Polarization of tightly focused laser beams

The polarization properties of monochromatic light beams are studied. In contrast to the idealization of an electromagnetic plane wave, finite beams which are everywhere linearly polarized in the same direction do not exist. Neither do beams which are everywhere circularly polarized in a fixed plane. It is also shown that transversely finite beams cannot be purely transverse in both their electric and magnetic vectors, and that their electromagnetic energy travels at less than c. The electric and magnetic fields in an electromagnetic beam have different polarization properties in general, but there exists a class of steady beams in which the electric and magnetic polarizations are the same (and in which energy density and energy flux are independent of time). Examples are given of exactly and approximately linearly polarized beams, and of approximately circularly polarized beams.Comment: 9 pages, 6 figure

The Rescue of the US Auto Industry, Module Z:Overview

In the fall of 2008, credit markets tightened amid a broader economic downturn that severely impacted the US auto industry, especially the three largest domestic manufacturers, General Motors (GM), Ford Motors, and Chrysler. The companies requested assistance from the government in a bid to stay afloat, but Congress declined to authorize funding. The Bush administration, however, provided bridge loans to GM and Chrysler under the Auto Industry Finance Program (AIFP), funded through the Troubled Assets Relief Program (TARP), to sustain them until the Obama administration was in place. Within months, the Obama administration decided that a speedy bankruptcy would be the appropriate way to restructure the companies into viable organizations. Treasury provided financing via TARP as Chrysler, then GM, went through an expedited bankruptcy process. In support of the restructurings, Treasury also provided assistance to GMAC, Chrysler Financial, and to certain auto suppliers, in addition to supporting consumer warranties, bringing total government support for the industry to approximately $80 billion. Although the government lost money on the AIFP, the rescue was relatively popular politically and both companies returned to profitability in 2011. This case discusses the various aspects of the government’s assistance to the auto industry that began in December 2008 and wound down in December 2013

2D and 3D cubic monocrystalline and polycrystalline materials: their stability and mechanical properties

We consider 2- and 3-dimensional cubic monocrystalline and polycrystalline materials. Expressions for Young's and shear moduli and Poisson's ratio are expressed in terms of eigenvalues of the stiffness tensor. Such a form is well suited for studying properties of these mechanical characteristics on sides of the stability triangles. For crystalline high-symmetry directions lines of vanishing Poisson's ratio are found. These lines demarcate regions of the stability triangle into areas of various auxeticity properties. The simplest model of polycrystalline 2D and 3D cubic materials is considered. In polycrystalline phases the region of complete auxetics is larger than for monocrystalline materials.Comment: 9 pages, 3 figures, in proceedings of the Tenth International School on Theoretical Physics, Symmetry and Structural Properties of Condensed Matter, Myczkowce 200

Low compressible noble metal carbides with rock-salt structure: ab initio total energy calculations of the elastic stability

We have systematically studied the mechanical stability of all noble metal carbides with the rock-salt structure by calculating their elastic constants within the density function theory scheme. It was found that only four carbides (RuC, PdC, AgC and PtC) are mechanically stable. In particular, we have shown that RuC, PdC, and PtC have very high bulk modulus, which has been remarkably observed by the most recent experiment for the case of PtC. From the calculated density of states, we can conclude that these compounds are metallic, like the conventional group IV and group V transition metal carbides.Comment: Appl. Phys. Lett. 89, 071913 (2006

‘Oh! - is that how you do it!’ Learning from cross-phase collaborative work in the Years 5-8 Continuity project

We report on work of the Years 5-8 Continuity project, which runs across England in Maths Hubs, led by the National Centre for Excellence in the Teaching of Mathematics (NCETM). The aim of this project is to strengthen primary to secondary transition by focusing on curriculum and pedagogical continuity in mathematics over Years 5 to 8. Promotion of cross-phase teacher collaboration is central. The vehicle for teacher development is the Work Group, whereby groups of teachers led by an experienced colleague work collaboratively over a period of time. In 2021-22 a small research team began to explore the impact of this project on teachers’ pedagogical practice and school policy/approaches. We discuss findings from internal evaluation, and from small scale case studies carried out by this team. Early findings indicate that cross-phase engagement with colleagues through focused and sustained professional activity is having a positive impact on practice and school/departmental approaches

A relativistic study of Bessel beams

We present a fully relativistic analysis of Bessel beams revealing some noteworthy features that are not explicit in the standard description. It is shown that there is a reference frame in which the field takes a particularly simple form, the wave appearing to rotate in circles. The concepts of polarization and angular momentum for Bessel beams is also reanalyzed.Comment: 11 pages, 2 fig

Conformality or confinement: (IR)relevance of topological excitations

We study aspects of the conformality to confinement transition for non-supersymmetric Yang-Mills theories with fermions in arbitrary chiral or vectorlike representations. We use the presence or absence of mass gap for gauge fluctuations as an identifier of the infrared behavior. Present-day understanding does not allow the mass gap for gauge fluctuations to be computed on R*4. However, recent progress allows its non-perturbative computation on R*3xS*1 by using either the twisted partition function or deformation theory, for a range of S*1 sizes depending on the theory. For small number of fermions, Nf, we show that the mass gap increases with increasing radius, due to the non-dilution of monopoles and bions, the topological excitations relevant for confinement on R*3xS*1. For sufficiently large Nf, we show that the mass gap decreases with increasing radius. In a class of theories, we claim that the decompactification limit can be taken while remaining within the region of validity of semi-classical techniques, giving the first examples of semiclassically solvable Yang-Mills theories at any size S*1. For general non-supersymmetric vectorlike or chiral theories, we conjecture that the change in the behavior of the mass gap on R*3xS*1 as a function of the radius occurs near the lower boundary of the conformal window and give non-perturbative estimates of its value. For vectorlike theories, we compare our estimates of the conformal window with existing lattice results, truncations of the Schwinger-Dyson equations, NSVZ beta function-inspired estimates, and degree of freedom counting criteria. For multi-generation chiral gauge theories, to the best of our knowledge, our estimates of the conformal window are the only known ones.Comment: 40 pages, 3 figures; modified various comments, reference adde

Student teachers’ beliefs about diversity: analysing the impact of a ‘diversity week’ during initial teacher education

This article reports findings from a week of enrichment placements framed around ‘diversity’ within a secondary Initial Teacher Education (ITE) programme in England. The authors outline the demographics of the county – a largely rural, White county in the East Midlands of England – and describe the challenges this presents for ITE. A mixed-methods approach was used to study student teachers’ (n = 56) beliefs about diversity, generating data through: pre- and post-survey of beliefs and attitudes; student-created reflective videos; journaling; and one pre- and post-diversity week interview. The findings reveal shifts in student teachers’ perceptions about gender, race and sexuality, and these attitudinal shifts were more significant in those attending all week than those attending only the first day. This is particularly interesting because for some topics the only formal input was on the first day, and so the authors argue for the importance of time and space for creative reflection in beginning teachers’ professional development

Non-linear optical susceptibilities, Raman efficiencies and electrooptic tensors from first-principles density functional perturbation theory

The non-linear response of infinite periodic solids to homogenous electric fields and collective atomic displacements is discussed in the framework of density functional perturbation theory. The approach is based on the 2n + 1 theorem applied to an electric-field-dependent energy functional. We report the expressions for the calculation of the non-linear optical susceptibilities, Raman scattering efficiencies and electrooptic coefficients. Different formulations of third-order energy derivatives are examined and their convergence with respect to the k-point sampling is discussed. We apply our method to a few simple cases and compare our results to those obtained with distinct techniques. Finally, we discuss the effect of a scissors correction on the EO coefficients and non-linear optical susceptibilities

Propagation of charged particle waves in a uniform magnetic field

This paper considers the probability density and current distributions generated by a point-like, isotropic source of monoenergetic charges embedded into a uniform magnetic field environment. Electron sources of this kind have been realized in recent photodetachment microscopy experiments. Unlike the total photocurrent cross section, which is largely understood, the spatial profiles of charge and current emitted by the source display an unexpected hierarchy of complex patterns, even though the distributions, apart from scaling, depend only on a single physical parameter. We examine the electron dynamics both by solving the quantum problem, i. e., finding the energy Green function, and from a semiclassical perspective based on the simple cyclotron orbits followed by the electron. Simulations suggest that the semiclassical method, which involves here interference between an infinite set of paths, faithfully reproduces the features observed in the quantum solution, even in extreme circumstances, and lends itself to an interpretation of some (though not all) of the rich structure exhibited in this simple problem.Comment: 39 pages, 16 figure