# High-resolution measurement of the five-dimensional phase space distribution
of a hadron beam

A. Hoover<EMAIL_ADDRESS>K. Ruisard<EMAIL_ADDRESS>A. Aleksandrov  A.
Zhukov  S. Cousineau  Oak Ridge National Laboratory, Oak Ridge, Tennessee
37830, USA

###### Abstract

We image the five-dimensional phase space distribution of a hadron beam in
unprecedented detail at the Spallation Neutron Source Beam Test Facility. The
measurement’s resolution and dynamic range are sufficient to resolve sharp,
high-dimensional features in low-density regions of phase space. We develop
several visualization techniques, including non-planar slicing, to facilitate
the identification and analysis of such features. We use these techniques to
examine the transverse dependence of longitudinal hollowing and the
longitudinal dependence of transverse hollowing in the distribution. Our
results strengthen the claim that low-dimensional projections do not
adequately characterize high-dimensional phase space distributions in low-
energy hadron accelerators.

††preprint: APS/123-QED

##  I Introduction

The beam intensity in hadron linear accelerators is limited by space-charge-
driven halo formation [1, 2] and consequent uncontrolled beam loss [3]. We
define the halo as the region of phase space in which the particle density is
(roughly) four to six orders of magnitude below the peak density in the core,
reflecting the typical fractional beam loss in megawatt-class accelerators
[4]. No simulation has reproduced measurements at this level of detail.
Although the relevant physics is assumed to be modeled correctly, there remain
significant uncertainties in the simulation inputs — the electromagnetic
fields throughout the accelerator and the initial distribution of particles in
six-dimensional phase space [5, 6].

We denote the six-dimensional phase space distribution by
$f(x,x^{\prime},y,y^{\prime},\phi,w)$; $x$ and $y$ are the transverse
positions, $x^{\prime}=dx/ds$ and $y^{\prime}=dy/ds$ are the transverse
slopes, $s$ is the position along the reference trajectory, $\phi$ is the
deviation from the longitudinal position of the synchronous particle (in units
of RF degrees), and $w$ is the deviation from the kinetic energy of the
synchronous particle. The initial six-dimensional distribution is typically
reconstructed from the set of measured two-dimensional projections
$\left\\{{f(x,x^{\prime}),f(y,y^{\prime}),f(\phi,w)}\right\\}$ [7], where each
projection is obtained by integrating over the unlisted coordinates:

$\displaystyle f(x,x^{\prime})$ $\displaystyle=\iiiint
f(x,x^{\prime},y,y^{\prime},\phi,w){dy}{dy^{\prime}}{d\phi}{dw},$ (1)
$\displaystyle f(y,y^{\prime})$ $\displaystyle=\iiiint
f(x,x^{\prime},y,y^{\prime},\phi,w){dx}{dx^{\prime}}{d\phi}{dw},$
$\displaystyle f(\phi,w)$ $\displaystyle=\iiiint
f(x,x^{\prime},y,y^{\prime},\phi,w){dx}{dx^{\prime}}{dy}{dy^{\prime}}.$

Given only this information, the reconstruction must take the following
maximum-entropy form [8]:

$f(x,x^{\prime},y,y^{\prime},\phi,w)=f(x,x^{\prime})f(y,y^{\prime})f(\phi,w).$
(2)

(This is easily achieved by sampling from each projection individually.
Tomographic methods must be employed if other projections are known.)

Direct evidence of nonlinear inter-plane correlations (and therefore the
inaccuracy of Eq. (2)) in real beams was presented in [9], which describes the
first six-dimensional phase space measurement. The measurement characterized a
2.5 MeV H- ion beam generated by a radio-frequency quadrupole (RFQ) at the
Spallation Neutron Source (SNS) Beam Test Facility (BTF) using transverse
slits, a dipole-slit energy spectrometer, and a bunch shape monitor (BSM). The
dynamic range ($\approx 10^{1}$) and resolution ($\approx 11$ points per
dimension) were relatively low, even with 32 hours of measurement time;
therefore, as part of a preliminary investigation, lower-dimensional scans
were used to examine smaller regions of phase space. Masking the beam in the
transverse plane before measuring the energy distribution — measuring $f(w\mid
x{=}x^{\prime}{=}y{=}y^{\prime}{=}0)$ — revealed a bimodal energy distribution
near the transverse core. Importantly, this feature was not visible in the
full projection $f(w)$, which was unimodal. The correlation’s five-dimensional
nature was explored by varying the number of slits inserted into the beam and
by varying the location of a single slit with the others held fixed; both led
to pronounced changes in the energy distribution. Repeating the measurement at
different beam intensities demonstrated that space charge drives this
dependence.

The transverse-longitudinal correlations observed in [9] were subsequently
studied in [10]. The dependence of the longitudinal phase space on $x$ and
$x^{\prime}$ was mapped by measuring $f(x,x^{\prime},\phi,w\mid\tilde{y}{=}0)$
(here $\tilde{y}$ is the BSM wire position, which roughly corresponds to
$y^{\prime}$ at the measurement plane). The measurements were also compared to
an RFQ simulation, which predicted a similar dependence of the energy
distribution on the transverse coordinates. Following the argument in [9] that
the longitudinal hollowing develops in the MEBT, particle-in-cell simulations
were used in [11] to explore the longitudinal hollowing of a Gaussian beam
during free expansion. These simulations illuminated the fact that hollowing
is a natural consequence of charge redistribution caused by nonlinear space
charge forces. However, in the “realistic” beam generated by the RFQ
simulation, the correlations were already present at the end of the RFQ and
showed little evolution in the MEBT. Therefore, it was concluded that this
feature likely develops in the RFQ.

In this paper, we do not address the impact of high-dimensional features on
the beam dynamics. Instead, we continue to refine our image of the initial
phase space distribution in the BTF. In particular, we provide a (nearly)
complete description of the distribution by measuring
$f(x,x^{\prime},y,y^{\prime},w)$. Such five-dimensional measurements capture
all significant inter-plane correlations in the initial beam 111The lack of
longitudinal focusing in the BTF results in rapid debunching and a large
linear $\phi$-$w$ correlation at the measurement plane. Thus, to an
approximation, the distribution is five-dimensional at the measurement plane.
and provide unprecedented detail: the resolution and dynamic range are
sufficient to resolve sharp, high-dimensional features in low-density regions
of phase space. We use our data to strengthen the claim that low-dimensional
projections do not adequately characterize high-dimensional phase space
distributions (in typical low-energy hadron accelerators). More generally, we
illustrate the complexity of visualizing high-dimensional distributions. (We
hope that our data [13] will facilitate the development of new approaches in
this area.)

Our experimental setup and measurement procedure are described in Section II.
In Section III.1, we develop several high-dimensional analysis and
visualization techniques, including non-planar slicing, and use them to re-
examine the longitudinal hollowing described above. In Section III.2, we focus
on the transverse phase space and its dependence on the longitudinal
coordinates, reporting a transverse hollowing that likely develops in the MEBT
and is independent of the longitudinal hollowing in the RFQ. In Section IV, we
discuss the use of five-dimensional measurements in future research.

##  II Five-dimensional phase space measurement

A detailed description of the BTF is available in [14]. The system consists of
an RF-driven H- ion source, 65 keV low-energy beam transport (LEBT), and 402.5
MHz radio-frequency quadrupole (RFQ), all identical to the components in the
SNS. These are followed by a 2.5 MeV medium-energy beam transport (MEBT) which
is longer than the SNS design and contains no re-bunching cavities. The
lattice ends with a 9.5-cell FODO transport line.

The BTF houses two measurement stations. The first is located 1.3 meters
downstream of the RFQ; the second is located after the FODO line. Each station
consists of four transverse slits (two horizontal, two vertical) and a
90-degree dipole bend followed by a scintillating screen. The screen at the
first station contains a vertical slit which is used in combination with the
downstream BSM to measure the longitudinal phase space. In this setup, it is
possible to measure the five-dimensional distribution
$f(x,x^{\prime},y,y^{\prime},w)$ using three slits: one horizontal slit
selects $y$; two vertical slits select $x$ and $x^{\prime}$; $y^{\prime}$ is a
function of $y$ and the vertical position on the screen, $w$ is a function of
$x$, $x^{\prime}$, and the horizontal position on the screen. The measurement
is efficient: two dimensions are measured in a single shot. The reduction in
the number of scanning slits allows a higher resolution ($>64$ points per
dimension) and dynamic range ($>10^{3}$) than the six-dimensional measurement.

We will primarily examine a single measurement in this paper. A rectilinear
scan pattern was employed with a linear correlation between $x$ and
$x^{\prime}$ to align with the $x$-$x^{\prime}$ distribution. The corners of
the $x$-$x^{\prime}$ grid were clipped, leading to a moderate reduction in
scan time. The scan was performed as a series of “sweeps” in which the
vertical slits were held stationary while the horizontal slit was moved
continuously across the beam. During each sweep, the screen image was saved on
each beam pulse (5 Hz repetition rate) in addition to scalar quantities such
as the slit positions and beam current. The camera integral and beam current
during the measurement are displayed in Fig. 1a.

(a)

(b)

Figure 1: (a) Camera integral (sum of all pixels) and beam current during the
five-dimensional measurement. (A noise floor was subtracted from the camera
integral before normalizing to the range [0, 1].) (b) Processed camera images
during one sweep.

(The small variation in the beam current during the 17-hour scan reflects the
stability of the distribution in the BTF; the $x$-$x^{\prime}$ and
$y$-$y^{\prime}$ projections from low-dimensional and high-dimensional scans
at the same beam current are typically in close agreement, even with weeks
between scans.)

Images from the sweep containing the maximum camera integral are shown in Fig.
1b, which corresponds to one spike in the inset panel of Fig. 1a. All images
were cropped, thresholded, and downscaled by a factor of three using local
averaging. The resulting points and scalar values in five-dimensional “slit-
screen space” were transformed to phase space as follows:

$\displaystyle x$ $\displaystyle=x_{1},$ (3) $\displaystyle y$
$\displaystyle=y_{1},$ $\displaystyle x^{\prime}$
$\displaystyle=\frac{x_{2}-x_{1}}{L_{1}},$ $\displaystyle y^{\prime}$
$\displaystyle=\frac{y_{3}-y_{1}}{L_{1}+L_{2}+\rho+L_{3}},$
$\displaystyle\delta$
$\displaystyle=\frac{1}{\rho+L_{3}}\left(x_{3}+\frac{L_{3}}{\rho}x-\left({\rho-\frac{(L_{1}+L_{2})L_{3}}{\rho}}\right)x^{\prime}\right),$

where $L_{1}$ is the slit-slit spacing, $L_{2}$ is the slit-dipole drift
length, $L_{3}$ is the dipole-screen drift length, $\rho$ is the dipole bend
radius, $x_{1}$ is the position of the first vertical slit, $x_{2}$ is the
position of the second vertical slit, $y_{1}$ is the position of the
horizontal slit, $y_{3}$ is the vertical position on the screen, $x_{3}$ is
the horizontal position on the screen, and $\delta$ is the fractional
longitudinal momentum deviation. The samples were then linearly interpolated
on a regular grid in five-dimensional phase space. After cropping, this
procedure yielded a five-dimensional image of shape 69 $\times$ 82 $\times$ 69
$\times$ 59 $\times$ 49, with pixel dimensions 0.22 mm $\times$ 0.21 mrad
$\times$ 0.37 mm $\times$ 0.21 mrad $\times$ 3.40 keV. This resolution
approaches the limit dictated by the 0.2 mm slit widths.

##  III Results

###  III.1 Revisiting the dependence of the energy distribution on the
transverse coordinates

Identifying and visualizing features in high-dimensional distributions is not
straightforward [15]. Although metrics are available to compare two
distributions to each other [16, 17, 18, 8], it can be difficult to correlate
these values with physical features. Visual inspection is a powerful tool but
requires the distribution to be projected onto a one- or two-dimensional
subspace.

The orthogonal one- and two-dimensional projections of the measured
distribution are shown in a corner plot in Fig. 2.

Figure 2: Projections of the measured five-dimensional phase space
distribution. One-dimensional projections are displayed on the diagonal
subplots. Two-dimensional projections are displayed on the off-diagonal
subplots. A $10^{-3}$ fractional threshold was applied to each image.

No sharp features are visible, and all linear inter-plane correlations are
negligible. One notable feature in the $x$-$x’$ and $y$-$y’$ projections is
that the Twiss parameters in the core and tails/halo are dissimilar, which
suggests that a matched core could lead to a mismatched halo. Some of these
projections can be examined with a much larger dynamic range, as demonstrated
in [19].

The projections in Fig. 2 represent averages over large regions of phase space
and do not fully describe the distribution. (The two-dimensional projections
are not recoverable from the one-dimensional projections, which suggests that
the information lost during the transition from five to two dimensions could
be significant.) It is therefore critical to observe partial projections [9,
10], where a partial projection is the projection of the distribution within
some constrained region of phase space. When the region is small, the
information loss is minimized, and a local description of the distribution
follows; many such regions must be compared to build a global description. The
selected region may generally be called a slice. Slices are typically planar;
in an $n$-dimensional space, a planar slice selects an ($n-m$)-dimensional
region defined by the intersection of $m$ orthogonal ($n-1$)-dimensional
planes. In practice, infinitely thin slices are not possible; for example, in
the measurement described here, the slice width is limited by the physical
slit widths. Thus, a planar slice is more practically defined as the
intersection of $m$ orthogonal $n$-dimensional slabs.

There is significant (largely unexplored) freedom here, both in the slice
construction and in the visualization of the resulting partial projections. We
will revisit the previously observed longitudinal hollowing in the transverse
core to accentuate this freedom. As mentioned in Section I, this feature has
thus far been examined by observing the energy distribution within a planar
slice centered on the origin in transverse phase space, collapsing the slice
dimensions one by one as in Fig. 3.

Figure 3: Energy distribution within planar slices in transverse phase space.
Each slice is obtained by fixing the indices along the specified axes of the
five-dimensional image. Each profile is normalized by area. (This figure
mirrors Fig. 5 in [9].)

We suggest two approaches to more comprehensively visualize this feature in
five-dimensional phase space.

The first approach leverages the fact that an $n$-dimensional image is an
($n-2$)-dimensional array of two-dimensional images. When $n=3$, the images
can be arranged in a row. When $n=4$, the images can be arranged in a two-
dimensional matrix [10]. In Fig. 4, we follow this approach to examine the
four-dimensional slice $f(x^{\prime},y,y^{\prime},w\mid x{=}0)$.

Figure 4: Dependence of the $y$-$w$ distribution on $x^{\prime}$ (columns) and
$y^{\prime}$ (rows) near $x=0$. Upper left: $f(x^{\prime},y,y^{\prime},w\mid
x{=}0)$; upper right: $f(y,y^{\prime},w\mid x{=}0)$; lower left:
$f(x^{\prime},y,w\mid x{=}0)$; lower right: $f(y,w\mid x{=}0)$. The color
scale is linear and is not shared between frames. The axis limits are shared.
13/82 indices are selected along the $x^{\prime}$ axis, and 13/59 indices are
selected along the $y^{\prime}$ axis.

In the main panel, the $y$-$w$ distribution is plotted as a function of
$x^{\prime}$ and $y^{\prime}$. The bimodal energy distribution is visible near
$x^{\prime}=y^{\prime}=0$ (seventh row/column) but quickly disappears as one
moves away from the sharp peak in the $x’$-$y’$ distribution. The $y$-$w$
distribution in these low-density regions is somewhat complex and challenging
to interpret. (Note that there is a linear correlation between $y$ and
$y^{\prime}$, which explains the shifting location of the first-order $y$
moment as $y^{\prime}$ varies.) We also display the three-dimensional and two-
dimensional marginal distributions on the bottom/right panels of the figure.
These marginal distributions highlight the information lost by integrating
over momentum space. Note that the energy hollowing is still present in the
marginal distributions, but is not as pronounced; this is consistent with Fig.
3.

Fig. 4, which we call a slice matrix plot, contains a significant amount of
information; however, it also excludes a significant amount of information.
First, only a small fraction of the indices along the sliced dimensions are
shown. Second, since the distribution is five-dimensional, one is tasked with
observing a three-dimensional array of $y$-$w$ images; thus, one should vary
the slice location along the fifth dimension ($x$, in this case). Third, a
separate set of figures can be produced for each of the ten pair-wise
relationships in the data set. These considerations can lead to a
proliferation of figures, and the problem is worse in six dimensions.
Nonetheless, the combination of several slice matrix plots can be very
informative.

A second approach is to utilize non-planar slices. Consider a slice of a
distribution $f(x_{1},x_{2},\dots,x_{n})$ defined by the intersection of $m$
perpendicular slabs, where $1<m<n$ and slab $i\in[1,m]$ is defined by
$|x_{i}|<=\Delta_{i}/2$ for finite width $\Delta_{i}$. Let us refer to the
$x_{1}$-$\dots$-$x_{m}$ plane as subspace A and the $x_{m+1}$-$\dots$-$x_{n}$
plane as subspace B. In subspace $A$, the intersection defines an
$m$-dimensional box of volume $V_{A}=\prod_{i}^{m}\Delta_{i}$. Instead of a
box, one might consider an ellipsoid (perhaps defined by the covariance matrix
of $f(x_{1},\dots,x_{m})$) or a more general boundary (perhaps defined by the
density contours of $f(x_{1},\dots,x_{m})$). It is also possible to nest two
such boundaries and select the region between them; we call this a shell
slice. Fig. 5 illustrates these options. In all cases, if the volume enclosed
by the boundary goes to zero, we recover an ($n-m$)-dimensional slice defined
by the intersection of $m$ planes. We also note that, for planar slices, it is
generally advantageous to minimize $V_{A}$, but it may be advantageous to
inflate the volume of non-planar slices.

Figure 5: Several possible slice geometries. Each slice selects the shaded
region of space.

Non-planar slices are well-suited to illuminate features in subspace $B$ that
depend on the distance from the origin in subspace $A$. In particular, they
are natural choices for demarcating the core and halo regions of the
distribution [20]. There are many possibilities when applying these slices in
six dimensions. (For example, one could select only those particles within the
root-mean-square (RMS) ellipse in the two-dimensional longitudinal phase space
and outside the $10^{-3}$ density contour in the four-dimensional transverse
phase space, isolating the transverse halo in the longitudinal core.) In the
case at hand, the energy distribution appears to have a radial dependence in
transverse phase space, but it is clear that the transverse distribution does
not have ellipsoidal symmetry. Therefore, we let the density contours of
$f(x,x^{\prime},y,y^{\prime})$ define the slices. Each curve in Fig. 6 is the
energy distribution within a shell defined by the region between two such
nested contours.

Figure 6: Bottom: energy distribution within contour shell slices in the
$x$-$x^{\prime}$-$y$-$y^{\prime}$ plane. The slice at level $l$ selects the
region $l\leq f(x,x^{\prime},y,y^{\prime})\leq l+0.01$, with
$f(x,x^{\prime},y,y^{\prime})$ normalized to the range [0, 1]. Top: two-
dimensional transverse projections of the first slice.

Fig. 6 is compact but useful in describing the extent of the hollow energy
core in the $x$-$x^{\prime}$-$y$-$y^{\prime}$ plane. The energy distribution
transitions smoothly from unimodal to bimodal when moving from the low- to
high-density contours. If the core is defined as the region in which
$f(x,x^{\prime},y,y^{\prime})>10^{-2}$, then the first slice selects the
region outside the core, and subsequent slices select regions within the core
in transverse phase space. (For reference, the 0.22 contour encloses encloses
one-fifth of the beam particles.) The two-dimensional projections of the first
slice are shown in the top half of Fig. 6; the projections appear to be
consistent with the shapes of the low-density contours in Fig. 2.

Since non-planar slices naturally identify the beam core and halo in high-
dimensional phase space, we suggest that they may be useful in future
analyses, especially when the distribution lacks ellipsoidal symmetry. (One
extension of the analysis shown here would be to vary the thickness of the
shells — averaging over a larger/smaller volume. Another extension would be to
define the slices in a three-dimensional space; for example, viewing the
$y$-$w$ distribution within contour slices in $x$-$x^{\prime}$-$y^{\prime}$
space.)

###  III.2 Charge redistribution and core hollowing in the transverse plane

The five-dimensional measurement has revealed an asymmetric, longitudinally
dependent hollowing of the transverse charge distribution, shown in Fig. 7.

Figure 7: Dependence of the $x$-$y$ distribution on $w$. The color scale is
linear and is not shared between subplots. Faint dashed lines indicate the
location of each slice on the energy axis. The full energy projection $f(w)$
is shown on the bottom subplot.

Some insight into the $x$-$y$ distribution can be gained by considering the
four-dimensional transverse phase space. To this end, Fig. 8 shows the
dependence of $x$-$x^{\prime}$ on the vertical coordinates, and Fig. 9 shows
the dependence of $y$-$y^{\prime}$ on the horizontal coordinates, both within
a central energy slice.

Figure 8: Dependence of the $x$-$x^{\prime}$ distribution on $y$ (columns) and
$y^{\prime}$ (rows) near $w=0$. Upper left: $f(x,x^{\prime},y,y^{\prime}\mid
w{=}0)$; upper right: $f(x,x^{\prime},y^{\prime}\mid w{=}0)$; lower left:
$f(x,x^{\prime},y\mid w{=}0)$; lower right: $f(x,x^{\prime}\mid w{=}0)$. The
one-dimensional projection onto the $x$ axis is plotted as a white line. The
color scale is linear and is not shared between frames. The axis limits are
shared. 13/69 indices are selected along the $y$ axis, and 13/59 indices are
selected along the $y^{\prime}$ axis. Figure 9: Dependence of the
$y$-$y^{\prime}$ distribution on $x$ (columns) and $x^{\prime}$ (rows) near
$w=0$. Upper left: $f(x,x^{\prime},y,y^{\prime}\mid w{=}0)$; upper right:
$f(x^{\prime},y,y^{\prime}\mid w{=}0)$; lower left: $f(x,y,y^{\prime}\mid
w{=}0)$; lower right: $f(y,y^{\prime}\mid w{=}0)$. The one-dimensional
projection onto the $x$ axis is plotted as a white line. The color scale is
linear and is not shared between frames. The axis limits are shared. 13/69
indices are selected along the $x$ axis, and 13/82 indices are selected along
the $x^{\prime}$ axis.

It is clear from these figures that a hollow $x$ or $y$ distribution is
associated with the nonlinear tails of an “s”-shaped $x$-$x^{\prime}$ or
$y$-$y^{\prime}$ distribution. The asymmetric $x$-$y$ hollowing is explained
as follows: after integration over $y^{\prime}$ (bottom row in Fig. 8), the
$x$-$x^{\prime}$ distribution near $y=0$ is oriented such that the $x$
projection is bimodal; after integration over $x^{\prime}$ (bottom row of Fig.
9), the $y$-$y^{\prime}$ distribution near $x=0$ is oriented such that the $y$
projection is not bimodal.

The main panels of Fig. 8 and Fig. 9 indicate that there are inter-plane
relationships in the transverse phase space distribution that are hidden by
full projections. The orientation of the $x$-$x^{\prime}$ distribution depends
on the vertical phase space coordinates, and vice versa: the vertical
distribution is diverging(converging) inside(outside) the $x$-$x^{\prime}$
core. The shape of the $x$-$x^{\prime}$ distribution depends on the vertical
phase space coordinates, and vice versa: the “s” shape in one phase plane is
most distinct near the origin in the other phase plane.

A more curious feature is the apparent “splitting” of phase space near the
beam edge. This is visible in both Fig. 8 and Fig. 9 (for example, the frames
at (row, column) = (2, 5), (3, 6), (4, 7) in Fig. 9). It should be noted that
this is a minor feature of the distribution, accentuated only by the variable
color scale per subplot: the peak density in frame (2, 5) is less than $1\%$
of the peak across all frames. Although this splitting appears exotic, it is a
straightforward consequence of using planar slicing to examine a four-
dimensional phase space distribution with nonlinear inter-plane correlations.

We suggest that the transverse hollowing in the BTF is driven by nonlinear
space charge forces in the MEBT, not the RFQ, and is independent of the
longitudinal hollowing that develops in the RFQ. This suggestion is based on
particle-in-cell simulations of the beam evolution, described below.

Our simulation procedure follows that in [10]: The input beam distribution at
the MEBT entrance was predicted using a PARMTEQ [21] model of the RFQ. The
input to the PARMTEQ simulation was based on two-dimensional measurements in
the LEBT at 50 mA beam current. The RFQ vane voltage was increased by 9% over
the SNS design value of 83 kV based on preliminary results from x-ray
spectrometry, which increased both transverse emittances by approximately 7%
at the RFQ exit. The predicted RFQ transmission was 84%, resulting in a 42 mA
beam current in the MEBT. The measured output current is lower than this
prediction, but the simulated current was kept at 42 mA to amplify the space-
charge-driven features of interest. A PyORBIT [22] model was used to propagate
the beam 1.3 meters from the RFQ exit to the first horizontal slit (HZ04), a
distance including four quadrupole magnets for which a hard-edge model was
used. Space charge kicks were applied every millimeter using an FFT Poisson
solver on a $64\times 64\times 64$ mesh with $8.6\times 10^{6}$ macro-
particles (for sufficient statistics in high-dimensional slices). Fig. 10
shows the simulated evolution.

Figure 10: Simulated transport of a 42 mA beam in the BTF MEBT (RFQ exit to
measurement plane). Top: snapshots of the $x$-$x^{\prime}$, $y$-$y^{\prime}$,
and $x$-$y$ distributions within a central longitudinal slice ($\phi\approx
0$). Each distribution was normalized before plotting such that
$\langle{xx}\rangle=\langle{yy}\rangle=\langle{\phi\phi}\rangle=1$ and
$\langle{xx^{\prime}}\rangle=\langle{yy^{\prime}}\rangle=\langle{\phi
w}\rangle=0$, where $\langle\dots\rangle$ represents the average over the
distribution. Faint gray lines indicate the position in the beamline. Bottom:
RMS emittance and beam size evolution.

A detailed study of the beam dynamics is beyond the scope of this paper; we
briefly note the following conclusions drawn from the simulations.

  1. 1.

The transverse hollowing is qualitatively reproduced, although the effect is
underestimated; lowering the beam current to 26 mA at the RFQ output (to match
the measured current) results in a less dramatic hollowing than in the
measurement.

  2. 2.

The transverse hollowing develops in the MEBT regardless of the correlations
that develop in the RFQ. This is supported by repeating the simulation after
decorrelating the initial bunch by randomly permuting $x$-$x^{\prime}$,
$y$-$y^{\prime}$, and $\phi$-$w$ coordinate pairs.

  3. 3.

The physical origin of the transverse hollowing is familiar. For example, the
simulated transport of an out-of-equilibrium four-dimensional Waterbag
distribution in an alternate-gradient focusing channel has produced similar
two-dimensional projections [23]. Of course, the details of the evolution
depend on the initial beam perveance, emittance, and focusing strength.

  4. 4.

The asymmetric $x$-$y$ hollowing is primarily due to the vertical beam waist
in the early MEBT. The round initial beam, which has an approximately Gaussian
charge distribution, is diverging horizontally and converging vertically. It
passes a vertical waist before the first quadrupole, then expands in both
planes. The horizontal emittance grows most rapidly just after this waist,
while the vertical emittance shrinks, presumably due to coupling between the
planes. If the initial $x$ and $y$ beam divergences are exchanged
($x^{\prime}\rightarrow-x^{\prime}$, $y^{\prime}\rightarrow-y^{\prime}$), the
hollowing is seen in $y$, not $x$, with an associated larger vertical
emittance growth. The dependence on the exact pattern of alternate-gradient
focusing is weak.

  5. 5.

The second-order moments disagree with the measurement; for example, the RMS
emittances differ by over 15%. This is expected based on previous longitudinal
benchmarks [10].

The simulations described above support the claim that the transverse
hollowing is driven by nonlinear space charge forces in the MEBT. It is
difficult to verify this experimentally without a current-attenuating grid at
the RFQ exit. Instead, we repeated a five-dimensional measurement at a lower
beam current extracted from the ion source. This mirrors previous measurements
to verify the space-charge-dependence of the longitudinal hollowing [9]. Fig.
11 shows that no transverse hollowing occurs at this lower beam current.

Figure 11: No transverse hollowing is apparent at the center of the energy
distribution in the low-current (7 mA) measurement. This figure is equivalent
to Fig. 7, which shows the 26 mA case.

(Although the low-current five-dimensional distribution is not hollow, it is
rich in structure, presumably due to the lack of smoothing by strong space
charge. We leave the investigation of this low-current distribution as future
work.)

##  IV Discussion

In summary, we have used five-dimensional measurements to enhance our image of
the initial phase space distribution in the SNS-BTF. We developed several
high-dimensional visualization techniques and used them to re-examine the
longitudinal hollowing in the transverse core. We also reported the existence
of transverse hollowing in the longitudinal core; simulations suggest that
this feature is driven by nonlinear space charge forces in the MEBT,
independent of the longitudinal hollowing that develops in the RFQ. We
examined both features in considerable detail, leveraging the resolution and
dynamic range of the five-dimensional measurement. Neither feature is visible
in the full two-dimensional projections of the distribution. Our data is
further evidence that the three phase planes — $x$-$x^{\prime}$,
$y$-$y^{\prime}$, $\phi$-$w$ — are (nonlinearly) correlated in real beams.

A longstanding goal in accelerator physics is to predict the beam evolution at
the halo level, which will require (i) an accurate simulation model and (ii) a
realistic initial bunch. Five-dimensional measurements at the end of the BTF
beamline will serve as precise benchmarks and help address (i). Addressing
(ii) will require knowledge of the initial six-dimensional phase space
distribution. Although six-dimensional measurements are the gold standard, the
demonstrated resolution and dynamic range are quite low 222Improved six-
dimensional measurements may be available in the future. First, we plan to
increase the beam repetition rate from 5 Hz to 10 Hz. Second, a smarter scan
pattern could provide a factor of 5 to 10 speedup, as roughly 10% of the
points in a five-dimensional grid measure signal (although this number depends
on the dynamic range of the measurement). Third, a two-dimensional BSM would
reduce the scan to four dimensions. Fourth, the dynamic range can be increased
by improving the light collection system [19]. With these improvements, a
medium-resolution measurement with 30 points per dimension in a 24-hour scan
is not out of the question.. In the BTF, five-dimensional measurements may be
able to serve as a proxy for six-dimensional measurements. For reasons
described in [12], it is likely that a reconstruction from
{$f(x,x^{\prime},y,y^{\prime},w)$, $f(\phi,w)$} would be quite accurate. The
reconstruction would ideally be treated using the principle of entropy
maximization (MENT) [25]; a six-dimensional MENT solver could be adapted from
one of several existing algorithms [26, 8]. Alternative reconstruction
approaches which incorporate low-resolution six-dimensional measurements may
also be possible [27]. In our case, it may suffice to sample from the five-
dimensional distribution, then assume a linear relationship between $\phi$ and
$w$, plus some phase width.

Our work may also be useful for high-dimensional phase space tomography — the
reconstruction of a four- or six-dimensional distribution from two-dimensional
projections. There are various challenges in extending tomographic algorithms
to six dimensions, mainly due to memory limitations, but also due to
uncertainty in the set of transformations necessary to accurately reconstruct
a high-dimensional distribution [28, 29, 30, 31, 32, 8, 33]. The accuracy of
reconstruction algorithms has primarily been evaluated by comparing the two-
dimensional projections of the reconstruction to the ground truth; it is an
open question whether the high-dimensional features presented herein can be
recovered. Direct measurements could serve as valuable benchmarks. Although
the manipulations necessary for six-dimensional tomography are not possible in
the BTF, our five-dimensional measurement data [13] could be used as a
benchmark in a simulated reconstruction.

##  V Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics. This manuscript has been
authored by UT Battelle, LLC under Contract No. DE-AC05-00OR22725 with the
U.S. Department of Energy. This research used resources at the Spallation
Neutron Source, a DOE Office of Science User Facility operated by the Oak
Ridge National Laboratory. The United States Government retains and the
publisher, by accepting the article for publication, acknowledges that the
United States Government retains a non-exclusive, paid-up, irrevocable, world-
wide license to publish or reproduce the published form of this manuscript, or
allow others to do so, for United States Government purposes. The Department
of Energy will provide public access to these results of federally sponsored
research in accordance with the DOE Public Access Plan
(http://energy.gov/downloads/doe-public-access-plan).

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