Analysis and methodology of determining the norm of eigenfunctions as a limit transition in the scalar product in the spectral Stourm-Louvile problem for a photonic one-dimensional crystal
Abstract
Relevance The last of the decades (approximately from the 90s of the 20th century) to rapid grow of photonics. That's why, firstly, relevance this work is related to relevance diffraction problems for the structures of optics ranges (photonic crystal). The problem of calculating the norm of eigenfunctions Stourm-Louvile problem, in particular, raised when a waves equations is solved by separating variables method, as well as when making the transition from one complete to another complete orthogonal system (when reducing to a common basis – the Fourier method). In addition, the significance of this work should be associated with the possibility of obtaining an analytical dependence, which gives a clear connection between the norm and its eigenfunctions.
The paper develops an approach to determining the norm of the eigenfunctions of the spectral Stourm-Louvile problem for a two-layer infinite one-dimensional photonic crystal. This approach is based on the limiting transition in the corresponding scalar product. The uncertainty arising at the limit transition is revealed using Lopital's rule.
The purpose of the work – Simplify the previously obtained marginal transformation of the norm (the transformation that directly occurs when the marginal transition is carried out in the corresponding scalar product). It is achieved mainly due to the fact that it is possible to find such a solution of a linear inhomogeneous differential equation (this inhomogeneous equation is obtained by taking the derivative of the spectral equation with respect to the spectral parameter) that satisfies the quasi-cyclic conditions on the period (the Floquet conditions). Also, the authors aimed to emphasize the advantages of the current approach to the calculation of the norm, because the latter gives the connection between the norm and the eigenfunction itself in an explicit form.
Materials and methods. The integral defining the norm (more precisely, the scalar product) is taken on a finite interval, therefore the inhomogeneous equation arising according to Lopital's is solved on a finite interval, that is, the solution of this inhomogeneous equation is sought as a solution of a boundary value problem with boundary conditions – by the conditions of Floquet. The spectral equation in the Stourm-Louvile problem is solved on an unlimited interval (-∞, +∞), therefore, in order to fit into the conditions of self-conjugation, the transfer matrix method is used.
Results. A solution was chosen that satisfies quasi-cyclic conditions on the period (Floquet conditions). The specified solution is selected from the set of all possible solutions of the inhomogeneous differential equation, which, according to Lopital's, arises at the limit transition. As a result of the substitution of this solution, the original marginal transformation of the norm is simplified.
Conclusion. The interest in the transformation of the norm, obtained as a result of the implementation of the limit transition in the corresponding scalar product, is rightly associated with the realized possibility of obtaining the dependence between the norm and the eigenfunction itself in analytical form. The main attention is paid to the case when it is possible to achieve the fulfillment of the conditions of Floquet, when obtaining the solution of the inhomogeneous equation required for finding the derivative in connection with Lopital's rule. In this case, the marginal transformation of the norm is simplified
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References
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