Two-chicane compressed harmonic generation of soft x rays

Seeding an electron bunch prior to compression simultaneously shifts the laser modulation to shorter wavelengths while decreasing the required modulation amplitude. The final x-ray wavelength is then tunable by controlling the compression factor with the rf phase. In this paper we describe a two-chicane scheme that allows for large modulation amplitudes, extending the method to photocathode beams with significant uncorrelated energy spreads. The downside of such compressed seeding is the need to maintain bunching across an extended accelerator region. We present analytical estimates and computer simulations to study tolerances for a sample lattice. We also note that transportation of the fine compressed modulation structure is helped by error self-correction in the second chicane, an effect that may be of more general interest.

Figure 1

Diagram of CHG scheme. (a) The first accelerator section raises the particle energy to $E_a$ and introduces a linear chirp, $h$. After modulating with a laser, the first dispersive section, BC1, compresses the beam and overbunches the modulation. A second rf section accelerates the particles to $E_b$, and cancels the chirp of the first section. The final dispersive section, BC2, unwinds the overbunching. (b) Operating BC1 in overcompression rotates the electron beam head to tail, allowing the use of a chicane for both BC1 and BC2. In both cases it is possible to add a third accelerator section to reach a final energy of $E_f$.

Figure 2

In the first step (upper left) we add a chirp and laser modulation. We show only one modulation wavelength, so the chirp effect is small. After the first chicane, the modulation is overbunched (zoom, upper right). A second chirp reverses the first chirp (zoom, middle left, shows reversed slope in phase space). After more acceleration, the second chicane revives bunching (middle right). Note the compression of the wavelength by $|\alpha |=10$, and the increase in modulation amplitude by $|\alpha |/g=2.5$. We find strong bunching past the 5th harmonic (bottom).

Figure 3

Courant-Snyder parameters for 3D elegant simulations showing $\beta$ functions (solid and dotted lines) and dispersion (${\eta }_x$, dash-dotted line). The weak second chicane is designed to minimize ISR effects. Following the second chicane, a third linac section increases the electron energy while simultaneously decreasing the $\beta$ function, enhancing the radiation power.

Figure 4

Particle phase space for elegant simulation of parameters in Table . The two horizontal stripes of higher density are signatures of a sinusoidal modulation. A third-harmonic accelerating cavity produces the flat central portion of the beam.

Figure 5

A zoom of Fig. 4 shows the compressed modulation bunched to optimize higher harmonics. The vertical stripes (“standing up” the modulation) in phase space produce sharp density spikes that drive harmonic generation. The $T_{566}$ component of the transfer matrix causes slight scalloping.

Figure 6

Bunching factor at two wavelengths with 0.15% bandwidth from elegant [16] simulation (parameters in Table ). Bunching is lower than for the 1D case because of 2nd order effects (e.g. emittance and $T_{566}$), ISR and finite laser radius.

Figure 7

To demonstrate the emittance cancellation effect, we track a longitudinal delta slice (zero length) following the laser modulation. Emittance effects increase the slice length before the second chicane, BC2 (top left, $x$ vs $z$), but the off-crest accelerating section introduces a chirp to the beam (bottom left, energy vs $z$). The second chicane then recompresses the bunch by the compression factor, $|\alpha |$ (center). Following the final accelerator section, L3, the slice starts to spread out again (right). The bunch head is to the left.

Figure 8

Wide bandwidth bunching for two simulations with a 0.01° phase shift (solid and dotted blue lines). An FEL would pick out a narrow $\Delta \lambda /\lambda =0.15\%$ bandwidth, leaving lower bunching (narrow peak, red dot-dashed line). To calculate the bunching factor at a position $z_0$ and bandwidth $\Delta \lambda /\lambda$, we sum the phases of all particles in the region $z_0-\lambda /2\Delta \lambda <z<z_0+\lambda /2\Delta \lambda$. The high baseline in the wide bandwidth curves is due to the relatively low number of particles per bunching calculation when $\lambda /\Delta \lambda$ is small. A bunched sine wave makes a sawtooth that may either be right leaning, as in Fig. 5, or left leaning, depending on the sign of the effective dispersion, $R_{56}^{(T)}$. In our simulation, $R_{56}^{(T)}$ changes as a function of longitudinal beam position, resulting in the double peak seen in the wide bandwidth bunching.

Figure 9

Double-horn energy distribution, left, from a sinusoidal modulation. Approximating a sawtooth with two frequencies gives the more uniform energy distribution at right.