si erf

Also at ]Technische Universität Darmstadt, Schloßgartenstr. 8, 64289
Darmstadt, Germany.

# Pickup concepts for ultra-low charged short bunches in X-Ray Free-Electron
Lasers

B. E. J. Scheible<EMAIL_ADDRESS>[  Technische Hochschule
Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany.  M. K.
Czwalinna  H. Schlarb  Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85,
22607 Hamburg, Germany.  W. Ackermann  H. De Gersem  Technische Universität
Darmstadt, Schloßgartenstr. 8, 64289 Darmstadt, Germany.  S. Mattiello  A.
Penirschke  Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13,
61169 Friedberg, Germany.

###### Abstract

The all-optical synchronization systems used in various X-ray free-electron
lasers (XFEL) such as the European XFEL observe the transient fields of
passing electron bunches coupled into one or more pickups in the Bunch Arrival
Time Monitors (BAM). The extracted signal is then amplitude modulated on
reference laser pulses in a Mach-Zehnder type electro-optical modulator. With
the emerging demand for future experiments with ultra-short FEL shots, fs
precision is required for the synchronization systems even with
$1\text{\,}\mathrm{pC}$ bunches. Since the sensitivity of the BAM depends in
particular on the slope of the bipolar signal at the zero-crossing and thus,
also on the bunch charge, a redesign with the aim of a significant increase by
optimized geometry and bandwidth is inevitable. In this contribution the
theoretical foundations of the pickup signal are aggregated and treated with a
focus on ultra-short bunches as well as a general formulation. A possible new
pickup concept is simulated and its performance is compared to the previous
concept. A significant improvement of slope and voltage is found. The
improvement is mainly achieved by the reduced distance to the beam and a
higher bandwidth.

##  I Introduction

Free-electron lasers (FEL) became an important light source for experiments in
various fields since they provide ultra-short pulses with extreme brilliance
in atomic length and time scales [1]. FEL are well suited for applications in
pump-probe experiments [1], where the timing jitter is specifically critical
[2], as well as for capturing image sequences with atomic resolution on
$\mathrm{fs}$-time-scales, even below the FEL repetition rate [1, 3, 4].

For the generation of ultra-short X-ray pulses, FEL with short and ultra-low
charge electron bunches ($\leq$$1\text{\,}\mathrm{pC}$) have been found as a
favorable option [5, 6]. Short bunches may shorten the X-ray pulse, reduce
timing jitter and lead to single-spike operation, if sufficiently small
compared to the cooperation length of the SASE process [2, 5, 6]. The European
XFEL (EuXFEL) was upgraded from initially $1\text{\,}\mathrm{nC}$ electron
bunches to cover a range from $0.02\text{\,}\mathrm{nC}1\text{\,}\mathrm{nC}$
[7] with a possible bunch length below $3\text{\,}\mathrm{fs}$ in the
undulator section [8]. Moreover, a decrease to ultra-low charges of
$1\text{\,}\mathrm{pC}$ is targeted.

The application in time-resolved experiments entails tight requirements for
the overall machine synchronization in order to reduce the timing jitter [2].
The timing information is also used for post-processing experimental data [1].
The synchronization concerns all critical subsystems, specifically in the
injector and if present the seeding and the pump laser [1]. Furthermore, the
instrumentation must be suited for a broad spectrum of operation modes with
different bunch properties even in a single bunch train [9, 10]. Besides,
bunch arrival time monitors (BAM) are installed throughout the whole facility,
thus experiencing different bunch properties.

A tremendous improvement in synchronization, exceeding RF techniques, and
reduction of arrival time as well as energy jitter was achieved by the
implementation of an all-optical synchronization system with two different
feedback loops [11]. Though some updates have been introduced [12, 13], the
basic scheme in use by the Deutsches Elektronen-Synchrotron (DESY) remained
unchanged.

In this contribution the all-optical synchronization system will be introduced
with special attention on the state-of-the-art cone-shaped pickups. The
schematic BAM description is followed by an in-depth analytical discussion of
the voltage signal, regarding different limiting cases as well as a general
solution, to identify principal parameters determining the BAM resolution and
their effect on the signal shape. For design purposes an additional numerical
treatment is introduced. These considerations are the basis for three designs,
which are presented at the end of this paper.

##  II All-optical Synchronization System

The all-optical synchronization system, as successfully tested at the Free-
Electron Laser in Hamburg (FLASH) by Löhl [11], mainly comprises of a mode-
locked reference laser, length stabilized fiber links and different end-
stations for synchronization and arrival time measurement [14, 11, 12, 13, 10,
15]. The arrival time is non-destructively measured with respect to the
reference laser in the BAM, which include high-bandwidth pickup electrodes in
the RF unit, a Mach-Zehnder type electro-optical modulator (EOM) and the data
acquisition system (DAQ) [10, 15].

###  II.1 Basic working principle of BAM

The transient electric fields of passing electron bunches are extracted in the
RF unit and, if foreseen, initially processed with analogue components like RF
combiners, limiters or attenuators [11, 12]. The received bipolar signal is
transmitted over radiation hard silicon dioxide coaxial cables to the EOM
[12], there it is probed at its zero-crossing by the reference laser [11]. Any
temporal deviation will lead to an additional voltage, which the EOM turns
into an amplitude modulation of the laser pulse. Therefore, the laser
amplitude holds the timing information, which can be retrieved in the DAQ
[11]. The signal slope at its zero-crossing strongly influences the BAM’s
temporal resolution. The minimum design requirement for the currently
installed BAM was $\geq$300\text{\,}\mathrm{mV}\text{\,}{\mathrm{ps}}^{-1}$$
with $20\text{\,}\mathrm{pC}$ bunches [12].

###  II.2 RF unit and pickups

The RF unit so far comprises four identical pickups mounted circularly around
the beam line. The combination of opposite pickup signals compensates for the
orbit dependency [16]. The original pickups used at FLASH were of button-type
and designed for a $10\text{\,}\mathrm{GHz}$ bandwidth [17]. This pickup
struggled with ringing and strong signal reflections at the alumina vacuum
feedthrough [17] degrading the signal strength and resolution for charges
below $150\text{\,}\mathrm{pC}$ [18]. Hence, a new design was required [18].

A novel pickup (Fig. 1, left), similar to those designed for the CERN linear
collider test facility [19], was proposed as a solution in FEL applications
[20]. The cone-shaped design, finalized in [12], with
$40\text{\,}\mathrm{GHz}$ bandwidth became a new standard device used at the
EuXFEL and at other FELs.

Figure 1: Cross section of the 1st generation cone-shaped pickup (left) and
the modified 2nd generation cone-shaped pickup for high peak-to-peak voltage
(right) adapted from [13].

Due to high losses in the RF path a design update was necessary (Fig. 1,
right). The second generation of cone shaped pickups were optimized towards a
maximum signal voltage at the cost of its slope by increasing the active
surface and letting the cone slightly protrude into the beam pipe [13]. A
limit was reached when a larger surface, e.g. more than a few bunch lengths,
further decreases the slope without a further increase in peak-to-peak voltage
[13]. The protrusion increases the inductance, which unfavorably deforms the
signal shape [13]. Nonetheless, a combination of both modifications increased
the peak-to-peak voltage sufficiently for $20\text{\,}\mathrm{pC}$ bunches
while maintaining an acceptable slope [13].

Recently the performance of the state-of-the-art system was evaluated at the
EuXFEL. The correlation of two adjacent monitors with less than
$1\text{\,}\mathrm{m}$ distance was analyzed. Examining the measured
arrivaltimes for a period of $1\text{\,}\mathrm{min}$ gave a timing jitter of
approximately $6\text{\,}\mathrm{fs}$ r.m.s. caused by the BAM resolution and
critical parts of the reference laser distribution system [21].

##  III RF signal

The limitations of current pickup structures are evident in the theoretical
consideration of the time domain voltage signal. The image charge on the
pickup surface is calculated by

$Q_{\mathrm{im}}\left(t\right)=\int_{-\infty}^{\infty}\frac{\lambda(z-c_{\mathrm{0}}t)w(z)}{2\pi
r_{\mathrm{p}}}\mathrm{d}z,$ (1)

with the line charge density $\lambda(t)$ of the ultra-relativistic bunch
($v\approx c_{0}$), pickup width $w(z)$ and the distance between pickup
surface and bunch $r_{\mathrm{p}}$ [22]. Because line charge density and
pickup width are real quantities, by substitution of $z$ into a moving frame,
Eq. eq:PUimageCharges can be transformed into the cross-correlation

$Q_{\mathrm{im}}\left(t\right)=\left(\lambda\star w\right)(t),$ (2)

which is readily solved numerically. This is true for any bunch and pickup
form. In the special case of an even charge density, Eq. eq:PUimageCharges is
the convolution

$Q_{\mathrm{im}}\left(t\right)=\left(\lambda^{\mathrm{even}}\circledast
w\right)(t).$ (3)

Eqs. eq:PUimageCharges-Corr and eq:PUimageCharges-Conv can be utilized in the
numerical treatment and also for analytical solutions, which are accessible
for many different cases. Some essential examples are covered in this section
followed by the numerical approach.

###  III.1 Analytical Model

Idealized representations of bunch and pickup surface, allow to find an
analytical solution for Eq. eq:PUimageCharges. A suitable representation that
leads to a substantial simplification is given by a Gaussian bunch and a
rectangular pickup surface. The stationary charge density of the Gaussian beam
centralized on the $z$-axis when is

$\rho(x,y,z)=\delta(x)\delta(y)\lambda(z).$ (4)

The line charge density is found by integration of the transverse area. A
Gaussian bunch is given by

$\lambda(z)=\lambda_{0}\exp\left(-\frac{z^{2}}{2\sigma_{\mathrm{z}}^{2}}\right)=Q_{\mathrm{b}}\
G(z,\sigma_{\mathrm{z}}),$ (5)

where $\lambda_{0}=Q_{\mathrm{b}}/\sqrt{2\pi\sigma_{\mathrm{z}}^{2}}$ is a
constant for normalization by the bunch charge, $Q_{\mathrm{b}}$ the total
charge of one bunch, $\sigma_{\mathrm{z}}$ the r.m.s. length and $G(\tau)$ the
normalized Gaussian distribution function

$G(\tau,\sigma_{\tau})=\frac{1}{\sqrt{2\pi}\sigma_{\tau}}\exp\left(-\frac{\tau^{2}}{2\sigma_{\tau}^{2}}\right).$
(6)

In the scope of this work, we denote

$\lambda(t):=\left.\lambda(z)\right|_{z=c_{0}t}=\frac{Q_{\mathrm{b}}}{c_{0}}\
G(t,\sigma_{\mathrm{t}}),$ (7)

using Eq. eq:GaussianBunchProfile, where the speed of light is incorporated
into the r.m.s length in units of time

$\sigma_{\mathrm{t}}=\frac{\sigma_{\mathrm{z}}}{c_{0}}.$ (8)

The rectangular pickup surface width is written as

$w(z)=w_{0}\ \Pi_{\ell}\left(z\right),$ (9)

where $\ell$ is the longitudinal extension of the pickup and

$\Pi_{\ell}(z)=\cases{1}&\left|z\right|\leq\frac{1}{2}\ell\\\
0\left|z\right|>\frac{1}{2}\ell$ (10)

the rectangular function.

####  III.1.1 Long Bunch Approximation

In many applications, specifically with hadrons, it is reasonable to assume a
bunch much longer than the pickup. This is treated in the long bunch
approximation (LBA). The pickup form is insignificant in the LBA, hence a more
general approach is used. For this purpose the parameterization of a finite
pickup profile is

$w\left(z\right)=\tilde{w}\left(z\right)\chi_{\left[a,b\right]}(z),$ (11)

where $\chi_{\left[a,b\right]}\left(z\right)$ is the indicator function of the
interval ${\left[a,b\right]}$, which bounds the pickup in longitudinal
direction, and $\tilde{w}\left(z\right)$ defines the pickup width at location
$z$. The rectangular pickup in Eq. eq:RectangularPickup is a special case of
Eq. eq:IndicatorPickup. The longitudinal characteristic dimension of the
pickup is defined by $\ell=\left|a-b\right|$ and its surface area
$A_{\mathrm{p}}$ by the integral

$A_{\mathrm{p}}=\int_{-\infty}^{\infty}\tilde{w}\left(z\right)\chi_{\left[a,b\right]}(z)\mathrm{d}z=\int_{a}^{b}\tilde{w}\left(z\right)\mathrm{d}z.$
(12)

The image charge found by introducing Eq. eq:IndicatorPickup in Eq.
eq:PUimageCharges is

$Q_{\mathrm{im}}\left(t\right)=\frac{1}{2\pi
r_{\mathrm{p}}}\int_{-\infty}^{\infty}\lambda(z-c_{\mathrm{0}}t)\tilde{w}\left(z\right)\chi_{\left[a,b\right]}(z)\mathrm{d}z$
(13)

which becomes

$Q_{\mathrm{im}}\left(t\right)=\frac{\lambda_{\mathrm{c}}}{2\pi
r_{\mathrm{p}}}\int\displaylimits_{a-c_{\mathrm{0}}t}^{b-c_{\mathrm{0}}t}\tilde{\lambda}(\zeta)\tilde{w}\left(\zeta+c_{\mathrm{0}}t\right)\mathrm{d}\zeta$
(14)

by substitution of $\zeta=z-c_{0}t$ and factorizing
$\lambda\left(\zeta\right)$ as the product of the constant term
$\lambda_{\mathrm{c}}$, which has the units of a line charge density, and a
dimensionless longitudinal distribution function $\tilde{\lambda}(\zeta)$.
Despite the explicit dependence of the bunch form, the following derivation of
the long bunch approximation only requires that the distribution
$\tilde{\lambda}(z)$ can be written as

$\mathrm{T}\tilde{\lambda}(z;c_{0}t)=\tilde{\lambda}(c_{0}t)+\tilde{\lambda}^{\prime}(c_{0}t)(z-c_{0}t)+...\
,$ (15)

$\forall z\in\mathbb{R}$ apart to a at most countable set of points, where
$\mathrm{T}\tilde{\lambda}(z;c_{0}t)$ is the Taylor expansion of
$\tilde{\lambda}(z)$ at $c_{0}t$. The image charge can be written as an
expansion as well,

$Q_{\mathrm{im}}\left(t\right)=\frac{\lambda_{\mathrm{c}}}{2\pi
r_{\mathrm{p}}}\ \sum_{n=0}^{\infty}q_{n}(c_{0}t),$ (16)

with

$q_{n}(c_{0}t)=\tilde{\lambda}^{(n)}(c_{0}t)\int\displaylimits_{a-c_{\mathrm{0}}t}^{b-c_{\mathrm{0}}t}\tilde{w}\left(\zeta+c_{\mathrm{0}}t\right)(\zeta-
c_{0}t)^{n}\mathrm{d}\zeta.$ (17)

For a long bunch, or equivalently a short pickup, it is justified to consider
only the leading term, with $n=0$, if the reminder term is negligible, which
implies the function to be constant in the rolling interval of length $\ell$.
This is applicable if the following limit is true

$\left|R_{0}\right|=\left|\tilde{\lambda}(z)-\tilde{\lambda}(c_{0}t)\right|\rightarrow
0\ \ \ \forall z\in\left[c_{0}t-a,c_{0}t+b\right]$ (18)

and justified for any given function at some point for
$\left|a-b\right|\rightarrow 0$. Therefore, it is useful to introduce a
scaling quantity $\mu$ for the bunch-pickup system, which controls the
approximation for any Lipschitz continuous function. For this purpose because
the derivative is bounded by
$M_{1}=\mathrm{sup}\left\\{\left|\tilde{\lambda}^{\prime}(t)\right|:t\in\mathbb{R}\right\\}$,
that is the Lipschitz constant of the distribution $\tilde{\lambda}(t)$, the
following inequality gives an upper limit for the reminder $R_{0}$

$\tilde{\lambda}(z)-\tilde{\lambda}(c_{0}t)\leq M_{1}\left|z-c_{0}t\right|\leq
M_{1}\left|b-a\right|.$ (19)

Therefore, the scaling is defined as

$\mu=M_{1}\left|b-a\right|=M_{1}\ell$ (20)

and the requirement in Eq. Eq:ReqLongBunchApprox is fullfilled for

$\mu\ll 1\ .$ (21)

Applied on a Gaussian bunch according to Eq. eq:GaussianBunchProfile it is

$\mu=\frac{1}{\sqrt{e}}\frac{l}{\sigma_{\mathrm{z}}}\ll 1\ .$ (22)

Under this assumption, the image charge is $q_{0}$ and thus

$Q_{\mathrm{im}}^{\mathrm{LBA}}\left(t\right)=\frac{\lambda(t)}{2\pi
r_{\mathrm{p}}}\int_{a}^{b}\tilde{w}\left(z\right)\mathrm{d}z\ ,$ (23)

which by Eq. Eq:PickupSurface-General is

$Q_{\mathrm{im}}^{\mathrm{LBA}}\left(t\right)=\frac{A_{\mathrm{p}}}{2\pi
r_{\mathrm{p}}}\lambda(t).$ (24)

This approximation is widespread and for example found in [23, 22]. Equation
eq:LongBunchApprox is valid for any pickup which is short enough to neglect
the change of line charge density along the pickup’s length, which is ensured
by requirement Eq:ReqLongBunchApprox, independent of pickup and bunch form.
The validity of this situation, i.e. of the long-bunch approximation, is
qualitatively controlled by condition Eq:ScalingConditionLBApprox. If this
condition is satisfied, it is ensured that the requirement
Eq:ReqLongBunchApprox is also met. Nonetheless Eq. Eq:ScalingConditionLBApprox
is a stricter condition and may be untrue for some functions that still are
sufficiently described by the LBA. These requirements are met in many
facilities, thus the theory is well developed and calculations for the image
current and output voltage discussed in various publications on beam
instrumentation and in particular beam position monitors as [23].

####  III.1.2 Short Bunch Approximation

Figure 2: Voltage signal in the SBA for two idealized filters. A rectangular
(left) and a Gaussian filter (right) have been utilized, each at
$100\text{\,}\mathrm{GHz}$ (blue) and $40\text{\,}\mathrm{GHz}$ (red) cut-off
frequency. Bunch and geometry parameters are in a reasonable range for XFEL
applications, e.g. $\sigma_{\mathrm{t}}=$200\text{\,}\mathrm{fs}$$,
$Q_{\textrm{b}}=$20\text{\,}\mathrm{pC}$$,
$\ell=w_{0}=$4.8\text{\,}\mathrm{mm}$$,
$r_{\mathrm{p}}=$20\text{\,}\mathrm{mm}$$ and
$R=$50\text{\,}\mathrm{\SIUnitSymbolOhm}$$.

In FEL applications as discussed in this work, the requirement of the long
bunch approximation $\mu\ll 1$ is not fulfilled. Instead the bunch length is
typically below $200\text{\,}\mathrm{fs}$ down to single digit $\mathrm{fs}$
and a standard pickup is in the range of some $\mathrm{mm}$, corresponding to
$\geq$3.3\text{\,}\mathrm{ps}$$. With the bunch more then one sometimes up to
three orders of magnitude shorter it is worthwhile to investigate the limiting
case of ultra short bunches, i.e.

$\mu\gg 1\ \Leftrightarrow\sigma_{\mathrm{z}}\ll\ell\ .$ (25)

Under this assumption in the short bunch approximation (SBA), because the
normalized Gaussian is a Dirac sequence, Eq. eq:GaussianBunchProfile is
described by a delta distribution with the total charge $Q_{\mathrm{b}}$
concentrated at one point in space

$\lim\limits_{\sigma_{\mathrm{t}}\rightarrow
0}\lambda(z-c_{\mathrm{0}}t)=Q_{\mathrm{b}}\ \delta\left(z-c_{0}t\right).$
(26)

Since the delta distribution is even, Eq. eq:PUimageCharges is treated as a
convolution, leading straightforward to

$Q_{\mathrm{im}}^{\mathrm{SBA}}\left(t\right)=\frac{Q_{\mathrm{b}}w_{0}}{2\pi
r_{\mathrm{p}}}\ \Pi_{2t_{0}}\left(t\right),$ (27)

where

$t_{0}=\frac{\ell}{2c_{0}}$ (28)

is the time it takes for the bunch center to pass half the pickup. The image
current found by differentiation of ${Q}_{\mathrm{im}}^{\mathrm{SBA}}(t)$ is
given by the following distribution

$I_{\mathrm{im}}^{\mathrm{SBA}}\left(t\right)=\frac{Q_{\mathrm{b}}w_{0}}{2\pi
r_{\mathrm{p}}}\left[\delta\left(t+t_{0}\right)-\delta\left(t-t_{0}\right)\right].$
(29)

The output voltage is simply derived for any given transfer function
$H\left(\omega\right)$ with corresponding impulse response function
$h\left(t\right)$ by the convolution with Eq. eq:ultraShortBunchCurrent

$U_{\mathrm{h}}^{\mathrm{SBA}}\left(t\right)=\left(h\circledast
I_{\mathrm{im}}\right)\left(t\right).$ (30)

Therefore, the voltage signal in time domain is

$U_{\mathrm{h}}^{\mathrm{SBA}}\left(t\right)=\frac{Q_{\mathrm{b}}w_{0}}{2\pi
r_{\mathrm{p}}}\left[h\left(t+t_{0}\right)-h\left(t-t_{0}\right)\right].$ (31)

The preceding derivation of the short bunch limit is possible for all line
charge density distributions in L1, thus with finite total bunch charge
$Q_{\mathrm{b}}$. The resulting voltage shape
$U_{\mathrm{h}}^{\mathrm{s}}\left(t\right)$ is certainly independent of the
infinitesimal short bunch form, but defined by the subtraction of the scaled
impulse response function $h\left(t\right)$ shifted parallel in different
directions.

In the following two idealized transfer functions are used to illustrate the
implications of the ultra-short bunch. The two applied low-pass filters are a
rectangular filter defined by

$H\left(\omega\right)=R\ \Pi_{2\Omega_{\mathrm{c}}}\left(\omega\right)$ (32)

and a Gaussian filter defined by

$H\left(\omega\right)=R\exp\left(-\frac{\ln(2)\omega^{2}}{2\Omega_{\mathrm{c}}^{2}}\right).$
(33)

The cut-off frequency is $\Omega_{\mathrm{c}}$ and the filters are terminated
with R, usually $50\text{\,}\mathrm{\SIUnitSymbolOhm}$. The corresponding time
domain response functions are

$h_{\Pi}(t)=\frac{R\Omega_{\mathrm{c}}}{\pi}\sinci\left(\Omega_{\mathrm{c}}t\right)$
(34)

and

$h_{\mathrm{G}}(t)=\frac{R\widetilde{\Omega}_{\mathrm{c}}}{\sqrt{2\pi}}\exp\left(-\frac{\widetilde{\Omega}_{\mathrm{c}}^{2}}{2}t^{2}\right),$
(35)

where $\widetilde{\Omega}_{\mathrm{c}}=\Omega_{\mathrm{c}}/\sqrt{\ln(2)}$ is
the cut-off frequency defined by a $1/e$ decrease. The voltage signals
$U_{\mathrm{\Pi}}^{\mathrm{s}}\left(t\right)$ respectively
$U_{\mathrm{G}}^{\mathrm{s}}\left(t\right)$ are found by substitution of the
response function in Eq. Eq:VoltageSignal-ArbFilter-ShortBunch. The resulting
curves are shown in Fig. 2 assuming dimensions in the range of existing bunch
arrival time monitors. The characteristic curve, resembling the derivative of
a Gaussian, is found at the center of both examples. An ideal rectangular
filter induces an infinite ripple, which in some cases also reflects on the
steep signal slope at the center, while the Gaussian is smooth.

The signal slope at the zero-crossing

$S_{\mathrm{h,ZC}}=\dot{U}_{\mathrm{h}}\left(t_{\mathrm{ZC}}\right)$ (36)

is the figure of merit in BAM applications. The significant ZC is located at
$t=0$ for both example filters. The respective slopes are

$S_{\mathrm{\Pi,ZC}}^{\mathrm{SBA}}=-\frac{Rw_{0}}{\pi^{2}r_{\mathrm{p}}}Q_{\mathrm{b}}\frac{\Omega_{\mathrm{c}}}{t_{0}}\left[\cos\left(\Omega_{\mathrm{c}}t_{0}\right)-\sinci\left(\Omega_{\mathrm{c}}t_{0}\right)\right]$
(37)

and

$S_{\mathrm{G,ZC}}^{\mathrm{SBA}}=-\frac{RA_{\mathrm{p}}}{2\pi
r_{\mathrm{p}}}\frac{Q_{\mathrm{b}}}{\sqrt{2\pi}}\frac{\widetilde{\Omega}_{\mathrm{c}}^{3}}{c_{0}}\left[\exp\left(-\frac{1}{2}\widetilde{\Omega}_{\mathrm{c}}^{2}t_{0}^{2}\right)\right].$
(38)

For the rectangular filter the slope is depending on the ratio of cut-off
frequency and pickup length, with a periodical behavior caused by the bracket
term. This term has local extrema of approximately $\pm 1$. For simplicity it
is sufficient to assume exactly $\pm 1$, found at
$\Omega_{\mathrm{c}}t_{0}=n\pi$, with $n\in\mathbb{N}$, although the real
maximum is located somewhat before that.

For the Gaussian filter the slope is also depending on the ratio of cut-off
frequency and pickup length. For an infinite as well as infinitesimal pickup
the slope is vanishing. The maximum slope is reached at
$\widetilde{\Omega}_{\mathrm{c}}=2c_{0}/\ell$. For a lower cut-off frequency
the signal width is increased, leading to distortion and partial annihilation
of the signal. If the bandwidth is increased further, two distinctive peaks
become visible leading to a lower gradient at zero-crossing. The optimum is

$S_{\mathrm{G,max}}^{\mathrm{SBA}}=-\frac{1}{\sqrt{e}}\frac{RA_{\mathrm{p}}}{2\pi
r_{\mathrm{p}}}\frac{Q_{\mathrm{b}}}{\sqrt{2\pi}}\frac{1}{c_{0}}\widetilde{\Omega}_{\mathrm{c}}^{3}.$
(39)

This equals the maximum slope received by the short pickup and high cut-off
frequency approximation, apart from the factor $1/\sqrt{e}$ and the
replacement of the reciprocal bunch width by the cut-off frequency, which are
the corresponding decisive widths.

Figure 3: Image charge by a Gaussian bunch on a rectangular pickup surface
according to the LBA in Eq. eq:LongBunchApprox (dashed red), the SBA in Eq.
Eq:ShortBunch-Q-im (dashed green) and the analytical solution in Eq.
Eq:General-Q-im (solid blue) for two different bunch lengths. One close to the
LBA regime, with $\mu\approx$0.65$$ (left) and one closer to the SBA regime,
with $\mu\approx$16.2$$ (right). Bunch and geometry parameters are in a
reasonable range for applications of button pickups, e.g.
$\sigma_{\mathrm{t}}=$0.6\text{\,}\mathrm{ps}$$ respectively
$\sigma_{\mathrm{t}}=$15\text{\,}\mathrm{ps}$$,
$Q_{\textrm{b}}=$20\text{\,}\mathrm{pC}$$,
$\ell=w_{0}=$4.8\text{\,}\mathrm{mm}$$ and
$r_{\mathrm{p}}=$20\text{\,}\mathrm{mm}$$. In the EuXFEL even shorter bunches
occur, further reducing the discrepancy between SBA and analytical solution at
the edges.

The real transfer function is neither rectangular nor Gaussian. In literature
the pickup is usually modeled by an equivalent circuit of a resistor R
parallel to the capacity C, with the image current $I_{\mathrm{im}}$ serving
as a current source [23, 24]. The expected low-pass filter then gives

$H(\omega)=\frac{R}{1+i\omega RC},$ (40)

with cut-off frequency $\Omega_{\mathrm{RC}}=1/\left(RC\right)$, as transfer
function from current source $I_{\mathrm{im}}$ to the output voltage
$U_{\mathrm{RC}}$. In time domain the impulse response function is

$h(t)=\frac{1}{C}\exp\left(-\frac{t}{RC}\right)\Theta\left(t\right),$ (41)

where $\Theta\left(t\right)$ denotes the Heaviside step function.

In order to calculate the output voltage with the equivalent circuit’s
transfer function the convolution with the image current in short bunch
approximation according to Eq. Eq:VoltageSignal-ArbFilter-ShortBunch gives

$\aligned
U_{\mathrm{RC}}^{\mathrm{SBA}}(t)=\frac{1}{C}\frac{Q_{\mathrm{b}}w_{0}}{2\pi
r_{\mathrm{p}}}\exp\left(-\frac{t+t_{0}}{RC}\right)\\\
\times\left[\Theta\left(t+t_{0}\right)-\exp\left(+\frac{2t_{0}}{RC}\right)\Theta\left(t-t_{0}\right)\right],$
(42)

which corresponds to the capacitor experiencing the complete electric field of
the infinitesimal bunch at once, leading to a compensating current, which is
equivalent to the discharging of a capacitor. The decay curve is observed
until the coasting beam reaches the end of the pickup. Than the effect of the
bunch’s electric field on the pickup suddenly ends and the initial voltage
jump is reversed. Afterwards the charges start flowing back, leading to a
continuation of the exponential voltage decay. Therefore slope at zero-
crossing, found at $t=t_{0}$, is an infinite decrease.

####  III.1.3 General Solution for the Image Current

In case none of the preceding approximations is sufficient a general solution
for Eq. eq:PUimageCharges is needed. It is found straightforward for Gaussian
bunches and rectangular pickups. Applying the shapes defined in Eqs.
eq:GaussianBunchProfile and eq:RectangularPickup without any approximation
leads to

$Q_{\mathrm{im}}(t)=\frac{w_{0}Q_{\mathrm{b}}}{2\pi
r_{\mathrm{p}}}\int_{-\ell/2}^{\ell/2}G\left(z-c_{\mathrm{0}}t,\sigma_{\mathrm{z}}\right)\mathrm{d}z.$
(43)

This integral is readily solved, giving

$Q_{\mathrm{im}}(t)=\frac{1}{2}\frac{w_{0}}{2\pi
r_{\mathrm{p}}}Q_{\mathrm{b}}\left[\erf\left(\frac{\frac{\ell}{2}-c_{0}t}{\sqrt{2}\sigma_{\mathrm{z}}}\right)+\erf\left(\frac{\frac{\ell}{2}+c_{0}t}{\sqrt{2}\sigma_{\mathrm{z}}}\right)\right]$
(44)

for the image charge induced on the pickup by the coasting beam.

Figure 3 shows exemplary results of the analytical description in Eq.
Eq:General-Q-im (blue) compared to the LBA, as in Eq. eq:LongBunchApprox
(dashed red), and the SBA, as in Eq. Eq:ShortBunch-Q-im (dashed green), for
two different values of the scaling factor $\mu$. In case of an ultra short
bunch, $\mu\approx 16.2$ (right), when the coasting bunch is almost completely
enclosed longitudinally by the pickup surface, a flat section is notable in
the analytical result, which is in accordance with the SBA, while the LBA
gives a sharp peak. In case of long bunches the curves found analytical and by
LBA converge, but the SBA in Eq. Eq:ShortBunch-Q-im still gives a flat center
with steep edges. Therefore both approximations are well suited for their
cases, but differ significantly apart from that.

Differentiation of Eq. Eq:General-Q-im gives the general image current

$I_{\mathrm{im}}(t)=\frac{w_{0}}{2\pi
r_{\mathrm{p}}}Q_{b}\left[G(t+t_{0},\sigma_{\mathrm{t}})-G(t-t_{0},\sigma_{\mathrm{t}})\right],$
(45)

with $t_{0}$ defined in Eq. Eq:Def-t0 and bunch width $\sigma_{\mathrm{t}}$
expressed in units of time according to Eq:Def-Sigt. The first Gaussian
represents the incoming charges introduced by the arriving bunch, the other is
caused by the bunch leaving the effective area. This result can be understood
as a generalization to the current found in the ultra short bunch
approximation. If the delta distributions are replaced by Gaussians, the
factor differs by the normalization $1/\sqrt{2\pi\sigma_{\mathrm{t}}^{2}}$.
Consistently the general solution is readily converted into Eq.
eq:ultraShortBunchCurrent by the limit of an ultra short bunch length.

Figure 4: Voltage signal according to Eq. Eq:VoltageSignal-Gerneral with
dimensions in the range of the LBA with $\mu\approx 0.7$ (left) and SBA with
$\mu\approx 24$ (right). The capacity is varied from almost the inductive
limit (red) up to the capacitive case (blue) with three intermediate steps
(dashed). Bunch and geometry parameters are in a reasonable range for XFEL
applications, e.g. $\sigma_{\mathrm{t}}=$7\text{\,}\mathrm{ps}$$ (left)
respectively $\sigma_{\mathrm{t}}=$0.2\text{\,}\mathrm{ps}$$ (right),
$Q_{\textrm{b}}=$20\text{\,}\mathrm{pC}$$, $\ell=$2.4\text{\,}\mathrm{mm}$$,
$w_{0}=$4.8\text{\,}\mathrm{mm}$$ and
$r_{\mathrm{p}}=$20\text{\,}\mathrm{mm}$$. In other applications, when the
capacitive case is required, the limiting case can be achieved by increasing
the terminating resistor R, to counter the $1/C$ dependence in Eq.
Eq:VoltageSignal-Gerneral, see Fig. 3 (a) in [24].

The general voltage signal measured at resistor R is found by the known
transfer function based on the pickups equivalent circuit, see Eq. Eq:RC-
Transfer-FD. It is determined by the convolution of the general image current,
Eq. Eq:GeneralImageCurrent, and the impulse response function, Eq. Eq:RC-
Transfer-TD. The result, also mentioned in [24], is

$\aligned U_{\mathrm{RC}}(t)=\frac{1}{2}\frac{1}{C}\frac{w_{0}}{2\pi
r_{\mathrm{p}}}Q_{\mathrm{b}}\exp\left(\frac{1}{2}\left(\frac{\sigma_{\mathrm{t}}}{RC}\right)^{2}\right)\left\\{\exp\left(-\frac{t+t_{0}}{RC}\right)\left[1+\erf\left(\frac{t+t_{0}}{\sqrt{2}\sigma_{\mathrm{t}}}-\frac{\sigma_{\mathrm{t}}}{\sqrt{2}RC}\right)\right]\right.-\\\
\left.\exp\left(-\frac{t-t_{0}}{RC}\right)\left[1+\erf\left(\frac{t-t_{0}}{\sqrt{2}\sigma_{\mathrm{t}}}-\frac{\sigma_{\mathrm{t}}}{\sqrt{2}RC}\right)\right]\right\\}.$
(46)

The voltage curve shows a very different behavior for different parameter
choice, which is not immediately accessible in Eq. Eq:VoltageSignal-Gerneral.
The two introduced limiting cases for $\mu\ll 1$ and $\mu\gg 1$ are in
accordance with the general solution. For an intermediate value of $\mu$, it
is possible to find an approximation of Eq. Eq:VoltageSignal-Gerneral by
assuming extreme values for the cut-off frequency or in particular the
capacity, as presented in the next section. The limiting cases and the
transitional behavior are exemplified in Fig. 4. On the left side bunch and
pickup length approach the LBA, $\mu\approx 0.7$. In contrast the right image
contains signals for a relatively short bunch with $\mu\approx 24$, looking
alike the SBA. In both plots the capacitance is varied starting at
$30\text{\,}\mathrm{fF}$, close to the so called inductive case (red) to
$1\text{\,}\mathrm{pF}$ respectively $4\text{\,}\mathrm{pF}$, which is around
the capacitive case (blue). The corresponding cut-off frequencies are
$\Omega_{\mathrm{RC}}=$0.67\text{\,}\mathrm{THz}$$ and
$5\text{\,}\mathrm{GHz}$ respectively $20\text{\,}\mathrm{GHz}$. The
transition between both cases is indicated with stripped lines.

The BAM resolution depends on the signal slope at the zero-crossing (ZC). The
time dependent signal slope found by derivation of Eq. Eq:VoltageSignal-
Gerneral

$\dot{U}_{\mathrm{RC}}(t)=\Omega_{\mathrm{RC}}\left[RI_{\mathrm{im}}(t)-U_{\mathrm{RC}}(t)\right].$
(47)

At the time of zero-crossing $t_{\mathrm{ZC}}$ the voltage
$U_{\mathrm{RC}}(t_{\mathrm{ZC}})$ is zero by definition. Therefore, the
signal slope at the zero-crossing is only defined by the image current

$S_{\mathrm{RC,ZC}}=\dot{U}_{\mathrm{RC}}(t_{\mathrm{ZC}})=\frac{1}{C}I_{\mathrm{im}}(t_{\mathrm{ZC}}).$
(48)

Because $I_{\mathrm{im}}(t=0)$ is zero, a zero-crossing at $t_{\mathrm{ZC}}=0$
would entail a gradient of zero, but the zero-crossing is located at
$t_{\mathrm{ZC}}\neq 0$ for any $C>0$.

####  III.1.4 Voltage by Frequency Regions

In many theoretical approaches the voltage signal is analyzed in two special
cases defined by frequency regions. In the so called capacitive limit, the
capacity C is large, respectively the cut-off frequency approaches zero
$\Omega_{\mathrm{RC}}\rightarrow 0$, hence all exponential functions approach
one and the first term of each error function is predominant. This leads to a
great simplification of the output voltage, which is

$U_{\mathrm{RC}}^{\mathrm{cap}}(t)=\frac{1}{C}Q_{\mathrm{im}}(t).$ (49)

Therefore, the signal in the limit of a infinitesimal cut-off frequency
resembles a charge source [24]. The limit for an infinitesimal capacity, the
inductive case [24] with $\Omega_{\mathrm{RC}}\rightarrow\infty$, is
straightforward for the frequency domain transfer function Eq. Eq:RC-Transfer-
FD, which becomes the constant R, and thus $h(t)=R\delta(t)$ in time domain.
The voltage is then

$U_{\mathrm{RC}}^{\mathrm{ind}}(t)=RI_{\mathrm{im}}(t),$ (50)

resembling a current source [24]. In this case, the zero-crossing is exactly
at $t=0$ and for a finite pickup length the signal slope is nonzero in a
seeming contradiction to Eq. Eq:SigSlope-General-I, but the denominator C is
zero as well.

####  III.1.5 Maximum Voltage and Signal Slope

For deployment in a BAM, maximum voltage and signal slope are the key features
of the pickup. Moreover a throughout understanding of the decisive parameters
is crucial. The general formulation in Eq. Eq:VoltageSignal-Gerneral confirms
the proportionality to the bunch charge and pickup width as well as reciprocal
to the distance between bunch and pickup

$U_{\mathrm{RC,max}}\propto Q_{\mathrm{b}}\frac{w_{0}}{r_{\mathrm{p}}}$ (51)

and

$S_{\mathrm{RC,max}}\propto Q_{\mathrm{b}}\frac{w_{0}}{r_{\mathrm{p}}}.$ (52)

The proportionality to the bunch charge was experimentally proven for the
signal slope of the cone shaped pickups at FLASH [16]. The inverse
proportionality to $r_{\mathrm{p}}$ has to be analyzed in simulations, since
these measurements are expensive and limited by the facilities design
parameters.

Another dependency observed in simulations, is on the ratio of pickup and
bunch lengths. The maximum voltage approaches its maximum asymptotically and
deviates only slightly from the value for a pickup longer than a few bunch
lengths [24, 13]. This also effects the signal slope, as the maximum voltage
barely changes in this region while both extrema drift apart, leading to a
decreasing slope [13]. In Eq. Eq:VoltageSignal-Gerneral it is readily shown,
that the voltage vanishes for $t_{0}=0$, which is identical to $\ell=0$. To
describe the behavior for any other ratio, the approximations must be
utilized. In case of the pure charge source the maximum voltage found at
$t_{\mathrm{max}}=0$ is

$\max\left(U_{\mathrm{RC}}^{\mathrm{cap}}\right)=\frac{1}{C}\frac{w_{0}Q_{\mathrm{b}}}{2\pi
r_{\mathrm{p}}}\erf\left(\frac{\ell}{\sqrt{8}\sigma_{\mathrm{z}}}\right)$ (53)

[24]. For a pure current source $t_{\mathrm{max}}$ must adhere to

$t_{0}=t_{\mathrm{max}}\tanh\left(\frac{t_{0}}{\sigma_{\mathrm{t}}}\frac{t_{\mathrm{max}}}{\sigma_{\mathrm{t}}}\right),$
(54)

which is found by setting $\dot{U}_{\mathrm{RC}}^{\mathrm{ind}}=0$. Two useful
approximations give the positions of extrema for a large or negligible
argument of the hyperbolic tangent, which are
$t_{\mathrm{max}}\approx\pm\sigma_{\mathrm{t}}$ for the LBA with
$t_{0}\ll\sigma_{\mathrm{t}}$ respectively $t_{\mathrm{max}}\approx\pm t_{0}$
for the SBA with $t_{0}\gg\sigma_{\mathrm{t}}$. The corresponding maximum
voltages are for a short pickup

$\max\left(U_{\mathrm{RC,t_{0}\ll\sigma_{\mathrm{t}}}}^{\mathrm{LBA,ind}}\right)\approx\frac{R}{\sqrt{e}}\frac{A_{\mathrm{p}}}{2\pi
r_{\mathrm{p}}}\frac{Q_{\mathrm{b}}}{\sqrt{2\pi}c_{0}\sigma_{\mathrm{t}}^{2}}=\mu\hat{U}_{0}^{\mathrm{ind}},$
(55)

with

$\hat{U}_{0}^{\mathrm{ind}}=R\frac{w_{0}}{2\pi
r_{\mathrm{p}}}\frac{Q_{\mathrm{b}}}{\sqrt{2\pi}\sigma_{\mathrm{t}}},$ (56)

and for a long pickup

$\max\left(U_{\mathrm{RC,t_{0}\gg\sigma_{\mathrm{t}}}}^{\mathrm{SBA,ind}}\right)\approx\hat{U}_{0}^{\mathrm{ind}}\left[1-\exp\left(-\frac{l^{2}}{2\sigma_{\mathrm{z}}^{2}}\right)\right].$
(57)

The first term is the maximum voltage for $\ell\rightarrow\infty$, which is
justified for $\ell>3\sigma_{\mathrm{z}}$ with the Gaussian bunch. In the
cases described by Eqs. Eq:Vmax-cap and Eq:Vmax-ind-longPU the maximum voltage
saturates for a pickup length of a few $\sigma_{\mathrm{z}}$, whereas in the
realms of LBA, with an infinitesimal pickup, the maximum voltage goes linear
with its length.

Regarding the BAM resolution a high signal slope is desired. The main
proportionality is already given by Eq. Eq:Slope-Dependencies, but in the
inductive limit a general solution is possible. The slope at the zero-crossing
is

$\left.S^{\mathrm{ind}}_{\mathrm{RC}}\right|_{t=0}=-RQ_{\mathrm{b}}\frac{w_{0}}{\pi
r_{\mathrm{p}}}\frac{t_{0}}{\sigma_{\mathrm{t}}^{2}}G\left(t_{0},\sigma_{\mathrm{t}}\right).$
(58)

This result is visualized in Fig. 5. For a fixed bunch length, the optimum is
reached at $t_{0}=\sigma_{\mathrm{t}}$, whereas for a fixed pickup length the
best result is found at $t_{0}=\sqrt{3}\sigma_{\mathrm{t}}$. At
$t_{0}=\sigma_{\mathrm{t}}=0$ is a pole, therefore reduction of both
dimensions is favorable in the case of a pure charge source, but in real
applications the bunch length is in a fixed range specific for the facility
and foreseen experiments.

Figure 5: Absolute signal slope at zero-crossing according to Eq.
Eq:ZCSlopeIndRC as a function of $t_{0}$ and $\sigma_{\mathrm{t}}$ in the
interval from $02\text{\,}\mathrm{ps}$. The dotted contours mark $1$, $10$ and
$100\text{\,}\mathrm{V}\text{\,}{\mathrm{ps}}^{-1}$ and dashed lines indicate
$t_{0}=\sigma_{\mathrm{t}}$ and $t_{0}=\sqrt{3}\sigma_{\mathrm{t}}$.

####  III.1.6 Bandwidth Limitation

The measured signal slope is limited by the bandwidth according to the
uncertainty principle. The product of a signal’s rise time $\tau$ as response
to a step function and the frequency bandwidth $\Delta f$ satisfies111Denote
that the right-hand side of the related uncertainty inequalities by Küpfmüller
[25] and Gabor [26] depend on the definitions of bandwidth and duration.

$\tau\Delta f=1.$ (59)

This rise time serves as an upper limit and can be expressed by the maximum
slope $S_{\mathrm{max}}$ and the peak-to-peak voltage $U_{\mathrm{PP}}$ with
$\tau=U_{\mathrm{PP}}/S_{\mathrm{max}}$ giving

$\aligned S_{\mathrm{max}}&\leq U_{\mathrm{pp}}\ \Delta f.$ (60)

This limitation affects the resolution of a transmitted signal. It determines
the highest possible slope allowed in the response function, but if the slope
of an incoming signal is well below this limit, a rise in bandwidth might even
have a negative effect on the outputs signal slope, as already shown for the
idealized Gaussian filter in the SBA.

####  III.1.7 Circular Pickup

In all prior considerations the pickup was assumed rectangular, which is not
in accordance with many real-world applications. From Fig. 4 and the
formulation of the SBA it is reasonable to assume that the rectangular model
is sufficient for long bunches, but its shape is significantly reflected in
the voltage signal for shorter bunches. A general analytical solution for the
circular surface of radius $r_{\mathrm{B}}$, defined by

$w(z)=\cases{2}\sqrt{r_{\mathrm{B}}^{2}-z^{2}}&|z|\leq r_{\mathrm{B}}\\\
0|z|>r_{\mathrm{B}}$ (61)

[22], is not available and numerical methods have to be utilized. Nonetheless,
some special cases are accessible by analysis. For the SBA, according to Eq.
eq:GaussianBunchProfileUltraShort, the image charge is proportional to the
pickup shape and thus the image current is

$I_{\mathrm{im}}^{\mathrm{SBA}}(t)\approx\cases{-}\frac{c_{0}^{2}Q_{\mathrm{b}}}{\pi
r_{\mathrm{p}}}\frac{t}{\sqrt{r_{\mathrm{B}}^{2}-\left(c_{0}t\right)^{2}}}&|t|<\frac{r_{\mathrm{B}}}{c_{0}}\\\
0|t|>\frac{r_{\mathrm{B}}}{c_{0}}$ (62)

with two poles at $t_{\pm\infty}=\pm r_{\mathrm{B}}/c_{0}$, which would be
softened by a broader bunch. In the inductive limit, interesting for
synchronization, the voltage is given by Eq. Eq:InductiveLimitVoltage, with
the zero-crossing exactly at $t=0$ and a signal slope of

$\left.S_{\mathrm{RC}}^{\mathrm{SBA,ind}}\right|_{t=0}=-R\frac{c_{0}^{2}Q_{\mathrm{b}}}{\pi
r_{\mathrm{p}}}\frac{1}{r_{\mathrm{B}}}.$ (63)

The slope is inversely proportional to the radius of the button pickup, hence
a smaller button is favorable, though this approximation is only valid for a
button much larger than the bunch.

####  III.1.8 Summary

An analytical solution for the voltage signal is readily found, which is valid
for any bunch or pickup length and for any value of the lumped elements in the
equivalent circuit representing the physical pickup. For different limiting
cases, as the SBA, LBA, capacitive as well as inductive limit, appropriate
expressions exist, which are easier to work with.

Nonetheless, some evaluations were carried out for the general solution as
well. First of all the highly relativistic bunch is assumed Gaussian and
centered on the longitudinal axis. Usually the pickup was treated as a
rectangular surface with the same curvature as the beam pipe, to keep the
radial distance to the beam constant. Fringing fields caused by the gap
between pickup surface and the beam pipe are neglected, but might be
considered sufficiently by a constant factor depending on the geometry. The
equivalent circuit was idealized as RC elements in parallel.

The most critical approximation is the rectangular pickup surface. The
solution is still useful in the range of medium to long bunches, but the form
becomes a significant factor for $\mu\gg 1$. There are numerical methods for
arbitrary pickup surfaces.

###  III.2 Numerical Solution

Figure 6: Voltage signal of a rectangular (dashed blue) and a round (solid
red) pickup surface calculated with Matlab® according to Eqs. Eq:NumQim-FD to
Eq:NumVo-TD. Analog to the analytic approach, a Gaussian bunch and an RC-
filter were used. The dimensions are in range of the SBA with $\mu\approx 49$
and at the capacitive limit (left) respectively inductive limit (right). Bunch
and geometry parameters, apart from the pickup length, are the same as in Fig.
4 (right), e.g. $R=$50\text{\,}\mathrm{\SIUnitSymbolOhm}$$,
$C=$1\text{\,}\mathrm{pF}$$ (left) respectively $C=$30\text{\,}\mathrm{fF}$$
(right), $\sigma_{\mathrm{t}}=$0.2\text{\,}\mathrm{ps}$$,
$Q_{\textrm{b}}=$20\text{\,}\mathrm{pC}$$,
$\ell=w_{0}=r_{\mathrm{B}}=$4.8\text{\,}\mathrm{mm}$$ and
$r_{\mathrm{p}}=$20\text{\,}\mathrm{mm}$$.

A numerical solution of Eq. eq:PUimageCharges is accessible for any bunch or
pickup shape, to study the signal in cases where the prior assumptions are not
valid. While the Gaussian bunch in Eq. eq:GaussianBunchProfile usually is a
good approximation, the button pickup has a circular surface with a width
given by Eq. eq:RoundPickup. In Fig. 4 the pickup form is apparent in the
signal shape in case of short bunches. An analytical solution was only
determined in the SBA with an inductive pickup. It is therefore crucial to
find a general description considering the pickup form, which is possible by
numerical methods. Acknowledging that for any even charge distribution, Eq.
eq:PUimageCharges corresponds to a convolution, the image charge is readily
calculated in frequency domain by the convolution theorem. It is

$Q_{\mathrm{im}}\left(\omega\right)=\mathrm{FFT}\left[\frac{\lambda(c_{0}t)}{2\pi
r_{\mathrm{p}}}\right]\mathrm{FFT}\left[w(c_{0}t)\right],$ (64)

where $\mathrm{FFT}$ is the fast fourier transform. In case of an uneven
charge density distribution it is necessary to exchange
$\mathrm{FFT}\left[\lambda(c_{0}t)\right]$ by the complex conjugate
$\overline{\mathrm{FFT}\left[\lambda(c_{0}t)\right]}$. By inverse FFT the time
domain image charge is

$Q_{\mathrm{im}}\left(t\right)=\mathrm{FFT}^{-1}\left[Q_{\mathrm{im}}\left(\omega\right)\right]$
(65)

[22]. The image current is

$I_{\mathrm{im}}\left(t\right)=\mathrm{FFT}^{-1}\left[i\omega
Q_{\mathrm{im}}\left(\omega\right)\right]$ (66)

and the voltage at the output of a system with transfer function $H(\omega)$

$U_{\mathrm{h}}\left(t\right)=\mathrm{FFT}^{-1}\left[H(\omega)I_{\mathrm{im}}\left(\omega\right)\right].$
(67)

This method works well for finite inputs. Residual values, e.g. of the
response function, at the end of the time interval lead to unphysical effects.
Therefore, the time interval and sample frequency must be chosen with care.

The voltage signal calculated with a parameter set in the range of the
operational parameters at the EuXFEL is pictured in Fig. 6. The dashed blue
line represents the numerical result for a rectangular pickup, which is in
good agreement with the analytical version in Fig. 4 (right). The red line
shows the signal by a circular pickup.

##  IV Ultra-low charge mode

For new experiments in the EuXFEL, with an ultra-low charge mode
($\leq$$1\text{\,}\mathrm{pC}$), the 2nd generation pickups are incapable of
providing the required signal for $\mathrm{fs}$ resolution. Therefore, the BAM
will be upgraded with a novel pickup structure and new EOM.

###  IV.1 Planned signal improvement

New layouts are restrained in each facility by design regulations and previous
design choices. For the next BAM upgrade in the EuXFEL a smaller pipe diameter
is now permitted. The possible reduction from $40.5\text{\,}\mathrm{mm}$ [12]
to $10\text{\,}\mathrm{mm}$ gives a potentially fourfold signal increase, by
relation Eq:Slope-Dependencies. When additionally the bandwidth is raised,
according to inequality eq:KupfmullerUP_Slope, one can expect a total
improvement by one order of magnitude. Moreover, it is possible to combine
multiple signals and to shorten the lossy RF path for further improvement.

The downside of these changes is a shorter dynamical range and a higher damage
risk due to increased likeliness of direct beam impacts and high voltages at
high charge mode. Special attention must be given to machine protection for
all subsequent components.

###  IV.2 Aperture reduction

A straightforward option for improvement is the sole reduction of the beam
pipe aperture. A diameter of $10\text{\,}\mathrm{mm}$ is possible in
combination with the smaller first-generation cone-shaped pickup. The second-
generation pickups cannot be installed in a four-pickup configuration with
this diameter due to their dimensions, as the cut-outs would overlap. A
simulation with the wakefield solver of CST PARTICLE STUDIO™ yields a slope of
$1746\text{\,}\mathrm{mV}\text{\,}{\mathrm{ps}}^{-1}$ with
$13.7\text{\,}\mathrm{V}$ peak-to-peak voltage and a dynamical range of
$12.4\text{\,}\mathrm{ps}$ at the nominal $20\text{\,}\mathrm{pC}$. The
ringing of the signal pictured in Fig. 7 initially is significantly higher but
decreases rapidly during the first $0.3\text{\,}\mathrm{ns}$.

Figure 7: Simulated signal in time domain (top) and its normalized spectrum
(bottom) taken at the end of the vacuum feedthrough of a single pickup with
$10\text{\,}\mathrm{mm}$ minimum distance, adapted from [27].

###  IV.3 90-GHz cone shaped pickup

We proposed a concept with pickups scaled to support up to
$90\text{\,}\mathrm{GHz}$ in 2019 with the dimensions specified in Table 1
[28]. This pickup was simulated afterwards in a BAM like setup, with four
identical pickups equally distributed around a pipe section, omitting the
proposed signal combination. In addition, the former bunch parameters have
been initially used. These are a super relativistic ($v=c_{0}$) Gaussian bunch
with $\sigma_{\mathrm{z}}=$1\text{\,}\mathrm{mm}$$ and a charge of
$20\text{\,}\mathrm{pC}$. Therefore, the voltage can be compared to the well-
described state-of-the-art pickups according to Angelovski et al. [13, 16].

Table 1: Specifications of the $90\text{\,}\mathrm{GHz}$-pickup [28], the modified pickup (2nd Gen.) [16] and the original (1st Gen.) [12]. | Draft’19 | 2nd Gen. | 1st Gen.
---|---|---|---
Cut-out dia. ($\mathrm{m}\mathrm{m}$) | $1.00$ | $1.62$ | $1.62$
Tapered cut-out dia. ($\mathrm{m}\mathrm{m}$) | $2.26$ | $13.6$ | $5.60$
Cone dia. ($\mathrm{m}\mathrm{m}$) | $0.45$ | $0.70$ | $0.70$
Tapered cone dia. ($\mathrm{m}\mathrm{m}$) | $1.02$ | $6$ | $2.42$
Cone height ($\mathrm{m}\mathrm{m}$) | \111Not specified in the publication | $6$ | $6$
Protrusion ($\mathrm{m}\mathrm{m}$) | \22footnotemark: 2 | $1$ | $0$
Relative permittivity | $3.75$ | $4.1$ | $4.1$
Line impedance ($\mathrm{\SIUnitSymbolOhm}$) | $50.0$ | $50.0$ | $50.0$

The bipolar signal pictured in Fig. 8 has a peak-to-peak voltage of
$3.52\text{\,}\mathrm{V}$. The peaks are separated by
$8.01\text{\,}\mathrm{ps}$, more than doubling the slope to
$722.4\text{\,}\mathrm{mV}\text{\,}{\mathrm{ps}}^{-1}$. Though the pipe radius
was decreased nearly by a factor of four, the gain is only by $2.4$. The
increased bandwidth leads to a reduced rise time of $7.9\text{\,}\mathrm{ps}$
but cannot compensate for the missing protrusion and the smaller active area.
Thus, the pickup cannot outperform the 1st generation, but a smaller peak-to-
peak voltage gives advantages in machine protection.

Reducing the bunch charge to $1\text{\,}\mathrm{pC}$ accordingly gives a slope
of $36.2\text{\,}\mathrm{mV}\text{\,}{\mathrm{ps}}^{-1}$, which undershoots
the minimum target by about a factor of $4$. For a further increase the
combination of more than two signals is planned. By the combination of $8$
pickups, without any phase shift and $3\text{\,}\mathrm{dB}$ attenuation at
each stage, a factor of $2.8$ might be possible. Eight pickups has been
determined as the limit, because $400\text{\,}\mathrm{\SIUnitSymbolOhm}$
pickups would be necessary due to the impedance change at a T junction type
combiner, to have a $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ connection to the
EOM. This is well above the vacuum impedance and requires tiny components. A
less radical option is the combination of $4$ pickups for a potential
improvement of a factor of $2$.

Compared to both preceding pickup generations, the voltage is approximately
proportional to the radius of the circular pickup surface. This may indicate a
bunch-pickup ratio where only the pickup width, but not its length is
relevant.

Figure 8: The signal in time domain (top) and its normalized spectrum (center)
taken at the end of the vacuum feedthrough of a single pickup as well as the
simulation model with $90\text{\,}\mathrm{GHz}$ cone-shaped pickups (bottom),
adapted from [27].

Figure 9: The signal in time domain (top) and its normalized spectrum (center)
taken at the end of the vacuum feedthrough of a single pickup as well as the
simulation model with $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ stripline pickups
(bottom), adapted from [27].

###  IV.4 Printed circuit board BAM

For ultra-short bunches the transition to a short rectangular pickup on a
printed circuit board (PCB) with a trace thickness still larger than the bunch
length appears possible. This concept possibly allows for a
$100\text{\,}\mathrm{GHz}$ pickup without the drawback of smaller dimensions.
Further benefits are the possibility to use well-known components with precise
production methods and well-described materials. The transmission lines (TL)
and combination network may be realized on the PCB reducing the RF path, which
is specifically important to prevent dispersion effects in broadband quasi-TEM
TLs. A microstrip is favorable for its width, but entails dispersion and is
less shielded. Therefore, it is planned to use a microstrip (MS) for coupling
to the field and a stripline (SL) for the combination network.

A preliminary simulation, shown in Fig. 9, of a PCB based pickup was done with
a $1.55\text{\,}\mathrm{mm}$ wide and about $15\text{\,}\mathrm{mm}$ long
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ SL in a $\epsilon_{\mathrm{r}}=$4.03$$
substrate disc with $10\text{\,}\mathrm{mm}$ aperture inside. For a
$20\text{\,}\mathrm{pC}$ bunch the simulation returns a slope of about
$1270\text{\,}\mathrm{mV}\text{\,}{\mathrm{ps}}^{-1}$. The SL pickup is
exceeding the current as well as the $90\text{\,}\mathrm{GHz}$ cone-shaped
pickups, but does not achieve the performance of a generation 1 pickup of
equal aperture. Furthermore, crosstalk and reflections at the vacuum
feedthrough as well as the open pickup end generate delayed but significant
ringing.

##  V Conclusion

A high bandwidth cone-shaped pickup with $10\text{\,}\mathrm{mm}$ aperture
leads to a significant improvement by reduction of the distance and increase
of the bandwidth. If a maximum voltage is of no concern, a configuration of
1st generation pickups in a $10\text{\,}\mathrm{mm}$ beam pipe is a simple
solution estimated sufficient for bunch charges down to
$4\text{\,}\mathrm{pC}$. In case of ultra-short bunches a PCB-type BAM may be
suited to support $100\text{\,}\mathrm{GHz}$ without the drawback of reduced
dimensions. With the current design restrictions, a signal combination is
inevitable. Further studies of an integrated combination network are required
to reduce signal reflections and losses. Furthermore, it is necessary to
investigate the properties of PCB boards regarding vacuum suitability and
radiation hardness. Specifically, potential damages caused by beam incidence
need to be assessed.

###### Acknowledgements.

This work is supported by the German Federal Ministry of Education and
Research (BMBF) under contract no. 05K19RO1. The contribution is a version of
[27], edited for publication in the Physical Review Accelerators and Beams
(PRAB) Special Edition for the 9th International Beam Instrumentation
Conference, IBIC 2020.

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# Measuring the intrinsic charge transfer gap using
K-edge X-ray absorption spectroscopy

C. Gougoussis, M. Calandra, A. Seitsonen, Ch. Brouder, A. Shukla and F. Mauri
CNRS and Institut de Minéralogie et de Physique des Milieux condensés, case
115, 4 place Jussieu, 75252, Paris cedex 05, France

###### Abstract

Pre-edge features in X-ray absorption spectroscopy contain key information
about the lowest excited states and thus on the most interesting physical
properties of the system. In transition metal oxides they are particularly
structured but extracting physical parameters by comparison with a calculation
is not easy due to several computational challenges. By combining core-hole
attraction and correlation effects in first principles approach, we calculate
Ni K-edge X-ray absorption spectra in NiO. We obtain a striking, parameter-
free agreement with experimental data and show that dipolar pre-edge features
above the correlation gap are due to non-local excitations largely unaffected
by the core-hole. We show that in charge transfer insulators, this property
can be used to measure the correlation gap and probe the intrinsic position of
the upper-Hubbard band.

###### pacs:

The description of electronic excitations in correlated materials is a
challenge since many relevant phenomena, such as magnetism in transition metal
(TM) compounds or high Tc superconductivity in doped Mott-insulators are a
consequence of strong electron-electron interaction. Recently, core-hole
spectroscopy unveiled unexpected electronic excitations in correlated
antiferromagnetic insulators Hasan ; Hayashi  as well as in high Tc
superconductors Collart ; Kim_Hill_PRL . In La2CuO4, K$\alpha$ resonant
inelastic X-ray scattering (RIXS) and theoretical calculationsShukla
demonstrated that dipolar pre-edge features just above the correlation gap are
due to intersite Cu 4p-3d hybridization. Since in correlated insulators the TM
empty d-states form the upper Hubbard bands, it is crucial to see (i) if off-
site pre-edge excitations occur in other correlated compounds Vanko , (ii) to
what extent their energy depends on the presence of a core-hole in the final
state.

The presence of a core-hole substantially complicates the interpretation of
X-ray absorption (XAS) spectra. Since core-hole attraction shifts the empty
d-states much more than the empty p-states, typically by several eV, it is
unclear to what extent the energy of the pre-edge features can provide useful
information on the position of the corresponding excitations in the material
in the absence of a core-hole. Furthermore, in correlated insulators core-hole
attraction and Hubbard repulsion partially compensate each other. Thus the use
of fitting parameters to describe these interactions, as is commonly done in
the literature Kotani_RMP , does not allow to distinguish between the two
effects. A treatment of core-hole attraction and Hubbard-repulsion from first
principles is needed.

In the case of K-edge XAS spectra, theoretical calculations are difficult
since excitations in a large energy window above the Fermi level need to be
described. Thus methods dealing successfully with correlation effects in NiO
in an energy window close to the Fermi level (such as DMFT KunesPRL07  or
cluster calculations TaguchiPRL08 ) cannot be used to describe pre-edge and
near-edge structures in K-edge XAS spectra. This is relevant since (i) low
energy pre-edge features provide relevant information on the local environment
of the absorbing atom Uozumi92 ; Joly99 ; Uozumi ; Shukla ; Vanko  and (ii) it
has been suggested that in correlated oxides “shakedown” excitations occur as
near-edge structures Tolentino .

In this work we present a first-principles approach to describe the K-edge XAS
spectra in correlated insulators. The method is based on a recently developed
DFT+U method and includes core-hole attractiontaillefumier . Differently from
other DFT+U schemes Anisimov , the U parameter is obtained by linear-response
cococcioni1 ; cococcioni2 . Consequently, U is not a fitting parameter but an
intrinsic linear response property since it measures the spurious curvature of
the energy functional as a function of occupation cococcioni1 . We apply the
method to the K-edge XAS of NiO, which is the prototype antiferromagnetic
correlated insulator and whose ground state and excitations are incorrectly
described by standard DFT approaches. We demonstrate that the pre-edge dipolar
features above the correlation gap are due to non-local excitations to second
nearest-neighboring Ni atoms reflecting the superexchange interaction. We show
that, due to its non-local nature, the dipolar pre-edge feature is unshifted
by core-hole attraction and it is, thus, a measure of the upper-Hubbard band
in the absence of a core-hole in the final state. Finally we exploit the
insensitivity of the pre-edge dipolar features on core-hole attraction to show
how to measure the charge transfer gap using K-edge XAS.

We use the NiO experimental crystal structure. The paramagnetic-cell group-
space is then Fm3̄m, Ni occupies the 4a position and O the 4b. The cubic
lattice parameter is $a=4.1788\AA$ for NiO srivastava . DFT calculations are
performed using the Quantum-ESPRESSO code PWSCF . We use Troullier-Martins
Troullier  pseudopotentials, the spin-polarized generalized gradient
approximation (GGA) PBE  and the recently developed DFT+U method of ref.
cococcioni1 ; cococcioni2 . The wave functions are expanded using a 140 Ry
energy cutoff for NiO. The calculated value of U for NiO is U=7.6 eV.

XAS spectra are computed in a supercell approach including core-hole effects
in the pseudopotential of the absorbing atom. From the self-consistent charge
density the XAS spectra are obtained using the continued fraction taillefumier
and the PAW method of ref. Blochl . We considered $2\times 2\times 2$
supercells of the magnetic cell (32 atoms). We used $4\times 4\times 4$
k-points grid both for the charge density and the continued fraction
calculation.

Figure 1: (Color online) Calculated and measured vedrinskii  Ni K-edge XAS of
NiO including partial fluorescence yield (PFY) (this work). The dashed (solid)
curves correspond to $e_{g}$ ($t_{2g}$) orientation.

In fig. 1 we show calculated NiO XAS spectra and their decomposition into
dipolar and quadrupolar parts using GGA and GGA+U, as compared to experimental
data vedrinskii  for single crystals. We consider the following two sets of
polarization ${\bm{\epsilon}}$ and wavevector ${\bf k}$: (i)
${\bm{\epsilon}}\parallel[110]$ and ${\bf k}\parallel[-110]$ , denoted by
$e_{g}$ orientation (dashed lines) and (ii) ${\bm{\epsilon}}\parallel[100]$
and ${\bf k}\parallel[010]$, denoted by $t_{2g}$ orientation (solid lines).
The directions are in terms of the paramagnetic cubic crystal cell. We note
that U has no effect on high-energy near-edge and far-edge features (labeled
C,D,E). On the contrary the energy and angular dependence of pre-edge
structures (see insets in Fig. 1) are incorrect in the framework of GGA. Peak
B is at too low energy and a too large mixing occurs between structures A and
B. In the GGA+U calculation the dipolar peak B is shifted to $\approx 2$ eV
higher energies and the angular dependence is in excellent agreement with
experimental data. The CGA+U calculation also shows that a very small dipolar
component is present in peak A, commonly interpreted as purely quadrupolar.
Detailed analysis of the angular dependence of the experimental NiO XAS
spectra vedrinskii  shows that even for the $t_{2g}$ orientation, where no
quadrupolar transition occurs because the t2g states are occupied, a small
peak is present ApeakNiO , in agreement with our findings. Thus, in correlated
insulators, the use of the DFT+U approximation is mandatory to obtain a
correct description of the pre-edge features.

To complete the understanding of the pre-edge features in NiO we resolve the
XAS spectrum in its spin dependence and calculate the density of states
projected over atomic orbitals using Löwdin projections. Without loss of
generality we have considered the absorbing Ni to be spin-up polarized. As can
be seen in Fig. 2, the quadrupolar part of peak A is mostly due to intra-site
excitations to d-states lowered by core-hole attraction. However, since the
hybridization between Ni and O is very strong, core-hole attraction lowers a
small portion of the O 2p states generating a small dipolar component in peak
A. In a atomic orbital picture, peak A is due to direct dipole transitions
from Ni 1s states to O 2p states. In our calculation the intensity of this
excitations is of the order of $1\%$ of the edge jump. In experiments it is
somewhat smaller because of the larger linewidth. This estimate is crucial for
the quantitative description of this off-site excitation in multiplet
calculation.

The spin-resolved dipolar spectrum shows that peak B is mostly due to
transition to up spin-polarized states. Since the absorbing atom has 5 up
electrons in d-states, the B excitation must have an off-site component.
Löwdin projections demonstrate that it is due to transition to on-site Ni
empty 4p-states hybridized to empty 3d-states of next-to-nearest neighbours Ni
atoms. This off-site excitation is then a fingerprint of the hopping process
leading to superexchange in NiO and a direct probe of the upper Hubbard band
in NiO.

Figure 2: (Color online) Comparison between GGA+U calculated K-edge Ni XAS and
Löwdin projected density of states.

To understand the role of core-hole effects on the A and B features we have
performed calculations with and without a core hole in the final state. We
found that peak A is shifted by 4 eV with respect to the edge by core-hole
attraction. On the contrary peak B is essentially unshifted. The difference in
the behavior of the two peaks is due to the non-local nature of peak B.

Figure 3: (Color online) Calculated pre-edge features of Ni K-edge XAS spectra
of LixNi1-xO with (a) and without (b) core-hole effects using a rigid doping
(solid line) approach or adding a compensating charged background (red-dashed
lines). Peaks labeling is the same as in Fig. 1, VB stands for Valence Band.
(c) Band structure of NiO along $\Gamma$X.

We now demonstrate that the non-local nature of the B excitation and its weak
dependence on core-hole effects can be used to measure the correlation gap
between the $|3d^{8}\underline{L}\rangle$ and the $|3d^{9}L\rangle$ states. We
consider hole-doped NiO. In practice this can be achieved through Li doping
ReinertZPHYSB1995 , as in the case of LixNi1-xO. A large range of doping can
be experimentally obtained with $0<x<0.7$. It has been shown KuiperPRL89 ;
vanElp  that, when NiO is doped with holes, the holes mainly reside on O
atoms, due to the charge transfer character of NiO. Thus the top of the NiO
valence band has mainly p character. This suggests that in Ni K-edge XAS of
LixNi1-xO, in the low doping regime, an additional dipolar peak should occur
in the pre-edge, resulting from the holes entering the top of the valence
band. This is confirmed by oxygen K-edge absorption where the top of the
valence band is seen even at dopings as low as $x=0.05$)KuiperPRL89 . At
somewhat larger doping, a similar peak due to Oxygen p-holes should also occur
in the dipolar part of Ni K-edge XAS. Since no experimental data are available
in the literature, we directly simulate Ni K-edge XAS in LixNi1-xO.

We calculate hole-doping of NiO in two different ways, namely (i) by rigid-
band doping of NiO and (ii) adding a compensating charge background and
recalculating self-consistently the charge-density and the XAS spectra. The
results of the Ni K-edge XAS in LixNi1-xO in the pre-edge region with and
without a core-hole in the final state are shown in Fig. 3. We find that the
top of the valence band should be visible in Ni K-edge XAS at dopings of
$x\approx 15-20\%$. In the calculated spectra, with or without a core-hole in
the final state, the energy position of the top of the valence band and of
peak B are independent on doping. The agreement between the rigid band picture
and the compensating-charge background calculation validates the rigid band
picture for $0<x<0.2$

In NiO the experimentally measurable gap is due to the excitation between the
top of the valence band and the empty d-states. In DFT+U Bengone  and GW
Faleev  calculations the lowest single particle excitation is between the
valence band and an s band occurring at energies lower than the empty d states
(see Fig. 3). This band is invisible in experiments since its optical matrix
elements are extremely weak Bengone . Moreover its also invisible in K-edge
XAS since it is highly dispersive and has mostly Ni s and O s components. Thus
in the absence of core-hole effects, the distance between the top of the
valence band and peak B is a measure of the correlation gap, being the
dependence of the energy position on doping very weak (see Fig. 3, right
panel). When core-hole attraction is considered (see Fig. 3, left panel), we
find that, despite the occurrence of a core-hole exciton (peak A), the
distance between the top of the valence band and peak B is very weakly
affected ($\approx 5\%$). Consequently in lightly hole-doped NiO the distance
between the top of the valence band and peak B is a measure of the correlation
gap even in the presence of a core-hole in the final state.

In our calculation the charge-transfer gap is 5 eV (peak-to-peak distance). In
NiO optical absorption starts at $3.1$ eV and reaches its maximum at $4.0$ eV
Sawatzky84 ; HuefnerReview . This value is reduced respect to our due to
excitonic effects. In photoemission and inverse photoemission Huefner86  the
gap (peak-to-peak distance) is roughly 5.5 eV, in fairly good agreement with
our value. Thus our proposal allows for an independent estimate of the optical
gap in NiO.

The procedure outlined for NiO can be used to measure the charge-transfer gap
and the upper Hubbard band in other charge-transfer insulators. In these
systems the top of the valence band is due to Oxygen p-states and it is thus
visible in dipolar TM K-edge XAS upon doping. If the doping necessary to
detect this feature is low enough not to affect substantially the electronic
structure then a rigid-band doping picture applies and the top of the valence
band is at the same energy as in the undoped system. Moreover, hole states
being dipolar in nature, the effects of core-hole attraction are weak. The top
of the valence band of the doped system can then be used as a reference energy
for the measurement of the excitations seen in TM K-edge XAS. The second
necessary condition is the occurrence of non-local dipolar features Shukla ;
Vanko  (peak B) in the pre-edge region due to transitions to d-states of
neighboring TM atoms promoted by hybridization with the TM absorbing atom
p-states. If these excitations are visible, we have shown that they represent
the upper Hubbard band of the material and the energy difference between this
excitation and the top of the valence band in the weakly hole-doped system is
a measure of the charge transfer gap. Thus, TM K-edge XAS in weakly hole-doped
charge-transfer insulators is unbiased experimental tool to measure the
charge-transfer gap.

In this work we have developed a new first-principles parameter-free method to
calculate K-edge XAS spectra including core-hole effects in the final state
and electronic correlation at the DFT+U level. We have shown that the method
provides spectra in excellent agreement with experimental data for NiO, the
prototype correlated insulator. We have interpreted all the pre-edge and near-
edge features in the experimental data. In particular, we have identified a
dipolar pre-edge feature as due to non-local excitations to d-states of the
second nearest-neighboring Ni atoms, namely the upper Hubbard band. We have
shown that, due to its non-local nature, this excitation is unaffected by the
presence of a core-hole in the final state. Starting from this result we have
proposed a new way to measure the intrinsic correlation gap in charge-transfer
insulators based on TM K-edge XAS. This method is complementary to optical
measurements and more straightforward than a combined photoemission-inverse
photoemission experiment.

We acknowledge useful conversation with A. Kotani, G. Dräger, O.Šipr , Ph.
Sainctavit, M. Arrio, D. Cabaret, A. Juhin, H. Kulik, N. Marzari and L.
Reining. Calculations were performed at the IDRIS supercomputing center
(project 071202).

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| $\rho$ | Probability a customer is willing to accept product A if product B is out of stock
Scenario C | $\tau_{t}$ |  Day of the week for day $t$
| Yt |  Vector of stock received to fill order $A_{t}$, ordered by ascending age
| $Y_{i,t}$ |  Element of Yt with $i$ periods of remaining useful life on arrival
| $Y_{t}$ |  Total stock received on day $t$
| $n$ | Target number of successes for the negative binomial distribution for daily demand
| $\delta$ | Mean of the negative binomial distribution for daily demand
| $c_{0}^{k}$ |  Log-odds ratio of receiving a unit with a remaining useful life of $k$ days versus
|  | 1 day when the uncertainty is exogenous.
| $c_{1}^{k}$ |  Increase in the log-odds ratio of receiving a unit with a remaining useful life of $k$
|  | days versus 1 day for each unit ordered when the uncertainty is endogenous
Table 17: Summary of notation used in this study.