  Exact method of analytical solution of flat, non-dispersive eigenstates in a
class of quasi-one dimensional structures is reported within the tight-binding
framework. The states are localized over certain sublattice sites. One such
finite size cluster of atomic sites is decoupled from the rest of the system by
the special non-permissible vertex having zero amplitude. This immediately
leads to the self-trapping of the incoming excitation. We work out an
analytical scheme to discern the localizing character of the diffraction free
dispersionless modes using real space renormalization group technique.
Supportive numerical calculations of spectral profile and transport are
demonstrated to substantiate the essence of compact localized states. Possible
experimental scope regarding the photonic analogue of the tight-binding
electronic case is also discussed elaborately. This eventually unfolds the
concepts of slow light and the related re-entrant mode switching from the study
of optical dispersion.
