Restrictions

The Modal methods have not any restriction on grating depth.
The common TMM and FMM problem is an electromagnetic field discontinuity at layer interfaces. Theoretically for infinite number of modes the fields at interfaces are matched correctly. Always there is a convergence with the truncated number of modes increasing. The convergence rate depends on diffraction structure refractive indexes contrast and on polarization type. The less contrast the larger convergence rate and this rate is also larger for the TE polarization than for the TM polarization.

The largest convergence problems arise for TM polarization and metal gratings. In that case besides the above mentioned electromagnetic field discontinuity problem there are an additional problems. In lamellar structures with metal so called plasmon modes may exist. As usual, these modes field has very sharp coordinate dependence. In this case also “hidden” modes⁶ exist with field distribution different from regular modes and spread as pairs along whole modes spectrum. These circumstances decrease convergence rate versus number of truncated modes and may lead to a non monotonic convergence.
Except the special case of highly conducting gratings under TM polarization, where the FMM needs a special treatment like modes filtering, both TMM and FMM methods have comparable rate of convergence.

Despite the problems described above these methods give practically acceptable results (accuracy is not less than fraction of percent) for moderate number of modes taken for calculation.

In case of the TMM the balance (sum of all diffraction orders power) is a good criterion of results reliability, but the FMM in case of lossless structures always gives balance equal to unity.