Evolution of density-modulated electron beams in drift sections

Initiating the amplification process in a short-wavelength free-electron laser (FEL) by external seed laser pulses results in radiation with a high degree of longitudinal and transverse coherence. The basic layout in seeded harmonic generation involves a periodic electron energy modulation by laser-electron interaction in a short undulator (the “modulator”), which is converted into a density modulation in a dispersive section immediately followed by a long FEL undulator (the “radiator”) tuned to a harmonic of the seed laser wavelength. With the advent of more complex seeding schemes, density-modulated beams may need to be transported in drift sections before entering the radiator. Long FEL undulators may also contain several drift spaces to accommodate focusing elements and diagnostics. Therefore, it is of general interest to study the evolution of density-modulated electron beams in drift sections under the influence of repulsive Coulomb forces. At FERMI, a seeded FEL user facility in Trieste, Italy, systematic studies of the impact of varying drift length on coherent harmonic emission were undertaken. In order to make the underlying physics transparent, the emphasis of this paper is on reproducing the experimental findings with analytical estimates and a simple one-dimensional numerical model. Furthermore, the Coulomb forces in a drift section may be employed to enhance the laser-induced energy modulation and yield an improved density modulation before entering the FEL radiator.

Figure 1

Schematic view of the seeded FEL lines of FERMI as of 2022 with modulators (mod), radiators, dispersive sections (DS), and a delay line (DL) for two-staged HGHG. Also indicated is the shortest and longest drift length between DS and a radiator in FEL-1 (not to scale, see also Table 1).

Figure 2

Top: electrons in longitudinal phase space after energy modulation followed by a dispersive section. Bottom: model of microbunches with charge $Q<0$ as disks (red) with area $a$ and thickness ${\sigma }_z$. The electric field $\mathcal{E}$ perpendicular to a short cylindrical box surrounding charge $Q$ (dashed lines) is given by Gauss’s law: $2a·\mathcal{E}=Q/{ϵ}_0$. The electric field of the almost homogeneous dilute charge between the microbunches (light red) is predominantly radial.

Figure 3

Relative energy offset in units of ${\sigma }_E$ (top) and electron density (bottom) as function of the longitudinal coordinate $z$ over one laser wavelength (here: ${\lambda }_{\mathrm{L}}=260\mathrm{nm}$) after energy modulation with amplitude $\mathrm{\Delta }E/{\sigma }_E=26$ and sections with different longitudinal dispersion: (a) $R_{56}=0\mathrm{\mu }\mathrm{m}$; (b) $R_{56}=42\mathrm{\mu }\mathrm{m}$, optimum density modulation; (c) $R_{56}=58\mathrm{\mu }\mathrm{m}$, density modulation with two peaks ${\lambda }_{\mathrm{L}}/10$ apart; (d) $R_{56}=70\mathrm{\mu }\mathrm{m}$, density modulation with two peaks $2{\lambda }_{\mathrm{L}}/10$ apart. Cases (b)–(d) give rise to the first three maxima of bunching factor $|b_{10}|$ as function of $R_{56}$.

Figure 4

Relative energy offset in units of ${\sigma }_E$ as function of the longitudinal coordinate $z$ over one laser wavelength (here: ${\lambda }_{\mathrm{L}}=260\mathrm{nm}$) with increasing drift distance $s$ after density modulation: (a) $s=0\mathrm{m}$; (b) $s=1.5\mathrm{m}$, correlated energy spread between the microbunches decreases; (c) $s=3.0\mathrm{m}$, minimum correlated energy spread; (d) $s=4.5\mathrm{m}$, overshoot of energy offset between the microbunches. While the energy spread within the microbunch steadily increases with drift distance, the squared bunching factor $|b_{10}{|}^2$ decreases from 0.078 to 0.068. In this example: electron energy $E=1300\mathrm{MeV}$, peak current $I=700\mathrm{A}$, and beam size ${\sigma }_{x,y}=60\mathrm{\mu }\mathrm{m}$.

Figure 5

Color-coded squared bunching factor ${|b_{10}|}^2$ of the 10th seed harmonic as function of $R_{56}$ and drift length. The white lines correspond to the drift between the dispersive section and the undulators of the FEL-1 beamline. Top: with negligible current, no LSC. Bottom: with a current of 700 A.

Figure 6

Measured single-undulator pulse energy at the 10th seed harmonic as function of longitudinal dispersion $R_{56}$ for drift lengths from 1.5 to 20.1 m (radiator 1 to 6) and estimated currents ranging from 350 to 1400 A.

Figure 7

Color-coded visualization of the pulse energy values shown in Fig. 6 as function of $R_{56}$ and drift length. With increasing drift, the centroid of the intensity distribution over $R_{56}$ (red dashed line) tends toward higher values while the approximate position of the first maximum (blue dashed line) shifts to lower $R_{56}$ consistent with the simulation result shown in Fig. 5.

Figure 8

Intensity in terms of pulse energy from radiator 1 to 6 (blue) and calculated values of $|b_{10}{|}^2$ (red) as function of $R_{56}$ of the dispersive section for a current of $I=700\mathrm{A}$. For each radiator, the bunching factor was scaled to the maximum of the measured pulse energy.

Figure 9

Left: pulse energy of the respective first maximum of the $R_{56}$ scans in Fig. 6 as function of drift length in $\mathrm{\mu }\mathrm{J}$ (top) and normalized to the value at drift length 1.5 m (bottom). Right: drift length at which the pulse energy reduces by a factor of 2 (interpolated from the left figure) as function of current. Dashed line: debunching length from Eq. (10).

Figure 10

Pulse energy from radiator 2 (left) and radiator 3 (right) under variation of the $K$ parameter of the preceding undulators. The red lines represent fits to the data points using Eq. (10).

Figure 11

Two examples of space-charge-enhanced modulation with relative modulation amplitudes $A=30$ (top row) and $A=10$ (bottom row). The initial maxima of the electron density are at $z=130$ and 390 nm. The drift lengths $s$ after the first dispersive section are 0.0 m (a,e), 6.3 m (b,f), and 17.0 m (c,g) followed by a second dispersive section (d,h) resulting in a new density maximum at $z=260\mathrm{nm}$.

Figure 12

Top: evolution of correlated energy spread (here: root mean square value of energy offset excluding a region of $\pm 10\mathrm{nm}$ around the microbunches) with initial modulation amplitude $A=10$ (left column) and $A=30$ (right column). Bottom: squared bunching factor $|b_{10}{|}^2$ of the 10th seed harmonic before (blue) and after (red) a second dispersive section DS2 as function of drift length between the two dispersive sections.

Figure 13

Left: Maximum drift as function of the initial modulation amplitude $A$ for a correlated energy spread limited to ${\sigma }_{\mathrm{\Delta }E}/{\sigma }_E\le 30$. Right: Squared bunching factor after the first dispersive section DS1 (blue) and after DS2 (red) placed at maximum drift according to the left graph.