A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains

This paper presents a review of the most widely-used methods in order to determine the structure of eigenmodes propagating in periodic materials. Both real and Fourier domain methods are outlined. The basic concepts such as eigensolutions and their k-labeling, reciprocal lattice, Brillouin zones, etc., are gradually introduced and explained. Special attention is devoted on the physical aspect and all non usual nomenclatures are defined. In a similar way, all indispensable mathematics are described and their physical content expounded. For completeness, all nonessential notions in a first reading are maintained but deferred in Appendix. Going on with the tutorial, we show how Brillouin exploited the correspondence between the real and Fourier domain representations in order to explain the band structure and especially its periodicity in the Fourier domain. Then, following the way paved by Brillouin, we show how the Bloch theorem may be deduced from general considerations concerning Fourier analysis. To this aim, using nowadays available mathematical tools inspired from the fields of discrete signal analysis we have built a formalism which allows a comprehensive vision of the two domain correspondence. This formalism, developed on mathematical tools well fitted to describe periodic media, introduces appreciable shortcuts and appears to be versatile and easily transposable to different physical domains. Formalism application examples are given in the case of solid-state, photonic and phononic crystals.