On the dispersion relation of radiation from charged particles in periodic structures

In this paper we considered analytically one of the key features describing the radiation from charged particles in periodic structures. It is shown that the dispersion relation in the radiation from periodic structures with a finite number of elements, both for polarization radiation mechanism and forbremsstrahlung-type mechanism, including those in undulators and wigglers, does not take into account all the details of the spectral-angular distribution. Based on the analysis of the asymptotic expression which allows one to represent the ratio of squares of sines as the sum of delta functions, the factor has been obtained that takes into account the contribution of additional peaks. The expression depends on the number of elements of the periodic structure, and in the limiting case of very large number of elements turns into well-known asymptotic expression. The resulting expression has been obtained in the form helpful for further mathematical analysis, including the use of saddle point technique.