[8] As detailed by *Lee et al.* [2003] and *Farle et al.* [2004], conventional field formulations for the modal analysis of EM waveguides suffer from low frequency instabilities, because the electric flux balance enters the eigenvalue problem in frequency dependent form. As a result, the underlying differential equations become linearly dependent when the frequency approaches zero, so that the formulation breaks down in the static limit. The formulation used in this work [*Farle et al.*, 2004] overcomes this deficiency by imposing the flux balance explicitly, with the help of a magnetic vector potential and a scaled electric scalar potential . Specifically, it employs the gauge *A*_{z} = 0 and a tree-cotree splitting that decomposes _{t} into components of nonvanishing circulation _{t}^{c} plus the transverse gradient _{t} of a scalar field *ψ*. Hence we have

and the EM fields are represented by

By plugging (1) and (2) into the time harmonic Maxwell equations, we arrive at the eigenvalue problem (EVP)

FE discretization results in the algebraic EVP

wherein **x**_{A}, **x**_{ψ}, and **x**_{V} denote the component vectors for _{t}^{c}, *ψ*, and (*jV*), respectively, and **S**_{0}, **S**_{1}, **S**_{2}, and **T** are sparse symmetric matrices, whose structure can be found in the work by *Farle et al.* [2004]. Note that equations (5a) and (5b) are satisfied not only by physical waveguide modes but also by a set of *null-field solutions*, i.e., nontrivial solutions with = 0 and = 0:

In the FE context (6), the null-field solutions **n** read

Equation (6) implies the generalized orthogonality equation

Hence any superposition of physical modes **p**(*k*) satisfies

which enables us to reconstruct **p** from given components **p**_{A} and **p**_{ψ}. The resulting equation takes the form

The structure of the matrices **P**_{0} and **P**_{1} can be found in the work by *Farle et al.* [2004].