Another function of the metal grating is to excite the graphene
surface plasmonic wave. Because the wavevectors between a graphene
plasmonic wave and an incident TM wave are in huge mismatch,
they cannot couple to each other unless to introduce the
grating to compensate the mismatch. Specifically, when a TM wave
with angular frequency x vertically incidents on the grating, its
nth diffraction order will gain an extra wavevector 2np/l with n
being an integer and l the grating period. Then, the nth order
graphene plasmonic wave can be excited by the diffraction order
provided Re [q(x)] = 2np/l, where q(x) is the dispersion relation
of the plasmonic wave with q being its wavevector. Typically, the
wavevector of a graphene plasmonic wave is an order higher than
that of the TM wave [4], which requires the grating period l to be
very small so that 2np/l is large enough to compensate the
mismatch.