Research and determination of the 1st and 2nd derivatives of the component terms of the dispersion equation for a flat two-layer one-dimensional periodic photonic crystal

Abstract

Actuality. Recent decades have seen a rapid development of photonics. Therefore, scientific interest in the optical range of electromagnetic radiation continues to be relevant. As a result, the problem of the scattering of electromagnetic waves (diffraction problem) on such objects as photonic crystals is presented as an urgent problem. It is about the solution of the wave equation with the subsequent application of the method of separation of variables and the transition to the Sturm-Liouville problem on the unbounded interval . For the diffraction structures considered in the paper, the specified method of separation of variables allows obtaining the solution of the wave equation (which in this case turns out to be an equation with periodic coefficients) in an explicit form. Another method - the method of the transfer matrix for the wave equation with periodic coefficients makes it possible to take into account the specificity of its solution on the unlimited interval  and to achieve the fulfillment of the component condition for the solvability of the Sturm-Liouville problem - the condition for the self-conjugation of the differential operator in this problem. Therefore, the transfer matrix method involves the construction and solution of the so-called dispersion equation - the equation that connects the parameters of the diffraction problem with the solvability conditions of the Sturm-Liouville problem. As a result, there is a need to study the components of such a dispersion equation. Namely, there is a need to understand the behavior of the solution of the spectral equation in this Sturm-Liouville problem depending on the spectral parameter. Therefore, according to the authors, the search for derivatives of this solution is relevant, since the derivative apparatus as a whole plays a rather important role in the study of any functional dependencies.

The purpose of the work. Determine the first and second derivatives of the spectral parameter from the solution of the spectral equation in the Sturm-Liouville problem for a flat two-layer one-dimensional periodic photonic crystal. And also show that each of the specified derivatives is linearly expressed through the solution itself and its derivative, but in terms of a spatial variable, and as a consequence, the possibility of having two linear dependencies, which makes it possible to obtain a linear homogeneous differential equation of the 2nd order with respect to of this solution. Further research of the specified equation in some perspective may serve the development of an alternative apparatus for understanding the behavior of this solution as a function of the spectral parameter.

Методи і методологія. Умова про самоспряженість диференціального оператора у проблемі Штурма-Ліувілля (складова умова розв’язності проблеми Штурма-Ліувілля) для плоского двошарового нескінченного одновимірно-періодичного фотонного кристала досягається шляхом застосування методу матриці перенесення (Transfer matrix method). Спираючись на принцип невизначених коефіцієнтів, автори використовують підставлення (що запропоновано у роботі) та здійснюють перехід від лінійного неоднорідного диференціального рівняння 2-го порядку, розв’язком якого є шукана похідна (2-га похідна), до системи рівнянь, котра розглядається як матричне рівняння. Для розв’язання матричного рівняння використовується метод варіації.

Methods and methodology. The condition for the self-conjugation of the differential operator in the Sturm-Liouville problem (a constituent condition for the solvability of the Sturm-Liouville problem) for a flat two-layer infinite one-dimensional periodic photonic crystal is achieved by applying the transfer matrix method. Based on the principle of undetermined coefficients, the authors use substitution (which is proposed in the paper) and make the transition from a linear inhomogeneous differential equation of the 2nd order, the solution of which is the sought derivative (2nd derivative), to a system of equations, which is considered as a matrix equation. The variational method is used to solve the matrix equation.

The results. In this work, the second derivative of the spectral parameter is determined from the solution of the spectral equation in the Sturm-Liouville problem for a flat two-layer one-dimensional periodic photonic crystal (unlimited along the periodicity). The defined derivative is linearly expressed in terms of the solution itself and its derivative, but in terms of the spatial variable. Also, in the work, a linear inhomogeneous differential equation of the 2nd order is solved, thus, in fact, the desired derivative is obtained. Such an equation can be solved on the basis of research and the results of previous works - works on the definition of the corresponding 1st derivative. However, it should be noted that a direct analogy between the method of determining the 1st and 2nd derivatives cannot be seen in this, in particular, and the meaningfulness of this paper is expressed.