The bidirectional mode expansion method for two-dimensional waveguides: the TM case

The mode expansion propagation method is a modelling technique for a large variety of waveguide components. Until now only the case with TE modes has been reported; we give details on the mode expansion propagation method for TM modes. Instead of one type of overlap integral in the TE case, a TM analysis requires two kinds of overlap integrals. The inclusion of radiation modes in this method is discussed. The modifications in the algorithm to include waveguides with gain or loss structures are also considered.As an example, the method was used for analysing a grating-assisted codirectional coupler. In particular, the radiation loss is calculated and compared with calculations when TE modes, instead of TM modes, propagate through the codirectional coupler. The radiation losses are found to be much higher for TM modes.