Concluding remarks

In this chapter, we presented a spatial coupled mode theory based model of 2-D bent-straight waveguide couplers. In this frequency domain approach, coupled mode equations are rigorously derived by a variational principle. Availability of 2-D analytical bent waveguide modes facilitates the implementation of the model consistent with standard physical notions. By solving the coupled mode equations numerically, and projecting the resulting coupled mode field on the modes of the straight waveguide, we obtained the scattering matrices of the couplers. In symmetrical coupler settings, the scattering matrices satisfy the reciprocity property, that permits to assess the validity of the simulation results.

Using the above coupled mode theory model, a detailed study of the effects of separation distance, the radius of the bent waveguide, and the wavelength on the scattering matrices has been carried out. For the couplers involving bent waveguides that support multiple whispering gallery modes, one can systematically investigate the significance of the individual modes. In combination with analytically computed bent mode propagation constants, the model allows to calculate rigorously the free parameters in the ``standard resonator model'' and to analyze the spectral response of the microresonators (Sections 1.4, 4.1). While the present study discusses only 2-D couplers, this theory can be extended in a straightforward way to the 3-D setting, as discussed in detail in Ref. [69].