Discussion Of Fourier Finite Element Modeling In Optical Waveguides

Controllably funneling emitted photons into certain modes has potential applications in developing single photon sources, low threshold lasers, optical nanoantennas, ultra-fast optical switches and many others. This imposes experimental challenges in controlling light emitters and waveguides, as well as theoretical difficulties in predicting light emission into complex photonic environment. Firstly, we consider a simple progress of dipole coupling with infinite nano-waveguides. There are already some analytical methods to solve the coupling efficiency of such a quantum dot coupled to the waveguide mode, such as the Green's function method. With the Finite Element Method(FEM) rapidly developing in the optical field, three-dimensional(3D) FEM has been ultilized to deal with such problem. However, both methods have limitations. It is difficult for them to deal with the coupling into a waveguide with a complex cross-section.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 coupled waveguide-emitter system can be decomposed into a series of independent 2D problems(2.5D), which significantly 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. Then 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 guided modes.Finally, we also study the Fourier Finite Element modeling of light emission in coupling between the dipole and optical resonator which has a rotational symmetry. A 2D weak form corresponding to a certain azimuthal mode order can be deduced by utilizing Fourier transform and FEM. Similarly, a 3D problem can be decomposed into a series of 2D decoupled problems(2.5D). These two-dimensional problems can be solved by parallel computer since they are decoupled, leading to great improvement in the computational efficiency. Finally, the weak form is applied in two different whispering gallery mode resonators, i.e., the toroidal microresonator and conical microdisk resonator, to calculate the corresponding field distribution and quality factor(Q).