Propagation modes are computed by solving the Helmholtz equation alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. z ). More specifically, the common approach is to first replace all unknown time-varying fields u ( x , y , z , t ) (assuming for simplicity to describe the fields in cartesian components) with their complex phasors representation U ( x , y , z ) , sufficient to fully describe any infinitely long single-tone signal at frequency f , (angular frequency ω ω = 2 π π f ), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like U ( x , y , z ) = U ^ ^ ( x , y ) e − − γ γ z , where the γ γ term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for γ γ and U ^ ^ ( x , y ) , yielding in the end an eigenvalue equation for γ γ and a corresponding eigenfunction U ^ ^ ( x , y ) γ γ for each solution of the former.