Development of a methodology for transitioning from a spectral equation relative to a spatial variable to a differential equation relative to a spectral parameter in the Sturm-Liouville problem for a one-dimensional periodic two-layer photonic crystal
Abstract
Relevance. In connection with solving the problem of scattering of electromagnetic waves (diffraction problem) on objects such as photonic crystals (one-dimensional periodic unbounded), it is important to study the dispersion relation. This involves solving the wave equation with subsequent application of the separation of variables method (For the diffraction structures considered in the paper, the specified method of separation of variables allows obtaining a solution to the wave equation, which in this case turns out to be an equation with periodic coefficients, in the explicit form) and transition to the Sturm-Liouville problem on an unbounded interval . The dispersion relation allows us to understand the conditions under which the Sturm-Liouville problem can be solved and connects these conditions with the parameters of the diffraction problem, and therefore becomes an indispensable step on the way to obtaining solutions to this wave equation. This work continues a series of previously published works on the development of approaches to studying the specified dispersion relation through understanding the behavior of the solution of the spectral equation in the Sturm-Liouville problem. The transfer matrix method for a wave equation with periodic coefficients makes it possible to take into account the specifics of its solution on an unlimited interval , and to achieve the fulfillment of a component condition under which the Sturm-Liouville problem is solvable – the condition of self-adjointness of the differential operator in this problem. This is on the one hand, and on the other hand, such a method clearly indicates the place occupied by the solution of the spectral equation in the dispersion relation.
The aim of the work is to simplify the previously obtained equation, which is a consequence of the spectral equation in the Sturm-Liouville problem for a one-dimensional periodic two-layer photonic crystal. In particular, to integrate some of the constituent terms of the linear representation of the 1st and 2nd derivatives of the solution of the spectral equation in the Sturm-Liouville problem.
Methods. The separation of variables method is used to solve the wave equation. The transfer matrix method for a wave equation with periodic coefficients makes it possible to take into account the specifics of its solution on an unlimited interval and achieve the fulfillment of a component condition under which the Sturm-Liouville problem is solvable – the condition of self-adjointness of the differential operator in this problem. To take the integral of the individual component terms of the linear representation of the 1st and 2nd derivatives of the solution of the spectral equation in the Sturm-Liouville problem, the method of integration by parts is used.
Results. The work showed that in the course of a series of successive transformations, the coefficient at zero derivative in the reduced differential equation (the equation relative to the spectral parameter to which the transition is made – according to the title of the work) is subject to simplification. In particular, the square of the coefficient of the linear representation of the 1st derivative of the solution of the spectral equation in the Sturm-Liouville problem was integrated.
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References
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