  Pendry and MacKinnon meaningful discretization of Maxwell's equations was put
forward specifically as part of a finite-element numerical algorithm. By
contrast with a numerical approach, in the same spirit evoked by the
relationships between the difference and the differential equations, we provide
an analytical solution for the transfer matrix elements generated by this
discretization of Maxwell's equations. The method, valid for all modes and any
non-magnetic pattern with frequency-dependent permittivities, is exemplified
for a bilaminar structure. The results from hundreds of path-operators simplify
to a small number of propagation channels, which transfer the electromagnetic
field between input and output. A specific bilaminar device is design-optimized
and used to generate a topological map from permittivities to the number of
modes and very high Q-factor resonances.
