  The relativistic wave equations of a charged particle propagating in a
classical monochromatic electromagnetic plane wave, in a medium of index of
refraction n_m < 1, have been studied. In the Dirac case the found exact
solutions [arXiv:1305.4370] are expressed in terms of new complex polynomials,
and in the Klein-Gordon case they are expressed in term of Ince polynomials
[arXiv:1306.0097]. In each case these solutions form a doubly infinite discrete
set, parametrized by quantized momentum components of the charged particle
along the polarization vector and along the propagation direction of the
electromagnetic radiation (which may be considered as a plasmon wave of
arbitrary high amplitude, propagating in an underdense plasma). These solutions
describe a high-contrast periodic structure of the particle density on the
plasma length scale, and they may have relevance in the study of novel
acceleration mechanisms.
