  A new algorithm based on auxiliary differential equation and finite
difference time domain method (ADE-FDTD method) is presented to model a
waveguide whose active layer is constituted of a silica matrix doped with
rare-earth and silicon nanograins. The typical lifetime of rare-earth can be as
large as some ms, whereas the electromagnetic field in a visible range and
near-infrared is characterized by a period of the order of fs. Due to the large
difference between these two characteristic times, the conventional ADE-FDTD
method is not suited to treat such systems. A new algorithm is presented so
that the steady state of rare earth and silicon nanograins electronic levels
populations along with the electromagnetic field can be fully described. This
algorithm is stable and applicable to a wide range of optical gain materials in
which large differences of characteristic lifetimes are present.
