Joule heating in a flat dechirper

We perform Joule power loss calculations for a flat dechirper. We consider the configurations of the beam on-axis between the two plates—for chirp control—and for the beam especially close to one plate—for use as a fast kicker. Our calculations use a surface impedance approach, one that is valid when corrugation parameters are small compared to aperture (the perturbative parameter regime). We find that most of the wake power lost by the beam is radiated out to the sides of the plates. For the case of the beam passing by a single plate, we derive an analytical expression for the broadband impedance. Our analytical results are also tested by numerical, time-domain simulations. While our theory can be applied to the LCLS-II dechirper with large gaps, for the nominal apertures we are not in the perturbative regime. With input from computer simulations, we estimate the Joule power loss for nominal apertures (assuming bunch charge of 300 pC, repetition rate of 100 kHz) is $21\mathrm{W}/\mathrm{m}$ for the case of two plates, and $24\mathrm{W}/\mathrm{m}$ for the case of a single plate.

Figure 1

Three corrugations of a vertical dechirper. A rectangular coordinate system is centered on the symmetry axis of the chamber. The blue ellipse represents an electron beam propagating along the $z$ axis.

Figure 2

Sketch of a vertical set of dechirper jaws seen end on: (a) in the real geometry the jaws have a finite width $w$; (b) in the model used in the calculation, the width is infinite, but the loss integration is performed only over a width $w$ (represented by the distance between the dashed lines). The purple lines represent the corrugated surfaces; the blue ellipses, the exciting particle’s transverse location.

Figure 3

The real part of the longitudinal impedance for the dechirper with plate width $w\to \infty$. This plot, in different units, and with the spike reaching to infinity, can also be found in Ref. [7].

Figure 4

The real part of the Joule heating impedance $\mathrm{Re}(Z_h)=\mathrm{Re}(Z_{hz})+\mathrm{Re}(Z_{hx})$ for the beam on axis in the dechirper with half aperture $a=0.7\mathrm{mm}$ and plate width $w=12\mathrm{mm}$.

Figure 5

Numerically calculated ratio of Joule loss into metal to wake loss of a point charge beam, $u_h/u_w=(u_{hz}+u_{hx})/u_w$ as a function of $w/(2a)$ (plotting symbols). The wake loss for a point particle is $u_w=Q^2{\pi }^2/(8a^2)$.

Figure 6

$\mathrm{Re}(Z)$ for beam passing by one, infinitely wide dechirper plate at a distance $b=0.25\mathrm{mm}$ (blue). The analytical results, Eq. (29), are also shown (red dashes).

Figure 7

The real part of the Joule heating impedance $\mathrm{Re}(Z_h)=\mathrm{Re}(Z_{hz})+\mathrm{Re}(Z_{hx})$ for the beam passing by a single dechirper plate, for the case of plate width $w\to \infty$ (blue curve), with the constituent parts given in dashes. The beam passes by at offset $b=0.25\mathrm{mm}$ from the plate. The area under $\mathrm{Re}(Z_{hz})$ [$\mathrm{Re}(Z_{hx})$] is 42% [58%] that under $\mathrm{Re}(Z)$.

Figure 8

The real part of the Joule heating impedance $\mathrm{Re}(Z_h)=\mathrm{Re}(Z_{hz})+\mathrm{Re}(Z_{hx})$ for the beam passing by a single dechirper plate, for the case of plate width $w=12\mathrm{mm}$ (blue curve), with the constituent parts given in dashes. The beam passes by at offset $b=0.25\mathrm{mm}$ from the plate. The area under $\mathrm{Re}(Z_{hz})$ [$\mathrm{Re}(Z_{hx})$] is 0.9% [2.4%] that under $\mathrm{Re}(Z)$, the beam impedance curve given in Fig. 6.

Figure 9

Model used in the single-plate, numerical (CST Studio) calculations. The line and symbols indicate the beam trajectory. The plate width is $w=12\mathrm{mm}$, and, nominally, the beam offset is $b=0.25\mathrm{mm}$.

Figure 10

The numerically obtained, longitudinal wake for the beam moving on axis of a double plate dechirper. Here half aperture $a=0.7\mathrm{mm}$ and dechirper length $L=135\mathrm{mm}$; the driving bunch is Gaussian with ${\sigma }_z=100\mu \mathrm{m}$. Note that the wake is normalized to structure length.

Figure 11

The real part of the longitudinal impedance for the beam moving on axis of a double plate dechirper with half aperture $a=0.7\mathrm{mm}$, obtained by taking the Fourier transform of the numerically obtained wake of Fig. 10.

Figure 12

Beam on axis between two plates: Joule power calculations obtained by time-domain simulation, for plate half aperture $a=0.7\mathrm{mm}$ (blue) and $a=1.4\mathrm{mm}$ (blue). The beam traverses the plate from its beginning, at $z=0$, to its end, at $z=0.135\mathrm{m}$. The dashed lines give extrapolation to steady state. Here $Q=300\mathrm{pC}$, $f_{\mathrm{rep}}=300\mathrm{kHz}$; the driving charge is Gaussian with ${\sigma }_z=100\mu \mathrm{m}$.

Figure 13

The numerically obtained, longitudinal wake for a beam moving past a single-plate dechirper. Here the beam offset $b=0.25\mathrm{mm}$ and dechirper length $L=115\mathrm{mm}$; the driving bunch is Gaussian with ${\sigma }_z=100\mu \mathrm{m}$. Note that the wake is normalized to structure length.

Figure 14

The real value of the impedance $\mathrm{Re}(Z)$ for the single-plate example (blue curve). The broadband impedance obtained from the same wake is given by green dashes. The analytical perturbation result (Fig. 6) is given by red dashes.

Figure 15

Single plate: numerically obtained, Joule power calculations for beam offsets $b=0.25\mathrm{mm}$ (blue) and $b=0.5\mathrm{mm}$ (red). The beam traverses the plate from its beginning, at $z=0$, to its end, at $z=0.115\mathrm{m}$. The dashed lines give extrapolation to steady state. Here $Q=300\mathrm{pC}$, $f_{\mathrm{rep}}=300\mathrm{kHz}$; the driving charge is Gaussian with ${\sigma }_z=100\mu \mathrm{m}$.

Figure 16

Integration path $C$ in the complex plane of variable $q$.