Design of general apochromatic drift-quadrupole beam lines

Chromatic errors are normally corrected using sextupoles in regions of large dispersion. In low emittance linear accelerators, use of sextupoles can be challenging. Apochromatic focusing is a lesser-known alternative approach, whereby chromatic errors of Twiss parameters are corrected without the use of sextupoles, and has consequently been subject to renewed interest in advanced linear accelerator research. Proof of principle designs were first established by Montague and Ruggiero and developed more recently by Balandin et al. We describe a general method for designing drift-quadrupole beam lines of arbitrary order in apochromatic correction, including analytic expressions for emittance growth and other merit functions. Worked examples are shown for plasma wakefield accelerator staging optics and for a simple final focus system.

Figure 1

Plot (a) shows width $w$ of three light beams vs the optical axis $s$ for a 3-color apochromat. Plot (b) shows transverse beam size $\sqrt{\beta }$ in $x$ and $y$ vs $s$ for a first-order apochromat. Both apochromats use the same principle: beams of different color/energy are focused differently through the system, but end up focused to the same point. However, the two examples are different in a subtle way: In plot (a) three distinct colors are focused to the same point, leaving intermediate colors slightly unfocused. This is how achromatic lenses for light optics are often designed. In contrast, in plot (b) the nominal energy is perfectly focused, and the focusing error is canceled to first order in $\delta$, leaving small-offset energies well focused. In this paper, we study arbitrary order apochromats for charged particle beams.

Figure 2

Plots of $\sqrt{\beta }$ (proportional to rms beam size) vs beam line axis $s$, and chromatic dependence of $\beta (\delta )$ and $\alpha (\delta )$ vs $\delta$, shown for different orders of apochromatic focusing. All solutions satisfy initial and final Twiss parameters $\beta =1\mathrm{m}$ and $\alpha =0$ in both planes, with a 1 m drift before and after the first and last quadrupoles respectively. The chromatic dependence of the lattice flattens progressively with higher orders of apochromatic focusing; No chromatic correction (a) results in chromatic amplitude $W\ne 0$ (a slope) at nominal energy $\delta =0$, whereas first order correction (b) removes this chromatic amplitude $W=0$ (no slope), and second order correction (c) flattens it further by removing second order chromatic errors (curvature) around $\delta =0$. Overall, the chromatic dependence can be decreased at the cost of longer lattices with more quadrupoles, where the appropriate order of the correction is determined by the energy spread of the beam.

Figure 3

Phase space plots with beam ellipses and a single tracked particle for several energy offsets $\delta$, after transport through the first order apochromatic lattice shown in Fig. 2. As the energy increases [$(\mathrm{a})\to (\mathrm{e})$], the particle experiences less phase advance ($\mathrm{\Delta }\mu$), indicating a nonzero chromaticity $\xi <0$ as defined by Eq. (2). However, since the distribution of particles retains the same shape around nominal energy [(b) and (d)], $\beta$ and $\alpha$ are unchanged to first order and therefore chromatic amplitude $W=0$. At larger energy offsets [(a) and (e)], the ellipse is distorted due to higher order errors.

Figure 4

Example A: PWFA staging optics, both using quadrupoles (a) and plasma lenses (b). Plots show $\sqrt{\beta }$ (proportional to rms beam size) vs beam line axis $s$, and chromatic dependence of $\alpha (\delta )$ and $\beta (\delta )$ vs offset $\delta$. Both solutions capture a 100 GeV beam exiting a plasma (with density ramps) matched to ${\beta }_0=32.5\mathrm{cm}$ and refocuses it back to 32.5 cm, with a 1 m drift space at the start and end for injection and extraction of drive beams. Solution (a) is a first-order apochromatic lattice using 8 quadrupoles with field gradient $160\mathrm{T}/\mathrm{m}$ are placed antisymmetrically (mirrored with polarity switched), whereas solution (b) is a third-order apochromatic lattice using 7 discharge capillary plasma lenses [18] with field gradient $3000\mathrm{T}/\mathrm{m}$ placed symmetrically. Transporting a beam with 1% rms energy spread leads to a projected emittance growth of 0.96% in lattice (a), and 0.000004% in lattice (b). Note the different $\delta$-scales in the two chromatic dependence plots.

Figure 5

Plot of relative projected emittance growth vs number of tracked particles, after elegant-tracking [19] a beam with 1% rms energy spread through the apochromatic lattice shown in Fig. 4. Approaching large particle numbers, the tracked emittance growth converges to the analytic estimate given by Eq. (18), with an expected statistical error ($\sim 1/\sqrt{N}$). Since the lattice is antisymmetric, the emittance growth is (very nearly) identical in both planes.

Figure 6

Example B: Final focus. A 166 m long first-order apochromatic lattice using 6 (thin) quadrupoles, focuses a beam of 0.5% rms energy spread with about 30% luminosity loss. The beam is focused from ${\beta }_{x0}=13.1\mathrm{m}$, ${\beta }_{y0}=15\mathrm{m}$ down to ${\beta }_y^{*}=16.3\mathrm{cm}$, ${\beta }_y^{*}=1.33\mathrm{cm}$, in total a $\beta$-demagnification of 80 in $x$ and 1125 in $y$. The $W$-function is intentionally increased, then transported and inverted, until it is finally canceled in the final doublet, which focuses the beam to the interaction point.