# Wakefield assisted bunch compression in storage rings

Sergey A. Antipov<EMAIL_ADDRESS>Ilya Agapov  Igor Zagorodnov
François Lemery<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

Equilibrium bunch lengths typical to modern day light sources of tens of
picoseconds limit the spectral reach of superradiant approaches for exploring
materials and biological samples. In particular, generation of terahertz
radiation seems appealing for pump-probe experiments. Taking advantage of
self-wakes generated by carefully chosen structure one can control the bunch
length and shape the bunch profile to increase significantly the high-
frequency spectral components. Numerical analysis indicates the presence of
self-consistent steady states that can be obtained with passive corrugated or
dielectric structures. This approach has potential applications for generating
superradiant terahertz radiation and pulse shaping in circular accelerators.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

Beyond the production of X-rays in storage rings, terahertz (THz) and mid-
infrared (MIR) radiation generation has been widely explored, especially in
the context of coherent synchrotron radiation. Early observations indicated
that by driving a machine toward the microwave instability (MWI), large
broadband and coherent bursting would occur under suitable conditions [1, 2,
3]. In parallel, the design of low-momentum-compaction (low-$\alpha$) storage
rings were developed to support bunch lengths comparable to the desired
radiation wavelength produced in a bending magnet [4]. In addition a low-
energy storage ring dedicated to THz generation was proposed [5]. Recently
feedbacks were employed to maintain a microbunching structure on the beam,
allowing the production of stable coherent radiation [6]. THz radiation
generation at an energy recovery linac has also been investigated; the smaller
achievable bunch lengths in such a platform are favorable to reach larger
frequencies and enable the production of large average powers [7].

These techniques all rely on bending magnets e.g. synchrotron radiation to
produce THz/MIR, and are generally broadband. Alternatively in the linear
accelerator community, there has been tremendous interest in Cherenkov
waveguides, e.g. dielectric-lined (DLW), corrugated, or bimetallic waveguides
for beam manipulation diagnostics and radiation generation. Beam manipulation
studies have especially shown and demonstrated techniques to microbunch beams
or to more generally synthesize longitudinal phase spaces [8, 9, 10, 11, 12].
These effects can be used to enhance the bunch form factor to couple more
strongly to similar waveguides which can in turn radiate more energy. Large
field gradients (GV/m) have been demonstrated in the superradiant regime where
the form factor is approximately unity [13]. For a comprehensive discussion on
superradiant radiation generation in Cherenkov structures, see Ref. [14]. We
note that a recent technique to characterize such waveguides has been
developed [15].

In this paper we investigate the use of Cherenkov waveguides in storage rings.
While it is well known that introducing impedances can modify the equilibrium
bunch length [16, 17, 18], here we further study this feature to
longitudinally squeeze the equilibrium bunch length by adiabatically
increasing the wakefield amplitude in a corrugated planar structure. The
scheme is investigated with analytical and numerical considerations and backed
with particle tracking simulations. We also discuss limitations with regards
to beam stability. Finally, we also investigate using a secondary wakefield
structure to produce radiation with the new equilibrium bunch length.

##  II Bunching for single-mode wakes

###  II.1 Stationary distribution for a single-mode wake

Consider a relativistic electron bunch orbiting in a storage ring. Let the
number of particles in a bunch be $N_{b}$, the revolution frequency $f_{0}$,
and $\gamma$ the Lorentz factor. Let us further assume a quadratic rf
potential well, in which the particles oscillate with an angular synchrotron
frequency $\omega_{s}$. Below the microwave instability threshold the
normalized stationary longitudinal bunch distribution $\rho$ in the presence
of a wakefield $W^{\prime}$ is described by the Haissinski integral equation
[16]:

$\begin{split}\frac{\rho(\xi)}{\rho_{0}(\xi)}&\propto\exp\left[\kappa\int_{0}^{\xi}\int_{\xi^{\prime\prime}}^{\infty}\rho(\xi^{\prime})w^{\prime}(\xi^{\prime\prime}-\xi^{\prime})d\xi^{\prime}d\xi^{\prime\prime}\right],\\\
\rho_{0}(\xi)&=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{\xi^{2}}{2}\right],\end{split}$
(1)

where $\xi=z/\sigma_{z0}$ stands for the normalized coordinate, $\sigma_{z0}$
is the zero intensity rms bunch length, and the normalization condition is
$\int\rho(\xi)d\xi=1$. The normalized wake force is
$w^{\prime}(\xi)=W^{\prime}(\xi)/W_{0}$, $W_{0}=W^{\prime}(+0)$. The coupling
strength is $\kappa=W_{0}\frac{\alpha
f_{0}c}{\gamma\omega_{s}^{2}}\frac{N_{b}r_{0}}{\sigma_{z0}}$; $c$ is the speed
of light and $r_{0}$ the classical electron.

Eq. (1) can be solved analytically for certain special wakes [19, 20, 17, 18].
One can see that some wakes can create additional longitudinal focusing,
reducing bunch length. Longitudinal focusing has been shown, for example, for
a purely inductive wake [18], a purely capacitative wake [20], and for a wake
of a ferrite-lined waveguide [21]. In this paper we consider short-range wakes
that can be easily generated in the ring with the help of corrugated or
dielectric waveguide structures. As an example consider a Gaussian
distribution $\rho(z)$ and a cosine wake $W^{\prime}(z)=W_{0}\cos(2\pi
z/\lambda)$. Then for $\lambda\sim\sigma_{z}$ the convolution of the wake with
the bunch density creates a longitudinal potential well with the dimensions
comparable to $\sigma_{z}$ (Fig. 1). The particles in the head lose energy,
while the particles in the tail gain, and the total energy loss by the bunch
is minimal. For negligibly weak wakes the solution of Eq. (1) is simply a
normal distribution:
$\rho(\xi)=\rho_{0}(\xi)=\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{\xi^{2}}{2}\right]$.
As $W^{\prime}$ increases it produces an extra focusing that reduces the bunch
length. In order to obtain the final stationary distribution one can start
with a normal distribution and solve Eq. (1) iteratively. Figure 2 presents
examples of resulting bunch shapes for different wake wavelengths.

Figure 1: Normalized wake potential for different ratios of the structures
frequency to the rms bunch length $\lambda/\sigma_{z}$ for a Gaussian bunch
profile (depicted by the dotted black line). Figure 2: Normalized bunch
distributions after compression with a cosine-like wake for different wake
wavelengths, found as solutions of Eq. (1). Normalized wake potential
$\kappa=18$. The unperturbed distribution is shown in black.

###  II.2 Stability limitations

Eq. (1) only holds below the MWI threshold. Above the threshold both the bunch
length and momentum spread increase thereby limiting the maximum bunch
compression that can be achieved with a passive structure. The MWI threshold
current for a Gaussian bunched beam can be found from [22]

$I_{th}=F\times\frac{\alpha(E/e)\delta_{p,FWHM}^{2}}{|Z/n|},$ (2)

where $E$ and $e$ stand for the beam energy and electron charge respectively,
$F$ is a form factor, $\alpha$ is the momentum compaction factor, and $|Z/n|$
is the effective longitudinal impedance. The second part of the product
represents the Keil-Schnell-Boussard stability criterion [23], which in case
of bunched beams has to be computed with local densities of bunch current $I$
and full-width-at-half-maximum relative energy spread $\delta_{p,FWHM}$. For
wakes with $\lambda\sim\sigma_{z}$, $F\approx 1$.

The effective impedance is defined as

$|Z/n|=\int_{-\infty}^{+\infty}Z(\omega)\frac{\omega_{0}}{\omega}h(\omega)d\omega,$
(3)

where $h(\omega)$ is the normalized beam power spectrum,
$\int_{-\infty}^{+\infty}h(\omega)d\omega=1$ and $\omega_{0}=2\pi f_{0}$. For
a Gaussian bunch spectrum
$h(\omega)\propto\exp(-\omega^{2}\sigma_{z}^{2}/c^{2})$ and an impedance
peaked at an angular frequency $\omega_{r}$ the $|Z/n|$ maximizes when
$\omega_{r}\sigma_{z}/c=1$ (Fig. 3). Thus, increasing the frequency of the
impedance seems beneficial for mitigating the instability, potentially
achieving larger bunch compression.

Figure 3: Effective impedance $Z/n$ of a narrow-band resonator as a function
of its normalized frequency.

It shall be noted as well that in a real machine the instability can be
mitigated with other measures, i.e. employing a longitudinal feedback or
increasing the momentum spread with an undulator. Thus, Eq. (2) does not
necessarily represent a strong limit on the practically achievable bunch
compression.

##  III Numerical simulation setup

In order to assess the effect numerically we performed particle tracking
studies using a 1D macroparticle tracking code, where the relativistic
electron beam is represented as an ensemble of $N=10^{5}$ macroparticles. The
code treats the ring as a transfer map with an rf and a wake kick. In order to
compute the wake kick the beam is sliced into 100 longitudinal slices, a
smooth current density function is computed from the discrete number of
particles in each slice using a 2nd order Savitzky–Golay filter, and finally a
convolution with the Green’s function of the wake is computed.

The initial distribution was assumed to be a 1D Gaussian. We neglected other
effects that might also affect the longitudinal distribution in a real
machine, like intra-beam scattering, synchrotron radiation damping, or
coherent synchrotron radiation. This was done with the sole purpose of
simplifying the model and assessing the limits of the proposed approach
separately.

##  IV Results and Discussion

In practice there are several ways of generating the desired wake potential in
the ring. Here we first consider a fixed aperture dielectric cylindrical
waveguide (DLW) that can be treated as a single-mode structure. We also
consider a variable gap corrugated structure.

###  IV.1 Single-mode structure

To gain insights into the beam dynamics and stability of this configuration,
we implemented a cylindrically-symmetric DLW with wake potential
$W^{\prime}(z)=W_{0}\cos(2\pi z/\lambda)$. Here the transverse parameters,
e.g. inner radius and dielectric thickness determine the spectral content of
the waveguide while the structure length determines the strength of the
produced wake [24, 25]. The wakefields were numerically implemented using
publicly available python scripts [26].

Considering $\lambda/\sigma_{z}=0.8$ and a moderately high normalized wake
strength $\kappa=36.12$ (Example parameters: radius of 5 mm, 7 $\mu$m coating
with a dielectric permittivity of 10 for a 400 GHz structure) we observe a
measurable bunching of the initial Gaussian distribution in our tracking (Fig.
4). The peak current density increases by $\sim$10%. This result is in
excellent agreement with the prediction of Eq. (1) for the considered wake.

Figure 4: Below the microwave instability threshold the predictions of the
Haisinski equation (blue dots) match well the tracking results for a single-
mode dielectric structure (orange line) and a corrugated slab structure that
creates generates the same wake potential (green line). Wavelength of the
1-mode structure $\lambda/\sigma_{z}=0.8$, normalized wake strength
$\kappa=36.12$. The initial distribution is shown in black.

Larger peak currents can be achieved by increasing the wakefield strength
until the MWI threshold. This limit was explored numerically, we first
implemented an unperturbed Gaussian distribution and then adiabatically
increased $W_{0}$ in small increments. The peak density increased quasi-
linearly until $\kappa\sim 75$, where the bunch started exhibiting unstable
behaviour and subsequently rapidly expanded and reduced its peak density; see
Fig. 5. Larger $\lambda$ helps bunching the beam more efficiently, requiring
less wakefield strength, but limits the maximum achievable bunch compression
to smaller values.

Figure 5: Normalized peak current (a) and bunch length (b) as a function of
the normalized strength parameter $\kappa$ for $\lambda/\sigma_{z,0}=1.2$,
$\lambda/\sigma_{z,0}=1.0$, and $\lambda/\sigma_{z,0}=0.8$. A shorter
wavelength requires a larger wake to achieve the same level of bunch
compression, but at the same time allows compressing the bunch further.

###  IV.2 Corrugated slab structure

In practice it is convenient to generate the required wakefield with a
variable-gap slab structure. In addition, slab-structures are advantageous,
especially with flat-beams, to mitigate dipole mode excitations which can
destabilize the beam [27]. In such a structure the period and the depth of
corrugations mainly govern the resonance frequency while the size of the
aperture largely affects the wake strength. Figure 6a shows an example of such
a structure, designed to have a resonance frequency $f_{r}$ about 400 GHz. The
corresponding longitudinal impedance, obtained from ECHO [28], is shown in
Fig. 6b; at lower frequencies it can be well approximated with a single-mode
broad-band resonator:

$Z(f)=\frac{R_{s}}{1+iQ(f/f_{r}-f_{r}/f)},$ (4)

where $R_{s}$ stand for the resonator shunt impedance and $Q$ for its quality
factor. Performing the particle tracking for the same wake strength
$\kappa=36.12$ as in the single-mode case we obtain an identical final beam
distribution (Fig. 4). Thus, for a realistic impedance under the MWI threshold
the results agree with the theoretical predictions and the simplified single-
mode-structure model.

Figure 6: (a) the profile of the corrugated structure (not to scale). Dashed
line shows the beam axis. (b) the longitudinal impedance per unit length. Blue
lines represent a fit with a single-mode broadband resonator model.

###  IV.3 KARA ring example

Table 1: Key parameters used for the study with a 400 GHz corrugated and a 100 GHz dielectric structures Structure | Corrugated | Dielectric
---|---|---
Ring circumference | 110 m | 110 m
Bunch energy | 1.3 GeV | 1.3 GeV
Bunch charge |  $300-700$ pC |  $300-700$ pC
Momentum compaction | $5\times 10^{-4}$ | $8\times 10^{-3}$
Bunch length, rms | 3 ps | 10 ps
Synchrotron tune | 0.0023 | 0.005
Structure length | 30 cm | 30 cm
Central frequency | 390 GHz | 100 GHz
Half-gap | 2.5 mm | 5 mm

As a practical example, we have studied numerically the extent of achievable
bunch shortening at a low-$\alpha$ KARA ring at KIT. The 110-m-long ring is
capable of circulating electron bunches of $370-740$ pC and rms bunch length
of $2-10$ ps (Table 1). It has a 40-cm-long straight section suitable for an
installation of a passive dielectric or corrugated structure. The
$\beta$-function in the straight section and orbit tolerances limit the
aperture of the structure to 5 mm. We have considered two cases: a machine
operating at $\alpha=5\times 10^{-4}$ with 3 ps rms bunches and a lattice with
a larger momentum compaction of $8\times 10^{-3}$ with 10 ps rms bunches.

Figure 7: a – longitudinal wake potential created by a dielectric slab
structure vs its half-opening $g$; b – particle tracking shows how, as the gap
of the structure is closed, the electron storage ring the bunches are
compressed by 20%; c – power spectrum density; the spectral density at the
structure’s eigenfrequency 100 GHz increases by an order of magnitude. Bunch
charge Q = 400 pC.

First, let us consider the later case of $\alpha=8\times
10^{-3},~{}\sigma_{z,0}/c=10$ ps (Table 1). A dielectric structure with the
dielectric thickness of 100 $\mu$m and a 5 mm half-gap would produce a
longitudinal wake with a characteristic frequency of 100 GHz, or
$\lambda/\sigma_{z}=1$. Assuming a 30 cm length the wake strength is
$W_{0}=4\times 10^{14}$ V/pC, or $\kappa\approx 30$. We simulated numerically
the process of slow adiabatic closure of the structure gap of 20 mm, where its
wake is negligible, to 5 mm (Fig. 7a). As the gap is closed an initially
Gaussian bunch transitions to a new equilibrium state with about 20% smaller
rms bunch length (Fig. 7b). This greatly boosts the bunch’s high frequency
spectral components. Figure 7c shows a comparison of the initial and the final
bunch power spectra: the components close to 100 GHz are enhanced by an order
of magnitude.

It is convenient to characterize the temporal structure of the bunch in terms
of the bunch form factor (BFF). The BFF is commonly used to characterize the
performance of accelerator-based radiation source [29]. In 1D it is computed
as

$\widetilde{F}(\omega)^{2}=\frac{1}{N^{2}}\left(\left|\sum_{i=0}^{N}\cos\frac{\omega
z_{i}}{c}\right|^{2}+\left|\sum_{i=0}^{N}\sin\frac{\omega
z_{i}}{c}\right|^{2}\right),$ (5)

where $z_{i}$ stand for individual positions of the macroparticles. The
observed shortening of the bunch greatly increases the BFF at high
frequencies. The boost is at least an order of magnitude at the frequencies
above 40 GHz (Fig. 8).

Figure 8: Bunch form factor can be increased at higher frequencies when using
the corrugated structure. Different lines correspond to different bunch
intensities.

A further decrease of the bunch length does not seem to be feasible as a
larger wake strength leads to a longitudinal instability. This results in an
increase of the bunch length, as seen, for example, by the drop of the BFF at
high frequencies when increasing the bunch charge above 400 pC (Fig. 8).

Accessing even higher frequencies, on the order of 100 GHz, requires shorter
bunches. Let us now consider the case of $\alpha=5\times 10^{-4}$, when the
rms bunch length $\sigma_{z,0}/c=3$ ps. A 30-cm-long corrugated structure can
be installed in the same straight section. Considering the profile and
impedance shown in Fig. 6 we obtain again a ratio $\lambda/\sigma_{z}\approx
1$. At the smallest gap of 5 mm the structure would generate a 5-10% increase
of the peak current for the operational range of bunch charges between 400 and
700 pC (Fig. 9a). If the length of the structure can be increased or,
alternatively, its gap reduced the amplification of the peak current can be
even larger (seen in the plot as a hypothetical 2000 pC line). The bunch
shortening, as expected, boosts the bunch form factor at higher frequencies,
above 100 GHz (Fig. 9b).

Figure 9: Bunch shortenings that can be obtained with a 30-cm-long 400 GHz
corrugated structure in the KARA ring for different bunch charges (a) and the
corresponding bunch form factors (b). The initial bunch length is assumed to
be 0.9 mm, rms (shown as black lines).

In all the simulations for the combination of available aperture and structure
length we stayed below the MWI threshold. On the other hand, our tracking
studies with the single mode structure indicates a potential 30% increase of
peak current before the impedance of the structure starts triggering the MWI
(Fig. 5). If the MWI can be avoided, for example, with the use of a
longitudinal feedback or damping wigglers, the bunching could potentially be
even greater.

In comparison, achieving a similar $20-30\%$ increase in the peak current
(shown in Fig. 5) by using a conventional rf-based approach would require
increasing the voltage by $40-60\%$, as the bunch length scales as
$\sigma_{z}\propto 1/\sqrt{V_{rf}}$. Alternatively, one can reduce the
momentum compaction, $\alpha$, of the lattice. On the other hand, this might
compromise the beam stability as the thresholds of the microwave instability
and the transverse mode coupling instability are proportional to $\alpha$,
while the rms bunch length – $\sigma_{z}\propto\sqrt{\alpha}$. Therefore,
employing passive structures for bunching might be an appealing option.

###  IV.4 PETRA IV ring example

As another practical example, let us consider a state-of-the art synchrotron
light source PETRA IV that is currently being planned at DESY [30]. This
2304-m-long ring has a beam energy of 6 GeV and is capable of circulating
trains of electron bunches with beam currents up to 200 mA. In the brightness
operation mode the ring stores trains of 40-ps-long (rms) 1 nC bunches with a
bunch spacing of 4 ns (Table 2). Thanks to its high energy physics heritage,
the ring has four long and four short straight sections suitable for
installation of additional components.

Table 2: PETRA IV parameters used for the study Ring circumference | 2304 m
---|---
Bunch energy | 6.0 GeV
Bunch charge | 400 pC
Momentum compaction | $3.33\times 10^{-5}$
Bunch length, rms | 5 ps
Momentum spread, rms | $9\times 10^{-4}$
Synchrotron tune | 0.005
Structure length |  $25-500$ cm
Central frequency | 160 GHz
Half-gap | 5 mm
Figure 10: Bunch compression (a) and bunch form factor (b) for different wake
strength in PETRA IV.

The equilibrium bunch length in the ring varies depending on the bunch
intensity and the 3rd harmonic cavity voltage. The shortest bunch length that
can be achieved in the ring without using its 3rd harmonics system is about 5
ps (1.5 mm) at low beam intensity [30].

For this example we considered a low intensity bunch of 400 pC and tracked it
for 20000 turns (about 100 synchrotron periods), slowly increasing the wake
strength. The wake was created by a hypothetical dielectric structure that
could be installed in one of the ring’s unused undulated cells. The length of
the structure was varied from 0 to 5 m (length of the standard undulator),
while its half-gap was fixed at 5 mm. We note that a similar wake can also be
produced with a corrugated structure (as shown in the previous section). As
the wake strength increased (with the 3rd harmonic cavity turned off) the
bunch length decreased until eventually the bunch became longitudinally
unstable. The maximum bunch compression was achieved for the structure length
of 2.25 m, corresponding to the wake strength $\kappa=13.2$. The maximum bunch
compression in this case amounted to a 10% increase in the peak current
density, consistent with the prediction of the compression limit for
$\lambda/\sigma_{z,0}=1.2$ (Fig. 5). A higher compression can be achieved with
a higher frequency structure, which would have to be longer, as discussed in
Sect. IV A.

##  V Radiation generation

After the bunch length has been reduced to increase the high-frequency
spectral components the radiation at the desired frequency can be out-coupled
using a separate, dedicated structure. According to [14], the energy radiated
at a given frequency $E_{rad}$ is proportional to the BBF:

$E_{rad}=N_{b}^{2}e^{2}\widetilde{F}^{2}LK_{||},$ (6)

where $L$ stands for the structure length and $K_{||}$ is the loss factor,
i.e. the total energy loss per meter of a charged particle travelling through
the structure. Thus, an order of magnitude increase in $\widetilde{F}^{2}$
results in a significant increase energy $E_{rad}$ radiated at that frequency.

To illustrate the benefit of the proposed bunch compression scheme let us
consider the hypothetical case depicted by green lines in Fig. (9): an
initially 3 ps rms bunch is compressed until its peak current density
increases by 20%. Let’s assume now there is a second structure to out-couple
the radiation at 100 GHz. Let it be a 10-cm-long dielectric structure similar
to the one considered in Sect. IV C. Initially, a part of the bunch loses the
energy to the wakefield, while an almost equal part gains, making the total
energy loss, and consequently the radiated power small (Fig. 11a). As the
current profile changes under the action of the wakefield, its peak density
aligns with the minimum of the wake potential and a larger fraction of the
beam enters the region of negative wake potential, where particles lose energy
(Fig. 11b) on each passage. This results in a four-fold increase of the beam
energy being emitted in the 100 GHz structure: from $8\times 10^{-8}$ to
$3\times 10^{-7}$ J per passage, or in terms of average radiation power, from
200 to 800 mW.

Figure 11: Wake potential of a hypothetical 100 GHz structure for the initial
Gaussian bunch (a) and after compression (b) together with the corresponding
current profiles. Before compression the energy loss per turn in the
structure, defined as the product of the two curves, is negligible and
increases dramatically as the bunch is compressed.

For the PETRA IV case described in Sect. IV D the increase in the radiated
power at 100 GHz would be four-fold: from $3\times 10^{-9}$ J to $2.3\times
10^{-8}$ J with the same 10-cm-long dielectric structure. The power radiated
at 100 GHz could thus reach as much as 1.4 W with the full filling of 1920
bunches, circulating at 130 kHz, providing an effective repetition rate of 250
MHz. In comparison, a dedicated linac to produce THz, TELBE [31], delivers
pulses of $\sim 1~{}\mu$J at $50$ kHz, or $\sim 0.05$ W.

##  VI Conclusion

We have demonstrated that a dielectric or a corrugated structure can be used
to compress the bunch length in a storage ring in a controllable way which is
more efficient than conventional RF-based approaches. To do this one can use a
structure with a characteristic wavelength $\lambda\sim\sigma$ that creates a
wake potential that produces longitudinal focusing. Numerical simulations
performed for the parameters of the KARA ring show that levels of 10-20% bunch
shortening can be realistically achieved before encountering the microwave
instability. Similar levels of additional bunch compression could be achieved
in 4th generation light sources, operating at low momentum compaction factors,
such as the planned PETRA IV. This decrease of bunch length can boost the
bunch form factor at the frequencies $\sim 0.1$ THz by up to an order of
magnitude. This results in a significant increase of radiation power that can
be extracted from the beam at those frequencies. Additional use of 3rd
harmonic cavities at 1.5 GHz could be explored in the PETRA IV case to further
reduce the bunch length in a similar fashion as the proposed CIRCE [5].

The required wakes can be realistically achieved with both dielectric and
corrugated structures. Slab structures are attractive in this context because
they can provide control over both the wake amplitude and partially, the mode
frequency. In addition, a more sophisticated corrugated waveguide with several
rows of corrugation placed in parallel would provide a tunable frequency by
exposing the beam to the different corrugated patterns. This could be
appealing for generating short wavelength microwave radiation in synchrotron
light sources. We also note that a reduction of bunch lengths can be
contemplated for luminosity improvement of future and existing colliders.

###### Acknowledgements.

We are grateful to the KARA team, in particular to Michael Nasse, Miriam
Brosi, Markus Schwarz, for multiple discussions of ring parameters that were
beneficial for producing realistic calculations of bunch shortening.

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