International audienceWe display a photonic embodiment of the Demkov-Ostrovsky solution to the crossing of two manifolds made of equidistant modes thanks to broad periodic waveguides. We find clearly narrowing resonances that signal the singular, flat dispersion case that we had termed "critical coupling." The reconciliation of band-edge confinement and cavity confinement, two pillars of photonics, appear from the guide length dependence of spectra. We suggest the generality of the Demkov-Ostrovsky or critical coupling flat dispersion across all kinds of waves, e. g., electronic and acoustic