Temporal derivation operator applied on the historic and school case of slab waveguides families eigenvalue equations: another method for computation of variational expressions
Université de Rennes, CNRS, Institut de Physique de Rennes − UMR 6251, 35000 Rennes, France
2 Université de Rennes, CNRS, IRMAR – UMR 6625, 35000 Rennes, France
3 Université du Maine, CNRS, Laboratoire d'acoustique de l'Université du Maine − UMR 6613, 72000 Le Mans, France
4 Université de Rennes, CNRS, Institut d'Électronique et de Télécommunications de Rennes − UMR S 6164, 35000 Rennes, France
* e-mail: email@example.com
Received in final form: 1 July 2019
Accepted: 15 July 2019
Published online: 6 August 2019
Starting from the well-known and historic eigenvalue equations describing the behavior of 3-layer and 4-layer slab waveguides, this paper presents another specific analytical framework providing time-laws of evolution of the effective propagation constant associated to such structures, in case of temporal variation of its various geometrical features. So as to develop such kind of time-propagator formulation and related principles, a temporal derivation operator is applied on the studied school case equations, considering then time varying values of all the geometrical characteristics together with the effective propagation constant. Relevant calculations are performed on three different cases. For example, 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 another approach yields rigorous new generic analytical relations, easily implementable and highly valuable to obtain and trace all the family of dispersion curves by one single time-integration and one way.
© L. Garnier et al., EDP Sciences, 2019
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