Thickness dependence of negative refraction in photonic crystals

The beam at the exit surface of a Photonic Crystal (PhC) slab can be periodically modulated, in positive or in negative direction, changing the slab thickness. The thickness in negative refraction on PhC's is not always appropriately considered, in spite of an always increasing literature in this subject. This effect is well known in x-ray diffraction, in the most comprehensive version: the Dynamical Diffraction Theory (DDT). Thickness dependence is a direct result of the so-called Pendellosung phenomenon and is linked to a periodic exchange, inside the crystal, of the energy among direct beam (or positively refracted) and diffracted beam (or negatively refracted). It represents an outstanding example of the application of the result of DDT as a tool for the analysis of s electromagnetic interaction in PhC's.