Negative refraction in Photonic Crystals: thickness dependence and Pendellösung phenomenon

We show that the refracted wave at the exit surface of a Photonic Crystal (PhC) slab is periodically modulated, in positive or in negative direction, changing the slab thickness. In spite of an always increasing literature, the effect of the thickness in negative refraction on PhC's does not seem to be appropriately considered. However such an effect is not surprising if interpreted with the help of Dynamical Diffraction Theory (DDT), which is generally applied in the x-ray diffraction. The thickness dependence is a direct result of the so-called Pendellösung phenomenon. That explains the periodic exchange, inside the crystal, of the energy among direct beam (or positively refracted) and diffracted beam (or negatively refracted). The Pendellösung phenomenon is an outstanding example of the application of the DDT as a powerful and simple tool for the analysis of s electromagnetic interaction in PhC's.