now at ]SLAC, Menlo Park, USA now at ]MAX IV Laboratory, Lund, Sweden now at
]DLR-VE, Oldenburg, Germany

# On the Perturbation of Synchrotron Motion in the Micro-Bunching Instability

Tobias Boltz<EMAIL_ADDRESS>[  Karlsruhe Institute of Technology,
Kaiserstraße 12, 76131 Karlsruhe, Germany  Miriam Brosi  [  Karlsruhe
Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany  Bastian
Haerer  Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe,
Germany  Patrik Schönfeldt  [  Karlsruhe Institute of Technology, Kaiserstraße
12, 76131 Karlsruhe, Germany  Patrick Schreiber  Karlsruhe Institute of
Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany
Minjie Yan  Karlsruhe Institute of Technology, Kaiserstraße 12, 76131
Karlsruhe, Germany  Anke-Susanne Müller  Karlsruhe Institute of Technology,
Kaiserstraße 12, 76131 Karlsruhe, Germany

###### Abstract

The self-interaction of short electron bunches with their own radiation field
can have a significant impact on the longitudinal beam dynamics in a storage
ring. While higher bunch currents increase the power of the emitted coherent
synchrotron radiation (CSR) which can be provided to dedicated experiments, it
simultaneously amplifies the strength of the self-interaction. Eventually,
this leads to the formation of dynamically changing micro-structures within
the bunch and thus fluctuating CSR emission, a phenomenon that is generally
known as micro-bunching or micro-wave instability. The underlying longitudinal
dynamics can be simulated by solving the Vlasov-Fokker-Planck (VFP) equation,
where the CSR self-interaction can be added as a perturbation to the
Hamiltonian. In this contribution, we focus on the perturbation of the
synchrotron motion that is caused by introducing this additional wake field.
Therefore, we adopt the perspective of a single particle and eventually
comment on its implications for collective motion. We explicitly show how the
shape of the parallel plates CSR wake potential breaks homogeneity in the
longitudinal phase space and propose a quadrupole-like mode as potential
seeding mechanism of the micro-bunching instability. Moreover, we consider
synchrotron motion above the instability threshold and thereby motivate an
approach to control of the occurring micro-bunching dynamics. Using
dynamically adjusted RF amplitude modulations we can directly address the
continuous CSR-induced perturbation at the timescale of its occurrence, which
allows for substantial control over the longitudinal charge distribution.
While the approach is not limited to this particular application, we
demonstrate how this can significantly mitigate the micro-bunching dynamics
directly above the instability threshold. The gained insights are supported
and verified using the VFP solver Inovesa and put into context with
measurements at the KIT storage ring KARA (Karlsruhe Research Accelerator).

##  I Introduction

In order to increase the emission of coherent radiation and to shorten the
generated light pulses, modern synchrotron light sources are deliberately
operating with short electron bunches. The Karlsruhe Research Accelerator
(KARA) thus has a dedicated short-bunch mode with picosecond-long bunches,
which provides emission of coherent synchrotron radiation (CSR) up to the
$\mathrm{THz}$ frequency range. Yet, due to self-interaction with its own
radiation field, the increased CSR also leads to complex longitudinal dynamics
within the electron bunch. At low bunch currents, the resulting perturbation
mainly causes a slight deformation of the still fairly stationary electron
distribution. However, above a particular threshold current $I_{\mathrm{th}}$
which depends on the specific machine settings of the accelerator, it leads to
the formation of dynamically changing micro-structures within the bunch. As
the longitudinal charge distribution varies over time, this in turn results in
major fluctuations of the emitted CSR power. Such fluctuations have been
measured at a wide range of facilities, e.g. Byrd  _et al._ (2002); Müller
(2011); Abo-Bakr  _et al._ (2002); Billinghurst  _et al._ (2016); Shields  _et
al._ (2012); Karantzoulis  _et al._ (2010); Andersson  _et al._ (2000); Wang
_et al._ (2006); Feikes  _et al._ (2011); Carr  _et al._ (2001); Evain  _et
al._ (2012); Walker  _et al._ (1997); Mochihashi  _et al._ (2006). The
phenomenon is generally referred to as micro-bunching or micro-wave
instability. The underlying longitudinal dynamics can be simulated to high
qualitative agreement by numerically solving the Vlasov-Fokker-Planck equation
(VFP) Warnock and Ellison (2000); Bane  _et al._ (2010); Brosi  _et al._
(2016); Schönfeldt  _et al._ (2017); Steinmann  _et al._ (2018).

In this contribution we focus on the perturbation of the synchrotron motion
that is caused by the CSR wake potential. Therefore, we adopt the perspective
of a single particle and consider its motion in the absence of collective
effects where the dynamics can be modeled by a simple one-dimensional harmonic
oscillator. By introducing CSR self-interaction as a perturbation (considering
the wake potential of a stationary charge distribution), the dynamics below
the threshold current can easily be illustrated in the single particle
picture. We will show how this perturbation breaks homogeneity in the
longitudinal phase space and leads to the formation of a quadrupole-like
modulation of the charge density, potentially acting as the seeding mechanism
for the micro-bunching instability. Furthermore, we consider single particle
motion above the instability threshold where the charge density is
continuously varying in time. Here, the single particle dynamics are largely
driven by a modulation of the effective potential’s slope caused by the
dynamic variation of the CSR wake potential. Eventually, we put these findings
into context with measurements taken at the KIT storage ring KARA and, based
on the gained insights, motivate an approach to control of the micro-bunching
dynamics using a dedicated RF modulation scheme.

##  II Longitudinal Beam Dynamics

As the CSR self-interaction predominantly affects the longitudinal motion of
particles, the micro-bunching instability can be described in good
approximation by modeling only the underlying longitudinal beam dynamics
Warnock and Ellison (2000); Bane  _et al._ (2010); Brosi  _et al._ (2016);
Schönfeldt  _et al._ (2017); Steinmann  _et al._ (2018). Due to the large
number of particles within a bunch (order of ${10}^{9}$ and higher),
collective effects such as CSR self-interaction are conveniently described by
modeling the charge distribution as a line charge and using a charge
distribution function $\psi(z,E)$ over the longitudinal position $z$ and the
particle energy $E$ instead of considering individual particles. The temporal
evolution of this charge distribution $\psi(z,E,t)$ is governed by the VFP
equation, which is introduced in the following subsection. In order to gain a
better understanding of the synchrotron motion of individual particles within
this distribution and the effects of the perturbation by the CSR wake
potential, single particle motion is considered directly afterwards.

###  II.1 Vlasov-Fokker-Planck Equation

Following the treatment and notation in Warnock and Ellison (2000); Bane  _et
al._ (2010), we introduce the generalized coordinates

$q=(z-z_{\mathrm{s}})/\sigma_{z,0}\quad\mathrm{and}\quad
p=(E-E_{\mathrm{s}})/\sigma_{E,0}~{}.$ (1)

where $z_{\mathrm{s}}$ and $E_{\mathrm{s}}$ denote position and energy of the
synchronous particle, $\sigma_{z,0}$ is the natural bunch length and
$\sigma_{E,0}$ is the natural energy spread. Thereby, the longitudinal charge
distribution $\psi$ can be described in the dimensionless phase space spanned
by $q$ and $p$, where the origin marks the position of the synchronous
particle. The temporal evolution of the normalized distribution
$\hat{\psi}(q,p,t)$ is described by the VFP equation

$\frac{\partial\hat{\psi}}{\partial\theta}+\frac{\partial\mathcal{H}}{\partial
p}\frac{\partial\hat{\psi}}{\partial q}-\frac{\partial\mathcal{H}}{\partial
q}\frac{\partial\hat{\psi}}{\partial
p}=\frac{1}{f_{\mathrm{s},0}\tau_{\mathrm{d}}}\frac{\partial}{\partial
p}\left(p\hat{\psi}+\frac{\partial\hat{\psi}}{\partial p}\right)~{},$ (2)

with time given in multiples of nominal synchrotron periods
$\theta=f_{\mathrm{s},0}\,t$, the Hamiltonian $\mathcal{H}$ and the
longitudinal damping time $\tau_{\mathrm{d}}$. The inhomogeneous part on the
right hand side describes the influence of radiation damping and diffusion. In
the absence of collective effects and assuming linear accelerating voltage
$V_{\mathrm{RF}}$ and linear momentum compaction factor $\alpha_{\mathrm{c}}$,
the Hamiltonian is given as

$\mathcal{H}_{0}(q,p,t)=\frac{1}{2}\left(q^{2}+p^{2}\right)~{}.$ (3)

The unperturbed system is thus a one-dimensional harmonic oscillator.
Collective effects such as CSR self-interaction can be included as a
perturbation to the Hamiltonian

$\mathcal{H}_{\mathrm{c}}(q,p,t)=\int_{q}^{\infty}Q_{\mathrm{c}}\,V_{\mathrm{c}}(q^{\prime},t)\mathrm{d}q^{\prime}~{},$
(4)

where $Q_{\mathrm{c}}$ denotes the charge involved in the perturbation and
$V_{\mathrm{c}}(q,t)$ is the potential due to collective effects. In order to
calculate the CSR-induced wake potential, it is useful to express the
potential in terms of an impedance $Z_{\mathrm{CSR}}(\omega)$

$V_{\mathrm{CSR}}(q,t)=\int_{-\infty}^{\infty}\widetilde{\rho}(\omega,t)Z_{\mathrm{CSR}}(\omega)e^{i\omega
q}\mathrm{d}\omega~{},$ (5)

where $\widetilde{\rho}(\omega)$ denotes the Fourier transformed longitudinal
bunch profile.

Figure 1: The process of CSR self-interaction can be described in good
approximation by modeling the shielding effect of the vacuum pipe by two
parallel plates. Note that the resulting impedance is frequency-dependent and
that higher frequencies generally correspond to a larger impedance and thus
stronger interaction. Shown is the CSR parallel plates impedance calculated
for the KIT storage ring KARA.

In general, the exact impedance of a storage ring is not known. However, in
the case of CSR-driven dynamics, the approximation of modeling the shielding
effect of the beam pipe by two parallel plates Murphy  _et al._ (1997) (see
FIG. 1) has proven to yield results which are quite comparable to experimental
data Brosi  _et al._ (2016); Schönfeldt  _et al._ (2017); Steinmann  _et al._
(2018). The full Hamiltonian is finally given by

$\mathcal{H}(q,p,t)=\mathcal{H}_{0}(q,p,t)+\mathcal{H}_{\mathrm{c}}(q,p,t)~{}.$
(6)

Figure 2: Shown is a snapshot of the charge distribution $\psi(q,p,t_{i})$ at
time step $t_{i}$ for a simulation of the longitudinal beam dynamics under CSR
self-interaction using the VFP solver Inovesa (left). By subtracting the
temporal average $\Delta\psi(q,p,t_{i})=\psi(q,p,t_{i})-\overline{\psi}(q,p)$
only the non-stationary part of the charge distribution remains, revealing
distinct micro-structures (right). Due to synchrotron motion the charge
distribution is rotating in phase space and evolving over time.

As of today, there is no analytic solution to the VFP equation for the full
Hamiltonian defined in Eq. (6) and arbitrary parameter settings. Following the
approach in Warnock and Ellison (2000) however, it can be solved numerically
on a discretized grid. To do so, we use the VFP solver Inovesa developed at
KIT Schönfeldt  _et al._ (2017). Inovesa is a parallelized open source
simulation code based on the mentioned approach, which enables us to do
extensive studies using merely standard desktop PCs. The generated simulation
data is thoroughly compared to measurements at the KARA storage ring and has
shown high qualitative agreement Steinmann  _et al._ (2018). As the micro-
bunching instability is reproduced with very similar characteristics, the
formation and evolution of these micro-structures can be studied in a fully
controlled environment facilitating a better understanding of the phenomenon.
An exemplary simulated charge distribution $\psi(q,p,t_{i})$ is shown in FIG.
2. By subtracting the temporal average

$\overline{\psi}(q,p)=\frac{1}{n}\sum_{i}^{n}\psi(q,p,t_{i})~{},$ (7)

the dynamically changing micro-structures become clearly visible. In the
following section, the single particle motion corresponding and leading to
this charge distribution is examined.

###  II.2 Single Particle Motion

In the absence of collective effects the Hamiltonian reduces to Eq. (3),
taking the form of a simple one-dimensional harmonic oscillator. This is
because the RF potential acts as a linear restoring force ($\sin(q)\approx q$
for $q$ close to zero)

$V_{\mathrm{RF}}(q)=-kq\quad~{},$ (8)

with the constant parameter $k$ describing the slope of the RF potential.
Neglecting radiation damping and diffusion, this leads to the simple equations
of motion

$\ddot{q}+\frac{k}{\xi}q=0\quad,\quad\ddot{p}+\frac{k}{\xi}p=0~{},$ (9)

with the constant scaling parameter $\xi$, and the solution

$\displaystyle q(t)$ $\displaystyle=a_{0}\cos(\omega t+\varphi_{0})$ (10)
$\displaystyle\dot{q}(t)$ $\displaystyle=p(t)=-a_{0}\,\omega\sin(\omega
t+\varphi_{0})~{},$ (11)

with $\omega=\sqrt{k/\xi}$, the amplitude $a_{0}$ and the initial phase
$\varphi_{0}$. Due to the specific choice of $q$ and $p$ in Eq. (1), the
expression $k/\xi$ simplifies to $1$, yielding the Hamiltonian in Eq. (3) and
perfectly circular trajectories in phase space. Let us now consider what
happens to these trajectories when we introduce a small perturbation to the
slope of the RF potential

$k^{\prime}=k-\varepsilon\quad\mathrm{with}\quad\varepsilon>0~{}.$ (12)

As the restoring force is still linear, the system remains a harmonic
oscillator, but now has the altered solution

$\displaystyle q^{\prime}(t)$
$\displaystyle=a_{0}\cos(\omega^{\prime}t+\varphi_{0})$ (13)
$\displaystyle\dot{q}^{\prime}(t)$
$\displaystyle=p^{\prime}(t)=-a_{0}\,\omega^{\prime}\sin(\omega^{\prime}t+\varphi_{0})~{},$
(14)

with $\omega^{\prime}=\sqrt{k^{\prime}/\xi}=\sqrt{(k-\varepsilon)/\xi}$. While
the maximum deviation in $q$ is unaffected

$\max\,\left\lvert q^{\prime}(t)\right\rvert=\max\,\left\lvert
q(t)\right\rvert=\left\lvert a_{0}\right\rvert~{},$ (15)

the maximum deviation in $p$ is decreased by the perturbation

$\max\,\left\lvert p^{\prime}(t)\right\rvert=\left\lvert
a_{0}\omega^{\prime}\right\rvert<\left\lvert
a_{0}\omega\right\rvert=\max\,\left\lvert p(t)\right\rvert~{}.$ (16)

Particle motion in the phase space spanned by the original definitions of $q$
and $p$ in Eq. (1) is thus elliptical, as illustrated in FIG. 3, and of
altered periodicity

$\left\lvert\omega^{\prime}\right\rvert<\left\lvert\omega\right\rvert~{}.$
(17)

Figure 3: A small perturbation of the RF slope, i.e. the strength of the
restoring force $k$ leads to an elliptical particle trajectory in the phase
space spanned by the generalized coordinates $q$ and $p$.

In order to examine the perturbation of this simple harmonic system by CSR
self-interaction, we need to consider the additional potential introduced in
Eq. (5).

##  III Particle Motion Below Threshold

Given the parallel plates impedance $Z_{\mathrm{CSR}}$ shown in FIG. 1, the
wake potential of a Gaussian bunch profile takes the form depicted in the
upper part of FIG. 4. While such a perfectly Gaussian electron distribution
only exists in the zero current limit, a higher bunch current leads to an
increased perturbation strength and thus distortion of the Gaussian shape.

Figure 4: CSR wake potential for a Gaussian bunch profile (red and blue, top)
and for the bunch currents
$I=($50$,$100$,$150$,$200$,$250$)\,$\mathrm{\SIUnitSymbolMicro A}$$ below the
instability threshold of
$I_{\mathrm{th}}=$260\text{\,}\mathrm{\SIUnitSymbolMicro A}$$ (bottom). Shown
are the respective temporal averages for simulations of the KIT storage ring
KARA using Inovesa (including damping and diffusion). Figure 5: Combining the
RF potential $V_{\mathrm{RF}}(q)$ and the CSR wake potential
$V_{\mathrm{CSR}}(q)$ yields an effective potential $V_{\mathrm{eff}}(q)$ that
the electrons are exposed to during their revolution in the storage ring.
Close to the synchronous position $q=0$, the potential can be approximated by
a linear function (shown as solid red line for an exemplary particle with an
amplitude of roughly $q_{\mathrm{max}}=$1.1$$). For larger deviations from
$q=0$ the linear approximation becomes less accurate, but still provides a
useful estimate of the perturbed potential.

Yet, below the threshold current, the distribution still remains fairly
stationary $\psi(q,p,t)\approx\psi(q,p)$, which corresponds to a stationary
wake potential as shown in the lower part of FIG. 4 for a range of different
bunch currents. It should be noted that the general shape of the wake
potential is still similar to that of a Gaussian shaped bunch up until right
below the threshold current of
$I_{\mathrm{th}}=$260\text{\,}\mathrm{\SIUnitSymbolMicro A}$$, where the wake
potential is no longer stationary. In order to investigate the effect of this
additional potential on the single particle motion, we introduce the effective
potential

$V_{\mathrm{eff}}(q)=V_{\mathrm{RF}}(q)+V_{\mathrm{CSR}}(q)~{},$ (18)

combining the linear RF potential $V_{\mathrm{RF}}(q)$ and the CSR wake
potential $V_{\mathrm{CSR}}(q)$. Thereby, we average over the full storage
ring and neglect the fact that these two types of interaction with external
fields are happening at different positions in the storage ring (e.g. RF
cavities and bending magnets). This simplification is reasonable as the
synchrotron motion happens at a much slower time scale than a single passage
of the storage ring ($f_{\mathrm{rev}}\gg f_{\mathrm{s},0}$).

A single particle moving in phase space is now subject to the effective
potential $V_{\mathrm{eff}}(q)$ over the interval
$\left[q_{\mathrm{min}},q_{\mathrm{max}}\right]$, where $q_{\mathrm{min}}$ and
$q_{\mathrm{max}}$ denote the maximum deviations from the longitudinal
position of the synchronous particle (see FIG. 3). By approximating
$V_{\mathrm{eff}}(q)$ as a linear function on the given interval

$V_{\mathrm{eff}}(q)\approx-k^{\prime}\,q~{},\quad
q\in\left[q_{\mathrm{min}},q_{\mathrm{max}}\right]~{},$ (19)

as illustrated in FIG. 5, the single particle motion is still harmonic below
the threshold current, with the strength of the restoring force $k^{\prime}$
being dependent on $q_{\mathrm{min}}$ and $q_{\mathrm{max}}$. While the linear
approximation seems quite suitable for particles close to $q=0$, the
approximation is getting inaccurate for oscillation amplitudes larger than
$\pm\sigma_{z,0}$. However, we’re mostly interested in describing the
potential near $q=0$ where the majority of the charge is located, and we will
see that this simple model yields a reasonable approximation nonetheless. As
is apparent from FIG. 5, the CSR wake potential acts as a perturbation of the
RF slope with the strength of the perturbation being dependent on the
amplitude of the particle’s oscillation. According to Eq. (12-17), this
results in a position-dependent ellipticity of particle trajectories in phase
space. Analogously, the oscillation frequency varies as a function of the
particle’s position.

Figure 6: In order to examine single particle trajectories, the initial charge
density $\psi(q,p,t_{0})$ (left) is modeled by a distribution of
$n=${10}^{5}$$ macro-particles (right). The single particle motion is then
simulated using the passive particle tracking implemented in Inovesa.

We can verify these insights using the passive particle tracking method111The
computation of the CSR wake potential is still based on the charge
distribution function $\psi(q,p,t)$, particles are tracked accordingly. that
was recently added to Inovesa Schönfeldt  _et al._ (2018). To that end, the
initial charge distribution $\psi(q,p,t_{0})$ is modeled by a particle
ensemble of $n=${10}^{5}$$ macro-particles (see FIG. 6). Subsequently, the
temporal evolution under the influence of the CSR wake potential is calculated
simultaneously for both, the charge and the particle distribution. The thus
obtained simulation results for the particle distribution are displayed in
FIG. 7. Here, the upper part shows the difference of the maximum amplitude in
$q$ and $p$

$\Delta
a_{\mathrm{max}}(q_{\mathrm{max}})=q_{\mathrm{max}}-p_{\mathrm{max}}~{},$ (20)

for the individual particles as a function of their maximum longitudinal
deviation $q_{\mathrm{max}}$. The particle trajectories clearly deviate from
$\Delta a_{\mathrm{max}}=0$, which would correspond to a circular trajectory,
and thereby show the expected position-dependent ellipticity. While the
trajectories are already elliptical close to the origin, the maximum
difference in amplitude is reached at values of $q_{\mathrm{max}}$ in the
range of $1.0$ to $1.5$ depending on the bunch current. Both the maximum value
of $\Delta a_{\mathrm{max}}$ and the corresponding longitudinal position are
increasing with higher bunch currents due to the increased strength of the
perturbation. For particles with larger deviation from the synchronous
particle the amplitude difference reduces again, indicating a trend to more
circular shaped trajectories for larger amplitudes.

Figure 7: Amplitude differences (top) and oscillation frequencies (bottom) of
$n=${10}^{5}$$ particle trajectories simulated with Inovesa for several bunch
currents below the instability threshold. Note that the oscillation
frequencies scatter significantly less than the amplitude differences. The
small oscillations of the frequencies in the range of
$$0$<q_{\mathrm{max}}<$1$$ are presumed numerical artifacts of the simulation
and data analysis. The solid red lines depict predictions based on the linear
approximation of $V_{\mathrm{eff}}(q)$ for the bunch current
$I=$250\text{\,}\mathrm{\SIUnitSymbolMicro A}$$.

Additionally, the lower part of FIG. 7 displays the corresponding oscillation
frequencies. As expected from Eq. (12-17), the individual synchrotron
frequencies of particles close to $q=0$ are significantly lower than the
nominal synchrotron frequency $f_{\mathrm{s},0}$ due to the perturbation of
the linear restoring force. Yet, this difference diminishes for particle
trajectories with larger amplitude yielding different oscillation frequencies
dependent on the position of the particles within the bunch. Finally, we can
directly compare these results with predictions based on the linear
approximation of $V_{\mathrm{eff}}(q)$. Therefore, we use the estimated value
of $k^{\prime}(q_{\mathrm{max}})$ to determine the oscillation frequency at
that position

$f_{\mathrm{s}}(q_{\mathrm{max}})=\omega_{\mathrm{s}}(q_{\mathrm{max}})/2\pi\approx\sqrt{k^{\prime}(q_{\mathrm{max}})}/\zeta~{},$
(21)

where $\zeta$ is just used for normalization purposes. Given an estimate of
the oscillation frequency and using the approximation $p_{\mathrm{max}}\approx
q_{\mathrm{max}}\omega_{\mathrm{s}}(q_{\mathrm{max}})$, we can easily
calculate the expected difference in amplitude

$\Delta a_{\mathrm{max}}(q_{\mathrm{max}})\approx
q_{\mathrm{max}}\left[1-\omega_{\mathrm{s}}(q_{\mathrm{max}})\right]~{}.$ (22)

The thus calculated estimates are shown as solid red lines in FIG. 7 for the
case of $I=$250\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. Clearly, the linear
approximation of the effective potential in Eq. (19) is already sufficient for
describing a major part of the perturbation by the CSR wake potential and its
effect on the synchrotron motion of single particles. As expected, the
estimates of $f_{\mathrm{s}}(q_{\mathrm{max}})$ and $\Delta
a_{\mathrm{max}}(q_{\mathrm{max}})$ deviate from the simulated trajectories
for larger values of $q_{\mathrm{max}}$ due to the inaccuracy of the linear
approximation of $V_{\mathrm{eff}}(q)$ for values further away from $q=0$. The
general shape however, can still be reproduced.

Figure 8: Visualization of the effect of the position-dependent elliptical
trajectories in phase space on the resulting charge density. Shown are
perfectly elliptical trajectories of ${10}^{3}$ particles with uniformly
distributed energies and an amplitude difference given by the estimate shown
as solid red line in FIG. 7. The large number of trajectories helps to
visualize the varying density of trajectories. In addition, the dashed red
line depicts a single particle trajectory with $q_{\mathrm{max}}\approx 1.1$.

In order to understand the implications of these modified single particle
trajectories, let us consider an ensemble of particles with uniformly
distributed energies and perfectly elliptical trajectories. Furthermore, let
the ellipticity be determined according to the position-dependent amplitude
difference $\Delta a_{\mathrm{max}}(q_{\mathrm{max}})$ shown as solid red line
in FIG. 7. As is apparent from the visualization in FIG. 8, this leads to a
non-uniform distribution of particle trajectories in phase space. In
particular, two distinguished locations of lower particle concentration are
visible close to the origin. Similarly, though harder to identify by eye,
there are two locations of higher particle concentration at larger oscillation
amplitudes (roughly at $q\approx\pm$2$$). This general pattern is a direct
consequence of the basic shape of the CSR wake potential shown in FIG. 4. The
CSR-induced perturbation of the RF potential thus breaks the homogeneity in
phase space and creates local particle densities that form a quadrupole-like
modulation of the longitudinal charge distribution. This inhomogeneity
introduces a higher frequency component to the longitudinal bunch profile and
may thereby initially seed the formation of micro-structures, i.e. kick off
the micro-bunching instability. Note that the general notion of dense particle
trajectories leading up to a distinct charge modulation within the bunch
resembles the caustic expression adopted for micro-bunching phenomena in
linear accelerators Charles  _et al._ (2016).

Figure 9: Shown is the magnitude of the Fourier transformed bunch profile
$\left\lvert\widetilde{\rho}(\omega)\right\rvert$ for a range of bunch
currents below the instability threshold of
$I_{\mathrm{th}}=$260\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. The red
rectangle marks the additional frequency component (at roughly
$85\text{\,}\mathrm{GHz}$) that is excited due to the perturbation by the CSR
wake potential.

We can further verify this initial excitation of a quadrupole-like mode by
examining the Fourier transformed longitudinal bunch profiles across different
bunch currents below the instability threshold. As is apparent from FIG. 9,
the current-dependent perturbation by the CSR wake potential introduces a
higher frequency component to the longitudinal charge distribution. For the
used parameter settings this frequency is found at about
$85\text{\,}\mathrm{GHz}$ corresponding to a modulation of the bunch profile
at the wavelength

$\lambda\approx c/$85\text{\,}\mathrm{GHz}$\approx$2.2$\,\sigma_{z,0}~{},$
(23)

which is roughly the distance of the two expected distinguished locations of
decreased charge density in phase space. Note that the micro-structures
occurring above the threshold current correspond to a significantly higher
frequency (here, roughly $150\text{\,}\mathrm{GHz}$), so the peak in FIG. 9
can clearly be attributed to the initial quadrupole mode.

Finally, we would like to conclude this section by pointing out that the
additional weak instability Bane  _et al._ (2010); Bane (2005); Brosi  _et
al._ (2019a), that occurs for specific parameter settings of the storage ring,
is usually observed with an instability frequency of $f_{\mathrm{inst}}\approx
2\,f_{\mathrm{s},0}$. This corresponds to a non-stationary quadrupole mode on
the longitudinal charge distribution, which aligns remarkably well with the
stationary deformation below the threshold current discussed here. Moreover,
measurements of such a quadrupole-like deformation of the longitudinal charge
distribution were already reported in Podobedov and Siemann (1997), albeit for
bunch currents above the instability threshold.

##  IV Particle Motion above Threshold

In the previous section we discussed the perturbation of single particle
synchrotron motion by the stationary CSR wake potential below the threshold
current. Here, the linear approximation of the effective potential in Eq. (19)
yields a simple model which sufficiently describes major aspects of the
resulting particle trajectories and thereby facilitates understanding the
implications of this perturbation. Essentially, the underlying longitudinal
dynamics can be considered a simple one-dimensional harmonic oscillator with a
position-dependent perturbation of the linear restoring force.

Figure 10: Shown are the amplitude difference (top) and oscillation frequency
(bottom) of particles trajectories for the bunch currents
$I=($250$,$260$,$270$,$280$,$290$)\,$\mathrm{\SIUnitSymbolMicro A}$$. As the
data scatters a lot, only a moving average over $q_{\mathrm{max}}$ is
displayed to enable a comparison between different currents. Figure 11:
Temporal evolution of the longitudinal bunch profile (top) and its difference
to the temporal average $\Delta\rho(q,t)=\rho(q,t)-\overline{\rho}(q)$
(bottom) for the bunch current $I=$290\text{\,}\mathrm{\SIUnitSymbolMicro
A}$$. In order to visualize the corresponding single particle motion
${10}^{3}$ particle trajectories are plotted on top with an opacity of $0.05$,
displaying a distinct modulation of the maximum deviation in the longitudinal
position. The maximum oscillation amplitude is reached for particles that are
exposed to the additional wake potential caused by the local charge modulation
(solid red curve).

Above the instability threshold, the longitudinal charge distribution as well
as the CSR wake potential are not stationary anymore, which makes this simple
model no longer applicable. Nevertheless, we will see how the notion of a
perturbed restoring force extends to the dynamics just above the instability
threshold and eventually motivates an approach to potential control of the
micro-bunching instability. The provided analysis builds upon extensive, prior
studies of the micro-bunching dynamics above the instability threshold at
KARA. It extends the gained understanding of the underlying longitudinal beam
dynamics and offers a new perspective on the interpretation of previous
observations.

In order to examine the single particle trajectories above the instability
threshold, the initial charge density $\psi(q,p,t_{0})$ is again modeled by a
distribution of macro-particles and passively tracked using Inovesa. The upper
part of FIG. 10 shows the amplitude difference $\Delta
a_{\mathrm{max}}(q_{\mathrm{max}})$ of the resulting particle trajectories for
a range of bunch currents in comparison to a current below the threshold
($I=$250\text{\,}\mathrm{\SIUnitSymbolMicro A}$$, violet line). Due to the
more complex dynamics and the variation of the individual particle
trajectories in time, the data scatters a lot. In order to enable a comparison
between multiple currents nonetheless, a moving average over
$q_{\mathrm{max}}$ is displayed. The elliptical shape is still apparent and
seems quite comparable to the results below the threshold current in FIG. 7.
The position of the maximum amplitude difference $\Delta
a_{\mathrm{max}}(q_{\mathrm{max}})$ however, is slightly shifting to larger
values. This implies that the local charge accumulation (see FIG. 8) also
shifts to a position further away from the origin. Particularly interesting is
the change in the distribution of the corresponding oscillation frequencies
that is displayed below. While the higher bunch current leads to an abrupt
change of the oscillation frequencies in the interval
$q_{\mathrm{max}}\in\left[$0.3$,$1.5$\right]$, it has a much smaller effect on
the oscillation frequencies of particles with larger amplitudes. This region
of $q_{\mathrm{max}}\in\left[$0.3$,$1.5$\right]$ is precisely where the micro-
structures occur in phase space and hints to the additional wake potential
caused by the corresponding charge modulation. To factor in the temporal
dynamics above the instability threshold, we need to find yet a different form
of visualizing the individual particle trajectories. Following the example of
prior work, e.g. Rota  _et al._ (2016); Funkner  _et al._ (2019); Brosi  _et
al._ (2019b), the upper part of FIG. 11 thus displays the temporal evolution
of the longitudinal bunch profile over a time period of two and a half
synchrotron periods for the bunch current
$I=$290\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. By close examination, the
periodic modulation of the charge density due to the occurring micro-
structures can already be identified. Nevertheless, in analogy to FIG. 2, the
lower part shows again the difference to the temporal average
$\overline{\rho}(q)$, which displays the micro-bunching dynamics much more
explicitly. In order to examine how the motion of single particles relates to
these micro-bunching dynamics ${10}^{3}$ particle trajectories are plotted on
top with an opacity of $0.05$. These trajectories are deliberately chosen to
have an average radius in phase space

$\overline{r}=\frac{1}{n}\sum_{i=1}^{n}r_{t_{i}}=\frac{1}{n}\sum_{i=1}^{n}\sqrt{q_{t_{i}}^{2}+p_{t_{i}}^{2}}~{},$
(24)

which is comparable to the estimated distance of the micro-structures from the
origin.

In the following subsections, we will examine different aspects of particle
motion above the instability threshold using the insights of previous
sections.

###  IV.1 Head-Tail Asymmetry

Particularly intriguing is the distinct sinusoidal modulation of the maximum
amplitude $q_{\mathrm{max}}$ that is visible at the head of the bunch ($q>0$).
This modulation is perfectly synchronized to the micro-structure dynamics in
the charge density and reaches its extrema at exactly the same positions in
time. However, similarly to the charge modulation, it predominantly occurs at
the head and diminishes towards the tail of the bunch. The maximum amplitude
in $q$ is reached when a particle travels on a trajectory that leads to its
exposure to an additional contribution in the CSR wake potential caused by the
local charge modulation. Particles traveling exactly along the position of the
maximum local charge density (red areas) while passing through
$q\in\left[0,2\right]$ are subsequently driven to the largest deviation in $q$
(illustrated by the solid red curve in FIG. 11). Similarly, particles passing
through the minima (blue areas) end up closest to the origin.

This asymmetry can be explained by the different effects a local structure at
different positions in the charge density has on the restoring force. For
$q>0$, a positive contribution to the effective potential $V_{\mathrm{eff}}$
results in a further decrease of the restoring force and thus drives particles
further outside in phase space. This amplifies the inhomogeneity and can
thereby drive and support the local charge modulation. In contrast, a positive
contribution at $q<0$ partially recovers the strength of the restoring force
and focuses the particles towards the center of the charge density, reducing
the inhomogeneity and damping the local structure.

###  IV.2 Formation of Micro-Structures

The previous subsection illustrated why the micro-structures are more
pronounced at the head of the bunch rather than the tail. Simultaneously, it
explains how single particle motion is leading up to these local charge
modulations. Particles are driven outside in phase space, cause an excess of
charge at that position and thereby form the occurring micro-structures.

Figure 12: Shown is the particle distribution at time step
$t=$0$\,T_{\mathrm{s}}$ for the bunch current
$I=$290\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. On the left hand side, each
individual particle is colored according to the corresponding relative charge
density $\Delta\psi(q,p,t_{\mathrm{color}}=$0$\,T_{\mathrm{s}})$ at that
particle’s location (opacity of $0.15$). On the right hand side, the color
assignment is adjusted to match
$\Delta\psi(q,p,t_{\mathrm{color}}=$1$\,T_{\mathrm{s}})$ instead. It thereby
illustrates which particles will form the micro-structures one synchrotron
period after this time step.

We would now like to further investigate how the motion of individual
particles relates to the motion of the observed micro-structures. Therefore,
we’ll take a look at the location of particles one synchrotron period before
they form a local structure. To do so, FIG. 12 displays the particle
distribution at time step $t=$0$\,T_{\mathrm{s}}$ in two different ways. On
the left hand side each individual particle is colored according to the
relative charge density at this time step

$\mathrm{Color}(n=i)\leftarrow\Delta\psi(q_{n=i},p_{n=i},t_{\mathrm{color}}=$0$\,T_{\mathrm{s}})~{},$
(25)

where $n=i$ is the particle index and $(q_{n=i},p_{n=i})$ denotes its location
in phase space at time $t_{\mathrm{color}}$. The color assignment on the right
hand side is adjusted to match the relative charge density one synchrotron
period afterwards
$\Delta\psi(q_{n=i},p_{n=i},t_{\mathrm{color}}=$1$\,T_{\mathrm{s}})$ instead.
The distribution on the right hand side thus shows where the particles forming
the micro-structures at time step $t=$1$\,T_{\mathrm{s}}$ were one synchrotron
period before. Thereby, we are able to analyze where the particles forming the
local structures come from and whether or not they stay within these
structures. Clearly, the two distributions look quite different. This implies
that the particles forming the structures at $t=$0$\,T_{\mathrm{s}}$ do not
necessarily participate in forming the same kind of structure one synchrotron
period later. They might e.g. travel from the position of a local maximum
(red) to position of a local minimum (blue) or vice versa. This effect was
already observed in previous studies Schönfeldt (2018).

FIG. 12 illustrates conceptually that the formation of these micro-structures
is not merely caused by the resonant motion of single particles, but rather by
the collective effect of many particles traveling on varying trajectories.
Moreover, during the analysis presented here, we found that the relation
between the motion of individual particles and the occurring micro-structures
can vary significantly across different bunch currents and parameter settings.

###  IV.3 Instability Frequency

As mentioned in the introduction, the occurrence of micro-structures in the
longitudinal phase space results in fluctuations of the emitted CSR power.
Right above the instability threshold, this fluctuation is typically dominated
by one characteristic frequency, which we will denote with
$f_{\mathrm{inst}}$. As reported by different facilities, e.g. Bane (2005);
Billinghurst  _et al._ (2016); Brosi (2020), this frequency is usually
observed close to an integer multiple of the nominal synchrotron frequency

$f_{\mathrm{inst}}\approx m\,f_{\mathrm{s},0}~{},$ (26)

but may deviate significantly depending on the parameter settings Billinghurst
_et al._ (2016); Brosi (2020). In this subsection, we aim to illustrate how
this frequency originates from the micro-bunching dynamics in phase space.

The integrated power of the emitted CSR $P_{\mathrm{CSR}}(t)$ is solely
determined by the projection of the charge density $\psi(q,p,t)$ on the
longitudinal axis, i.e. the longitudinal bunch profile $\rho(q,t)$ and the CSR
impedance. Different charge distributions in phase space at different time
steps $t_{i,j}$ thus result in the exact same value of the radiated power if
they correspond to identical longitudinal profiles
$\rho(q,t_{i})\equiv\rho(q,t_{j})$. Consequently, the periodic structure of
the fluctuating CSR power can be explained by a repetitive sequence of the
longitudinal profiles. The most trivial way to produce similar longitudinal
profiles at different time steps is by having the charge densities in phase
space be similar as well

$\psi(q,p,t_{i})\simeq\psi(q,p,t_{j})~{}.$ (27)

In the simulations with Inovesa we find this to always be the case. The charge
densities separated by one period of the instability frequency $\Delta
t=1/f_{\mathrm{inst}}$ are nearly indistinguishable by eye and only differ by
small numeric values. The observed instability frequency is thus directly
determined by the periodic behavior of the micro-bunching dynamics and the
corresponding time interval. However, as established above, the propagation of
these micro-structures in phase space and in time is a non-trivial subject.
With the varying oscillation frequencies of the single particle trajectories
(e.g. illustrated in FIG. 10) and the collective formation of the occurring
micro-structures, very little can be deduced about the oscillation frequency
of the micro-structures themselves. Empirically, we found that this can
deviate up to $$20$\,\%$ from the nominal synchrotron frequency. In that case,
the simple estimate in Eq. (26) is no longer applicable. In particular, trying
to estimate the integer $m$ via

$m\approx f_{\mathrm{inst}}/f_{\mathrm{s},0}$ (28)

will yield inconsistent and unreliable results as the micro-structures may
propagate at differing frequencies.

Lastly, note that the interpretation of $m$ as a simple azimuthal mode number
is difficult to align with the head-tail asymmetry of the CSR self-interaction
discussed in subsection IV.1. This asymmetry explains why the micro-structures
always form at the head of the bunch and are more pronounced there. The
instability frequency observed in the emitted CSR power is simply determined
by the repetition rate of this formation process.

Figure 13: Modifying the CSR shielding by varying the vacuum gap leads to
altered micro-bunching dynamics in the longitudinal phase space. While the
single particle frequency at the micro-structure position (blue squares)
increases smoothly with reduced vacuum gap, the oscillation frequency of the
micro-structures themselves (red circles) grows much faster and shows an
abrupt change when an additional structure is observed.

###  IV.4 Dependence on Shielding

Figures 2 and 6 illustrate the distinct charge modulation that forms under the
influence of CSR self-interaction in the micro-bunching instability.
Evidently, the modulation pattern is asymmetric as the micro-structures change
shape and notably amplitude depending on their position in phase space.
Nonetheless, one might be willing to identify an integer number
$n_{\mathrm{str}}$ of maxima constituting the charge modulation. Modifying Eq.
(26), we can relate this to the instability frequency

$f_{\mathrm{inst}}=n_{\mathrm{str}}\,f_{\mathrm{str}}~{},$ (29)

where $1/f_{\mathrm{str}}$ denotes the time it takes the micro-structures to
perform one full revolution in phase space. As discussed above, this is only
loosely related to the oscillation frequency of single particles and may
deviate from the nominal synchrotron period.

As reported in Boltz  _et al._ (2018), the integer $n_{\mathrm{str}}$ changes
if the shielding by the vacuum pipe is altered by varying its height. Within
this subsection, we are interested in examining the corresponding synchrotron
motion across these different parameter settings. Figure 13 thus displays the
oscillation frequencies of the micro-structures (red) and the single particle
frequencies at their positions in phase space (blue). All frequencies are
estimated directly above the instability threshold $I_{\mathrm{th}}$, which
itself is changing due to the varying shielding Bane  _et al._ (2010). The
dashed red line connects data points with the same number of maxima
$n_{\mathrm{str}}$ in the charge modulation pattern, which was estimated by
examining the relative charge density $\Delta\psi(q,p,t)$ by eye.

With increased shielding (reduced vacuum gap) the particle frequencies are
growing quite smoothly. The oscillation frequencies of the micro-structures,
however, show a differing behavior. While they take on very similar values for
the vacuum gaps $($19$,$22$,$24$,$27$,$32$)\,$\mathrm{mm}$$, the micro-
structure oscillation frequencies are growing much faster with decreasing
vacuum gap. This can be observed up until the point where an additional
extremum is identified and $n_{\mathrm{str}}$ is incremented. Afterwards, the
frequencies of particles and micro-structures are found at very similar values
again (e.g. transition from $28\text{\,}\mathrm{mm}$ to
$27\text{\,}\mathrm{mm}$). Figure 13 thereby illustrates again the partial
decoupling of the micro-structure propagation in phase space from single
particle motion.

The transitions in the observed number of micro-structures and the associated
changes in their oscillation frequency are of particular interest as they
provide additional insight into the formation process of the micro-structures
under different boundary conditions. We consider this a promising starting
point for further studies.

###  IV.5 Amplitude and Position of Micro-Structures

As explained in subsection IV.1, the additional wake potential caused by the
micro-structures at the head of the bunch can support and drive the micro-
bunching dynamics. Larger local charge modulations lead to a larger
perturbation by the additional wake potential which then results in the
individual particles being driven to larger oscillation amplitudes. Following
this chain of thought, naturally we expect a correlation between the maximum
amplitude of the occurring micro-structures and their maximum longitudinal
deviation from the origin in phase space.

Figure 14: Correlation between the maximum amplitude and maximum longitudinal
position of the occurring micro-structures. Shown is simulation data for bunch
currents between $260\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
$1000\text{\,}\mathrm{\SIUnitSymbolMicro A}$. The dashed red line illustrates
the clear linear correlation with a correlation coefficient of $0.91$.

In order to verify this, FIG. 14 displays the maximum amplitude and maximum
longitudinal position of the occurring micro-structures for a range of bunch
currents between $260\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
$1000\text{\,}\mathrm{\SIUnitSymbolMicro A}$. With increasing current the
strength of the perturbation caused by the CSR self-interaction increases,
which leads to larger amplitudes of the micro-structures within the bunch.
Figure 14 illustrates that this corresponds to a larger deviation from the
origin on the longitudinal axis as the particles are exposed to a stronger CSR
wake potential. For the simulation data considered here, we find a clear
linear correlation (as illustrated by the dashed red line) with a correlation
coefficient of $0.91$.

###  IV.6 Interpretation of Measurements

Finally, we discuss the implications of the previously gained insights on the
interpretation of the measurements taken at the KIT storage ring KARA.

One convenient way to observe the micro-bunching dynamics at the storage ring
is by measuring the emitted CSR power in the THz frequency range. At KARA, the
in-house developed KAPTURE system Caselle  _et al._ (2017) allows for a
continuous readout of the THz detectors on a turn-by-turn basis. The
measurement of this signal over decaying bunch current provides information
about the current dependent dynamics of the micro-bunching instability. As an
efficient and concise way to display the results, the data is typically post-
processed and visualized in CSR power spectrograms Brosi (2020); Steinmann
(2019) as shown in FIG. 15. Here, each horizontal line displays the Fourier
transformed CSR power signal measured at the corresponding bunch current with
the color code indicating the spectral intensity. Although the exact details
of the image depend on the used machine parameters (e.g. momentum compaction
factor or acceleration voltage), several features are always present and are
considered characteristic for the micro-bunching instability. Right above the
instability threshold, here at roughly $210\text{\,}\mathrm{\SIUnitSymbolMicro
A}$, the emitted CSR power fluctuates with a single distinct frequency
introduced as $f_{\mathrm{inst}}$ earlier. For higher currents this dominant
frequency increases slightly until it fans out at about
$230\text{\,}\mathrm{\SIUnitSymbolMicro A}$. Simultaneously, additional
contributions in the frequency range between $0\text{\,}\mathrm{kHz}$ and
$5\text{\,}\mathrm{kHz}$ emerge at the left edge of the figure. This general
pattern is always observed and indicates the transition from the longitudinal
dynamics directly above the instability threshold to the sawtooth bursting, as
e.g. initially observed at the SLC damping rings Krejcik  _et al._ (1993), at
higher bunch currents.

Figure 15: Measurement of the micro-bunching instability at the KIT storage
ring KARA (data already published in Schönfeldt  _et al._ (2017)). Shown is a
CSR power spectrogram in which each horizontal line displays the Fourier
transform of the CSR power signal at this particular bunch current. While the
specifics of the image may change dependent on machine parameters, some
characteristic features are always observed.

In the previous subsections, we discussed how the propagation of the micro-
structures in phase space may deviate from the single particle motion.
Nonetheless, given the general distribution of particle oscillation
frequencies in Figures 7 and 10, is seems reasonable to assume that also the
collectively formed structures propagate faster if they are located further
away from the origin in phase space. Based on this hypothesis we can derive a
basic reasoning which explains the occurrence of some characteristic features
in the CSR power spectrogram.

The increasing bunch current leads to a stronger perturbation by the CSR wake
potential and thus to a larger amplitude of the occurring micro-structures. As
established in subsection IV.5, this corresponds to a drift of the micro-
structure position towards larger longitudinal deviations in phase space.
Granted that this leads to a faster oscillation of the structures in phase
space ($f^{\prime}_{\mathrm{str}}>f_{\mathrm{str}}$), the instability
frequency is, according to Eq. (29), increasing as well. While this explains
the slight shift of the instability frequency across current, the simple
initial dynamics are only observed in a comparably small current range. In
FIG. 15, the dominant frequency fans out at roughly
$230\text{\,}\mathrm{\SIUnitSymbolMicro A}$ marking a clear transition in the
occurring dynamics. At these higher bunch currents, the micro-structures reach
an amplitude that can no longer be supported by the corresponding CSR wake
potential. Reaching this amplitude, the bunch blows up in size and the
structures smear out in phase space. As the increased bunch length leads to a
reduced perturbation by CSR, the bunch is mainly shrinking due to radiation
damping afterwards. Once the bunch length is short enough, the micro-
structures emerge again and continuously grow in amplitude. When forming
initially, the structures are located close to the origin in phase space and
propagate rather slowly, which corresponds to a low instability frequency
observed in the CSR power signal. With increasing amplitude the micro-
structures are then driven further outside in phase space and accelerate in
oscillation frequency. This leads to a sawtooth-shaped burst of CSR emission
up until the point where the micro-structure amplitude becomes too large and
the cycle starts anew. The spread out instability frequency is thus caused by
the varying oscillation frequencies of the micro-structures during their
continuous growth in phase space. These considerations motivate a future study
that concentrates on a more detailed analysis of the temporal evolution of the
instability frequency in both, simulations and measurements. The rather low
frequencies at the edge of the figure, however, correspond to the repetition
rate of the described bursting cycle and are thus related to the longitudinal
damping time as shown in Brosi  _et al._ (2018).

In order to approach control of the instability, we will thus concentrate on
time scales comparable to the formation process of the micro-structures, which
is governed by the synchrotron period.

Figure 16: Dynamic modulation of the RF amplitude in order to counteract the
perturbation by the CSR wake potential. Careful adjustment of the amplitude
and frequency of the RF modulation (top) mitigates the micro-bunching in the
longitudinal charge distribution (bottom). This reduces the fluctuation of the
emitted CSR power (blue, middle) compared to the natural behavior of the beam
(gray, middle). Note that this looks slightly different from FIG. 11 as the
simulations were done with a lower accelerating voltage.

##  V Control via dynamic RF Modulation

As discussed in the previous section, the micro-bunching dynamics above the
instability threshold correspond to a modulation of the restoring force, i.e.
the slope of the effective potential $V_{\mathrm{eff}}(q)$. That naturally
motivates an approach to control of the micro-bunching dynamics by aiming to
counteract this perturbation with an RF amplitude modulation

$\displaystyle V_{\mathrm{RF}}(t)$ $\displaystyle=\hat{V}(t)\sin(2\pi
f_{\mathrm{RF}}\,t)~{},$ (30) $\displaystyle\hat{V}(t)$
$\displaystyle=\hat{V}_{0}+A_{\mathrm{mod}}\sin(2\pi
f_{\mathrm{mod}}\,t+\varphi_{\mathrm{mod}})~{}.$ (31)

While the perturbation by the CSR wake potential cannot be compensated in its
entirety, this aims at stabilizing the strength of the restoring force and
thereby mitigating the micro-bunching dynamics. By choosing the natural
frequency of the occurring instability (e.g. observed in the emitted CSR
signal) as the modulation frequency $f_{\mathrm{mod}}=f_{\mathrm{inst}}$ and
carefully adjusting the amplitude $A_{\mathrm{mod}}$ and phase
$\varphi_{\mathrm{mod}}$, the RF modulation can be expected to partially
compensate the perturbation by the CSR wake potential. There is however one
major complication which has to be considered. Once we start interfering with
the natural beam dynamics, the evolution of the longitudinal charge
distribution and thus the CSR wake potential is changing as well. Particularly
the deliberately chosen modification of the restoring force leads to an
altered oscillation frequency and thus significantly affects the synchrotron
motion of the present micro-structures. Continuously applying this RF
amplitude modulation may therefore initially have the desired effect, but will
eventually run out of sync with the perturbation we are aiming to counteract.
In that case, the RF modulation may no longer stabilize the restoring force,
but actually further drive the instability. Hence, the RF modulation has to be
adjusted over time according to the altered micro-bunching dynamics

$\displaystyle A_{\mathrm{mod}}$ $\displaystyle\rightarrow
A_{\mathrm{mod}}(t),\quad f_{\mathrm{mod}}\rightarrow f_{\mathrm{mod}}(t)~{},$
(32) $\displaystyle\hat{V}(t)$
$\displaystyle=\hat{V}_{0}+A_{\mathrm{mod}}(t)\sin(2\pi
f_{\mathrm{mod}}(t)\,t+\varphi_{\mathrm{mod}})~{}.$ (33)

Figure 16 illustrates the results that can be achieved by applying such a
dynamic RF amplitude modulation scheme. The modulation of the RF amplitude
(top) clearly results in a mitigation of the micro-structures in the
longitudinal charge distribution (bottom) and stabilizes the emitted CSR power
(middle). It is worth pointing out that the RF modulation is neither reducing
the RF amplitude (on average) nor is it lengthening the bunch. These would
both be trivial ways to reduce the strength of CSR self-interaction we are not
interested in. In fact, the bunch length is even slightly reduced as indicated
by the increase of the CSR power signal towards the end of the displayed time
period ($$15$\,T_{\mathrm{s}}<t<$20$\,T_{\mathrm{s}}$). In order to achieve
these results the modulation amplitude
$A_{\mathrm{mod}}\in\left[0,4.5\right]${10}^{-4}$\,\hat{V}_{0}$ and frequency
$f_{\mathrm{mod}}\in\left[0.233,0.245\right]\,f_{\mathrm{s},0}$ were both
adjusted at a step width of $\Delta t=$0.25$\,T_{\mathrm{s}}$. While this step
width might possibly be relaxed to larger values, it has to be chosen small
enough to be able to react to the altered dynamics of the micro-bunching. As
the synchrotron period $T_{\mathrm{s}}$ governs the time scale of these
dynamics, the step width $\Delta t$ has to be chosen in the same order of
magnitude.

For the results shown in FIG. 16, the sequence of modulation amplitudes
$A_{\mathrm{mod}}(t_{i})$ and frequencies $f_{\mathrm{mod}}(t_{i})$ was found
via empirical testing. However in general, finding this dynamic sequence of RF
amplitude modulations is a non-trivial task that depends on the bunch current
and the specific machine parameters of the storage ring. As illustrated in
Boltz  _et al._ (2019), we can view this as a sequential decision problem
where the amplitudes $A_{\mathrm{mod}}(t_{i})$ and frequencies
$f_{\mathrm{mod}}(t_{i})$ need to be chosen iteratively, one step after
another. Informing these decisions with diagnostics about the current state of
the micro-bunching yields a closed feedback loop and provides the basis for
the reinforcement learning approach discussed in Boltz  _et al._ (2019).

Finally, it should be mentioned that the bunch currents considered here are
mostly below or directly above the instability threshold. While the
characteristic sawtooth-like bursting only occurs at higher bunch currents,
this behavior is still caused by very similar micro-structure dynamics within
the bunch (as discussed in Boltz (2017)). Controlling these micro-bunching
dynamics should thus imply control over the instability in the sawtooth regime
as well.

##  VI Summary and Outlook

In order to better understand the longitudinal dynamics underlying the micro-
bunching instability at storage rings we discussed different aspects of the
synchrotron motion under CSR self-interaction. To develop some intuition, we
focused on single particle motion and its perturbation by the CSR wake
potential. We showed that this system, to good approximation, can be viewed as
harmonic oscillator with perturbed linear restoring force. Using this simple
model we predicted a quadrupole-like charge modulation below the instability
threshold, which was verified using Inovesa simulations. Furthermore, we
considered synchrotron motion above the instability threshold focusing on how
the single particle motion relates to the formation and propagation of the
occurring micro-structures in phase space. We found that the observed charge
modulation is not merely caused by resonant motion of single particles, but
rather is a collective effect of many particles traveling on varying
trajectories. Moreover, structures are formed at the head of the bunch and are
located further away from the synchronous position the larger their
amplitudes. Eventually, we elaborated on how these insights help interpret
measurements taken at KARA and motivated an approach to control of the micro-
bunching dynamics.

Following the outlined concept of a longitudinal feedback loop, we propose a
method to control the micro-bunching dynamics to an extent that can enable new
operation modes at modern storage rings. If the longitudinal charge
distribution can be kept stable at these small bunch lengths, it would
significantly extend the limits of the longitudinal beam properties that can
be provided to experiments. Given the highly developed diagnostics at KARA,
the storage ring seems particularly well-suited for these efforts and first
studies are ongoing.

##  VII Acknowledgements

T. Boltz and P. Schreiber acknowledge the support by the DFG-funded Doctoral
School “Karlsruhe School of Elementary Particle and Astroparticle Physics:
Science and Technology (KSETA)”.

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