Daubechies wavelets as basis functions for the vectorial beam propagation method

ABSTRACT

We analyze the propagation of electromagnetic signals in optical waveguides using the Daubechies wavelets as basis functions applied in a mathematical model based in finite element method (FEM). The Daubechies wavelets have the peculiar property of having the compact support, which allows describing the behavior of functions with discontinuities or some abrupt variations of values in frequency and/or time domain. Then, the basis functions and the Galerkin method (GM) are applied together with the vector beam propagation method (VBPM) to develop a new mathematical approach. In order to validate the proposed formulation performance, we analyze a new polished-type optical fiber coupler, for which two cores are curved in a parabolic shape, and between them, a leaky-waveguide with an index-matching liquid layer is included. The distance between cores has variable parameters. Due to the inclusion of the leaky-waveguide, the coupling distance drastically reduces.

KEYWORDS:

• Daubechies wavelets
• VBPM
• Galerkin method
• finite element method

Acknowledgements

The authors are grateful for the support of CCEN/UFERSA for the release for professional qualification, PPGEEC/UFRN for the technical support for conducting this research and Anibal Mascarenhas Filho by revising the text.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

P. C. L. da Silva

P. C. L. da Silva was born in Fortaleza, Ceará, Brazil, on 7 February 1979. He graduated in Mathematics from the Federal University of Ceara (UFC), Ceara, Brazil in 2003, and master's degree in computer science by the same university in 2009. Currently is a doctoral student in Electrical Engineering and Computation by Federal University of Rio Grande do Norte (UFRN) Brazil. Actually, he is Assistant Professor of the Department of Exact and Natural Sciences Mathematics and Statistics of the Federal University of the Semi-arid (UFERSA), Mossoró, Brazil. His research includes analytical combinatorics and wavelets.

J. P. da Silva

J. P. da Silva was born in São João do Sabugí, Rio Grande do Norte, Brazil, on 6 March 1970. He received the title of Electrical Engineer and the MSc degree in electrical engineering from Federal University of Rio Grande do Norte (UFRN), Brazil, in 1996 and 1999, respectively, and the PhD degree in Telecommunications from the School of Electrical and Computer Engineering, University of Campinas (UNICAMP), São Paulo, Brazil. Actually, he is Associate Professor of the Department of Electrical Engineering of the UFRN. His research includes applied electromagnetism, wave propagation, integrated optics, antennas, and photonics devices.

A. R. G. Garcia

A. R. G. Garcia Graduated in Licentiate with Habilitation in Mathematics at State University of Rio Grande do Norte in 1993, Master in Mathematics at University of São Paulo (IME-USP) in 2001 and Doctor in Mathematics by the same University in 2006. He is currently Associate Professor III at Federal Rural University of the Semi-Arid Region. He has experience in Mathematics, with emphasis on Algebra of the Generalized Functions of Colombeau and comes working with Control Theory and Automation in projects linked to PPGEE/UFERSA.