Cálculo en la estructura de bandas de cristales fotónicos

Abstract

In this work, a brief review of the theory supporting the occurrence of bands of frequencies in so-called photonic crystals is made, as well as the calculation of the band diagram of some purely theoretical structures that may have no affinity with the structures that It can be carried out in a laboratory. First, it reviews the theory that justifies the definition of electrical permittivity over a material medium under considerations or approximations. The starting points are, of course, Maxwell's equations, since the periodic variation of the electrical permittivity is the main responsible for the appearance of the prohibited bands, it is necessary to review from these equations as one can characterize the fields in a medium . Second, to review a set of analytical methods that reduce the problem of finding the frequencies and fields that can be propagated by these structures; That the methods are not, not formally developed, how the complexity of the problem can be reduced by the use of symmetries and the use of Bloch's theorem. Finally, using the revised tools, we obtain the equation that describes how the electromagnetic field propagates in these structures, as well as the frequencies that satisfy this equation. This equation is emphasized as a problem of eigenvalues ​​and eigenvectors, which implies a solution to a finite set of first-degree algebraic equations with the main difficulty of finding the representation of a function of type 1 / f (r) Is a basis of the function space associated with the Fourier series. It is shown in that it consists of a band diagram, how it can be constructed and what relevant information can be extracted from it. The master equation for one-dimensional structures is solved by usual methods and a couple of examples are shown. He solves the master equation for structures in two and three dimensions using the software developed in the MIT of the MIT Photonic bands by which the numerical solution of the master equation is solved and the data can be obtained to graph the band diagram and The distribution of the fields in one, two and three dimensions.