A method of variation of boundaries for waveguide grating couplers

Abstract

We describe a method for calculating the solution of the electromagnetic field in a non-rectilinear open waveguide by using a series expansion, starting from the field of a rectilinear waveguide. Our approach is based on a method of variation of boundaries. We prove that the obtained series expansion converges and we provide a radiation condition at infinity in such a way that the problem has a unique solution. Our approach can model several kinds of optical devices which are used in optical integrated circuits. Numerical examples will be shown for the case of finite aperiodic waveguide grating couplers.

Keywords:

• wave propagation
• Helmholtz equation
• optical waveguides

AMS Subject Classifications:

• 78A40
• 78A50
• 78M35
• 35J05
• 35A05

Acknowledgements

This work has been partially supported by the PRIN project ‘Equazioni alle derivate parziali e disuguaglianze funzionali: aspetti quantitativi, proprietà geometriche e qualitative, applicazioni’, financed by MIUR. The author wish to thank Prof. Rolando Magnanini for his constant support in writing this article. The author is also indebted to Prof. Fadil Santosa and Prof. Fernando Reitich, who suggested several topics studied in this work while he was visiting the Institute for Mathematics and its Applications (IMA) at University of Minnesota.

Notes

Note

1. Here and in the rest of the section, we use the following notation: by a subscript, as in L _ϵ u, we denote functions of the variables (x, z), whereas a superscript, as in L ^ϵ w, indicates (the corresponding) functions of the variables (t, s).