 
Summary: Variational Effective Index Mode Solver
O.V. Ivanova, M. Hammer, R. Stoffer, E. van Groesen
MESA+ Institute for Nanotechnology, University of Twente, The Netherlands
A variational approach for the modal analysis of dielectric waveguides with arbitrary piecewise constant rectangular 2D crosssection is developed. It is based on a representation of a mode profile
as a superposition of modes of the constituting slab waveguides times some unknown continuous coefficient functions, defined on the entire lateral coordinate axis. The propagation constant and the
lateral functions are found from a variational principle. It appears that this method, while preserving the computational efficiency of the standard effective index method ([1], [2]), provides more accurate
estimates for propagation constants, as well as welldefined continuous approximations for mode profiles.
Finding guided modes
Given
· a zinvariant waveguide with refractive index n(x, y),
· the semivectorial TE and TM mode equations for the do
minant electric E = Ey and magnetic H = Hy field compo
nents at vacuum wavelength = 2/k0 ,
searching for
· square integrable profiles E(x, y) and H(x, y), propagating
in the z  direction with propagation constant ,
leads to the following eigenvalue problems
E+k2
0n2
E = 2
