Modeling chromatic emittance growth in staged plasma wakefield acceleration to 1 TeV using nonlinear transfer matrices

A framework for integrating transfer matrices with particle-in-cell simulations is developed for TeV staging of plasma wakefield accelerators. Using nonlinear transfer matrices in terms up to ninth order in normalized energy spread $\sqrt{⟨\delta {\gamma }^2⟩}$ and deriving a compact expression for the chromatic emittance growth in terms of the nonlinear matrix, plasma wakefield accelerating stages simulated using the three-dimensional particle-in-cell framework osiris 4.0 were combined to model acceleration of an electron beam from 10 GeV to 1 TeV in 85 plasma stages of meter scale length with long density ramps and connected by simple focusing lenses. In this calculation, we find that for initial relative energy spreads below ${10}^{-3}$, energy-spread growth below ${10}^{-5}$ of the energy gain per stage and normalized emittance below mm-mrad, the chromatic emittance growth can be minimal. The technique developed here may be useful for plasma collider design, and potentially could be expanded to encompass nonlinear wake structures and include other degrees of freedom such as lepton spin.

Figure 1

Schematic showing $N$ stage plasma accelerator, with (chromatic) focusing lenses between plasma accelerating stages with $2f$ focusing throughout. $f_s$ is the plasma stage focal length and $f_{Ls}$ is the beam focusing optic focal length (at the design energy) of the $s$th accelerating stage. ${\mathrm{FPP}}_s$ and ${\mathrm{SPP}}_s$ are the primary and secondary principal planes of the accelerating stage, respectively.

Figure 2

Phase-space coordinates of 100 000 particles for an initial beam matrix are shown in blue. Red indicates particle coordinates having propagated through a transfer matrix and green shows the coordinates for the corresponding nonlinear matrix with randomly sampled $\delta \gamma$.

Figure 3

Normalized emittance growth through a nonlinear matrix calculated by summing over a large number of particle tracks (blue) compared the calculation using Eq. (36) (red), as a function of $\sqrt{⟨\delta {\gamma }^2⟩}$.

Figure 4

Fields taken from the 3D simulation at ${\omega }_{p0}t=9000$ in the $z-x$ plane at $y=0$. (left) Accelerating ($E_z$) field and (right) focusing ($E_x$) field.

Figure 5

(left) The axial accelerating field $E_z$ and (right) focusing gradient on an ultrarelativistic particle propagating in the $z$ direction, $(e/mc{\omega }_{p0})\partial (E_x-c{B}_y)/\partial x{|}_{x=0}$ both extracted as a lineout in the $z$ direction along the axis and plotted as a time series. The black dashed line indicates the wake phase chosen for the beam transport.

Figure 6

Representative particle track through fields taken from the 3D simulation. The design (normalized) energy is initially $\gamma =19500$, and it is accelerated to $\gamma =42891$, as indicated by the blue curve. The particle undergoes betatron oscillations indicated by the red and black dashed curves. The red curve shows the track for a beam $\delta \gamma$ from the design energy, with $\delta \gamma =1950$. The black and red circled dots at the end show the result of a calculation from the initial coordinates using the combined matrix $\mathcal{M}$. Colorscale is the accelerating field (see Fig. 5).

Figure 7

Relative emittance growth $\mathrm{\Delta }{\epsilon }_N/{\epsilon }_{N0}$ through 85 accelerating stages as a function of initial relative energy spread $\sqrt{⟨\delta {\gamma }^2⟩}/{\gamma }_0$ and initial normalized emittance ${\epsilon }_{N0}$. The colormap/contours show the base-10 logarithm of $\mathrm{\Delta }{\epsilon }_N/{\epsilon }_{N0}$.

Figure 8

Relative emittance growth $\mathrm{\Delta }{\epsilon }_N/{\epsilon }_{N0}$ through 85 accelerating stages as a function of constant relative energy spread $\sqrt{⟨\delta {\gamma }^2⟩}/{\gamma }_0$ and initial normalized emittance ${\epsilon }_{N0}$. The colormap/contours show the base-10 logarithm of $\mathrm{\Delta }{\epsilon }_N/{\epsilon }_{N0}$.

Figure 9

Betatron oscillations modeled using the expanded nonlinear transfer matrix in Eq. (b4). (a) Oscillations with an increasing $\gamma$ factor from ${\gamma }_0=100$ to ${\gamma }_0=500$ corresponding to linear acceleration. The dashed lines correspond to analytic (WKB) solutions for betatron oscillations for the design $\gamma ={\gamma }_0$ (blue) and $\gamma ={\gamma }_0+\delta \gamma$ (black), for a energy deviation $\delta \gamma =10$. The colored solid lines indicate solutions using the nonlinear matrix with different orders in $\delta =\delta \gamma /\gamma$ up to the ${\delta }^9$th term, i.e., $m=9$. (b) The error in the solution for oscillations with fixed $\gamma =100$, defined as $|x_{n{\delta }^m}-x{|}^2$, where $x$ is the analytic solution and $x_{n{\delta }^m}$ is the matrix solution including terms up to $m$. The black dashed lines show the thresholds ${|\frac{{\psi }_0}{2}\frac{\delta \gamma }{\gamma }|}^m/m!=1$ corresponding to Eq. (b5).

Figure 10

Demonstration of method accuracy; Betatron oscillations for particle linearly accelerated with design energy $\gamma =1000$ to $2\times {10}^6$ over a total time of $\omega t={10}^7$, where the betatron frequency is $\omega /\sqrt{\gamma }$, with a deviation from the design energy of $\delta \gamma =20$, i.e., corresponding to an initial relative energy spread $\delta \gamma /\gamma =2\%$. The main panel shows the (unresolvable) oscillations for a WKB solution compared with the nonlinear matrix solution including terms up to $m=9$ over the full range. The inset panels show expanded regions at the beginning, middle and end.