Six-dimensional phase space preservation in a terahertz-driven multistage dielectric-lined rectangular waveguide accelerator

Staged acceleration, driven by terahertz (THz) frequency radiation pulses in a lattice with alternating orientation dielectric-lined waveguides and intervening matching optics, is shown to mitigate transverse emittance and energy spread growth, opening a route to multistage THz linacs. Decomposition of the longitudinal THz field into the multipolar components reveals a quadrupole field component with strong radial dependence. As such, it induces a transverse energy correlation in the beam during acceleration due to the large variation in the electric field with radius and azimuthal position of the electrons. An alternating orientation of stages separated by a matching section provides a compensation of transverse energy spread correlation induced in the beam during its interaction with the THz field. Furthermore, the monopolar component of the accelerating ${\mathrm{LSM}}_{11}$ mode was found to be constant with respect to transverse position, entailing zero monopolar transverse voltage and preventing emittance growth, unlike conventional radio-frequency structures. We demonstrate in a rectangular dielectric-lined waveguide structure that, when used for the acceleration of relativistic electrons, the slice transverse emittance is conserved and the growth in the slice energy spread is reduced by 70%–80% simultaneously over a system of two stages, each providing an interaction length of 4 mm and an energy gain of up to 2 MeV.

Figure 1

Schematic diagram of the rectangular dielectric-lined waveguide (DLW) structure and the alternating orientation of two waveguides used for staged acceleration with separation $d$. The DLWs were designed for synchronism to relativistic electrons therefore, operating at 0.465 THz with $a=250\mu \mathrm{m}$, $b=300\mu \mathrm{m}$ and $w=1.2\mathrm{mm}$. The permittivity of the vacuum and dielectric slab are denoted by ${\epsilon }_0$ and ${\epsilon }_r$, respectively.

Figure 2

Dispersion plot for ${\mathrm{LSM}}_{11}$ mode synchronous to a relativistic electron beam ($v_p\approx c$).

Figure 3

Pure monopole, quadrupole and octupole components of the longitudinal potential and their sum in (a) for 0.465 THz and the geometry given in Fig. 1. (b) The corresponding transverse voltage for quadrupole and octupole components according to the Panofsky-Wenzel theorem for an amplitude of $100\mathrm{MV}/\mathrm{m}$. The radially constant longitudinal monopole field component yields a zero transverse monopolar voltage.

Figure 4

(a) The rms beam energy as a function of transverse coordinates in a rectangular waveguide correlated with the transverse dependency of quadrupole component of $E_z$. Results are for a slice electron beam with $100\mu \mathrm{m}$ radius resulting in a $7\mathrm{MV}/\mathrm{m}$ rms field deviation. (b) The energy distribution for the same beam after propagating through the orthogonal multistage arrangement. In the latter, the correlated rms spread in the field is reduced down to $0.5\mathrm{MV}/\mathrm{m}$.

Figure 5

The spatial profile of the THz field amplitude due to finite bandwidth as it copropagates synchronously with relativistic electrons.

Figure 6

(a) The flat initial energy surface around 45 MeV without uncorrelated energy spread. (b) The transverse energy distribution after interaction with the THz field correlated with the dominant quadrupole component.

Figure 7

Slice energy spread strongly correlated to the quadrupole component of $E_Z$ for an initially 45 MeV beam. (a) Propagated through a single DLW, (b) two DLW stages separated with a 10 mm drift section without matching and (c) after the drift length is removed and matching algorithm employed. Results are for 400 keV energy gain per stage.

Figure 8

(a) Beam envelope evolution through an orthogonal two-stage system with and without a pseudoinverse matching between stages. (b) The energy spread profile across a multistaged system. Entrance and exit of DLWs are marked with vertical dashed lines where the entrance of the first DLW is located at zero.

Figure 9

The transversely correlated energy spread as a function of incoming rms beam size at the entrance of the first DLW. Results are presented at the exit of ${\mathrm{DLW}}_1$ and at the exit of ${\mathrm{DLW}}_2$ for a matched and an unmatched beam. No drift section is introduced between the stages to monitor the pure response of staging without the increase in beam size due to a drift section.

Figure 10

Elements of nonperiodic transfer matrix over a range of phase advances for a beam with an rms radius of $24\mu \mathrm{m}$ and an rms angle of 4 mrad focused at the entrance of ${\mathrm{DLW}}_1$. Horizontal dashed lines indicate the values predicted by the pseudoinverse matching.

Figure 11

The energy spread as a function of the most sensitive transfer matrix element ${\mathrm{R}}_{12}$. Data points from Table 1 are interpolated to determine the ${\mathrm{R}}_{12}$ value yielding the minimum energy spread of 0.31 keV at the end of ${\mathrm{DLW}}_2$.

Figure 12

(a) The beam envelope focused at the entrance of ${\mathrm{DLW}}_1$ and across ${\mathrm{DLW}}_2$ after the matching section resulting from each solution presented in Table 1. (b) The increase of energy spread due to the correlation with the longitudinal quadrupole component of the THz field in ${\mathrm{DLW}}_1$ and recovery of this correlated energy spread in ${\mathrm{DLW}}_2$ corresponding to a different solution for the matching section.

Figure 13

Evolution of energy spread as a function of (a) and (b) focusing hence $\alpha$, and (c) and (d) $\beta$ for an initial emittance of 0.1 mm mrad and 400 keV energy gain per DLW. The vertical columns (a) and (c) present results for rms bunch lengths of 1 and 50 fs, respectively (reading left to right for the bunch lengths). The results are generated under the optimized physical matching matrix described in Sec. 3a.

Figure 14

Evolution of transverse beam emittance for (a) a slice (1 fs) and (b) a 50 fs long bunch as a function of focusing conditions imposed while the beam is incident at the first structure entrance.

Figure 15

Scaling of single-stage and multistage performance with energy gain per stage. (a) Slice energy spread and emittance, (b) projected energy spread and emittance for a 50 fs beam ($\alpha =40$). $\delta E_0$ and ${\epsilon }_0$ are initial energy spread and emittance at injection.

Figure 16

(a) Transverse voltage regarding the longitudinal monopole component of the THz field for synchronous frequency, sidebands and their sum. (b) Effect of the total transverse voltage due to bandwidth on energy spread and emittance in a single structure. A beam with a geometric emittance of 0.2 mm mrad was tracked for both single frequency and broadband cases for an energy gain of 400 keV.

Figure 17

(a) The 30 mm long rectangular structure with the 23 mm coupler section used in a cst benchmarking study. (b) Longitudinal electric field excited in the structure. The beam travels from right to left.

Figure 18

Temporal profile of 0.465 THz excitation signal with a bandwidth of 100 GHz.

Figure 19

A transverse central cross section of an accelerating phase of $E_z$ in cst that is dominated by the quadrupole component. The saddle point is denoted with intercepting axial lines.

Figure 20

The beam energy distribution in the $x\text{−}y$ plane after 4 mm copropagation with the accelerating polarity of the THz field (a) using multipole based tracking and (b) as a result of the cst PIC solver. The particle distributions with rms radius of $10\mu \mathrm{m}$ and slice length of 25 fs in the transverse plane are denoted with gray dots and the initial rms total energy is 45 MeV for both cases. The cst PIC solver presents only the kinetic energy. The transverse phase space distributions are presented in (c).