How to obtain a shortest mode converter based on periodic waveguide with limited index contrast?

Mode converter is one of most significant elements in photonic integrated circuits. It relies on increasing index contrast to shorten its length. However, index contrast is limited for technology. In addition, an overlarge index contrast leads to some disadvantages, such as large scattering loss, reflection loss, and small tolerance for manufacturing. Thus, an approximate scheme to design a mode converter is manipulating the transverse distribution of index to achieve the minimum length when the index contrast is given. We have analytically deduced the theoretical maximum coupled efficiency in periodic waveguide, which determines the minimum coupling length of mode converter. What is more, we have demonstrated how to construct a distribution function of indices in a cross section of waveguide to achieve the minimum length and a case is also given to illustrate the process. Proofs, based on both mathematic derivation and numerical simulation, have been exhibited in the paper.