# Timing resolution performance of Timepix4 bump-bonded assemblies

R. Bolzonella 11footnotetext: Corresponding author. J. Alozy  R. Ballabriga
M. van Beuzekom  N. V. Biesuz  M. Campbell  P. Cardarelli  V. Cavallini  V.
Coco  A. Cotta Ramusino  M. Fiorini  V. Gromov  M. Guarise  X. Llopart Cudie
S. Okamura  G. Romolini  A. Saputi  A. Vitkovskiy

###### Abstract

The timing performance of the Timepix4 application-specific integrated circuit
(ASIC) bump-bonded to a $100\;\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ thick
n-on-p silicon sensor is presented. A picosecond pulsed infrared laser was
used to generate electron-hole pairs in the silicon bulk in a repeatable
fashion, controlling the amount, position and time of the stimulated charge
signal. The timing resolution for a single pixel has been measured to
$107\;\mathrm{p}\mathrm{s}$ r.m.s. for laser-stimulated signals in the silicon
sensor bulk. Considering multi-pixel clusters, the measured timing resolution
reached $33\;\mathrm{p}\mathrm{s}$ r.m.s. exploiting oversampling of the
timing information over several pixels.

##  1 Introduction

Timepix4 is the latest generation application-specific integrated circuit
(ASIC) of the Timepix family [1, 2], mainly targeted for single particle
detection in hybrid pixel detectors [1].

The Timepix4 will be used in a novel device for the detection of single
visible photons combining excellent timing and spatial resolution with high
rate capabilities and large active area, currently under development within
the 4DPHOTON project [3, 4]: it is based on a vacuum tube, with transmission
photocathode, a microchannel plate (MCP) for electron multiplication and the
Timepix4 ASIC used as pixelated anode. This kind of approach, exploiting a
bare ASIC used inside a vacuum tube, has been already proved on other works
[5, 6]. A detector with such performance would have a fundamental role in
improving state-of-the-art imaging technology, with applications in many
fields such as high-energy physics, life sciences or quantum physics to quote
a few examples.

A detailed characterization of the Timepix4 ASIC, especially in terms of
timing resolution, is necessary to correctly assess the performance of the
MCP-based hybrid photon detector, which is currently under construction. This
paper presents the timing characterization of the available assembly of the
Timepix4 bump-bonded to a $100\;\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ thick
n-on-p Si detector illuminated by a picosecond infrared laser.

##  2 Timepix4 ASIC

Timepix4 is a CMOS ASIC in $65\;\mathrm{n}\mathrm{m}$ developed by the
Medipix4 Collaboration for the readout of pixel detectors, which is able to
reach high spatial and timing resolutions together with high rate
capabilities.

The Timepix4 consists of a $512\times 448$ array with square pixels with a
$55\;\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ pitch, for a total active area of
about $7\;\mathrm{c}\mathrm{m}^{2}$. The Timepix4 front-end can be configured
to be sensitive either to electrons or holes. Pixel electronics amplifies and
discriminates the signals in each pixel, as shown in Figure 1. Signals which
exceed the threshold are processed by the digital front-end. A 5-bit Digital-
to-Analog Converter (DAC) per pixel allows to equalize the threshold value.
A Time to Digital Converter (TDC) in each pixel uses a Voltage-Controlled
Oscillator (VCO) to measure the Time-of-Arrival (ToA) and Time-over-Threshold
(ToT) for each signal above threshold. The TDC nominal bin size is
$195\;\mathrm{p}\mathrm{s}$ that corresponds to a $56\;\mathrm{p}\mathrm{s}$
r.m.s. resolution.
The pixels are organized in groups of 8 (2 columns of 4 pixels), called
SuperPixels (SP), sharing the same TDC and the same VCO. 4 SuperPixels in one
double column form a SuperPixel Group (SPG) and 16 SPGs form a full double
column.

Figure 1: Simplified block diagram of the pixel front-end electronics and the
pixel, SP and SPG structure. [1]

Timepix4 can be operated in data-driven or frame-based mode. The measurements
presented in this paper are based on the data-driven mode: each single pixel
hit produces a 64-bit data word, 64b/66b encoded, which is transmitted off-
chip by 16 high speed programmable links running at a maximum
$10\;\mathrm{G}\textrm{b}\mathrm{/}\mathrm{s}$ data bandwidth. Therefore, the
Timepix4 ASIC can reach a maximum bandwidth of
$160\;\mathrm{G}\textrm{b}\mathrm{/}\mathrm{s}$ corresponding to a maximum hit
rate of $\sim 2.5\;\mathrm{G}\textrm{Hit}\mathrm{/}\mathrm{s}$.

The Timepix4 ensures also a low equivalent noise charge ($\sim
80\;\textrm{e}^{-}$) per pixel, which, combined with a threshold dispersion
after equalization of $\sim 40\;\textrm{e}^{-}$, allows to set a low threshold
($\sim 800\textrm{e}^{-}$) to the whole ASIC masking just few noisy pixels
[1].

External digital signals can be sent to the Timepix4 through two single-ended
inputs (called "digital pixels") in the top and the bottom peripheries. At one
periphery one external input is routed to the even columns and the other to
the odd columns. One SP (at the end of the column) is chosen to receive each
of the external input signals. In the chosen SP signals for all 8 front-ends
are ignored. This permits up to 4 external inputs to be included in the output
data stream at the expense of losing 8 pixel inputs per external input.

###  2.1 Time measurement

The time-to-digital conversion in the Timepix4 ASIC happens in three steps
[7]. The first one consists of the determination of the
$40\;\mathrm{M}\mathrm{Hz}$ clock cycle in which the preamplifier output
signal rises above threshold. When this happens, a clock, generated by the
SuperPixel VCO, is enabled. This clock has a nominal frequency of
$640\;\mathrm{M}\mathrm{Hz}$, providing a fine-Time-of-Arrival (fToA), as
shown in Figure 2(a). This clock is active until the following
$40\;\mathrm{M}\mathrm{Hz}$ clock rising edge, so the number of
$640\;\mathrm{M}\mathrm{Hz}$ clock cycles provides a finer measurement of the
ToA, improving the bin width from $25\;\mathrm{n}\mathrm{s}$ to
$1.56\;\mathrm{n}\mathrm{s}$. Moreover, when the VCO is activated, 4 internal
phases of the oscillator can be latch to further divide the
$1.56\;\mathrm{n}\mathrm{s}$ bins into 8 bins with nominal width of
$195\;\mathrm{p}\mathrm{s}$ and providing an Ultra-fine-Time-of-Arrival
(UfToA), as shown in Figure 2(b).

(a)

(b)

Figure 2: (a) Operation of the ToA and fToA measurement. (b) Copies of
$640\;\mathrm{M}\mathrm{Hz}$ clock shifted returning UfToA measurement [7].

##  3 Experimental setup

The measurements presented in the paper consist of sending simultaneous
signals to different pixels and measuring the time-difference between them to
estimate the timing resolution. In order to do that, a pulse generator (Active
Technologies PG-1072) with a measured interchannel jitter of
$7\;\mathrm{p}\mathrm{s}$ r.m.s. has been used to produce two synchronized
pulses with a period of $5\;\mathrm{m}\mathrm{s}$.

The first pulse is sent directly to one of the Timepix4 digital pixels. The
second pulse is used to trigger a pulsed diode laser (PicoQuant PDL-800 B
driver with LDH-P-1060 laser head), which produces an infrared laser pulse
that is then attenuated and guided to the Timepix4 through an optical fiber
patch cord. This reaches a microcollimator, which is positioned using through
linear translation stages with micrometric precision. The overall setup is
shown in Figure 3.

Figure 3: Laser setup used for the described measurements.

###  3.1 DAQ system: Timepix4 assembly and SPIDR4 control board

The Timepix4 ASIC used in this measurements is bump-bonded to a
$100\;\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick n-on-p silicon detector,
and glued and wire-bonded onto a carrier board developed by Nikhef. The
silicon sensor has a breakdown voltage of $-180\;\mathrm{V}$, as shown in
Figure 4(a), and it is biased at $-150\;\mathrm{V}$ for the measurements
presented here. The current flowing in the sensor is higher if the Timepix4 is
switched on, due to a higher temperature, but the breakdown voltage is not
affected by that. The sensor has a back-side metallization with a pattern of
circular openings with 300 micron diameter (visible in Figure 4(b)) which
allow the laser light to enter the silicon bulk. The silicon attenuation
coefficient for a wavelength $1060\;\mathrm{n}\mathrm{m}$ is about 900
microns, which ensures an almost uniform generation of electron-hole pairs
through the silicon bulk thickness.

(a)

(b)

Figure 4: (a) I-V dependency of the Si sensor bonded to the Timepix4. (b)
Holes pattern in the metallization coated on the Si sensor.

In the measurements described in this paper, the Timepix4 is operated in
electron-collecting mode with a threshold set to $1000\;\textrm{e}^{-}$ after
pixel equalization. The measurements here described don’t require to reach
very low energies, so this allows to avoid noisy events by choosing a higher
threshold. The DACs configuration set, reported in Table 1, leads to a pixel
current of $7.67\;\mathrm{\SIUnitSymbolMicro}\mathrm{A}$. By increasing the
power per pixel the timing resolution would improve at low input charges, were
the resolution is dominated by the front-end contribution [7].

DAC name | DAC value set | DAC gain set
---|---|---
VcascPreamp | 170 | 5
VFBK | 128 | 2
VCascDisc | 130 | 5
VControlVCO | 0 | 2
VThreshold | 8192 | 5
VTPCoarse | 128 | 2
VTPFine | 512 | 5
(a) Bias voltage DACs.

DAC name | DAC value set
---|---
VBiasPreamp | 100
VbiasDiscTailNMOS | 60
VBiasIkrum | 5
VBiasLevelShift | 90
VbiasDiscPMOS | 40
VBiasDiscTRAFF | 50
VBiasADC | 128
VBiasDAC | 50
(b) Bias current DACs.

Table 1: DAC settings of the Timepix4.

The Timepix4 carrier board is connected through a flat cable to a SPIDR4
control board, developed by Nikhef, used both to configure the ASIC and to
transfer the output data to a data acquisition PC. During the measurements,
two high-speed links were used (one link per half matrix) both running at
$2.56\;\mathrm{G}\textrm{b}\mathrm{/}\mathrm{s}$.

###  3.2 Laser setup and configuration

The infrared laser pulse has a full-width-half-maximum time duration of about
$36\;\mathrm{p}\mathrm{s}$, and a jitter with respect to the laser trigger and
synch-output signals lower than $20\;\mathrm{p}\mathrm{s}$ r.m.s. from
datasheet. The amount of light per single pulse is tuned using a variable
fiber optics attenuator (VOA Electronics V1000F). The light is then collimated
and focused in order to illuminate the smallest possible region onto the
sensor back-side. Precise positioning and focusing of the laser spot in the
center of the metallization openings are achieved thanks to three remotely
controlled linear translation stages with micrometric precision mounted in a
3D configuration.

After focusing, the standard deviation of the laser spot size was measured to
about 1.4 pixels r.m.s., as shown in Figure 5. In this way, a wide Time-over-
Threshold range has been covered. The laser has been focused on a pixel close
to the center of the bottom half matrix, illuminating the same region of
pixels during the entire measurement. Due to the limited time chipboard has
been available, the measurements and analysis had to focus only on this pixel
region; further measurements will be performed to study the timing resolution
performance on different regions of the matrix at a later date.

Figure 5: ToT distribution among the pixel on a laser pulse event.

In order to estimate the timing resolution contribution due to the laser setup
and the pulse generator alone, the jitter of the laser pulse, triggered by one
output of the pulse generator, has been measured at the oscilloscope
(Tektronix MSO 70804C, $8\;\mathrm{G}\mathrm{Hz}$,
$25\;\mathrm{G}\textrm{S}\mathrm{/}\mathrm{s}$) with respect to the second
output of the pulse generator (that is usually sent to a Timepix4 digital
pixel for the other measurements presented in this work). The laser pulse has
been measured with an Ultrafast Photodiode (ALPHALAS InGaAs Photodetector
UPD-35-UVIR-P). The measured jitter amounts to $(10.9\pm
0.1)\;\mathrm{p}\mathrm{s}$ r.m.s., which is negligible compared to the other
contributions to the detector timing resolution, as described in section 7.

##  4 Measurements and analysis description

The typical measurement consists of illuminating the Timepix4 assembly in a
fixed position with a variable amount of light per pulse, at a fixed
repetition rate ($200\;\mathrm{Hz}$).

The Timepix4 output data received by the data acquisition PC are decoded,
assigning to each event a spatial coordinate ($X$ and $Y$), a timestamp
($ToA$) and a Time-over-Threshold value ($ToT$).
Afterwards, the decoded events are clustered using a common Density Based
algorithm (Density-Based Spatial Clustering of Applications with Noise or
DBSCAN), which consists of searching for hits which neighbour is a seed hit in
space and in time, thereby forming clusters [8].

Once each cluster has been created, the ToA of each pixel in the cluster is
compared with the ToA of the reference signal ($ToA_{ref}$), which is the one
sent to the digital pixel input.

A priori each pixel in the cluster is expected to have a different time walk
contribution, and different ToT vs charge ($Q$) calibration.

For this reason, the analysis has been performed initially on single pixels.
However, a number of corrections and calibrations still have to be performed
in order to obtain the best performance, as shown in the next sections. After
all corrections and calibrations have been applied, cluster-related
quantities, like cluster time and cluster charge, have been introduced.

##  5 Procedure for precise VCO calibration

During the measurements it has been observed that the different VCOs were
oscillating at different frequencies. This effect can be quantified by
studying the length of a full fToA cycle:

$UfToA_{cycle}=fToA_{rise}\cdot 8+UfToA_{stop}-UfToA_{start}$ (5.1)

where $fToA_{rise}$, $UfToA_{start}$ and $UfToA_{stop}$ are the fine and
ultra-fine corrections of the ToA described in paragraph 2.1. Figure 6
represents a simplified schematic of the $fToA_{rise}$ and $UfToA$ codes
measurements. The $fToA_{rise}$ is the number of $640\;\mathrm{M}\mathrm{Hz}$
clock cycles between the discriminator output rising edge and the following
$40\;\mathrm{M}\mathrm{Hz}$ clock cycle rising edge. The $UfToA_{start}$ and
$UfToA_{stop}$ codes are obtained as shown in Figure 2(b), respectively from
the discriminator output rising edge and the following
$40\;\mathrm{M}\mathrm{Hz}$ clock cycle rising edge.

Figure 6: Simplified clock schematic of the $fToA_{rise}$ and $UfToA$
measurements.

The quantity defined in Equation (5.1) represents the number of ultra-fine
bins that would fit between the signal rising edge and the next reference
clock rising edge.
The range of $UfToA_{cycle}$ is obtained for each TDC by plotting a histogram
of the values measured when the pixel is triggered with externally applied
digital test pulses which are not synchronised to the
$40\;\mathrm{M}\mathrm{Hz}$ clock, as shown in Figure 7.

Figure 7: $UfToA_{cycle}$ distribution obtained measuring laser pulses on a
single pixel.

Knowing that this range would always cover a period of
$25\;\mathrm{n}\mathrm{s}$, depending only on the stability of the
$40\;\mathrm{M}\mathrm{Hz}$ clock, in nominal conditions, i.e. the VCO
oscillating at $\nu_{VCO}=640\;\mathrm{M}\mathrm{Hz}$, this range would be

$range(UfToA_{cycle})=8\cdot
nbins(fToA)=8\cdot\frac{\nu_{VCO}}{\nu_{refclk}}=8\cdot\frac{640\;\mathrm{M}\mathrm{Hz}}{40\;\mathrm{M}\mathrm{Hz}}=128$
(5.2)

Observing the histogram for different pixels not sharing a common VCO, it is
possible to see that different VCOs oscillate at different frequencies.
If the frequency is not correctly taken into account both the UfToA and the
fToA bins width would be badly estimated. This turns into non-negligible
errors when the rising edge is far enough from the next reference clock rising
edge (when the VCO would stop), since there would be a large amount of fToA
and UfToA bins contributing, and the bins width error would have to be
considered for all of them.
As an example, if the range of $UfToA_{cycle}$ is 131 instead of 128,
inverting Equation (5.2), the resulting frequency of the VCO is
$655\;\mathrm{M}\mathrm{Hz}$, which turns into a UfToA bin width of
$191\;\mathrm{p}\mathrm{s}$ instead of $195\;\mathrm{p}\mathrm{s}$. A signal
measured on the $UfToA_{cycle}$ bin number 3 would be measured with a decoded
ToA mistaken by just $\epsilon_{3}=ToA_{3}-ToA_{3}^{*}=3\cdot
4\;\mathrm{p}\mathrm{s}=12\;\mathrm{p}\mathrm{s}$, but the error on a signal
measured on the bin 130 would be
$\epsilon_{131}=ToA_{131}-ToA_{131}^{*}=520\;\mathrm{p}\mathrm{s}$.

Therefore, it is mandatory to measure and calibrate the frequency for the TDCs
in the whole matrix. To perform this calibration, internal digital test pulses
have been sent to one pixel per each TDC to measure the VCO frequency.

An internal test pulse synchronized with the $40\;\mathrm{M}\mathrm{Hz}$
reference clock is available, and it is possible to set its phase $\phi$ with
respect to the reference clock. Fixing the phase with respect to the clock,
the populated $UfToA_{cycle}$ bin is always the same. Varying the phase,
different bins are populated, defining pairs $(\phi,UfToA_{cycle})$.
The 16 pairs $(\phi,UfToA_{cycle})$ have then been fitted with a linear
function $UfToA_{cycle}=p_{0}+p_{1}\cdot\phi$, as shown in Figure 8, whose
slope $p_{1}\;\left[\mathrm{Hz}\right]$ is associated to the VCO frequency
through the relation

$\nu_{VCO}=\frac{p_{1}}{8}$ (5.3)

where the coefficient 8 is the ratio between the width of the $fToA$ bins and
the one of the $UfToA$ bins.

Figure 8: $UfToA_{cycle}$ dependence on VCO for different pixels.

By measuring this frequency for each TDC a map of the VCO frequencies has been
produced, as shown in Figure 9(a).

(a)

(b)

Figure 9: (a) VCO frequency bi-dimensional distribution over the whole matrix.
(b) VCO frequency values distribution.

In Figure 9(b), which is an histogram of the values in Figure 9(a), it can be
noticed that the frequency distribution is peaked at the nominal frequency
$640\;\mathrm{M}\mathrm{Hz}$, but it covers a range of $\sim
40\;\mathrm{M}\mathrm{Hz}$; the circuit allows to adjust the VCO frequency on
a range of $\sim 20\;\mathrm{M}\mathrm{Hz}$, so it would not be possible to
perform measurements adjusting all the VCOs in order to make them run at the
same frequency. However, this will not be an issue, since correcting their
contribution on the offline analysis would preserve the timing resolution, as
described in subsection 7.1.
The Timepix4 on-pixel TDC’s VCO frequency is locked by a PLL running with the
same VCO placed at the center of the chip. The distributed control voltage
suffers from local GND reference mismatch due to internal power supply drops.
Figure 9 is therefore is an indirect measurement of the internal digital power
supply which corresponds approximately to $\sim
1\;\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{M}\mathrm{Hz}$.

##  6 Time-over-Threshold calibration

A per pixel calibration of the ToT measured against the deposited charge has
been performed on the whole matrix, using the Timepix4 internal analog test
pulse, that is a signal with known and selectable amplitude injected in a
capacitor placed at the input of the front-end chain. This allows to inject on
each pixel several pulses carrying different charges. The capacitances can
vary by about $1\;\%$ in different pixels, but for the calibration they have
been assumed as equal. This may slightly degrade the calibration performance.
The ToT depends non-linearly on the deposited charge, so for each pixel the
pairs $\left(ToT,Q\right)$ have been fitted using the function:

$Q(ToT)=p_{0}+p_{1}\cdot ToT-\frac{p_{2}}{ToT-p_{3}}$ (6.1)

where Q is in $\mathrm{k}\textrm{e}^{-}$ and ToT is in ns. A plot showing the
measured ToT calibration for a pixel is shown in Figure 10, with superimposed
fitting function. The maximum injected charge corresponds to about
$14\;\mathrm{k}\textrm{e}^{-}$ due to the test pulse saturation for the
particular DAC parameters used in this measurement, presented in Table 1.

Figure 10: Example of the charge non-linear calibration against ToT of a
single pixel.

By repeating this procedure on all Timepix4 pixels ($\sim 230$ thousands) a
bi-dimensional distribution of the fit parameters has been obtained, allowing
a per-pixel calibration covering the whole matrix. As an example, Figure 11
presents the distribution across the matrix of the $p_{1}$ parameter. A
structure due to the gain difference between top and bottom matrix is clearly
visible, as well as vertical structures due to different behaviour of
different double columns.

Figure 11: Example of the fit parameter $p_{1}$ distribution over the matrix.

The resulting calibration has been validated measuring the X-rays spectrum
produced by ${}^{241}\textrm{Am}$ and ${}^{137}\textrm{Cs}$ sources used to
simultaneously irradiate the whole detector matrix from a few cm distance. All
the pixels, except for the noisy ones, have been enabled and irradiated, as
shown in Figure 12(e). The injected charge expressed in
$\mathrm{k}\textrm{e}^{-}$ is converted in $\mathrm{k}\mathrm{eV}$ when using
the sources by multiplying by a factor
$3.6\;\mathrm{eV}\mathrm{/}\textrm{e}^{-}$, which is the average energy to
produce $e-h$ pairs in Si. The improvement given by the per pixel calibration
to the resolving power is clearly visible in the difference between the
uncalibrated cluster ToT spectrum (Figure 12(a)) and the energy calibrated one
(Figure 12(b)). The ${}^{241}\textrm{Am}$ peaks at
$13.9\;\mathrm{k}\mathrm{eV}$, $17.8\;\mathrm{k}\mathrm{eV}$,
$20.8\;\mathrm{k}\mathrm{eV}$, $26.35\;\mathrm{k}\mathrm{eV}$ and
$59.5\;\mathrm{k}\mathrm{eV}$ are well resolved and their energy is correctly
estimated, as well as the ${}^{137}\textrm{Cs}$ peak at
$32.0\;\mathrm{k}\mathrm{eV}$. Figures 12(c) and 12(d) show a spectrum
comparison with respect to the cluster size considered, respectively for the
whole spectrum up to $70\;\mathrm{k}\mathrm{eV}$ and only for the peak at
$59.54\;\mathrm{k}\mathrm{eV}$. An offset between the peaks related to single
pixel clusters and the ones related to two-pixels clusters is present, due to
a small calibration error introduced by the uncertainty on the capacitors used
to convert the test pulse DAC voltage into a charge on each pixel.

(a)

(b)

(c)

(d)

(e)

Figure 12: (a) ${}^{241}\textrm{Am}$ and ${}^{137}\textrm{Cs}$ ToT spectrum
measured on the whole matrix before the per-pixel calibration, considering all
clusters. (b) ${}^{241}\textrm{Am}$ and ${}^{137}\textrm{Cs}$ energy spectrum
measured on the whole matrix after the per-pixel calibration, considering all
clusters. (c) Comparison of the calibrated spectrum due to 1-pixel clusters
and 2-pixels clusters. (d) Comparison of the peak at
$59.54\;\mathrm{k}\mathrm{eV}$ resulting from clusters with size 1, 2 or
greater. (e) Hitmap of the events measured irradiating the matrix with
${}^{137}\textrm{Cs}$ and ${}^{241}\textrm{Am}$.

In Table 2 the energy values corresponding to the radioactive source
measurements are summarised: the expected energies of the measured peaks,
together with the measured ones and the full-width-half-maximum values of the
Gaussian fit used to extract these measurements.

Nuclide | $\mathbf{E_{expected}[\mathrm{k}\mathrm{eV}]}$ | $\mathbf{E_{measured}[\mathrm{k}\mathrm{eV}]}$ | $\mathbf{FWHM_{measured}[\mathrm{k}\mathrm{eV}]}$
---|---|---|---
$\textrm{Np}\;\textrm{L}\alpha$ | 13.9 | 14.3 | 1.6
$\textrm{Np}\;\textrm{L}\beta_{1}\;+\;\textrm{Np}\;\textrm{L}\beta_{2}$ | 17.7 | 17.8 | 2.0
$\textrm{Np}\;\textrm{L}\gamma_{1}$ | 20.7 | 21.3 | 2.4
${}^{241}\textrm{Am}$ | 26.3 | 26.5 | 2.6
$\textrm{Ba}\;\textrm{K}\alpha_{1}+\textrm{K}\alpha_{2}$ | 32.0 | 32.0 | 2.7
${}^{241}\textrm{Am}$ | 59.5 | 59.4 | 4.7
Table 2: Measured energy of ${}^{137}\textrm{Cs}$ and ${}^{241}\textrm{Am}$
main emissions compared with expected energies, considering all clusters.

##  7 Timing resolution contributions

The achievable timing resolution obtained from laser illumination is affected
by several contributions, described in this section. Some of these
contributions, i.e. the TDC resolution, are intrinsic and cannot be removed,
while others, i.e. the time walk effect or the VCO frequency variations, can
be corrected for.

The impact of effects like time walk and jitter can be reduced by maximising
the charge collected in a single pixel. The measurements presented in this
section are performed under this assumption. The measurements are based on the
central (most illuminated) pixel in the cluster. The laser attenuation was set
in order to inject a charge included in
$\left[50\;\mathrm{k}\textrm{e}^{-};60\;\mathrm{k}\textrm{e}^{-}\right]$ in
the central pixel, in order to minimise time walk and jitter.
Further measurements performed with a variable attenuation will be described
in Sections 8 and 9, together with a description of the time walk correction.

###  7.1 Measured variation of the Voltage Controlled Oscillator frequency

Precise timing measurements require to take into account and correct for the
different VCO frequencies, either in the Timepix4 configuration or in the
offline analysis, as demonstrated in Section 5.

Without taking into account VCO frequency variation, the measured timing
resolution is about $200\;\mathrm{p}\mathrm{s}$ r.m.s.. After correcting the
timing measurements using the proper VCO oscillation frequency, the standard
deviation of the ToA difference between the central pixel and the reference
pixel is greatly improved:

$\sigma_{VCO}=126\pm 1\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ (7.1)

as shown in Figure 13.

Figure 13: Distribution of $ToA-ToA_{ref}$ after the VCO frequency correction.

The measurement above is based on a VCO frequency correction applied at the
analysis level after data acquisition, during which the VCOs were running at
different frequencies.

A second measurement has been performed by equalising the VCO frequencies at
the level of the Timepix4 configuration, before the measurement, setting the
VCO frequencies of the two pixels to the nominal $640\;\mathrm{M}\mathrm{Hz}$
value. The results is $\sigma_{adj-VCO}=132\pm 1\;\mathrm{p}\mathrm{s}$
r.m.s., slightly worse than the previous measurement.

The calibration of the VCOs frequencies, done so far only on the two pixels
under test, has been extended to the whole matrix as described in Section 5 to
allow precise timing measurements on the whole Timepix4 active area.
All measurements presented in the following sections are corrected for the VCO
frequency variation at the analysis level, after data acquisition.

###  7.2 Reference signal resolution

The reference signal resolution has been estimated through externally
generated test pulses. A periodic pulse has been generated and sent to the
pixel used as reference through the digital pixels, and the difference between
the ToA of each pulse with the previous one has been computed. The external
pulses are not synchronous to the Timepix4 reference clock at
$40\;\mathrm{M}\mathrm{Hz}$, and they are generated with a period
($T=(1165.000\pm 0.007)\;\mathrm{n}\mathrm{s}$) not multiple of
$25\;\mathrm{n}\mathrm{s}$, so they cover the entire range of fToA and UfToA
bins. Thus, the resulting resolution already takes into account different bin
widths. The standard deviation of the $(ToA_{n+1}-ToA_{n})$ distribution
amounts to $\sigma(ToA_{n+1}-ToA_{n})=102\pm 1\;\mathrm{p}\mathrm{s}$ r.m.s.,
as shown in Figure 14. Being this result related to the difference between two
signals registered by the same TDC, the resolution of the reference signal
itself can be estimated, to a first approximation, as:

$\sigma_{REF}=\sigma(ToA_{n+1}-ToA_{n})/\sqrt{2}=72\pm
1\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ (7.2)

The above result is slightly higher than the resolution expected from a
$195\;\mathrm{p}\mathrm{s}$ time bin
($195\;\mathrm{p}\mathrm{s}/\sqrt{12}=56\;\mathrm{p}\mathrm{s}$). This
overestimation may be due to the contribution given by the generation of the
external test pulses and their distribution to Timepix4 and within it.

Figure 14: Distribution of the period $ToA_{n+1}-ToA_{n}$ of external periodic
test pulses sent to a single pixel through digital pixels.

###  7.3 Residual contributions to the timing resolution

The timing resolution obtained so far ($\sigma_{VCO}$) represents the error
associated to the difference $ToA-ToA_{ref}$. The timing resolution of a
single Timepix4 pixel for a charge included in
$\left[50\;\mathrm{k}\textrm{e}^{-};60\;\mathrm{k}\textrm{e}^{-}\right]$ can
therefore be calculated subtracting in quadrature the contribution due to the
reference signal ($\sigma_{REF}$):

$\sigma(ToA)_{final}=\sqrt{\sigma_{VCO}^{2}-\sigma_{REF}^{2}}=103\pm
2\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ (7.3)

The value above includes, to a first approximation, the contributions due to
the signal generation in the silicon sensor, the front-end jitter and the TDC
bin width.

##  8 Time walk correction and single pixel timing resolution charge
dependence

To obtain precise timing measurements, it is necessary to correct for the time
walk effect on single pixels. To estimate this effect, a set of measurements
has been performed by varying the laser attenuation. In particular, the
intensity was varied with fine steps. This allowed to cover charge values in
the range of $\left[\sim
10\;\mathrm{k}\textrm{e}^{-}\;-\;200\;\mathrm{k}\textrm{e}^{-}\right]$ with a
continuous-like distribution, and to obtain the time walk spectrum shown in
Figure 15. As shown the distribution has been fitted using the function:

$TW(ToT)=ToADiff(ToT)=ToA-ToA_{ref}=\frac{p_{0}}{Q^{p_{1}}+p_{2}}+p_{3}$ (8.1)

Figure 15: Single pixel time walk distribution superimposed with the fitting
function.

Afterwards, the time walk effect has been corrected for each measured ToA
value, by subtracting the charge-dependent correction given by Equation (8.1).
The resulting time walk corrected plot is shown in Figure 16.

Figure 16: Single pixel $\left(Q,ToA-ToA_{ref}\right)$ distribution after time
walk correction.

The distribution in Figure 16 was divided in several "vertical slices", each
one selecting a narrow charge range of $\sim 4\;\mathrm{k}\textrm{e}^{-}$. The
timing resolution has been estimated for each charge slice as the standard
deviation of the distribution of the slice’s projection on the $ToADiffCorr$
axis. The resulting resolutions are plotted in Figure 17.

Figure 17: Time resolution dependency with respect to the calibrated charge
for a single pixel. Timing resolution of the reference pixel has not been
subtracted.

Starting from low input charges the values the timing resolution difference
improves, reaching a value of $\sigma_{ToA-ToARef}=124.4\pm\
0.8\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ for charge values
$Q\in\left[52\;\mathrm{k}\textrm{e}^{-};60\;\mathrm{k}\textrm{e}^{-}\right]$.
At higher input charge values the timing resolution difference degrades again,
reaching a saturation at $\sigma_{ToA-ToARef}=129\pm\
2\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ for charge values higher than
$80\;\mathrm{k}\textrm{e}^{-}$. The single pixel timing resolution at
$80\;\mathrm{k}\textrm{e}^{-}$ input charge is estimated by subtracting in
quadrature the contribution due to the reference signal: this results to a
single pixel timing resolution of $\sigma_{ToA}=107\pm
3\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$.

The charge distribution (ToT) obtained when measuring using a laser with fixed
intensity is gaussian-like, with a spread due both to the electronic noise
introduced by the Timepix4 and the intrinsic spread given by the silicon
detector. At fixed laser intensity, the timing distribution should also
resemble a gaussian distribution. However, as the arrival time measurement is
quantised into bins of $195\;\mathrm{p}\mathrm{s}$ one might expect the hits
to be spread over a few bins of $195\;\mathrm{p}\mathrm{s}$, in the ideal case
where frequencies of the reference VCO and the VCO of laser-excited pixel are
identical. However in reality the oscillation frequency of the VCO’s are
slightly different and this leads to a finer quantisation, but still with
peaks, as shown in Figure 13. This timing quantisation generates the
horizontal structures visible in Figure 15, which become tilted bands in
Figure 16 after subtracting the time walk contribution.

With higher injected charges, the spread of the measured charge is wide enough
to cause an overlap of the distributions at different laser intensities.

Due to these effects, the $ToA-ToARef$ distributions related to different
laser intensities will overlap when selecting the events using the "vertical
slice" method described above. This overlap and the distribution quantisation
will result on a spread larger than the one that would be obtained performing
a single measurement at a fixed charge corresponding to the considered slice
charge. This explains the degradation of the resolution observed in Figure 17
at high charges ($Q\gtrsim 70\;\mathrm{k}\textrm{e}^{-}$).

##  9 Cluster timing resolution

When considering a cluster of more than one pixel, the information from the
different pixels can be combined to extract more precise timing information,
provided that the time walk effect at the single pixel level has been taken
into account.

The ToA for each pixel in a cluster is corrected for the time walk effect
using Equation (8.1). Afterwards, the “cluster ToA” ($ToA_{cluster}$) is
calculated as the weighted-average of the ToA of individual pixels in the
clusters. The used weights are the charge values for each pixel. Instead, the
“cluster charge” is simply computed as the sum of the individual pixels’
charge.

The cluster ToA as a function of the cluster charge is shown in Figure 18.

Figure 18: Cluster ToA distribution as a function of the cluster charge.

The slice analysis described in Section 8 was repeated over the complete
cluster charge range. The distribution shown in Figure 19, where the “cluster
ToA” resolution ($\sigma_{ClusterToA}$), evaluated as the standard deviation
of the $ToA_{cluster}-ToA_{ref}$ distribution, is plotted as a function of the
cluster charge.

Figure 19: Cluster timing resolution
($\sigma_{ClusterToA}=\sigma(ToA_{cluster}-ToA_{ref})$) as a function of the
calibrated cluster charge. Timing resolution of the reference pixel has not
been subtracted.

As shown, the timing resolution distribution reaches a minimum for cluster
charge $Q\in\left[\sim 900\;\mathrm{k}\textrm{e}^{-};\sim
1300\;\mathrm{k}\textrm{e}^{-}\right]$, and the value at the minimum is:

$\sigma_{ClusterToA}=79\pm 1\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ (9.1)

Once the reference signal contribution ($\sigma(ToA_{ref})$) is subtracted in
quadrature, the minimum of cluster timing resolution distribution is:

$\sigma_{True\;ClusterToA}=33\pm 3\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$
(9.2)

Measurement fluctuations of the order of a few $\mathrm{p}\mathrm{s}$ in
Figure 19 are due to the intrinsic overlap among $ToA$ distributions centered
on different timing bins explained in Section 8.

###  9.1 Timing resolution dependence on the cluster size

The laser spot position onto the silicon sensor is fixed in all presented
measurements, as shown in Figure 5. Around the central pixel, the most
illuminated one and therefore the one with the largest charge value, there is
a number of pixels which have an induced signal either due to the optical
spread of the laser beam (and possible internal reflections) or charge sharing
effects.

It is important to estimate the minimum cluster size in order to reach the
best resolution results obtained in Section 9. To do that, the analysis
described above has been repeated creating “artificial” clusters with
different sizes, removing selected pixels via software from the original
physical cluster as shown in Figure 20.

Figure 20: Sketch showing clusters of different size (n) artificially created
via software. Figure 21: Cluster timing resolution as a function of the
cluster charge, divided by the cluster size. Different cluster sizes are
presented as different colors. Timing resolution of the reference pixel has
not been subtracted.

For each artificial cluster, the timing resolution distribution has been
computed as a function of the cluster charge, as shown in Figure 21. In order
to better compare the results, cluster charge has been re-scaled dividing it
by the cluster size. The distribution corresponding to a single-pixel cluster
is consistent with the one described in Section 8. Figure 21 shows two
important contributions to the cluster timing resolution. The first is the
dependence of the resolution on the cluster charge: increasing the total
charge improves initially the timing resolution, as expected from the front-
end jitter contribution, but for large charge values it leads to a worsening,
explained in Section 8. The second contribution is due to the cluster size. A
significant improvement in the timing resolution has been observed passing
from a single-pixel cluster to a 5-pixel cluster, while the improvement given
by a further increase in cluster size is modest. Thus, this proves that it’s
possible to reach the resolution obtained in the previous paragraph even with
a lower multiplicity cluster.

In order to decouple the cluster charge and the cluster size contributions,
clusters with different sizes but with “fixed” charge have been analysed, as
shown for example in Figure 22, where only clusters with charge in the range
$\left[360\;\mathrm{k}\textrm{e}^{-},380\;\mathrm{k}\textrm{e}^{-}\right]$
have been included. This is equivalent to selecting a fixed total amount of
charge spread over a different number of pixels. Increasing the cluster size
has two counteracting effects: first, the charge is spread among a greater
number of pixels, thus each of them receives less charge, resulting in a
worsening of the timing resolution due to the increased front-end jitter;
second, the statistical factor due to the average improves the resolution of
the cluster, as already demonstrated.

In Figure 22 the first factor is dominant for larger cluster sizes, because
the charge would spread more, and the average factor would not be able to
compensate for the loss of resolution in each pixel due to the increased
jitter. On the contrary, the improvement due to the average is dominant for
low cluster sizes, because each pixel collects more charge and therefore the
jitter contribution is much reduced.

Figure 22 has been obtained from a "vertical" slice of Figure 21 for a given
cluster charge range. Therefore, depending on the actual sensor coupled to the
Timepix4 ASIC and its geometry, it will be possible to find the optimal
cluster size in order to have the best timing resolution, depending on the
expected total amount of charge that will be spread onto the Timepix4 matrix.

Figure 22: Example of the cluster timing resolution dependence on the cluster
size for a fixed range of cluster charge. A cut between
$360\;\mathrm{k}\textrm{e}^{-}$ and $380\;\mathrm{k}\textrm{e}^{-}$ on the
cluster charge has been placed. Timing resolution of the reference pixel has
not been subtracted.

##  10 Conclusions

The timing performance of the Timepix4 ASIC has been extensively characterized
using a picosecond laser setup illuminating a bump-bonded assembly based on
$100\;\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ n-on-p silicon sensor.

Reaching the expected time performance on every pixel of the whole matrix
requires careful calibration and correction procedures. The Timepix4 timing
resolution is heavily affected by the different VCO oscillation frequencies,
and an automated calibration procedure had been implemented to correct for
these variations on the whole pixel matrix. Another automated correction is
implemented to correct for the time walk effect in each individual pixel.

The single pixel timing resolution amounts to $\sigma_{pixel}=107\pm
3\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ for signals above
$50\;\mathrm{k}\textrm{e}^{-}$. When considering clusters of sufficient size,
the best resolution of $\sigma_{cluster}=33\pm
3\;\mathrm{p}\mathrm{s}\;\textrm{r.m.s.}$ has been obtained. For input charges
$Q_{in}\gtrsim 20\;\mathrm{k}\textrm{e}^{-}$ the timing resolution is almost
constant, excluding the contribution due to the measurements overlap, proving
that the system resolution is dominated by the TDC resolution, and no more by
the front-end one.

We have demonstrated that cluster timing resolutions of few tens of
picoseconds could be reached even when small size clusters are used, provided
that the average charge received per pixel is $\gtrsim 20\;\mathrm{k}e^{-}$,
therefore validating the concept of the 4DPHOTON detector.

## Acknowledgments

This work was carried out in the context of the Medipix4 Collaboration based
at CERN, and in the framework of the MEDIPIX4 project funded by INFN CSN5.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 819627, 4DPHOTON project).

## References

  * [1] X. Llopart, J. Alozy, R. Ballabriga, M. Campbell, R. Casanova, V. Gromov et al., _Timepix4, a large area pixel detector readout chip which can be tiled on 4 sides providing sub-200 ps timestamp binning_ , _Journal of Instrumentation_ 17 (2022) C01044.
  * [2] R. Ballabriga, M. Campbell and X. Llopart, _Asic developments for radiation imaging applications: The medipix and timepix family_ , _Nuclear Instruments and Methods A_ 878 (2018) 10.
  * [3] M. Fiorini, J. Alozy, M. Bolognesi, M. Campbell, A.C. Ramusino, X. Llopart et al., _Single-photon imaging detector with o(10) ps timing and sub-10 $\mu$m position resolutions_, _Journal of Instrumentation_ 13 (2018) C12005.
  * [4] J.A. Alozy, N.V. Biesuz, M. Campbell, V. Cavallini, A. Cotta Ramusino, M. Fiorini et al., _Development of a single-photon imaging detector with pixelated anode and integrated digital read-out_ , _Journal of Instrumentation_ 17 (2022) C06007 [2112.10636].
  * [5] J. Vallerga, J. McPhate, A. Tremsin and O. Siegmund, _Optically sensitive MCP image tube with a Medipix2 ASIC readout_ , _Proc. SPIE Int. Soc. Opt. Eng._ 7021 (2008) 702115.
  * [6] J. Vallerga, A. Tremsin, J. DeFazio, T. Michel, J. Alozy, T. Tick et al., _Optical MCP image tube with a quad Timepix readout: initial performance characterization_ , _JINST_ 9 (2014) C05055.
  * [7] K. Heijhoff, K. Akiba, R. Ballabriga, M. van Beuzekom, M. Campbell, A. Colijn et al., _Timing performance of the timepix4 front-end_ , _Journal of Instrumentation_ 17 (2022) P07006.
  * [8] M. Ester, H.-P. Kriegel, J. Sander and X. Xu, _A density-based algorithm for discovering clusters in large spatial databases with noise_ , in _Proceedings of the Second International Conference on Knowledge Discovery and Data Mining_ , KDD’96, p. 226–231, AAAI Press, 1996.