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Fernandez-Guasti M.,University tropolitana Iztapalapa
Journal of Modern Optics | Year: 2015

Reflection and refraction at an abrupt dielectric interface at normal incidence are evaluated using an amplitude and phase (amph) formalism. The optical properties of a stack with two quarter-wavelength layers are then calculated. The characteristics of a mirror constructed with quarter-wavelength layers are discussed using the amplitude and phase representation. Floquet nonlinear theorem is invoked to describe the multi-layered system. Results are consistent with Fresnel formulae and conventional matrix methods for stratified periodic media. However, the amph formalism offers several advantages: (i) it is capable of showing the field properties as it propagates though the stack, (ii) it gives a lucid physical insight because the variables involved have a clear physical meaning, and (iii) the mathematical description is simple. © 2014 Taylor & Francis.

Fernandez-Guasti M.,University tropolitana Iztapalapa
International Journal of Bifurcation and Chaos | Year: 2016

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in 1 + 2 dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period m, are shown to be necessarily surrounded by points that produce a divergent magnitude after m iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the mth iteration, have preperiod 1 and period m. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary. © 2016 World Scientific Publishing Company.

Fernandez-Guasti M.,University tropolitana Iztapalapa
Journal of Optics (United Kingdom) | Year: 2015

The solution to a non-autonomous second order ordinary differential equation is presented. The real function, dependent on the differentiation variable, is a squared hyperbolic tangent function plus a term that involves the quotient of hyperbolic functions. This function varies from one limiting value to another without having any singularities. The solution is remarkably simple and involves only trigonometric and hyperbolic trigonometric functions. The solution is analyzed in the context of wave propagation in an inhomogeneous one-dimensional medium. The profile is shown to act as a perfect anti-reflection interface, providing a possible alternative route to the fabrication of reflectionless surfaces. © 2015 IOP Publishing Ltd.

Fernandez-Guasti M.,University tropolitana Iztapalapa
Proceedings of SPIE - The International Society for Optical Engineering | Year: 2011

Wave phenomena involve perturbations whose behavior is equivalent in space and time. The perturbations may be of very different nature but they all have to conform with the notion of a field, that is, a scalar or vector quantity-defined for all points in space. Some wave phenomena are described in terms of only one field. For example water waves where the perturbation is the level above or below from the equilibrium position. Nonetheless, electromagnetic waves require the existence of two fields. I shall argue that in fact, all wave phenomena involve two fields although we sometimes perform the description in terms of only one field. To this end, the concept of cyclic or dynamical equilibrium will be put forward where the system continuously moves between two states where it exchanges two forms of energy. In a mechanical system it may be, for example, kinetic and potential energy. Differential equations that form an Ermakov pair require the existence of two linearly-independent fields. These equations possess an invariant. For the time dependent harmonic oscillator, such an invariant exists only for time dependent potentials that are physically attainable. According to this view, two fields must be present in any physical system that exhibits wave behavior. In the case of gravity, if it exhibits wave behavior, there must be a complementary field that also carries energy. It is also interesting that the complex cosmic tension field proposed by Chandrasekar involves a complex field because complex functions formally describe two complementary fields. © 2011 SPIE.

Fernandez-Guasti M.,University tropolitana Iztapalapa
European Physical Journal Plus | Year: 2014

The invariant transformations of a deformed Lorentz metric are explored. These transformations are described by the product operation with a unit magnitude element in hyperbolic scator algebra. The real scator set forms a group under the addition and product operations in a restricted space. However, the product is not distributive over addition. The restricted space condition is equivalent to the time-like subspace in special relativity. In 1+1 dimensions (time and one spatial variable), the deformation vanishes and the scator metric becomes identical to the Lorentz metric. In higher dimensions, time dilation and parallel space contraction are preserved albeit with slight quantitative modification. However, the deformed transformation also exhibits a transverse spatial elongation. © 2014, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.

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