From the notion of dispersion diagram of a protonic crystal, an analytic study of ultra-refraction and negative refraction phenomena is achieved. In these phenomena, a protonic crystal simulates a material having an optical index that is positive and less than unity (ultra-refraction) or negative (negative refraction). The study is made in both harmonic and time-domain regimes in conditions close to experimental ones: a beam or a light pulse illuminates a semi-infinite protonic crystal. the form of the transmitted light as well as its phase and group velocities are described analytically. Special attention is paid to frequencies very close to the edge of a gap, where an effective index independent of the incidence angle can be defined. The present analysis extends to the case of the so-called 'left-handed materials' recently reported in the literature.
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