Variational Method to Shape Analytical Expressions of Time Dependent Eigenvalue Equations: Slab Waveguides Families

Starting from the eigenvalue equations describing the behavior of 3-layer and 4-layer slab waveguides, we will present a global analytical framework providing laws of evolution of the effective propagation constant associated to such structures, in case of temporal variation of its geometrical features. So as to develop such a formulation and related principles, a temporal derivation operator is applied on the studied eigenvalues equations, considering then time varying values of the geometrical characteristics together with the effective propagation constant. Relevant calculations are performed on three different cases. We first investigate the variation of the height of the guiding layer for the family of 3-layer slab waveguides: then, considering the 4-layer slab waveguide's family, we successively address the variation of its guiding layer and of its first upper cladding. As regards the family of 4-layer waveguides, calculations are performed for two different families of guided modes and light cones. Such an approach yields rigorous generic analytical relations, easily implementable on a personal computer and highly valuable to obtain and trace all the dispersion curves by single integration. In conclusion, this work presents the derivation of generic and rigorous analytical expressions evolutive in time regarding the propagation constant for various families of slab waveguides in case of temporal variation of their geometrical characteristics. This analysis provides a valuable analytical link between the variation of geometrical features of the global dimensions structures and the correlated changes of all the effective propagation constants describing their associated modes.