an increasing interest in the use of agent-based simulation and some discussion of the relative merits of this approach as compared to discrete-event simulation. There are differing views on whether an agent-based simulation offers capabilities that discrete-event cannot provide or whether all agent-based applications can at least in theory be undertaken using a discrete-event approach. This paper presents a simple agent-based NetLogo model and corresponding discrete-event versions implemented in the widely used ARENA software. The two versions of the discrete-event model presented use a traditional process flow approach normally adopted in discrete-event simulation software and also an agent-based approach to the model build. In addition a real-time spatial visual display facility is provided using a spreadsheet platform controlled by VBA code embedded within the ARENA model. Initial findings from this investigation are that discrete-event simulation can indeed be used to implement agent-based models and with suitable integration elements such as VBA provide the spatial displays associated with agent-based software.

Information theory analysis of regenerative channels

In this paper we summarize our recently proposed work on the information theory analysis of regenerative channels. We discuss how the design and the transfer function properties of the regenerator affect the noise statistics and enable Shannon capacities higher than that of the corresponding linear channels (in the absence of regeneration).

Engineering education research - the UK perspective at a time of change

Engineering education in the United Kingdom is at the point of embarking upon an interesting journey into uncharted waters. At no point in the past have there been so many drivers for change and so many opportunities for the development of engineering pedagogy. This paper will look at how Engineering Education Research (EER) has developed within the UK and what differentiates it from the many small scale practitioner interventions, perhaps without a clear research question or with little evaluation, which are presented at numerous staff development sessions, workshops and conferences. From this position some examples of current projects will be described, outcomes of funding opportunities will be summarised and the benefits of collaboration with other disciplines illustrated. In this study, I will account for how the design of task structure according to variation theory, as well as the probe-ware technology, make the laws of force and motion visible and learnable and, especially, in the lab studied make Newton's third law visible and learnable. I will also, as a comparison, include data from a mechanics lab that use the same probe-ware technology and deal with the same topics in mechanics, but uses a differently designed task structure. I will argue that the lower achievements on the FMCE-test in this latter case can be attributed to these differences in task structure in the lab instructions. According to my analysis, the necessary pattern of variation is not included in the design. I will also present a microanalysis of 15 hours collected from engineering students' activities in a lab about impulse and collisions based on video recordings of student's activities in a lab about impulse and collisions. The important object of learning in this lab is the development of an understanding of Newton's third law. The approach analysing students interaction using video data is inspired by ethnomethodology and conversation analysis, i.e. I will focus on students practical, contingent and embodied inquiry in the setting of the lab. I argue that my result corroborates variation theory and show this theory can be used as a 'tool' for designing labs as well as for analysing labs and lab instructions. Thus my results have implications outside the domain of this study and have implications for understanding critical features for student learning in labs. Engineering higher education is well used to change. As technology develops the abilities expected by employers of graduates expand, yet our understanding of how to make informed decisions about learning and teaching strategies does not without a conscious effort to do so. With the numerous demands of academic life, we often fail to acknowledge our incomplete understanding of how our students learn within our discipline. The journey facing engineering education in the UK is being driven by two classes of driver. Firstly there are those which we have been working to expand our understanding of, such as retention and employability, and secondly the new challenges such as substantial changes to funding systems allied with an increase in student expectations. Only through continued research can priorities be identified, addressed and a coherent and strong voice for informed change be heard within the wider engineering education community. This new position makes it even more important that through EER we acquire the knowledge and understanding needed to make informed decisions regarding approaches to teaching, curriculum design and measures to promote effective student learning. This then raises the question 'how does EER function within a diverse academic community?' Within an existing community of academics interested in taking meaningful steps towards understanding the ongoing challenges of engineering education a Special Interest Group (SIG) has formed in the UK. The formation of this group has itself been part of the rapidly changing environment through its facilitation by the Higher Education Academy's Engineering Subject Centre, an entity which through the Academy's current restructuring will no longer exist as a discrete Centre dedicated to supporting engineering academics. The aims of this group, the activities it is currently undertaking and how it expects to network and collaborate with the global EER community will be reported in this paper. This will include explanation of how the group has identified barriers to the progress of EER and how it is seeking, through a series of activities, to facilitate recognition and growth of EER both within the UK and with our valued international colleagues.

One-dimensional optical wave turbulence:experiment and theory

We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT theory predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long- and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).