Study of Flocke-Bloch waves depending on the spectral parameter for a one-dimensional photonic crystal - Kazanko, O.V.; Penkina , O.Ye. (Radiotekhnika, 2025)
The presence of forbidden and permitted bands in the parameters of diffraction processes on crystal structures is one of the main phenomena, thanks to which the latter have mostly proven their practical application. An understanding of the nature of the appearance of such forbidden bands for one-dimensional photonic crystals is provided by dispersion equations. One of the forms of this dispersion equation can be written through the solution of the spectral equation taken in the context of the Sturm-Liouville problem, posed accordingly for a one-dimensional photonic crystal (the Sturm-Liouville problem arises in connection with the application of the method of separation of variables to the wave equation). The current work is a continuation of the work on finding the 1st and 2nd derivatives by the spectral parameter from the solution of the spectral equation. In previous works, it was found that the 1st derivative of the solution of the spectral equation in terms of the spectral parameter is represented linearly through the solution itself and its derivative, but in terms of the spatial variable. This circumstance prompted us later to think about the possibility of writing down the 2nd derivative in a similar form. In this work, the dispersion equation for a one-dimensional photonic crystal is derived. According to the form of the equation obtained in such aoway, it is fair to assign an important role to the functions that are the solution of the spectral equation in connection with the desire to develop an alternative view of the dispersion equation itself. It is also shown that it is possible to represent linearly the 2nd derivative of the solution of the spectral equation. The authors state that the coefficients of the 2nd derivative are relatively uncomplicated for functions bearing linearity (the function itself and its derivative with respect to the spatial variable). Finally, a linear differential equation of the 2nd order is written with respect to the solution of the spectral equation of the function as a function of the spectral parameter.