Estimate of Joule Heating in a Flat Dechirper

We have performed Joule power loss calculations for a flat dechirper. We have considered 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). In our model we ignore effects of field reflections at the sides of the dechirper plates, and thus expect the results to underestimate the Joule losses. The analytical results were also tested by numerical, time-domain simulations. 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 broad-band impedance, and---in Appendix B---numerically confirm recently developed, analytical formulas for the short-range wakes. 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 and the reflection contribution to Joule losses is not negligible. With input from computer simulations, we estimate the Joule power loss (assuming bunch charge of 300 pC, repetition rate of 100 kHz) is 21~W/m for the case of two plates, and 24 W/m for the case of a single plate.


INTRODUCTION
A corrugated structure device-a so-called dechirper [1]-is being proposed for installation after the end of the linac and before the undulator regions of LCLS-II.
Such a device has been installed in LCLS-I [2], where it has been used for energy chirp control and as a fast kicker for self-seeding and two color operation of the FEL [3]. Because of the high repetition rate in LCLS-II compared to LCLS-I, Joule heating of the device by the beam's wakefields may now become significant and a cooling system may be required. Thus, it is important to estimate the amount of Joule heating power that is deposited in the corrugated plates.
In previous work, the surface impedance approach to obtaining the impedances and wakes of a short, high energy bunch in a flat dechirper was derived [4][5][6].
The resulting formulas were shown to be approximate and valid in the perturbative regime, i.e. when h/a 1, with h the corrugation depth and a the half aperture of the dechirper. In this regime (assuming also that h/p 1, with p the corrugation period) the impedance can be described by a single mode and the longitudinal wake by a damped cosine function [7]. The RadiaBeam/SLAC dechirper at nominal parameters (h = 0.5 mm and a = 0.7 mm, thus h/a = 0.7), however, is not in the perturbative regime. In this non-perturbative regime, it has been shown that the impedance consists of more than one mode [8] and the wake begins with a droop [9].
However, even for this regime analytical fitting functions for the short-range wakes have been derived and also verified by numerical simulation [10].
In previous work on the dechirper, the effect of the corrugations alone was considered, and the effect of the resistance of the boundary metal was ignored. For Joule power loss calculations, however, one needs to include both contributions.
Also, in most of the previous work on the impedance of a flat dechirper, the case of the beam passing between two corrugated plates was considered. However, when used as a fast kicker, with the beam passing close to one plate, the second plate no longer influences the beam nor needs to be in the problem. Note however, that analytical fitting formulas for the short-range wakes for this case have also been recently derived [11], though without numerical verification.
In the present report, by applying the surface impedance approach to both double and single plate dechirpers, we obtain analytical estimates of the Joule heating power. The surface impedance approach is used where it is not quite applicable (h/a = 0.7 and is not small), introducing an error. In addition, our estimates ignore the effect of reflections from the side edges of the dechirper plates. We thus expect our results to underestimate the Joule power losses. Finally, we test the accuracy of these calculations, by performing numerical simulations using the time-domain Maxwell equation solver in CST Particle Studio (PS) [12], and also (for verification purposes) with the program PBCI [13]. These calculations are themselves quite challenging, since a fine mesh is needed to resolve the short bunch, and runs need to be performed over a large mesh domain for a relatively long time. In Appendix B we perform numerical simulations to test the accuracy of the single plate, short-range wake formulas derived in [11].
The RadiaBeam/SLAC dechirper that is installed in LCLS-I consists of two modules. Each module has two corrugated plates, with the beam passing between them (see Fig. 1). Two modules were chosen in order to partially cancel the unavoidable quadrupole wakefield that is induced by the beam; one has plates parallel to the x-z axis (horizontal-longitudinal) plane and is adjustable vertically (the "vertical dechirper"), and the complimentary one is adjustable horizontally (the "horizontal dechirper"). For LCLS-II, the corrugation parameters are period p = 0.5 mm,  The equations in this report are given in cgs units. To convert an impedance or wake into MKS units, one multiplies the cgs result by Z 0 c/(4π), with Z 0 = 377 Ω.

JOULE HEATING ESTIMATES
In a round corrugated structure of a finite length (a dechirper), the wakefield energy loss experienced by a relativistic beam of charged particles is partly absorbed in the walls as Joule heating and partly generates a THz pulse that leaves the structure just behind the driving particle. In the flat geometry of dechirpers like those that have actually been built-like the RadiaBeam/LCLS dechirper-some of that energy can also escape through the aperture to the side. The energy per unit length lost by the beam to the wake is then given by the sum with u h the energy generating Joule heating in the metal walls, (u rad ) z the energy in the THz pulse that leaves the end of the structure following the driving particle, and (u rad ) x the energy radiating out the sides of the structure.
A particle of charge Q moves at the speed of light c on the axis of a structure.
The Joule energy loss into the walls per unit length is given by with S the Poynting vector, dA the incremental surface area vector (into the wall is positive), and B represents the metallic boundary. The calculation is performed at time t = 0 when the particle is at z = 0, and the transverse coordinate is r. The particle is assumed to be moving to the left; the fields are zero ahead of the particle (for z < 0).
Let us begin by sketching how we would solve the case of a dechirper with round geometry. The walls are located at radius r = a, and the Poynting vector at the walls is given by S = −( c 4π )E z H φ , with E z (r, z) the longitudinal component of the electric field and H φ (r, z) the azimuthal component of the magnetic field. The Joule energy loss into the walls becomes The fields can be written in terms of their Fourier transforms; for example, for the electric field where ω is the frequency and a tilde indicates the Fourier transform of a field (and similarly for H φ ). Substituting into Eq. 3 and changing the order of integrations we The last integral on the right equals 2πcδ(ω + ω ). Thus, we obtain where in the last integral * indicates the complex conjugate of a function. To obtain the last integral we used the relationH φ (−ω) =H φ (ω) * ; such a relation holds for the fields in the frequency domain, since the same fields in the time domain must be real quantities.
On the metallic surface we have the relatioñ with ζ z (ω) the surface impedance of the structure walls, which includes both the contributions of the corrugations and the wall resistance (the subscript z indicates that the surface currents move in the longitudinal direction). Substituting into Eq. 6, and noting the symmetry of the integrand, we finally obtain If we write the Joule loss as u h = ∞ −∞ dũ dω dω, then we can define an effective Joule heating impedance Z h (ω), which has a real part defined as In this round example it turns out that all the wake losses end up in the walls, since they have nowhere else to go. It is easy to see that Z h (ω) = Z(ω), with Z(ω) the normally defined impedance of the corrugated structure, given by Z(ω) = −Ẽ z /Q.
For a bunch of particles, the Joule heating energy is given by withĨ(ω) the Fourier transform of the current, I(z) = Qcλ(z), and λ(z) the longitudinal bunch distribution. However, in following energy and power calculations we will let |Ĩ(ω)| 2 → 1, since for the small bunch lengths considered and the frequency reach of the dechirper impedance, this approximation is good.
The Joule power loss is simply given by P = u h f rep , with f rep the bunch repetition rate. However, we need to make one more adjustment. In the round case, since all the beam energy loss becomes Joule heating, we obtain u h = 2Q 2 /a 2 . However, our Joule energy loss calculations are perturbative calculations. Since in our calculations the corrugation parameter ratio (h/a) is not small we are not in the perturbative regime and there will be wake droop. To better estimate the Joule power loss we take with κ(σ z ) the loss factor of a Gaussian bunch of length σ z . The point charge loss factor κ(0) = 2/a 2 (with a the pipe radius) in the round case, and κ(0) = π 2 /(8a 2 ) (with a the half aperture) in the flat case. The loss factor κ(σ z ) includes the effect of the wake droop.  Table I we see that this is not true, so we will lose accuracy here, too. The final contributor to inaccuracy is that the perturbation calculation assumes that the structure is infinitely long, which does not allow for part of the wake loss to contribute to the generation of a THz pulse in the z direction. However, this final contribution can be shown to be small for the parameters of Table I.
Let us consider a vertical dechirper with plate walls of width w located at y = ±a with respect to the axis. The Poynting vector at the walls Note that in this case, in addition to a surface current in the z direction, corresponding to the surface impedance ζ z , there is a surface impedance in the x direction, corresponding to surface impedance ζ x . The surface impedance in the z direction is given by the sum of the resistive wall and the corrugation impedance contributions, ζ z = ζ rw + ζ corr , with [16], [4], and σ c is the conductivity of the metal walls. In the x direction, we take the surface impedance to be ζ x = ζ rw , since the horizontal surface currents are not impeded by the corrugations. Note that an anisotropic surface impedance was not used before for modeling the impedance of the corrugated structure; this form, however, is important in our application here. Note also that with only corrugations and no wall resistance, the Joule heating is zero.
The Joule wall energy loss per unit length becomes (an overall factor of 2 is added because there are two plates). There are two contributions. Let us take u h = u hz + u hx , with u hz the part that depends on E z H x at the walls; u hx the part that depends on E x H z at the walls. Following a calculation similar to that for the round case above, we can rewrite the equation for u z in the frequency domain as To perform the calculation, we follow the procedure described in Ref. [5], which explicitly gives the fields and wakefields in structures with flat geometry for which the effect at the boundaries can be approximated by a surface impedance. First the frequency representation of the fields are Fourier transformed in x as: Substituting into Eq. 15, changing the order of integration, and noting the symmetry of the integrand with respect to ω, we find that It is this triple integral that we solve numerically to estimate the fraction of wakefield losses that end up as Joule heating in the walls. Note that for the special case width w → ∞, the equation simplifies to (we have used the fact that the integrand is symmetric with respect to q).
Performing a similar calculation for u x , we obtain, for finite w, and for infinite w The general form of the impedances for a flat structure, one that can be described using the surface impedance concept, is developed in Refs. [5,6]. A conclusion of this work was that the final results-the impedances-were approximate and valid, provided that the frequency k = ω/c 1/a-which is satisfied-and that |ζ| 1 (or h/a 1), which is not. The expressions forĤ x (q, a, ω),Ĥ z (q, a, ω), that we need here were not given in the earlier reports. Following the correct derivation of the fields, we obtained the general form ofĤ x (q, a, ω) andĤ z (q, a, ω) as functions of beam offset (not shown). For the special case of the beam on axis, the fields on the boundary at y = a arê with k = ω/c.
In the case of flat geometry, with corrugated plate width w → ∞, the total Joule energy loss on the walls, u z + u x , must equal a point charge beam's energy loss on the axis, u w . In this case the point charge loss factor with the on-axis electric field The impedance is given by Substituting from Eq. 23 and integrating numerically we obtain the impedance. The real part, for this case (a = 0.7 mm and w → ∞), is shown in Fig. 3. We see that the impedance is a highly spiked function, with the peak location at ka ≈ √ 2a/h ≈ 1.673 [7]. Note that a different, earlier perturbation analysis, one that ignored wall resistance, gave essentially the same result except that the spike reached to infinity [7]. This implies that the short-range wake is essentially independent of the boundary conductivity. The Joule heating energy has two components, u hz and u hx , corresponding to contributions from ζ z and ζ x , respectively. Numerically solving Eqs. 18 and 20 for the infinitely wide plates, we find that u hz = 0.8u w , u hx = 0.2u w , and, to good . This is what we expect: for plates of infinite width, the sum of the two Joule energy contributions should equal the energy loss of the on-axis point charge beam.
Performing the numerical integrals for finite plate width w = 12 mm, we obtain u h = u hz + u hx using Eqs. 17 and 19. For a = 0.7 mm, we find that the total Joule loss is a small part of the beam energy loss, u h = 0.03u w (with u hz = 0.91u h ). Then using Eq. 9 we obtain the real part of the Joule heating impedances, Z hz and Z hx .
In Fig. 4 we plot the Joule impedance sum, Re(Z h ) = Re(Z hz ) + Re(Z hx ). Note that on the scale of the plot, Re(Z h ) ≈ Re(Z hz ) and Re(Z hx ) ≈ 0. As a spot check on sensitivity to conductivity, we reduced σ c by a factor of 4, repeated the calculation, and found that (u h /u w ) increased by 35%. We repeat the numerical calculation for several values of plate width w. In Fig. 5 we plot the Joule energy loss vs. plate width, normalized to the point charge loss of the beam, u w = Q 2 π 2 /(8a 2 ). We see that, even for w ∼ 100a, only a small part of the beam energy loss ends up as Joule heating. For our final estimate of Joule power loss of a uniform beam of full length = 60 µm we need to use the short-range, point charge wake of a beam on the axis of a flat dechirper [10] For the parameters   of Table I, the scale factor s 0 = 434 µm. For a bunch with uniform distribution of full length , the loss factor is given by Here κ = 19 kV/(pC*m), and the loss compared to a point charge beam is κ/κ(0) = 0.82.
For the beam parameters of Table I

BEAM NEAR ONE PLATE
There is interest in streaking the beam by inducing the transverse wakes of the dechirper, by passing the beam close to one jaw. With the beam a distance from the near wall of b ∼ 0.25 mm and from the far wall by 5 mm, the second wall will no longer affect the results. The physics will be quite different than before: with two plates the impedance has a narrow resonance whose frequency depends on the plate separation 2a; in the single plate case this parameter no longer exists. We present more details of this case, since the analysis of it is a relatively new topic. Note that in Ref. [11] the wakes for a short beam passing by a single plate of a dechirper are obtained, and in Appendix B of this report we use the time domain solver CST PS to test the accuracy of these results.
For the case of an ultra-relativistic beam passing by one plate of a corrugated structure, one question that comes to mind is: will the radiation generated be the same as Smith-Purcell (SP) radiation [14]? The answer is that the radiation is not "classical" SP radiation. With classical SP radiation, the beam energy is relatively low (a few MeVs); the radiation wavelength λ = (β −1 − cos θ)p/n [β is the ratio of the electron speed and that of light, θ is observation angle, p is corrugation period, and n is an integer greater than zero], i.e. the radiated wavelength is short compared to the corrugation period; and the coupling is sensitive to the detailed shape of the corrugations. For our case, in contrast, the radiation is not sensitive to β (we can and do let β = 1), the radiation period covers several corrugation periods, and the coupling is insensitive to the details of the corrugation shape.
For the Joule heating calculation we start with the equations for the magnetic fields on the wall, for a beam offset by y from the axis of a two-plate dechirper (equations not shown here). We let y = a − b, and then let a → ∞, to obtain the fields on the walls of one plate due to a bunch passing by at distance b. The fields on the wall (at y = b) are given bŷ Meanwhile the electric field at the particle location (here, at y = 0) iŝ First, note that while the point charge energy loss per length of the two-plate system was u w = Q 2 π 2 /(8a 2 ), with a the half gap, for the single plate it is only , with b the distance between the beam's path and the plate. In Fig. 6 we plot Re(Z), where Z = −Ẽ z /Q (in blue). Instead of the narrow spike of the two-plate case, we now find a relatively broad peak.
Actually, Fig. 6 can be obtained analytically. If we let the resistive terms of both ζ z and ζ x be zero, the integral over q that needs to be performed to obtain the impedance becomes singular. However, the integral can be performed as a Cauchy integral, yielding a finite result (see Appendix A for details). The result is where h is the depth of corrugation. Note that the frequency at the peak can be approximated as (k) peak = √ 3/(2hb), which is similar to the perturbation peak frequency formula for the two plate case, (k) peak = √ 2/(ha), with a the half gap. The result of Eq 29 is given by the red dashes in Fig. 6; we see that this function is almost identical to the earlier, numerical result that included wall resistance.
We next insert the single plate magnetic fields, Eq. 27, into Eqs. 18, 20, (but divided by 2, since there is only one plate) and numerically integrate to obtain the Joule heating impedance for the single, infinitely wide plate example. In Fig. 7 we show the real part of the Joule heating impedances Re(Z hz ), Re(Z hx ) (the components with wall currents aligned in, respectively, the z and x directions) that we obtain.
Again the beam is assumed to pass by at offset b = 0.25 mm from the plate. As expected, the total Joule heating impedance curve obtained numerically is the same, to good accuracy, as the impedance curve of Fig. 6  The short-range, point charge wake of a beam passing by a single plate of a flat dechirper at offset b is [11] w z (s) with s 0l = 2b 2 t/(πα 2 p 2 ) and α = 1 − 0.465 t/p − 0.070(t/p). In Appendix B we compare this formula with time-domain simulations, and find that the agreement is good. For corrugation parameters of Table I dechirper plate is the fraction (u h /u w ) = 0.033 of this, or P h = 14 W/m.

NUMERICAL TIME-DOMAIN COMPARISONS
The analytical model for the short-range wakes of an LCLS-type beam passing between two jaws of the RadiaBeam/LCLS dechirper has been verified in Ref. [10] using the finite difference, time-domain, wakefield solving program ECHO(2D) [15]. However, in our analytical Joule heating calculations, for the case of finite-width plates, we assumed that reflections from the side walls (in x for a vertical dechirper) are negligible. To see how good this approximation is, we have performed numerical, time-domain calculations using the program CST PS, both for an example with the beam on axis between two dechirper plates, and for an example with the beam passing by a single plate. For the case of two dechirper plates, the accuracy of CST PS simulations was verified by cross-checking with results using the wakefield code PBCI [13]. Since the bunches are short (z rms ∼ 20 µm), the catch-up distance [z cu ∼ a 2 /(2σ z )] is large (on the order of cm's). However, the most challenging simulation issue is the long damping time of the fields that need to be followed for the Joule loss calculation. In the double plate case this time is on the order of cm's/c. Figure 9 shows the geometry of the single corrugated plate and the (nominal) beam path used in the simulations. For modeling the lossy metal, a resistive wall impedance boundary condition for aluminum is applied. The corrugated plate structure is enclosed within a larger computational box that has free-space boundary conditions applied on all sides. This is necessary for modeling field radiation and thus, for the proper computation of the Joule losses. Such boundary conditions are provided by CST PS. This is why all Joule loss results in the following were obtained using this program. To confirm that the choice of bounding box does not affect the results, we performed one calculation with changed bounding box dimensions and found that the wake obtained was left unchanged.
The simulations are performed in the time domain by tracking a single bunch along the beam path until the wake potential and Joule loss per unit length saturate to steady state. Typical plate lengths considered in the simulations are some tens of cm's. To make the calculations manageable, for both double-and single-plate cases, we use Gaussian bunches with σ z = 100 µm, i.e. significantly longer than our nominal bunch length. The resulting mesh with the necessary numerical resolution consists of up to 1.5 × 10 9 mesh points, and it typically takes several days to complete just one simulation. Since the typical bunch frequency is much higher than the structure frequency, the ratio (u h /u w ) will be about the same for this bunch length as for the target rms bunch length (see Table I), σ z = 17 µm.

Beam on Axis between Two Dechirper Plates
For the two-plate case, the nominal half aperture is a = 0.7 mm. The numerically obtained, steady-state bunch wake, for the Gaussian bunch,w z (s) is given in Fig. 10.
We see that the wake damps away on a scale of s ∼ 150 mm. Fourier transforming the wake, we obtain the impedance (not shown). We find that Re(Z) is given by a collection of spikes (dominated by the first one) beginning with ka = 1.36, 1.53, 1.77; note that these values agree with results of mode matching calculations, applied to the same geometry [8]. This behavior is quite different than our analytical, perturbative solution, with its one spike at ka = 1.67 (see Fig. 3). This mismatch was expected.
For the numerical Joule losses calculations, the beam passed by a two-plate dechirper of length L = 135 mm. From the magnetic fields at the plate surfaces, the Joule heating power at each time step was obtained. Fig. 11 shows the results for the nominal a = 0.7 mm case (blue, solid curve); the steady-state was obtained by extrapolation using a double exponential fitting function (the dashes). The final result is P h = 13.5 W/m. The wake power loss for the σ z = 100 µm Gaussian bunch the driving bunch is Gaussian with σ z = 100 µm. Note that the wake is normalized to structure length. P w = 108 W/m. Thus, the ratio of Joule heating and wake energy, according to the numerical calculation, is (u h /u w ) num = P h /P w = 0.125, which is a factor of 4 larger than the analytical result obtained above, (u h /u w ) ana = 0.030. Our best estimate for the Joule power loss for the LCLS-II beam,P h , is obtained taking (u h /u w ) num and multiplying it with the wake power (analytically obtained above) for the short, uniform LCLS-II bunch shape, (P w ) ana ; i.e. and we find that the impedance (not shown) is closer to the analytical one. The simulation finds that the Joule power P h = 2.64 W/m (see Fig. 11, the red curve).
The energy ratio (u h /u w ) num = 0.065, which is a factor of 2.5 larger than the analytical value (u h /u w ) ana = 0.025, rather than the factor of 4 we had before. For a = 1.4 mm, (P w ) ana = 46 W/m, and our best estimate of Joule power loss becomes P h = (0.065)(46 W/m) = 3.0 W/m.

Beam Passing by a Single Corrugated Plate
The numerically obtained wake for a 100 µm Gaussian bunch passing at distance b = 0.25 mm from the plate is shown in Fig. 12. Here we see that the wake dies out after s ∼ 120 mm and that there are reflections from the sides of the plate. Note that since the period in the reflections is ∼ 13.5 mm (and not equal to the plate width, w = 12 mm), we infer that the group velocity of the waves moving sideways is The real part of the impedance Re(Z) for the single plate example is given in Fig. 13 (blue curve). To obtain this, the longitudinal wake (Fig. 12) was Fourier transformed and multiplied by e k 2 σ 2 z /2 . The narrow, evenly-spaced spikes in the impedance [with spacing ∆ f ≈ c/(13.5 cm)] are due to the reflections in the wake.
A short bunch, however, cannot resolve these spikes. To generate a "broad-band impedance", one that is easier to compare with our analytical result, we multiplied the wake by a Gaussian form factor, with rms length σ = 5 mm, before Fourier transforming. The resulting impedance is given by green dashes in the figure. Our analytical result (Fig. 6) is given in red dashes. Although the low and higher frequency behavior of the red and green curves agree well, the numerical peak is narrower and the frequency of the peak is lower than the analytical one. This disagreement appears to be a consequence of the corrugation parameters not being in the perturbative regime: here (h/b) = 2, which is not small compared to 1. The broad-band impedance obtained from the same wake is given by green dashes. The analytical perturbation result (Fig. 6), is given by red dashes.
For Joule loss simulations, the beam was passed by a single-plate dechirper of length L = 115 mm. Fig. 14 Table II.   TABLE II. Summary of Joule heating calculations for the LCLS-II dechirper, giving case; wake power lost by beam, (P w ) ana ; ratio of energy in Joule heating and beam energy loss, analytical calculation, (u h /u w ) ana ; Joule power loss, analytical calculation, (P h ) ana ; energy ratio, according to numerical calculation, (u h /u w ) num ; and our best estimate of Joule losses, P h = (u h /u w ) num (P w ) ana . Both cases assume the high charge scenario, with Q = 300 pC and f rep = 100 kHz; the bunch shape is taken as uniform, with total length = 60 µm. Here Q = 300 pC, f rep = 300 kHz; the driving charge is Gaussian with σ z = 100 µm.

CONCLUSIONS
We have performed Joule power loss calculations for the RadiaBeam/LCLS-II dechirper, whose engineering details-for example concerning the cooling confirmed that the analytical expressions for short-range wakes found in Ref. [11] are valid when (h/b) 1, where h is depth of corrugation and b is distance of beam from plate. In fact, we showed that the longitudinal and dipole (but not quadrupole) wakes agree well even for (h/b) 2.
We need to calculate the following integral (where we have used the fact that the integrand is symmetric with respect to q). In the limit when the resistive term ζ rw → 0, we have ζ x = 0 and ζ z = − 1 2 ihk and the integral reduces to .
It is easy to see that the denominator of the integrand vanishes at and the integrand has a pole at this point. This pole has to be bypassed in the complex plane (of variable q) or, equivalently, the integration path needs to be shifted from the real axis. The direction of the shift can be found by analyzing the position of the pole when ζ rw is small, but not equal to zero. This analysis shows that for non-zero ζ rw the pole has a positive imaginary part, which means that in the limit ζ rw → 0 the integration path should be modified as shown in Fig. 15.
We are interested in calculating the real part of the impedance, Re(Z). Because

Appendix B CONFIRMATION OF SHORT-RANGE WAKE FORMULAS FOR THE CASE OF A BEAM PASSING BY A SINGLE CORRUGATED PLATE
The short-range wake formulas for a beam passing between the two corrugated plates of a dechirper were verified by time-domain, finite difference simulations using the computer program ECHO(2D) [10]. Then the short-range wake formulas for a beam passing by a single corrugated plate were derived in Ref. [  with s 0y = 8b 2 t/(9πα 2 p 2 ). Here s 0y = 44 µm. The quad short-range wake is given by w yq (s) = 3 2b w yd (s). The wake of a Gaussian bunch of length σ z can be obtained from the point-charge wake w(s) by convolution; e.g. 3) The Gaussian bunch wakesw yd (s) andw yq (s) are obtained in the analogous manner, except without the overall minus sign.
In Fig. 16 we plot the longitudinal, dipole, and quad wakes of a Gaussian bunch (with σ z = 100 µm) passing by a single dechirper plate at distance b = 250 µm, as obtained numerically (in blue) and compare with the analytical results (in red). The bunch distribution is also shown, with the bunch head to the left. We see that the longitudinal and dipole bunch wakes agree well over the bunch; the quad analytical wake, however, is significantly larger than the numerical result.
The numerical simulations were repeated for bunch offsets b = 0.5, 1, 1.5 mm.
Averaging the bunch wake over the Gaussian bunch distribution we obtain the loss factor or kick factors. In Fig. 17  ) We see that κ and κ yd agree well for all offsets. However, κ yq , at b = 250 (500) µm has an analytical solution that is 90% (45%) larger than the numerical one; at the larger two offsets, however, the agreement is again quite good.
In conclusion, we have verified that the short-range longitudinal, dipole, and quad analytical formulas for wakes of a single plate agree well with the results of numerical simulations, provided we are in the perturbative regime, i.e. (h/b) 1.