Cbn 11-5

We represent results of calculations of coherent synchrotron radiation (CSR) of the relativistic bunch in an undulator with a vacuum chamber of arbitrary cross section with a new algorithm. This algorithm associated with direct calculations of electric field rather than the vector potential. CSR normalized to the incoherent one and compared with analytical calculations for a free space. INTRODUCTION In our previous publication [1] we introduced the method of calculation of coherent radiation in an undulator. The method described in [1] dealt with calculations of vector and scalar potentials with appropriate boundary condition on the surface of the vacuum chamber of arbitrary cross section. Later we changed algorithm slightly, so the scalar potential calculated from the Lorentz condition rather than from a wave equation for a scalar potential. In the current publication we describe some details of newest algorithm for numerical calculation of power radiated coherently. In these new calculations of power we use the direct equations for the electric field, rather than equations for the vector (and scalar) potential. This delivers faster algorithm and much smoother solutions. With the help of this new algorithm we investigated CSR in an undulator suggested for ERL (see details of this undulator in [11]). Let we briefly remind the results obtained in [1] first. CR IN A FREE SPACE The particle in an undulator performs transverse oscillations with a spatial period of undulator field, which is u u D π λ 2 = . While oscillating in the undulator field, a particle radiates on harmonics [2]-[4] θ β ω Cos n n − Ω = 1 , (1) where n=1,2,3..., numerates the harmonics of frequency u c D / β = Ω , β β ≅ = c v / , v is a particle’s average longitudinal velocity in the undulator. For a helical undulator one revolution around the axis corresponds to a single oscillation in the longitudinal direction also, so the harmonics numerator is the same for the transverse and longitudinal oscillations. Formula (1) corresponds to the Doppler-shifted frequency Ω. Accordingly, the wavelength of radiation is β θ β n Cos u n ) 1 ( − ⋅ = D D (2)