# Longitudinal charge distribution measurement of non-relativistic ion beams
using coherent transition radiation

R. Singh1, T. Reichert1
1GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Darmstadt, Germany
<EMAIL_ADDRESS>
(June 2021)

##  1 Abstract

Longitudinal charge profile measurements or bunch shape measurements is a
challenge for temporally short non-relativistic bunches. The field profile has
a larger longitudinal extent compared to the charge profile for such beams.
This affects the ability of field sensing devices such as phase pick-ups or
wall current monitors to measure charge distribution. Here we evaluate the
feasibility for usage of coherent transition [1, 2] and diffraction radiation
from non-relativistic beams for bunch shape measurements.

##  2 Introduction

Charge profile measurements are essential for verification of LINAC beam
dynamics, commissioning and optimization of LINACs. This has been historically
challenging for non-relativistic beams since the non-destructive field
monitors could not be used for this purpose. Meanwhile there are other devices
such as fast Faraday cup designs [3, 4] (FFCs) which use the ground plates to
the shield the detection plate to avoid early arrival of beam field. However,
the signal generation process including the role of secondary electron
emission is not fully understood . Further, Faraday cups are interceptive
measurements which makes their usage unattractive with high intensity beams as
well as unfeasible for phase space measurements. The commonly used alternative
is the bunch shape monitor [5] or ”Feschenko monitor”, which relies on
electron emission from a wire after the beam interaction. This device is
however limited to a temporally averaged measurement due to small interaction
of electron emitting wire with beam and inherently relies on stable beam
conditions. Therefore such averaged measurements can be misleading when there
are shot- to-shot current fluctuations or any other effects which lead to non-
reproducible charge distribution in the duration of measurements.
Interestingly, there is no bench-marking of the aforementioned devices
available in literature.

In this report, we study a non-destructive bunch-by-bunch longitudinal charge
profile measurement alternative based on coherent transition [1, 2] and
diffraction radiation. Transition radiation from its very nature provides a
measure of relative charge variation. The range of interest in terms of beam
velocities is $\beta\in[0.05,0.9]$ and bunch lengths between 50-500 ps
(1$\sigma$). In the first section, we will discuss the generation process,
angular distribution of the radiated field, formation zone, target size
dependence as well as provide signal estimates using available analytical
models for transition radiation (TR). The non-destructiveness of the
measurement principle by means of a hole in the target will also be discussed.
In the second section, we compare the analytical results with electromagnetic
particle-in-cell simulations. Few examples for relativistic beams i.e.
$\beta=0.99$ are shown to connect the results in this report to the electron
beam examples found extensively in literature as well as to discuss the
feasibility of our proposal for relativistic beams. In the final section,
first results from a prototype installed at GSI UNILAC are shown.

##  3 Analytical estimates

###  3.1 Electric field calculations with ideal set-up

Figure 1 shows a potential set-up for the proposed bunch shape measurement
using transition radiation. A perfectly conducting ring-shaped target
(inner/outer radius $a$ , $b$ respectively) is located at $z=0$ so that its
surface normal is given by $\vec{n}_{s}=(0,0,1)^{T}$. The charged particle
beam originates from infinity and travels through vacuum approaching the
target with perpendicular incidence with velocity $\vec{v}_{e}=-\beta
c_{0}\hat{e}_{z}$. The target can be made of standard metals since they can
practically all be considered perfect electric conductors (PEC) in the
frequency regime of interest ($\leq 30$ GHz). The hole of radius $a$ in the
target is for non-destructive measurements.

Figure 1: Coordinate system and the target plane

As depicted in Fig.1, the field monitor or observation point is located in the
$x-z$ plane sensitive to radiation emitted in that plane
$k_{x}\hat{e}_{x}+k_{z}\hat{e}_{z}$.

$\displaystyle\vec{R}=\left(\begin{matrix}x\\\ 0\\\
z\end{matrix}\right)=\left(\begin{matrix}R\sin(\theta)\\\ 0\\\
R\cos(\theta)\end{matrix}\right)$ (1)

and the co-ordinates on the target plate are given by,

$\displaystyle\vec{r}_{s}=\left(\begin{matrix}x_{s}\\\ y_{s}\\\
0\end{matrix}\right)=\left(\begin{matrix}r_{s}\cos(\phi_{s})\\\
r_{s}\sin(\phi_{s})\\\ 0\end{matrix}\right)$ (2)

A comprehensive and modern review of the generation process of transition
radiation is given in [6] which is a pre-requisite for the reader of this
report. In this report, we make an attempt to use the same formalism as [6]
and restrict ourselves to the relevant and special case of normal incidence of
beam on the target. The Fourier transform of the ”source” field of the moving
charged particle is given as,

$\displaystyle E_{s,x}(z=0,\omega)$
$\displaystyle=G(r_{s},\omega)\cos(\phi_{s})$ $\displaystyle
E_{s,y}(z=0,\omega)$ $\displaystyle=G(r_{s},\omega)\sin(\phi_{s})$
$\displaystyle\text{with}\ \ G(r_{s})$
$\displaystyle:=\frac{e\alpha}{4\pi\epsilon_{0}\pi\beta c}K_{1}(\alpha r_{s})$
(3)

where $K_{1}$ represents the modified Bessel’s function of the second kind and
$\alpha=\omega/{\beta\gamma c}$. In [6] the scattered/radiated transition
radiation field originates from a virtual magnetic surface current induced on
the target due to the source field,

$\displaystyle\vec{j}_{vm}$
$\displaystyle=\frac{c}{4\pi}\hat{n}_{s}\times\vec{E}_{s}$
$\displaystyle\Rightarrow\vec{j}_{vm}$
$\displaystyle=\frac{c}{4\pi}\left(\begin{matrix}-E_{s,y}\\\ E_{s,x}\\\
0\end{matrix}\right)$ (4)

As evident in Eq. 4, for the case of normal incidence, only the transverse
components $E_{s,x}$ and $E_{s,y}$ are of relevance in formation of virtual
magnetic surface current. The associated vector potential can be calculated by
inserting (3) and (4) in Eq.(19) of [6],

$\displaystyle\vec{A}$
$\displaystyle=\frac{2}{c}\int_{a}^{b}dr_{s}\int_{0}^{2\pi}d\phi_{s}r_{s}\vec{j}_{vm}\frac{\exp(ikR_{s})}{R_{s}}$
$\displaystyle=\frac{1}{2\pi}\int_{a}^{b}dr_{s}\int_{0}^{2\pi}d\phi_{s}r_{s}G(r_{s})\frac{\exp(ikR_{s})}{R_{s}}\left(\begin{matrix}-\sin(\phi_{s})\\\
\cos(\phi_{s})\\\ 0\end{matrix}\right)$ (5)

where $R_{s}$ is the norm of the difference vector between $\vec{R}$ and
$\vec{r}_{s}$

$\displaystyle R_{s}$ $\displaystyle=\sqrt{(x-x_{s})^{2}+y_{s}^{2}+z^{2}}$
$\displaystyle=\sqrt{x^{2}+z^{2}+r_{s}^{2}-2xr_{s}\cos(\phi_{s})}$ (6)

Since $R_{s}$ is an even function of $\phi_{s}$ and gets multiplied by an odd
function in $A_{x}$ the symmetric integration over $\phi_{s}$ vanishes.
Analogously the same integration from $A_{y}$ can be limited to the interval
$[0,\pi]$ with a doubled integrand resulting in,

$\displaystyle\vec{A}$
$\displaystyle=-\frac{\hat{e}_{y}}{\pi}\int_{a}^{b}dr_{s}\int_{0}^{\pi}d\phi_{s}r_{s}\cos(\phi_{s})G(r_{s})\frac{\exp(ikR_{s})}{R_{s}}$
(7)

It should be noted that Eq. 7 implies that only the source fields in the plane
of observation contribute to the vector potential perpendicular to it. From
the vector potential, the electric fields can be obtained by,

$\displaystyle\vec{E}=-\nabla\times\vec{A}$ (8)

Thus the generated transition radiation field in the plane of observation is
the result of source fields only in that same plane. A quasi-spherical
approximation (QSA) for calculation of electric fields is made ($R_{s}\approx
R$ and $1/kR\ll 1$)in [6] due to reasons to computational complexity, i.e.
$\vec{E}=-i\vec{k}\times\vec{A}$ and the validity of this QSA is discussed and
quantified. For our purpose of bunch length measurements, the target size,
wavelength of interest and monitor distance are of the same order and the
validity of quasi-spherical approximation is not clear. Therefore, we
calculate the radiated electric field without the quasi-spherical
approximation. As there is only a $y$-component in $\vec{A}$ we can write,

$\displaystyle\vec{E}=-\nabla\times\vec{A}=\left(\begin{matrix}\partial_{z}A_{y}\\\
0\\\ -\partial_{x}A_{y}\end{matrix}\right)$

For this purpose we show

$\displaystyle\partial_{z}\frac{\exp(ik\tilde{R_{s}})}{\tilde{R_{s}}}$
$\displaystyle=\frac{z\exp(ik\tilde{R_{s}})}{\tilde{R_{s}}^{3}}\left(ik\tilde{R_{s}}-1\right)$

and

$\displaystyle\partial_{x}\frac{\exp(ik\tilde{R_{s}})}{\tilde{R_{s}}}$
$\displaystyle=\frac{(x+r_{s}\cos(\tilde{\phi_{s}}))\exp(ik\tilde{R_{s}})}{\tilde{R_{s}}^{3}}\left(ik\tilde{R_{s}}-1\right)$
(9)

and thus we arrive at,

$\displaystyle\vec{E}$
$\displaystyle=\frac{1}{\pi}\int_{a}^{b}dr_{s}\int_{0}^{\pi}d\tilde{\phi_{s}}G(\alpha,r_{s})r_{s}\cos(\tilde{\phi_{s}})\frac{\exp[ik\tilde{R_{s}}]}{\tilde{R_{s}}^{3}}(1-ik\tilde{R_{s}})\left(\begin{matrix}z\\\
0\\\ -x-r_{s}\cos(\tilde{\phi_{s}})\end{matrix}\right)$

Inserting Eq.(3) we have the expression for the special case of normal
incidence radiation (NIR) which should be valid at any position from the
target,

$\displaystyle\vec{E}=\frac{2e}{(4\pi\epsilon_{0})\pi\beta^{2}\gamma
c\lambda}\int_{a}^{b}dr_{s}\int_{0}^{\pi}d\tilde{\phi_{s}}K_{1}(\frac{2\pi
r_{s}}{\beta\gamma\lambda})r_{s}\cos(\tilde{\phi_{s}})\frac{\exp[ik\tilde{R_{s}}]}{\tilde{R_{s}}^{3}}(1-ik\tilde{R_{s}})\left(\begin{matrix}z\\\
0\\\ -x-r_{s}\cos(\tilde{\phi_{s}})\end{matrix}\right)$

It is possible to further simplify the result in Eq. (LABEL:eq15) to a single
integral in case of quasi-spherical approximation as discussed in the
Appendix.

(a)  (b)

Figure 2:  (a) Comparison of Normal incident radiation (NIR), quasi-spherical
approximation (QSA) and far field approximation (FFA) for $\beta=0.3$ for
target radius $b$ = 0.4 m, monitor distance $R$ = 0.4 m and frequency = 0.7
GHz. (b) Comparison of NIR, QSA and FFA for target radius $b$ = 1.2 m, monitor
distance $R$= 1.2 m at $\beta=0.99$.

In Fig. 2 we compare the transition radiation angular distribution of three
cases, the traditional far field result (FFA) [2], quasi spherical
approximation (QSA) [6] and the ”exact” estimate for the special case of
normal incidence (NIR) shown in Eq. LABEL:eq15. There is a significant
discrepancy between the QSA and NIR for lower frequencies and distances of
interest especially for smaller detection angles $\theta$. Based on this
observation, we will utilize NIR as the basis for our analytical transition
radiation field calculations in this report.

###  3.2 Target size dependence

It is well known from literature [9, 10, 6], that the target size should be
larger than the transverse source size in order to avoid deformations in the
generated transition radiation fields with respect to the far field angular
distribution. The effective source field extent is given by
$r_{\text{eff}}=\beta\gamma\lambda$ for generation of radiation at wavelength
$\lambda$. Beyond this radius the incident field’s contribution to the
radiation field becomes very small. This dependence can be readily seen in the
argument of modified Bessel’s function shown in Eq. 3. A finite target size
thus introduces a ”high pass filter” behavior at a frequency $f_{cut}$
$\approx\frac{b}{\beta\gamma c}$ where $b$ is the target radius. The
calculated frequency responses are shown in Fig. 3 for various target sizes in
units of $\beta\gamma\lambda$ for a monitor placed at the angle of 60°at a
distance of 2 m for two beam velocities. The $\lambda$ chosen is 1 m
corresponding to 300 MHz in vacuum. The high pass behavior in the frequency
response is evident in the plots.

(a)  (b)

Figure 3:  (a) Frequency response as a function of target size for
$\beta=0.05$ at monitor distance $R$ = 2.0 m (b) Frequency response as a
function of target size for $\beta=0.9$ at monitor distance $R$ = 2.0 m.

Figure 4 shows the electric field amplitude at a detector angle of
$\theta=70\degree$ for different particle velocities as a function of target
size. The field reaches the nominal far field values for
$\beta\gamma\lambda>0.8$. This could be used as a rule of thumb for
determining the smallest target size required to avoid target size dependence
on the radiation for a given reference wavelength $\lambda_{ref}$.

Figure 4: Electric Field spectral amplitude dependence on target size.

###  3.3 Formation zone and monitor placement

Ginzburg and Tsytovich [1] reviewed theoretical aspects of the transition
radiation phenomenon discussed the concept of radiation formation zone. It is
defined as the distance travelled by the charged particle while the radiation
”emitted” by the charged particle between the start and end points accumulates
a phase larger than $\pi$. This can be understood as the distance travelled by
the charged particle where any remnants of the preceding radiation due to
interaction with the surroundings is separated. In other words, the charge
requires a certain ”formation” length after its interaction with a medium to
reach its equilibrium or ”proper” field again. For the forward transition
radiation, this effect is evident as the distance required for the charged
particle ”proper” coulomb field or ”direct field” to separate from the
radiated transition radiation. For the backward transition radiation, the
question of proper coulomb field and transition radiation does not arise,
however there are other practical constraints. Typically it is assumed that
charged particle is travelling from infinity towards the target which is not
true for most practical cases. Therefore the remnant of the interaction with
the boundary (e.g. a beam pipe) which the charge exits will still interfere
with the backward transition radiation if a certain minimum separation is
unavailable between the boundary and the target. Already from this simple
argumentation, it is clear that the formation zone will be a function of
observation angle $\theta$. The formation zone for forward radiation when
traversing from a medium to vacuum was derived by Garibian [7] and is given
as,

$\displaystyle L_{formation}=\frac{\lambda\beta}{2\pi(1-\beta\cos{\theta})}$
(12)

. For relativistic beams, the relevant ”formation length” is defined with
respect to the angle where peak power is radiated,
$\theta\approx\frac{1}{\gamma}$ and it can be shown to reduce to this simple
widely used expression $L_{formation}=\gamma^{2}\lambda/2\pi$ for the radiated
wavelength $\lambda$. The detector distance $r$ should be much greater than
$L_{formation}$, i.e. $r>>L_{formation}$ to observe the far field radiation.
Formation length plays a fundamental role in radiator design for high energy
physics detectors [8].

The discussion on radiation formation zone was furthered by Verzilov [9],
where it was shown that the effective transverse size of the incident field
$r_{\text{eff}}$ is linked to a characteristic distance around the target
referred to as ”prewave” zone. Therein the source has to be considered
distributed continuously over the target plane and thus the radiation field
will exhibit interference patterns resulting from contributions due to
different parts of the target being lit by the incident field. The qualitative
arguments given in [9] are for the small angle approximation and the extent of
prewave zone for a beam energy corresponding to a $\gamma$ at the peak of TR
again turns out to be $\gamma^{2}\lambda$. Again, in Fig. 3 [9] it is shown
that the interferences are stronger towards smaller angles and the transition
radiation peak is shifted in the prewave zone while these interference effects
subside for large observation angles. Both of the aforementioned effects
appear different in their origin, yet yield a similar tendency for transition
radiation angular distribution as the function of beam parameters. The
analytical results discussed in [9] only addresses the source size effects and
do not consider the ”Ginzburg” formation zone mentioned in [7]. A detailed
account of these formation zones and its implications for different detector
sizes is given in [10] and experimentally verified in [11]. The Ginzburg
formation zone for backward transition radiation is partially addressed in
[15] via line diffraction sources however the authors of this report could not
reproduce the results due to numerical difficulties. In this section, we only
consider the Verzilov prewave zone in context of backward transition radiation
and show the angular distributions calculated using NIR (Eq. LABEL:eq15) and
its deviation from the far field distribution. With reference to the
discussion of detector size in [10], our detectors can be understood as
arbitrarily small or ”dotted”. We will resort to full electromagnetic
simulations in the next section which should include the effects of both
Ginzburg ”formation zone” and Verzilov ”prewave zone”.

Figure 5 shows the angular field distribution for two wavelengths $\lambda=1$
m and $\lambda=0.055$ m for the upper end of our beam energy consideration
i.e. $\beta=0.99$ ($\gamma=7.08$) as a function of monitor distance and
observation angle with respect to the TR target. Let us consider to
frequencies 0.3 GHz and 5.7 GHz as the extremas for a 100 ps ($\sigma$)
Gaussian bunch. The target size chosen is 1.4$\beta\gamma\lambda$ with the
reference wavelength $\lambda_{ref}=1$ m in order to discount the target size
effects at frequencies above $300$ MHz. Note this leads to $b=\approx 20$ m
target radius for $\beta=0.99$ case. The Verzilov prewave zone formula
predicts a $L_{formation}$ of $\approx 50$ m for $\lambda=1$ m and $\approx
2.5$ m for $\lambda=0.055$ m. One should note that Ginzburg ”formation zone”
is $2\pi$ times smaller and its angular dependence is marked in Fig. 5. For
$\lambda=0.055$ m case, the far field angular field distribution is obtained
(for all angles) from a monitor distance of 2.5 m (and onwards) in accordance
to the estimate. It is also seen that the deviation from the far-field field
distribution is a strong function of the angular placement of the monitor.
Figure 6 shows the relative error between the field angular distribution for
the two frequencies. If we tolerate a 5% relative error, already at $R>2$ m
and $\theta>60$ degrees, the near field effects are negligible and the monitor
for the full range of frequencies in an 100 ps bunch and the monitor can be
considered in the wave zone. Thus, the field monitors can be placed much
closer in comparison to the relativistic approximation $\gamma^{2}\lambda$ at
steeper angles. This is a useful observation in context of measurement of
longitudinal charge profiles since monitors can be placed relatively close to
target for relevant energy range $\beta<0.99$ without relevant distortions of
the near field effects. This also allows for higher signal intensity and
flexibility in the set-up design.

(a)  (b)

Figure 5:  The angular distribution 0.3 GHz and 5.7 GHz for $\beta=0.99$ as a
function of monitor distance from the target. The target size is set to
1.4$\beta\gamma\lambda$ for the reference wavelength of $\lambda_{ref}=1$ m.
Eq. 12 scaled with $2\pi$ is shown as dotted lines which separate wave and
prewave zone. Figure 6: Percentage error between 0.3 GHz and 5.7 GHz for
$\beta=0.99$ as a function of monitor angle and distance from the target. The
target size is set to 1.4$\beta\gamma\lambda$ for the reference wavelength of
$\lambda_{ref}=1$ m. Above 2 m radial distance, the radiation has a far field
like distribution for the relevant frequency range for angles larger than 60°.

###  3.4 Effect of hole in the target

Another important design consideration for any beam diagnostics device is its
potential usage without interfering with the beam. Diffraction radiation is a
closely related concept to transition radiation which can be utilized for non
destructive measurements. Unlike transition radiation, the charges need not
”hit” the second medium, but only pass close enough, such that the effective
source size has significant interaction with the second medium. In our setup,
a hole through the target provides such a possibility. A transversely small
beam charge distribution will pass though the hole of diameter $d_{hole}$
($2a$ in Fig. 1), however the effective size of the incident field
$\beta\gamma\lambda$ will interact with the target to produce diffraction
radiation. It is clear, for a wavelength $\lambda$ where
$\beta\gamma\lambda<d_{hole}$, there is barely any radiation generated. The
hole thus introduces a low pass behavior in the generation process. Figure 7
(a) shows the frequency response calculated from the angular field
distribution of $\beta=0.15$ beam at monitor distance of $R=1$m and
$\theta=85$ degrees and target size $\beta\gamma\lambda=1$ for different hole
sizes. In addition, the frequency content of a 100 ps bunch is plotted for
comparison. Figure 7(b) shows convolution of NIR frequency response with a 100
ps bunch for upto 10 mm holes are there is no significant widening of the
bunch observed.

(a)  (b)

Figure 7:  (a) Frequency response as a function of hole size for $\beta=0.15$
at monitor distance $R$ = 1.0 m. Frequency spectrum of a 100 ps is marked with
dashed lines. (b) Measured electric field at 85 degrees for an input of 100 ps
bunch for different hole sizes.

###  3.5 Signal estimates

An estimate of the expected induced voltages at a monitor placed at a distance
of $R=1.0$ m and angle $\theta=$ 60 °for a 100 ps ($\sigma$) Gaussian bunch
length containing 10 pC charge at the energy corresponding to $\beta=0.15$ is
sought in this section. Aforementioned charge per bunch coming at 36 MHz
repetition corresponds to an average current of 0.37 mA. The target diameter
should be larger than the effective source field, e.g.
$d_{target}>2\beta\gamma\lambda_{ref}$ for $\lambda_{ref}=1$ m. This gives the
lower bound on the target diameter of 0.3 m. The field estimate per charge per
unit frequency can be obtained by inserting these parameters in Eq.
LABEL:eq15, which results in $\frac{dE_{TR}}{d\omega dq}=4.0\cdot 10^{-18}$
V/(m$\cdot$Hz). The field obtained per charge should be scaled with the number
of charges $N=10pC/(1.6\cdot 10^{-19})=6.25e7$. For a bunch of charges with
standard deviation $\sigma_{t}=100ps$ of the longitudinal profile, most of the
power lies below $2\sigma_{f}=\frac{1}{\pi\sigma_{t}}\approx 3.2$ GHz. The
peak field estimate thus obtained is 400 mV/m for 10pC charge in Gaussian 100
ps ($\sigma$) bunch. The antenna factor for a commercially available biconical
antenna [18] $\approx 30dB$, i.e. $\frac{E_{field}}{V_{induced}}=30$ and the
12 mV peak voltage should be induced. As seen from Figure 7, a hole of
diameter 10 mm will reduce the signal by a factor 2 and 5 mV peak voltage is
available for non destructive charge profile measurements. In comparison, a
typical amplifier [17] with $50$ $\Omega$ input impedance has a noise of
$\approx 1$ $nV/\sqrt{Hz}$ which for a 10 GHz bandwidth should produce an rms
noise of 0.1 mV. Thus a signal to noise ratio (SNR) of 50 is available for 10
pC charge in a 100 ps bunch measured with a commercially available biconical
antenna place at $\theta=60\degree$ at 1 m distance between target and
detector.

###  3.6 Applicability range for beam and target parameters

The longest bunches which can be measured by this method is limited by the
target size which in turn is given by $\beta\gamma\lambda$ as well as the
lower frequency cut-off of the commercially available linear phase broadband
antennas. Both of the above conditions converge to a similar limit, i.e. a
wavelength of $\lambda=2$ m corresponding to a frequency of 150 MHz. This
means that a bunch length of 500 ps (1 $\sigma$) could still be faithfully
reconstructed. The shortest bunches which can be measured is primarily limited
by the sampling speed and analog bandwidth of acquisition electronics and the
current technology can allow 20 ps bunches to be measured. Since the
transition radiation signal is proportional to beam velocity and current,
beams upto $\beta=0.01$ should be measurable provided there is enough beam
current. Our range of application is $\beta=0.05$ to $0.15$ at GSI UNILAC.

##  4 CST simulations and comparison

Transition radiation process was simulated using the particle-in-cell (PIC)
solver of the electro-magnetic simulations software CST. Here a collection of
macro-particles with common mass $m_{q}$, total charge $q$ and velocity
$\beta$ form a bunch with user-defined transversal as well as longitudinal
shape. Figure 8 shows the simulation domain. The transverse beam size is fixed
to 5 mm $(\sigma)$. This charged particle bunch is introduced into and
propagated through a rectangular calculation domain (here: vacuum) which is
spatially discretized by a regular mesh. The temporal discretization complies
to the Courant–Friedrichs–Lewy criterion [13] and is provided by the solver
algorithm. After propagation through the calculation domain, the bunch is
normally incident on a target made of a perfect electric conductor (PEC)
spanning fully over one of the boundary planes. Optionally it has a hole in
the center to let the bunch particles pass through. Here the wanted transition
or diffraction radiation is formed since the self-field of the bunch has to
comply dynamically to the PEC interface as discussed earlier in this report.
All remaining boundaries are set to be open so that time-dependent electro-
magnetic fields are absorbed with a low level of residual reflections which is
realized by imposing perfectly matched layers (PML). A drawback of the
simulation via the PIC solver is given by the fact that the insertion of a
charge into the simulation domain results in prompt bremsstrahlung which we
refer to as ”domain entry radiation” for the lack of a better term. It is
briefly discussed in the next section. The electric fields produced during the
simulation are collected with ideally broadband field monitors radially ($R$)
and azimuthally $\theta$ around the impact point. We choose a Gaussian
longitudinal distribution of the bunch charge with tails cut at $4~{}\sigma$
and uniform velocity $\beta\in[0.15,0.99]$. The range of bunch lengths is
given as $\sigma_{t}\in[100,400]$ ps thus a conservative upper cutoff
frequency of $6.5$ GHz is used to contain the frequency span of the shortest
bunch. Behind the entry pipe, an electrically large problem is faced here and
the mesh cell size is controlled via the spatial sampling of the smallest
wavelength. The relevant parameter lines per wavelength (lpw) is set to at
least lpw $=11$ to avoid numerical dispersion.

Figure 8: The Simulation model

To ignore any potential space charge induced bunch blow-up the specific charge
of the macro-particles is set to (unphysical) $10^{-4}~{}$C/Kg. In accordance
with the simulated velocity and thus the effective source size the domain size
has been varied between 0.5 m and 2 m in transversely and between 1 m and 3 m
in longitudinally marked as $L_{tr}$ and $L_{z}$ in Fig. 8. Figure 9(a) shows
the output of the field monitors placed at $\theta=70\degree$ and distance of
$R=1$ m. The beam velocity corresponds to $\beta=0.15$ and the bunch length is
set to 100 ps $\sigma$. The domain parameters are $L_{tr}=1.25$ m and
$L_{z}=1.0$ m .$\vec{E_{x}}$, $\vec{E_{z}}$ as well as absolute field value is
shown. Figure 9(b) shows a comparison of the peak fields seen by a perfect
broadband monitor at 1 m distance obtained by CST simulations from 0 to 90°in
comparison to angular distribution from the NIR (Eq. LABEL:eq15). For NIR,
angular distributions of radiated electric field for 100 ps bunch is
calculated as follows,

$\displaystyle Ef(\theta,R)=\frac{\sum_{f=1}^{N}W_{f}\cdot
NIR(\theta,R,f)}{\sum_{f=1}^{N}W_{f}}$ (13)

where f depicts the frequency from 0.05 to 6 GHz with a resolution of 0.05
GHz. $NIR(\theta,R,f)$ can be obtained from Eq. LABEL:eq15 for a given angle,
distance and frequency. The weights $W_{f}$ are according to relative
amplitudes at the given frequency for $\sigma=100$ ps bunch, i.e. if
$g(t)=\exp{-\frac{t^{2}}{2\sigma^{2}}}$, and $W_{f}=DFT[g(t)]$ The absolute
estimates obtained for Eq. 13 is 30% lower than the peak values obtained in
CST. The peak values obtained in CST are scaled before plotting for easier
comparison of relative strengths. However the relative agreement between the
angular distributions obtained via CST and NIR is rather good.

(a)  (b)

Figure 9:  (a) Time domain snapshot of all three components of the transition
radiation electric field for beam velocity $v=\beta\cdot c$ where $\beta=0.15$
(b) Field amplitudes as a function of monitor angle obtained from CST in
comparison with NIR.

Figure 10 shows the absolute electric fields at 4 monitors placed at
incremental distances from the target at 45°and 80°angles for $\beta=0.6$. The
first peak at 1 ns is the diffraction radiation generated when the charge
exits the pipe. Following that, direct fields of the beam are seen until the
beam crosses the target at 4.5 ns when the transition radiation is generated
and detected at various monitors placed at incremental distances. The direct
field is stronger for monitors placed at 45°due to proximity to the traversing
charge. This proximity effect and direct field estimates are shown in the
appendix. It should be noted that at 85°the peak heights at different
distances closely follow the $1/r$ dependence while at 45°there is a slight
deviation. This is in line to the discussion in earlier sections concerning
the dependence of formation zone on detection angle $\theta$.

(a)  (b)

Figure 10: Absolute TR electric fields at 4 monitors placed at incremental
distances from the target at (a) 45°and (b) 80°angles for $\beta=0.6$.

###  4.1 ”Entry” radiation

Unlike in the most theoretical approaches where the charge usually comes from
infinity onto the target, there is a the production of ”entry” radiation when
charge enters computational domain in our PIC simulations. This is thus seen
as an artifact and complicates comparison between simulation and theory. The
production of this kind of bremsstrahlung on domain entry in terms of
amplitude and phase is expected to be dependent on the combination of mesh
cell size and velocity which was experienced in our preliminary simulations.
To prevent the propagation of the radiation generated at charge entry, the
bunch is introduced into the domain in a small pipe of length $250$ mm as
shown in Fig. 8. The bunch exiting the pipe produces diffraction radiation
which is still an unwanted artifact yet more stable w.r.t. mesh cell size and
beam velocity. The choice of the pipe diameter is a tradeoff, it should be
small such that the excitable waveguide modes are well above the set frequency
cutoff of the simulation, but if its too small, significant diffraction
radiation is generated which interferes with the transition radiation for
small angles $\theta$ with respect to beam axis. We use the pipe diameter to
30 mm in most cases except for one case where 100 mm was chosen to suppress
diffraction radiation.

###  4.2 Angular distribution at different beam velocities and comparison with
analytical estimates

Figure 11 shows the angular distribution of absolute value of E-fields
calculated using NIR for 100 ps bunch in the same way as discussed in previous
section (Eq. 13 and the peak field recorded on the CST broadband monitors. The
dimensions of the domain were made longitudinally large and transversally
asymmetric in order to have a reasonable calculation time, i.e. $L_{z}=3.2$ m,
while $L_{tr,x}=2.0$ m and $L_{tr,y}=0.6$ m. As we know from the first
section, the TR detected in horizontal plane is produced by the corresponding
vector currents in the same plane and the orthogonal components make no
contribution to each other. The entry pipe radius used in this case was chosen
to be of 100 mm diameter because the diffraction radiation at the exit of a
smaller pipe for higher betas $\beta=0.9,0.99$ was significantly distorting
the angular distribution at smaller angles. The disadvantage of larger pipes
is that waveguide modes (here predominantely $TE_{11}$ with $f_{cut}\approx
1.76$ GHz) are excited in the pipe and make it to the monitors which are seen
as wiggles in the angular distribution.

Figure 11: Comparison of analytical and simulation angular field distribution
for different betas at $R=1.0m$ distance.

(a)  (b)

Figure 12:  The angular distribution 0.3 GHz and 5.7 GHz for $\beta=0.99$ as a
function of monitor distance from the target. The target size is set to
1.4$\beta\gamma\lambda$ for the wavelength of $\lambda=1m$.

Figure 12 shows the comparison of the angular distribution for the NIR and CST
monitors at three distances for $\beta=0.9,0.99$ and the dependence is similar
as a function of monitor distance. The deviation for $\beta=0.99$ for smaller
angles $\theta<40\degree$ is primarily due to the influence of ”entry”
radiation.

###  4.3 Effect of transverse target and hole size on radiation pattern and
field intensity

(a)  (b)

Figure 13:  (a) Obtained radiation (Absolute values of E field) as a function
of target size for $\beta=0.6$ at monitor distance $R$ = 0.4 m at 45 degrees.
(b) Selected region to highlight the differences.

The transverse target size was varied from 0.3 m to 1.0 m (half width) for the
beam velocity of $\beta=0.6$. The target size was varied along with the
transverse domain size in order for the target to touch the open boundaries
and avoid the possibility of build up of a floating potential on the target.
The absolute value of field measured at a monitor placed and 40°and 0.4 m is
shown in Fig. 13 (a). Figure 13 (b) shows the radiation peak without the entry
radiation and the direct field parts for better visibility of the radiation
difference due to target size differences. Minor differences in the shape of
the measured bunch are seen, and it appears that some low frequencies cut off
for target size $b=0.3$ m which is inline with the expectation.

(a)  (b)

Figure 14:  (a) Obtained radiation (z component of E field) as a function of
hole size for $\beta=0.15$ at monitor distance $R$ = 1.0 m at 85 degrees for
an input of 100 ps $\sigma$ bunch for different hole sizes.(b) Zoomed out view
of the plot on left.

Similarly the size of the hole in the center of the target for the non-
intrusive passage of the beam is varied. Figure 14(a) shows the $z$ component
of the electric field observed at R= 1.0 m at 85 °for a 100 ps $(\sigma)$
bunch. The no-hole case is compared with the NIR estimate of 6 mm and 14 mm
hole (Same as presented in fig. 7) as well as CST calculation of 5 mm and 14.5
mm hole radius. NIR estimated temporal profile is obtained by convolving a 100
ps bunch with the frequency response at the given $\theta$ and $R$. There is a
good agreement between NIR and CST calculations and the charge distribution is
faithfully reproduced even for the largest hole size chosen.

##  5 First prototype and results

The GSI UNILAC facility provides wide range of ions from 1.4 MeV/u to 11.4
MeV/u. The expected bunch lengths range from 100-500 ps ($\sigma$) with a
repetition rate of 36 MHz or 108 MHz depending on the injector used. The
currents are in the order of 50 uA to few mA.

(a)

(b)

Figure 15: (a) The Simulation model of the set-up. (b) Photo of the first
prototype set-up.

The main consideration of the TR set-up design is the dimensions and material
of the RF window, transition section between the beam pipe and the glass, as
well as target size and hole size. On the detector side, the Antenna gain and
phase response are important to evaluate.

###  5.1 Set-up details and simulations

Figure 15(a) shows the photo of the first prototype of the set-up at the end
of X2 experiment cave beamline at GSI. The last quadrupole was about 3m
upstream, current measurement device was 1 m upstream and profile grids 1.5 m
upstream. We chose the largest fused silica (glass) available off-the-shelf as
the RF window with some custom connectors for a vacuum tight assembly with the
beam pipe and the TR target. A Tantalum target (GTR target) with 3 mm hole in
the center was used as target plate. A sensitive Faraday cup was installed
just behind the hole to ensure that the beam was hitting the target since the
alignment of the beam was tricky due to a small flange between beam-pipe and
RF window. Figure 15(b) shows CST model of the proposed schematic for a set-
up. Fig. 16 (a) shows the $\vec{E_{z}}$ component detected at the monitor for
a 100 ps bunch, with $\beta=0.15$ at 1.0 m distance. Direct field is observed
between 2 and 9 ns ending with the TR field peaking at 9.4 ns. The enlarged
plot around the TR radiation in Fig. 16(b) and the TR reflection from the
glass chamber is also seen at 10 ns.

(a)  (b)

Figure 16:  (a) CST simulation output showing $\vec{E_{z}}$ component at the
broadband monitor of the model shown in Fig. 15 (b) for 100 ps Gaussian charge
distribution traversing with velocity $\beta=0.15$. (b) Enlarged section
around the bunch signal.

The detector for the generated pulse was a Biconical antenna [18]. Relevant
characteristics of the antenna were measured as shown in the Appendix. A ring
pick-up similar to shown in Figure 5.7 in [23] is installed 2m upstream of the
GTR target which was used to trigger the digitizer. The pick-up signal and the
Antenna output were connected to the same 80 GSa/s digitizer with an analog
bandwidth of 20 GHz. Full macropulse of 100 $\mu$s signal was recorded for
many successive pulses.

###  5.2 Pick-up vs GTR measurement

Figure 17(a) shows two consecutive single shot bunches measured by the pick-up
and the GTR antenna. The pick-up signal is divided by 10 for fitting on the
same y-axis and time (x-axis) is translated by 42.5 ns to account for the
particle traversal from pick-up to GTR target along with the signal
propagation from GTR target to the antenna. Figure 17(b) shows the zoomed view
of the second bunch and an additional structure on the pulse which is clearly
visible on the GTR signal which is also hinted on pick up signal (but not
resolved). The signals measured are about factor 5 smaller than expected from
analytical estimates. Our suspicion in hindsight is that; most of the beam was
lost at the boundary of beam pipe and RF window.

(a)  (b)

Figure 17:  (a) Two consecutive bunches measured by a pick-up and GHz
transition radiation monitor spaced apart by 2 m (b) Enlarged view focussing
on the second bunch.

###  5.3 Shot-to-shot fluctuations in charge distribution

Figure 18: Bunch length measurement for three consecutive macropulses by GTR
(top row) and Pick-up (bottom row).

Figure 18 shows three consecutive macropulse images from GTR (top row) and BPM
(bottom row). The x-axis is a region selected within the rf period where the
pulses are present while on the y-axis the bunch length evolution along the
macropulse is shown. There seems to be slight energy shift as well as
significant shot-to-shot bunch shape variation. This is currently attributed
to the beam loading in the LINAC and the detailed and precise cause is under
investigation and outside the scope of this report. These measurements also
highlight that an averaged measurement for such pulses could result into
spurious measurements and bunch-by-bunch shot-by-shot non-destructive bunch
length measurement could be important for high intensity LINAC optimization.

##  6 Summary

We have presented a concept for time domain monitoring for longitudinal charge
profiles for non relativistic beams based on coherent transition and
diffraction radiation in GHz regime. We have extended the quasi spherical
approximation presented in [6] to an exact expression in near field for
normally incident beam. Signal estimates for typical beam intensities using
commercially available detectors are provided. The derived near field
expressions were compared with CST simulations and a good agreement is found.
Finally, first prototype measurements were performed which confirm the signal
estimates and proves applicability of this method for non destructive
longitudinal charge profile monitoring. This method forms a building block for
a compact 6D phase space measurement system when combined with non-destructive
transform profile monitoring using transition radiation in optical regime
[21]. Although we have focussed on non relativistic bunch charge distribution
in this case, variants of this technique could also be helpful in storage
rings for very small bunches i.e. where the pick-up bandwidth pose
limitations.

##  7 Acknowledgment

B. Walasek-Hoehne is gratefully acknowledged for the support and discussions.
P. Forck and C. Krueger are acknowledged for providing support in data
acquistion part. M. Mueller, A. Abou Abed are acknowledged for help with the
GTR mechanical set-up.

##  8 Appendix

###  8.1 Direct field estimates on the monitor

Vacuum Conductor $\vec{\beta}c$ $q$ $\vec{r}$ $\vec{R}$ $M$ $\theta$ $\hat{x}$
$\hat{z}$ Figure 19: Schematic of a charge passing by a monitor towards a
target where it is normally incident.

The direct field of the charge travelling at a velocity $\beta$ as shown in
Fig. 19 is given as[22],

$\displaystyle E_{direct}=\frac{q}{4\pi\epsilon_{0}}\cdot\frac{\gamma
R\hat{e}_{x}+\gamma\beta ct\hat{e}_{z}}{\Big{[}R^{2}+(\gamma\beta
ct)^{2}\Big{]}^{3/2}}$ (14)

where $t=0$ when the charge has smallest Euclidean distance to the monitor
$R$. The peaks for $E_{direct,x},E_{direct,z}$ occurs at $t=0,\pm
R/\sqrt{2}\gamma\beta c$ respectively.

The charge is incident normally at the target and monitor distance is $r$ as
shown in Fig. 19, while $R=r\sin{\theta}$. The transition radiation field per
unit frequency originating at the target is given by Eq. LABEL:eq15. In the
far field, Eq. 15 reduces to a simpler result [14].

$\displaystyle\frac{dE_{TR}}{d\omega}=\frac{q\beta}{2\pi\epsilon_{0}rc}\cdot\frac{\sin{\theta}\cos{\theta}\hat{e}_{x}+\sin^{2}{\theta}\hat{e}_{z}}{1-\beta^{2}\cos^{2}{\theta}}\cdot\delta(t-(r\cos{\theta})/\beta
c-r/c)$ (15)

The ratio of peaks $E_{TR}/E_{direct}$ and time of peak field as a function of
$r$ and $\theta$ register at the monitor $M$ is shown in Fig. 20.

Figure 20: Ratio of direct field against transition radiation field as a
function of monitor distance from target for two monitor angles.

Since our frequencies of interest are far below the plasma frequency, $E_{TR}$
can be simply obtained by integrating over the frequency range of interest,
inversely related to the bunch length. For $\sigma_{t}=100$ ps bunch length,
$d\omega=1/2\pi\sigma_{t}$ and

$\displaystyle\frac{E_{direct,x}}{E_{TR,x}}=\frac{\pi\gamma
c\sigma_{t}(1-\beta^{2}\cos^{2}{\theta})}{\beta
r\sin^{3}{\theta}\cos{\theta}}$ (16)
$\displaystyle\frac{E_{direct,z}}{E_{TR,z}}=\frac{\pi\gamma
c\sigma_{t}(1-\beta^{2}\cos^{2}{\theta})}{\sqrt{2}\cdot 1.5^{1.5}\beta\cdot
r\sin^{4}{\theta}}$ (17)

and shortest time between the peaks of both components;

$\displaystyle{t_{direct,x}}-{t_{TR,x}}$ $\displaystyle=(r\cos{\theta})/\beta
c-r/c$ (18) $\displaystyle t_{direct,z}-t_{TR,z}$
$\displaystyle=(r\cos{\theta})/\beta c-r/c+(R/(\sqrt{2}\gamma\beta c))$ (19)

###  8.2 Measured frequency response of Bikon Antenna

Fig. 21 shows the frequency response measured of the Biconical antenna used
for these measurements. The measurements were not performed in an anechoic
chamber. Further characteristics of the antenna can be found here [18].

(a)  (b)

Figure 21:  (a) Magnitude response obtained using S11 measurement. (b) Phase
response.

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