Investigation of the Abraham-Minkowski dilemma in Smith-Purcell radiation from photonic crystals

The debate over whether light's momentum within a medium is accurately described by Abraham or Minkowski formulation has persisted for over a century. To our knowledge, this dilemma has not been explored within the context of Smith-Purcell radiation. This is because, in conventional Smith-Purcell radiation scenarios, the refractive index is equal to one, leading both the Abraham and Minkowski formulations to yield identical results. Here, we investigate the Abraham-Minkowski dilemma within the realm of Smith-Purcell radiation from photonic crystals, where the refractive index deviates from one. In particular, we find that 3 MeV free electrons impinge a photonic crystal with a grating length of 2.1 mu m, resulting in the emission of red light when analyzed based on Abraham's momentum and blue light when analyzed based on Minkowski's momentum. In addition, our findings reveal that the disparity in wavelength as predicted by Abraham's momentum and Minkowski's momentum depends on the grating length and the refractive index. Our findings offer a method to address the Abraham-Minkowski dilemma within the context of Smith-Purcell radiation, thereby enhancing our understanding of both the Abraham-Minkowski dilemma and Smith-Purcell radiation.