Comment on 'Exact solution of resonant modes in a rectangular resonator'

We comment on the recent Letter by J. Wu and A. Liu [Opt. Lett. 31, 1720 (2006)] in which an exact scalar solution to the resonant modes and the resonant frequencies in a two-dimensional rectangular microcavity were presented. The analysis is incorrect because (a) the field solutions were imposed to satisfy simultaneously both Dirichlet and Neumann boundary conditions at the four sides of the rectangle, leading to an overdetermined problem, and (b) the modes in the cavity were expanded using an incorrect series ansatz, leading to an expression for the mode fields that does not satisfy the Helmholtz equation

Elliptical beams

A very general beam solution of the paraxial wave equation in elliptic cylindrical coordinates is presented. We call such a field an elliptic beam (EB). The complex amplitude of the EB is described by either the generalized Ince functions or the Whittaker-Hill functions and is characterized by four parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the EB are the standard, elegant, and generalized Ince-Gauss beams, Mathieu-Gauss beams, among others

Normalization of the Mathieu-Gauss optical beams

A series scheme is discussed for the determination of the normalization constants of the even and odd Mathieu-Gauss (MG) optical beams. We apply a suitable expansion in terms of Bessel-Gauss (BG) beams and also answer the question of how many BG beams should be used to synthesize a MG beam within a tolerance. The structure of the normalization factors ensures that MG beams will always be normalized independently of the particular normalization adopted for the Mathieu functions. In this scheme, the normalization constants are expressed as rapidly convergent series that can be calculated to an arbitrary precision

Airy-Gauss beams and their transformation by paraxial optical systems

We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The model of the AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation

Conocimientos matemáticos presentes en las prácticas propias y habituales de un grupo de danza folclórica y su circulación al interior del grupo

Este documento contiene los hallazgos de pensamiento matemático encontrados en un grupo de danza folclórica, la manera como éstos son comunicados y desarrollados en la cotidianidad del grupo y la creación de coreografías. La investigación se centró en la identificación, descripción y clasificación cualitativa de los conocimientos matemáticos inmersos en las prácticas del grupo, analizados bajo la óptica de la Etnomatemática. Este enfoque postula que todo grupo social desarrolla habilidades matemáticas en contextos diferentes a las aulas habituales; también afirma que es posible identificar seis actividades matemáticas (contar, medir, jugar, diseñar, explicar y localizar), las cuales son el filtro para esta experiencia. Así pues, se eligió el momento de creación, este se dividió en tres fases: creación propia, modificación de lo creado y creación bajo condiciones especiales diferentes a las habituales. Finalmente se mostrarán hallazgos encontrados respecto a cada actividad básica y la validación de los mismos a través del proyecto

Lorenz-Mie scattering of focused light via complex focus fields: an analytic treatment

The Lorenz-Mie scattering of a wide class of focused electromagnetic fields off spherical particles is studied. The focused fields in question are constructed through complex focal displacements, leading to closed-form expressions that can exhibit several interesting physical properties, such as orbital and/or spin angular momentum, spatially-varying polarization, and a controllable degree of focusing. These fields constitute complete bases that can be considered as nonparaxial extensions of the standard Laguerre-Gauss beams and the recently proposed polynomials-of-Gaussians beams. Their analytic form turns out to lead also to closed-form expressions for their multipolar expansion. Such expansion can be used to compute the field scattered by a spherical particle and the resulting forces and torques exerted on it, for any relative position between the field's focus and the particle.Comment: 11 pages, 7 figure

The sustainability of public pensions in a State of social well-being

El envejecimiento poblacional es un problema demográfico de carácter estructural que afecta en mayor o menor medida a todos los Estados de la Unión Europea. En España, este problema se agudizará dentro de diez años con la jubilación de la generación conocida como el baby-boom lo que ha puesto en duda la viabilidad del sistema de reparto de la seguridad social diseñado en los años 1963 a 1966 cuando la situación social de España y las necesidades de protección eran muy distintas a las que imperan actualmente. Además con carácter coyuntural, desde el año 2006 España ha estado inmersa en una crisis económica como consecuencia de los efectos de la globalización económica en el ámbito financiero y por la caída del sector inmobiliario debida a una subida desproporcionada en los precios de este sector secundario. Este hecho ha provocado un descenso de cotizantes que ha suscitado que los partidos políticos se conciencien en la necesidad de que el sistema de reparto de la seguridad social se modifique para que sea sostenible en lo que respecta al pago de las pensiones públicas. A partir del año 2011, se han introducido reformas en relación a las pensiones públicas; así, el retraso en la edad de jubilación, el envejecimiento activo, la regulación del factor de sostenibilidad, el índice de revalorización y una recomendación hacia los instrumentos de previsión social privados como complemento de las pensiones públicas pretenden asegurar el principio constitucional de la suficiencia económica de los ciudadanos.The aging of the population is a demographic problem of a structural character which affects in a large or small measure all members of the European Union. In Spain, this problem will enlarge within the next 10 years with the retirement of former baby boom generation, which has put in doubt the viability of the social security economic distribution, designed in the years between 1963-1966, when the state of affairs of Spain and their protective needs were very distinct to the current existing ones. Furthermore, with transitory character, Spain has been immersed in an economic crisis since 2006, due to the effects of the economic globalization in the financial field especially by the fall in the real estate sector caused by the disproportionate rise in the prices of this secondary sector. This fact has provoked a decline in contributors which has raised a necessary awareness in political parties that requires a sustainable modification in respect to public pension payments. From the year 2011 reforms have been introduced in relation to public pensions, so the delay in retirement age, the active aging, the regulation of the sustainability factor, the revaluation indicator and recommendation towards private social pension plans as a complement of public pensions, are aiming to secure the constitutional principle of economic sufficiency for citizens