tokens: third harmonic. Figure 8 presents the temporal dependence of the radiation field E x in units, recorded at one point on the boundary of the computational domain for run 1. The emission spectra recorded at the points indicated by the star in Figures 4 – 6 are shown in Figure 9 . The emission line width near the second harmonic of the plasma frequency for all regimes is less than or of the order of 2%. Figure 4. Efficient regime. (a) The map of the electric field E x in the moment (b) radiation spectra of electric field components for the entire simulation time at the outer boundary of the vacuum region as functions of the longitudinal coordinate x ; (c) theoretical growth rate along the line of the local maximum. Figure 5. Weak field. . Figure 6. Unmagnetized. . Figure 7. Strong field. t = 728 ω p −1 . (b) Phase space ( x , v x ) of the beam. Figure 8. The temporal dependence of the E x field in the single point indicated by the white star in Figure 4 . Figure 9. The frequency spectrum of the radiation produced in the single point indicated by the white star on the corresponding Figures 4 – 6 and 10 . Figure 10. Eff. regime + ions. . (a) Ion density in plasma column and radiation field E x in vacuum. (b) Frequency spectrum of radiation field along the boundary of the simulation box. (c) Theoretical growth rate along the line of its local maximum. The beam instability becomes purely longitudinal in the strong magnetic field (Figure 7 ). In this regime, we also observe a noticeable extension of the beam relaxation length. Therefore, we have extended the total system length in order to overlap beam relaxation regions more efficiently. The overlapping degree can be estimated from the image of the phase space (Figure 7 (b)). Figure 7 (a) shows that, in the case of a sufficiently strong external magnetic field, EM emission disappears. Let us consider the efficiency of EM radiation generation in these regimes quantitatively. Figure 11 presents the fraction of the total beam power P b converted into the radiation power P rad for run 1 as a function of time (efficient regime). In this graph, we separate contributions from various radiation polarizations (TM or O mode includes the fields E x , E z , and B y ; TE or X mode— B x , B z , E y ). It is seen that the ordinary TM mode dominates and the efficiency of its generation reaches 4% of the beam power. The total beam-to-radiation energy conversion efficiency is ≈10 −2 . The duration of the main emission can be estimated as an FWHM width of the plot shown in Figure 11 . In all simulations performed for the efficient regime, this duration lies in the range . Figure 11. The efficiency of beam-to-radiation power conversion as a function of time for run 1. The radiation efficiency depends not only on the amplitudes of the plasma oscillations excited by the beams, but also on the degree of their overlap. For this reason, the dependence of the emission level on time has a pronounced maximum corresponding to
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