A theoretical analysis of Smith-Purcell radiation at very high energies is presented. The energy per unit frequency and solid angle is expressed in closed form as a function of the grating geometry, beam energy, and viewing angles. A certain choice of grating geometry is shown to optimize the output energy for a particular order of radiation. Scaling laws are derived for the energy emitted into all orders of radiation in the relativistic limit. It is shown that the total energy emitted into each order scales as the three-halves power of the beam voltage.
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Trotz et al. (2000) studied this question.