Cascaded modulator-chicane modules for optical manipulation of relativistic electron beams

A sequential arrangement of three pairs of modulators and dispersive sections that performs precise manipulation of a relativistic electron beam’s longitudinal phase space is described. We show that using only a single laser wavelength, this scheme acts as a waveform synthesizer through linearization of local regions of phase space to generate sawtooth, triangular, and square wave-type distributions. It also acts as an optical analog of an rf function generator to generate intense coherent radiation that has periodic triangle and square field profiles at the optical wavelength. The same setup can also be used to improve the high-harmonic bunching factors in echo-enabled harmonic generation schemes up to 25% and to produce a bunching factor above 90% at the laser fundamental wavelength for high-efficiency capture in inverse free electron laser acceleration applications.

Figure 1

Triple modulator and chicane layout. The timing of single laser in each section can be adjusted by adjustable phase delay. Variable quadrupoles between bends in the chicanes (yellow diamonds) permits reversal of the sign of the momentum compaction.

Figure 2

Waveform generation through harmonic modulation. Different waveforms are shown in independent rows. The left column is the modulation by one laser, the middle column is with two lasers, and the right column is with three lasers. Top: Sawtooth wave generation with modulations of wavelengths ${\lambda }_1$, ${\lambda }_2={\lambda }_1/2$, and ${\lambda }_3={\lambda }_1/3$. Here $A_1=10$, $A_2=A_1/2$, $A_3=A_1/3$, $B_1=B_2=0$, ${\varphi }_2={\varphi }_3=0$. Middle: Square wave generation with ${\lambda }_2={\lambda }_1/3$, ${\lambda }_3={\lambda }_1/5$, $A_1=10$, $A_2=A_1/3$, $A_3=A_1/5$, $B_1=B_2=0$, ${\varphi }_2={\varphi }_3=0$. Bottom: Triangle wave with ${\lambda }_2={\lambda }_1/3$, ${\lambda }_3={\lambda }_1/5$, $A_1=10$, $A_2=A_1/9$, $A_3=A_1/25$, $B_1=B_2=0$, ${\varphi }_2=\pi$, and ${\varphi }_3=0$.

Figure 3

Local linearization with a dispersive section between two modulations at the same wavelength, ${\lambda }_1$. The beam is first modulated with amplitude $A_1$ (a), and then longitudinally dispersed by $B_1=1/A_1$ (b). This distributes particles in the region $-{\lambda }_1/2<z_1<{\lambda }_1/2$ along a sinusoid with wavelength $2{\lambda }_1$ (dashed line). A subsequent smaller modulation $A_2$ shifted by $\pi$ (solid line), removes the curvature in the decompressed region (c).

Figure 4

Synthetic triangle or sawtooth wave generation with three modulations of wavelength ${\lambda }_1$. The initial modulation (top left) is weakly dispersed in the first chicane (top right). Compensation of the curvature takes place in the second modulator (middle left). Reversing the dispersion (middle right) then enables correction of the remaining portion in the third modulator (bottom left). The last dispersion section sets the final waveform shape (bottom right). Here $A_1=10$, $A_2=A_1/16$, $A_3=A_2$, $B_1=\pi /2A_1$, $B_2=-2B_1$, $B_3=-B_2/2$, ${\varphi }_2=\pi$, and ${\varphi }_3=\pi$.

Figure 5

Synthetic sawtooth wave generation with three modulations of wavelength ${\lambda }_1$. Here $A_1=10$, $A_2=3A_1/4$, $A_3=A_1$, $B_1=-1/2A_1$, $B_2=-2B_1$, $B_3=2B_1$, ${\varphi }_2={\varphi }_3=\pi$.

Figure 6

Synthetic square wave generation with three modulations of wavelength ${\lambda }_1$. Here $A_1=10$, $A_2=A_1/4$, $A_3=A_2/16$, $B_1=\sqrt{3}\pi /2A_1$, $B_2=-3B_1$, $B_3=-3B_2/4$, ${\varphi }_2=0$, and ${\varphi }_3=\pi$.

Figure 7

Three modulator generation of odd-harmonic bunching that goes like $b[(2h-1)k_1]=b(k_1)/(2h-1{)}^2$ for the emission of triangular radiation fields. Top: Modified phase space and current distribution with $A_1=10$, $A_2=1.934$, $A_3=0.287$, $B_1=0.013$, $B_2=-0.294$, $B_3=-0.233$, ${\varphi }_2={\varphi }_3=\pi$. Bottom: Bunching factors compared with $1/(2h-1{)}^2$ dependence of sawtooth amplitudes.

Figure 8

Two-modulator generation of odd-harmonic bunching with amplitudes $b[(2h-1)k_1]=b(k_1)/(2h-1)$ for the emission of square-wave fields. Top: Modified phase space and current distribution with $A_1=10$, $A_2=1.388$, $B_1=0.295$, $B_2=-0.551$, ${\varphi }_2={\varphi }_3=0$. Bottom: Bunching factors compared with $1/(2h-1)$ dependence of square-wave amplitudes.

Figure 9

i-EEHG compensation in third modulator.

Figure 10

I-EEHG compensation (red solid line) in third modulator for (a) 50th and (b) 150th harmonic. Standard EEHG is shown as a dashed line. Here $A_1=A_1^0=2$, $A_2=A_2^0=10$, $A_3=A_2/16$, ${\varphi }_2=0$, ${\varphi }_3=\pi$, $B_2\simeq -1/A_2$, $B_3\simeq B_2^0-B_2$.

Figure 11

Bunching factor tolerances of i-EEHG in third modulator for 50th harmonic (top) and 150th harmonic (bottom) using parameters given in Fig. 10.

Figure 12

Adiabatic buncher with three modulations of wavelength ${\lambda }_1$. Here $A_1=3$, $A_2=3A_1$, $A_3=3A_2$, $B_1=1.8/A_1$, $B_2=2.2/A_2$, $B_3=1.55/A_3$, ${\varphi }_2={\varphi }_3=0$.

Figure 13

Current (left) and harmonic bunching factors (right) in the adiabatic bunching scheme.

Figure 14

Tolerance of adiabatic buncher to phase jitter in second and third lasers.