How short can an adiabatic mode transformer be in a coupled waveguide system?

@article{Sun2009HowSC,
  title={How short can an adiabatic mode transformer be in a coupled waveguide system?},
  author={Xiankai Sun and H Liu and Amnon Yariv},
  journal={2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum electronics and Laser Science Conference},
  year={2009},
  pages={1-2}
}
We derive the shortest length of adiabatic mode transformers which keep the power scattered into unwanted modes below a certain level. The theory is well supported by the numerical results from a transfer matrix method.