Numerical computation of high-order transfer maps for rf cavities

Modern map-based accelerator beam-dynamics codes model magnetic elements so as to include nonlinear effects and realistic fringe fields, but they persist in modeling rf cavities as either energy kicks or linear maps. This work presents a method for including the nonlinear effects of rf cavities in a map-based code.

Figure 1

Simple five-cell rf cavity with bore radius $a$, gap length $g$, and cell length $L$.

Figure 2

On-axis longitudinal field for a simple five-cell cavity, as given by (35) with $n=0$, $L=0.2125\mathrm{m}$, $g=L/2$, $a=0.085\mathrm{m}$, and $E_g=1\mathrm{V}/\mathrm{m}$.

Figure 3

First six derivatives (left-to-right, then top-to-bottom) of the on-axis longitudinal field shown in Fig. 2, computed using (35).

Figure 4

Discrete longitudinal field values $E_z(\rho ,z)$, as a function of $z$, at radius $\rho =0.075\mathrm{m}\approx 0.88a$.

Figure 5

Comparison of on-axis longitudinal field values computed analytically (blue) and from data with 10% relative noise (red). The latter curve is just barely visible above the leftmost peak.

Figure 6

Comparison of the first six derivatives (left-to-right, then top-to-bottom) of the on-axis longitudinal field computed analytically (blue) and from data with 10% relative noise (red).

Figure 7

Semilog plot of the relative errors ${\epsilon }_z^{(n)}(z)$, see (37), as functions of $z$ for $n=0$ (black), 1 (red), 2 (orange), 3 (green), 4 (light blue), 5 (dark blue), and 6 (violet). Here the noisy values ${\widehat{E}}_z^{(n)}$ derive from surface field values with 10% relative noise.

Figure 8

Semilog plot of the absolute difference, as a function of $z$, between the analytic computation of the longitudinal electric field, (20c) with $m=0$ and (34) with all cavity parameters as in Fig. 2, and an approximate computation, the first three terms of (29b). Results are shown for various fractions of the bore radius $a$: $\rho =0$ (black), $0.1a$ (red), $0.2a$ (orange), $0.3a$ (green), $0.4a$ (light blue), $0.5a$ (dark blue), and $0.6a$ (violet).

Figure 9

Semilog plot of the relative difference $|\Delta E_{\rho }|/\max {}_z|E_{\rho }|$, as a function of $z$, between the analytic computation of the radial electric field, (20a) with $m=0$ and (34), and an approximate computation, the first three terms of (29a). Results are shown for various fractions of the bore radius $a$: $\rho =0.1a$ (red), $0.2a$ (orange), $0.3a$ (green), $0.4a$ (light blue), $0.5a$ (blue), and $0.6a$ (violet).

Figure 10

Same as Fig. 8, except that here the surface field data used for the approximate computation includes 0.1% relative noise.

Figure 11

Same as Fig. 9, except that here the surface field data used for the approximate computation includes 0.1% relative noise.

Figure 12

Semilog plot of the absolute difference, as a function of $z$, between the SuperFish values of the longitudinal electric field with a peak of about $1.4\mathrm{V}/\mathrm{m}$, and an approximate computation, the first three terms of (29b). Results are shown for radii $\rho =0$ (black), $1\mathrm{cm}$ (red), $2\mathrm{cm}$ (orange), $3\mathrm{cm}$ (green), $4\mathrm{cm}$ (light blue), $5\mathrm{cm}$ (dark blue), and $6\mathrm{cm}$ (violet).

Figure 13

Semilog plot of the relative difference $|\Delta E_{\rho }|/\max {}_z|E_{\rho }|$, as a function of $z$, between the SuperFish values of the radial electric field, and an approximate computation, the first three terms of (29a). Results are shown for radii $\rho =1\mathrm{cm}$ (red), $2\mathrm{cm}$ (orange), $3\mathrm{cm}$ (green), $4\mathrm{cm}$ (light blue), $5\mathrm{cm}$ (dark blue), and $6\mathrm{cm}$ (violet).

Figure 14

On-axis longitudinal field for a SuperFish cavity, computed using (22, 30) with $m=0$.

Figure 15

First six derivatives (left-to-right, then top-to-bottom) of the on-axis longitudinal field of the SuperFish cavity.