This paper describes a simulation method for 2D frequency domain scattering problems in photonics. The technique reduces the spatial dimensionality of the problem by means of global, continuous mode expansion combined with a variational formalism; the resulting equations are solved using a finite element method. Transparent influx boundary conditions and perfectly matched layers are employed at the computational window boundaries. Numerical examples validate the method.

The spatial dimensionality of vectorial 3D frequency domain optical scattering problems is reduced by means of a global expansion of the field in one direction in slab modes of some reference slice(s). A variational formalism yields the equations in the other two directions. These coupled partial differential equations are solved using a Finite Element Method with modified Transparent Influx Boundary Conditions with PMLs.

Recent Near-field Scanning Optical Microscopy (NSOM) experiments with slit metal coated probes claim to measure the out-of-plane optical magnetic field around a dielectric sample waveguide . The observations can also be explained by mode overlap calculations.

In order to reduce the computational effort we develop a method for 3D-to-2D dimensionality reduction of scattering problems in photonics. Contrary to the 'standard' Effective Index Method the effective parameters of the reduced problem are always rigorously defined using the variational technique, based on the vectorial 3D Maxwell equations. Results for a photonic crystal slab waveguide show that this approach predicts the location of the bandgap and other spectral features much more precisely than any 'standard' EIM approximation.

A variational approach for the semivectorial modal analysis of dielectric waveguides with arbitrary piecewise constant rectangular 2D cross-sections is developed. It is based on a representation of a mode profile as a superposition of modes of constituting slab waveguides times some unknown continuous coefficient functions, defined on the entire coordinate axis. The propagation constant and the lateral functions are found from a variational principle. It appears that this method with one or two modes in the expansion preserves the computational efficiency of the “standard” effective index method while providing more accurate estimates for propagation constants, and well-defined continuous approximations for mode profiles. By including a larger number of suitable trial fields, the present approach can also serve as a technique for rigorous semivectorial mode analysis.