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

The Vera C. Rubin Observatory’s LSST Camera (LSSTCam) 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. This laboratory study using artificial stars projected on the sensors shows overall small effects. The residual effects on point-spread function (PSF) size and shape are below 0.1%, meeting the ten-year LSST survey science requirements. However, the CCD mid-line presents distortions that can have a moderate impact on PSF measurements. This feature can be avoided by masking the affected regions. Effects of tree-rings are observed on centroids and PSFs of the artificial stars and the nature of the effect is confirmed by a study of the flat-field response. Nevertheless, further studies of the full-focal plane with stellar data should more completely probe variations and might reveal new features, e.g., wavelength-dependent effects. The results of this study can be used as a guide for the on-sky operation of LSSTCam.


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. 2003Holland et al. , 2009Holland et al. , 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 3×3 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 brighterfatter effect (Gruen et al. 2015).Park et al. (2017Park et al. ( , 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(Bosch et al. , 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 system-atic 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 pointspread 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.

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 singleblade beam shutter ("Thorlabs 1").The spot pattern is set by a photographic mask (HTA Photomask photolithographic) on the filter wheel.The commercial lens (Nikon 105mm f/2.8 Al-s Micro-Nikkor) is used to reimage 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 49×49 spots as shown in Figure 1.

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 electrooptical 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 ∼50000 e-/pixel, which is bright relative to the readout noise, ∼10 e-, and well below sensor full well, ∼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 ±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 ± 5mm to cover the entire CCD.
For Run 5 we also replaced a manual adjustable Zstage on the XY stage with a remotely controlled Zstage 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.

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 × 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 ∼ 5 pixels, corresponding to ∼ 1 ′′ in the LSSTCam focal plane.With ellipticity of ∼ 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.

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. 2018Bosch et al. , 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 or I yy < 20.0 px 2 in order to exclude sources that do not correspond to points on the 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.
projected grid.The remaining sources were then fit to an ideal grid model of 49 × 49 spots, with three free parameters corresponding to the x/y grid center position and θ 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.

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 δ ⃗ ℓ between the position of an ideal grid spot ⃗ s and the position of the corresponding detected spot ⃗ d is (Snyder et al. 2021b): where ⃗ ϵ is a random error term, δ ⃗ ℓ optical represents displacements that can be caused by the optical setup, including the mask used to generate the spots.Suppose we average the residual δ ⃗ ℓ 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: where δ ⃗ ℓ 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): The second moments of brightness are a building block for the PSF shape and size measurements (Kaiser et al. 1995;Schneider 2005): 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: if the profile is Gaussian.
The resulting residuals of the PSF shape and size are: We computed the fractional flux residuals in terms of the ratio: For small fluxes residuals (δf /f ) can be expressed as a multiplicative factor, where δm ≈ −1.08573δf /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 , ⃗ ℓ, and I xx , I xy , I yy , respectively.

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 δI 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 δℓ, δI ij , and δf 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.

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.

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 superpixels of size 10 × 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.
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 ad-ditional 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 (×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 pixel = 100 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 δy 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 righthand 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 midline 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.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.
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 ycomponent 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 × 10 −5 − 3 × 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.

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

Scientific Goal
Requirements < 10.0 Residual Impurities 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.

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 → r 0 (Plazas et al. 2014a,b;Okura et al. 2015): At the first order one can show: 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 δe and relate them to the area perturbation w(r).The perturbation on T 1/2 can be written as (Okura et al. 2015): (14) For a first order expansion, Thus, the fractional change in PSF size is equal to the opposite of the flat-field distortion: Similarly, the change in shear due to the tree-rings γ T R (Okura et al. 2015) in conjunction with equation 13 can be shown to be: (17) Our ellipticity definition, Equation 6, is a factor 2 times the shear (Schneider 2005); thus we can write: Equations 16, 18 equate the PSF size and shape distortions to the opposite of the flat-field distortion w(r).

Tree-Rings Coordinate System
The centroid and shape tree-rings distortions depend on the coordinate system.In contrast, the PSF-size and In the right, the e1 and δx 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 δy, with a signal level 100× higher than the noise.
the flat-field distortions are invariant quantities.To measure the tree-ring distortion effects, we transform (ℓ x , ℓ 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: e r e θ = cos 2ϕ sin 2ϕ sin 2ϕ − cos 2ϕ where ϕ = tan −1 y−y0 x−x0 .The factor 2ϕ is a consequence of the invariance of ellipticities against 180 • rotations.The center of this coordinate system (x 0 , y 0 ) is the center of the tree-rings circles, which is outside the CCD.

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. 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 8×8 pixels to increase the signal-tonoise 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.

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, θ) 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 θ direction would be tilted or disturbed.
3. Extracting the profile: We average the (r, θ) image over θ 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. (2017Park et al. ( , 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.

Measured Distortions Due to Tree Rings
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).

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 Tree Ring Signal -ITL: R03-S12 parameter that depends on the filter band (see Plazas et al. (2014a,b); Bernstein et al. (2017aBernstein et al. ( , 2018Bernstein et al. ( , 2017b))).This approach could be incorporated into the pipelines (Bosch et al. 2018(Bosch et al. , 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.

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

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 ∼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.

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 ×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 nonuniform 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 subpercent 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.

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).
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.

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.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.

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 treering 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 δe, 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 treerings 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 nonexistent 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.

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 δT /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 ≥ 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).

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× 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.

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.

Figure 2 .
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.

Figure 4 .
Figure4.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 .

Figure 5 .
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.

Figure 7 .
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.

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

Table 2 .
(Juramy et al. 2014)is 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.APPENDIX A. OPERATION VOLTAGESThe LSSTCam operation voltags setups using the during the data acquisitions are presented in Table2.Nominal operational parameters for both e2v and ITL sensors in the different run campaigns for which we acquired spot projector data.See Figure2inSnyder et al. (2018)for the schematic diagram of voltages.RC and Gain are configurable settings in the readout electronics boards(Juramy et al. 2014)