  We study transmission of the terahertz radiation through a two-dimensional
electron gas with a concentration controlled by grating gate electrodes.
Voltage applied to these electrodes creates a lateral plasmonic crystal with a
gate-tunable band structure. We find that only a part of plasmonic modes of
such a crystal is seen in the transmission spectrum for the case of homogeneous
excitation (so-called bright modes), while there also exist dark modes which
show up only in a case of inhomogeneous excitation. We develop a theory that
describes both weak- to strong- coupling transition in the crystal with
increasing depth of the density modulation and a transition from resonant to
super-resonant regime with increasing quality factor of the structure. We
discuss very recent experiment, where transmission of the terahertz radiation
through GaN/AlGaN based grating gate periodic structures was studied. We argue
that this experiment represents an evidence of formation of the lateral
plasmonic crystal with the band structure fully controlled by the gate
electrodes and magnetic field, in a full agreement with developed theory.
