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A comparison of the new exact solutions of the relativistic wave equations of a charged particle propagating in a strong laser field in an underdense plasma

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.