Using the Beam-Echo Effect for Generation of Short-Wavelength Radiation

We propose to use the mechanism of an echo effect previously observed in hadron accelerators for up-frequency conversion of density modulation in an electron beam. We show that, for generation of high harmonics, this method is much more efficient in comparison with the currently used approach. A one-dimensional model of the effect is developed which allows us to optimize the amplitude of the modulation for a given harmonic number.

Figure 1

A traditional system (a) consists of an undulator and a dispersive section. The proposed new scheme (b) includes two consecutive modulators.

Figure 2

The bunching factor for $k=10$ as a function of parameter $B_1$ for four different values of $B_2$: $1-B_2=1.18$, $2-B_2=1.25$, $3-B_2=1.3$, $4-B_2=1.35$.

Figure 3

The phase space of the beam after the first undulator (top left), the first dispersive element (top right), the second undulator (bottom left), and the second dispersive element (bottom right). Horizontal axes in the plots are $\zeta$, and the vertical axes are $p$.

Figure 4

FEL power versus the undulator length $L_u$.