Frequency domain methods manipulate the frequency spectrum of the input laser
pulse to control the time profile of the output laser pulse. There are two
approaches to frequency domain-based laser shaping that have been applied to
PE cathode guns: acousto-optic programmable dispersive filter and Fourier
Transform pulse shaping.

Figure 12: Fourier Transform laser pulse shaping system. A laser pulse (INPUT
PULSE) enters a 4-f grating stretcher and its frequency spectrum is exposed at
the Fourier Transform Plane. A mask (MASK) described by a frequency domain
transfer function, $H(\omega)$, is placed at this plane to modify the spectrum
of the input laser pulse resulting in a modified time structure of the output
laser (OUTPUT PULSE). From [307].

The acousto-optic programmable dispersive filter (AOPDF) relies on a
longitudinal interaction between a polychromatic acoustic wave and a
polychromatic optical wave in the bulk of a birefringent crystal. It controls
the IR spectrum by controlling the group delay versus wavelength with a
programmable acoustic wave in the birefringent crystal. Optical signals in the
hundreds of Terahertz range are controlled with RF signals in the tens of MHz
range. This is a compact IR device, installed between the photocathode laser
oscillator and amplifier, with high temporal resolution, which makes it
suitable for the shaping of femtosecond IR pulses. A widely utilized
commercial AOPDF—the dazzler™—can shape pulse length over a 6-ps duration at a
maximum repetition rate close to a MHz [297, 304].

In Fourier Transform pulse shaping, a 4-f grating stretcher is used to expose
the spectrum of the input laser pulse in the spectral Fourier Transform plane
of the stretcher (see Fig. 12). The spectrum can be controlled with an optical
element placed in this plane. Let the spectrum of the input laser pulse be
given by $X(\omega)$ and the frequency-domain transfer function of the optical
element at the Fourier plane be represented by, $H(\omega)$, then the spectrum
of the output laser pulse is given by their convolution,

$Y(\omega)=X(\omega)H(\omega),$ (184)

so that the output laser pulse in the time-domain is just the inverse Fourier
Transform of this convolution,

$y(t)=F^{-1}[Y(\omega)],$ (185)

The first systems used for shaping with an RF photocathode gun [216, 215] used
a fixed mask, described by $H(\omega)$, located in the Fourier Transform plane
to modify the amplitude of the input spectrum. However, this method can be
based on the modification of any of the amplitude, phase, or polarization of
the input spectrum. described by the appropriate $H(\omega)$. After the
spectrum is modified, the second grating is used to bring the spectrum back to
a line in the time domain to generate the desired temporal pulse shape of the
output pulse. The laser community has achieved both laser pulse trains and
temporal flattop laser pulses (Fig. 12) using masks to modify the amplitude of
the spectrum [307]. Recent Fourier Transform pulse shaping approaches are
based on programmable spatial light modulators (SLMs) due to their superior
resolution and flexibility. The first beam physics applications used an SLM
system to convert a Gaussian pulse of 9 ps FWHM to a flattop pulse of the same
length to drive a photocathode RF gun [325]. Gaussian pulses were transformed
into both triangular and super-triangular laser pulses with an SLM-based
system [156] as shown in Fig. 14.

Figure 13: Demonstrated laser pulse temporal shapes generated with the Fourier
Transform laser pulse shaping system of Fig. 12 using an amplitude mask. Laser
pulse train (left) and quasi-flattop laser pulse (right). Courtesy [307].
Figure 14: Comparison of simulated and demonstrated laser pulse temporal
shapes achieved with a Fourier Transform shaping system using SLMs. (a)
unmodified laser pulse shape; (b) spectrum optimized for a right triangle; (c)
spectrum optimized for a right triangle with the leading edge replaced by a
super-Gaussian section. Courtesy [156].

The advantage of the frequency domain method is its high resolution and
flexibility to produce various pulse shapes while the downside is its
complexity and stability. In addition, the need to transmit the IR laser pulse
shape through the laser amplifier and harmonic conversion crystals introduces
non-linearities causing differences between the IR and UV laser pulse shapes.
Recent trends point toward further development of programmable Fourier
Transform pulse shaping applied directly in the UV. Another solution would be
the development of PE cathodes that can respond to IR; this is another active
area of research.

#####  III.4.3.2 Time domain laser shaping

Time domain methods are based on laser pulse stacking. This is where a series
of short laser pulses are longitudinally combined (i.e., stacked) to form the
desired longitudinal profile. There are a number of methods that have been
tried in the past. An early method used for PE cathode guns [278] was based on
a Michelson interferometer in which $n$ 50/50 beam splitters, embedded in a
set of optical delay legs, were used to split a single input laser pulse into
a series of $2^{n}$ output laser pulses. Their device was used to generate a
train of 16 pulses at THz spacing with very low loss. More recently, pulse
stacking with birefringent crystals is almost exclusively used. It exploits
the group velocity mismatch between the ordinary and extraordinary axes of a
birefringent crystal (e.g. $\alpha$ -BBO) [335, 312, 243]. When a single
Gaussian input pulse passes through a uniaxial birefringent crystal rotated at
an angle relative to the crystal axis, two output pulses will emerge projected
onto the ordinary and extraordinary axes; see Fig. 15. The time delay
difference between these two components is due to the differing group
velocities along the ordinary and extraordinary axes

$\Delta t=L_{Xtal}\left(\frac{1}{v_{g,e}}-\frac{1}{v_{g,o}}\right),$ (186)

where $L_{Xtal}$ is the length of the crystal, and $v_{g,e}$ and $v_{g,o}$ are
the group velocities associated with the extraordinary and ordinary optical
axes, respectively. The crystal length controls the delay between the
different polarizations while crystal angle controls the relative intensity of
the different polarizations. If the angles are set to $45^{\circ}$, a flattop
laser pulse is obtained [312], as shown in Fig. 15. However, it is also
possible to obtain more complicated shapes (e.g., a double triangle) by
passing the input laser pulse through a stack of crystals rotated at optimized
angles relative to the incident laser pulse. In [175, 174], they used this
method to generate a double-triangular laser pulse which was then used in PE
cathode gun to generate the corresponding double-triangular electron bunch as
shown in Fig.16.

Figure 15: Flattop laser pulse generated with $\alpha$ -BBO laser pulse
stacking. From [312] Figure 16: Measured double-triangular electron bunch
generated with $\alpha$ -BBO laser pulse stacking. (top) x-z projection of the
bunch on a screen. (bottom) corresponding current profile. From [175].

The advantage of the time domain method lies in its simplicity. Moreover, if
the pulse stacker is implemented in the UV [243] the distortion of the pulse
due to nonlinearities arising during the frequency-conversion process is
completely averted. Its downside is its limited flexibility since there is no
way to change the crystal length on the fly though some control over the pulse
shape can be achieved by rotating the crystals.

In addition to producing shaped the “continuous” distributions shown above
(i.e. triangle and flattop), birefringent crystals have also been used to
produce bunch trains. Reference [167] proposes a simple and more compact
system specifically adapted to bunch train generation. The setup consists of a
concentric stair-step echelon combined with a focusing lens. The echelon
consists of a series of concentric flat zones with different thicknesses, so
as to introduce a discrete delay correlated to the transverse radius. At the
focal point of a downstream lens, such a configuration produces pulses that
are delayed in time. The nature of the echelon design introduces small delays
so that the method is well adapted to the generation of pulses with ps-scale
temporal separation. In [167] the technique is numerically investigated for
the formation of a train comprising 20- to 100-fs bunches with an associated
bunching factor peaked at $\sim 0.5$ THz for application to a coherent Smith-
Purcell THz source based on a 50-kV electron beam.

####  III.4.4 PE based spatio-temporal (3D) shaping

A natural continuation of the progress with programmable 2D transverse and 1D
longitudinal laser shaping are methods for shaping the complete 3D spatio-
temporal distribution of the laser. This method gives the electron source
designer complete control over the initial electron distribution in a PE
cathode gun. Recall, however, that this does not mean complete control over
electron distribution in general, since, once it leaves the cathode it is
subject to nonlinear space-charge forces that will distort the initial
distribution. Interestingly, one of the main motivations for this line of R&D
is to generate electron bunches with uniform 3D ellipsoid distributions since
these are the only distributions whose space-charge fields (i.e., internal
force fields) are linear functions of position [178]. This gives rise to
particularly simple dynamical behavior: a uniform ellipsoid under the
influence of its self-fields (electrostatic) will change its size but retain
its shape; a uniform ellipsoid with linear internal fields. In addition to the
3D ellipsoid laser pulse shapes, other distributions of interest is the 3D
cylindrical distribution and distributions with 2D circular transverse
distributions but combined with triangular 1D longitudinal distributions.

Recent progress in 3D laser shaping with IR laser pulses has been achieved by
several groups [197, 158, 157]. For example, researchers used multiple
programmable SLMs (Fig. 17) to generate both a 3D quasi-cylinder and 3D quasi-
ellipsoidal IR laser distribution [197]. In addition to SLM-based techniques,
3D shaping can also be accomplished via control of chromatic aberration. as
discussed by [166], where a dazzler™ system is used to introduce a complex
spectral structure that, in combination with a highly dispersive section,
resulted in the generation of a 3D ellipsoidal bunch [168].

Figure 17: Demonstrated 3D laser distributions based on SLMs. (a) measured
transverse intensity distributions of quasi-cylindrical pulses; (b)
reconstructed 3D distribution of quasi-cylindrical; (c) reconstructed 3D
distribution of quasi-ellipsoidal. From [197].

Despite the successful demonstration of these 3D IR laser distributions, these
have not yet been used to extract electrons from a photocathode gun. The next
steps require converting the IR pulse shapes to the UV where they can be used
to exicte PE cathodes. Nonetheless, this is an exciting and important area of
research. Further, it is interesting to note that the demand for 3D laser
pulse shaping, and its subsequent progress, has been lead by the PE cathode
gun community, not the laser community.

##  IV Beam control within one degree of freedom using external fields

This section discusses the use of externally applied fields to shape the beam
distribution. We introduce the general formalism and discuss various methods
commonly employed or recently proposed to control the beam distribution in
longitudinal or transverse directions and some of the associated projections.

###  IV.1 General considerations

We categorize beam-shaping methods using external fields into three main
approaches as depicted in Figure 18. The first approach consists of finding a
phase-space transformation that maps a given initial distribution to a final
target distribution; see Fig. 18(a). A second approach introduces a local
coupling between two variables so that the overall transformation is still
within one degree of freedom (DOF) but the local coupling enables access to
another variable Fig. 18(b). A common example is the use of dispersive
collimation where particles with given energies are removed via the
introduction of a local correlation between energy and position. Finally,
another technique employs an interceptive mask that modifies the momentum or
affects the particles’ transmission according to their transverse position
Fig. 18(c).

The use of an external electromagnetic field to impart correlation within one
DOF is commonly employed to control the beam properties along any of the DOFs.
For instance time-dependent fields are commonly used to introduce, e.g., a
linear correlation in the longitudinal phase space (LPS) such as $\delta=hz$,
that can later be exploited using a longitudinally dispersive beamline to
alter the final bunch length via $\sigma_{z,f}=C^{-1}\sigma_{z,0}$, where the
compression factor is defined in Eq. (100) and $h$ and $R_{56}$ are,
respectively, the chirp in the incoming ($z,\delta$) LPS and the longitudinal
dispersion associated with the dispersive section. It should be pointed out
that the same mechanism applies to the other DOFs, e.g., in the transverse
planes to focus the beam.

Generally, this type of manipulation can be split into two stages referred to
as a modulator and a convertor. The modulator introduces a position-dependent
momentum whereas the convertor consists of a beamline providing a momentum-
dependent change in the position. For the sake of simplicity, we consider the
case of uncoupled motion between the three DOFs, and focus on one of the DOFs,
with particle conjugated variables $(\zeta_{i},\zeta_{i+1})$ [where
$i\in[1,3,5]$]. We consider the effect of the modulator to impact a position-
dependent external force along $\hat{\zeta}_{i+1}$ of the form
${F}_{i+1}(\zeta_{i})$, resulting in the coordinate transformation

$\displaystyle\zeta_{i,0}\rightarrow\zeta_{i,m}$ $\displaystyle=$
$\displaystyle\zeta_{i,0},$
$\displaystyle\zeta_{i+1,0}\rightarrow\zeta_{i+1,m}$ $\displaystyle=$
$\displaystyle\zeta_{i+1,0}+{F}_{i+1}(\zeta_{i,0}),$ (187)

under an impulse approximation (where we ignore change in the particle
position). In the latter equation, the subscripts 0 and m respectively refer
to the values before and after the modulator section. The downstream converter
beamline introduces a momentum-dependent change in position so that the final
position $\zeta_{i,f}$ is related to the upstream coordinate as

$\displaystyle\zeta_{i,m}\rightarrow\zeta_{i,f}$ $\displaystyle=$
$\displaystyle\zeta_{i,m}+{G}_{i}(\zeta_{i+1,m}),$
$\displaystyle\zeta_{i+1,m}\rightarrow\zeta_{i+1,f}$ $\displaystyle=$
$\displaystyle\zeta_{i+1,m}.$ (188)

Overall, the transformation associated with the beamline takes the coordinate
$(\zeta_{i,0},\zeta_{i+1,0})$ and transforms it to the final coordinate

$\displaystyle\begin{pmatrix}\zeta_{i,f}\\\
\zeta_{i+1,f}\end{pmatrix}=\begin{pmatrix}\zeta_{i,0}+{G}_{i}[\zeta_{i+1,0}+{F}_{i+1}(\zeta_{i,0})]\\\
\zeta_{i+1,0}+{F}_{i+1}(\zeta_{i,0})\end{pmatrix},$ (189)

and a transfer map defined such that $(\zeta_{i,f},\zeta_{i+1,f})={\cal
M}(\zeta_{i,0},\zeta_{i+1,0})$ can be formally associated with the
transformation.

Considering an initial phase-space-density distribution
$\Phi_{0}(\zeta_{i,0},\zeta_{i+1,0})$ and invoking Liouville’s theorem II.3.1
$\Phi_{f}(\zeta_{i,f},\zeta_{i+1,f})d\zeta_{i,f}d\zeta_{i+1,f}=\Phi_{0}(\zeta_{i,0},\zeta_{i+1,0})d\zeta_{i,0}d\zeta_{i+1,0}$,
we can write the final phase-space distribution as

$\displaystyle\Phi_{f}(\zeta_{i,f},\zeta_{i+1,f})=\Phi_{0}[{\cal
M}^{-1}(\zeta_{i,0},\zeta_{i+1,0})],$ (190)

since the Jacobian of the transformation is unity. We now consider the
projection along the position direction $\widehat{\zeta_{i}}$ defined as

$\displaystyle P_{i}(\zeta_{i})=\int d\zeta_{i+1}\Phi(\zeta_{i},\zeta_{i+1}).$
(191)

By virtue of the charge-conservation we now have

$\displaystyle
P_{f}(\zeta_{i,f})=P_{0}[\zeta_{i,0}(\zeta_{i,f})]\frac{\partial\zeta_{i,0}}{\partial\zeta_{i,f}},$
(192)

where the RHS can be written solely in term of $\zeta_{i,f}$ via inversion of
the map described in Eq. (189). Therefore, by properly tailoring the
transformation ${\cal M}$, one can modify the shape of the projection along
any direction. To illustrate the set of derived equations we first consider
the simple example of the bunch compression discussed in Section II.3.1. Given
the incoming-LPS coordinate $(z_{0},\delta_{0})$, the correlation introduced
by the linear accelerator and the energy-dependent path length from the
compression can be described, respectively, by $F(z)=hz$ and
$G(\delta)=R_{56}\delta$. Here for sake of simplicity we take both $F$ and $G$
to be linear functions of $s$ such that the overall transformation is
described by

$\displaystyle\begin{pmatrix}z_{f}\\\
\delta_{f}\end{pmatrix}=\begin{pmatrix}z_{0}+R_{56}[\delta_{0}+h(z_{0})]\\\
\delta_{0}+h(z_{0})\end{pmatrix},$ (193)

so that the final longitudinal charge distribution is

$\displaystyle\lambda_{f}(z_{f})=C\lambda_{0}\left(\frac{z_{f}-R_{56}\delta_{0}}{{C}}\right),$
(194)

where ${C}$ represents the compression factor defined in Eq. (100). Taking
$\lambda_{0}(z_{0})=1/\sqrt{2\pi\sigma_{z,0}^{2}}\exp[-z_{0}^{2}/(2\sigma_{z,0}^{2})]$
and assuming that $\delta_{0}$ represents a random uncorrelated fraction
energy spread, we obtain

$\lambda_{f}(z_{f})=\frac{C}{\sqrt{2\pi}\sigma_{z,0}}\exp\left(-\frac{C^{2}z^{2}_{f}}{2\sigma_{z,0}^{2}}\right)\\\
\times\exp\left(-\frac{C^{2}R^{2}_{56}\delta_{0}}{2\sigma_{z,0}^{2}}\right),$
(195)

which showcases the well-known results associated with linearized bunch
compression previously mentioned in Section II. A simple extension is to
consider the case where nonlinearities play a role in the compression. For
instance, owing to RF curvature in the linac employed to impart the chirp, a
quadratic dependence on $\zeta_{5,0}$ is also introduced when the bunch length
does not strictly verify the condition $\sigma_{z,0}\ll\lambda$ (where
$\lambda$ is the wavelength of the accelerating mode in the linac) so that
$F(z)=hz+h_{2}z^{2}$. Likewise, standard four-bend bunch compressors are known
to introduce a second-order longitudinal dispersion $T_{566}$, and
consequently $G(\delta)=R_{56}\delta+T_{566}\delta^{2}$. Following the same
approach as before yields a final distribution along the longitudinal axis to
be of the form [164]

$\displaystyle\lambda_{f}(z_{f})=\frac{1}{\sqrt{2\pi}\sigma_{z,f}}\frac{\exp({z_{f}/(\sqrt{2}\sigma_{z,f}})}{[-z_{f}/(\sqrt{2}\sigma_{z,f})]^{1/2}}\Theta(-z_{f}),$
(196)

where $\Theta()$ is the Heavide function. Thus, the nonlinear transformation
introduced by the functions $F$ and $G$ now results in a change of the bunch
current profile commonly encountered in, e.g., magnetic compression, subjected
to strong nonlinearities; see for instance [59].

Figure 18: Overview of phase-space shaping method discussed in Section IV with
the labels “initial” and ”final” referring to the initial and final
distributions. Shaping via non-linear fields and transport (a): a momentum
dependence on the coordinate is introduced via the function $F$ and combined
with a downstream beamline introducing a momentum-dependent position via $G$
so that $\zeta_{i}\mapsto G\circ F(\zeta_{i})$. Shaping using a local
correlation between two DOFs (b): one of the coordinates $\zeta_{i}$ is
coupled to another DOF coordinates $\zeta_{j}$, via the transformation
$\zeta_{j}=H(\zeta_{i})$, $\zeta_{j}$ and the coupling removed via $H^{-1}$.
Shaping through collimation (c): an intercepting mask in the $(x,y)$ plane
shown as a shaded area applies a transmission function $T(x,y)$ resulting in a
shaped transverse distribution.

The example considered so far can be generalized via the introduction of
arbitrary nonlinear functions $F$ and $G$ to control the final phase space
$(\zeta_{i},\zeta_{i+1})$ correlations. By controlling the degree of the non-
linearity introduced, one can, in principle, tailor the correlations within
the phase space to produce the desired profile along each of the phase-space
directions.

In beam physics, it is customary to expand the transport map via a truncated
Taylor series and consider each nonlinear order separately; see Section
II.3.5. This is a typical process in optics where, e.g., the energy-dependent
path length introduced by a dispersive section, e.g., a bunch compressor as
discussed above, is now written as
$\zeta_{5,f}=\sum_{i=1}^{6}R_{5i}\zeta_{i,0}+\sum_{i=1}^{6}\sum_{j\geq
1}^{6}T_{5ij}\zeta_{i,0}\zeta_{j,0}+{\cal O}(\zeta_{0}^{3})$ and likewise for
any beamline; see Eq. (133). For sake of simplicity we assume a pencil bunch
so that high-order coupling between the longitudinal and transverse phase
spaces can be neglected in the expansion of the transfer map.

So far we have considered the modulator function to be a continuous function
of the coordinate over a finite interval. Another approach is to consider a
periodic function of the form $F(u)=F_{0}\cos(ku+\phi)$. In such a case, a
series of concatenated modulators described with a function
$F_{i}(u)=F_{0,i}\cos(k_{i}u+\phi_{i})$ could be used to synthesize the
desired final distribution by controlling the term of the Fourier series
associated with the final distribution. Such a description can be employed,
for example, to describe a chain of linacs operating at different frequencies.
By shaping the energy spectrum and using a transformation in $z={\cal
M}(\delta)$ one could tailor the energy spread or current profile. In practice
introducing an arbitrarily high harmonic may be challenging, especially for
the time-dependent field given, e.g., the limited set of klystron frequencies.

####  IV.1.1 Interceptive beam shaping

The shaping techniques described so far combine nonlinear external fields with
the beamline providing nonlinear correlation between the position and momentum
of particles. A straightforward shaping technique consists of intercepting the
beam with a mask that has a given “transmission” function $T(x,y)$ so that the
mask can only affect the distribution in the transverse spatial coordinates;
see Fig. 18(c). The incoming phase-space distribution
$\Phi_{i}(\zeta_{1}...\zeta_{4})$ is then simply transformed as

$\displaystyle\Phi_{f}(x,y)=\Phi_{i}(x,y)T(x,y),$ (197)

where we assume the mask to be thin and only affecting the beam according to
the transverse coordinate (and not the momentum). Consequently, the transverse
profile along, e.g., $\zeta_{1}$ can be found from

$\displaystyle P_{f}(x)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\Phi_{i}(x,y)T(x,y)dy$ (198)
$\displaystyle=$
$\displaystyle\int_{\Upsilon_{-}(x}^{\Upsilon_{+}(x)}\Phi_{i}(x,y)dy,$

where we have taken the mask to be a binary function with unity value in the
domain $y\in[\Upsilon_{-}(x),\Upsilon_{+}(x)]$.

A drawback of such a masking technique is its intrusive nature, which may
hinder its application to high-power or high-repetition-rate beams, as the
beam loss associated with the shaping process could result in radiological
activation or hardware damage.

####  IV.1.2 Manipulation with local coupling

Another class of transformation involves local correlations between
coordinates within two DOFs; see Fig. 18(b). In such a transformation, an
external field introduces the required correlations $\zeta_{j}=H(\zeta_{i})$
and the coordinate $\zeta_{j}$ is manipulated (e.g., via a function similar to
$F$ or $G$). Finally, the inverse transformation $H^{-1}$ to remove the
correlation between $\zeta_{j}$ and $\zeta_{i}$. In the process the shaping
imparted to $\zeta_{i}$ via shaping of $\zeta_{j}$ is preserved; see Fig.
18(b). A simple example of implementation of a manipulation based on local
coupling regards dispersive collimation where a local dispersion bump locally
introduces a correlation between transverse position and energy where a
collimator is used to tailor the energy distribution (e.g. remove energy
tail). One advantage of local-coupling methods combined with a mask is its
simple implementation while providing a high degree of control over the beam
shape (via precise shaping of the intercepting mask [182]). However, the mask
can result in significant particle losses.

###  IV.2 Generation of shaped current distributions

One important aspect of LPS control resides in the ability to control the beam
current profiles for application in beam-driven wakefield accelerators and
light sources. Historically, current beam shaping has been an integral part of
electron injectors based on continuous wave (CW) electron sources where a
combination of masks and RF cavities $-$ often dubbed “chopping” systems $-$
are commonly employed to form bunches for injection in the subsequent linear
accelerators [282, 296].

####  IV.2.1 Local coupling combined with transverse masking

Current shaping techniques were initially discussed as a means to prebunch the
beam for free-electron laser (FEL) [217] applications. The method was
eventually demonstrated at the ATF facility [201] where it was also extended
to shaping beyond microbunch generation [275]. In brief, the method combined
local coupling between the transverse (horizontal) and longitudinal phase
spaces with masking. To explain the technique we consider an incoming bunch
with an LPS chirp $h_{0}$ sent to a dispersive section with a transfer matrix
producing the horizontal and longitudinal dispersions $\eta$ and $\xi_{-}$,
respectively. Downstream of such a dispersive section the final horizontal and
longitudinal positions of an electron is

$\displaystyle x_{m}$ $\displaystyle=$ $\displaystyle
x_{\beta}+\eta\delta_{0},$ $\displaystyle z_{m}$ $\displaystyle=$
$\displaystyle z_{0}+\xi_{-}\delta_{0},$ (199)

where $x_{\beta}$ is the geometric contribution to the beam spot. An
intercepting mask in $(x_{m},y_{m})$ with transmission function dependent on
the horizontal coordinate $f(x_{m})$ will tailor the transverse profile. For a
beam with correlated momentum spread $\delta_{0}=h_{0}z_{0}$ the mask will
alter the shape of the longitudinal distribution following $f[x_{m}/(\eta
h_{0})]$. A downstream dispersive section with longitudinal dispersion
$\xi_{+}$ and designed to suppress the dispersion introduced by the section
upstream of the mask results in the overall shaping function $f[x_{m}/(\eta
h_{0})]$. This technique bears similarities with the frequency-domain temporal
shaping method commonly encountered in ultrafast laser shaping, where a
frequency-chirped laser is dispersed, its spectrum modified with a mask, and
subsequently recombined [156, 307]; see Section III.4.3.1.

The method was implemented in [201] to form train of sub-ps bunch. In such a
case the mask consists of a set of $N$ slits so that the transmission can be
approximated as $f(x_{m})=\sum_{\ell=1}^{N}\delta(x_{m}-X_{\ell})$ where
$X_{\ell}$ are the horizontal positions of the vertical slits. Considering
$X_{\ell}=\ell D$, with $D$ being the inter-slit spacing on the mask, the
final modulation period is shown to be [201, 125]

$\displaystyle\Delta z$ $\displaystyle\simeq$ $\displaystyle
D\frac{1+\xi_{+}h_{0}}{\eta h_{0}}.$ (200)

A practical implementation of the method used for a proof-of-principle
experiment at the Accelerator Test Facility (ATF) appears in Fig. 19(a). The
beam was locally dispersed in a dogleg beamline with vanishing net dispersion.
The dogleg included an optical lattice providing an antisymmetric dispersion
function with maximum value attaining $\eta\simeq 1.5$ m in close proximity to
the mask location. The mask consisted of sub-mm tungsten wire; see inset
photograph in Fig. 19(a). The produced bunch was modulated in energy and time
($z$) owing to the final LPS chip. Given the fixed dispersion, the mask sets
the energy modulation period and ultimately the final current modulation,
which can be controlled via the incoming chirp and $\xi_{+}$ function of the
beamline. It was demonstrated that the technique enables some control over the
final temporal period downstream of the dogleg beamline: for $\sim 10$ pC
modulations with sub-mm periods were produced, consistent with the resonant
excitation of wakefield in plasmas [199]. Likewise, this method was extended
to produce bunches with triangular beam distribution at the ATF [275] for
wakefield excitation in dielectric-lined waveguides [11].

Figure 19:  Experimental generation of a sub-picosecond bunch train at the ATF
using a dogleg beamline (a) with final beam distribution in a downstream
spectrometer (b) and measured temporal distribution via interferometry of
coherent transition radiation (c). Figure adapted from [201] and [200].

More recently, a current-shaping method based on directly introducing a
spatio-temporal coupling was proposed using a pair of transverse-deflecting
cavities [154, 333]. Such a concept is similar to the one commonly employed in
DC electron injectors as part of the low-energy “chopping” system required to
form the bunched beam before injection in an RF linac [314].

The main advantage of this shaping method compared to the one based on
dispersive coupling stems from its ability to directly couple $z$ to one of
the transverse coordinates. In the method does not suffer from CSR-induced
phase-space degradation. It can also provides control over the LPS chirp. In
its simplest implementation, the beamline consists of two transverse
deflecting cavities (TDCs) separated by a beamline described by a transfer
matrix $M$. The overall matrix of the system in $(x,x^{\prime},z,\delta)$ is

$\displaystyle
S=R_{TDC}(\kappa_{d})MR_{TDC}(\kappa_{u})\equiv\begin{pmatrix}A&B\\\ C&D\\\
\end{pmatrix},$ (201)

where $R_{TDC}(\kappa)$ represents the matrix of a TDC with strength $\kappa$,
see Section (II.2.3). In order for the beamline to be globally uncoupled, the
$2\times 2$ block matrix should vanish: $B=C=0$. Solving for $B=0$ yields the
condition on the elements of $M$ and cavity strength to

$\displaystyle m_{12}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{L\left(\kappa_{d}+\kappa_{u}\left(Lm_{21}+m_{11}+2m_{22}\right)\right)}{2\kappa_{u}},$
$\displaystyle\kappa_{d}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{\kappa_{u}\left(Lm_{21}+2m_{22}\right)}{2},\mbox{and}$
(202) $\displaystyle m_{11}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{Lm_{21}}{2}-\frac{\kappa_{u}}{\kappa_{d}}.$

These matching conditions ensure $S$ becomes block diagonal. The
$(x,x^{\prime})$ block matrix is independent of the TDC strengths, and the
overall effect of the combined TDCs is to modify the LPS chirp
$h\equiv\mbox{$\langle{\zeta\delta}\rangle$}/\mbox{$\langle{\zeta^{2}}\rangle$}^{1/2}$
as $h\rightarrow h=h_{0}-S_{65}$ with

$\displaystyle
S_{65}=\displaystyle\frac{L\kappa_{u}^{2}\left[(Lm_{21}+2m_{22})^{2}+4\right]}{16}.$
(203)

We have $S_{65}>0$ for a cavity length $L\neq 0$. It is interesting to note
that the chirp vanishes in the thin-lens model of the TDC ($L=0$). Likewise
the block matrix becomes

$\displaystyle M_{\perp}=\begin{pmatrix}m_{22}^{-1}&0\\\ m_{21}&m_{22}\\\
\end{pmatrix},$ (204)

consistent with Ref. [154] where the choice $m_{11}=m_{22}=-1$ was made so
that $\kappa_{u}=\kappa_{d}$ (given that decoupling forces $m_{12}=0$). The
shaping mask is located between the two TDCs and with sufficient phase advance
from the upstream TDC to ensure the beam has a significant correlation
$\langle{xz}\rangle$ at its location. The profile of the mask $T(x,y)$ is then
tailored to provide the desired temporal shape. The application of this method
to produce a modulated relativistic electron bunch was first discussed in [66]
in the context of superradiant THz radiation, and further investigated
numerically in [114] to support the formation of various current shapes to
support a beam-driven wakefield accelerator with enhanced transformer ratio.
Figure 20 presents numerical simulation results for a minimal system composed
of two TDCs and a quadrupole magnet with a different type of mask to produce
doorstep distributions [Fig. 20(a,b)] for beam-driven acceleration and sub-ps
bunch trains with variable microbunch spacing [Fig. 20(c,d)].

Figure 20:  Numerical modeling of current-profile shaping showcasing the
generation of ramped (a), reverse-triangle (b) and modulated (c,d) bunch
distributions using local correlations imparted by a pair of transverse-
deflecting cavities (TDC) combined with a mask (e) (adapted from [114]).

The masks described so far stop a fraction of the incident beam. An
alternative solution is to use a spoiling mask made of a thin foil to alter
the beam properties associated with a fraction of the beam. An example of
implementation uses a slotted foil at the Linac Coherent Light Source (LCLS)
[75] to selectively spoil the transverse emittance of an electron beam (the
fraction of the beam intercepted by the foil undergoes multiple scattering and
suffer an emittance growth). The method was employed to ultimately control the
duration of X-ray pulses in FELs, as only the unscattered beam population
contributed to the lasing mechanism and supported the generation of isolated
femtosecond X-ray pulses along with twin variable-delay X-ray pulses using a
dual-slot foil [58].

####  IV.2.2 Modulators combined with longitudinally dispersive sections

A simple implementation of a current-shaping technique consists of introducing
an energy modulation using RF accelerating cavities to properly control the
longitudinal dispersion in the downstream beamline. The energy modulation is
introduced as a time-varying field with accelerating voltage of the form
$V(z)=\widehat{V}\cos(kz+\varphi)$, where $\widehat{V}$, $\varphi$, and $k$
are, respectively, the peak accelerating voltage and phase, and the
accelerating mode wavevector. The LPS transformation associated with the
downstream dispersive sections is described via the Taylor expansion of the
longitudinal map with, e.g., first- and second-order the coefficients
$R_{56}\equiv\frac{\partial z}{\partial\delta}$,
$T_{566}\equiv\frac{\partial^{2}z}{\partial\delta^{2}}$ as detailed in Section
II.3.5. The two widely-used approaches are to either ($i$) introduce a linear
energy modulation and controls the nonlinear dispersion in the dispersive
section, or to ($ii$) control the nonlinear modulation using accelerating
sections operating at multiple frequencies.

Figure 21:  Experimental setup for phase-space shaping via introduction of
nonlinear longitudinal dispersion at the NEPTUNE facility (a) and resulting
LPS (b,d) with associated current distribution (c,e) experimentally generated.
Figure adapted from [79].

A technique to control the nonlinear longitudinal dispersion to shape the beam
current distribution was introduced in [78]. It was observed that compression
of a “long” incoming bunch accelerated in an S-band linac such that $kz\ll 1$
is not strictly valid. As a result, the final fractional energy offset is

$\displaystyle\delta\simeq\frac{e\widehat{V}}{\bar{E}}\left[-kz\sin\varphi+\frac{(kz)^{2}}{2}\cos\varphi\right],$
(205)

(where $\bar{E}$ is the final beam energy) resulting in significant quadratic
correlation in the LPS with associated current profile described by an
equation similar to Eq. (196). It was recognized that the addition of
sextupole magnets in the dispersive section (arranged as a dogleg beamline)
provided a sufficient handle over the $T_{566}$ coefficient to ultimately
control the current distribution. The experimental implementation of this
technique at the NEPTUNE facility demonstrated the generation of $\sim 12$-MeV
bunches with linearly ramped current profiles [79]; see Fig. 21. The technique
was also shown to provide some control over the ramp shape and direction.

The introduction of higher-order longitudinal dispersion (up to third order)
was also proposed to suppress current spikes arising from collective effects
in the LCLS accelerator [45]. At LCLS, extreme current values at the head and
tail of the electron bunch are responsible for substantial CSR-induced
projected emittance growth. The nominal operating mode of LCLS consits of
truncating the head and tail to suppress the current spikes to improve the FEL
lasing performances. It was shown that current spikes can be strongly
suppressed via control of the
$U_{5666}\equiv\frac{1}{6}\frac{\partial^{3}z}{\partial\delta^{3}}$
longitudinal-dispersion term using an octupole magnet without a significant
increase of the horizontal slice emittance.

Further improving the precision of shaping methods discussed in this section
can be accomplished using multiple energy modulators operating at different
frequencies. Such an approach allows for the incoming LPS to be an arbitrary
polynomial function $\delta(z)$ with its coefficients controlled by the
settings of the modulator accelerating-voltage amplitude and phase. The
solution was investigated using a dual frequency modulator operating at the
frequencies $f_{1}$ and $f_{n}\equiv nf_{1}$ with total accelerating voltage
$V(z)=V_{1}\cos(k_{1}z+\varphi_{1})+V_{n}\cos(k_{n}z+\varphi_{n})$, where
$\widehat{V_{1,n}}$ and $\varphi_{1,n}$ are, respectively, the accelerating
voltages and operating phases of the two linac sections, and $k_{1,n}\equiv
2\pi f_{1,n}/c$. Assuming $k_{1,n}z_{0}\ll 1$, the electron’s LPS coordinates
downstream of the linac are
$(z_{l}=z_{0},\delta_{l}=a_{l}z_{0}+b_{l}z_{0}^{2})$, where $a_{l}\equiv
a_{0}-e(k_{1}V_{1}\sin\varphi_{1}+k_{n}V_{n}\sin\varphi_{n})/\bar{E}_{l}$,
$b_{l}\equiv
b_{0}-e(k_{1}^{2}V_{1}\cos\varphi_{1}+k_{n}^{2}V_{n}\cos\varphi_{n})/(2\bar{E}_{l})$
with $e$ being the elementary charge and $\bar{E}$ the beam’s average energy
downstream of the linac. The passage of the bunch through a longitudinally
dispersive section results in an electron final coordinate to be given as a
function of the initial coordinates following
$z_{f}=a_{f}z_{0}+b_{f}z_{0}^{2}$ with $a_{f}\equiv 1+a_{l}R_{56}$ and
$b_{f}\equiv b_{l}R_{56}+a_{l}^{2}T_{566}$. Taking the initial current to
follow the Gaussian distribution
$I_{0}(z_{0})=\hat{I_{0}}\exp[-z_{0}^{2}/(2\sigma_{z,0}^{2})]$ (where
$\hat{I_{0}}$ is the initial peak current), and invoking the charge
conservation gives the final current distribution

$\displaystyle I^{u}_{f}(z_{f})$ $\displaystyle=$
$\displaystyle\frac{\hat{I}_{0}}{\Delta^{1/2}(z_{f})}\exp\left[-\frac{(a_{f}+\Delta^{1/2}(z_{f}))^{2}}{8b_{f}^{2}\sigma_{z,0}^{2}}\right],$
(206) $\displaystyle\times\Theta[\Delta(z_{f})],$

where $\Delta(z_{f})\equiv a_{f}^{2}+4b_{f}z_{f}$ and $\Theta()$ is the
Heaviside function.

An example of the method discussed above was experimentally implemented in the
FLASH facility at DESY [234], where a linac composed of 1.3-GHz and 3.9-GHz
superconducting-linac modules and two magnetic bunch compressors was used; see
Fig. 22(a). Specifically, the accelerator settings were optimized to form a
linear-ramped current profile; see Fig. 22(b,c). The technique provided ample
control over the shape owing to the number of variables available and produced
a shaped beam at 300 MeV for future injection in the FLASH-Forward facility
[17].

Figure 22:  Experimental results on phase-space shaping using a dual-frequency
linac at the FLASH facility at DESY (a) with corresponding nominally
compressed beam (b) and linearly-ramped bunch generation (c) obtained via
proper control of the accelerating voltage phase and amplitude of the linac
sections ACC1 (1.3 GHz) and ACC39 (3.9 GHz) prior to the bunch compressor 1
(BC1) (adapted from Ref. [234]).

This method could be generalized in principle by introducing an arbitrary
number of accelerating sections operating at different frequencies to
ultimately synthesize any correlation in the LPS via a Fourier series. The
generalized scheme is challenging in practice due to the limited number of RF
sources and associated prohibitive cost. However, it could be implemented
passively as discussed in Section V.

As a final note, we should point out that most of the time, current shaping
relies on an integration of different shaping techniques as explored in [48,
161]. For instance, the generation of uniform beams required for cascaded
harmonic lasing in X-ray FELs combines photocathode laser shaping to precisely
precompensate for nonlinearities introduced in the compression process or
arising from collective effects [48]. This work especially demonstrated the
generation of uniform current distribution based on numerical simulations; see
Fig. 23(a,b). Likewise, numerical simulations indicate that a properly shaped
photocathode laser pulse combined with a multi-frequency linac and a two-stage
nonlinear compression process could provide precise control over the LPS and
current distribution [294], ultimately realizing the sought-after door-step
distribution without the requirement for any collimation; see Fig. 23(c,d).

Figure 23:  Examples of LPS (a,c) and associated current distribution(b,d)
obtained via a fully integrated combining the photocathode-laser shaping
method discussed in Section III with multi-frequency linacs and nonlinear
dispersive sections. Plots (a,b) are reproduced from [48], which simulated a
possible operation mode of the FERMI@ELETTRA FEL. Plots (c,d) are extracted
from [294], which investigated the generation of shaped bunches for efficient
acceleration in a corrugated-waveguide wakefield accelerator. In plot (c), the
superimposed red trace corresponds to the slide RMS energy spread (right
label).

###  IV.3 Realizing ultra-low energy spread

A recurrent topic associated with the use of bright electron beams regards the
generation of ultra-low longitudinal emittance. One application of such a
capability is the production of a low-energy-spread beam, ultra-short electron
bunch for use in ultra-fast electron scattering experiments. There has been a
renewed interest in meV energy-spread, sub-MeV electron bunches for
applications in electron energy-loss spectroscopy [73].

In principle, the method described above and detailed in [329] could be
tailored to produce low energy spread. However, most of the low-energy-spread
instruments have been based on dispersive collimation implementing an ‘omega
filter’ [300]. One issue associated with this class of monochromator commonly
used in electron-scattering instrument is the limited beam transmission. To
palliate this limitation, a lossless monochromator was proposed [69] and
demonstrated via numerical simulation the ability to produce 200-keV electron
bunch with meV-level energy spread. The setup combines a photoemission
electron source with a pair of accelerating cavities. The bunch-length
lengthening ultimately set the final fraction energy spread
$\sigma_{\delta,f}$ under Liouville’s theorem as
$\sigma_{\delta,f}=\sigma_{\delta,0}\frac{\sigma_{z,0}}{\sigma_{z,f}}\geq\frac{\hbar}{2}$
where 0 refers to the initial bunch length and fractional energy spread. [69]
note that the final kinetic energy $E_{K}$ distribution along the bunch is
quadratically dependent on the particles’ time of arrival and radial position
within the bunch $E_{K}(t,r)=at^{2}+br^{2}$ such a dependence can be
compensated by the pair of accelerating cavities.

###  IV.4 Controlling longitudinal-phase-space (LPS) correlations

As discussed so far, the shaping of the bunch’s current profile is often
implemented by controlling correlation in the LPS. In this section, we focus
on methods that have been implemented with the primary purpose of controlling
correlation in the LPS. An example is the removal of correlated energy spread
downstream of an accelerator to produce, e.g., a more monoenergetic bunch.

Figure 24:  A TDC-based beamline for the generation of strong LPS chirps (a).
The diagrams (b-d) illustrate the $x-y$ electron-bunch distribution inside
each TDC with the green arrows gives the direction of the transverse kick
$x^{\prime}$. The color map indicates the energy distribution with larger and
lower energies respectively appearing as red and blue colors. Adapted from
[324].

Although linear energy-spread control is conventionally achieved using an
accelerating cavity operated off the crest, the method introduced in [324]
combines several deflecting cavities to provide a power-efficient control of
the LPS chirp. Specifically, the technique considers three horizontally
deflecting cavities separated by a distance $D$ and with respective deflection
strengths $\kappa$, $-2\kappa$, and $\kappa$ to produce an uncoupled transport
matrix in $(x,x^{\prime},z,\delta)$ with longitudinal $2\times 2$ block given
by

$\displaystyle R_{z|\delta}=\begin{pmatrix}1&0\\\
-\frac{2}{3}\kappa^{2}(3D+2L_{c})&1\end{pmatrix},$ (207)

where $L_{c}$ is the transverse-deflecting cavity length; see Fig. 24. The
latter equation reveals the main advantage associated with the proposed TDC-
based scheme over conventional off-crest acceleration. The introduced chirp
$R_{65}$ scales quadratically with the cavity’s strength $\kappa$ while the
off-crest acceleration scales linearly with the accelerating field. Although
this simple model is not capable of altering the polarity of the chirp,
introducing focusing elements between the TDC does enable such control.
Additionally, the focusing elements can also be optimized to mitigate
transverse-emittance degradation throughout the beamline [324].

Figure 25:  Example of longitudinal phases-space linearization at the LCLS
facility (adapted from [4]. Shown is the LPS measurement without (a) and with
(b) operation of the fourth-harmonic linearizing cavity (operating at 11.424
GHz).

Accessing higher-order correlation in the LPS is more challenging especially
when dealing with short (picosecond-scale) bunches, as the required RF
wavelength needs to be comparable to the bunch length. However, some
correction can be achieved with proper choice and interplay between RF systems
operating at different frequencies or through non-relativistic effects. For
example, correcting or imposing quadratic nonlinearities in the LPS can be
achieved by combining a harmonic RF field to the fundamental mode, as
introduced in [282] and later adapted for the LCLS and FLASH FELs [86]. The
method relies on introducing a harmonic frequency so that the final fractional
energy offset associated with an electron located at $z$ takes the form

$\displaystyle\delta(z)$ $\displaystyle=$
$\displaystyle\delta_{0}\frac{E_{0}}{E_{f}}+\frac{eV_{1}}{E_{f}}[\cos(k_{1}z+\varphi_{1})-\cos\varphi_{1}]$
(208)
$\displaystyle+\frac{eV_{n}}{E_{f}}[\cos(k_{n}z+\varphi_{n})-\cos\varphi_{n}]$
$\displaystyle\simeq$
$\displaystyle\delta_{0}\frac{E_{0}}{E_{f}}+\left(\frac{eV_{1}}{E_{f}}k_{1}\sin\varphi_{1}+\frac{eV_{n}}{E_{f}}k_{n}\sin\varphi_{n}\right)z$
$\displaystyle-\left(\frac{eV_{1}}{2E_{f}}k_{1}^{2}\cos\varphi_{1}+\frac{eV_{n}}{2E_{f}}k^{2}_{n}\cos\varphi_{n}\right)z^{2},$

where $\delta_{0}$ is the initial relative energy offset, $\varphi_{1}$ and
$V_{1}$ (resp. $\varphi_{n}$ and $V_{n}$) are, respectively, the phase and
amplitude of the fundamental (resp. harmonic) section of the linac, and
$E_{f}$ the final energy. Considering the simple case of on-crest operation
$\varphi_{1}=\varphi_{n}=0$, we note that cancellation of the second-order
term can be accomplished when the harmonic section is operating on the
deceleration phase with voltage amplitude
$V_{n}=V_{1}\left(\frac{k_{1}}{k_{n}}\right)^{2}$. Such an approach was first
demonstrated at the Free Electron LASer in Hamburg (FLASH) facility in DESY
[117] by adding a third-harmonic linac ($f_{n}=3.9$-GHz) to the fundamental-
frequency linac ($f_{1}=1.3$ GHz). The same scheme was implemented at the
European X-ray FEL [53]. Figure 25 showcases a measurement of the LPS-
linearization process at the LCLS where a fourth-harmonic cavity
($f_{n}=11.424\mbox{\leavevmode\nobreak\ GHz}=4\times 2.856$ GHz) is employed
[4]. Similar, methods were investigated in the context of ultrafast electron-
diffraction setups [85].

It has also been recognized that this type of LPS linearization can be
implemented without the need for a harmonic field when dealing with non-
ultrarelativistic bunches. Such a scheme was introduced in [153] were it was
shown via numerical simulations and experimentally confirmed that a proper
control of the phase and amplitude of a buncher and capture cavities could
control the $R_{56}$ and $T_{566}$ in the CEBAF injector [306]. A similar
technique was explored theoretically and via simulation for an ultra-fast
electron source using an RF photoinjector; see [329]. Specifically, the
authors show that expanding the beam after the electron source enabled a
higher-order correction of the longitudinal focus by a subsequent accelerating
cavity that is operated at the same frequency as the electron gun. Although
the method was implemented as part of a ballistic-compression scheme to
demonstrate the generation of sub-fs bunches at low energy, it could in
principle be extended to higher energy when combined with standard compression
beamlines.

Additionally, introducing high-order longitudinal dispersion, as discussed
previously in Section IV.2 can also be used to control LPS nonlinearity. Such
a method was successfully implemented in a high-power energy-recovery linac
[236] and supported the first demonstration of same-cell energy recovery of a
high-power electron beam with increased energy-spread after its interaction in
an FEL oscillator [211]; see Fig. 26. Higher-order correction using octupole
magnets was also implemented in the JLab 10-kW FEL facility [212].

Figure 26:  Nonlinear control of LPS with high-order longitudinal dispersion
in JLab IR-FEL Demo for two tuning of the longitudinal lattice along with
resulting energy spread measured after energy recovery [images (g) and (h)].
The simulated LPS are displayed at acceleration-module exit (a), before(b) and
after (c) the FEL iteration, and after energy recovery for the cases of linear
(e) and nonlinear (f) correction (via independent control of the $R_{56}$ and
$T_{566}$ of the recirculator) of the LPS. . The measured $(\delta E,y)$
distributions shown in images (g) and (h) correspond respectively to simulated
LPS presented in (e) and (f). The images are recorded on the final beam dump
window and the horizontal axis is representative of the energy spread after
deceleration in the acceleration module. Adapted from Ref. [236].

Finally, a laser-pulse shaping similar to the one discussed in Sections II and
IV.2 initially proposed in [48] can also be combined with a beamline with
nonlinear longitudinal dispersion to control the final LPS of a relativistic
bunch. An experimental demonstration of the concept was performed at the
FERMI@ELETTRA accelerator using a 1.4-GeV beam [230]. The photocathode laser
was temporally shaped to precompensate nonlinear correlations nominally
accumulated during the acceleration and two-stage bunch compression processes.
The measured LPS upstream of the FEL beamline is uncorrelated and linearized;
see Fig. 27. In the process, the authors demonstrated that the beam dynamics
of the ramped bunch did not significantly affect the beam transverse emittance
compared to their nominal operating point.

Figure 27:  Experimental demonstration of control of LPS correlation at at the
FERMI@ELETTRA accelerator using a 1.4-GeV beam. Electron-bunch LPS (image)
with superimposed current profile (white line, right axis). These data were
obtained using a shaped photocathode laser with ramped temporal profile
configuration. The 300-fs bunch core has a nearly constant incoherent energy
spread of about $\sigma_{E}\simeq 150$ keV. From [230].

###  IV.5 Transverse phase-space control

We now examine techniques to transversely shape the beam. This type of shaping
has commonly been implemented via transverse shaping of the photocathode-laser
pulse in a photoemission source or by proper shaping of the potential in a
thermionic-cathode-based electron source as detailed in Section II. This type
of shaping is ultimately impacted by collective effects so that devising
shaping methods implementable at high energy is beneficial. For instance,
producing uniform electron beams with uniform transverse distribution is
critical to a broad range of applications including beam irradiation of a
target, e.g., to produce X rays, as it provides a uniform dose and mitigates
thermo-mechanical stress on the target [228]. Likewise, using bright electron
beams with uniform transverse distribution offers an effective way of
improving the performance and output power of tapered X-ray FELs [74].
Finally, transversely shaped beams combined with the mask-based shaping
techniques discussed earlier can lower beam losses on the mask. These
transverse-shaping methods have gained considerable interest, as their
possible combination with the phase-space-exchanging techniques discussed in
Section VI could enable control of the current distribution with unprecedented
precision and versatility.

####  IV.5.1 Nonlinear transformations

The idea of using nonlinear transformations parallels the early discussion
related to LPS manipulation. Such an approach was first explored in [194] to
remove the beam tail via nonlinear focusing (implemented with sextupole
magnets) as part of a “nonlinear collimation” scheme proposed for future
linear colliders. Likewise, the use of nonlinear focusing to form uniform beam
distributions was discussed in [192]. Numerical simulations [137] demonstrated
the formation of uniform distribution using an octupole magnet implemented in
practical beamline. Likewise, [22] derives the nonlinear force required to
redistribute an incoming beam non-uniform distribution into a uniform
distribution using a beamline composed of a multipole lens followed by a drift
space. Further work discussed in [193] analytically shows that the odd-order
multipole fields, such octupole and dodecapole components, are required for
transverse uniformization. Efforts to produce transversely uniform
distributions over appreciable distances along an accelerator beamline has
been investigated for possible use in combination with undulator tapering in
X-ray FELs [133]. Over the years several experiments have been conducted. An
early demonstration experiment using a 200-MeV $H^{-}$ beam was performed at
BNL [299], and an example of distribution measured at the NASA Space Radiation
Laboratory (NSRL), using an electron beam at BNL [298], appears in Fig. 28.
Similarly, uniform beam distributions are sometime used in medical
accelerators.

Figure 28:  Measured beam distribution obtained downstream of a uniformizing
beamline composed of octupole magnets. Adapted from [298].

Reference [328] develops a theoretical framework to understand the formation
of shaped beam profiles using a nonlinear focusing beamline. Following this
work, the phase-space coordinates downstream of the beamline under
consideration can be expanded as a truncated power series of the initial
phase-space coordinate upstream of the beamline ($x_{0},x_{0}^{\prime}$) via a
transformation of the form

$\displaystyle\begin{pmatrix}x_{t}\\\
x^{\prime}_{t}\end{pmatrix}=\begin{pmatrix}(M_{11}-\frac{\alpha_{0}}{\beta_{0}}M_{12})x_{0}-M_{12}\sum_{n=3}^{\infty}\frac{K_{2n}}{(n-1)!}x_{0}^{n-1}\\\
(M_{21}-\frac{\alpha_{0}}{\beta_{0}}M_{22})x_{0}-M_{22}\sum_{n=3}^{\infty}\frac{K_{2n}}{(n-1)!}x_{0}^{n-1}\end{pmatrix}.$

Imposing Liouville’s theorem as discussed in Section II.3.1 indicates that the
final distribution along one of the position coordinates can be written as

$\displaystyle\rho_{t}=\frac{\rho_{0}}{M_{11}-\frac{\alpha_{0}}{\beta_{0}}M_{12}-\sum_{n=3}^{\infty}\frac{K_{2n}}{(n-2)!}x_{0}^{n-1}}.$
(209)

This equation provides the values of the magnetic multipole strength $K_{2n}$
required to transform the initial distribution into the desired one. The work
also extends previous methods by investigating the effect of the even-order
multipole fields on beam uniformization and demonstrating the feasibility of
uniformization of a Gaussian beam for even-order fields, or considering
incoming transversely asymmetric beams. The authors conclude that using a
combination of even-order multipoles may ultimately provide uniform beam over
a larger transverse section; see Fig. 29(b).

Figure 29:  Example of computed transverse profiles for a nonlinear-transport
beamline composed of odd-order fields (a,b) and two sextupole magnets (c,d).
Adapted from [328].

Finally, although the emphasis has often been on the generation of uniform
beams, the technique can also produce beams with transversely shaped
distribution. For instance, Fig. 29 demonstrates the generation of a ramped
horizontal distribution obtained as an intermediary step to generate a uniform
distribution. Furth development demonstrated the generation of patterned
hallow beams [327]. This latter capabilities could be taken advantage of and
combined with phase-space-exchange methods discussed in Section VI.

Figure 30:  Configuration for transverse beam shaping using a
“transverse”-wiggler magnet (left) and horizontal phase-space (right)
immediately downstream (upper plot) the wiggler and after a subsequent drift
(lower plot). Adapted from [108].

Another method toward shaping the transverse phase space is to use a
transversely periodic magnetic field to modulate the electron divergence
across the transverse beam distribution as proposed in [113] and illustrated
in Fig. 30. Taking the example of the horizontal phase space $(x,x^{\prime})$
passing through a set of transverse wigglers $-$ a wiggler oriented transverse
to the beam direction $-$ with wiggler parameter $K_{w,i}$, the divergence of
an electron at initial position $x_{0}$ will be given by

$\displaystyle x^{\prime}=x_{0}^{\prime}+\sum_{i=1}^{n}K_{w,i}(\sin k_{i}x),$
(210)

where $k_{i}\equiv 2\pi/\lambda_{i}$ is the transverse wavevector with
$\lambda_{i}$ being the $i^{th}$ wiggler period. A proper choice of $k_{i}$
and $K_{w,i}$ allows the synthesis of any correlation in the $(x,x^{\prime})$
phase space that could produce a tunable profile along the horizontal spatial
or divergence direction.

####  IV.5.2 Interceptive beam shaping: beyond binary masks

As discussed in Section IV.2, an interceptive “binary” mask with optimized
contour provides a versatile and simple tool to shape the beam transverse
distribution but is rarely used owing to the accompanying losses. These losses
are especially problematic when the shaping technique is implemented in a
high-current accelerator as they can lead to, e.g., radiological activation
and damage of beamline hardware.

For ultra-low emittance beams where quantum coherence is achieved and the beam
can be described by its wavefunction, optical (photon) techniques have been
adapted to shape the beam with a high degree of control, e.g., in electron
microscopes [208, 276]. An example of such a manipulation is the generation of
“vortex” electron beams carrying orbital angular momentum [301] using a nano-
engineered spiral-like phase plate made of stacked graphite thin film.

Figure 31:  Principle of diffraction modulation to produce a nano-patterned
beam (a) with diffraction pattern downstream of the grating (b) and final
transverse distribution of the transmitted beam (c) with associated spatial
spectrum along the vertical coordinate (d). Figure adapted from [102].

Although electron beams produced in conventional accelerators are incoherent,
ultralow-emittance relativistic electron beams can be manipulated with
variable-transmission masks, as discussed in [210], provided their transverse
coherent
$L_{\perp}=\frac{\hbar}{mc}\frac{\sigma_{\perp}}{\varepsilon_{\perp}}$ [224]
is comparable or larger than the intercepting-mask crystalline structure
(where $\sigma_{\perp}$ and $\varepsilon_{\perp}$ respectively refer to the
transverse beam size and emittance). In this reference, a diffraction contrast
modulation technique was proposed to realize nano-patterned electron beams.
The method relies on electron diffraction in a transmission grating with
periodically variable thickness. The grating structure spatially modulates the
fraction of electrons diffracted into a particular Bragg peak. An aperture
selecting the transmitted beam or one of the Bragg peaks then results in a
final modulated beam. The method was demonstrated at SLAC using a 2.3-MeV 1-pC
electron beam from an ultrafast electron-diffraction setup [102]. The beam was
diffracted through a thin lithographically patterned Si membrane [Fig. 31(a)]
to produce a vertically modulated beam distribution [Fig. 31(d)] with a
spatial period of $\sim 400$ nm and an associated bunching factor of 0.5 [see
spatial spectrum in Fig. 31(b)]. In this experiment, the unwanted electrons
were scattered into the (220) Bragg peak, and the transmitted electron
associated with the (000) peak was used to produce the final 300-fC modulated
beam [Fig. 31(c,d)]. This masking technique has some limited tunability
accomplished by varying the tilt angle of the grating and selecting the Bragg
peak to diffract the beam where desired. This method ultimately suffers from
the mask’s limited lifetime due to atomic displacement over long periods of
time. The achieved sub-micrometric spatial period combined with demagnifying
optics could produce beams with nanometric modulations.

##  V Longitudinal shaping with beam self-generated field

As described in the previous section, the rule of thumb in beam shaping is to
introduce controlled electromagnetic fields, and external fields are easy-to-
control sources for beam shaping. However, we can also use the beam’s self-
generated fields to shape the beam.

Self-generated fields are usually considered to be an obstacle to improving
the beam quality. While fully decoupled Gaussian beam or beams having linear
correlations are desirable, most self-generated fields introduce non-linear
correlations on the beam. The non-linearity on the beam’s phase space not only
increases the emittance, but it introduce intrinsic limitations on the
manipulation of the beam. Overcoming the degradation of the beam quality
caused by the self-fields is one of the major research topics in accelerator
physics. In the best case, self-fields might enable manipulation of the beam
and avoid its degradation simultaneously.

This section focuses on shaping mechanisms that are based on beam-generated
fields. Most of the methods introduced in this section are for longitudinal
shaping because the use of self-generated fields for transverse shaping is
scarce. We first discuss longitudinal profile shaping using the space-charge
field and CSR in Sec. V.1 and Sec. V.2. Then we describe the mechanism using
wakefields in Sec. V.3. Each subsection provides a short description of the
results and their applications.

###  V.1 Shaping profiles using space-charge field

The space-charge field has two characteristics. The first is that it depends
on the beam’s spatial distribution. If the beam has a symmetric profile, the
field strength is symmetric to the beam center. The second is that it acts in
a way to defocus or lengthen the bunch when the beam is not density modulated
because electrons repel each other. These two characteristics give us several
methods for using the space-charge field for shaping. The following
subsections describe these shaping methods and give relevant results.

####  V.1.1 Space-charge field with a single bunch

Due to the field’s dependence on the beam’s spatial distribution, the
usefulness of the space-charge field from a single bunch is limited. However,
we can imagine two different uses of the space-charge field from a single
bunch.

First, the space-charge force applied to the bunch is not linear except for a
few profiles. Thus, similar to the transverse shaping using an octupole magnet
[328] (see Sec. IV.5.1 for further details), a symmetric-nonlinear space-
charge field can induce a symmetric change in the profile. For example, in a
cylindrical beam with a uniform transverse distribution and a longitudinal
distribution of $\lambda(z)$, the space-charge field can be expressed as in
Eq. (149). If the initial longitudinal distribution is Gaussian, then the
axial space-charge field will be

$E_{z}(z)={e\over
4\pi\epsilon_{0}\gamma^{2}}{N_{b}z\over\sigma_{z}^{3}}\left(1+2\ln{b\over
a}\right)\exp\left[-{z^{2}\over 2\sigma_{z}^{2}}\right].$ (211)

The Gaussian profile and corresponding space-charge field in Eq. (211) are
visualized in Fig. 32(a). In the case of the example shown in Fig. 32, all
particles move outward due to the space-charge field as the beam drifts, and
the bunch length will increase. The space-charge field in this example is
nonlinear and has its maximum strength at around $\pm 0.5$ mm. Thus, particles
initially located at $\pm$0.5 mm gain more momentum than particles in
$|z|>0.5$ mm, and eventually overtake these particles in $|z|>0.5$ mm. This
situation is very similar to octupole-based transverse shaping, see Fig. 28.
Thus, if the beam traverses a long enough distance (i.e., large enough
$R_{56}$), then this Gaussian distribution would evolve into a uniform
distribution. Figure LABEL:sec5_fig_scshaping.eps(b) shows a particle tracking
simulation result, which demonstrates the idea. A Gaussian profile with a
charge of 1 nC and energy of 3 MeV evolves into a uniform profile due only to
its space-charge field as it drifts for 3 m. As we can see from Eq. (211), the
space-charge field gets weaker as the beam energy increases. Thus, the change
arising from the space-charge field requires a large R56 to convert momentum
change to position change. This requirement limits the use of the space-charge
field to near the electron gun; thus, so far this type of shaping is not
demonstrated experimentally.

Figure 32: Current profile shaping using space-charge force. (a) Gaussian
longitudinal profile (blue) and corresponding longitudinal space-charge field
from Eq. (211). (b) Longitudinal profile of the initial particle distribution
(blue), and the longitudinal profile after a 3-m-long drift (red). (b) is a
simulation result of a beam with 1-nC charge and 3-MeV energy.

We can consider a similar type of shaping for generating asymmetric bunches,
but it is necessary to introduce an originally asymmetric profile. Thus, using
the space-charge field for asymmetric control is not an attractive option, but
the understanding of this mechanism may help to improve the shaping quality of
emission-based shaping methods introduced in Section III.

The other usage is exploiting the symmetric lengthening feature of the space-
charge field. In [178] they suggested a new concept, the so-called blow-out
regime, to generate a 3D ellipsoidal distribution with uniform density. In
this concept, an ultra-short laser pulse with an appropriate radial
distribution ($\lambda_{\perp}(r)\sim\sqrt{1-(r/A)^{2}}$ where A is the
maximum radius) shines on a photocathode. This generates an ultra-thin sheet
of electrons that evolves into a uniform ellipsoidal shape. Here, the ultra-
short pulse length plays a key role because it allows that all particles in
the bunch to experience a similar space-charge field strength. This allows the
beam to evolve into a uniform ellipsoid regardless of the original pulse
shape. The imperfection here will appear as a soft edge of the ellipsoid. In
[178] they ran simulations using a 100-pC beam with 1-mm radius and 0.4-eV
energy at the cathode surface. The electric field at the cathode surface is
assumed to be 100 MV/m, and the incident laser pulse is assumed to have a
Gaussian profile with 30-fs FWHM. Figure 33 shows a simulated particle
distribution 50 ps after the laser illuminates the cathode. The original disk-
shaped distribution has evolved into a nearly elliptical one that is slightly
less than ideal (black curve) due to the imperfection.

Figure 33: Blow-out regime concept and its demonstration using simulation. The
figure shows simulated particle distribution in x-z. The image is taken 50 ps
after the laser illuminates the photocathode. The black circle represents the
theoretically ideal result. From [178].

In [256], further theoretical work was performed to combine the blow-out
regime and emittance compensation. Later, this blow-out regime was
experimentally demonstrated by [206, 223, 240]. The first experiment in [206]
demonstrated the blow-out regime using 15-pC beam; see Fig. 34. Later, the
experiment in [240] demonstrated the blow-out using a higher charge level
(0.5-nC) with a Cs2Te cathode, and it showed well-shaped bunches. However, the
method required a disk-shaped beam, making it difficult to reduce the laser
spot size at the cathode to obtain target charge. This limits the intrinsic
emittance, which scales with laser spot size, Eq. 181. The emittance
demonstrated so far does not show a clear advantage of this method. There is
another emission scheme, based on a cigar-beam, introduced in [255] that
achieved lower emittance. While the blow-out regime uses a short but large
radius laser profile, which produces a longitudinal expansion of the beam, the
cigar-beam regime uses a long but small radius laser profile which produces
radial expansion of the beam. Simulations showed a noticeable improvement in
beam brightness of the cigar compared to the blow-out regime. Due to these
limitations, research efforts were moved to using ellipsoidal laser profiles
instead of the blow-out regime to generate an ellipsoidal beam; see [140]. See
Section III for further details of laser shaping.

Figure 34: Experimental demonstration of ellipsoidal beam generation using
blow-out method. The beam images were measured after a deflecting cavity. The
beam was focused vertically, and the deflecting cavity kicked the beam
vertically. x and y axes correspond to the beam’s horizontal and longitudinal
distributions, respectively. The beam’s energy was 3.75 MeV and the charge was
15 pC. From [206].

####  V.1.2 Space-charge field with a few bunches

When more than a single bunch exists, the bunches mutually repel each other
via the space-charge field. If there are extra bunches in front of and behind
the target bunch, the electromagnetic field felt by the target bunch will
depend on these extra bunches and the target bunch’s self-field. Here, the
charge ratio of each bunch and the spacing between the bunches can be used as
knobs to control the space-charge field applied to the target bunch.

In [176], this concept was used to compress an electron bunch. During the
experiment, they generated a low-charge main bunch and two extra bunches for
space-charge field shaping. The bunches were generated by a photocathode gun
and using a laser beam splitter and $\alpha$-BBO crystals to generate a laser
pulse train. The main bunch was located between two field-shaping bunches.
These two field-shaping bunches push the target bunch, which generates a
negative longitudinal chirp of the target bunch’s longitudinal phase space
(LPS). Note that the negative chirp means the head of the bunch has lower
energy. Because the space-charge field outside of the bunch shows the usual
1/$r^{2}$ tendency, they located the target bunch at the middle so that the
nonlinearity of the space-charge field coming from the field-shaping bunches
almost cancels out. This provided a nearly linear longitudinal chirp to the
target bunch; the relative separation was adjusted by using a laser delay
line. The laser was injected into the RF photocathode gun that accelerated the
beam to about 3.4 MeV. Due to the beam’s low energy, a few meters of drift
provided enough $R_{56}$ for ballistic bunching. Ballistic bunching means that
the bunch compression is accomplished by the velocity difference of the
particles in a drift. Velocity bunching is a more general term, but the
ballistic bunching is used when acceleration is not included in the
compression.

During the experiment, they fixed the target bunch’s charge level to 50 fC
while varying the field-shaping bunches’ charges from 0.2 to 6.7 pC. Here, the
charges of the field-shaping bunches were used to control the field strength
applied to the target bunch. They observed bunch compression with an LPS
measurement as shown in Fig. 35. Also, the main bunch length was controlled
from 220 fs to 109 fs by varying the charge. As we can see from Fig. 35(d),
there is still room for further compression.

Another interesting point of this work is stability. In the case of chicane
compression, the RF jitter of the cavity controlling the chirp affects the
longitudinal chirp or the beam energy. This RF jitter, in turn, result in an
increase of the bunch length jitter and arrival time jitter. On the other
hand, all three bunches originated from a single laser pulse, and there are no
RF cavities other than the gun in this scheme. Thus, it can provide better
stability than other methods. This is an interesting method for applications
requiring a short bunch with good stability such as ultra-fast electron
diffraction (UED), ultra-fast electron microscopy (UEM), or FELs.

Figure 35: Space-charge field shaping for bunch compression. Measured
longitudinal phase spaces are displayed. Each panel corresponds to different
charge level of field-shaping bunches. (a-d) correspond to charges of 0.2,
3.2, 4.5, and 6.7 pC, respectively. $t>0$ corresponds to the head. From [176].

A similar method was also experimentally demonstrated for transverse phase
space. In [253], they introduced extra bunches surrounding a central main
bunch in the transverse space; see Sec. III.4.2. Due to the interaction via
the space-charge force and the beam’s distribution change by solenoid
focusing, these extra bunches either introduced extra focusing to the main
beam or changed the beam’s distribution to a shape that was other than a
Gaussian distribution. This could be an interesting option for a system having
difficulty using magnets. For example, locating a solenoid near the cathode of
superconducting guns is difficult, but transverse control is necessary to
preserve the beam’s quality. These extra bunches may provide all-beam-focusing
to the target beam, so it may be able to locate a solenoid further downstream
while preserving the beam quality.

Although the experimental demonstrations discussed in this section are limited
to one-dimensional density control, we can imagine appling this method to more
general bunch shaping applications. For example, nonlinearity is a key for
asymmetric bunch shaping, as we saw in Section IV. We can imagine exploiting
nonlinearities that space-charge force introduces. Bunch charges and spacing
can be used to control the shape of LPS. Combining this method with laser
shaping, may be an opportunity to overcome the high-charge limitation of the
laser shaping method. Laser shaping methods are limited to low-charge bunches
because of the strong space-charge force of the main bunch. However, the
space-charge field of the main bunch can be controlled by using additional
bunches which may provide a new way to apply a laser-shaping technique to
high-charge bunches.

We should note that beam shaping with extra bunches always has the problem of
eliminating the extra bunches after they are no longer needed for shaping. The
elimination process may introduce additional disadvantages that make the
method less beneficial than described so far. A beam cutting process may
happen using a mask and an additional beamline element such as chicane or RF
deflecting cavities; [114]. However, these beamlines can easily add extra
timing jitter to the beam, negating one of the benefits of this method. Thus,
it is necessary to consider this elimination process during the design.

####  V.1.3 Space-charge field with multiple bunches: space-charge oscillation

We now consider more than three bunches by providing a density modulation at
the beginning. This initial density modulation introduces oscillatory behavior
in the longitudinal phase space, which is similar to the plasma oscillation
that ions and electrons form. In the electron bunch case, the initial density
modulation and bunch-to-bunch interaction via the space-charge force
introduces the same oscillation.

This behavior is shown in the left panel of Fig. 36. The initial density
modulation (black) lengthens each micro-bunch, due to the space-charge force,
so that the modulation becomes weaker as the beam drifts until it finally
disappears at a certain distance (red). Once the beam passes this point, the
particles in each micro-bunch continue to move in the same direction, due to
their momentum. These particles build another density modulation whose phase
is 180∘ off from the original modulation, thus the direction of the space-
charge force now is reversed. At a certain point (blue), the momentum
direction changes, due to the reversed space-charge force, and this density
modulation starts to oscillate. The oscillation frequency of this oscillation
is equal to the plasma oscillation’s frequency in Eq. (159). The change of the
relative energy spread induced by the longitudinal space-charge force can be
written as

$\Delta\delta_{LSC}=4{\gamma^{3}\omega_{p}b\over ck},$ (212)

where k is wave number ($={2\pi\over\lambda}$), and $b$ is bunching factor
$=|\int\lambda(z,0)e^{ikz}dz|$; see [205].

Figure 36: Nonlinear plasma oscillation of the beam to generate a bunch train
with a high bunching factor. Beam’s longitudinal density profiles and their
evolution are displayed. The left panel started with a bunching factor of
0.01, and the right panel started with a bunching factor of 0.24. These are
numerical solutions of the system of equations describing a coasting beam’s
density profile evolution. From [204].

In the linear oscillation situation described above, a sine-like shape on the
phase space does not break and keeps oscillating because the energy modulation
induced by the initial density modulation is small so that all particles
cannot arrive at the density peak area before the momentum is reversed (i.e.,
$R_{56}\delta\ll\lambda/4$). However, when the modulation becomes big enough,
the space-charge force can induce wave-breaking where the momentum modulation
is large enough to cause all particles to reach the density peak area (i.e.,
$R_{56}\delta\approx\lambda/4$).

It is also possible to understand this process as the summation of harmonics.
In the case of linear oscillation, only the fundamental mode governs the
oscillation. However, as the initial density modulation gets larger, the
oscillation process picks up more harmonic components of the fundamental
wavelength. After half of the plasma period, these harmonic components sum in-
phase and generate current spikes. This viewpoint can be seen from the
equation below showing the electron density. More details are described in
[205].

$\lambda(z,t)=\left[1+\cos\left(\omega_{p}t\right)\sum_{m=1}^{\infty}mc_{m}(t)\cos\left(mkz\right)\right],$
(213)

where $c_{m}(t)={(-1)^{m+1}\over
m}{2\over\alpha(t)}J_{m}\left[m\alpha(t)b\right]$, and
$\alpha(t)=2\sin^{2}(\omega_{p}t/2)$.

Although this concept can be applied to any coasting beam, it is impractical
for a high-energy beam because of its long plasma wavelength. Thus, it was
applied to an electron gun by several groups [216, 118, 215]. Here we describe
more details of the experiment done by [204], which successfully demonstrated
the generation of current spikes.

They used three $\alpha$-BBO crystals to generate eight equally spaced laser
pulses to form a 1-THz ($\lambda=300$ $\upmu$m) pulse train; see Sec. III.4.3.
They tested the bunch charge up to 40 pC, and the gun solenoid was used to
focus the beam. To demonstrate the method, they varied the laser intensity,
which controls the charge, to change the phase advance of the oscillation. The
phase advance determines the final modulation density. The modulation was
measured with a deflecting cavity at the end of an experimental beamline. The
solenoid field was fixed, and the gun phase was set to 35∘ to preserve the
original modulation period. The result is summarized in Fig. 37.

According to their simulation, the phase advance of the lowest charge case
(1.6 pC) was about 0.18$\pi$; see Fig. 37. Thus, it still showed the original
modulation and all eight peaks appeared at the end of the beamline. When the
charge was increased to 3.9 pC, the modulation was washed out; this charge
level corresponds to the phase advance of 0.25$\pi$ in their simulation. As
the charge increased further, the phase advance became slightly lower or
higher than 0.5$\pi$, so a density modulation appeared again, but now there
were only seven peaks because of the $\pi$ phase shift of density peaks in a
given total bunch length.

Figure 37: Experimental demonstration of bunch train generation using
nonlinear plasma osciallation. (a-d) shows temporally streaked beam images,
and corresponding longitudinal profiles are shown in (e). Streaking direction
for beam images is vertical. From [204].

A few years later, this method was revisited and experimented at a higher
charge level to consider an application to terahertz (THz) radiation. In [330]
they set up an experimental beamline similar to [204] and added a chicane to
broaden the frequency tuning range. They used a charge up to 1 nC, and results
were very similar to the ones from [204]. They controlled both the charge
level and the solenoid focusing strength to control the phase advance. The
initial modulation was washed out with a low phase advance while the
modulation was shifted by 180∘ when the phase advance was 0.25$\pi$.

In addition to confirming the oscillation at a higher charge level, they also
tried to generate THz coherent transition radiation (CTR) using a foil. They
adjusted two tuning parameters to control the modulation frequency and
measured the spectrum of CTR using a Michelson interferometer equipped with a
Golay cell [99]. The first parameter was the launching phase of the gun.
Depending on the launching phase, the bunch had different longitudinal chirps.
This eventually introduced different compression ratios for ballistic bunching
in the low energy area. They varied the phase from 25∘ to 50∘, and the
frequency of CTR was varied from 1 THz to 0.7 THz, correspondingly. The second
parameter was the chicane’s bending angle, which controlled $R_{56}$. Compared
to the launching phase, this second parameter provided a much wider tuning
range. They changed the current to the dipole magnet from 0 A to 30 A, which
corresponded to $R_{56}$ values from 0 to 18 mm, and it changed the frequency
from 0.8 THz to 1.6 THz.

Because of the high charge level, they achieved 2 $\upmu$J of THz energy from
CTR. They expected that a 30-mm-long quartz tube with 0.3- and 0.4-mm inner
and outer diameters, respectively, would provide 8-MW, 1.4-mJ THz radiation at
0.7-THz frequency. This intense and tunable THz radiation from a compact
beamline can be useful for a typical spectroscopy-type equipment, pump-probe
measurements in XFELs, or THz wakefield accelerators.

####  V.1.4 Space-charge field with multiple bunches: longitudinal cascade
amplifier

When a momentum modulation from an initial density modulation is strong
enough, the initial density modulation can be amplified with an appropriate
$R_{56}$. This phenomenon was studied extensively in the early 2000s to
understand the significant beam quality degradation and radiation generated
from the beam. This is called the microbunching instability and many theories
and experimental papers were published to explain and suppress this
phenomenon; for example, [264, 261, 262, 119, 123, 318, 179, 122, 186, 284,
250, 244]. One of them, [250], showed a picture of the longitudinal phase
space with various conditions, and Fig. 38 clearly shows the modulation
amplified by the gain process; see Sec.II.4.5.

Figure 38: Measurement of microbunching instability. The left panel shows the
measured longitudinal phase space without a laser heater. The strong
modulation originated from the shot noise and its amplification. The right
panel shows the corresponding current profile. Adapted From [250].

As described in Sec. II.4.5, [261] derived a simple equation to estimate the
gain ($G$) of the density modulation, which is the ratio of final to initial
modulation amplitude. Longitudinal space-charge fields introduced energy
modulation to the beam, as appeared in Eq. (169) as an impedance term. Then,
$R_{56}$ of the given beamline converted an energy modulation to a density
modulation. For the microbunching instability, the longitudinal space-charge
force from the shot noise makes the gain higher than 1 and amplified this
unwanted shot noise. Thus, an effort, such as using a laser heater, was made
to reduce the gain [see the $\sigma_{\delta}$ term in the exponent of Eq.
(169)].

On the other hand, [267] introduced a concept to use this amplification of
shot noise as a way for generating a high-frequency bunch train; it was called
the longitudinal space-charge cascade amplifier (LSCA). In this concept, they
used a drift with a focusing channel to introduce a momentum modulation from
the density modulation, then a chicane is used to introduce an $R_{56}$
optimized for the modulation conversion with maximum gain (see the top panel
of Fig. 39). This focusing channel and chicane form a single cell of the
amplifier, and each cell provides gain to the density modulation. Thus, they
used a cascade of cells to saturate the density modulation. The numerical
example they provided shows that the gain per cell reaches more than 40 for a
beam with an energy of 3 GeV, a peak current of 2 kA, a normalized emittance
of 2 $\upmu$m, and an energy spread of 0.3 MeV, which is close to typical FEL
parameters. Thus, two cascades would provide a $G>1,000$. This is high enough
to saturate the shot noise for this example case. Here the amplitude of shot
noise ($A_{sh}\simeq\sqrt{ec\over I\lambda}$) is around 0.001 for $\lambda=15$
nm getting optimal gain.

The LSCA method was experimentally demonstrated in 2013 by [185]. The
experiment amplified shot noise using three chicanes with drifts between them.
The amplified modulation was used to generate radiation by using an undulator,
with an undulator parameter of 0.58, a period of 1.9 cm, and a total of 11
periods. Each chicane provided $R_{56}$ of 4 mm, 2.5 mm, and 1.5 mm, in order,
and the chicanes were 2 m apart. The initial momentum modulation was
accumulated for a 10-m-long drift before the first chicane, and the beam
energy was 72 MeV. The spectrum of undulator radiation was measured. They used
a photodiode detector to measure the integrated intensity gain. With a 12-pC
bunch, the average gain in the integrated intensity over the incoherent
background was 600. When they measured the gain of on-axis radiation and only
considered the bunch charge contributing to the coherent radiation, the gain
went up to 1.5$\times 10^{4}$, which was in good agreement with their
estimated gain of 2.5$\times 10^{4}$ from the linear theory.

Figure 39:  Measurement of radiation spectrum from a bunch train generated by
a longitudinal cascade amplifier. The top panel shows a configuration of a
longitudinal cascade amplifier with two cells. The bottom panel shows a
measured spectrum of undulator radiation with the cascade amplifier. The blue
corresponds to the on-axis radiation spectrum while the red shows the
integrated one. The bottom figure is adapted from [185].

This method is considered for FEL facilities to generate high-power coherent
radiation, large radiation bandwidth, or attosecond pulses as described in
[267] and [60]. On the other hand, another use of this cascade method was
proposed by [249] wherein they suggested using the cascade method to achieve a
high cooling rate for coherent electron cooling. Here, the modulation
imprinted by an hadron beam onto the electron beam is amplified with the
cascade amplifier, and the field from the amplified modulation of electron
beam is then used to cool the hadron beam.

####  V.1.5 Space-charge field with multiple bunches: plasma cascade amplifier

While the previous LSCA did not consider transverse behavior much, [171]
introduced a new method called the plasma-cascade amplifier (PCA), which
generates modulation at the plasma frequency using solenoid focusing. As
previously mentioned, $G>1$ occurs when the space-charge induced momentum is
strong enough, which is converted to the density by an appropriate $R_{56}$.
It is hard to expect a high gain in a drift because the $R_{56}$ of a drift
($L_{d}/\gamma_{0}^{2}$) is much lower than the one from a chicane
($2L_{dogleg}\theta^{2}$, $\theta\ll 1$). However, the final density
modulation is governed by how many particles gather together at one spot, so
$R_{56}\delta$ is the critical term. Thus, even though a drift provides a low
$R_{56}$, providing a stronger momentum modulation from the space-charge force
can make a similar amplification. One of the clear ways to achieve it is to
increase the density by focusing the beam, as we can infer from Eq. (212).

The plasma cascaded instability can also be understood as a parametric
resonance of harmonic oscillators as described by [171]. In the harmonic
oscillator

$\ddot{x}+\omega_{0}^{2}x=0,$ (214)

the oscillation can increase exponentially when the oscillation frequency is
modulated at a certain frequency (i.e.,
$\omega_{0}(t)=\omega_{0}\left[1+A\cos(\omega t)\right]$). This resonance
happens when $\omega\simeq 2\omega_{0}$. In the case of the PCA, $\omega_{0}$
corresponds to the plasma oscillation frequency. Thus, if the plasma
oscillation frequency can be modulated, then a parametric resonance will occur
in the beam’s longitudinal density modulation. Here, a series of focusing
solenoids are used to modulate the plasma oscillation frequency, as shown in
Fig. 40. Due to the required relationship between the plasma oscillation
frequency and its modulation frequency, it is clear that system parameters
such as the beam envelope, solenoid-to-solenoid distance, and charge level
should be carefully designed. Thorough theoretical work has been done by
[171], and further work is on-going [see [28]].

The method was experimentally demonstrated by [173, 231]. A long bunch (400
ps) with low energy (1.76 MeV) was used for the experiment, and the charge was
varied. They used 5 cells to amplify the shot noise, and the solenoids were
not placed periodically, which works equally well as a periodic setup. From
the simulation, they expected a gain of 400-500 for 0.4 THz, and a gain around
200 for 0.6 THz. Figure 40 shows measured longitudinal beam profiles. The
density modulation is very clear, and it shows a charge dependency, which is
expected because $G>1$ occurs with a strong enough momentum modulation from
the space-charge force. The spectrum of these modulated bunches is broadband
with a peak around 0.4 THz as [173, 231] expected from their simulation. See
[172] for further details.

As LSCA had applications to radiation and coherent electron cooling, the PCA
can be applied for the same purposes. While the LSCA requires a dispersive
beamline such as a chicane, the PCA does not introduce any dispersion to the
beam. This may provide a relative benefit in terms of beam quality
preservation due to the lack of CSR and timing issues for the electron
cooling.

Figure 40: Experimental demonstration of plasma cascade amplification. The
conceptual figure on top shows the configuration of a PCA with corresponding
plasma wavelength modulation (top), and the measured longitudinal density
distribution (bottom). Adapted from [231].

###  V.2 Shaping profiles using coherent synchrotron radiation

CSR is another beam-generated field that changes a longitudinal beam momentum
distribution along time. Similar to the space-charge field, a CSR field is
determined by the bunch’s longitudinal profile, and the field strength is the
only controllable knob, as we can see from Eq. (155). CSR cannot be generated
without a dispersive element such as a dipole magnet. This means that the
transverse and longitudinal phase space will be correlated along the path, and
it makes the analysis and control more difficult than the space-charge field
of a single bunch. Thus, using CSR directly for shaping is challenging, and no
direct use of it has been proposed so far.

However, CSR is indeed a field that can change the longitudinal profile. It is
normally considered an obstacle for longitudinal beam manipulation. There have
been several research efforts that tried to preserve the shape through a
beamline with CSR. In the rest of this section, we describe efforts to
suppress CSR’s impact on the profile. Studying these methods will tell us how
to use CSR as a shaping tool.

[198] showed theoretically that a certain beam shape can flatten the CSR wake,
and it can minimize the CSR’s impact on transverse emittance growth through
the chicane. The beam profile is given by

$\lambda(z)={4\over 3}{(z-a)^{1/3}\over(b-a)^{4/3}},$ (215)

where $a$ and $b$ are the limits of the longitudinal bunch domain ($a\leq
z\leq b$). This shape is given on the left of Fig. 41, and the corresponding
CSR field is on the right.

Figure 41: Theoretical estimation of aberrations of ideal current profile
flattening CSR wake and aberration’s impact on the CSR wake. Current profile
with aberrations are displayed in the left panel. The right panel shows
corresponding CSR wakes. The dashed curve in the left panel shows the ideal
profile from Eq. (215). Solid curves show profiles aberrated by the chicane.
Each curve corresponds to a different $\mathcal{D}$ in Eq. (216);
$\mathcal{D}/L=$ 0.02, 0.04, 0.06, 0.08, and 0.10, where L is the initial
bunch length. $L_{f}$ ($=L/C$) is the final bunch length after compression.
Adapted from [198].

Here, preserving the initial shape is critical to minimize CSR’s impact on the
emittance, so [198] derived an equation for the longitudinal profile starting
from a decoupled 6D distribution. The profile at distance $s$ can be written
as

$\lambda(z;s)={C\over\sqrt{2\pi}\mathcal{D}}\int_{-\infty}^{\infty}\lambda(\zeta;0)\exp\left(-{(\zeta-
zC)^{2}\over 2\mathcal{D}^{2}}\right)d\zeta,$ (216)

where $C$ is the compression factor, $\mathcal{D}$ is given by

$\mathcal{D}^{2}(s)=C^{2}\left(R_{56}^{2}(s)\sigma_{\delta}^{2}+\mathcal{H}(s)\right),$
(217)

and $\mathcal{H}$ is given by

$\mathcal{H}(s)={\left[\sigma_{x}^{2}R_{51}(s)-\sigma_{xx^{\prime}}R_{52}(s)\right]^{2}+\varepsilon_{x}^{2}R_{52}^{2}(s)\over\sigma_{x}^{2}}.$
(218)

The change of the profile depends on $C$ and $\mathcal{D}$. When $\mathcal{D}$
is close to zero, $\lambda(z;s)=C\lambda(zC;0)$, which means the final profile
has the same shape as the initial profile but is compressed. Thus, the profile
would not change significantly when $\mathcal{D}$ is small. Figure 41, which
shows the longitudinal profile for different values of $\mathcal{D}$ and their
corresponding CSR wakes, proves that CSR has a small impact on the profile
when $\mathcal{D}$ is small. This tells us that a small $R_{56}$, initially
low energy spread, and a small transverse emittance are preferred. Also, beam
matching into the beamline is another important factor.

[106] tracked CSR effects on the beam in an EEX beamline to find a way to
preserve the shaping quality of EEX-based shaping (see Sec. II.3.3 and Sec.
VI.3.1). The longitudinal shift due to CSR can be written as

$\Delta
z_{f}=\xi_{2}\Delta\delta_{2}^{\eta=const}+\xi_{2}^{*}\Delta\delta_{2}^{\eta\neq
const}+\xi_{2}(1+\kappa\eta^{*})\Delta\delta_{1}^{\eta\neq const},$ (219)

where the $\Delta\delta$ terms are integrated momentum changes made by CSR
from each section of the EEX beamline, $\kappa$ is the deflecting cavity’s
kick strength, $\eta$ is the dispersion of a dogleg, and $\xi_{2}$ is the
$R_{56}$ of the second dogleg. $\eta^{*}$ and $\xi_{2}^{*}$ are effective
dispersion and $R_{56}$ terms to account for their effects when dispersion is
not a constant. This simple equation shows that $R_{56}$ is the carrier of
CSR-induced momentum change. Thus, reducing $R_{56}$ is clearly a good way to
minimize CSR’s impact on the profile.

Figure 42: Correction of CSR-induced aberration in start-to-end simulation.
The x-axis is the beam’s longitudinal position after a EEX beamline, and the
y-axis is the normalized longitudinal density. The colors represent charges of
1.1, 3.3, 5.5, 7.7, and 11 nC with blue corresponding to a low charge and red
corresponding to a high charge. The left panel was simulated with a 20∘ bend
for the dogleg, and the right panel used a 12∘ bend. Adapted from [106].

Figure 42 shows current profiles at the exit of an EEX beamline for several
different charge levels. The left figure corresponds to an EEX beamline with a
20∘ bend, and the right figure corresponds to a bending angle of 12∘. The
$R_{56}$ of the doglegs were 0.29 m and 0.18 m, respectively. The profile of
11 nC case (red) has several clear differences from the 1.1 nC case (blue)
when $R_{56}$ is high. For example, the tail is lengthened, the peak is
rounded, the linear ramp is changed to a concave curve, and the bunch length
is significantly lengthened. On the other hand, the low $R_{56}$ case
preserved all features (tail, peak, ramp, and bunch length) even for 11 nC.

[294] performed an optimization by using a reverse tracking simulation to find
a beamline setting and an initial beam condition for providing the desired
final profile. By optimizing the entire beamline, nonlinearities introduced by
each part of the beamline could cancel each other out. The beamline they used
in the simulation consisted of an SRF gun, 650-MHz linac cavities, a 3.9-GHz
linac cavity, and two chicane compressors. Optimization variables included
accelerating gradient, phase, frequencies of 650-MHz linac cavities, $R_{56}$
and $T_{566}$ of each chicane, and parameters defining the initial profile.

Even though the charge of the beam they used was 10 nC, they found that a
reasonable beamline setting provided the desired final longitudinal profile.
Even this setting included quite a high $R_{56}$ for both chicanes, 0.129 m
and 0.131 m, which can introduce a strong CSR effect on the profile. However,
the work was done with a simulation code with several simplified physics
(e.g., beam propagation using $R_{56}$ and $T_{566}$, steady-state CSR only,
etc.). Thus, more work is required to reach a definite conclusion. However, it
is promising that control of nonlinearities of the beamline may compensate a
strong CSR’s impact on the final profile.

From the study of this research on CSR suppression, we can infer how to make
CSR’s impact on the final profile stronger. Strong CSR, due to high charge or
short bunch, coupled with high $R_{56}$ may provide the CSR dominant control
over the final profile. This would induce a profile change along the beamline;
thus the final profile could be made to have completely different features
than the initial one. However, the final profile that the CSR can generate
would be limited because the CSR field is determined by the beam’s profile
along the beamline. This means that we need to either shape the initial
profile or create other knobs to control the beam’s longitudinal profile
inside the beamline. $R_{51}$, $R_{52}$, and transverse phase space may be
good candidates for the additional knob, and we may be able to use
nonlinearities from other parts of the beamline to manipulate the final
profile further.

###  V.3 Shaping profiles using wakefields

The longitudinal wakefield inside a given structure can be expressed as the
convolution of current distribution and wake function, which is the wakefield
from a single particle; see Eq. (135). Here, the longitudinal wake function
can be written as the summation of each mode excited in the structure:

$w_{z}(z)=\sum_{i}2\mathcal{K}_{i}cos(k_{i}z),$ (220)

where $k_{i}$ is the wavenumber of the $i$th mode, and $\mathcal{K}_{i}$ is
the loss factor of $i$th mode which is defined as the energy lost by the
particle exciting the mode per unit charge squared. While the space-charge
field and CSR did not provide an additional knob other than its strength, in
the case of the wakefield, the material or geometry of the vacuum pipe
determines the wake function (i.e., loss factor and fundamental frequency).
Thus, manipulation using the wakefield provides higher degrees of freedom than
other beam-generated fields.

In this section, we discuss the use of the wakefield in shaping the
longitudinal density profile. Similar to methods described in Section IV,
wakefields control time-energy correlation and the subsequent beamline (e.g.
drift and anisochronous beamline) make the correlation that determines the
longitudinal density distribution. Thus, we can easily imagine applications
such as bunch compression, bunch train generation, and single-bunch shaping.
These applications will follow the same principle as the ones in Section IV,
but the wakefield-based techniques could be more compact and cost effective
than external devices.

In addition to profile shaping, the degrees of freedom that wakefields provide
enable further control over the longitudinal phase space. Therefore, we also
discuss manipulations of the energy distribution and longitudinal phase space
using wakefields.

####  V.3.1 Bunch compression

A wakefield can work with two compression mechanisms; ballistic bunching and
magnetic bunching. Both mechanisms require control over the longitudinal
chirp, which can be provided by the wakefield. The chirp can be produced in
two different ways. If there is a wakefield-driving bunch, the target bunch
can be placed at the zero-crossing phase of the wakefield, to provide either a
negative or a positive chirp. If there is no driving bunch, then the self-
wakefield inside the target bunch always makes the tail lose more energy than
the head of the bunch (with an appropriate choice of frequency). Thus, the
target bunch always has a positive chirp. The positive chirp can work with
magnetic bunching, but it is not appropriate for ballistic bunching because a
drift has $R_{56}>0$.

Ballistic bunching using a wakefield was experimentally demonstrated by [331].
They split a Ti:sapphire laser at 800 nm into three pulses. Two pulses were
used to generate the drive and target bunch while the third pulse was used for
diagnostics purposes by converting it to THz radiation through optical
rectification. A 5-cm-long quartz tube with an inner diameter of 400 $\upmu$m
was used to generate a wakefield, and a drive bunch charge ranging from 0.6 to
1.3 pC was used to generate different longitudinal chirps. Thus, for a given
$R_{56}\approx 3.07$ cm, they generated different compression ratios. The
results are displayed in Fig. 43. An uncompressed bunch length of 150 fs rms
was compressed to 2.8 fs rms in the case of (c).

Figure 43: Experimental demonstration of wakefield-based bunch compression
using THz structure with drive bunch. The figures show measured longitudinal
phase space of the target bunch and corresponding projections. Each figures
corresponds to different drive bunch charges. The charges were (a) 0, (b) 0.6,
(c) 0.9, and (d) 1.3 pC. Adapted from [331].

While a similar compression can be easily accomplished with conventional RF
cavities, the wakefield-based ballistic bunching offers a few advantages.
First, this compression requires a driving bunch that can easily be generated
by splitting UV laser pulses. Therefore, it does not require extra cavities
for chirping and corresponding RF power sources. Second, RF sources always
introduce jitter on the beam energy. Because the wakefield fully relies on the
given structure and the beam, it can provide better energy stability. [331]
measured energy stability of 2.4$\times 10^{-4}$ rms for wakefield-based
bunching, while it was 1.5$\times 10^{-3}$ rms for bunching with a buncher
cavity. Third, this bunching also reduces arrival time jitter. In [331] they
measured arrival timing jitter using a THz streaking method proposed by [332].
The measured arrival time jitter for wakefield-based bunching was about 60 fs
rms, while the buncher provided a jitter of 170 fs rms.

Similarly, magnetic bunching with a wakefield can also have the benefit of
reduced timing jitter. Small timing jitter is highly demanded in many modern
accelerators. Especially, it is important for wakefield accelerators.
Wakefield accelerators usually operate at very high frequency, so placing a
short target bunch at the right time is critical because it determines the
beam’s energy spread and energy gain. However, controlling the separated
bunch’s timing in a few ps or lower is not straightforward.

[331] proposed a scheme to use the wakefield to reduce the timing jitter
between an externally injected drive bunch and main bunch. The proposed
concept uses the wakefield from a drive bunch to produce a negative chirp on
the main bunch’s longitudinal phase space. This chirping process is followed
by a chicane to compress the main bunch. If the main bunch is a little late
compared to the ideal time, it will be accelerated by the driving bunch’s
wakefield. In the following chicane, the main bunch will take a shorter path
than the path it would take when it arrives at the ideal time (i.e., zero-
crossing of the wakefield). This path length difference compensates the
original timing error after the compression. Similarly, if the main bunch
arrives earlier than the ideal time, the main bunch will be decelerated by the
wakefield and this bunch will take a longer path than the bunch at the right
timing. Thus, the timing errors can be compensated.

####  V.3.2 Bunch train generation

The bunch compression discussed in the previous section requires a condition
${\sigma_{z}\over\lambda_{wake}}\ll 1$. If we go to the other regime
(${\sigma_{z}\over\lambda_{wake}}\gg 1$), the wakefield can imprint a
sinusoidal energy modulation on the beam, which can be used for bunch train
generation. Once again, we can use either a drift or other anisochronous
beamline as a momentum-to-density modulation converter. A drift requires a
low-energy beam to provide sufficient $R_{56}$, and the overall beamline setup
can be simple and compact. On the other hand, an anisochronous beamline, such
as a chicane, can generate a high $R_{56}$ that is compatible with any beam
energy. This was experimentally demonstrated by [162] using a low-energy beam
with drift and by [9] using a compact chicane.

In [162], they used a beam with charge of 1 nC and an energy of 6.2 MeV. Two
structures were prepared (Inner diameter: 450, 750 $\upmu$m, 5 and 8 cm long),
followed by an accelerating cavity providing 14 MeV of energy gain. For both
structures, downstream diagnostics measured the phase space, and the
modulation was successfully imprinted in the LPS. The measured wavelength of
the final modulation was 1.01, and 1.81 mm for each structure. They showed
good agreement with the expected fundamental wavelengths of each structure.

On the other hand, [9] used a chicane consisting of four permanent magnets to
provide a high $R_{56}$ ($\approx 4.9$ cm). Here the beam’s charge and energy
were 0.5 nC and 57 MeV, respectively. A 5-cm-long Kapton capillary with an
inner diameter of 300 $\upmu$m was used. The corresponding fundamental
frequency was around 0.8 THz. During the experiment, they controlled the
incident longitudinal chirp by controlling the phase of the linac before the
compressor. This macro-chirp on top of sinusoidal modulation, can provide
either a frequency upshift or downshift. The modulation frequency was measured
from the autocorrelation of the coherent transition radiation, and the result
is shown in Fig. 44. When there was no chicane, the modulation conversion to
the density modulation was not observed. When a chicane was introduced, a
density modulation appeared, and its frequency was close to the expected 0.8
THz (see the 189 keV/mm case). Also, depending on the linac phase, the chirp
was changed from -567 keV/mm to +567 keV/mm. Although the initial modulation
frequency was about 0.8 THz, the final density modulation frequency could be
tuned from 0.68 to 0.9 THz.

Equation (167) is applicable to both [162] and [9] methods. In the case of the
wakefield, the modulation amplitude induced by the initial energy modulation
and $R_{56}$ was $A_{ind}=Ck|R_{56}|\Delta\delta\exp\left(-{1\over
2}C^{2}k^{2}R_{56}^{2}\sigma_{\delta}^{2}\right)$, where $\Delta\delta$ is the
amplitude of the initial energy modulation [see [261]]. Note that the required
$R_{56}\Delta\delta$ for the same final modulation amplitude decreases as
modulation frequency increases. $R_{56}\Delta\delta$ is the particle’s travel
distance. To build a density spike from the sinusoidal energy modulation,
particles need to travel a little less than $\lambda/4$. Thus, high-frequency
modulation naturally requires either a small $R_{56}$ or a small initial
modulation amplitude. Although the beam energy was 57 MeV for [9], the total
length of the chicane was less than 50 cm, and $R_{56}$ was only around 0.049
m. Their high modulation frequency (0.8 THz) enabled the use of a compact
chicane. Also, as the modulation frequency increases, the initial uncorrelated
energy spread becomes more important. The exponent of $A_{ind}$ includes a
$k\sigma_{\delta}$ term. As the frequency increases, the modulation amplitude
decays exponentially. Thus, a higher frequency requires a lower initial spread
to keep the exponential decay at the same level.

Finally, we note that modulation does not occur if the bunch’s longitudinal
profile’s frequency spectrum does not overlap with the structure’s frequency
spectrum. Thus, imprinting multi-period modulation is not possible for a
Gaussian profile. Both experiments described above used flat-top profiles.

Figure 44: Experimental demonstration of bunch train generation using a
dielectric structure followed by a chicane. The figure shows measured CTR
signals, which are autocorrelation results. Each curve corresponds to a
different initial energy chirp as described in the label next to the curve.
The squares are the data points, and the red curves are the smoothed fits.
From [9].

####  V.3.3 Single-bunch profile shaping

The current profile of a single bunch can be shaped by introducing an
appropriate nonlinearity or by masking. As we saw from transverse manipulation
by [328], introducing an appropriate nonlinearity on the phase space can
change the profile. Figure 45 shows an example for a triangular profile. A
wakefield introduces a nonlinear curvature in the linear phase space. In this
example, particles in the tail are accelerated due to the wakefield, while
particles near the center lose their energy. Thus, when the beam enters a
chicane, particles in the tail overtake the leading particles and build up a
density spike. At the same time, due to the nonlinearity in the energy gain
along the longitudinal position, the tail becomes sharp. The head of the
current profile smears out for the same reason. Note that this happens when
the wavelength of the wakefield is comparable to the bunch length so that the
appropriate nonlinearity is introduced to the phase space.

In [6] this method was experimentally demonstrated using an 80-pC, 50-MeV beam
and a compact chicane with $R_{56}=9.2$ mm. They used a dielectric wakefield
structure having an inner diameter of 200 $\upmu$m and a length of 5 cm. The
fundamental mode frequency was 0.39 THz, which is about a quarter of
$1/\sigma_{z}$. The beam’s profile was measured with and without the structure
to compare the effect of the wakefield. The reconstructed profile from CTR
interferometry is given in Fig. 45(d).

Figure 45: Experimental demonstration of triangular current profile generation
using a dielectric structure followed by a chicane.(a-c) are simulation
results, and (d) is reconstructed profile from interferometer measurements
with and without the structure. (a-c) corresponds to longitudinal phase space
before the structure, after the structure, and after the chicane,
respectively. The initial phase space (a) is generated artificially using beam
parameters used for the experiment. The modulation applied to this phase space
is calculated from a formula using structure parameters used for the
experiment. Adapted from [6].

The method can be extended to generate arbitrary current profiles when we use
a series of structures with appropriate frequencies. Similar to [234] and
[108], each structure represents a Fourier component of the target
correlation. With an appropriate $R_{56}$, the correlation can change the
initial profile to the desired profile. A scheme using transverse wigglers was
proposed by [108] and can be used for the longitudinal phase space directly by
simply replacing transverse wigglers with wakefield structures. Simulations in
[189] showed the feasibility of using several structures with different
frequencies. They also showed the feasibility of structure fabrication.

A masking-based technique has not been proposed yet. However, combining
wakefield-based shaping with methods introduced this paper could provide
advantages. For example, one can adopt a wakefield-based deflector (see [222])
instead of RF deflecting cavities for the method in [114]; see Sec. IV.2.1.
The wakefield deflector uses a transverse wakefield that kicks the trailing
particle transversely. Thus, it generates time and transverse correlation just
like RF deflecting cavities. While this passive device may provide a big
advantage on timing jitter, the nonlinearity of the transverse wake will
require more extensive analysis to design the mask and eliminate the
correlation after chopping [24, 52, 273].

####  V.3.4 Control over the energy distribution

Because the wakefield controls a correlation between the longitudinal position
and energy, it is also applicable for controlling the beam’s energy
distribution. For example, most modern accelerators use bunch compression. The
bunch compression requires a longitudinal chirp, and it is a negative chirp in
most cases. Usually, the chirp remaining after the compression is eliminated
by a combination of the linac’s RF phase control and wakefields in the linac.
However, it is a costly method since it requires operating the linac off-
crest. Alternatively, methods using dielectric and corrugated metallic
structures, which generate a strong wakefield, can remove the chirp
inexpensively. In the case of profile shaping in Fig. 45, the wakefield
frequency is chosen so that the particle in the tail is accelerated. If we
make the frequency even lower, it is possible to make the beam only see a
decelerating field. Because this wakefield always makes a positive chirp, the
remaining negative chirp after the bunch compression can be compensated. A
device using the wakefield to compensate a positive longitudinal chirp is
called dechirper.

Both [10] and [77] experimentally demonstrated the effectiveness of the
wakefield dechirper. Both experiments used a slab structure, which consists of
two jaws. Thus, the wakefield strength was adjusted by the gap size. When the
operating conditions, such as a beam’s profile, are known, the structure can
be optimized for the operating condition. However, these proof-of-principle
experiments used the slab structure with an adjustable gap to provide
flexibility to the experiment. Figure 46 shows measured beam images after the
spectrometer and corresponding spectrum from [10]. As the gap size decreased,
a quasi-linear wakefield amplitude increased, and it successfully reduced the
energy spread.

The low correlated energy spread from a dechirper can benefit all modern
accelerator applications. For example, the X-ray free-electron oscillator
(XFELO) requires energy spread of $<1\times 10^{-4}$ [147], which is
challenging to achieve with existing methods. However, a recent study by [245]
showed its feasibility. Optimization of most of the linac parameters (phase,
gradient, $R_{56}$, etc.) was performed, and the beamline had a dechirper at
the end. This optimization in the simulation generated the core of the beam
($\sim$400 fs) having a relative energy spread of $\sim 2.6\times 10^{-5}$
(105 keV rms).

Figure 46: Experimental demonstration of energy spread reduction by wakefield
from dielectric structure. The left panel shows beam images on the
spectrometer with different structure gap sizes. Corresponding profiles are
shown in the right panel. From [10].

Although the methods just introduced demonstrate the compensation of the chirp
by experiments or simulations, there was a common limitation. The methods
introduce a nonlinearity in phase space that increases the correlated energy
spread. The linearity in the core of the phase space can be preserved by
designing a structure having a low-frequency single-mode, whose wavelength is
much longer than the bunch length. However, preserving the linearity in the
periphery is challenging. When $k\sigma_{z}\ll 1$, Eq. (135) and Eq. (220) can
be simplified to

$W_{z}(z)\approx 2\mathcal{K}Q_{b}\int_{-\infty}^{z}\lambda(\zeta)d\zeta,$
(221)

where $Q_{b}$ is the total charge, and $\lambda(z)$ is the normalized density
profile. Thus, only a uniform profile provides $W_{z}(z)\sim z$, which
preserves both the core and the periphery’s linearity. All other profiles
would would have a nonlinear periphery in their phase space. .

A simulation in [10] showed the feasibility of extending linear region of the
wakefield by using a multi-mode structure. However, this approach does not
provide any tunability after the fabrication of a structure. [10] also
remarked about the use of two or more dechirpers to control nonlinearity
issues. This multi-structure approach was recently simulated by [189]. It
provided high-quality dechirping, but the dechirping for the periphery was
still limited due to the limitation of Fourier synthesis known as the Gibbs
phenomenon.

####  V.3.5 Control over longitudinal phase space

As we notice with the previous single-bunch shaping and energy distribution
control, a wakefield applied to the beam changes the longitudinal phase space,
and it can be used to control the correlation of the phase space. The most
demanding control over the correlation is linearization, and it is
experimentally demonstrated by [54] and [93]. For a simple description, if we
assume that only the RF curvature from the linac generates nonlinearities and
the beam’s energy gain from the linac can be written as $E_{RF}\cos(k_{RF}z)$,
this gain can be approximated to $E_{RF}\left[1-{(k_{RF}z)^{2}\over 2}\right]$
for $k_{RF}z\ll 1$. Similarly, the wakefield from a structure having only a
single mode and a uniform profile can be written as ${n_{0}W_{0}\over
k_{w}}\sin\left(k_{w}(z+\Delta z/2)\right)$, where $\Delta z$ is the width of
the uniform profile, and $n_{0}$ is the density. Because $k_{w}t$ should be
small while the wavelength of the wakefield should be shorter than the
wavelength of the RF field in the linac, we can approximate it as
$n_{0}W_{0}\left[-{\Delta zk_{w}^{2}\over 4}z^{2}+\left(1-{k_{w}^{2}\Delta
z^{2}\over 8}\right)z+{\Delta z\over 2}\right]$. Thus, it is possible to
eliminate the quadratic term that generates the curvature in the phase space
by choosing the proper strength of the wakefield. This can be simply done by
adjusting the gap size in the case of slab structures.

The top panel of Fig. 47 shows the simulation by [54]. The blue corresponds to
the longitudinal phase space before a corrugated waveguide, and the red shows
the corrected phase space after using the wakefield. It is clear that most of
the region is linearized and the periphery is not affected due to a very
strong initial nonlinearity. During the experiment, they generated undulator
radiation to estimate the impact of the linearization on the radiation
quality. They expected to see a significant reduction of the radiation
bandwidth due to the linearization and a shift of the central wavelength. The
bottom panel shows the measured spectrum of the radiation. Here, the blue
spectrum corresponds to a gap size of 6 mm for the corrugated waveguide while
the red corresponds to a 2-mm gap. As the wakefield strength increased, the
bandwidth became narrower. The bandwidth from the 6-mm gap was measured to 7.8
nm. On the other hand, the 2-mm gap provided 3.7-nm bandwidth.

This experiment was designed to observe the effect of linearization on the
radiation. However, we can imagine that the bunch compression will benefit
from this linearization too. Currently, most of the linearization is done with
harmonic cavities, which require additional RF power sources. However,the
beam-generated field may be able to manage the nonlinearity with lower cost.
[229] used wakefields from a high-impedance linac and a dielectric waveguide
structure to replace high-harmonic cavities. The result was similar to the
linearization result by high-harmonic cavities.

Figure 47: Demonstration of phase space linearization and its impact on the
radiation spectrum. The top panel shows simulated longitudinal phase space
before (red) and after (blue) a wakefield structure. The bottom panel shows
the measured spectrum of undulator radiation. Here the red and the blue
corresponds to results from different structure gap sizes. 2-mm gap used for
the red, and 6-mm gap used for the blue. Adapted from [54].

Although Fig. 47 shows noticeable linearization accomplished with a single
structure, a series of structures with different frequencies may enable
further control of the phase space such as correlation control of the phase
space for an arbitrary current profile or control of longitudinal chirp with
the linearization. The wakefield from each structure will work as a Fourier
component and can generate arbitrary correlation in the phase space; see
[108]. This concept was demonstrated in a simulation [189] using a series of
dielectric structures. Because the choice of parameter set becomes too
complex, they used an optimization algorithm to find the best result. Figure
48 shows two of their simulation results. The top panel shows the longitudinal
phase space with and without linearization by wakefields, and the bottom panel
shows the corresponding structure geometries. They used a total of 10
structures to correct the shape and control the chirp. Although this approach
requires experimental demonstration and further study on beam transport,
instability, fabrication error, etc., this is indeed a powerful method that
would enable us to optimize the longitudinal phase space for each application.

Figure 48: Longitudinal correlation control via a series of dielectric
structures. The top panels show simulated longitudinal phase spaces before and
after the correction using 10 dielectric structures, and the bottom panels
show the corresponding structure geometries for simulations. The left column
shows an example of linearization while the right column shows dechirping.
Adapted from [189].

##  VI Coupling between degrees of freedom for phase-space tailoring

###  VI.1 Introduction

This section discusses shaping methods that rely on the coupling between phase
spaces associated with different degrees of freedom (DOF). In Section IV, the
introduction of local coupling was shown to provide some control over a
coordinate not usually accessible in an uncoupled beamline. In that case,
shaping is performed within “coupling bumps” and the correlation is removed
downstream. Ultimately, the fact that the coupling is partial limits the
precision of that shaping method. This section focuses on beamlines that have
strong coupling or can swap phase-space coordinates. This class of phase-space
manipulation enables precise shaping of the beam phase space and opens a path
to emittance repartition among the different DOFs.

###  VI.2 Coupling between the two transverse degrees of freedom

####  VI.2.1 Producing beams with canonical angular momentum

Canonical angular momentum (CAM)-dominated, or “magnetized”, beams have
important applications in electron cooling of heavy-ion beams [33, 55, 227].
In such a scheme, a cold electron beam co-propagates with the ion beam at the
same average velocity. Collisions between ions and electrons transfer thermal
motion away from the ion to the electron beam. The cooling efficiency can be
greatly improved by using a magnetized beam. More recently, the use of a CAM-
dominated beam was also considered for mitigating a resonance-driven
instability in long periodic focusing channels [46]. A simple technique for
forming CAM-dominated beam consists of immersing a cathode in an axial
magnetic field $B_{z}$, i.e., inside a solenoid magnet. In such a case, the
canonical momentum $P_{\theta}=eA_{\theta}=\frac{eB_{z}}{2}r$ is proportional
to $B_{z}$. In a rotationally invariant system, the conservation of angular
momentum implies that the beam acquires a mechanical angular momentum (MAM)
equal to the CAM once it exits the solenoidal field as $A_{\theta}$ vanishes;
henceforth the beam’s motion in the two transverse planes becomes coupled.

As discussed in Section- II, beam dynamics is determined by several factors
including space charge, thermal emittance, angular momentum, and external
fields. However, beam dynamics differs drastically when one factor dominates
over the others; see Fig. 49. In Fig. 49(a) the dynamics is dominated by the
thermal emittance, and the electrons have random momentum direction; in (b)
the beam is dominated by its angular momentum, and the electrons shear in a
vortex pattern; and in (c) the beam is dominated by space charge, and the
electrons repel each other by the Coulomb force and move outwards.

Figure 49: Motions of the electrons when the beam is dominated by (a)
emittance, (b) angular momentum or (c) space charge. Each dot represents an
electron in $(x,y)$ space, and the arrows show the magnitude and direction of
the electrons’ velocities. From [286].

Taking into account all if these contributions, the rms transverse envelope
equation for an electron bunch propagating in a drift space is described by
[251] as

$\displaystyle\sigma^{\prime\prime}-\frac{K_{p}}{4\sigma}-\frac{\varepsilon_{u}^{2}}{\sigma^{3}}-\frac{{\cal
L}^{2}}{\sigma^{3}}=0,$ (222)

where $\sigma$ is the transverse rms size,
$K_{p}=\frac{2I}{I_{0}\beta^{3}\gamma^{3}}$ is the generalized perveance, $I$
is the absolute value of the peak beam current,
$I_{0}=4\pi\epsilon_{0}mc^{3}/e\approx 17$ kA is the Alfvén current,
$\varepsilon_{u}$ is the uncorrelated transverse rms emittance, and
$\mathcal{L}$ is related to the average canonical angular momentum
$\langle{L}\rangle$ (see Section II.2.4) and the longitudinal momentum $p_{z}$
of the beam via the magnetization

$\mathcal{L}=\frac{\mbox{$\langle{L}\rangle$}}{2p_{z}}.$ (223)

The second, third, and fourth terms of Eq. (222) represent the effects due to
space charge, emittance, and angular momentum, respectively. When the fourth
term is much greater than the second and the third terms, the beam is angular-
momentum-dominated.

If there is external electromagnetic linear focusing, an extra term in the
form of $k_{0}\sigma$ can be added to the envelope equation, where $k_{0}$ is
related to the strength of the external focusing force.

The magnetic field on the photocathode is normally zeroed to minimize the
projected emittances. This can be seen from Eq. (222), where the canonical
angular momentum term $\cal{L}$ plays the same role as the emittance term in
the beam-envelope equation so that it can be introduced to an effective
emittance $\varepsilon_{eff}=\sqrt{\varepsilon_{u}^{2}+{\mathcal{L}}^{2}}$, as
noted in Section II. However, a large magnetic field is required to produce an
angular-momentum-dominated beam in order that the correlation between the two
transverse degrees of freedom dominates, i.e., ${\cal L}\gg\varepsilon_{u}$.