# Reversible electron beam heating for suppression of microbunching
instabilities at free-electron lasers

Christopher Behrens1, Zhirong Huang2, and Dao Xiang2 1 Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
2 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

###### Abstract

The presence of microbunching instabilities due to the compression of high-
brightness electron beams at existing and future X-ray free-electron lasers
(FELs) results in restrictions on the attainable lasing performance and
renders beam imaging with optical transition radiation impossible. The
instability can be suppressed by introducing additional energy spread, i.e.,
“heating” the electron beam, as demonstrated by the successful operation of
the laser heater system at the Linac Coherent Light Source. The increased
energy spread is typically tolerable for self-amplified spontaneous emission
FELs but limits the effectiveness of advanced FEL schemes such as seeding. In
this paper, we present a reversible electron beam heating system based on two
transverse deflecting radio-frequency structures (TDSs) up and downstream of a
magnetic bunch compressor chicane. The additional energy spread is introduced
in the first TDS, which suppresses the microbunching instability, and then is
eliminated in the second TDS. We show the feasibility of the microbunching
gain suppression based on calculations and simulations including the effects
of coherent synchrotron radiation. Acceptable electron beam and radio-
frequency jitter are identified, and inherent options for diagnostics and on-
line monitoring of the electron beam’s longitudinal phase space are discussed.

###### pacs:

29.27.-a, 41.60.Cr, 41.85.Ct

##  I Introduction

X-ray free-electron lasers (FELs) provide an outstanding tool for studying
matter at ultrafast time and atomic length scales LCLSnature1 , and have
become a reality with the operation of the Free-Electron Laser in Hamburg
(FLASH) FLASHAnature , the Linac Coherent Light Source (LCLS) LCLSnature2 ,
and the SPring-8 Angstrom Compact Free Electron Laser (SACLA) SACLAnature .
The required high transverse and longitudinal brightness of the X-ray FEL
driving electron bunches may encounter various degradation effects due to
collective effects like coherent synchrotron radiation (CSR) or microbunching
instabilities (e.g., Refs. CSR ; CSR-ub ; lsc-ub ), and need to be preserved
and controlled. In order to suppress a microbunching instability associated
with longitudinal bunch compression that deteriorates the FEL performance, the
LCLS uses a laser heater system to irreversibly increase the uncorrelated
energy spread within the electron bunches, i.e., the slice energy spread, to a
level tolerable for operation of a self-amplified spontaneous emission FEL LH
; LCLSheater . For future X-ray FELs that plan to use external quantum lasers
(seed lasers) to seed the FEL process in order to achieve better temporal
coherence and synchronization for pump-probe experiments, a smaller slice
energy spread is required to leave room for the additional energy modulation
imprinted by the seed laser. Thus, the amount of tolerable beam heating is
more restrictive and the longitudinal phase space control becomes more
critical (e.g., Refs. NLS ; flatflat ). The same strict requirement on small
slice energy spreads is valid for optical klystron enhanced self-amplified
spontaneous emission free-electron lasers okly .

Originally designed for high-energy particle separation by radio-frequency
(rf) fields LOLA , transverse deflecting rf structures (TDSs) are routinely
used for high-resolution temporal electron beam diagnostics at present X-ray
FELs (e.g., Refs. Kick ; Roehrs ; Filippetto ; xtcav ) and are proposed to use
for novel beam manipulation methods (e.g., Refs. psxray ; exchange1 ;
exchange2 ; mapping ; ramp ; psex ). Recently, a TDS was used to increase the
slice energy spread in an echo-enabled harmonic generation FEL experiment
Xiang ; Xiang2 . In this paper, we present a reversible electron beam heating
system that uses two TDSs located up and downstream of a magnetic bunch
compressor chicane. The additional slice energy spread is introduced in the
first TDS, which suppresses the microbunching instability, and then is
eliminated in the second TDS.

The method of reversible beam heating is shown in Sec. II by means of linear
beam optics and a corresponding matrix formalism. In Sec. III, we show the
feasibility of this scheme to preserve both the transverse and longitudinal
brightness of the electron beam, and discuss the impact of coherent
synchrotron radiation. Section IV covers the gain suppression of microbunching
instabilities by analytical calculations and numerical simulations, and in
Sec. V we discuss the impact of beam and rf jitter, and show inherent options
for diagnosis and on-line monitoring of the electron beam’s longitudinal phase
space. The results and conclusions are summarized in Sec. VI.

##  II Method

Figure 1: Layout of a reversible electron beam heater system including two
transverse deflecting rf structures located up and downstream of a magnetic
bunch compressor (BC) chicane, and longitudinal phase space diagnostics using
screens and synchrotron radiation monitors (SRM). Parameters related to the
reversible beam heater system are denoted in curly brackets.

In this and the following sections, we consider a linear accelerator (linac)
employing a single bunch compressor for a soft X-ray FEL, such as the proposed
linac configuration for the Next Generation Light Source (NGLS) at LBNL
Corlett . The choice of a single magnetic bunch compressor simplifies our
consideration and analysis, although the concept is also applicable for
typical bunch compressor arrangements with multiple stages. We note that a
single bunch compressor arrangement has also been considered for the
FERMI@Elettra FEL in order to minimize the impact of microbunching
instabilities Venturini .

The generic layout of the reversible electron beam heater system is depicted
in Fig. 1. It consists of linac sections providing and accelerating high-
brightness electron beams, a magnetic bunch compressor chicane in order to
achieve sufficient peak currents to drive the FEL process, and two transverse
deflecting rf structures located up and downstream of the bunch compressor. An
additional higher-harmonic rf linearizer system (Linearizer), like at the LCLS
or FLASH Lin , can be used to achieve uniform bunch compression by means of
longitudinal phase space linearization upstream of the bunch compressor. The
whole system can be supplemented by dedicated longitudinal phase space
diagnostics (see Sec. V), and except for the two TDSs, the layout is commonly
used for bunch compression at present and future X-ray FELs.

The principle of the reversible electron beam heater relies on the physics of
TDSs arising from the Panofsky-Wenzel theorem Panowsky ; Bro , which states
that the transverse momentum gain $\Delta\vec{p}_{\perp}$ of a relativistic
electron imprinted by a TDS is related to the transverse gradient of the
longitudinal electric field $\nabla_{\perp}\mathcal{E}_{z}$ inside the TDS,
and yields

$\Delta\vec{p}_{\perp}=-i\frac{e}{\omega}\int_{0}^{L}\nabla_{\perp}\mathcal{E}_{z}d\tilde{z}\,,$
(1)

where $\omega/(2\pi)$ is the operating rf frequency, $e$ is the elementary
charge, $L$ is the structure length, and $\tilde{z}$ is the longitudinal
position inside the structure (not to be confused with the beamline
coordinate, which is given by $s$ in the following). Operating a TDS with
vertical deflection, i.e., in y-direction, near the zero-crossing rf phase
$\psi=\omega/c\,z$, electrons experience transverse kicks Kick

$\Delta y^{\prime}=\frac{e\omega V_{y}}{cE}z=K_{y}z$ (2)

and relative energy deviations ($\delta=\Delta E/E$) IES1 ; IES2

$\Delta\delta=K_{y}\frac{1}{L}\int_{0}^{L}{y(s)ds}=K_{y}\overline{y}\,,$ (3)

where $K_{y}=e\omega V_{y}/(cE)$ is the vertical kick strength, $V_{y}$ is the
peak deflection voltage in the TDS, $c$ is the speed of light in vacuum, $E$
is the electron energy, and $\overline{y}$ is the mean vertical position over
the structure length $L$ along the beamline relative to the central axis
inside the finite TDS. Here, $z$ is the internal bunch length coordinate of
the electron relative to the zero-crossing rf phase. Both the additional
transverse kicks and relative energy deviations are induced by the TDS
operation itself and generate correlations within an electron bunch. In fact,
near the zero-crossing rf phase (see Eq. (2)), the induced transverse kick
correlates linearly with the internal bunch length coordinate ($z=ct$) and
enables high-resolution temporal diagnostics (e.g., Refs. Kick ; Roehrs ;
Filippetto ), whereas the induced relative energy deviation correlates with
the vertical offset inside the TDS and results in an induced relative energy
spread $\Delta\sigma_{\delta}=K_{y}\sigma_{y}$. Here, the symbol $\sigma$
denotes the root mean square (r.m.s.) value, and $\sigma_{y}$ is the vertical
r.m.s. beam size. This additional energy spread (cf. laser heater LH ;
LCLSheater ), in combination with the momentum compaction $R_{56}$ of a bunch
compressor chicane, is able to smear microbunch structures, and
correspondingly suppresses the associated instability as is shown in Sec. IV.
The effect of induced energy spread (“beam heating”) is generated by off-axis
longitudinal electric fields, related to the principle of a TDS by the
Panofsky-Wenzel theorem, and has been observed experimentally at FLASH IES3
and the LCLS emma . The induced energy spread is uncorrelated in the
longitudinal phase space $(z,\delta)$, but shows correlations in the phase
space $(y,\delta)$, which is the reason that it can be eliminated (“beam
cooling”) with a second TDS in a reversible mode as is shown in the following
by two different approaches.

###  II.1 Linear beam optics

The transverse betatron motion of an electron passing through a TDS with
vertical deflection (in $y$) is given by

$y(s)=y_{0}(s)+S_{y}(s,s_{0})z$ (4)

with the vertical shear function (e.g., Refs. Kick ; Roehrs ; IES2 )

$S_{y}(s,s_{0})=R_{34}K_{y}=\sqrt{\beta_{y}(s)\beta_{y}(s_{0})}\mathrm{sin}(\Delta\phi_{y}(s,s_{0}))\frac{e\omega
V_{y}}{cE}\,,$ (5)

where $R_{34}$ is the angular-to-spatial element of the vertical beam transfer
matrix from the TDS at $s_{0}$ to any position $s$, $\beta_{y}$ is the
vertical beta function, $\Delta\phi_{y}$ is the vertical phase advance between
$s_{0}$ and $s$, and $y_{0}$ describes the vertical beam offset independent of
any TDS shearing effect. Referring to the layout depicted in Fig. 1 and taking
bunch compression into account, the induced vertical beam offset ($\Delta
y=y-y_{0}$) downstream of the second TDS becomes (omitting the subscript $y$
in $S_{y}$)

$\displaystyle\Delta y(s)=$ $\displaystyle
S_{1}(s,s_{1})z_{1}+S_{2}(s,s_{2})z_{2}$ $\displaystyle=$
$\displaystyle(CS_{1}(s,s_{1})+S_{2}(s,s_{2}))z_{2}$ (6)

with the bunch compression factor $C=z_{1}/z_{2}$ and the shear functions
$S_{1,2}(s,s_{1,2})$ of the corresponding TDSs. Here, $S_{1}(s,s_{1})z_{1}$
describes the vertical beam offset induced by the first TDS located at $s_{1}$
that is independent of the second TDS. In order to cancel the spatial chirp
induced by the combined TDS operation, the beam offset $\Delta y$ in Eq. (6)
must vanish for any $z_{2}$. Hence, using Eq. (5) for $S_{1,2}$ in Eq. (6) and
taking acceleration from $E_{1}$ to $E_{2}$ in Linac2 into account by making
the replacement
$\beta_{y}(s)\beta_{y}(s_{1})\rightarrow\beta_{y}(s)\beta_{y}(s_{1})E_{1}/E_{2}$
Kick , we get

$\displaystyle
C\sqrt{\beta_{y}(s_{1})}\mathrm{sin}(\Delta\phi_{y}(s,s_{1}))\sqrt{E_{1}}K_{1}$
$\displaystyle+\sqrt{\beta_{y}(s_{2})}\mathrm{sin}(\Delta\phi_{y}(s,s_{2}))\sqrt{E_{2}}K_{2}=0\,,$
(7)

where $K_{1,2}$ are the vertical kick strengths of the corresponding TDSs, and
$\Delta\phi_{y}(s,s_{1,2})$ describes the vertical phase advances between
$s_{1,2}$ and $s$, respectively. As a consequence, the phase advance between
both TDSs is
$\Delta\phi_{y}(s_{2},s_{1})=\Delta\phi_{y}(s,s_{1})-\Delta\phi_{y}(s,s_{2})$.
A general solution, valid for any position $s$ downstream of the second TDS,
is only possible for a phase advance difference of

$\Delta\phi_{y}(s_{2},s_{1})=n\cdot\pi$ (8)

with $n$ being integer, and the kick strength

$K_{2}=\pm
C\sqrt{\frac{\beta_{y}(s_{1})}{\beta_{y}(s_{2})}}\sqrt{\frac{E_{1}}{E_{2}}}K_{1}\,,$
(9)

where the sign depends on the actual phase advance, i.e.,
$\Delta\phi_{y}(s_{2},s_{1})=\pi+n\cdot 2\pi$ for ($+$) and
$\Delta\phi_{y}(s_{2},s_{1})=n\cdot 2\pi$ for ($-$). The different sign of $K$
can technically be achieved by changing the rf phase in the TDS by
$180^{\circ}$ which results in a zero-crossing rf phase with opposite slope
and deflection. Besides cancelation of the induced spatial chirps, the induced
energy spread of the first TDS needs to be eliminated in the second structure
in order to have a fully reversible electron beam heater. Applying Eq. (3)
similar to Eq. (6), the relative energy deviation downstream of the second TDS
for finite structure lengths become (omitting the argument in $y(s)$ and
$S(s)$)

$\Delta\delta=K_{1}\overline{y_{1}}C\frac{E_{1}}{E_{2}}+K_{2}\overline{(y_{2}+S_{1}z_{1})}$
(10)

with the mean vertical offsets $\overline{y_{1}}$ and
$\overline{(y_{2}+S_{1}z_{1})}$ inside the TDSs. For constant vertical offsets
inside the TDSs or short structure lengths, the mean vertical offsets can be
replaced by the actual offsets, i.e., $\overline{y_{1}}\rightarrow y_{1}$ and
$\overline{(y_{2}+S_{1}z_{1})}\rightarrow(y_{2}+S_{1}z_{1})$. The latter
describes the offset in the second TDS and involves the spatial chirp induced
by the first TDS with $S_{1}\sim\mathrm{sin}(\Delta\phi_{y}(s_{2},s_{1}))$,
which vanishes in the case of spatial chirp cancelation given by Eq. (8). In
order to cancel the relative energy spread induced by the combined TDS
operation, it follows

$K_{1}\overline{y_{1}}C\frac{E_{1}}{E_{2}}+K_{2}\overline{y_{2}}=0\,.$ (11)

The general transverse beam transport optics with the vertical phase advance
condition in Eq. (8) gives
$\overline{y_{2}}=\pm\overline{y_{1}}\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$,
and taking $\beta_{y}(s_{2})\rightarrow\beta_{y}(s_{2})E_{1}/E_{2}$ (see prior
Eq. (7)) into account yields exactly the same condition as in Eq. (9).
Simultaneous spatial chirp and energy spread cancelation in the second TDS is
the basic principle for reversible electron beam heating and enables local
increase of slice energy spread. The additional energy spread in the bunch
compressor, which is added in quadrature by the first TDS, can be controlled
by the kick strength $K_{1}$ and the vertical beam size $\sigma_{y}(s_{1})$.

In the following, a complementary approach to discuss the reversible beam
heating system is shown. It uses the matrix formalism for beam transport and
provides an analytical way to show microbunching gain suppression and to
discuss the impact of beam and rf jitter.

###  II.2 Matrix formalism

We adopt the beam transport matrix notation of a 6x6 matrix for
$(x,x^{\prime},y,y^{\prime},z,\delta)$ but leaves $(x,x^{\prime})$ out for
simplicity, i.e., $(y,y^{\prime},z,\delta)$ is used in the following. The 4x4
beam transport matrix for a vertical deflecting TDS in thin-lens approximation
reads (e.g., Refs. exchange1 ; IES1 ; IES3 )

${\mathbf{R}}_{T}^{thin}=\begin{pmatrix}1&0&0&0\\\ 0&1&K&0\\\ 0&0&1&0\\\
K&0&0&1\end{pmatrix}.$ (12)

As discussed above, the main components of the given reversible heater system
shown in Fig. 1 consist of TDS1 with the kick strength $K_{1}$, a bunch
compressor with the momentum compaction factor $R_{56}$, and TDS2 with the
kick strength $K_{2}$. Including the momentum compaction factor $R_{56}$ and
acceleration in Linac2 ($E_{1}\rightarrow E_{2}$), the 4x4 beam matrix between
the two TDSs is given by

${\mathbf{R}}_{C}=\begin{pmatrix}R_{33}&R_{34}&0&0\\\ R_{43}&R_{44}&0&0\\\
0&0&1&R_{56}\\\ 0&0&0&\frac{E_{1}}{E_{2}}\end{pmatrix}\,.$ (13)

In order to allow the energy change in the first TDS to be compensated for in
the second TDS, we require the point-to-point imaging from TDS1 to TDS2 (i.e.,
$R_{34}=0$), which corresponds to an equivalent vertical phase advance of
$\Delta\phi_{y}(s_{2},s_{1})=n\cdot\pi$ with $n$ being integer (see Eq. (8)).
Then we get the magnification factor
$R_{33}=\pm\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$ and $R_{44}=1/R_{33}$.

The linear accelerator section with higher-harmonic rf linearizer (Linac1 and
Linearizer) upstream of the first TDS introduces an appropriate energy chirp
$h$ for uniform bunch compression. Without loss of generality, we neglect
acceleration between the two TDSs, i.e., we do not consider Linac2 anymore.
Including Linac2 would simply result in a correction term $\sqrt{E_{1}/E_{2}}$
(cf. Eqs. (9) and (15) below) but would leave the physics unchanged. Then the
entire 4x4 beam transport matrix from the beginning of TDS1 to the end of TDS2
becomes

$\begin{pmatrix}R_{33}&0&0&0\\\
R_{43}+K_{1}K_{2}R_{56}&\frac{1}{R_{33}}&\frac{K_{1}}{R_{33}}+K_{2}(1+hR_{56})&K_{2}R_{56}\\\
K_{1}R_{56}&0&1+hR_{56}&R_{56}\\\
{K_{1}}+R_{33}K_{2}&0&0&\frac{1}{1+hR_{56}}\end{pmatrix}\,.$ (14)

Cancelation of the induced spatial chirp ($\Delta y^{\prime}\sim z_{0}$, cf.
Eq. (2)) requires $R_{45}=0$ (6x6-matrix notation), i.e.,

$K_{1}/R_{33}+K_{2}(1+hR_{56})=0\,,$ (15)

where $R_{45}$ describes the coupling between $y^{\prime}$ and $z_{0}$. We
note that the coupling between $\delta$ and $y_{0}$ (i.e., $R_{63}$ element)
is nonzero in Eq. (14) because the bunch is energy-chirped after compression
($\delta\sim z\sim y_{0}$), which can be removed by Linac3 downstream of TDS2.
For uniform bunch compression with $C^{-1}=(1+hR_{56})$, no acceleration in
Linac2, i.e., $E_{2}=E_{1}$, and taking into account that
$R_{33}=\pm\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$, Eq. (15) is identical to
Eq. (9). Thus, both formalisms yield the same result.

Since the kick strength of the first TDS is very weak, it can be implemented
by means of a short rf structure and the thin-lens approximation is still
valid. However, the kick strength of the second TDS is usually stronger, and
the effect of the finite structure length should be taken into account. The
symplectic beam transport matrix of a finite TDS with the length $L_{2}$ is
given in Ref. exchange1  by

${\mathbf{R}}_{T}^{thick}=\begin{pmatrix}1&L_{2}&K_{2}L_{2}/2&0\\\
0&1&K_{2}&0\\\ 0&0&1&0\\\ K_{2}&K_{2}L_{2}/2&K_{2}^{2}L_{2}/6&1\end{pmatrix}.$
(16)

In this case, we require the point-to-point imaging is from the first TDS to
the middle of the second TDS in order to have a complete cancellation. The
overall matrix from TDS1 to the end of TDS2, when Eq. (15) is fulfilled,
becomes more complicated. A few correction terms containing the length $L_{2}$
of TDS2 appear, which however does not change the working principle of the
reversible beam heater system. It should be pointed out that downstream of the
reversible heater system, the beam is slightly coupled in
$y^{\prime}-\delta_{0}$ and $y-z_{0}$, which results in a small growth of the
projected emittance given by

$\epsilon_{y,z}^{2}=\epsilon_{y0,z0}^{2}+\epsilon_{y0}\epsilon_{z0}\frac{\beta_{y0}\gamma_{z0}K_{1}^{2}R_{56}^{2}}{(1+hR_{56})^{2}}\,,$
(17)

where $\epsilon_{y0,z0}$ is the initial vertical (longitudinal) emittance, and
$\beta_{y0}$ and $\gamma_{z0}$ are the initial Twiss parameters. As is shown
in the following section, this projected emittance growth is typically
negligible.

##  III Reversible heating and emittance preservation

We demonstrate the feasibility of the reversible beam heater system by
numerical simulations using the particle tracking code elegant Elegant , and
the simulations in the following include $5\,\times\,10^{5}$ particles. Table
1 summarizes the main parameters used in the simulations, and the adopted
accelerator optics model, including the positions of the TDSs, is shown in
Fig. 2. The magnetic bunch compressor chicane is assumed to bend in the
horizontal plane, and the TDSs are oriented perpendicularly with vertical
deflection. In the previous section,

Table 1: Parameters of the electron beam, of the bunch compressor system, and of the transverse deflecting rf structures. Parameter | Symbol | Value | Unit
---|---|---|---
Beam energy at TDS1/2 | $E$ | 350 | MeV
Lorentz factor at TDS1/2 | $\gamma$ | 685 |
Initial transverse emittance | $\gamma\epsilon_{x,y}$ | 0.6 |  $\mu$m
Initial slice energy spread | $\sigma_{E}$ |  $\sim$ 1 | keV
Momentum compaction factor | $R_{56}$ | $-138$ | mm
Compression factor | $C$ |  $\sim$ 13 |
Final bunch current | $I_{f}$ |  $\sim$ 520 | A
TDS1/2 rf frequency | $\omega/2\pi$ | 3.9 | GHz
Voltage of TDS1 | $V_{1}$ | 0.415 | MV
Voltage of TDS2 (without CSR) | $V_{2}$ | 5.440 | MV
Length of TDS1 | $L_{1}$ | 0.1 | m
Length of TDS2 | $L_{2}$ | 0.5 | m
Figure 2: Relevant accelerator optics (Twiss parameters) and positions of the
transverse deflecting rf structures used to numerically demonstrate the
reversible beam heater system.

we included Linac2 for a general derivation of the method, but in practice,
due to wakefield concerns, we recommend putting TDS2 right after the bunch
compressor. In order to show numerical examples based on this approach, Linac2
is not considered anymore throughout the rest of this paper. Except for the
TDSs, the parameters are similar to the magnetic bunch compressor system
discussed for the Next Generation Light Source at LBNL Corlett ; Venturini2 .

The initial longitudinal electron bunch profile is assumed to be flat-top with
a peak current of $\sim$ 40 A and a slice energy spread of $\sim$ 1 keV
(r.m.s.). The initial linear and quadratic chirp is set for a uniform
compression factor $C$ of about 13 across the entire bunch length. This is
possible even with bunch compressor nonlinearities by using a higher-harmonic
rf linearizer upstream of the bunch compressor Lin  and needed to achieve
uniform cancelation of the induced energy spread downstream of TDS2.

Figure 3 shows the principle of the reversible beam heater system by means of
simulation of the longitudinal phase space at different positions along the
beamline.

(a) Upstream of TDS1.

(b) Downstream of TDS1.

(c) Upstream of TDS2.

(d) Downstream of TDS2.

Figure 3: Simulation of the longitudinal phase space after removing the
correlated energy chirp: (a) upstream of the first TDS, (b) directly
downstream of the first TDS, (c) directly downstream of the bunch compressor
and upstream of the second TDS, and (d) downstream of the second TDS. The axes
scales change from (b) to (c) when bunch compression takes place. The bunch
head is on the left, i.e., where $z/c<0$.

The impact of CSR is not taken into account (cf. next subsection for CSR
effects). The initial slice energy spread is heated up to $\sim$ 10 keV
(r.m.s.) in the first TDS, increased by the compression factor in the bunch
compressor to $\sim$ 130 keV (r.m.s.), and finally cooled down to $\sim$ 13
keV (r.m.s.) by the second TDS (see Figs. 3(a)-3(d)). The plot in Fig. 4(b)
shows that the heating induced by the first TDS is perfectly reversible, and
the final slice energy spread is simply the initial slice energy spread scaled
with the compression factor, which would be exactly the same like in the case
without using the reversible beam heater system. Figure 4(a) shows the heater
system impact on both the projected emittance (horizontal and vertical) and
the core energy spread, i.e., the slice energy spread in the center of the
bunch, for different voltages in the second TDS. The minimum of the vertical
emittance is related to the cancelation of the spatial chirp and energy spread
induced by the first TDS. The horizontal emittance is not affected at all, and
the small projected emittance growth (6 %) in the vertical plane at the
minimum is due to residual coupling generated by the system that is described
by Eq. (17). Nevertheless, as shown in Sec. III.1, even in the case with CSR
effects, the horizontal slice emittance stays unaffected at all and the
vertical slice emittance exhibits only deviations in the bunch head ($z/c<0$)
and tail ($z/c>0$).

(a) Projected emittances and core energy spread.

(b) Slice energy spread.

Figure 4: Simulations without CSR effects on the impact of the reversible
heater system on projected emittances, core energy spread, and slice energy
spread : (a) Projected emittances (normalized) and core energy spread, and (b)
slice energy spread for $V_{\mathrm{2}}$ at minimum emittance (see Fig. 4(a)).
The longitudinal coordinate is normalized to the bunch length.

###  III.1 Impact of coherent synchrotron radiation

The previous results undergo small modifications when including CSR effects,
which is shown in Fig. 5. The voltage of the second TDS for minimum projected
emittance in the vertical is shifted by about 0.2 MV to lower values which is
due the additional energy chirp induced by CSR. In comparison to the case
without any CSR effects (cf. Fig. 4), the projected emittance in the vertical
plane is slightly increased and the slice energy spread is not perfectly
canceled in the head and tail. The slice energy spread in the core part of the
bunch is also slightly increased to 17.5 keV (r.m.s.) (instead of 13.5 keV
(r.m.s.) in the absence of CSR). The projected emittance in the horizontal is
about 1.7 larger which is independent of the reversible beam heater operation.
This horizontal emittance growth can further be reduced by minimizing the
horizontal beta function in the last dipole of the chicane where the bunch
length becomes the shortest. This optimization is independent of the relevant
motion in the vertical and does not affect the results of the reversible
heater system.

(a) Projected emittances and core energy spread.

(b) Slice energy spread.

Figure 5: Simulation on the impact of the reversible beam heater system on
projected emittances, core energy spread, and slice energy spread: (a)
Projected emittances (normalized) and core energy spread, and (b) slice energy
spread for $V_{\mathrm{2}}$ at minimum emittance (see Fig. 5(a)). CSR effects
are included by means of the 1-dimensional model in elegant Elegant .

Albeit the fact that the projected emittances are increased, the horizontal
slice emittance stays unaffected and the vertical slice emittance exhibits
only deviations in the head and tail due to CSR effects as is shown in Fig. 6.
Thus, the core emittances are well preserved. We note that vertically streaked
bunches in the bunch compressor chicane may change the impact of CSR effects
but require a 3-dimensional “point-to-point” tracking which is not available
neither in elegant nor in CSRtrack CSRtrack , and is beyond the scope of this
paper.

Figure 6: Simulation of the normalized slice emittance for both the vertical
and horizontal upstream of the first and downstream of the second TDS. CSR
effects are included.

##  IV Microbunching gain suppression

The principle of the microbunching gain suppression with the reversible beam
heater system is shown by an analytical treatment following Ref. LCLSheater
and by using the beam transport matrix in Eq. (14). Then we show the
feasibility of the reversible heater system to suppress microbunching
instabilities by means of particle tracking simulations with initial density
and energy modulations.

###  IV.1 Analytical calculations

Using the vector notation $(y_{0},y_{0}^{\prime},z_{0},\delta_{0})$ for
particles in the first linac upstream of the first TDS, the longitudinal
position downstream of the second TDS is given by

$z=K_{1}R_{56}y_{0}+(1+hR_{56})z_{0}+R_{56}\delta_{0}\,.$ (18)

Suppose that $\delta_{0}=\delta_{u}+\delta_{m}$, where $\delta_{u}$ is the
uncorrelated relative energy deviation, and $\delta_{m}(z_{0})$ is the
relative energy modulation accumulated before and in the first linac (Linac1).
Following Ref. LCLSheater , the initial energy modulation at the wavenumber
$k_{0}$ is converted into additional density modulation at a compressed
wavenumber $k$. For a 4-dimensional (4-D) distribution function
$F(y,y^{\prime},z,\delta)$, the bunching factor $b(k)$ is given by

$\displaystyle b(k)=$ $\displaystyle\int dydy^{\prime}dzd\delta
e^{-ikz}F(y,y^{\prime},z,\delta)$ $\displaystyle=$ $\displaystyle\int
dy_{0}dy_{0}^{\prime}dz_{0}d\delta_{u}e^{-ikK_{1}R_{56}y_{0}-ik(1+hR_{56})z_{0}}$
$\displaystyle
e^{-ikR_{56}(\delta_{u}+\delta_{m}(z_{0}))}F_{0}(y_{0},y_{0}^{\prime},z_{0},\delta_{u})\,,$
(19)

where $F_{0}(y_{0},y_{0}^{\prime},z_{0},\delta_{u})$ is the initial 4-D
distribution. If the induced energy modulation is small such that
$|kR_{56}\delta_{m}|\ll 1$, we can expand the exponent of Eq. (19) up to the
first order in $\delta_{m}$ to obtain

$\displaystyle b(k)\approx$ $\displaystyle~{}b_{0}(k_{0})-ikR_{56}\int
dz_{0}\delta_{m}(z_{0})e^{-ik_{0}z_{0}}$ $\displaystyle\times$
$\displaystyle\int
dy_{0}d\delta_{u}e^{-ikK_{1}R_{56}y_{0}-ikR_{56}\delta_{u}}U(y_{0})V(\delta_{u})\,,$
(20)

where $k=Ck_{0}$, $C=1/(1+hR_{56})$, $U(y_{0})$ describes the transverse
profile, and $V(\delta_{u})$ is the initial energy distribution. For both
Gaussian profiles ($U$ and $V$), we have

$\displaystyle b(k)=b_{0}(k_{0})-$ $\displaystyle
ikR_{56}\delta_{m}(k_{0})\exp\left[-(k^{2}R_{56}^{2}K_{1}^{2}\sigma_{y1}^{2}/2)\right]$
$\displaystyle\times$ $\displaystyle\exp\left[-(k^{2}R_{56}^{2}\sigma_{\delta
u}^{2}/2)\right]\,.$ (21)

Here, we denote the Fourier transform of $\delta_{m}(z_{0})$ as
$\delta_{m}(k_{0})$, which is the accumulated energy modulation at the
wavenumber $k_{0}$ in the first linac due to longitudinal space charge and
other collective effects. The initial energy spread is given by
$\sigma_{\delta u}$, and $\sigma_{y1}$ is the vertical beam size in the first
TDS. We see that $K_{1}\sigma_{y1}$ acts like effective energy spread for
microbunching gain suppression.

###  IV.2 Numerical simulations

(a) Downstream of TDS2: Heater system off.

(b) Downstream of TDS2: Heater system on.

Figure 7: Simulation on suppression of microbunching instabilities due to an
initial density modulation, i.e., simulating CSR-driven microbunching. The
entire longitudinal phase space, after removing the correlated energy chirp,
is shown.

Suppression of microbunching instabilities is demonstrated by using both a
pure initial density modulation with 5 $\%$ peak amplitude and
$100\,\mathrm{\mu m}$ modulation wavelength ($\lambda_{m}$), and a pure
initial energy modulation with 3 keV peak amplitude and
$\lambda_{m}=50\,\mathrm{\mu m}$. Whereas the case with initial energy
modulation is immediately consistent with the previous analytical treatment
and describes the longitudinal space charge driven microbunching instability
lsc-ub , the initial density modulations need to be converted into energy
modulations by longitudinal CSR-impedance which expresses the consistency and
describes the CSR-driven microbunching instability CSR-ub . The simulations
were performed using the code elegant with $1\,\times\,10^{6}$ particles.
Figure 7 shows the longitudinal phase space downstream of the second TDS,
after removing the correlated energy chirp (linear and quadratic chirp), for
both the reversible beam heater system switched off (Fig. 7(a)) and on (Fig.
7(b)). In the case without reversible beam heater, energy and density
modulations at the compressed modulation wavelength $\lambda_{m}/C$ appear,
i.e., CSR-driven microbunching becomes visible. When switching the reversible
beam heater on, the microbunching instability disappears and the resulting
longitudinal phase space remains smooth. The reason is that the microbunches
at the compressed wavelength are smeared due to $R_{56}K_{1}\sigma_{y1}$ (cf.
Eq. (21)), and accordingly, the modulations appear as correlations in the
phase spaces $(y,z)$ and $(y^{\prime},\delta)$. The same effect of
microbunching suppression is given for initial energy modulations as shown in
Fig. 8.

(a) Downstream of TDS2: Heater system off.

(b) Downstream of TDS2: Heater system on.

Figure 8: Simulation on suppression of microbunching instabilities due to an
initial energy modulation, i.e., simulating longitudinal space charge driven
microbunching. For the sake of clarity, only the core of the longitudinal
phase space, after removing the correlated energy chirp, is shown.

The effect of the microbunching instability appears even stronger compared to
the simulations case with initial density modulations, but the performance of
the reversible heater system is the same with a smooth residual longitudinal
phase space when the reversible beam heater is switched on (see Fig. 8(b)).

Figures 7 and 8 are obtained for a magnetic bunch compressor system as shown
in Fig. 1. The electron bunch will be further accelerated and transported
throughout the rest of the accelerator to reach the final beam energy and peak
current in order to drive an X-ray FEL (not studied in this paper). A
microbunched electron beam as illustrated in Figs. 7(a) and 8(a), i.e., when
the reversible beam heater system is switched off, will accumulate additional
energy and density modulations, which would lead to unacceptable longitudinal
phase space properties for an X-ray FEL such as a large slice energy spread.

##  V Practical considerations

The previous sections covered the principle of reversible electron beam
heating and microbunching gain suppression by means of analytical calculations
and numerical simulations. In real accelerators, we also have to deal with
imperfections, jitter and drifts of various parameters, and accordingly
supplementary studies with respect to sensitivity on jitter sources and
tolerances have to be performed. In the following, we discuss the impact of
beam and rf jitter on the reversible beam heater system, and also point out
the inherent possibility of longitudinal phase space diagnostics and on-line
monitoring.

###  V.1 Jitter and tolerances

The impact of beam and rf jitter on the reversible beam heater method can
effectively be discussed using the Eqs. (2) and (14) with the condition in Eq.
(15). Deviations from the conditions in Eq. (15) can appear from jitter of the
individual peak deflection voltages $V_{1}$ and $V_{2}$ of the TDSs, and lead
to growth of the projected vertical emittance as is shown in Fig. 5(a), where
the voltage of the second TDS is varied. Even in the case of a large TDS
voltage jitter of 1 %, the vertical projected emittance growth is less than 2
$\%$ (see Fig. 5(a)). In the case of acceleration between the first and second
TDS, also energy jitter, which is similar or smaller than TDS voltage jitter,
due to this intermediate acceleration leads to deviation of the condition in
Eq. (15). The choice of superconducting accelerator technology even provide
much better rf stability HS ; CS . Pure arrival time jitter upstream of the
first TDS has no impact as long as the condition in Eq. (15), which describes
the coupling between $y^{\prime}$ and $t=z/c$, is fulfilled. In the case that
Eq. (15) is not exactly fulfilled, e.g., due to TDS voltage jitter which is on
the percent-level, the impact of typical arrival time jitter well below 100
fs, like at the LCLS LCLSnature2  or FLASH CS , is negligible. The most
critical jitter sources arise from energy jitter upstream of the bunch
compressor chicane and from rf phase jitter in the TDSs. The momentum
compaction factor translates energy jitter into arrival time jitter, which
leads to vertical kicks in the second TDS. The same effect of additional
vertical kicks is generated by rf phase jitter in the TDSs. In order to have
small impact of vertical kicks on the remaining beam transport, we demand
$\Delta\sigma_{y^{\prime}}\ll\sigma_{y^{\prime}}$ directly downstream of the
second TDS with the induced vertical r.m.s. kick $\Delta\sigma_{y^{\prime}}$
and the intrinsic beam divergence $\sigma_{y^{\prime}}$. The relevant total
vertical r.m.s. kick is given by

$\displaystyle\Delta\sigma_{y^{\prime}}=$
$\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\sigma_{\varphi_{2}}\right)^{2}+\left(\frac{K_{1}}{R_{33}}\frac{c}{\omega}\sigma_{\varphi_{1}}\right)^{2}}$
$\displaystyle=$
$\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\right)^{2}\left(\sigma_{\varphi_{2}}^{2}+\frac{1}{C^{2}}\sigma_{\varphi_{1}}^{2}\right)}$
$\displaystyle\approx$
$\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\right)^{2}\sigma_{\varphi_{2}}^{2}}$
(22)

with the energy jitter $\sigma_{E}/E$ upstream of the bunch compressor, the rf
phase jitter $\sigma_{\varphi_{1,2}}$ of the TDSs, the magnification factor
$R_{33}$ from the first to the second TDS (see Eq. (13)), and using Eq. (15)
with the compression factor $C=(1+hR_{56})^{-1}$. We see that the vertical
r.m.s. kick due to rf phase jitter in the first TDS scales with $C^{-2}$ and
can be neglected compared to the vertical r.m.s. kick induced by the second
TDS when we assume the same amount of rf phase jitter in both TDSs. The
condition for trajectory stability
$\Delta\sigma_{y^{\prime}}\ll\sigma_{y^{\prime}_{2}}=\sqrt{\epsilon_{y_{2}}/\beta_{y_{2}}}$
with the intrinsic (uncorrelated) r.m.s. beam divergence
$\sigma_{y^{\prime}_{2}}$ downstream of the bunch compressor at TDS2, where
$\epsilon_{y_{2}}$ is the geometrical emittance, can be restated as

$\sqrt{\left(R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(\frac{c}{\omega}\sigma_{\varphi_{2}}\right)^{2}}\ll\frac{\epsilon_{y_{2}}}{K_{2}\sqrt{\beta_{y_{2}}\epsilon_{y_{2}}}}=\frac{\sqrt{\epsilon_{y_{2}}\epsilon_{y_{1}}}}{C\Delta\sigma_{\delta_{1}}}\,.$
(23)

Here, $\Delta\sigma_{\delta_{1}}$ is the additional relative energy spread
induced by the first TDS for suppression of microbunching instabilities, and
$\epsilon_{y_{1}}$ denotes the geometrical emittance upstream of the bunch
compressor at TDS1.

For the example parameters discussed throughout this paper (see also Table 1),
i.e., $C=13$, $\gamma\epsilon_{y_{1}}=0.6\,\mathrm{\mu m}$,
$\gamma\epsilon_{y_{2}}=0.72\,\mathrm{\mu m}$ (see Fig. 5(a)), and
$\Delta\sigma_{\delta_{1}}E\approx 10\,\mathrm{keV}$ with
$E=350\,\mathrm{MeV}$ ($\gamma=685$), the stability condition in Eq. (23)
yields pure relative energy jitter (neglecting rf phase jitter) of
$\sigma_{E}/E\ll 1.9\cdot 10^{-5}$ or pure rf phase jitter (neglecting energy
jitter) of $\sigma_{\varphi_{2}}\ll 0.012^{\circ}$. A combination of both will
obviously tighten the acceptable jitter. This level of rf stability is
difficult to achieve in normal conducting linacs with single bunch operation,
but might be achieved with superconducting accelerator technology like at
FLASH or as planned for NGLS, where many bunches can be accelerated in a long
rf pulse, i.e., in a bunch train. Currently, several rf feedforward and
feedback controls are able to stabilize the bunches at FLASH to
$\sigma_{E}/E=3.0\cdot 10^{-5}$ and $\sigma_{\varphi}=0.007^{\circ}$ at 150
MeV HS ; SP , and further improvements towards $\sigma_{E}/E\leq 1.0\cdot
10^{-5}$ are planned using a fast normal conducting cavity upstream of the
bunch compressors CS ; HS . With perfect scaling of rf jitter from several
independent rf power stations that adds uncorrelated, we would expect an
improvement of $\sqrt{150\,\mathrm{MeV}/350\,\mathrm{MeV}}\approx 0.66$
compared to the results at FLASH with $150\,\mathrm{MeV}$ and assuming the
beam energy of $350\,\mathrm{MeV}$ in the bunch compressor of the NGLS design.
Continuous-wave rf operation, as planned for the NGLS design Corlett , and a
proper choice of rf working points for FEL operation might improve the
stability further.

###  V.2 Integrated longitudinal phase space diagnostics

A practical spin-off of the reversible beam heater system is the availability
of longitudinal phase space diagnostics. The vertical betatron motion of
electrons passing through a TDS is described by Eq. (4), which enables a
mapping from time (longitudinal coordinate) to the vertical Kick ; Roehrs ;
Filippetto , and finally a possibility to obtain temporal bunch information by
means of transverse beam diagnostics. In a similar manner, the relative energy
deviation is mapped to the horizontal in the presence of horizontal momentum
dispersion, like in a magnetic bunch compressor chicane (see, e.g., Refs.
Roehrs ; Filippetto ). The combined operation makes single-shot measurements
of the longitudinal phase space possible, and in the case of the generic
layout of a reversible electron beam heater system as depicted in Fig. 1,
longitudinal phase space measurements become feasible using the first TDS and
observation screens in the dispersive section of the bunch compressor chicane.
In order to get information of the bunch length after the bunch compression,
the second TDS can be used with downstream observation screens (not shown in
Fig. 1).

In addition to invasive longitudinal phase space measurements of a single
bunch using observation screens, even fully noninvasive measurements utilizing
incoherent synchrotron radiation, emitted in the bunch compressor bending
magnets, are possible (see, e.g., Ref. Gerth ). When using a fast gated
camera, the implication will be the possibility of on-line monitoring the
longitudinal phase space of individual bunches in multi-bunch accelerators.

##  VI Summary and conclusions

Our studies show that the reversible beam heater system proposed here can
suppress microbunching instabilities and preserve the high beam brightness at
the same time. Due to CSR effects, some vertical emittance degradation in the
head and tail region of the bunch occurs, but the core emittances are well
preserved. In the numerical demonstrations using the code elegant, the first
TDS generates about 10 keV (r.m.s.) slice energy spread, which is similar to
the laser heater but with a more Gaussian energy distribution (cf. laser
heater). The bunch compression process increases the slice energy spread to
$\sim$ 130 keV (r.m.s.), which is then reversed to $\sim$ 17 keV (r.m.s.)
after the second TDS in the presence of CSR effects. Without CSR effects, the
slice energy spread is reversed to $\sim$ 13 keV (r.m.s.), which demonstrates
perfect cancelation. The simulations also show that initial bunching in energy
and density in the beam can be smeared out during the process in the
reversible beam heater system, i.e., microbunching instabilities can be
suppressed. The resulting smooth beam can then propagate through the remaining
accelerator without further generation of much additional energy spread and is
advantageous for any kind of laser seeding manipulations and experiments. For
example, this scheme significantly loosen the required laser power for short-
wavelength HHG seeding NLS  and may strongly impact the design of future
seeded FELs. In addition, the reversible beam heater system exhibits
integrated options for diagnosis and on-line monitoring of the longitudinal
phase space applicable for multi-bunch machines, which is also the preferred
type of accelerator for the reversible heater system due to large
sensitivities on energy and rf jitter. Linear accelerators based on
superconducting rf technology might be able to match the strict tolerances in
order to keep vertical kicks small and to achieve a sufficient trajectory
stability in the downstream undulators.

###### Acknowledgements.

We would like to thank P. Emma, Ch. Gerth, A. Lumpkin, H. Schlarb, and J.
Thangaraj for useful discussions and suggestions. This work was supported by
Department of Energy Contract No. DE-AC02-76SF00515.

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