Sensitivity of echo enabled harmonic generation to sinusoidal electron beam energy structure

We analytically examine the bunching factor spectrum of a relativistic electron beam with sinusoidal energy structure that then undergoes an echo-enabled harmonic generation (EEHG) transformation to produce high harmonics. The performance is found to be described primarily by a simple scaling parameter. The dependence of the bunching amplitude on fluctuations of critical parameters is derived analytically, and compared with simulations. Where applicable, EEHG is also compared with high gain harmonic generation (HGHG) and we find that EEHG is generally less sensitive to several types of energy structure. In the presence of intermediate frequency modulations like those produced by the microbunching instability, EEHG has a substantially narrower intrinsic bunching pedestal.

Figure 1

$|{\overline{b}}_{n,m}(k_E)|$ % for $a_E=50\mathrm{th}$ harmonic of 260 nm lasers with $n=-1$ and $A_{1,2}=3$. The dashed line is ${\xi }_E=0$, and the ${\xi }_E=\pm 1/2$ optima (dots) lie at the centers of the bunching peaks in their respective regions, with a difference in dispersion $\mathrm{\Delta }{B}_1=-2{\xi }_E/n$.

Figure 2

Variation of bunching with deviations in $B_1$ from the approximate optimum in Eq. (4). $A_1=5$, $A_2=3$, $a_E=100$ is assumed. Solid lines are from Eq. (7), dashed lines are from exact solutions.

Figure 3

Variation of bunching with deviations in $A_1$ from that used to calculate Eq. (4). $A_1=5$ is assumed. Solid lines are from Eq. (9), dashed lines are from exact solutions.

Figure 4

Variation of bunching with deviations in $A_2$ from $j_{m,1}^{\prime }/a{B}_2$. Solid lines are from Eq. (7), dashed lines are from exact solutions.

Figure 5

Harmonic bunching bandwidth as a function of linear chirp. For EEHG $a_E=10$, $m=11$, $K=1$, $B_1=3.73$, and $A_{1,2}=3$. For HGHG $a_H=A_1=10$ and $B_1=0.11$. Numerical particle simulations written in MATLAB [25].

Figure 6

Harmonic bunching bandwidth as a function of quadratic chirp. Same parameters as Fig. 5.

Figure 7

Harmonic bunching spectrum envelopes (blue) of EEHG, and bunching harmonic spike (red), for $a_E=100$ with $A_{1,2}=3$, $n=-1$. The excitation bandwidth $\delta k/k_E$ is the width of the envelope, which shifts position depending on the sign of optimal ${\xi }_E$.

Figure 8

Top: Bunching and sideband spectra for different $A_0$ with $a_E=100$, $A_{1,2}=3$, $n=-1$, and $k_0=|k_1{\xi }_E/2B|$. Bottom: Bunching amplitudes as a function of $A_0$. Solid lines are predictions from the analytic model, squares are from numerical simulations.

Figure 9

EEHG bunching simulations for the $a_E=100$ harmonic of 266 nm lasers in the presence of broadband MBI energy modulations with rms energy spread (a) ${\sigma }_{p_0}=1$, (b) ${\sigma }_{p_0}=\sqrt{3}$, (c) ${\sigma }_{p_0}=3$. The beam is modeled after the parameters of LCLS-II; $E=4\mathrm{GeV}$, ${\sigma }_E=500\mathrm{keV}$, and ${\sigma }_z/c=100\mathrm{fs}$ (${\sigma }_0={10}^{-5}$). The EEHG energy modulations are $A_{1,2}=3$ and the dispersions are $R_{56}^{(1)}=12\mathrm{mm}$ ($B_1=35.46$) and $R_{56}^{(2)}=118\mu \mathrm{m}$ ($B_2=0.35$). At these settings ${\xi }_E=-0.53$.

Figure 10

FEL output spectrum for EEHG with ${\xi }_E<0$ (red) shows reduced spectral pedestal than ${\xi }_E>0$ (blue) on beam with ${\sigma }_{p_0}=\sqrt{3}$ broadband MBI noise. $A_{1,2}=3$, $a_E=75$, $\rho =1.58\times {10}^{-3}$, ${\sigma }_E/E={10}^{-4}$.