# Photometry, Centroid and Point-Spread Function Measurements in the LSST
Camera Focal Plane Using Artificial Stars

Johnny H. Esteves Department of Physics, University of Michigan, Ann Arbor,
Michigan, United States  Yousuke Utsumi Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California, United
States  SLAC National Accelerator Laboratory, Menlo Park, California, United
States  Adam Snyder Department of Physics and Astronomy, University of
California/Davis, Davis, California, United States  Theo Schutt Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California, United States  SLAC National Accelerator Laboratory,
Menlo Park, California, United States  Alex Broughton Department of Physics &
Astronomy, University of California/Irvine, Irvine, California, United States
Bahrudin Trbalic  Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford, California, United States  Sidney Mau Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California, United States  Andrew Rasmussen  Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford,
California, United States  SLAC National Accelerator Laboratory, Menlo Park,
California, United States  Andrés A. Plazas Malagón Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, Stanford,
California, United States  SLAC National Accelerator Laboratory, Menlo Park,
California, United States  Andrew Bradshaw Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California, United
States  SLAC National Accelerator Laboratory, Menlo Park, California, United
States  Stuart Marshall L. Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California, United States  SLAC
National Accelerator Laboratory, Menlo Park, California, United States  Seth
Digel Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, California, United States  SLAC National Accelerator
Laboratory, Menlo Park, California, United States  James Chiang Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California, United States  SLAC National Accelerator Laboratory,
Menlo Park, California, United States  Eli Rykoff SLAC National Accelerator
Laboratory, Menlo Park, California, United States  Chris Waters Department of
Astrophysical Sciences, Princeton University, Princeton, New Jersey, United
States  Marcelle Soares-Santos Department of Physics, University of Michigan,
Ann Arbor, Michigan, United States  Aaron Roodman Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California, United
States  SLAC National Accelerator Laboratory, Menlo Park, California, United
States

###### Abstract

The Vera C. Rubin Observatory’s LSST Camera pixel response has been
characterized using laboratory measurements with a grid of artificial stars.
We quantify the contributions to photometry, centroid, point-spread function
size, and shape measurement errors due to small anomalies in the LSSTCam CCDs.
The main sources of those anomalies are quantum efficiency variations and
pixel area variations induced by the amplifier segmentation boundaries and
“tree-rings” — circular variations in silicon doping concentration. We studied
the effects using artificial stars projected on the sensors and find that the
resulting measurement uncertainties pass the ten-year LSST survey science
requirements. In addition, we verify that the tree-ring effects can be
corrected using flat-field images if needed, because the astronomic shifts and
shape measurement errors they induce correlate well with the flat-field
signal. Nevertheless, further sensor anomaly studies with on-sky data should
probe possible temporal and wavelength-dependent effects.

Rubin Observatory; Rubin Observatory Legacy Survey of Space and Time (LSST);
LSST Camera; CCDs; Tree-rings; Shear

##  1 Introduction

The Vera C. Rubin Observatory is a next-generation optical and near-infrared
observatory currently under construction in Cerro Pachón, Chile. The Rubin
Observatory will conduct the Legacy Survey of Space and Time (LSST), an
unprecedented galaxy survey of 18000 sq-deg of the southern sky that will
revisit each area over 825 times in 10 years and in six photometric bands,
_ugrizy_. The four science pillars of LSST main are to probe the nature of
dark energy and dark matter, take an inventory of the solar system, explore
the transient optical sky, and study the evolution and structure of the Milky
Way (Ivezić et al., 2019). To achieve these goals, the Rubin Observatory will
use the 3.2 gigapixel LSST Camera (LSSTCam), mounted on the 8.4-meter Simonyi
Survey Telescope. The LSSTCam has a large field of view of approximately 10
sq-deg, with a focal plane of 201 4k by 4k thick (100 microns), fully-
depleted, back-illuminated charge-coupled devices (CCDs; Holland et al., 2003,
2009, 2014).

The focal plane of the LSSTCam is populated by 189 science CCDs, 8 CCDs for
auto-guiding, and 4 split CCDs for wavefront measurements. The focal plane
consists of 25 sub-assemblies, 21 Science Rafts (O’Connor et al., 2016) with a
3$\times$3 mosaic of science CCDs and 4 Corner Rafts (Arndt et al., 2010) each
with two guiders and one split wavefront sensor. For LSST custom CCDs, 100
micron thick back-illuminated deep-depletion devices, were developed. These
sensors feature 16 amplifier segments, each 2k by 0.5k, arranged in two rows
of eight segments and separated by a mid-line break, to enable low-noise, two-
second readout via parallel readout of the segments. The CCD sensors were
fabricated by two vendors: Imaging Technology Laboratory (ITL) and Teledyne
e2v (e2v). Both vendors met the LSST CCD requirements Radeka et al. (2009);
Doherty et al. (2014); Kotov et al. (2016) but there are subtle differences
between those sensors, as well as from sensor to sensor from each vendor.

Thick, high-resistivity CCDs have been used by other wide-field imagers such
as the Dark Energy Camera (DECam, Flaugher et al. (2015a)) and the Hyper
Suprime-Cam (HSC, Miyazaki et al. (2018)), in part due to their high quantum
efficiency (QE) at longer wavelengths (near infrared). However, these types of
detectors have been found to imprint subtle but significant undesirable
characteristics that impact centroid, photometric, flux, and shape
measurements (Stubbs, 2014; Astier, 2015; Mandelbaum, 2015). Study and
characterization of any source of systematic errors will be crucial to
achieving the required accuracy to achieve the scientific goals of a survey
such as the LSST.

An initial study characterizing prototype LSST sensors was performed to ensure
that the CCDs met basic LSST performance requirements for, e.g., read noise,
quantum efficiency, charge transfer efficiency, diffusion and full well
(Doherty et al., 2014). Snyder et al. (2018) studied optimization of
operational voltages for sensors from ITL. Snyder et al. (2021a, b) performed
measurements of effects of sensor anomalies such as deferred charge
distortions, centroid shifts, and the brighter-fatter effect (Gruen et al.,
2015). Park et al. (2017, 2020) studied the imprints of circular patterns due
to silicon dopant concentration variation (tree-rings effect) on flat-field
images for the full set of LSST CCDs. In addition to these studies, Juramy et
al. (2020) identified an anomaly in response at the amplifier boundaries of
the e2v sensors. The bias and clock voltages as well as the CCD controller
sequence were optimized in the course of individual Raft and focal plane
testing. Comprehensive results of the overall testing campaign will be
described in a future publication. In addition, we anticipate that full
characterization of each individual sensor will be necessary to achieve the
desired level of precision from the LSSTCam Bond et al. (2018); Roodman et al.
(2018); Snyder et al. (2021a)

The focus of this paper is a dedicated image campaign projecting a grid of
artificial stars on a subset of LSST CCDs in the assembled focal plane, to
study the detailed response of the sensors and in particular to evaluate the
uniformity of the photometry, centroid and measured shapes of these artificial
stars. We identify and evaluate anomalies in response observed from these
measurements. We analyze the images using pipeline software developed for LSST
(Bosch et al., 2019, 2018), and make response maps of the deviations from
nominal in each of our quantities of interest. We quantify the observed
anomalies in terms of the desired systematic requirements of the LSST, paying
special attention to effects occurring at certain locations on the CCDs. The
tree-rings signal is observed on centroid and point-spread function (PSF) and
it is compared with flat-field response confirming the nature of the effect.
The analysis focused on the Brighter-Fatter effect will be presented in
Broughton et al. in prep.

##  2 Focal Plane: Spot Grid Test

This section first describes the laboratory test setup and the data
acquisitions, then the imaging data collection and the post-processing
methodology, which includes the source detection, the grid fitting algorithm,
and the calibration of the spot-measured quantities: flux, centroid, PSF
shape, and size.

###  2.1 Camera Bench For Optical Testing: Spot Projector

The focal plane in the Camera cryostat was mounted on the top of the Bench for
Optical Testing (BOT), facing downward. The BOT was designed to achieve a dark
environment for electro-optical testing of the focal plane. The background
light is reduced to below 0.01 electrons per second per pixel, an ideal
environment to perform optical tests without light contamination. For a
complete description of the BOT design, assembly, and requirements, see Newbry
et al. (2018); Snyder et al. (2021b).

Underneath the BOT, we mounted the spot projector, which creates the
artificial star grid. Basically, the spot projector projects the image of a
spot mask onto the focal plane. The spot projector comprises a light source, a
shutter, an integrating sphere, a photographic mask on a controllable filter
wheel, and a commercial lens. In particular, the shape and size of the
artificial stars are set by the photographic spot mask characteristics.

The spot projector experimental apparatus has a 450 nm light-emitting diode
(LED) light source fed by an optical fiber into the integrating sphere gated
by a single-blade beam shutter (“Thorlabs 1”). The spot pattern is set by a
photographic mask (HTA Photomask photo-lithographic) on the filter wheel. The
commercial lens (Nikon 105mm f/2.8 Al-s Micro-Nikkor) is used to re-image the
integrating sphere’s 1” exit port. The F-stop ring was set to be closed as
much as possible. The entire image of the mask is about the size of an LSST
CCD.

The magnification of the system is close to 3 (75 pixels between spots on the
focal plane, 0.25mm spacing at the mask) the final beam is about F/100 in the
image plane. The spot projector is placed on a remotely controlled XY stage.
This XY stage allows the translation of the projector to point at any location
of the focal plane. In this work, we use a grid pattern of 49$\times$49 spots
as shown in Figure 1.

Figure 1: Example images of (left) the 49x49 spot grid and (right) a close up
of the highlighted region on the left, respectively. The star-like spots are
approximate point sources, with FWHM 5.2 pixels. The grid allow probing much
of the sensor area with each acqusition. Note that the sources located in the
corners are masked due to their diminished flux, a result of the vignetting
effect produced by the commercial lens in our setup.

###  2.2 Data Acquisition

We collected images for this analysis in two series, Run 3 and Run 5. We began
with a randomly selected sensor from each vendor: R22-S11 (e2v) and R02-S02
(ITL), as a pilot study in phased third round of electro-optical testing
campaign at SLAC (Run 3; 2019/10/4–2019/11/5). Then we extended the scope to
four other sensors in Run 5 (2021/11/4 and 2022/1/6) with some improvements in
the acquisition procedure: two ITL sensors (R03-S12, R10-S11) and two e2v
sensors (R24-S11, R32-S01). The nominal flux of the spots was set at
$\sim$50000 e-/pixel, which is bright relative to the readout noise, $\sim$10
e-, and well below sensor full well, $\sim$100000 e-. The sensor voltages
readout sequence parameters are different between those two data acquisition
campaigns. The specific parameters are tabulated in Appendix A.

We collected 1600 images with the spot projector position dithered around the
center of each of the studied CCDs, spanning $\pm$5 mm in Run 3. However, we
found two major technical issues with the Run 3 images: (a) the projected spot
grid did not cover an entire CCD; (b) the images were not sharply focused.
Therefore, in Run 5, we changed the acquisition to 2000 dithered images at the
center and four quadrants of the studied CCDs, spanning $\pm$ 5mm to cover the
entire CCD.

For Run 5 we also replaced a manual adjustable Z-stage on the XY stage with a
remotely controlled Z-stage to address the focus issue. This updated setup
significantly improved the focusing process. However, the structure of the
Z-stage introduced vibrations that were significant at the physical size of
the projected spots. Such effects were corrected through post processing since
we realized this issue during data analysis after Run 5 was completed. In
Section 2.5, we describe our corrections.

###  2.3 Artificial Stars Image Collection

An example of a single exposure taken during the imaging acquisition process
is shown in Figure 1. The left-hand panel illustrates our star-like field, a
square grid with $49\times 49$ optical spots projected on the CCD. Note that
the grid fills a large part of the detector in a single exposure. The star-
like physical size and Gaussian shape of the spots may be discerned from the
figure. The spots on the grid are equally spaced and have FWHM of $\sim 5$
pixels, corresponding to $\sim 1^{\prime\prime}$ in the LSSTCam focal plane.
With ellipticity of $\sim 0$, the spots are quite similar to point sources,
and their size was chosen to be equivalent to the PSF of the camera.

The optical setup creates image artifacts as shown on Figure 1. For instance,
the commercial lens vignettes the image. The spots on the edges are affected
and have significantly lower flux. In addition, the spots have a
characteristic shape and size that vary across the grid. The setup design can
then impact our analysis if not treated correctly. Therefore, having many
exposures and different spots for each CCD pixel is desirable to mitigate
effects from the experimental setup. Section 2.5 further describes our
statistical treatment of the experimental setup imperfections.

###  2.4 Single Image Processing And Grid Characterization

Here we describe the analysis for a single exposure, from raw image to the
resulting catalog of identified sources. The spot exposures presented in
Section 2.3 were analyzed using version 21.0.0 of the LSST Science Pipelines
(hereafter pipelines; https://pipelines.lsst.io; Bosch et al., 2018, 2019).
The pipelines is a set of data processing tasks actively being developed to
process the LSST data. The raw images taken of the spot grid were processed
using the standard pipelines instrument signature removal (ISR) task. An
extension to the standard pipelines for these lab spot images was developed
and used (mixcoatl). This procedure includes a bias level subtraction using
the row-by-row median value of the overscan region, 2D structure in the bias
using a medianed overscan-subtracted bias image, masking of pixel defects, and
applying gain correction as derived from the photon-transfer curves (PTC).

After ISR processing, the identification of sources and measurement of source
properties was performed using a custom source detection task
mixcoatl.characterizeSpots.CharacterizeSpotsTask. This task detects sources by
applying a maximum filter to the image and identifying peaks above a pixel
value threshold of 200 electrons. This methodology was needed because the
spatial variation of the projector’s scattered light background was ill-suited
for the background modeling and subtraction performed by the standard
pipelines source detection task. The detected sources’ fluxes, positions, and
second moments of brightness were measured using the SDSS HSM algorithm
(Hirata & Seljak, 2003; Mandelbaum et al., 2005).

The next step was to derive the properties of the projected grid from the set
of detected sources, including the overall magnification, the row/column
spacing, and the rotation of the grid with respect to the pixel array. We
filtered the outlier sources with the following threshold $2.0<I_{xx}~{}\text{
or}~{}I_{yy}<20.0\text{ px}^{2}$ in order to exclude sources that do not
correspond to points on the projected grid. The remaining sources were then
fit to an ideal grid model of $49\times 49$ spots, with three free parameters
corresponding to the x/y grid center position and $\theta$ the grid angle. We
employed a least-squares minimization approach to minimize the distances
between detected and grid model source positions. To initiate the grid model
fitting step, we utilized a convex hull technique to provide an initial guess.
The convex hull method proved to be more robust than using the commanded grid
center as an initial value. After determining a best-fit model grid, each
detected source was assigned an index label corresponding to its row and
column number in the projected grid; this identification allowed for tracking
individual sources across exposures. The per-source position residuals from
the ideal grid model were then calculated and recorded in the source catalog
for the measurement of optical and sensor distortions, as described in Section
2.5.

###  2.5 Measurement Calibrations: Residuals

Here we describe how we calibrate our measurements.

As shown in Section 2.3, the spots are not ideal point sources, and their
locations on the grid, shape deviations, and projector lens aberrations, for
example, impact their measured quantities. In order to remove effects of the
optical aberrations, we compute the residuals of a measured and ideal spot
property for a large collection of exposures centered at different positions.
The residual vector $\delta\vec{\ell}$ between the position of an ideal grid
spot $\vec{s}$ and the position of the corresponding detected spot $\vec{d}$
is (Snyder et al., 2021b):

$\delta\vec{\ell}=\delta\vec{\ell}_{optical}+\delta\vec{\ell}_{sensor}+\vec{\epsilon}\;,$
(1)

where $\vec{\epsilon}$ is a random error term, $\delta\vec{\ell}_{optical}$
represents displacements that can be caused by the optical setup, including
the mask used to generate the spots. Suppose we average the residual
$\delta\vec{\ell}$ of one single spot measured over several CCD pixels. In
that case, the sensor anomalies should average out, and only the constant
displacements from the optical setup should remain. For a large set of
residual measurements, the sensor anomalies should be:

$\delta\vec{\ell}_{sensor}=\delta\vec{\ell}-\left<\delta\vec{\ell}\right>\;,$
(2)

where $\left<\delta\vec{\ell}\right>$ is a constant value for one spot since
it is the average residual vector over the CCD pixels. For randomly
distributed errors, the expectation value of the error term is zero. The
statistical error can be reduced by increasing the sample size.

Similarly, the residuals of the PSF shape and size are computed in terms of
the residuals of the second moments of brightness (Bernstein & Jarvis, 2002):

$\delta I_{ij}=I_{ij}-\langle I_{ij}\rangle\;,$ (3)

The second moments of brightness are a building block for the PSF shape and
size measurements (Kaiser et al., 1995; Schneider, 2005):

$\displaystyle T$ $\displaystyle=I_{xx}+I_{yy}$ (4) $\displaystyle e_{1}$
$\displaystyle=\left(I_{xx}-I_{yy}\right)/T$ (5) $\displaystyle e_{2}$
$\displaystyle=2I_{xy}/T$ (6)

where $T$ is the PSF-size, and $e_{1}$ and $e_{2}$ are the $x$ and $y$
components of ellipticity. We note that others definitions of ellipticity from
the second moments of brightness are possible (Schneider, 2005). Also, the
above definition of PSF-size ($T$) relates to the PSF full width at half
maximum (FWHM) as:

$\text{FWHM}\equiv 2.355\sigma=2.355\sqrt{T/2}\text{ ,}$ (7)

if the profile is Gaussian.

The resulting residuals of the PSF shape and size are:

$\displaystyle\delta T$ $\displaystyle=\delta I_{xx}+\delta I_{yy}$ (8)
$\displaystyle\delta e_{1}$ $\displaystyle=(\delta I_{xx}-\delta
I_{yy})/\langle T\rangle$ (9) $\displaystyle\delta e_{2}$
$\displaystyle=2\delta I_{xy}/\langle T\rangle\;.$ (10)

We computed the fractional flux residuals in terms of the ratio:

$\frac{\delta f}{f}=\frac{f-\langle f\rangle}{\langle f\rangle}\;,$ (11)

For small fluxes residuals ($\delta f/f$) can be expressed as a multiplicative
factor, where $\delta m\approx-1.08573\delta f/f$.

For the record, we used the following measurements quantities of the
pipelines: base_SdssShape_instFlux, spotgrid_x/y, base_SdssShape_xx/xy/yy.
These are the mixcoatal outputs for $f$, $\vec{\ell}$, and $I_{xx}$, $I_{xy}$,
$I_{yy}$, respectively.

###  2.6 Vibration Correction

After completion of Run 5 data taking, a vibration of the spot projector setup
was identified in the collected images. The vibration appeared to be induced
when the stage translating the projector in x and y stopped moving, and it had
a long settling time. As a result, the spot images were slightly enlarged. The
effects were significant on a sub-percent level for the positions, sizes and
shapes of the spots.

he effects were removed by the following procedure. We assumed that the
measurement residuals, e.g., second moments of brightness $\delta I_{ij}$, are
to be zero on average. The measurement residuals for $k$-th exposure were
fitted with a plane $a_{k}x+b_{k}y+c_{k}$ by varying $a_{k},b_{k},c_{k}$ for
each exposure so that we minimize the summed square of the difference between
the measurement residual and the plane model. If the effect was constant
across the exposure subtracting off $c_{k}$ should be enough. However, the
subtraction of a plane from each exposure was needed empirically. The source
could be the combination of the effects of vibrations and the tilt of the
projector with respect to the focal plane that are different in different
dithered exposures. The operation was applied for the $\delta\ell$, $\delta
I_{ij}$, and $\delta f$ quantities exposure-by-exposure. In particular, this
calibration had to be done in the second moments of brightness rather than the
final products, such as ellipticities and shear, because they are not linear
quantities under this operation.

##  3 Distortions of Photometry, Centroid, PSF Size and Shape

In this section, we report our measurements of the intrinsic sensor distortion
in photometry, centroid, and PSF shape and size for six LSSTCam sensors. Then
we describe the most-important distortions revealed by the residual maps of
the two CCD designs, ITL and e2v. The physical nature of the effects is
discussed in Sections 5, 6.

###  3.1 Effects of Sensor Anomalies on measurements

We show the intrinsic sensor pixel response distortion maps in Figure Figure
2, which were created using calibrated flux, centroid, PSF size, and shape
deviations measurements to reveal sensor anomalies as described in Section
2.5.

We used a large collection of star-like exposures to obtain highly precise
measurements. The faints sources were masked by selecting only the top 80%-th
percentile of flux (see Figure 1). In addition, we binned the deviation
measurements by pixel location and used super-pixels of size 10 $\times$ 10
pixels to create the deviation maps, which were then stretched back to their
original size.

For instance, in Run 5, which consisted of 2000 exposures, we achieved a noise
level of $10^{-4}$ pixel (physical size $=1$ nm), and a centroid shift and
shape measurement noise level of $10^{-5}$. Each exposure contained up to 2401
sources, resulting in an average of 10 sources per super-pixel.

Figure 2: Maps of deviations of measurement quantities for two LSSTCam
sensors, top: E2V R32_S01 and bottom: ITL R10_S11. The measured quantities are
flux, PSF size, centroid, and shape. Shifts of centroid and shape shown in $x$
and $y$ components separately. The photometric map corresponds to a flat image
for the sensor but made by stitching measurements of spots. Quantum efficiency
variations across amplifiers are evident features in the these maps. The
structures in the PSF size, position, and shape residual maps are caused by
pixel-area variation effects. Some striking examples are the tree-ring pattern
and the amplifier boundary effects. Note: the lower-right corner of R03_S12 is
an image artifact, the spot images in this region are likely affected by stray
light reflections from the side of the cryostat window.

The deviation maps for an E2V and an ITL sensor are presented in Figure 2,
with six different measurements shown for each sensor. Additional deviation
maps for other sensors can be found in Appendix B. To relate to the electric
fields in the serial and parallel transfer directions of a sensor, we present
the deviation maps separately for the x and y components. A brief description
of each deviation map is provided.

Photometry deviations: This map corresponds to a “flat” image but is made by
stitching the deviations of flux measurements from the calibrated artificial
stars. The e2v and ITL sensors display two distinct features in the maps: a
rectangular shape associated with the 16 CCD segments, and irregular patterns
resembling “coffee stains” in the ITL sensors and an “annealing pattern” in
the e2v CCDs. The e2v sensor exhibits an additional radial gradient feature.
The lower right corner of the ITL sensor exhibits an important feature, which
is likely attributed to inconsistent illumination from the cryostat entrance
window. In addition, the right-most segment of the lower half of the sensor
has a bad amplifier likely due to poor charge transfer efficiency. These
features are discussed further in Section 6.1.

PSF size deviations: The maps on the lower left in each group present the
measured variations in the size of the PSF. The variations are at the sub-
percent ($\times 10^{-3}$) level. They are related to effects that increase or
decrease the size of a point source. One of the interesting observations is
that we do not see the annealing pattern for e2v or the coffee stain feature
for ITL in PSF size variations. We do see a circular ripple pattern centered
on the outside of the sensor. It appears to be matched with the “tree-rings”.
We discuss this effect in Figure 2. An unexpected structured variation in PSF
size is also observed on the ITL sensors, with circles in three corners and a
square in the middle. This pattern matches the hardware CCD frame that holds
the detector. We investigate this feature further in Section 6.3.

Centroid deviations: The two maps in the middle column show the $x$ and $y$
centroid deviations. The measurement displacement from the calibrated source
is of order $0.01\,\text{ pixel}=100\,{\rm nm}=2$ mas on the LSST focal plane.
The circular pattern of the tree-ring effect is the most noticeable feature in
the centroid residual maps. We analyze this effect further in Section 5. In
the ITL sensors, we do not see features related to the CCD shapes other than
the mid-line break on the $\delta y$ map. In contrast, a strong mid-line
signal is evident for e2v along with strong contrast between the CCD segments’
edges in both directions ($x$ and $y$). Beyond those effects, we see irregular
spatial feature variations in both CCDs.

PSF shape deviations: The two maps in the right-hand column show the
variations of measured ellipticities from the calibrated spots. Although less
pronounced, the PSF shape residual map has similar features to the centroid
residual map. Specifically, the shared distortions are the tree-ring features,
the mid-line break, and the global variation. In contrast, the segment edges
are not prominent in the ellipticity variations except for the segment of
R03_S12 that has poor serial charge transfer efficiency. In Section 4, we
investigate further the distortions related to the CCD segment boundaries.

In summary, Table 1 lists the most-prominent features. The table displays the
maximum absolute deviation values measured in the six CCDs examined in this
study.

Table 1: Summary of the LSSTCam sensor effects limits on photometry (phot),
centroid (center) and PSF-size and shape.

Origin | Name | Level | CCD Type
---|---|---|---
|  | phot. res [mmag] | centr. res [pixel] | psf-size frac. dev | psf-shape dev |
Scientific Goal | Requirements | $<10.0$ | $<5\times 10^{-2}$ | $<1\times 10^{-3}$ | $<1\times 10^{-3}$ | Both
| References |  Section 5.5(a,b) |  Section 7.3(a,b) |  Section 7.3(c) |  Section 7.3(a,c) |
| Mid-Line Break | – | $<4\times 10^{-2}$ |  | $<4\times 10^{-3}$ | Both
Pixel-Area Varations | Hardware Imprints | – | $<1\times 10^{-2}$ | $<1\times 10^{-3}$ |  | Both
| Amplifier Boundaries | – | $<4\times 10^{-3}$ |  | $<4\times 10^{-4}$ | Both
| Tree–Rings | – | $<4\times 10^{-3}$ | $<5\times 10^{-4}$ | $<5\times 10^{-4}$ | Both
| Radial Gradient | $<25.0$ | – | – | – | E2V
Photometric Response | Residual Impurities | $<4.0$ | – | – | – | Both
| Amplifiers Gain variation | $<1.0$ | – | – | – | Both

references: (a)LSST Science Collaboration et al. (2009); (b)Ivezić et al.
(2019); (c) The LSST Dark Energy Science Collaboration et al. (2018)

##  4 Effects of edges, amplifier boundaries and mid-line break

This section closely examines the anomalies at the CCD sensors amplifier
segments and edges.

The deviation maps have global features that can mask effects at the edges,
amplifier boundaries, and the mid-line break. We remove such features by
applying a high pass filter to the maps of variations with pixel periods
higher than 250 pixels. Figure 3 presents the cleaned residual map of the e2v
R32-S01 sensor on the left. We compute the signal profile in the $x$ and $y$
directions for bands of 500 pixels width as indicated in the left-hand plot.
The middle and the right-hand columns show the resulting vertical and the
horizontal signal profiles, respectively. In the signal profile plots, the
amplifier boundaries and the mid-line break are represented by grey dashed
lines. We define the signal noise (red dashed lines) as the standard deviation
of the signal.

Figure 3: Distortion maps (left) and signal distortion profile across the
horizontal (middle) and vertical (right) directions. A high-pass filter was
applied to the images in the left-hand column to highlight the high-frequency
distortions. The profiles are the map signal computed over the regions of the
gray bands; the red dashed lines indicate the error level (rms). In the right,
the $e_{1}$ and $\delta x$ horizontal profiles show the effects at the
amplifier boundaries (gray dashed lines). In the middle column, the mid-line
break is the most noticeable feature, especially for $\delta y$, with a signal
level $100\times$ higher than the noise.

The e2v sensor exhibits significant distortions in the vertical profiles,
particularly at the mid-line break. This feature is evident in all residual
quantities, the peak is about 100 times greater than the noise for the
y-component maps. Away from the mid-line, we see other patterns in the flux
profile, for instance, the residual surface features (periodic
stripes/rectangles, etching) and increased distortion at the edges.

The photometric deviations have a pattern of distortions at the amplifier
boundaries with amplitude on the order $10^{-3}$. The observed sharp constants
on flux, despite gain corrections being applied to each segment, could
potentially be attributed to residual mismatches between the amplifiers.
Alternatively, they might be artifacts originating from the high-pass filter.
To fully understand this characteristic, a more comprehensive evaluation is
required.

In the other sensors of this study, we see some diversity in the values
measured for the mid-line break and amplifier boundary features. For example,
while all the e2v sensors studied here have mid-line break peaks of amplitude
$>10^{-4}$ on PSF-shape, the ITL sensors show peak amplitudes between $6\times
10^{-5}-3\times 10^{-2}$, see Appendix B. The amplitudes at amplifier
boundaries are also different among sensors from the same vendor. For R03_S12
Figure 8, the amplifier boundaries in the horizontal profile has peaks for
certain segments. The ITL sensors have a problematic lower right segment with
distortions in flux and y-components of the centroid and PSF-shape, possibly
due to serial charge-transfer inefficiency and vignetting by the entrance
window.

##  5 Tree-rings

In this section we focus on the signatures of tree-rings. We first present the
relation between the flat-field distortion due to tree rings and centroid and
PSF shape and size changes. Then, we describe our method to measure the
tangentially averaged effects in the polar coordinate system. Finally, we
present the tree-rings oscillatory radial distortions and compare them with
the flat-field signal.

###  5.1 Pixel Area Variation

In Plazas et al. (2014a, b), the tree-ring effect in the DECam camera sensors
is interpreted as effective changes of pixel area caused by a lateral electric
field. Any quantities dependent on the pixel area are affected by this
distortion. Here we follow the same approach to interpret the tree-ring effect
in the LSST sensors in terms of pixel area changes.

The centroid shifts are modeled as the displacement $d(r)$ of the centroid of
photons incident at a radial distance $r$ from the inferred center of the
tree-rings compared to the actual radial displacement from the center
$r_{0}=r-d(r)$. Then, the corresponding area distortion, $w(r)$, can be
calculated as the Jacobian determinant of the coordinate transformation
$r\rightarrow r_{0}$ (Plazas et al., 2014a, b; Okura et al., 2015):

$1+w(r)=\left|\frac{r_{0}dr_{0}d\theta}{rdrd\theta}\right|\approx\left|\frac{(r-d(r))(dr-\partial_{r}d(r)dr)}{rdr}\right|\;,$
(12)

At the first order one can show:

$\frac{\partial d(r)}{\partial r}=-w(r)-\frac{d(r)}{r}\approx-w(r)\;,$ (13)

we ignore $d(r)/r$ as the centroid shift is sufficiently small compared to the
radius from the center of the tree-rings. Note that we follow the definition
of Okura et al. (2015) for the distortion of $d(r)$. As a result, equation
Equation 13 has sign opposite to the convention in Plazas et al. (2014a).

Further, we can define magnification in size $T^{1/2}$ and change in
ellipticity $\delta e$ and relate them to the area perturbation $w(r)$. The
perturbation on $T^{1/2}$ can be written as (Okura et al., 2015):

$\frac{(T+\delta T)^{1/2}-T^{1/2}}{T^{1/2}}=-\frac{1}{2}\left(\frac{\partial
d(r)}{\partial r}+\frac{d(r)}{r}\right)=-\frac{1}{2}w(r)\;.$ (14)

For a first order expansion,

$\frac{(T+\delta T)^{1/2}-T^{1/2}}{T^{1/2}}\approx\frac{\delta T}{2T}\;.$ (15)

Thus, the fractional change in PSF size is equal to the opposite of the flat-
field distortion:

$\frac{\delta T}{T}=-w(r)\;,$ (16)

Similarly, the change in shear due to the tree-rings $\gamma_{TR}$ (Okura et
al., 2015) in conjunction with equation 13 can be shown to be:

$\displaystyle\gamma_{TR}(r)$ $\displaystyle=\frac{1}{2}\left(\frac{\partial
d(r)}{\partial
r}-\frac{d(r)}{r}\right)=-\frac{1}{2}\left(w(r)-2\frac{d(r)}{r}\right)$ (17)
$\displaystyle\approx-\frac{1}{2}w(r)\;.$

Our ellipticity definition, Equation 6, is a factor 2 times the shear
(Schneider, 2005); thus we can write:

$\delta e_{r}\approx 2\gamma_{TR}\approx-w(r)\;.$ (18)

Equations 16, 18 equate the PSF size and shape distortions to the opposite of
the flat-field distortion $w(r)$.

###  5.2 Tree-Rings Coordinate System

The centroid and shape tree-rings distortions depend on the coordinate system.
In contrast, the PSF-size and the flat-field distortions are invariant
quantities. To measure the tree-ring distortion effects, we transform
$(\ell_{x},\ell_{y})$ and $(e_{1},e_{2})$ to polar coordinates by applying the
rotation matrix. It is worth noting that the rotation matrix used for
ellipticities is slightly different since they are pseudo-vectors:

$\left(\begin{array}[]{c}e_{r}\\\
e_{\theta}\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\phi&\sin 2\phi\\\
\sin 2\phi&-\cos 2\phi\end{array}\right)\left(\begin{array}[]{c}e_{1}\\\
e_{2}\end{array}\right)\;,$ (19)

where $\phi=\tan^{-1}\left(\frac{y-y_{0}}{x-x_{0}}\right)$. The factor $2\phi$
is a consequence of the invariance of ellipticities against $180^{\circ}$
rotations. The center of this coordinate system $(x_{0},y_{0})$ is the center
of the tree-rings circles, which is outside the CCD.

###  5.3 Algorithm

As we saw in the previous sections, our residual maps have the tree-rings and
other features, such as global variations or structures associated with the
amplifier boundaries. Therefore, to determine the components due to tree-
rings, we perform image processing described in the following steps:

  1. 1.

Image pre-processing: We clean the residual images. As other effects impact
the residual maps at larger angular scales, we apply a high-pass filter to
highlight the tree-ring effects. First, we binned the maps by 8$\times$8
pixels to increase the signal-to-noise ratio of the features and apply a high-
pass filter to remove the global variation with pixel frequency higher than
250 pixel, as described in (Park et al., 2020). During this process, we also
identify bad pixels and mask them. Finally, the image is stretched to the
original size.

  2. 2.

Polar Transformation: We convert the original rectangular CCD pixel image
(x,y) to the polar coordinate system with respect to the tree-rings center. We
use (warpPolar from opencv) and the wafer center (Park et al., 2020). In this
operation, the output is an image in $(r,\theta)$ coordiinates where $r$ is
the distance from the tree-rings center. In this step, we check whether the
steps above are successful by examining the transformed image. If the tree-
rings center is misidentified, the straight line of the tree-rings along with
the $\theta$ direction would be tilted or disturbed.

  3. 3.

Extracting the profile: We average the ($r,\theta$) image over $\theta$ to
evaluate the profile in $r$.

The tree-rings center can be approximated by the center of the silicon wafer
in most cases (see their figure 2. Park et al., 2017). However, we noticed a
center mismatch in some cases after visually inspecting the polar images.
Since these mismatches affected the tree-rings signal, we measured the tree-
rings center on flat-field images with a back-bias voltage equal to zero, as
the signal is more prominent at this voltage setup (Park et al., 2017).
Following Park et al. (2017, 2020) algorithm, we fit tree-rings center values
using 30 flat-field images. This verification test showed that the differences
between the silicon wafer center and the actual center have an r.m.s of $75$
pixel. For the cases where the signal was affected by the center mismatch we
use the fitted values.

###  5.4 Measured Distortions Due to Tree Rings

Figure 4: Distortions due to tree-rings for PSF size (left), shape (middle)
and centroid (right) deviations as a function of distance from the the tree
rings center for the sensors R32_S01 (top) and R03_S12 (bottom). The radius
range was limited to stress the oscillatory features of the signal. The
distortion $w(r)$ inferred from the flat fields (black dashed line) is
correlated with the measured quantities. Note that the centroid shift is
related to the derivative of the tree-rings distortions .

Using the procedure described in Section 5.3, we extracted the one-dimensional
profiles of distortions due to tree-rings, finding amplitudes at the $10^{-4}$
level for centroid, PSF shape, and size distortions. Figure 4 shows the radial
profiles of the oscillating distortions. For the centroid shift, we show the
derivative of the centroid shift with respect to the radius.

As described in the previous Section, the tree-ring distortions of centroid
and PSF shape and size are directly related to the distortions inferred from
flat-field images; see, e.g., Equation 18. For this reason, on each plot we
overlay the distortions inferred from flats. Although inferred distinctly, the
distortions that we infer from tree-ring effects are very close to the direct
measurement of the flat-field image distortions presented in Section 3. The
amplitudes and the phases of the oscillations matches qualitatively the flat-
field signal. The correlation coefficients close to unity (see Figure 4)
confirm the visual similitude and validate the relation between the tree-ring
effects on pixel-area quantities as expressed by equations 13, 16, 18. Also,
the amplitudes of the oscillating patterns increase with distance from the
center as studied in (Park et al., 2017).

###  5.5 Corrections: Tree-Rings

The centroid displacements due to tree-ring effects are smaller than needed to
satisfy the LSST science requirements on centroid (The LSST Dark Energy
Science Collaboration et al., 2018). Nevertheless, given the exquisite control
on systematics demanded by LSST, the tree-ring effects could be corrected by
measuring their radial profiles from flats (and photometry) and star flats
(centroid; or using the formula 13 to extract the profiles of centroid shifts
from flats). The profiles would be incorporated as templates in the centroid
and photometric solution optimization during image reduction, with an
amplitude parameter that depends on the filter band (see Plazas et al. (2014a,
b); Bernstein et al. (2017a, 2018, b)).

This approach could be incorporated into the pipelines (Bosch et al., 2018,
2019) for correcting centroid distortions. However, it does not apply to PSF
size. As presented in Jarvis et al. (2021), the DECam detectors show PSF size
distortions similar to the tree-rings even after full centroid corrections.
The remaining distortions of PSF size are suspected to be due to diffusion
(Magnier et al., 2018). Therefore, a model for the PSF size distortions is
required. For example, if these residual patterns are caused by charge
diffusion, they could be incorporated directly into the PSF model as a
diffusion component. This model can be a static map for a given sensor that
sets the size distortion depending on the pixel location.

##  6 Other effects

In addition to the tree-ring effects, we discuss a couple of other effects
that we observed in Section 4.

###  6.1 Quantum Efficiency Variations

The patterns in the photometric residual maps for e2v and ITL are different.
The detailed patterns can be interpreted as quantum efficiency variations that
are caused by the back-side surface finish in the manufacturing process. The
ITL pattern in the flux distortion map (coffee stains) is due to a layer of
non-stoichiometric oxidized silicon and cleaning residue of acid on the
silicon surface right after etching, creating some non-uniformity in backside
charging (Bajat et al., 2020). Instead, the e2v sensors have a regular striped
pattern in the flux distortion map which appears to be caused by the laser
annealing process after the thinning process in the CCD fabrication (Burke et
al., 2004; Radeka, 2006; Bender et al., 2014). This residual surface effect is
generally greater at shorter wavelengths since the blue photons are converted
close to the CCD back-side. The amplitude of this effect should be at most
$10^{-2}$ and almost undetectable in redder wavelengths than $\sim$500 nm from
the verification tests on flat-fields (Park et al., 2017; Roodman et al.,
2018).

The origin of the radially symmetric gradients in e2v maps is not clear. We do
not see a similar pattern in regular flat images taken with a flat
illuminator. The dithering of the spot projector might cause this pattern;
however, the fact that the maps for ITL based on the same projector dithering
pattern do not have the same radially symmetric pattern suggests otherwise.
Further investigation using on-sky images will give more understanding of this
effect.

###  6.2 Mid-line Break & Amplifier Boundaries

The mid-line break, which divides the top and bottom halves of the CCDs, is
the most striking sensor feature, clearly visible in the map of both types of
sensor Figure 2. Also evident are features associated with the readout
amplifier boundaries, which divide the sensors vertically. These effects are
stronger for the e2v CCDs and can be up to $\times 10$ greater than for ITL
sensors. This difference stems from design differences for the electronic
readouts. For instance, e2v CCDs have a physical boundary that blocks electron
flow between the upper and lower halves. On the other hand, ITL sensors have
less-prominent mid-line effects because they have channel stops in the
electronics readout similar to the amplifier boundary structures. The
importance of the mid-line can vary between the sensors of the same design and
can be greater than reported here. Mid-line break distortions can be explained
by the pixel-area variation associated with the electric field distortions
created at the boundaries between channels, caused by the non-uniform
distribution of holes around the channel stop (Tearing, Juramy et al., 2020,
see their section 2.2.3). We discuss this in Section 7.1.

The photometric distortion maps show sharp contrasts at the segments’ edges;
see Figure 2. The flux values might have been affected by the gain (e-/ADU)
measurement. Although during our imaging processing (see Section 2.4) we
accounted for the gain differences between CCD segments we still see a
residual gain at a sub-percent level between the segments. As the PTC gain
determination is sensitive to additional sources of variance (Stubbs, 2014), a
possible explanation is that the distortions at amplifier boundaries might
have caused the gain mismatch between segments.

In addition, there are extreme examples, for instance the R03_S12 sensor, for
which the amplifier at the bottom right shows a percent-level gain contrast.
This segment suffered by stray light reflections from the side of the cryostat
window. As a result, the shadow made a significant impact on the PTC
determination.

These results show that the accuracy of the gain determined from PTC flat pair
measurement that we currently achieve still results in a sub-percent
discontinuities between amplifiers by the current implementation of the ISR.
This suggests that further gain adjustment across amplifiers or more
sophisticated PTC gain determination is needed.

###  6.3 Imprints From Hardware Structure: CCD Frame

Multiple effects leave their imprints on the centroid and PSF-size maps of
several sensors, including ITL’s R03/S12, R10/S11, and e2v’s R22/S11. In
Figure 5, the PSF size map of R10/S11 is shown alongside a photo of the ITL
CCD support structure. Interestingly, the structure in the residual maps
aligns with the location of the alignment pins and the hold-downs (Lesser &
Ouellette, 2017).

Figure 5: Left: The back structure of the ITL CCD (gold) is seen with two
alignment pins (green) and three hold-downs (red). The connector to the flex
cable is within the center box, where no gold metal support is present. Middle
and right: PSF size residual map without and with a CCD frame drawing (red).
The distortions of PSF size follow the shape of the CCD support frame.

For the DECam CCDs, Bernstein et al. (2017a, see Figure 8) showed that the
connectors impact their astrometry (centroid) when stacking all CCD images
with different wavelengths (in the gri bands). They concluded this is likely
results from stresses induced in the CCD lattice by the connector or the hole
in the mounting board. In addition, they also identified a wavelength-
dependent feature related to the CCD frame metallic structure (Bernstein et
al., 2018). The effect was strongest for the Y filter owing to the
reflectivity of the metal structure, since a significant fraction of the
infrared photons pass through the sensor. However, the ITL sensors for LSST
have a highly IR-absorbing material “lithoblack” deposited on the sensor
wafer’s front side surface to prevent such an effect (Lesser & Ouellette,
2017). In any case reflection of blue photons that would have passed through
the CCD is implausible.

###  6.4 Artifacts

The photometry distortion map in Figure 7 has a noisy pattern. We do not have
a clear explanation for this effect. It is likely caused by code failures
related to determining the flux normalization since the other maps do not have
corresponding effect.

There are shadows features in photometry distortion maps and similar features
in the size, centroid and shape maps at $x=2500,y=1000$ Figure 2 (both
sensors) and at $x=2500,y=1500$ in Figure 7. In a single raft testing setup,
we confirm these features are caused by fingerprints on the aperture window.

##  7 Discussion

###  7.1 Tearing mitigation

Juramy et al. (2020) discussed tearing patterns associated with the mid-line
break and amplifier boundaries in e2v flat-field images. The tearing is
suggested to be caused by the non-uniform distribution of the holes along the
channel stops and the mid-line break. A significant effort to mitigate the
tearing feature has been made since then. For Run 3, we operated the e2v
sensors with a unipolar voltage setup while for Run 5 we operated the e2v
sensors with a bipolar voltage setup (see Table 2) as recommended by Juramy et
al. (2020). This change in operation voltage made a significant impact on the
tearing features, mostly invisible to the eye. However, we did not find any
obvious improvement or decrease of the effects at the amplifier boundaries and
at the mid-line break. When we compare two set of profiles of R32_S01 (Run5)
or R24_S11 (Run 5) and R22_S11 (Run 3) in the Appendix, the mid-line break and
amplifier boundaries distortion are essentially the same.

This fact indicates that we need either 1) to mask out the adjacent pixels
nearby the amplifier boundaries or the mid-line break, or 2) to develop
further improvement in the ISR. As the amplitude of theses segments effect is
at same level as the photometry residual maps, the electric field distortions
is likely the cause. The same correction algorithm as for the tree-rings
effect could be applied; evaluation is underway.

###  7.2 Tree-Rings In Other CCD Devices

The presence of tree-ring features is reported in other large CCDs cameras
such as DECam (Flaugher et al., 2015b), Hyper Suprime-Cam (Kamata et al.,
2014a), and PanSTARRS GPC1 (Magnier et al., 2018). The relative amplitudes of
the effect range from 0.1% to 10% in those systems.

Plazas et al. (2014b) reported the amplitude of tree-ring effects in DECam
flat images is at the 1–10% level and concluded their nature is pixel size
variations by comparing their photometric and centroid deviations. A similar
effect has also been reported in the Hyper Suprime-Cam (Kamata et al., 2014b).

In contrast, a unique effect, referred to as ‘charge diffusion’, was
identified in the GPC1 detectors (Magnier et al., 2018). Although this effect
displays similarities to the tree-ring signal in terms of photometry and PSF
size, it is actually driven by variations in the rate of vertical charge
transportation. The primary outcome is a charge diffusion of variable length
that predominantly affects the PSF size, but not the shape.

Later, the tree-rings effect on PSF-size was also seen in a few CCDs in DECam,
after application of a correction for tree-ring distortion based on the pixel
size variations (Jarvis et al., 2021). The residual amplitude is much more
prominent in the blue band, which indicates it occurs at the surface where
light enters the CCD. This residual could be interpreted as being due to
charge diffusion. In the LSSTCam, further on-sky studies should quantify the
dependence of the effect with wavelength.

In this study, we observed strong correlations between size $(dT/T)$, shape
$\delta e$, and centroid shift $d(r)$ distortions and the flat signal $w(r)$,
as predicted by equations 13, 16, 18. Consistent with the findings of (Plazas
et al., 2014b), we propose that the distortions caused by tree-rings are due
to changes in the effective pixel area. These changes are likely the result of
shifts in the lateral electric field, which are caused by variations in doping
and charge diffusion effects. Notably, the impact of charge diffusion
variability on PSF size distortions is comparable to the influence of tree-
ring effects on centroid and PSF shape distortions. Such distortions are
greater at bluer wavelength and should be reduced or almost non-existent in
the redder filters (Park et al., 2017). It is likely that no additional
component in the correction is needed beyond 450 nm, which supports that our
sensors are less affected by the diffusion effect.

###  7.3 Survey Requirements

The LSST science goals (LSST Science Collaboration et al., 2009; The LSST Dark
Energy Science Collaboration et al., 2018) guide and set the necessary
accuracy of the corrections for distortions due to sensor anomalies.
Uncertainties in the final LSST galaxy shear catalog are to be dominated by
the statistical error, $10^{-3}$, which sets an upper limit on contributions
from systematic errors (LSST Science Collaboration et al., 2009). In addition,
the maximum acceptable PSF size bias $\delta T/T$ is $10^{-3}$ for the ten-
year Rubin/LSST survey (The LSST Dark Energy Science Collaboration et al.,
2018). In addition to PSF size and shape, photometric accuracy is essential to
several scientific applications; the photometric error is expected to be below
10 mmag (Ivezić et al., 2019). Furthermore, the solar system bodies’ parallax
and proper motion measurement should have errors below 10 mas (Ivezić et al.,
2019).

Most of the sensor anomalies reported in this study do not violate the
inferred performance requirements, as presented in Table 1. The mid-line break
is the only distortion with an error on the order or greater than the centroid
and shape requirements. Fortunately, the effects can be easily corrected.
Below, we discuss approaches to mitigate the sensor effects:

  * •

Mid-line break: This is the most pronounced annomaly, and the simplest
solution is to mask the region, albeit at the cost of losing approximately
0.5% of the sensor area. An alternative solution is to flag the sources
detected within these areas as lower quality. An elaborated approach is to
model the distortion signal with a functional form, assuming the effect is
static. This solution is somewhat similar to the brighter-fatter effect
correction (e.g. Lage et al., 2017). Further assessment of the feasibility of
such corrections is needed.

  * •

Amplifier boundaries: A solution similar to that employed for the mid-line
break could be implemented.

  * •

Tree-rings: The correction is discussed in Section 5.5; however, the
modulation of $10^{-4}$ level in a flat due to the tree-ring effect is
negligible for cosmology with LSST (Okura et al., 2016).

  * •

Hardware imprints: The effect is unlikely to have an easy solution. For
instance, Bernstein et al. (2017a) masked the regions that had imprints from
the metallic structure. A similar solution could be employed, but the affected
regions of the sensors in the LSSTcam is significant. Given the effect can be
up to $10^{-3}$ on PSF-shape further assessment of science impact is needed.

  * •

Edge effects: Bradshaw et al. (2018) reported on CCD edge effects. The
metallization around the edges of the CCDs is set at a positive potential that
induces lateral electric field shift up to $\geq$ 10 pixels into the bulk. The
footprint of the artificial stars is not extended enough to cover the CCD
edge, so we cannot provide any guideline from this work. However, CCD edge
effect also should be taken into account in image reduction.

On-sky validation of the CCD sensors should be performed, and the 189 LSSTCam
science sensors should have their distortions quantified and compared to the
LSST science requirements. As shown in the distortion maps (Appendix B), the
level of the sensor distortions varies even among sensors of the same type.

Furthermore, the wavelength dependence of sensor anomalies should be
evaluated, particularly their effects on PSF size and shape (Meyers & Burchat,
2015; Kamath et al., 2020). Also, fringing effects in the y-band should be
quantified and the proposed corrections for this effect validated (Guo et al.,
2022).

##  8 Conclusion

This work represents the first probe of the impact of CCD anomalies on
photometry, centroid and PSF measurements of the LSSTCam. We analyze the
impacts using measurements artificial stars on a subsample of six LSSTCam
sensors. We classify a variety of distortions according to their source: CCD
mid-line break, hardware imprints, amplifier boundaries and tree-rings. We
report our main findings below.

  * •

The centroid distortions are much less than the LSST Year 10 limits, below
0.04 pixel which is equivalent to 8 mas. The greatest centroid distortions are
due to the mid-line break, a design feature present only on the e2v sensors in
LSSTCam. In contrast, the ITL sensors have an unexpected hardware imprint from
the metallic structure of the CCD frame with a maximum distortion of 0.02
pixel. The centroid distortions due to amplifier boundaries and tree-rings,
are 10$\times$ smaller.

  * •

The shape distortions presented here are similar to the centroid ones.
However, the amplitude of the mid-line break distortion in the shape
measurements is slightly greater than the LSST requirement. Since the area
affected is less than $0.5\%$ of the CCD area we suggest masking a region
around the mid-line of e2v CCDs during image reduction.

  * •

The photometric distortions originate with spatial variation of quantum
efficiency; their effect is not higher than 3 mmag. The main features observed
are due to gain variations between the sixteen CCD segments and global pattern
features. The global features distortions have distinct visual appearance
between the two CCD designs. For ITL sensors, the distortions have the
appearance of ‘coffee stains’ while for e2v sensors the appearance is due to
laser annealing. The differences may be traced to differences in the
manufacturing processes, in particular, the CCD back-side silicon treatment
procedure.

  * •

The tree-rings distortion effect measured for centroid shift is of the order
of $10^{-4}$ pixel and $10^{-4}$ for PSF size and shape. These changes can be
related to the distortions measured in flat-fields through the transformation
defined in equations 13, 16, 18. Therefore, if necessary the effect can be
corrected with the use of flat-fields.

We find variations in distortion even among sensors from the same vendor.
Further on-sky studies are needed to probe the variation of these effects for
the 189 LSSTCam science sensors, as well as to study their dependence on
wavelength. The laboratory study we present here provides a foundation for
understanding those effects in the entire focal plane using on-sky data.

##  9 Acknowledgments

This material is based upon work supported in part by the National Science
Foundation through Cooperative Agreement AST-1258333 and Cooperative Support
Agreement AST-1202910 managed by the Association of Universities for Research
in Astronomy (AURA), and the Department of Energy under Contract No. DE-
AC02-76SF00515 with the SLAC National Accelerator Laboratory managed by
Stanford University. Additional Rubin Observatory funding comes from private
donations, grants to universities, and in-kind support from LSSTC
Institutional Members. We sincerely thank Christopher Waters and Eli Rykoff
for their thorough review of this manuscript and their invaluable feedback.

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##  Appendix A Operation voltages

The LSSTCam operation voltags setups using the during the data acquisitions
are presented in Table 2.

| e2v (Run 3) | e2v (Run 5) | ITL (Run 3) | ITL (Run 5)
---|---|---|---|---
Output Drain [V] | 24.4 | 23.4 | 25.0 | 26.9
Reset Drain [V] | 12.7 | 11.6 | 13.0 | 13.0
Guard Drain [V] | 26.0 | 26.0 | 20.0 | 20.0
Output Gate [V] |  $-$2.2 |  $-$3.4 |  $-$2.0 |  $-$2.0
Backbias [V] |  $-$50.0 |  $-$50.0 |  $-$50.0 |  $-$50.0
Parallel Clock High [V] | 3.4 | 3.3 | 2.0 | 2.0
Parallel Clock Low [V] |  $-$5.8 |  $-$6.0 |  $-$8.0 |  $-$8.0
Serial Clock High [V] | 4.7 | 3.9 | 5.0 | 5.0
Serial Clock Low [V] |  $-$4.2 |  $-$5.4 |  $-$5.0 |  $-$5.0
Reset Gate High [V] | 6.4 | 6.1 | 8.0 | 8.0
Reset Gate Low [V] |  $-$3.4 |  $-$4.0 |  $-$2.0 |  $-$2.0
Gain | 0 | 0 | 0 | 0
RC | 3 | 14 | 3 | 14
sequencer file | FP_E2V_2s_ir2_v2.seq | FP_E2V_2s_ir2_v26.seq | FP_ITL_2s_ir2_v3.seq | FP_ITL_2s_ir2_v26.seq
Table 2: Nominal operational parameters for both e2v and ITL sensors in the
different run campaigns for which we acquired spot projector data. See Figure
2 in Snyder et al. (2018) for the schematic diagram of voltages. RC and Gain
are configurable settings in the readout electronics boards (Juramy et al.,
2014)

##  Appendix B Supplemental Figures

We show the figures 2, 3, and 4 for the other sensors studied: E2V - R24_S11
and R22_S11, and ITL - R10_S11 and R02_S02.

Figure 6: Same as Figure 2 caption.

Figure 7: Same as Figure 2 caption. The data was collected during the Run 3
testing period. In this period the coverage of the fiducial spot projector
position didn’t extended to cover the entire CCD, which results in the
circular shape map in the center of the CCD. Figure 8: Same as Figure 3
caption. Figure 9: Same as Figure 3 caption. Figure 10: Same as Figure 3
caption. Figure 11: Same as Figure 3 caption. The data was collected during
the Run 3 period. Figure 12: Same as Figure 3 caption. The data was collected
during the Run 3 period.

Figure 13: Same as Figure 4 caption.