#  Landau damping of transverse head-tail instabilities with a pulsed electron
lens in hadron synchrotrons

Vadim Gubaidulin<EMAIL_ADDRESS>Technische Universität
Darmstadt, Schlossgartenstraße 8, 64289 Darmstadt, Germany  Vladimir Kornilov
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291
Darmstadt, Germany  Oliver Boine-Frankenheim  Technische Universität
Darmstadt, Schlossgartenstraße 8, 64289 Darmstadt, Germany  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291
Darmstadt, Germany  Elias Métral  CERN, CH-1211 Geneva 23, Switzerland

###### Abstract

A pulsed electron lens produces a betatron tune shift along a hadron bunch as
a function of the longitudinal coordinates, which is a longitudinal detuning.
An example of transverse detuning are the tune shifts due to octupole magnets.
This paper considers a pulsed electron lens as a measure to mitigate the head-
tail instabilities. Using a detailed analytical description within a Vlasov
formalism, the coherent properties of the longitudinal and transverse detuning
are presented. The analytical predictions are compared with the results of the
particle tracking simulations. A pulsed electron lens is demonstrated to be a
source of tune spread with two components: a static one, leading to Landau
damping; and a dynamic one, leading to an effective impedance modification,
which we demonstrate analytically and in our particle tracking simulations.
The effective impedance modification can be important for beam stability due
to devices causing a longitudinal detuning, especially for nonzero head-tail
modes. The Vlasov formalism is extended to include the combination of
longitudinal and transverse detuning. As a possible application at the SIS100
heavy-ion synchrotron ( Facility for Antiproton and Ion Research (FAIR) at GSI
Darmstadt, Germany), a combination of a pulsed electron lens with octupole
magnets is considered.

††preprint: APS/123-QED

##  I Introduction

Transverse collective instabilities, induced by the beam coupling impedances,
can limit the beam intensity in synchrotrons. The head-tail instability occurs
without a threshold in the beam intensity and it is observed or expected in
many synchrotrons (in the heavy ion synchrotron SIS100 of the Facility for
Antiproton and Ion Research (FAIR) accelerator complex [1, 2] or the Large
Hadron Collider (LHC) [3], for example). Instability mitigation methods
(linear chromaticity adjustments, Landau damping, transverse feedback system,
linear coupling, and others) play a crucial role in the operation of
synchrotrons for high beam intensities [4].

This contribution focuses on Landau damping, an effect caused by energy
exchange between coherent and incoherent motion, in bunched beams in ring
accelerators [5, 4, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This effect
requires an incoherent betatron tune spread in the bunch. The tune spread can
depend on longitudinal or transverse single-particle amplitudes. An incoherent
betatron tune shift as a function of the longitudinal amplitude is referred to
as the longitudinal detuning. The transverse detuning is the tune shift as a
function of the transverse action variable (for example, octupole magnets).
The dispersion relation analytically relates the incoherent tune spread to
Landau damping.

This paper introduces a longitudinally pulsed electron lens (PEL) as a source
of the longitudinal detuning and demonstrates its ability to mitigate
transverse instabilities via Landau damping. We compare PEL PEL with other
possible devices ( radio frequency quadrupole cavity (RFQ) [8], DC electron
lens (DC EL) [9], and Landau octupoles (LO) [10]) and investigate their
effects on the coherent beam stability with an SIS100 application in mind. The
Vlasov formalism is extended to include the linear combination of longitudinal
and transverse detuning. Particle tracking simulations are used to verify the
analytical results. As a possible application at the SIS100, a combination of
PEL PEL with LO is considered.

PEL PEL affects the hadron bunch via the electromagnetic field of a co-
propagating (or counter-propagating) electron beam, similar to DC EL DC EL. It
relies on pulsing the electron beam current in a way that each slice of the
hadron bunch receives a longitudinal position dependent kick. The longitudinal
pulsing has a Gaussian profile with the peak current occurring in the
longitudinal center of the bunch. The pulsing profile is matched to the bunch
profile if their root mean squared (rms) widths are equal. In this paper, the
transverse distribution of the electron beam is assumed to be homogeneous.
This PEL was proposed in [16] for space-charge compensation in the SIS18
[17]/SIS100 [18]. Instead DC EL DC EL relies on the transverse nonlinearity of
the electron beam’s electromagnetic field providing tune shifts depending on
the transverse single-particle amplitudes. In the SIS18, a prototype PEL will
be installed [19] and the resulting increase of the space charge limit in the
SIS18/SIS100 is under investigation [19, 20]. The effect of PEL PEL on
coherent beam dynamics and on transverse beam stability are the topic of the
this contribution.

Dispersion relations for Landau damping of transverse instabilities in bunches
were given in Eq. (1, 2) of [7] for two particular cases: only transverse
detuning or only longitudinal detuning. Combined dispersion relation for
arbitrary longitudinal and transverse bunch profiles have not been given
before. Such dispersion relations could be used to estimate stability
boundaries from the following devices and their combinations.

LO are a standard source of the transverse detuning to mitigate transverse
instabilities in several accelerators, e.g. the LHC [10], SIS100 [5] and the
proposed Future Circular Collider (hh) (FCC-hh) [11]. In recent years, several
alternative to LO methods of Landau damping were proposed, mainly for high
energy hadron colliders. The authors of [9] proposed DC EL DC EL as a source
of Landau damping due to the transverse detuning. They analytically estimated
and compared the resulting stability boundary with LO for the LHC and FCC-hh
using Eq. (1) from [7]. Both the magnitude and the shape of the stability
boundaries given in [9] for DC EL DC EL were not confirmed with particle
tracking simulations for different head-tail modes.

Authors of [8] introduced an RFQ as an alternative to LO for Landau damping.
An RFQ, like PEL PEL, provides Landau damping due to the longitudinal
detuning. Studies of an RFQ using the dispersion relation (Eq. (2) in [7]) and
particle tracking were performed in [8, 12]. In [8] a combination of LO with
an RFQ for the instability mitigation was proposed and verified in [12] using
particle tracking. However, no analytical expression for this combination was
given via a dispersion relation for a Gaussian bunch.

In [21, 13] the second order chromaticity was studied in an experiment and,
analytically, using a Vlasov formalism, where its effects on the coherent beam
stability were linked to an RFQ. A new dispersion relation (see Eq. (31) in
[13]) was derived including only the longitudinal detuning. Authors of [13]
establish two separate effects for an RFQ and for the second order
chromaticity $\xi^{(2)}$: Landau damping and an effective impedance
modification. These effects were studied separately. The effective impedance
modification was demonstrated in a case of an airbag bunch, with no incoherent
tune spread, which means no Landau damping. Whereas Landau damping was
demonstrated in a regime where the effective impedance modification was shown
to be weak. Therefore, it was impossible to establish the relative strength of
Landau damping and the effective impedance modification for various parameter
regimes. In this work we use the Vlasov formalism to derive a dispersion
relation that includes a linear combination of transverse and longitudinal
detuning.

Landau damping of nonzero head-tail modes was investigated in detail for
transverse detuning [22] due to octupoles. Studies of DC EL DC EL, an RFQ and
the second order chromaticity $\xi^{(2)}$ neglected to discuss Landau damping
of nonzero head-tail modes in detail using either particle tracking or
analytical expressions. While in the operation usually the instability is
shifted to nonzero head-tail modes by adjusting the linear chromaticity.

The main results of the paper are structured in the following manner. In
Section II a new dispersion relation for the linear combination of
longitudinal detuning and transverse detuning is derived. Furthermore, we
demonstrate that the longitudinal detuning induced by PEL PEL leads to Landau
damping of a transverse head-tail instability, similarly to higher-order
chromaticity and to an RFQ. The effective impedance modification and change of
the instability coherent tune shift are related to the head-tail mode spectrum
distortion by the longitudinal detuning.

In Section III zero head-tail mode stability boundaries are reconstructed for
PEL PEL, DC EL DC EL, LO and an RFQ with particle tracking using an antidamper
[23] as a rigid mode kick. Furthermore, with this method, we validate the
dispersion relation for a combination of LO and PEL PEL, for the FAIR SIS100.
These simulation results are compared to the respective dispersion relations
from Section II.

In Section IV we investigate Landau damping of nonzero head-tail modes using a
resistive wall impedance model. Particle tracking simulation results are
compared with analytical formulae of Section II for PEL PEL, DC EL DC EL, LO,
an RFQ. We discover that the effective impedance modification for nonzero
head-tail modes raises the threshold for Landau damping due to the
longitudinal detuning. In Section V our results are summarized.

##  II Vlasov description of the longitudinal detuning due to the pulsed
electron lens

The present derivation for PEL PEL applies the general perturbation formalism
presented in [13, 24, 25, 26]. Additionally, it expands to both transverse
$\Delta Q_{y}^{\perp}(J_{x},J_{y})$ and longitudinal $\Delta
Q_{y}^{\parallel}(J_{z},\varphi)$ detuning. ($J_{u}$ is used for the single-
particle amplitudes.) At the end, new dispersion relations are derived and two
effects of PEL PEL on the coherent beam stability are discussed. For the
longitudinal detuning, the dependency on the longitudinal phase $\varphi$ is
included, because the time scale of the head-tail instability development is
the timescale of the synchrotron motion, determined by the synchrotron tune
$Q_{\text{s}_{0}}$. The transverse detuning, on the contrary, can be averaged
over significantly faster betatron motion $Q_{{x_{0}},{y_{0}}}\gg
Q_{\text{s}_{0}}$. Therefore, the transverse detuning is justified to be
independent of the betatron phases $\theta_{x}$, $\theta_{y}$.

Using a perturbation method, let us assume that the distribution function
$\Psi$ has a small ($\epsilon\ll 1$) perturbation $\epsilon\Psi_{1}$:

$\Psi=f_{0}(J_{x},J_{y})g_{0}(J_{z})+\epsilon\Psi_{1}(J_{x},\theta_{x},J_{y},\theta_{y},J_{z},\varphi;t).$
(1)

Here we assume that the unperturbed distribution function can be factorized
into a longitudinal $g_{0}(J_{z})$ and a transverse distribution
$f_{0}(J_{x},J_{y})$. Time $t$ is taken in the revolution turns,
$(J_{u},\theta_{u})$ are action-angle variables. The Vlasov equation with a
dipolar wake $H_{\text{wake}}$ and the PEL $H_{\text{PEL}}$ contributions is

$\frac{d\Psi}{dt}=\frac{\partial\Psi}{\partial
t}+\left[H_{0}+H_{\text{PEL}}+\epsilon H_{\text{wake}},\Psi\right]=0,$ (2)

where $H_{0}=Q_{y_{0}}J_{y}+Q_{x_{0}}J_{x}-Q_{\text{s}_{0}}J_{z}$ is the
Hamiltonian of the unperturbed system, $\left[H,\Psi\right]$ is a Poisson
bracket notation.

This study is limited to the instabilities driven by dipolar wakefields. The
unperturbed distribution has no dipolar moment, therefore
$H_{\text{wake}}=\frac{F_{\text{wake}}(z,t)}{\omega_{0}\gamma
mv}\sqrt{2J_{y}\hat{\beta}_{y}}\sin{\theta_{y}}$ is of the order $\epsilon$
($\hat{\beta}_{y}$ is the average beta function). Thus, at the perturbation
order $\epsilon$, the Vlasov equation is:

$\frac{\partial\Psi_{1}}{\partial
t}+[H_{0},\Psi_{1}]+[H_{\text{PEL}},\Psi_{1}]=-[H_{\text{wake}},\Psi_{0}].$
(3)

Hamiltonians are normalised by the revolution frequency $\omega_{0}$ and by
the particle momentum $\gamma mv$. Let us consider PEL PEL with a homogeneous
transverse profile and a modulated current with a Gaussian longitudinal
profile ($z$ is the longitudinal coordinate)
$I_{e}(z)=I_{\text{max}}e^{-(z/\sqrt{2}\sigma_{e_{\parallel}})^{2}}$, then:

$H_{\text{PEL}}=\Delta
Q_{\text{max}}J_{y}e^{-(z/\sqrt{2}\sigma_{e_{\parallel}})^{2}},$ (4)

where $\Delta Q_{\text{max}}$ is given in the Table 1.

If the ratio of transverse and longitudinal (geometrical) rms emittances
satisfies $\varepsilon_{y}/\varepsilon_{z}\ll 1$ then the only significant
effect from PEL PEL is the longitudinal detuning $\Delta
Q_{y}^{\parallel}(J_{z},\varphi)=\frac{\partial H_{\text{PEL}}}{\partial
J_{y}}$, the same is true for an RFQ as a longitudinal detuning device.
Additionally, all synchrobetatron resonances are considered to be weak. The
same assumptions are made for the head-tail instability theory in [27] that
does not account for either transverse or longitudinal detuning. From the
Hamiltonian, we can find that PEL PEL tune shift is:

$\displaystyle\Delta Q_{y}^{\parallel}(J_{z},\varphi)/\Delta
Q_{\text{max}}=\underbrace{I_{0}^{(e)}\left(\frac{J_{z}}{2\varepsilon_{z}}\frac{\sigma_{z}^{2}}{\sigma_{e_{\parallel}}^{2}}\right)}_{\text{static}}$
$\displaystyle+\underbrace{2\sum_{n=1}^{\infty}I_{n}^{(e)}\left(\frac{J_{z}}{2\varepsilon_{z}}\frac{\sigma_{z}^{2}}{\sigma_{e_{\parallel}}^{2}}\right)\cos{(2n\varphi)}}_{\text{dynamic}},$
(5)

where $I^{(e)}_{n}(x)=e^{-x}I_{n}(x)$ stands for the exponentially scaled
modified Bessel function of the first kind. This tune shift has two
components. The static component is independent of the longitudinal phase. It
is similar to LO but depending on the longitudinal amplitude instead of the
transverse amplitudes. The dynamic component depends on the even harmonics
$\cos{2n\varphi}$. It is closer to the linear chromaticity $\xi^{(1)}$, its
average tune shift over a synchrotron period is zero. A similar Fourier
decomposition exists for any longitudinal detuning $\Delta
Q_{y}^{\parallel}(z,\delta)$ because it is guaranteed to be periodic in
$\varphi$. Chromatic tune shifts depend on the energy spread
$\delta^{n}\propto\sin^{n}{\varphi}$, and longitudinal position based kicks
(an RFQ or PEL PEL) depend on $z^{n}\propto\cos^{n}{\varphi}$. Thus, it is
generally not accurate to consider tune shifts only varying with the
longitudinal amplitude ignoring their dependency on the phase $\varphi$.

Assuming a dipolar motion only in the vertical plane in Eq. (3), the perturbed
distribution function has the following two terms
$\Psi^{\pm}_{1}(J_{x},\theta_{x},J_{y},\theta_{y},J_{z},\varphi,t)=\psi_{1}(J_{x},J_{y},J_{z},\varphi)e^{\pm
j\theta_{y}}e^{-jQ_{\text{coh}}t}$. Knowing that the instability coherent tune
shift is small $Q_{\text{coh}}\approx Q_{y_{0}}$, only one term
$\Psi^{-}_{1}\propto e^{-j\theta_{y}}$ is significant. The result, after
inserting the Hamiltonians and the perturbed distribution function in Eq. (3)
is a partial differential equation w.r.t $\varphi$ only:

$\displaystyle\\{Q_{\text{coh}}-Q_{y_{0}}-\Delta
Q^{\perp}_{y}(J_{x},J_{y})-\Delta Q_{y}^{\parallel}(J_{z},\varphi)$
$\displaystyle-j[Q_{\text{s}_{0}}+\Delta
Q_{\text{s}}(J_{z},\varphi)]\partial_{\varphi}\\}\psi_{1}(J_{x},J_{y},J_{z},\varphi)$
$\displaystyle=\sqrt{2J_{y}\hat{\beta}_{y}}g_{0}(J_{z})\frac{\partial
f_{0}(J_{x},J_{y})}{\partial
J_{y}}\frac{F_{\text{wake}}(z,t)}{2\omega_{0}\gamma mv},$ (6)

where the transverse detuning $\Delta Q_{y}^{\perp}(J_{x},J_{y})$ (e.g. LO) is
introduced as a small addition to the betatron tune $Q_{y_{0}}$. A small
synchrotron detuning $\Delta Q_{\text{s}}(J_{z},\varphi)\ll Q_{\text{s}_{0}}$
is also included here. Let us find the eigenfunction expansion of
$\psi_{1}=\sum_{l}\psi_{1}^{l}(J_{x},J_{y},J_{z},\varphi)$ and its eigenvalues
$Q^{l}$ assuming periodic boundary conditions
$\psi_{1}^{l}(J_{x},J_{y},J_{z},\varphi=0)=\psi_{1}^{l}(J_{x},J_{y},J_{z},\varphi=2\pi)$
and considering the differential operator in the left hand side of Eq. (6):

$\displaystyle Q^{l}=-Q_{\text{coh}}+Q_{y_{0}}+lQ_{\text{s}_{0}}$
$\displaystyle+\langle\Delta Q_{y}^{\parallel}\rangle_{\varphi}(J_{z})+\Delta
Q_{y}^{\perp}(J_{x},J_{y})+l\langle\Delta
Q_{\text{s}}\rangle_{\varphi}(J_{z}),$ (7)
$\displaystyle\psi_{1}^{l}(J_{z},\varphi)=A_{l}(J_{x},J_{y},J_{z})e^{jl\varphi}e^{-jB(J_{z},\varphi)},$
(8) $\displaystyle B(J_{z},\varphi)=\int_{0}^{\varphi}\left[\Delta
Q_{y}^{\parallel}(J_{z},\varphi^{\prime})-\langle\Delta
Q_{y}^{\parallel}\rangle_{\varphi^{\prime}}(J_{z})\right]\frac{d\varphi^{\prime}}{Q_{\text{s}}},$
(9)

Figure 1:  Comparison between the betatron phase factors $B(J_{z},\varphi)$
for PEL PEL, $\xi^{(2)}$ and an RFQ. The PEL and the RFQ are matched to the
rms bunch length, the value of $\xi^{(2)}$ corresponds to the same rms tune
spread, $J_{z}=\varepsilon_{z}$. The case of a linear chromaticity $\xi^{(1)}$
is shown for reference.

where $\langle\Delta Q_{y}^{\parallel}\rangle_{\varphi^{\prime}}(J_{z})$
indicates averaging over a synchrotron period and thus depends only on the
longitudinal amplitude $J_{z}$. The function $B(J_{z},\varphi)$ generalises
the betatron phase factor $\chi$ of the linear chromaticity $\xi^{(1)}$ (see
Eq. (6.185, 6.187) in [27]) for arbitrary longitudinal detuning. It has been
modified w.r.t Eq. (20) in [13] to include $Q_{\text{s}}(J_{z},\varphi)$ in
the integral to account for $\Delta Q_{\text{s}}(J_{z},\varphi)$ detuning and
its effect on the betatron phase factor. The function $B$ specifies for a
given particle with an amplitude $J_{z}$ and a phase $\varphi$, by how much it
heads or trails the head particle in the betatron phase. Figure 1 demonstrates
the variation of the betatron phase factor over a single synchrotron period:
for the linear chromaticity $\xi^{(1)}$, there is a phase difference between
the head and the tail of the particle reflected by the $B$-function; for PEL
PEL or the second order chromaticity $\xi^{(2)}$, bunch head and tail
particles have the same phase, but particles near bunch ends still have
different phases. Specific formulae for the betatron phase factor
$B(J_{z},\varphi)$ and tune shifts $\Delta Q_{y}$ are given in Table 1 for PEL
PEL, an RFQ, DC EL DC EL, and LO. One can observe that the $B$-function of PEL
PEL and an RFQ has only even harmonics of $\varphi$, and only the $J_{z}$
dependency is different. For PEL PEL and for an RFQ, the first even harmonic
$\sin{2\varphi}$ is the strongest one, corresponding to the second order
chromaticity $\xi^{(2)}$ tune shift (see Eq. (33) in [13]). The $B$-function
of PEL PEL is described well by the first two even $\varphi$ harmonics, see
Fig. 1. Figure 1 illustrates that the
$B(J_{z}=\varepsilon_{z},\varphi)$-function of PEL PEL is similar to that of
the second order chromaticity for $\sigma_{z}=\sigma_{e_{\parallel}}$ (from
Table 1) or to an RFQ when the wavelength is matched to the bunch length.
Landau damping with the transverse detuning, in contrast, does not affect
betatron phase relation between the head and the tail of the bunch.

| LO | RFQ | PEL | DC EL
---|---|---|---|---
$\Delta Q_{\text{max}}$ | $\propto I_{\text{LO}}\frac{J_{x}}{\varepsilon_{x}}$ | $\frac{\hat{\beta}_{y}}{2\pi}\frac{qv_{2}}{p_{0}\omega_{\text{RFQ}}}$ | $\frac{Z}{A}\frac{I_{e}}{I_{a}}\frac{m_{e}}{m_{p}}\frac{gL_{e}}{4\pi\varepsilon_{n_{x}}}\frac{\sigma_{x}^{2}}{\sigma_{e}^{2}}\frac{1\pm\beta_{e}\beta_{i}}{\beta_{p}\beta_{i}}$
$\langle\Delta Q_{y}/\Delta Q_{\text{max}}\rangle_{\phi}$ | $a_{xx}\frac{J_{x}}{\varepsilon_{x}}+a_{xy}\frac{J_{y}}{\varepsilon_{y}}$ | $J_{0}\left(\frac{\sigma_{z}}{\lambda_{\text{RFQ}}}\sqrt{2\frac{J_{z}}{2\varepsilon_{z}}}\right)$ | $I^{(e)}_{0}\left(\frac{J_{z}}{2\varepsilon_{z}}\frac{\sigma_{z}^{2}}{\sigma_{e\parallel}^{2}}\right)$ | $\begin{aligned} &&\int_{0}^{1}\left[I^{(e)}_{0}\left(uk_{y}\right)-I^{(e)}_{1}\left(uk_{y}\right)\right]\\\ &&\times I^{(e)}_{0}\left(uk_{x}\right)du\end{aligned}$
$\Delta Q_{\text{rms}}/\Delta Q_{\text{max}}$ | 0.24 | 0.2 | 0.14 | 0.16
$B(J_{z},\varphi)/\frac{\Delta Q_{\text{max}}}{Q_{\text{s}}}$ | 0 | $\sum_{n=1}^{\infty}J_{2n}\left(\frac{\omega_{\text{rfq}}\sigma_{z}}{\beta_{\text{r}}c}\frac{J_{z}}{\varepsilon_{z}}\right)\frac{\sin{(2n\varphi)}}{n}$ | $\sum_{n=1}^{\infty}I^{(e)}_{n}(\frac{J_{z}}{2\varepsilon_{z}}\frac{\sigma_{e_{\parallel}}^{2}}{\sigma_{z}^{2}})\frac{\sin{(2n\varphi)}}{n}$ | 0
$H_{l}(z^{(1)},z^{(2)},..z^{(k)})$ | $j^{-l}J_{l}(z^{(1)})$ |  $\approx j^{-l}\sum_{n=-\infty}^{\infty}J_{l+2n}(z^{(1)})J_{n}(z^{(2)})$ | $j^{-l}J_{l}(z^{(1)})$
Table 1:  Tune spreads, betatron phase factors $B(J_{z},\phi)$ and head-tail
spectrum functions $H(z^{(1)},z^{(2)})$ for electron lenses, an RFQ and LO.
$k_{x,y}=\frac{J_{x,y}}{2\varepsilon_{x,y}}\frac{\sigma_{x,y}^{2}}{\sigma_{e_{\perp}}^{2}}$.

After substituting $\Psi_{1}$ using the eigenfunction expansion of Eq. (8)
into the Vlasov equation Eq. (6) it reduces to an equation for the function of
the transverse and longitudinal amplitudes $A_{l}(J_{x},J_{y},J_{z})$:

$\displaystyle\sum_{l^{\prime}=-\infty}^{\infty}\frac{A_{l^{\prime}}(J_{x},J_{y},J_{z})}{\sqrt{2J_{y}\hat{\beta}_{y}}\frac{\partial
f_{0}(J_{x},J_{y})}{\partial
J_{y}}}e^{jl^{\prime}\varphi}e^{-jB(J_{z},\varphi)}Q^{l^{\prime}}$
$\displaystyle=-\frac{F_{\text{wake}}(J_{z},\varphi;t)}{2\omega_{0}\gamma
mv}g_{0}(J_{z}).$ (10)

The right hand side of Eq. (10) does not depend on the transverse amplitudes
$\\{J_{x},J_{y}\\}$, thus the left hand side must be constant w.r.t. these
coordinates. The amplitude dependency can be redefined as
$A_{l^{\prime}}(J_{x},J_{y},J_{z})=R_{l^{\prime}}(J_{z})I_{l^{\prime}}(Q_{\text{coh}})\sqrt{2\hat{\beta}_{y}/J_{y}}$,
where $R_{l^{\prime}}(J_{z})$ is the longitudinal mode. By substituting the
expression for the amplitude dependency in Eq. (10) we find the dispersion
integrand $I_{l}(Q_{\text{coh}})$:

$I_{l}(Q_{\text{coh}})=\frac{\frac{\partial f_{0}(J_{x},J_{y})}{\partial
J_{y}}J_{y}}{Q_{\text{coh}}-Q_{y_{0}}-\Delta Q_{y}^{\perp}-\langle\Delta
Q_{y}^{\parallel}\rangle_{\varphi}-lQ_{\text{s}}}.$ (11)

This separation of pure longitudinal amplitude dependency
$R_{l^{\prime}}(J_{z})$ from $A_{l^{\prime}}(J_{x},J_{y},J_{z})$ is only
possible because we consider a specific combination of transverse and
longitudinal detuning $\Delta Q_{y}=\Delta
Q_{y}^{\parallel}(J_{z},\varphi)+\Delta Q_{y}^{\perp}(J_{x},J_{y})$ and not a
more general case of detuning $\Delta Q_{y}(J_{x},J_{y},J_{z},\varphi)$. In
the latter case one would need to solve an integral equation for the
amplitudes and the betatron phase factor $B$ will also be a function of the
amplitudes $J_{x},J_{y}$ (in addition to $J_{z}$).

Now everything about the perturbed distribution function $\Psi_{1}$ is known,
except its longitudinal modes $R_{l}(J_{z})$:

$\displaystyle\Psi_{1}=$ $\displaystyle
e^{-jQ_{\text{coh}}t}e^{-jB(J_{z},\varphi)}\sum_{l=-\infty}^{\infty}R_{l}(J_{z})e^{jl\varphi}$
(12) $\displaystyle\times
e^{-j\theta_{y}}I_{l}(Q_{\text{coh}})\sqrt{2\hat{\beta}_{y}/J_{y}}.$

In order to find the longitudinal modes we need to solve the Vlasov equation
perturbed by a wakefield force, expressed as

$\displaystyle
F_{\text{wake}}=\frac{q^{2}Q_{\text{s}_{0}}\omega_{0}}{Q_{y_{0}}\eta
R}\sum_{p=-\infty}^{\infty}Z_{y}^{\perp}(Q_{p})e^{jQ_{p}\frac{r}{R}\cos{\varphi}}\sum_{l^{\prime}=-\infty}^{\infty}\lambda_{l}(Q_{p}).$
(13)

For comparison, see Eq. (6.173) in [27] with $r=\sqrt{2J_{z}\hat{\beta}_{z}}$,
where $\hat{\beta}_{z}$ is the longitudinal beta function. The frequency of
the impedance $Q_{p}=Q_{y_{0}}+lQ_{\text{s}_{0}}+p$ is normalized by the
revolution frequency $\omega_{0}$, $R=C/(2\pi)$ is the accelerator effective
radius, $\eta$ is the slip factor, $q$ is the ion charge. The line density of
the dipolar moment $\lambda_{l}(Q_{p})$ of the distribution function $\Psi$
for the azimuthal mode $l$ in the frequency domain is

$\lambda_{l}(Q_{p_{0}})\doteq\int_{0}^{\infty}H^{p_{0}}_{l}(r)R_{l}(r)\left[\iint
I_{l}(Q_{\text{coh}})dJ_{x}dJ_{y}\right]rdr.$ (14)

$H$-functions have the physical meaning of an airbag bunch head-tail mode
spectrum. In general, the head-tail mode spectrum is $|\lambda_{l}|^{2}$ and
it depends on both the spectrum functions $H$ and the longitudinal
distribution function $g_{0}$. Integral representation of $H$-functions was
first defined by the authors of [13] in Eq. (22):

$H^{p}_{l}(J_{z})\doteq\int_{0}^{2\pi}e^{-jQ_{p}\frac{r}{R}\cos{\varphi}}e^{-jB_{\text{PEL}}}e^{jl\varphi}\frac{d\varphi}{2\pi}.$
(15)

Using Jacobi-Anger expansion [28], we would introduce a simpler sum
representation, considering only the first even harmonic
($\propto\sin{2\varphi}$) of $B_{\text{PEL}}$, see Table 1, and the
contribution of the wakefield, and the linear chromaticity
$\xi^{(1)}\propto\cos{\varphi}$:

$H_{l}^{p}(z^{(1)},z^{(2)})=j^{-l}\sum_{n=-\infty}^{n=+\infty}J_{l+2n}(z^{(1)})J_{n}(z^{(2)}),$
(16)

Figure 2:  (top) Head-tail mode spectra $|H_{l}^{p}(z^{(1)},z^{(2)})|^{2}$ for
an airbag bunch, unperturbed (solid lines) and with the effect of PEL PEL
(dashed lines) for $\Delta Q_{\text{max}}/Q_{\text{s}_{0}}=0.5$. (bottom)
Dependency of mode power spectrum on the PEL strength $\Delta Q_{\text{max}}$
at points where modes $|l|=\\{0,1\\}$ are near their maxima.

where $J_{n}(x)$ is the Bessel function of the first kind [28]. The arguments
of the $H$-function are defined as

$\displaystyle z^{(1)}$
$\displaystyle=(Q_{p}-Q_{y_{0}}\xi^{(1)}/\eta)\frac{2\sigma_{z}}{R}\sqrt{\frac{J_{z}}{2\varepsilon_{z}}},$
(17) $\displaystyle z_{\text{PEL}}^{(2)}$ $\displaystyle=\frac{\Delta
Q_{\text{max}}}{Q_{\text{s}_{0}}}I_{1}^{e}\left(\frac{J_{z}}{2\varepsilon_{z}}\frac{\sigma^{2}_{z}}{\sigma^{2}_{e_{\parallel}}}\right),$
(18)

where $z^{(1)}$ describes the first harmonic effects (wakefield and
$\xi^{(1)}$), and $z_{\text{PEL}}^{(2)}$ describes the second harmonic effects
(from PEL PEL in Eq. (18), an RFQ or the second order chromaticity $\xi^{(2)}$
have a similar formula). This expression can be extended to include other
higher harmonics as well. The sum representation in Eq. (16) has advantages
for faster numerical calculations and for understanding the effect of PEL PEL
on the head-tail mode spectrum. Indeed, if $z^{(2)}\approx 0$, only $n=0$ term
is nonzero, and we have a known expression $j^{-l}J_{l}(z^{(1)})$ (for
example, Eq. (6.177) and discussions therein in [27]). As $z^{(2)}$ rises,
$n=\pm 1$ becomes significant and the mode spectrum mixes with $J_{l\pm
2}(z^{(1)})$ terms and $J_{n=0,1}(z^{(2)})$ serve as the weight functions. The
head-tail mode spectrum distortion by PEL PEL, determined by
$z_{\text{PEL}}^{(2)}$, scale with the parameter $\Delta
Q_{\text{max}}/Q_{\text{s}_{0}}$. Additionally, like for the Bessel functions
[28], from Eq. (16) $\sum_{l}|H^{p}_{l}(z^{(1)},z^{(2)})|^{2}=1$ for any $p$,
$z^{(1)}$, $z^{(2)}$. Meaning that the second harmonic effect $z^{(2)}$
redistributes the energy between different modes, modifying the shape of the
mode spectrum by making some modes stronger and others weaker. Distortion of
the mode spectrum by the longitudinal detuning is the origin of the effective
impedance modification.

An example for changes in the mode spectrum is demonstrated in Fig. 2 (top
plot) for a moderately strong PEL $\Delta
Q_{\text{max}}/Q_{\text{s}_{0}}=0.5$. The weak distortion of the zero mode is
evident from Eq. (16), because the sum becomes symmetric for $\pm n$ terms and
odd terms cancel each other. The $l$-modes are no longer degenerate, the
$l=1,2$ modes become weaker, and the $l=-1,-2$ modes become stronger, see Fig.
2 (top plot). In Fig. 2 (bottom plot), we consider two different frequencies
and demonstrate how the mode spectrum at these frequencies is modified
depending on the strength of the longitudinal detuning $\Delta
Q_{\text{max}}/Q_{\text{s}_{0}}$.

After pasting Eq. (13) into Eq. (10), multiplying both sides by
$e^{-jl\varphi}$ and integrating over $\varphi$, a single mode $l$ is
determined,

$R_{l}(J_{z})=jKg_{0}(J_{z})\sum_{p=-\infty}^{\infty}Z_{y}^{\perp}(Q_{p})H_{l}^{p^{\dagger}}(J_{z})\sum_{l^{\prime}=-\infty}^{\infty}\lambda_{l^{\prime}}(p),$
(19)

with $K=q^{2}Q_{\text{s}_{0}}/(2Q_{y_{0}}E\eta T_{0})$ ($1/\text{Ohm}$), where
$E$ is the energy and $T_{0}$ is the revolution period. This extends the
Sacherer’s integral equation [29]. Using the Laclare’s approach [30], an
expression for $\lambda_{l}(p_{0})$ is obtained by multiplying both sides of
Eq. (19) by $H^{l}_{p_{0}}\left[I_{l}(Q_{\text{coh}})dJ_{x}dJ_{y}\right]rdr$
and integrating:

$\displaystyle\lambda_{l}(p_{0})=-jK\sum_{p=-\infty}^{\infty}\sum_{l^{\prime}=-\infty}^{\infty}\lambda^{l^{\prime}}(p)Z_{y}^{\perp}(Q_{p})$
$\displaystyle\times\int_{0}^{\infty}\left[I_{l}(Q_{\text{coh}})dJ_{x}dJ_{y}\right]H^{p^{\dagger}}_{l}(r)H_{l}^{p_{0}}(r)g_{0}(r)rdr.$
(20)

Consequently, this general integral equation simultaneously includes Landau
damping with transverse and longitudinal detuning. If the transverse detuning
is negligible $\Delta Q_{y}^{\perp}(J_{x},J_{y})=0$, the results of Eq. (23)
in [13] are recovered. If the longitudinal detuning is negligible
$\langle\Delta Q_{y}^{\parallel}\rangle_{\varphi}(J_{z})=0$, the results of
the dispersion relation Eq. (1) in [7] are recovered. Thus, our results for
the linear combination of transverse and longitudinal detuning are general and
in the limiting cases converge to the known results. Equation (20) can be
written in the matrix form, and the solutions to this eigenvalue problem would
be the coherent tune shifts and the eigenmodes. Dedicated Vlasov solvers are
usually used to solve this type of an eigenvalue problem numerically [31, 14].
However, the present semi-analytical Vlasov solvers do not account for
arbitrary longitudinal detuning. Under the assumption of a narrow band
impedance (see [13]) and only one excited azimuthal mode, we obtain the
following dispersion relation:

$\Delta Q^{-1}=\frac{1}{N_{1}}\int\dfrac{\dfrac{\partial\Psi_{0}}{\partial
J_{y}}J_{y}|H_{l}^{p_{0}}(z^{(1)},z^{(2)})|^{2}dJ_{z}dJ_{x}dJ_{y}}{Q_{\text{coh}}-Q_{y_{0}}-\langle\Delta
Q_{y}^{\parallel}\rangle_{\varphi}(J_{z})-\Delta
Q_{y}^{\perp}(J_{x},J_{y})-l[Q_{\text{s}_{0}}+\langle\Delta
Q_{\text{s}}\rangle_{\varphi}(J_{z})]},$ (21)

where $N_{1}=\int\dfrac{\partial\Psi_{0}}{\partial
J_{y}}J_{y}|H_{l}^{p_{0}}(z^{(1)},z^{(2)})|^{2}dJ_{x}dJ_{y}dJ_{z}$ is the
normalisation found from no Landau damping case. $\Delta Q$ is the coherent
tune shift in the absence of Landau damping ($\langle\Delta
Q_{y}^{\parallel}\rangle_{\varphi}(J_{z})=\Delta
Q_{y}^{\perp}(J_{x},J_{y})=\langle\Delta
Q_{\text{s}}\rangle_{\varphi}(J_{z})=0$). $Q_{\text{coh}}$ is the coherent
tune in the presence of Landau damping. This case corresponds to the static
component of the tune shifts being zero, but the normalisation $N_{1}$ is
still affected by the dynamic component of the tune shift via $H^{l}_{p_{0}}$
function.

Linear chromaticity $\xi^{(1)}$ is a special case of the longitudinal detuning
that has no static component and, thus, no Landau damping. In this case the
effective impedance changes and the mode spectrum modification is a shift by
the chromatic frequency $Q_{y_{0}}\xi^{(1)}/\eta$. In Eq. (22), no assumptions
about the accelerator impedance and the arguments of the $H$-function are
necessary for the case of the transverse detuning only.

If we assume a small first argument $z^{(1)}\ll 1$ (Eq. (17)) of the
$H$-function in Eq. (16), this is a further assumption on the impedance,
meaning that the frequency of the narrow band impedance (shifted by the
chromatic frequency) is much smaller than the spread of frequencies in the
bunch. Additionally, we assume a small second argument $z^{(2)}\ll 1$ (Eq.
(18)) such that $J_{0}(z^{(2)})\approx 1$ and other weight functions in Eq.
(16) are approximately zero—the head-tail mode spectrum is unperturbed by the
longitudinal detuning. The dispersion relation simplifies to

$\Delta Q^{-1}=\frac{1}{N_{2}}\int\dfrac{\dfrac{\partial\Psi_{0}}{\partial
J_{y}}J_{y}J_{z}^{|l|}dJ_{x}dJ_{y}dJ_{z}}{Q_{\text{coh}}-Q_{y_{0}}-\langle\Delta
Q_{y}^{\parallel}\rangle_{\varphi}(J_{z})-\Delta
Q_{y}^{\perp}(J_{x},J_{y})-l[Q_{\text{s}_{0}}+\langle\Delta
Q_{\text{s}}\rangle_{\varphi}(J_{z})]},$ (22)

where $N_{2}=\iiint\frac{\partial\Psi_{0}}{\partial
J_{y}}J_{y}J_{z}^{|l|}dJ_{x}dJ_{y}dJ_{z}$. This corresponds to Eq. (1, 2) of
[7] if we set either the longitudinal $\Delta Q_{y}^{\parallel}$ or the
transverse detuning $\Delta Q_{y}^{\perp}$ to zero. Our results are valid for
arbitrary distributions $\Psi_{0}=f_{0}(J_{x},J_{y})g_{0}(J_{z})$, account for
the dynamic part of the longitudinal detuning $\Delta
Q_{y}^{\parallel}(J_{z},\varphi)$ and quantify the head-tail mode spectrum
distortion by the longitudinal detuning.

##  III  Stability diagrams and Landau damping of the zero head-tail mode

In this section, the rigid mode kick model reconstructs the stability
boundaries in the simulations. This model corresponds to a constant wake force
acting on the beam (a delta-function impedance
$Z_{y}^{\perp}\propto\delta(Q)$) [14]. Therefore, it drives a rigid mode
oscillations with a specific coherent tune shift, corresponding to the zero
head-tail mode. A similar method was described and employed in a proof-of-
principle experiment in [23] using a transverse feedback system (as an
antidamper) in the LHC study and it agreed with the known stability boundaries
for LO.

The rigid mode of bunch oscillations is driven in the simulations by the
following kick:

$\Delta y^{\prime}\propto\Im\Delta Q\overline{y^{\prime}}+\Re\Delta
Q\overline{y}/\hat{\beta}_{y}$ (23)

where $\overline{y}$, $\overline{y^{\prime}}$ are the beam offset and its
derivative; $\Re\Delta Q$, $\Im\Delta Q$ are real and imaginary parts of the
coherent tune shift caused by the kick in the absence of Landau damping,
$\hat{\beta}_{y}$ is the average beta function of the ring. The present
implementation of the rigid mode kick is only for the zero linear chromaticity
$\xi^{(1)}=0$ case. Thus, the parameter $z^{(1)}=0$. Using a weak longitudinal
detuning $\Delta Q_{\text{max}}/Q_{\text{s}_{0}}\ll 1$, we ensure that the
underlying assumptions of the dispersion relation Eq. (22) are fulfilled.

In the simulations, 2D scans over the complex coherent tune shift are
performed. All 2D scans are made with the same beam and accelerator parameters
using $10^{5}$ macroparticles over $10^{5}$ turns ($\gtrsim 150$ synchrotron
periods), only changing the source of the detuning (PEL PEL, DC EL DC EL, LO,
an RFQ). In this section the bunch is taken to be Gaussian transversely and
longitudinally. Separating this 2D plane into a stable and an unstable areas
results in the reconstruction of the stability boundaries. We reconstruct
stability boundaries from the simulation data by identifying all points (on
the coherent tune shift complex plane) with exponentially growing beam offset
and extrapolating their growth rates to the zero growth rate isoline—obtained
stability boundaries are defined using the same criterion as the analytical
ones.

The tracking code PyHEADTAIL [32] is employed for the simulations. The
implementations of LO and of an RFQ are identical to [13], both DC EL DC EL
and PEL PEL are implemented as slice-by-slice localized kicks. For DC EL DC
EL, each particle of the ion beam receives a kick from the field of a
transversely Gaussian electron beam, with a constant longitudinal profile. For
PEL PEL, the electromagnetic field of the electron beam is transversely
homogeneous, and the kick amplitude is modulated along the bunch length
matching the ion beam profile.

First, our particle tracking simulation should verify that a tune shift from
PEL PEL leads to Landau damping. Then we compare PEL PEL to the other means of
Landau damping (LO, an RFQ, DC EL DC EL) using the same simulation framework,
the same criteria for beam stability and the same rms tune spread from all
four devices. Finally, we demonstrate the stability boundary from a
combination of longitudinal detuning and transverse detuning, using PEL PEL
and LO as an example. These simulation results are used to verify our
analytical results from Section II and specifically the dispersion relation
Eq. (22).

Figure 3 (top) compares simulated (green histograms) tune distributions for LO
(yellow), DC EL DC EL (red), PEL PEL (blue), and an RFQ (light-blue) to the
corresponding analytical distributions (coloured histograms). The vertical and
horizontal tune spreads and the complex coherent tune shift axes in Fig. 3 are
normalised by the rms betatron tune spread $\Delta Q_{\text{rms}}$. Using this
normalisation allows us to compare various tune spread distributions from PEL
PEL, LO, DC EL DC EL and an RFQ.

Figure 3:  (top) Incoherent tune spreads and average tune shifts (green dot).
Left to right: LO (yellow), DC EL DC EL (red), PEL PEL (dark-blue), an RFQ
(light-blue). (bottom) Stability boundaries for the head-tail mode $l=0$ from
Eq. (22) compared to the results of the simulation scans with the rigid mode
kick, Eq. (23).

Figure 3 (bottom) shows the reconstructed stability areas (solid green lines)
of all four devices together with the analytical results from Eq. (22)
(colored lines). For equal rms tune spreads, the stability area is roughly the
same. An RFQ and LO tend to have a wider stability boundary. This is
attributed to larger tails in the tune distribution for LO and an RFQ. Tune
shifts from PEL PEL are the same in the vertical and the horizontal planes,
but not for an RFQ. Thus, PEL PEL, contrary to an RFQ, ensures the same
stability area in both the horizontal and the vertical planes. An RFQ would
require a two-family scheme or a combination with LO (see [12]) because the
instability coherent tune shift is typically similar in both planes.

Figure 3 (bottom right) illustrates a near-perfect agreement with analytical
stability boundaries for the weak longitudinal detuning. Furthermore,
comparisons with Landau damping with the transverse detuning reveal that the
stable area is roughly the same for equal rms tune spreads. As a rule of
thumb, $\Delta Q_{\text{rms}}$ defines the stability boundary ’height’,
meaning that the fastest instability that is damped has a growth rate roughly
equal to the rms tune spread $\approx\Delta Q_{\text{rms}}$. The full tune
spread $\Delta Q_{\text{full}}\approx$4-6$\Delta Q_{\text{rms}}$ determines
the ’width’ of the stability boundary. The shape of the stability boundary is
related to the incoherent tune distribution.

In Fig. 4 analytical estimations are compared with our simulation results for
the transverse Gaussian distribution and a round electron beam for DC EL DC EL
for several values of the transverse beam size ratio
$r=\sigma_{e_{\perp}}/\sigma_{{x_{0}},{y_{0}}}=\\{0.7,1.0,1.4,1.8\\}$. Similar
ratios were considered only analytically in [9]. Therefore, our simulation
results confirm that DC EL DC EL has a possibility to slightly adjust the
stability boundary, depending on this ratio.

Figure 4:  DC EL stability diagrams from Eq. (22) (red lines) for the head-
tail mode $l=0$ depending on the electron to ion rms beam transverse size
ratio $r=\sigma_{e_{\perp}}/\sigma_{{x_{0}},{y_{0}}}$. It is compared to the
results of simulation scans (green lines) with the rigid mode kick (defined in
Eq. (23)).

Finally, Fig. 5 shows Landau damping from a combination of PEL PEL and LO from
the results of the simulation scans (green) and from the dispersion relation
Eq. (22) (dark-blue). The light-blue and the light-yellow lines show the
stability diagram from the dispersion relation for the cases with PEL PEL only
and LO only. For both devices, the same settings as in Fig. 3 are applied. The
stability boundary (if compared to PEL PEL one only) increases in the tails
and becomes wider. This device combination helps to mitigate instabilities for
a broader range of real coherent tune shifts $\Re\Delta Q_{\text{coh}}$ but
not the instabilities with a higher growth rate $\Im\Delta Q_{\text{coh}}$. In
a comparison with the LO only case, the device combination mitigates the
instabilities with the nearly doubled growth rate, and in the similar range
for the real coherent tune shifts.

Figure 5:  Stability diagrams for the head-tail mode $l=0$ from Eq. (22) for
LO (transverse detuning $\Delta Q_{y}^{\perp}(J_{x},J_{y})$, light-yellow
line), for PEL PEL (longitudinal detuning $\langle\Delta
Q_{y}^{\parallel}(J_{z})\rangle_{\varphi}$, light-blue line), and for the
related combination (dark-blue line). The corresponding simulation results for
the device combination is shown by the green curve.

##  IV  Landau damping of nonzero head-tail modes

This section investigates the Landau damping of wakefield driven instabilities
from PEL PEL, with the focus on nonzero head-tail modes. Thresholds of the
instability suppression of PEL PEL are compared with the ones for LO, DC EL DC
EL and an RFQ. Threshold of the instability suppression, or threshold for
Landau damping is an rms tune spread value at which the beam is stabilised.
The resistive wall impedance $Z_{y}^{\perp}\propto 1/\sqrt{Q_{p}}$, relevant
for many hadron accelerators (e.g. [11, 33]), is used in this section.

Relevant accelerator and beam parameters for the simulation results of this
section are summarized in Table 2.

$\xi^{(1)}$ | $l$ |  $\Re\Delta Q_{\text{inst}}$/$Q_{\text{s}_{0}}$ |  $\Im\Delta Q_{\text{inst}}$/$Q_{\text{s}_{0}}$ | Accelerator Parameters
---|---|---|---|---
-0.02 | 0 | -0.530 | 0.017 | $\eta$ | $3.45\cdot 10^{-4}$
0.1 | -1 | -0.052 | 0.017 | $Q_{\text{s}_{0}}$ | $1.74\cdot 10^{-3}$
0.5 | -2 | -0.017 | 0.009 |  $N_{\text{macro}}$/$N_{\text{turns}}$ |  $10^{5}$/$10^{5}$
Table 2:  Simulation and accelerator parameters for the wakefield driven
instability simulations. The coherent tune shifts are for the case with no
Landau damping.

The bunch is taken to be Gaussian transversely and longitudinally. A linear
radio frequency (RF) bucket is used in the simulations, thus the synchrotron
frequency detuning is not taken into account. We identify several linear
chromaticity settings where the head-tail modes $l=\\{0,-1,-2\\}$ are
unstable. The instability coherent tune shifts and chromaticity settings are
given in Table 2.

Frequency spectra and characteristic intrabunch motion of these instabilities
are illustrated in Fig. 6.

Figure 6:  Spectra and the trace plots of the centroid motion for head-tail
instabilities for three head-tail modes $l=0$ (right curves), $l=-1$ (middle
curves), $l=-2$ (left curves). Those are the instabilities when both the
longitudinal and the transverse detuning are zero.

The mode spectra maxima are near the related synchrotron sidebands, indicating
the azimuthal mode number $l$ for each mode. A characteristic head-tail
pattern with the $|l|$ number of nodes in the offset trace plots is observed.
Each instability has an exponential growth of the transverse offset. The
instability parameters are chosen to be well below the threshold of the
transverse mode coupling instability (TMCI).

Our particle tracking simulation results (solid lines) for Landau damping of
head-tail modes $l=\\{0,-1,-2\\}$ are demonstrated in Figs. 7, 8, 9; where we
compare it to the analytical predictions (dashed lines) of Eq. (22).

Figure 7: Instability growth rate dependency on the strength of Landau damping
$\Delta Q_{\text{rms}}/Q_{\text{s}_{0}}$ for head-tail mode $l=0$. PEL PEL
(dark-blue), LO (yellow), DC EL DC EL (red), an RFQ (light-blue) results are
compared with respective dispersion relations Eq. (22) (dashed lines).  Figure
8:  Instability growth rate dependency on the strength of Landau damping
$\Delta Q_{\text{rms}}/Q_{\text{s}_{0}}$ for head-tail mode $l=-1$. PEL PEL
(dark-blue), LO (yellow), DC EL DC EL (red), an RFQ (light-blue) results are
compared with respective dispersion relations Eq. (22) (dashed lines).  Figure
9:  Instability growth rate dependency on the strength of Landau damping
$\Delta Q_{\text{rms}}/Q_{\text{s}_{0}}$ for head-tail mode $l=-2$. PEL PEL
(dark-blue), LO (yellow), DC EL DC EL (red), an RFQ (light-blue) results are
compared with respective dispersion relations Eq. (22) (dashed lines).

Equation 22 does not account for the effective impedance modification by the
longitudinal detuning. Therefore, if this effect is significant for the head-
tail modes of a Gaussian bunch, we will observe a disagreement between Eq.
(22) and the simulation results. Also, four betatron frequency detuning
devices are compared to each other: PEL PEL (dark-blue), an RFQ (light-blue),
DC EL DC EL (red), LO (yellow). For our particle tracking scans, we
iteratively increase the strength of Landau damping for each identified head-
tail mode setting from Table 2. The strength of Landau damping is expressed by
the rms betatron tune spread $\Delta Q_{\text{rms}}$ to allow a comparison
between the devices. It is normalised by the synchrotron tune
$Q_{\text{s}_{0}}$. The instability growth rate is normalised by its value in
the absence betatron frequency detuning $\Im\Delta Q_{\text{inst}}$, see Table
2.

The simulation results for the head-tail mode $l=0$ are presented in Fig. 7.
We observe a good agreement with Eq. (22) both for the stability threshold and
for the evolution of the instability growth rate. No significant effective
impedance modification due to the longitudinal detuning (PEL PEL, an RFQ) is
observed for the zero head-tail mode.

Figures 8, 9 demonstrate qualitatively the same particle tracking results for
head-tail modes $l=\\{-1,-2\\}$. For the transverse detuning (LO, DC EL DC EL)
the threshold for the instability suppression agrees with the dispersion
relation Eq. (22). For the longitudinal detuning (PEL PEL, an RFQ), in the
simulation the threshold of the instability suppression is higher than
expected from Eq. (22). For example, for PEL PEL the threshold of $\Delta
Q_{\text{rms}}/Q_{\text{s}_{0}}\approx 0.2$ is observed in the simulation,
whereas analytically we expect a lower value $\Delta
Q_{\text{rms}}/Q_{\text{s}_{0}}\approx 0.05$. This indicates that there is a
destabilising effect from PEL PEL and an RFQ for the head-tail modes
$l=\\{-1,-2\\}$. In Section II this was identified as the effective impedance
modification due to the longitudinal detuning. The stronger effective
impedance modification for nonzero head-tail modes was already indicated by
Fig. 2. In contrast, transverse detuning from LO or DC EL DC EL agrees with
the dispersion relation Eq. (22) regardless of the head-tail mode number.

Figure 10:  Stability boundaries for two PEL strengths for the mode $l=-1$
(solid and dashed lines respectively) from Eqs. (21, 22) and respective
coherent tune shifts of the instability (stars) from Eq. (20). Accelerator and
beam parameters are the same as in Fig. 8. Two Landau damping strengths
$\Delta Q_{\text{rms}}/Q_{\text{s}_{0}}=\\{0.05,0.07\\}$ are considered
(magenta and cyan colors respectively).

The interplay between the effective impedance modification and Landau damping
for the head-tail mode $l=-1$ is illustrated in Fig. 10. The same accelerator
and beam parameters as in Fig. 8 are used for two PEL strengths. This figure
illustrates two effects: first, stability boundary (solid and dashed lines for
Eqs. (21, 22) respectively) is increasing due to Landau damping; second, the
instability coherent tune shift from Eq. (20) gets further away from this
boundary due to the effective impedance modification. In the calculation we
used the airbag bunch head-tail mode spectrum as an approximation. In Fig. 10
the stability boundaries for two PEL rms tune spread values are presented.
First, at $\Delta Q_{\text{rms}}/Q_{\text{s}_{0}}\approx 0.05$ (magenta
color), is the stability threshold estimated from Eqs. (21, 22). In this
situation the instability coherent tune shift, in the absence of Landau
damping (grey star), lies inside the stability boundary. But the instability
coherent tune shift from Eq. (20) (magenta star) does not, due to the
effective impedance modification. The new stability threshold (cyan star) is
then at the PEL rms tune spread of $\Delta
Q_{\text{rms}}/Q_{\text{s}_{0}}=0.07$. In this situation the effective
impedance modification, illustrated as a shift of the coherent frequency, is
$\approx 62\%$. At the same PEL strength, but for the head-tail mode $l=0$ it
would be $\approx 1\%$. The instability suppression threshold for the head-
tail mode $l=-1$ in the simulation is significantly higher at $\Delta
Q_{\text{rms}}/Q_{\text{s}_{0}}\approx 0.2$. This discrepancy could be due to
several approximations that were made. First, we approximated the head-tail
mode spectrum with the one of an airbag bunch. Second, dispersion relations
Eqs. (21, 22) were derived for a delta-function like narrow band impedance.
But the resistive wall impedance has the frequency dependency $\propto
1/\sqrt{Q_{p}}$. Thus, for more accurate description of the effective
impedance modification and Landau damping one needs to solve Eq. 20 without
assuming an airbag bunch or narrowband impedance or employ particle tracking
simulations. But the above analytical approach is sufficient to illustrate the
interplay between the effective impedance modification and Landau damping.

##  V Conclusions

The PEL as a source of the longitudinal detuning has been introduced and its
effects on transverse beam stability have been demonstrated. A Vlasov
formalism was used to derive new dispersion relations Eqs. (21, 22) for a
linear combination of the transverse and the longitudinal detuning. There are
two distinct effects of the PEL (or other longitudinal detuning) on the
transverse beam stability: Landau damping due to the static component of the
detuning; and the head-tail mode distortion due to the dynamic component of
the detuning, see Eq. (5).

The head-tail mode distortion is the origin of the effective impedance
modification [13], as it is explained in Section II, see Fig. 2 and Eqs. (9,
16). In contrast, for a transverse detuning there is no effective impedance
modification and only Landau damping is affected. These analytical results
were confirmed for a Gaussian bunch with particle tracking in Sections III, IV
for three different head-tail modes and compared to the stability boundaries
of LO, DC EL DC EL, an RFQ.

The betatron phase factor was demonstrated to vary along the bunch in the
presence of the longitudinal detuning, see Eq. (9) and Fig. 1. This variation
relates the PEL dynamic longitudinal detuning to the dynamic longitudinal
detuning due to higher-order chromaticity [13] and the RFQ [12].

Fig. 2 displays how the head-tail mode spectrum is affected depending on the
strength of the longitudinal detuning from PEL PEL. The head-tail modes in the
presence of the longitudinal detuning are described by the sum in Eq. 16,
similar sum representation exists for any longitudinal detuning.

The ratio $\Delta Q_{\text{max}}/Q_{\text{s}_{0}}$, determines the strength of
the effective impedance modification by the longitudinal detuning. In the weak
regime $\Delta Q_{\text{max}}/Q_{\text{s}_{0}}\lesssim 1$, only the zero head-
tail mode is described well by the dispersion relation Eq. (22), see Fig. 7.
The rigid mode kick model was shown to reconstruct the shape and the magnitude
of the stability boundaries, see Fig. 3. For the zero head-tail mode, Landau
damping either with transverse or longitudinal detuning is equally effective
for a given rms tune spread $\Delta Q_{\text{rms}}$, see Fig. 3. The
simulation results for the zero head-tail mode (see Fig. 4) indicate that with
DC EL DC EL it is possible to adjust the stability boundary by changing the
electron to ion beam size ratio.

The nonzero head-tail modes are described by Eqs. (20, 21) which include the
effective impedance modification, leading to the instability amplification,
see Figs. 2, 8. For nonzero modes this is a significant effect, neglected by
Eq. (22). The instability becomes stronger because some modes are getting
weaker and other modes are amplified , see Fig. 2. For the nonzero head-tail
modes Landau damping with longitudinal detuning requires rms tune spreads
$\Delta Q_{\text{rms}}$ up to 2-5 times larger than predicted by the
dispersion relation Eq. (22) due to the destabilising effect of the effective
impedance modification (see Figs. 8, 9). Landau damping with transverse
detuning, on the contrary, does not have this effect and is described by the
dispersion relation Eq. (22).

In the strong lens regime $\Delta Q_{\text{max}}/Q_{\text{s}_{0}}\gg 1$, the
instability is shifted towards higher-order modes with a smaller growth rate,
see Fig. 2. This is the regime foreseen for PEL PEL in the SIS100 as a space-
charge compensation device [16]. Due to the large mode spectrum modification
by PEL, expressions of Eq. (20, 21) should be used instead of Eq. (22). Landau
damping is stronger for a larger rms tune spread. Therefore, both Landau
damping and the effective impedance modification from PEL PEL are stabilising
in this regime.

The complex interplay between PEL PEL and space-charge is beyond the scope of
the present work. In [34] the feasibility of Landau damping with DC EL DC EL
in space-charge dominated beams was demonstrated. For PEL PEL the same
argumentation should be valid for the static component of the detuning. While
the dynamic component of the detuning should lead to a weaker head-tail
instability, see Fig. 2 (bottom).

Combinations of transverse and longitudinal detuning devices are expected in
several hadron accelerators. SIS100 is foreseen [2] to have both, PEL PEL and
LO. For the LHC, a combination of LO and an RFQ was recently proposed [8]. We
have derived new integral equations Eqs. (19, 20) and dispersion relations
Eqs. (21, 22) for the combinations of transverse and longitudinal detuning and
for arbitrary bunch profiles. Those dispersion relations are an extension of
Eq. (31) in [13], where the transverse detuning is not included. Our
dispersion relation Eq. (21) simplifies to Eq. (22) for a situation with the
assumptions discussed at the end of Section II. Finally, our dispersion
relation Eq. (22) reduces to the Eqs. (1, 2) from [7] for either longitudinal
or transverse detuning. Stability boundaries due to PEL PEL, LO and their
combination are given and verified with simulation results, see Figs. 3, 5. A
transversely non-linear PEL is not covered in this contribution, see section
II, Eq. (10) and discussion therein. One can presume that Landau damping will
stay approximately the same for equivalent rms tune spreads regardless if the
lens is pulsed or not. The effective impedance modification or head-tail modes
is however not trivial in this case. Nonlinear synchrotron oscillations were
not taken into account in this contribution but could be relevant for SIS100
operation with longer bunches.

In summary, we presented an analytical description of Landau damping due to a
linear combination of transverse and longitudinal detuning. Coherent
properties of longitudinal detuning and transverse detuning, and Landau
damping due to four devices (PEL, DC EL, LO, RFQ), were compared in detail.
The application of PEL PEL in different settings for Landau damping was
demonstrated. We showed that the effective impedance due to longitudinal
detuning has a significant effect on the beam stability, especially for
nonzero head-tail modes. The results from analytical considerations are
verified using particle tracking simulations.

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  *[FAIR]: Facility for Antiproton and Ion Research
  *[LHC]: Large Hadron Collider
  *[PEL]: pulsed electron lens
  *[RFQ]: radio frequency quadrupole cavity
  *[DC EL]: DC electron lens
  *[LO]: Landau octupoles
  *[rms]: root mean squared
  *[FCC-hh]: Future Circular Collider (hh)
  *[RF]: radio frequency
  *[TMCI]: transverse mode coupling instability


# Fast acquisition of spin-wave dispersion by compressed sensing

Ryo Kainuma1 Keita Matsumoto1,2 Takuya Satoh1 1Department of Physics, Tokyo
Institute of Technology, Tokyo 152-8551, Japan
2Department of Physics, Kyushu University, Fukuoka 819-0385, Japan

(February 11, 2021)

###### Abstract

For the realization of magnonic devices, spin-wave dispersions need to be
identified. Recently, the time-resolved pump-probe imaging method combined
with the Fourier transform was demonstrated for obtaining the dispersions in
the lower-wavenumber regime. However, the measurement takes a long time when
the sampling rate is sufficiently high. Here, we demonstrated the fast
acquisition of spin-wave dispersions by using the compressed sensing
technique. Further, we quantitatively evaluated the consistency of the
results. Our results can be applied to other various pump-probe measurements,
such as observations based on the electro-optical effects.

††preprint: APS/123-QED

Spin waves are collective modes of spin precession in magnetically ordered
materials. They are considered promising information carriers in the field of
magnonics, because they can propagate over a long distance without Joule
heating [1, 2, 3, 4]. Various devices such as spin wave switches [5],
magnonic-logic circuits [6, 7], spin wave-assisted recorders, [5, 8], and low-
magnetic-field sensors [9] require the spatial control of spin waves.

The propagation characteristics of spin waves are manifested in their
dispersion relation. The higher-wavenumber regime is governed by exchange
interactions. In contrast, lower-wavenumber spin waves are governed by
magnetic dipole interactions and are called magnetostatic waves [10, 11, 12,
13]. Since the magnetostatic waves are suitable for long-distance propagation,
further investigation of dispersion relations in the lower-wavenumber regime
is indispensable [2].

Experimental techniques for acquiring dispersion relations of spin waves are
being actively studied. For example, inelastic neutron scattering [14, 15] and
spin-polarized electron energy loss spectroscopy [16] have been demonstrated.
However, these methods are suitable for observing the higher-wavenumber
region, rather than observing the lower-wavenumber region of the dispersion.

Recently, a method called spin-wave tomography (SWaT), which used time-
resolved pump-probe measurements and the Fourier transform to visualize the
dispersion relations of spin waves in the lower-wavenumber region was
demonstrated [17, 18]. Further, similar measurements in metals were performed
using the magneto-optical Kerr effect [19]. In the SWaT method, a pump pulse
is used to impulsively excite the spin wave, and a probe pulse with a time
delay is used to detect the change in magnetization. By using an ultrashort
pulsed laser as a pump pulse, spin waves in a wide frequency range can be
excited simultaneously. Moreover, by focusing the pulses, spin waves in a wide
wavenumber range can be excited. Wavenumber-resolved measurements can be made
by spatially scanning a sample with the focused probe pulses [19] or imaging a
large area without focusing the probe pulses [17, 18]. Therefore, this method
is useful for observing the dispersions over a wide region in the wavenumber
($k$)–frequency ($f$) space.

For observing the dispersion, a sufficiently high sampling rate must be
maintained for the time-resolved measurement. This is because of the risk of
folding noise due to the nature of the discrete Fourier transform (DFT). As a
result, the measurement time can range from 10 hours to several days. To
search for novel photo-induced dynamics in innumerable materials, the
measurement time must be reduced.

In recent years, in experiments such as the terahertz imaging [20], the NMR
spectroscopy [21], and the scanning tunneling microscopy and spectroscopy
[22], it has been shown that a method called compressed sensing can reduce the
measurement time. Compressed sensing is a signal processing technique that
allows the estimation of a signal from a small amount of data [23]. The signal
estimation can be achieved by the least absolute shrinkage and selection
operator (LASSO) method, a commonly used form of sparse regression [24].

In this letter, we demonstrate the fast acquisition of the spin-wave
dispersions by time-resolved pump-probe measurements using compressed sensing.
Moreover, we quantitatively evaluated the effect of reducing the number of
measurements in compressed sensing on the results of observations.

Our experimental setup is shown in Fig. 1. Our sample was a single crystal of
(111)-oriented 150 $\mu\mathrm{m}$ thick bismuth-doped rare earth iron garnet
(Gd3/2Yb1/2BiFe5O12) grown on a gadolinium gallium garnet substrate by the
liquid-phase epitaxy method. This magnetic material has been widely used for
investigating laser-induced spin dynamics due to its strong magneto-optical
coupling [25, 26, 27, 28, 29, 30, 31]. The magnetic field of
$H_{\mathrm{ext}}=2440\ \mathrm{Oe}$ was applied in the $x$-direction.

The light pulse for this pump-probe measurement was generated by a Ti:sapphire
regenerative amplifier with a pulse duration of $70\ \mathrm{fs}$ and a
repetition rate of $1\ \mathrm{kHz}$. A circularly polarized pump pulse with a
central wavelength of $1300\ \mathrm{nm}$ was focused along a line parallel to
the $y$-axis with a cylindrical lens with a fluence of $80\ \mathrm{mJ\
cm^{-2}}$. This pump pulse produced an effective magnetic field in the
$z$-direction via the inverse Faraday effect [32], and the magnetization
saturated in the $x$-direction tilted in the $y$ direction. Since the
effective magnetic field was instantaneous, the magnetization then began to
precess in the $y$-$z$ plane. The spin precession excited along the line-
shaped pumping spots propagated perpendicular to the line via magnetic dipole
interactions. A linearly polarized pulse with a central wavelength of 800 $\
\mathrm{nm}$ was used to probe the $z$-component of magnetization
$m_{z}(\mathbf{r},t)$ via the Faraday effect. The Faraday rotation angle was
determined from the angle of the analyzer, which minimized the intensity of
the transmitted probe pulse detected by a complementary metal–oxide
semiconductor (CMOS) camera. The time delay between the pump and probe pulses
was achieved by a variable optical path difference using a delay stage. This
was changed in increments of $\Delta t=0.01\ \mathrm{ns}$. The maximum value
of the delay was set to $T=3.60\ \mathrm{ns}$. Then, we obtained the
spatiotemporal waveform of the spin wave. The waveform was integrated along
the $y$-direction to create a one-dimensional waveform $m_{z}(x,t)$. The power
spectrum that depicts the dispersion curve in the $k$–$f$ space was obtained
by the two-dimensional DFT of the spatiotemporal waveform. Further,
micromagnetic simulations were performed to confirm our experimental results
(See supplementary data).

Figure 1: Setup of our pump-probe experiment. The circularly polarized pump
pulse was focused along a line by a cylindrical lens. The linearly polarized
probe pulse was irradiated on the entire sample without focusing. HWP: half-
wave plate, GTP: Gran Taylor prism, and QWP: quarter-wave plate.

Let $N=T/\Delta t$ be the number of samples in the time domain of the time-
resolved pump-probe measurement. Owing to the nature of the DFT, the frequency
resolution of the spectrum is $\Delta f=1/T$. In addition, according to
Nyquist’s theorem, the frequency components above $1/(2\Delta t)$ cannot be
observed, and they appear as folding noise. Therefore, with regard to reducing
the measurement time, decreasing $T$ leads to a poor resolution, and
increasing $\Delta t$ increases the risk of the appearance of folding noise.

Compressed sensing is a method of reducing $N$ by taking $\Delta t$ randomly,
without changing $T$, and estimating the spectrum from such data. Since
$\Delta t$ is not a constant, the DFT does not work well, resulting in
spectral leakage. Instead, the spectrum can be estimated by the LASSO method.

In the LASSO method, the spectrum estimation is treated as an inverse problem.
Let $\mathbf{y}=(y(t_{1}),y(t_{2}),\cdots,y(t_{N}))^{\mathrm{T}}$ be a
waveform sampled with discrete time $t_{i}\ (i=1,2,\cdots,N)$ and
$\mathbf{x}=(a(\omega_{1}),a(\omega_{2}),\cdots,a(\omega_{M}),b(\omega_{1}),b(\omega_{2}),\cdots,b(\omega_{M}))^{\mathrm{T}}$
are the cosine and sine components of the spectrum for discrete frequencies
$\omega_{j}\ (j=1,2,\cdots,M)$. The spectrum to be estimated is the solution
$\hat{\mathbf{x}}$ of the following minimization problem

$\displaystyle\hat{\mathbf{x}}=\mathop{\rm
arg~{}min}\limits_{\mathbf{x}}\left\\{\frac{1}{2N}||\mathbf{y}-A\mathbf{x}||_{2}^{2}+\lambda||\mathbf{x}||_{1}\right\\}.$
(1)

Arg min$\\{\cdot\\}$ denote the argument of the minimum, an element that
minimizes the value in the brackets. $||\cdot||_{p}$ is a term called the
$l_{p}$ norm of the vector and is defined as
$||\mathbf{x}||_{p}=\left(\sum_{i}x_{i}^{p}\right)^{1/p}$. The first term on
the right-hand side corresponds to the method of least squares. $A$ is a
$N\times 2M$ matrix, which corresponds to the inverse Fourier transform:

$\displaystyle
A_{ij}=\begin{cases}\cos(\omega_{j}t_{i})\qquad&(j=1,2,\cdots,M)\\\
\sin(\omega_{j-M}t_{i})\qquad&(j=M+1,M+2,\cdots,2M).\end{cases}$ (2)

The second term on the right-hand side in Eq. (1) imposes a sparsity
constraint on the solution $\hat{\mathbf{x}}$, and $\lambda$ is a parameter
that adjusts the sparsity. We determined $\lambda$ via five-fold cross-
validation [33]. In this method, the elements of $\mathbf{y}$ were randomly
divided into five data sets. Four of these sets were used to obtain
$\hat{\mathbf{x}}$, and the other set was used to evaluate the waveform
reproduced from $\hat{\mathbf{x}}$.

Generally, the output of the LASSO is not strictly unique, and it depends on
the random sampling [34]. Cosine similarities (CS) were used to evaluate the
consistency of results from different dataset. The CS of two dispersions is
given by

$\displaystyle\mathrm{CS}=\frac{\mathbf{f}\cdot\mathbf{g}}{|\mathbf{f}||\mathbf{g}|}$
(3)

where $\mathbf{f}$ and $\mathbf{g}$ are vectorized data of the dispersion
relations. Here, $\mathbf{f}$ is the dispersion calculated by LASSO from the
$N=361$ data as the most ideal dispersion available from the present data and
$\mathbf{g}$ is the dispersion with reduced $N$ to be compared.

For the DFT method, $N$ was reduced by taking $\Delta t$ as $0.01\
\mathrm{ns}$ multiplied by the divisors of 360 without changing $T$. For the
LASSO method, random sampling with $N=10,\ 20,\ \cdots,\ 360$ was performed in
ten ways each, and the mean and standard deviation of the CSs were calculated.
Moreover, the points $t=0,\ 0.01,\ 3.60\ \mathrm{ns}$ were always sampled to
maintain the frequency resolution and the Nyquist frequency.

Figure 2(a) shows the entire spatiotemporal waveform of the spin wave that we
observed in our experiments with $N=361$ at increments of $\Delta t=0.01\
\mathrm{ns}$. Figure 2(b) shows a waveform dataset with $N=46$ by taking
$\Delta t=0.08\ \mathrm{ns}$, and Fig. 2(c) represents a dataset with $N=46$
by taking $\Delta t$ at random. Further, Figs. 2(d)–(f) shows the simulated
waveforms correspond to Figs. 2(a)–(c).

Figure 2: Spatiotemporal waveforms of spin wave observed by time-resolved
pump-probe magneto-optical imaging method. (a) the whole data with $N=361$.
(b) waveform sampled from (a) with $N=46$ by taking $\Delta t=0.08\
\mathrm{ns}$. (c) waveform sampled from (a) at random in time domain with
$N=46$. (d)–(f) waveforms simulated via MuMax3 corresponding to (a)–(c),
respectively.

Figure 3 shows the dispersion relations corresponding to the dataset shown in
Fig 2. Figure 3(a) was obtained from the data shown in Fig. 2(a) by DFT in the
time and space domain. Moreover, Fig. 3(b) was obtained by DFT from the data
shown in Fig. 2(b). The data in Fig. 3(c) were estimated via the LASSO method
instead of DFT in the time domain. Figure 3(b) shows that the information was
only available up to $6\ \mathrm{GHz}$ due to the insufficient sampling rate
of the data in Fig. 2(b). Therefore, signals appearing to exhibit the
dispersion relation are folding noises bounded by the Nyquist frequency. In
Fig. 3(c), the same curve can be observed as in Fig. 3(a), which implies that
sufficient information could be extracted from random sampling as shown in
Fig. 2(c). Figures 3(d)–(f) show the dispersion relations calculated from the
data shown in Figs. 2(d)–(f), the waveforms calculated by MuMax3. The results
of the simulation and the experiment were found to be in good agreement.

The excited modes were the backward volume magnetostatic waves, which are
mainly dominated by magnetic dipole interactions, and they have a negative
gradient of dispersion [10, 12, 13]. The observed dispersion was in good
agreement with the lowest order mode shown in Fig. 3 with red lines, and there
were no peaks corresponding to the higher order modes. This is because the
higher order modes have nodes in the thickness direction, and the Faraday
rotation caused by the higher modes was mostly cancelled out through the light
transmittance. The multiple branches seen in Figs. 2(a) and (d) are due to the
spin-wave echoes [35].

Figure 3: (a)–(c) Dispersion relations obtained from spatiotemporal waveforms
shown in Figs. 2(a)–(c). (d)–(f) dispersion relations obtained by analyzing
the simulated waveforms corresponding to Figs. 2(d)–(f). The red lines are the
theoretical curves of the dispersion relation of the backward volume
magnetostatic wave. The horizontal axis is the wavenumber multiplied by the
thickness of the sample. Figure 4: Cosine similarities of the dispersions at
reduced $N$ to the dispersion at $N=361$. Blue and orange dots represent the
results of LASSO method and DFT, respectively. The error bars of blue dots
represent the standard deviations of the CSs of 10 different sets of random
sampling.

Figure 4 shows the $N$-dependence of the CSs between the dispersion obtained
from the data with $N=361$ and the dispersions obtained from the data with
reduced $N$. For the DFT method, the points at $N\leq 46$ correspond to the
conditions with $\Delta t\geq 0.08\ \mathrm{ns}$, and their CSs were almost
zero because the Nyquist condition was not satisfied. In contrast, the CSs of
the results of the LASSO method were above $0.90$, even when $N$ was reduced
to $50$. Note that in the LASSO method, $N$ was reduced without decreasing the
Nyquist frequency.

It is necessary to discuss in what systems LASSO can be applied. Empirically,
samplings 2–5 times the number of sparse coefficients is sufficient to
reconstruct the spectrum using the $l_{1}$ norm [36]. Based on this, even for
a system with multiple modes, the number of required samples can be roughly
estimated from the number of predicted peaks. Furthermore, for a system with
strong damping, the linewidth of the spectrum may be underestimated. This can
be improved by setting the sampling time range to $T\approx 1/(\alpha f_{0})$
where $f_{0}$ is the center frequency.

In conclusion, we demonstrated the fast acquisition of spin-wave dispersion by
using compressed sensing. Further, we quantitatively evaluated the effects of
random sampling on the results of the LASSO method. This technique
significantly reduced the measurement time for acquiring the dispersion
relations. Moreover, this method of applying compressed sensing to time-
resolved pump-probe measurements is not limited to the magneto-optical imaging
of spin waves, to various experiments based on pump-probe measurements, such
as observations via the electro-optical effects and the refractive index
modulations.

## Acknowledgments

We would like to thank K. Hukushima and T. Ishikawa for valuable suggestions.
This study was supported by the Japan Society for the Promotion of Science
(JSPS) KAKENHI (Grants No. JP19H01828, No. JP19H05618, No. JP19J21797, No.
JP19K21854, and No. JP26103004). K. M. would like to thank the Research
Fellowship for Young Scientists by the JSPS.

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