  We present a Fourier finite element modeling of light emission of dipolar
emitters coupled to infinitely long waveguides. Due to the translational
symmetry, the three-dimensional (3D) coupled waveguide-emitter system can be
decomposed into a series of independent 2D problems (2.5D), which reduces the
computational cost. Moreover, the reduced 2D problems can be extremely
accurate, compared to its 3D counterpart. Our method can precisely quantify the
total emission rates, as well as the fraction of emission rates into different
modal channels for waveguides with arbitrary cross-sections. We compare our
method with dyadic Green's function for the light emission in single mode
metallic nanowire, which yields an excellent agreement. This method is applied
in multi-mode waveguides, as well as multi-core waveguides. We further show
that our method has the full capability of including dipole orientations, as
illustrated via a rotating dipole, which leads to unidirectional excitation of
guide modes. The 2.5D Finite Element Method (FEM) approach proposed here can be
applied for various waveguides, thus it is useful to interface single-photon
single-emitter in nano-structures, as well as for other scenarios involving
coupled waveguide-emitters.
