Efficient computation of coherent synchrotron radiation in a rectangular chamber

We study coherent synchrotron radiation (CSR) in a perfectly conducting vacuum chamber of rectangular cross section, in a formalism allowing an arbitrary sequence of bends and straight sections. We apply the paraxial method in the frequency domain, with a Fourier development in the vertical coordinate but with no other mode expansions. A line charge source is handled numerically by a new method that rids the equations of singularities through a change of dependent variable. The resulting algorithm is fast compared to earlier methods, works for short bunches with complicated structure, and yields all six field components at any space-time point. As an example we compute the tangential magnetic field at the walls. From that one can make a perturbative treatment of the Poynting flux to estimate the energy deposited in resistive walls. The calculation was motivated by a design issue for LCLS-II, the question of how much wall heating from CSR occurs in the last bend of a bunch compressor and the following straight section. Working with a realistic longitudinal bunch form of r.m.s. length $10.4\mu \mathrm{m}$ and a charge of 100 pC we conclude that the radiated power is quite small (28 W at a 1 MHz repetition rate), and all radiated energy is absorbed in the walls within 7 m along the straight section.

Figure 1

The simulated form of the LCLS-II bunch at the end of the second bunch compressor. It has r.m.s. length ${\sigma }_z=10.34\mu \mathrm{m}$ and zero mean. This is a smoothed version of a histogram with 100 bins.

Figure 2

Fourier transform $\widehat{\lambda }(k)$ of the simulated form of the LCLS-II bunch at the end of the second bunch compressor, compared to that of a Gaussian with the same ${\sigma }_z$.

Figure 3

The first four modes of the initial field ${\widehat{E}}_{yp}(x)$ and the sum over all modes, at $y=g$. The factor $\widehat{\lambda }(k)$ is omitted. $p=1$ (blue); $p=3$ (brown); $p=5$ (red); $p=7$ (magenta); Sum on p (black).

Figure 4

Left: the first four modes of the initial field $Z_0{\widehat{H}}_{yp}(x)$. The factor $\widehat{\lambda }(k)$ is omitted. $p=1$ (blue); $p=3$ (brown); $p=5$ (red); $p=7$ (magenta). Right: sum over all $p$ of $Z_0{\widehat{H}}_{yp}(x)$.

Figure 5

The sum to $p=9$ of ${\widehat{E}}_{yp}(k,s,x)$ at $y=g$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 6

The decomposition $\mathrm{Re}{\widehat{E}}_{yp}={\xi }_E+\mathrm{Re}{u}_E$ of Eq. (54) for the case of Fig. 5, on two different scales. The black curve is $\mathrm{Re}{u}_E$, the orange curve is ${\xi }_E$, and the blue curve is $\mathrm{Re}{\widehat{E}}_{yp}$ (equal to $\mathrm{Re}{u}_E$ for $x<0$).

Figure 7

The sum to $p=9$ of ${\widehat{E}}_{yp}(k,s,x)$ (real part) at $y=g$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=2\times 10^6$.

Figure 8

The sum to $p=9$ of $Z_0{\widehat{H}}_{yp}(k,s,x)$ at $y=0$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 9

The sum to $p=9$ of ${\widehat{E}}_{sp}(k,s,x)$ at $y=0$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 10

The sum to $p=9$ of $Z_0{\widehat{H}}_{xp}(k,s,x)$ at $y=g$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 11

The sum to $p=9$ of $Z_0{\widehat{H}}_{sp}(k,s,x)$ at $y=g$, at the end of the bend ($s=0.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 12

The sum to $p=9$ of ${\widehat{E}}_{sp}(k,s,x)$ at $y=0$, at 1 m into the straight after the bend ($s=1.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 13

The sum to $p=9$ of ${\widehat{E}}_{sp}(k,s,x)$ at $y=0$, at 5 m into the straight after the bend ($s=5.55\mathrm{m}$) for $kR=5\times 10^5$.

Figure 14

${\widehat{E}}_{sp}(k,s,x)$ vs. $x$ for $s=0.55\mathrm{m}$ at the end of the bend, $p=9$, comparing values of $kR$ around the shielding cutoff at $kR=2.92\times 10^5$.

Figure 15

The total energy radiated in perfectly conducting model (blue); energy absorbed in perturbative model (red); energy absorbed in horizontal walls, perturbative model (magenta); energy absorbed in vertical walls, perturbative model (brown). The beginning of the bend is at $s=0$, where the fields have the steady-state values for an infinite straight waveguide.

Figure 16

Contour plots of $H_x(s,x,y,t)$ in the $(x,y)$-plane at the instant the bunch is at location $s$. Plots in the upper $1/4$ of the chamber only, in arbitrary units.

Figure 17

Surface plots of $H_x(s,x,y,t)$ as a function of $(x,y)$ at the instant the bunch is at location $s$. Plots in the upper $1/4$ of the chamber only, in arbitrary units. The $(x,y)$-domain is the same as that in Fig. 16.

Figure 18

Time-integrated energy flow to the upper horizontal surface, per unit area, as a function of $x$ at various $s$.

Figure 19

Wakefield $W(z,s)$ as a function of $z=s-\beta ct$ at $s=0.55\mathrm{m}$ (end of bend). Graph on left is for the simulated bunch form of Fig. 1, that on right for a Gaussian bunch with the same ${\sigma }_z$.

Figure 20

Fourier transforms of the two wakefields of Fig. 19.

Figure 21

Wakefield $W(z,s)$ as a function of $z=s-\beta ct$ at $s=1.2375\mathrm{m}$ (left) and at $s=1.65\mathrm{m}$ (right).

Figure 22

Evolution of the wakefield $W(z,s)$ of a Gaussian bunch versus $s$ in a bend of length $s_b=0.825\mathrm{m}$.

Figure 23

Wakefield (blue) to compare with Fig. 3(b) of [4] (red).

Figure 24

The ratio $r_E=\parallel {\partial }^2{\widehat{E}}_{yp}/\partial s^2\parallel /2k\parallel \partial {\widehat{E}}_{yp}/\partial s\parallel$ which should be small compared to 1 to justify the SVA approximation.

Figure 25

$Z_0|{\widehat{H}}_x(k,s_b,0,g)|$ as a function of $k$ (left) and $Z_0{\widehat{H}}_x(k,s_b,x,g)$ as a function of $x$ at $k{\sigma }_z=8$ (right).