High ?delity point-spread function retrieval in the presence of

electrostatic, hysteretic pixel response

Andrew Rasmussen

a

, Augustin Guyonnet

b

, Craig Lage

c

, Pierre Antilogus

b

, Pierre Astier

b

,

Peter Doherty

d

, Kirk Gilmore

a

, Ivan Kotov

e

, Robert Lupton

f

, Andrei Nomerotski

e

, Paul

O'Connor

e

, Christopher Stubbs

d

, Anthony Tyson

c

, and Christopher Walter

g

a

SLAC National Accelerator Laboratory, Menlo Park, CA, United States

b

LPHNE/IN2P3/CNRS, UPMC, France

c

University of California at Davis, Davis, CA, United States

d

Harvard University, Cambridge, MA, United States

e

Brookhaven National Laboratory, Upton, NY, United States

f

Princeton University, Princeton, NJ, United States

g

Duke University, Durham, NC, United States

ABSTRACT

We employ electrostatic conversion drift calculations to match CCD pixel signal covariances observed in ?at ?eld

exposures acquired using candidate sensor devices for the LSST Camera.

1

,

2

We thus constrain pixel geometry

distortions present at the end of integration, based on signal images recorded. We use available data from several

operational voltage parameter settings to validate our understanding. Our primary goal is to optimize ?ux point-

spread function (FPSF) estimation quantitatively, and thereby minimize sensor-induced errors which may limit

performance in precision astronomy applications. We consider alternative compensation scenarios that will take

maximum advantage of our understanding of this underlying mechanism in data processing pipelines currently

under development.

To quantitatively capture the pixel response in high-contrast/high dynamic range operational extrema, we

propose herein some straightforward laboratory tests that involve altering the time order of source illumination

on sensors, within individual test exposures. Hence the word hysteretic in the title of this paper.

Keywords: CCDs, drift ?elds, charge collection, ?at ?eld statistics, pixel size variation, imaging nonlinearities,

brighter-fatter e?ect, instrument signature removal

1. INTRODUCTION

The literature already includes several instances of how ?at ?eld correlations may be used to correct or compensate

astronomical data for the so called brighter-fatter (BF) e?ect.

3

{

5

In a separate work, Niemi et al.

6

opted to favor the information available from direct, focused spot measure-

ments over the indirect information from ?at ?eld correlations, and that the ?ux level dependence to measured

spot sizes was described as an intrinsic CCD PSF that depends on intensity and wavelength only. A key piece

of the puzzle that con?icts with this picture is that the BF e?ect nearly vanishes when a single pixel's center is

illuminated with a spot of sub-pixel diameter:

7

the instrument's signature that contributes to systematic errors

in turn must depend on the incident ?ux distribution at the sensor's entrance window as well as the instanta-

neous recorded signal distribution as illumination progresses. Consequently, we argue that contributions from

the instrument cannot be separated from the contributions of the incident ?ux, unless the integration of the

recorded image is also considered in the process.

In this contribution, we approach the issue by calculating changes to pixel areas based on families of electro-

static solutions to Poisson's equation in the drift region.

8

A separate e?ort, not discussed in detail here, applies

a Poisson solver to sensor's semiconductor properties informed by detailed fabrication steps and lithographic

information, provided by the vendor.

9

Send correspondence to A.R. { E-mail: arasmus@slac.stanford.edu; Tel: +1 650 926 2794; Fax: +1 650 926 5566

Full electronic version of this paper with color ?gures is available on arXiv.org [astro-ph.IM]

2. DYNAMIC CHANGES TO PIXEL AREAS AND THEIR RELATIONSHIP TO

MEASURED COVARIANCES

Direct pixel boundary calculations, if they indeed reproduce available characterization data, are likely to be

preferable to pixel border shift models because they provide a two-dimensional point-to-pixel partition (they do

not over- or under-count area elements), and also o?er chromaticity information (pixel boundaries as a function

of conversion depth) in the dynamic pixel geometric response. Direct boundary calculations properly handle

built-in nonlinearities in pixel geometries as aggressor signals, due to accumulated conversions, approach full

well. In the context of the rolled-up model we describe, the recorded signal distribution of an image is used to

calculate the self-consistent, distorted pixel boundaries in e?ect at the end of the exposure. Better constraints

on the incident ?ux distribution would naturally result from knowledge of those boundaries.

In the following, we provide some calculations that extend work discussed in the aforementioned references

to connect pixel correlations to their theoretical signal level dependence, whether for ?at ?eld or focused spot

applications. We compute pixel area response to aggressor signal level.

2.1 Flat ?eld statistical ?uctuations and signal expectation values as a function of lag

In this discussion, the aggressor is a statistical ?uctuation ? in a recorded ?at ?eld image, that occurs about the

mean ? and induces changes in neighboring pixels' areas. A neighboring pixel is indicated by its lag from the

aggressor using two indices, ij, where i and j are the lags along the serial and parallel directions, respectively.

Because we calculate area variations due to drift during collection and not transfer statistics due to trap pop-

ulations and channel occupancy, full descriptions of area variations are captured by considering only positive i

and j. Correspondingly, ij = 00 indicates the aggressor pixel and q

00

is the charge signal accumulated there.

Considering the direct aggressor{victim channel only, a nearby pixel with lag ij has an area at the end of

integration:

?lna

ij

(q

00

= ?+?j?) ˇ

d ln a

ij

dq

00

?;

(1)

and on average this pixel would contain a signal level that is systematically biased by the area distortion. While

the area distortion is zero at the beginning of the integration (?a

ij

= 0 @ t = 0), it should be ?nite by integration

end. Averaged over all possible trajectories ?a

ij

(t), the in?uence of the statistical ?uctuation in pixel ij = 00

on the bias h?q

ij

i is readily isolated:

h?q

ij

(q

00

j?)i =

?

2

(exp(?lna

ij

(q

00

j?))? 1) ˇ

1

2

??

d ln a

ij

dq

00

=??

d ln a?

ij

dq

00

(2)

where for convenience, we also use the exposure time averaged pixel area a?

ij

. The expression for the covariance

Cov

ij

may also be simpli?ed by using the same approximation, and the variance Var ? Cov

00

appears:

?

Cov

ij

=

P

kl

?

kl

P

h?q

k + i;l + j

i

kl

1

=

?

2

P

kl

?

kl

(exp(?ln

P

a

k + i;l + j

(q

kl

j?)) ? 1)

kl

1

(3)

ˇ

?

2

P

kl

?

2

kl

d ln a

k + i;l + j

dq

kl

P

kl

1

=

?

2

Cov

00

d ln a

ij

dq

00

:

(4)

The correlations are then the covariances divided by the variance, and following through with the same approx-

imation, they are proportional to the area response d ln a

ij

=dq

00

and the scaling term ?=2:

Corr

ij

?

Cov

ij

Cov

00

ˇ

?

2

d ln a

ij

dq

00

?

For consistency in nomenclature, we de?ne the zero lag covariance to be equal to the variance:

Cov

00

=

P

kl

?

k

P

l

?q

k +0 ;l +0

kl

1

=

P

kl

?

2

kl

P

kl

1

? Var:

Now irrespective of ?nite correlations in the ?at ?elds, Poisson statistics are recovered by re-binning images

(in the case of data frames), or in the case of ?nite pixel area distortions:

P

kl

?

?

kl

P

ij

?

k + i;l + j

?

P

kl

P

ij

1

=

P

kl

?

2

kl

P

kl

1

+

P

kl

P

ij 6=00

?

kl

?

k + i;l + j

P

kl

P

ij 6=00

1

= Cov

00

+

X

ij 6=00

Cov

ij

? ?:

Because the expressions for Cov

ij

are symmetric under exchange i ! ? i and j ! ? j, the above expression may

be further simpli?ed to include only the unique quantities Cov

ij

for i ? 0;j ? 0:

?=

X

ij

Cov

ij

= Cov

00

+ 2

X

i ?0 ;j ?0

(1? ?

0 i

?

0 j

)(1+?

ij

)Cov

ij

;

(5)

where ?

ij

is the Kronecker delta. Finally, area is conserved, such that any area lost (or gained) by pixel ij = 00

is recovered (or ceded) by others:

X

ij

?a

ij

= 0 = (exp(?lna

00

(q

00

j?))? 1)+

X

ij 6=00

(exp(?lna

ij

(q

00

j?))? 1):

(6)

Equations

5

&

6

provide a way to express the shape of the mean-variance curve. Solving for Cov

00

:

Cov

00

= ?

0

@

1?

X

ij 6=00

Cov

ij

1

A

=

?

?

1+

P

ij 6=00

Covij

Cov

00

?

(7)

ˇ

?

1+

?

2

P

ij 6=00

d ln a

ij

dq

00

=

?

1?

?

2

d ln a

00

dq

00

=?

X

1

n =0

?

?

2

d ln a

00

dq

00

?

n

:

(8)

Evidently, the observed deviation from Poissonian behavior in the mean-variance curve indicates

d ln a

00

dq

00

<0

3

,

8

and the approximate relation is valid if j

?

2

d ln a

00

q

00

j ˝ 1. In general however, Equation

7

should remain valid

independent of this assumption. The general form of this is Equation

9

. Routine, accurate gain determination

may be enabled by ?tting functions of this form to gain-variance measurement pairs that show this curvature.

10

2.2 Application to measured covariances

We apply the equations above to the speci?c regime where statistical ?uctuations ? are much smaller than (and

tied to) the ?at ?eld ?ux ?. Recall that in the Poisson limit, ?

2

? ?, but this is apparently not correct in

actual photon transfer curves. High-quality ?at ?eld data sets can be used to generate a pattern of lag (ij)

speci?c covariances Cov

ij

, which are in turn converted into correlations via Corr

ij

? Cov

ij

=Cov

00

. Generally,

Cov

ij 6=00

scale as ?

2

, while Corr

ij 6=00

scale as ?. From the modeling side, a statistical ?uctuation translates to

an aggressor amplitude p? which in turn produces the pattern of area distortions ?a

ij

(p?j?) that govern the biases

in the expression for Cov

ij

(Eq.

2

). Taking the rms exposure averaged aggressor to be p? ? z

chan

?q

e

, we write

the following:

Cov

ij

= ???a

ij

(p?)

Cov

00

= ?

2

=

?

1?

?

?

?a

00

(p?)

(9)

Corr

ij

=

?

?

?a

ij

(p?) = (? ? ??a

00

(p?)) ?a

ij

(p?):

(10)

In the above, Corr

ij

, ? and ? are measurements and ?a

ij

(p?) are compiled from results of the drift calculation.

A measured correlation pattern Corr

ij

, together with estimates for ? and ? for a speci?c ?at ?eld illumination

were used to ?t an electrostatic drift model for its undetermined parameters. The data were acquired from a

candidate sensor prototype for the LSST Camera manufactured by e2v: it was described previously in Ref.

4

.

˜

2

was minimized using the Nelder-Mead method with results shown in Figure

1

(left). The best-?t parameter

list is given in Table

1

. Four free parameters were jointly estimated in the process. The area distortion model

appears to provide enough detail to reproduce the measured, anisotropic correlation pattern and fallo? with

separation. In addition to constraining magnitudes of the periodic barrier dipole moments, the ?t also favors a

speci?c value for the impurity concentration in the silicon bulk, N

a

. The goodness of ?t was acceptable when

using estimated ? and ?, so the process completed without invoking a gain error parameter in Eq.

10

.

A secondary result of this ?tting procedure is a value for the channel depth. For ? ˘ 65ke

?

, we ?nd

z

chan

= p=? (?q

e

) ˇ 2:37?m. This result is shown below (x

4.1

) to constrain other physical properties of the sensor.

1

10

100

10

−4

10

−3

0.01

lag specific correlation coefficients Corr

ij

radial lag (i

2

+j

2

) [pix

2

]

measured_vs_computed_corr_ij_comparision.qdp

electrostatic computed (surface)

measured correlations

electrostatic computed (@ saddle locus)

0

5×10

−4

10

−3

1.5×10

−3

0

5×10

−4

10

−3

1.5×10

−3

parallel coordinate [cm]

serial coordinate [cm]

contour_map_surface_deep.qdp

p = 2.8189 p

0

δ

ln a

00

= −0.306626

δ

ln a

00

= −0.130224

δ

ln a

10

= 0.0094869

δ

ln a

10

= 0.0239552

δ

ln a

01

= 0.032465

δ

ln a

01

= 0.021333

δ

ln a

11

= 0.0097589

δ

ln a

11

= 0.0047034

Figure 1. Pixel area variations. Left: a comparison of the best-?t electrostatic drift model to measured correlation

coe?cients for a speci?c ?at ?eld illumination. Measurements of Corr

ij

are the dots (black) with error bars, ?lled

triangles (red) are for shallow conversions, and open triangles (blue) are limiting cases for the conversions occurring near

the saddle locus along the drift lines. Any observed chromaticity in the ?at ?eld correlations should produce numbers

that lie somewhere between the shallow and deep limits corresponding to the same lag ij. Best-?t parameters, along with

quantities that a?ect the interpretation of this model are given in Table

1

. Right: a graphical representation of the same

electrostatic model's pixel boundary distortions, for the nearest neighbors (the aggressor is denoted lag ij = 00). The solid

(red) lines show the pixel boundaries for cold electrons very close to the backside surface, while the dotted (blue) lines

show a two-dimensional projection of the saddle locus, where adjacent drift lines diverge to feed the channels belonging to

adjacent pixels. The dashed (black) lines, together with the plot frame, show positions of the undistorted pixel boundaries,

i.e., for zero aggressor amplitude. For the purpose of this graphic, the aggressor dipole moment amplitude p was increased

by a factor of ˘ 500 as compared to the best-?t on the left (while holding all other parameters ?xed).

3. TUNABLE ELECTROSTATIC INFLUENCES TO PIXEL AREAS

We have a starting point for more detailed modeling. It is a relatively straightforward task to reproduce ?xed

pattern features seen in the sensors that would be categorized as cosmetics { but can in fact be traced to pixel

area distortions. In previous work we have demonstrated success in reproducing observed features seen in ?at

?eld illumination via electrostatic modeling: edge rollo?, midline charge redistribution, bright and dark column

pairs identi?ed as tearing features,

11

and bamboo,

12

each with self-consistent pixel shifts and elongations that

accompany the pixel area distortions revealed in the ?at ?elds.

8

The important di?erence between this and prior

work is that the statistical properties of the ?at ?eld illumination, Cov

ij

, are also reproduced at the same time.

Table 1. Parameter list for the best-?t electrostatic drift model (for cold carriers)

parameter

value

units

comments

N

a

1:11?10

12

cm

? 3

acceptor density in depleted Si

t

Si

100

?m

sensor thickness (?xed)

BSS

? 78

Volts

backside bias (?xed)

a

˘

cs

12:407

˘

0

b

channel stop 2-D dipole moment

˘

ck

2:6425

˘

0

b

clock barrier 2-D dipole moment

p?

0:0057208

p

0

c

aggressor dipole moment

d

?

65230

e

?

mean signal level in ?at

?

2

58429

(e

?

)

2

variance in ?at

d

a

constrained by measured X-ray di?usion variation with BSS on a

similar device

b

˘

0

?10

? 6

q

e

c

p

0

? 10

5

?mq

e

d

exposure averaged, rms aggressor moment is p? ? z

chan

? q

e

The best-?t electrostatic parameters given in Table

1

can represent ?ducial performance, and small changes

in their values will consequently a?ect the dynamic, hysteretic response of the sensor. For example, a change in

clock rail voltage di?erential would induce a proportional change in ˘

ck

, a change in backside bias (BSS) would

induce a change in E~

BD

(z) according to Appendix

B

, Equation

13

, and operation with tearing present (betrayed

by darker column pairs straddling segment boundaries in ?at ?eld response) would alter ˘

cs

11

and boost the

BF e?ect due to reduced barriers there. Moreover, parameterization of the aggressor moment p, its dependence

on signal ? and the geometric response of pixel boundaries (see x

3

) can provide an informed process by which

recorded images can be used to constrain ?ux distributions incident at the sensor entrance window. A check for

how Corr

ij

vary with BSS between measurement and calculation is given in Figure

2

. It demonstrates, via a

blind test, that counterintuitive dependencies in the measured Corr

ij

are reproduced using the simple, far-?eld

approximate, multipole expansion drift model that explicitly satis?es Poisson's equation. The Dirichlet solution

approach

9

to describe the sensor's photosensitive volume from the clocks and channel stops toward the backside

surface, may still be required to accurately model or understand other details of sensor operation, but these are

not de?ned or addressed here. This serves as a proof of concept that the drift model may already be adopted

for more widespread application in data analysis to reduce sensor systematics.

3.1 The brighter-fatter template

Several e?orts could bene?t from this detailed and robust modeling. These include astronomical data reduction

pipelines and simulation tools (based on image, table, or ray tracing). E?cient implementations require that

results of the time consuming electrostatic drift calculation be ported to faster simulation or analysis frame-

works. This was discussed previously (Ref.

8

, x5.3) but is brie?y summarized and expanded for completeness in

Appendix

D

. A single calculation result, in the format of a BF template could be applied over broader range of

aggressor amplitudes, indeed up to the canonical full well depth for these sensors, using the linear perturbation

model. The work addressed in x

3.2

tests this notion.

3.2 Pixel area dependence on aggressor dipole moment

By evaluating pixel border distortions for di?erent aggressor strengths p and comparing against a linear scaling

of the BF template, we essentially test the validity of the following expressions (cf. Appendix

D

), where p

t

is the

aggressor dipole moment for which the template was generated:

?~c

t ( i;j )

k

(p) =

?

p

p

t

?

?~c

t ( i;j )

k

(p

t

); 8k

?d

t ( i;j )

l

(p) =

?

p

p

t

?

?d

t ( i;j )

l

(p

t

); 8l

d ln(a

10

)/dq

00

ij=01

2.6E−7

2.8E−7

3.0E−7

3.2E−7

d ln(a

10

)/dq

00

ij=10

0.0E−7

1.0E−7

2.0E−7

−1.0E−7

0

20

40

60

80

d ln(a

ij

)/dq

00

aligned |BSS| (V)

ij=02

ij=20

0.0E−7

0.5E−7

1.0E−7

1.5E−7

Figure 2. A blind, qualitative comparison between measured correlation coe?cients (left) and computed area distortions

(right) as the backside bias is varied. Left: this plot was reproduced from Fig. 8 of Guyonnet et al.

4

and represents ?at ?eld

correlations measured for several lags ij 2 f01; 10; 02; 20g for ? ˘ 100 ke

?

. Right: computed per-electron area distortions

for the same lag selection, for the model summarized in Table

1

, ?t for jBSSj = 70V (\aligned" jBSSj = ? 78V

12

). While

even the Corr

ij

pattern shown here (for jBSSj = 70V) isn't entirely consistent with the Corr

ij

pattern shown in Figure

1

(left) collected for ? ˘ 65 ke

?

, this imperfect comparison shows that counterintuitive results seen in the data are readily

reproduced in the drift model. These include changes in sign of Corr

ij

and their derivatives with respect to jBSSj. The

ordinate scales on the right hand plots were adjusted to compare directly to the corresponding plots on the left.

where the instantaneous p=p

t

may be as large as twice the ratio between the canonical full well (100ke

?

) to

the rms statistical ?uctuation level ? given in Table

1

. In the present case, p

max

˘ (2FW=?)p? ˘ 826p?, many

times the ?uctuation levels sampled by ?at ?eld correlations. Figure

3

provides a quantitative comparison of

the evolution of pixel area variations with aggressor p and shows that the scaled template/linear perturbation

approach su?ers from signi?cant error. It should be emphasized that the apparent (per lag) nonlinearities

revealed here are entirely inaccessible to con?rmation via ?at ?eld correlations, unless the operational sensor full

well is closer to 10

7

ke

?

: a natural consequence of ?at ?eld statistics we used to probe the BF e?ect in the ?rst

place.

4. POSSIBLE NONLINEARITIES MEASURABLE IN FLAT FIELD COVARIANCES

Although the nonlinearities suggested in x

3.2

are not measurable using ?at ?elds for the reasons described, we

consider other terms here that could alter parameterization of the electrostatic elements in the drift model, or

would otherwise a?ect charge accumulation in the receiving channels beyond the cold carrier approximation.

4.1 Aggressor dipole moment dependence on accumulated conversions

The connection between charge collected and e?ective aggressor dipole moment in this drift model is the depth

of the buried channel, z

chan

(cf. x

2.2

). A strawman model for the buried channel depth may be constructed

by superposing two contributors of the axial component of the drift ?eld, one from the bound charge density

(depleted n-type Si) and one from the in?uence of the free charges in the conductive polysilicon gates, which can

be treated as an image charge. With z

0

equal to z

chan

in the limit of an unpopulated channel, we do not allow

0

0.1

0.2

0.3

0.4

pixel_area_evolution.qdp

−Δa

00

(p)

drift calc

simple linear

linear perturbation

0

0.02

0.04

+Δa

01

(p)

0

5×10

−3

0.01

0.015

+Δa

10

(p)

0

1

2

3

4

5

0

5×10

−3

0.01

0.015

aggressor dipole moment p [p

0

]

+Δa

11

(p)

0

1

2

3

4

5

1

1.2

1.4

Δ

a (drift calc) /

Δ

a (linear perturbation)

aggressor pixel dipole moment p [p

0

]

pixel_area_ratio_evolution.qdp

ij=00

ij=01

ij=10

ij=11

(p

FW

)

0

5×10

−4

10

−3

1.5×10

−3

2×10

−3

0

5×10

−4

10

−3

1.5×10

−3

2×10

−3

parallel transfer coordinate [cm]

serial transfer coordinate [cm]

compare_pixel_contour_maps.qdp

ij=00

ij=01

ij=10

ij=11

direct calculation (p/p

0

= 4.65)

p/p

0

= 5.72E−3 template

scaled to p/p

0

=4.65

dir:

δ

ln a

01

= 5.442E−2

sca:

δ

ln a

01

= 4.229E−2

dir:

δ

ln a

11

= 1.607E−2

sca:

δ

ln a

11

= 1.352E−2

dir:

δ

ln a

00

= −0.5776

sca:

δ

ln a

00

= −0.4955

dir:

δ

ln a

10

= 1.551E−2

dir:

δ

ln a

10

= 1.061E−2

Figure 3.

A study of pixel area distortion dependence on aggressor amplitude p. Upper left: for each of 4 lags,

ij 2 f00; 01; 10; 11g, measures of the pixel area distortions described in the text are given { the linear perturbation using

the BF template (dotted line), the direct drift calculation (solid line), and a linear area model where the area distortion

scales directly with aggressor amplitude p (dashed line). All curves in each plot coincide where p

t

= p? = 0:00572 p

0

from

the ?t to ?at ?eld correlations, and are tabulated out to p ˘ 4:7p

0

, which corresponds approximately to 100ke

?

in the

aggressor. In the case of ij = 00 (top tier), a negative sign has been applied to ?a for a better comparison to the other

curves. Upper right: ratios are plotted to compare the direct drift calculation to the linear perturbation approach using

the template for each of the lags considered here. While the two approaches mismatch by about 10% for ij = 00 near

the full well scale, the discrepancies are typically much larger, as much as 50%, for the other lags there. Bottom: pixel

boundary maps for an aggressor at full well, 100ke

?

(p = 4:65p

0

) to further compare the two methods for these lags.

The ˘50% discrepancy between the two methods shown for ij = 10 may appear counterintuitive because the area gained

appears larger than it really is. In fact, ?a

10

ˇ ? 0:027?a

00

, based on the drift calculation shown here.

for any spatial extent to the charge cloud and treat only its centroid:

E~(z

chan

) ? z^ =

?

E~

B

(z

chan

jz

0

)+E~

image

(z

chan

jz

0

)

?

? z^ = 0

z^? E~

B

(z

chan

jz

0

) =

N

d

2?

0

?

Si

(z

chan

? z

0

)

z^? E~

image

(z

chan

jz

0

) =

?+?

4ˇ?

0

?

Si

(2z

chan

)

2

z~ ? z

chan

=z

0

z~

3

? z~

2

= ?

?+?

8ˇN

d

z

3

0

:

The preceding equations describe two real solutions for ?nite z~ (one stable solution for each charge sign) in terms

of physical properties of the channel, channel depth for zero signal z

0

and the signal occupancy ? + ? (mean plus

aggressor). As the channel becomes populated, the solutions draw closer to one another until they coalesce as an

in?ection point. Beyond this there is no ?nite, real solution and the channel should empty as quickly as it ?lls

to expose gate structure layers with conversions. We may de?ne this measure of the full well, FW ˇ

32 ˇ

27

N

d

z

3

0

(to potentially constrain physical parameters of the channel using linearity measurements). Solutions for z~ for

?+?˝

32 ˇ

27

N

d

z

3

0

follow the approximately linear trend:

z~? 1ˇ?

?+?

8ˇN

d

z

3

0

=?

?+?

3:3 ? 10

6

?

N

d

10

16

cm

3

?

? 1

?

z

0

2:36 ?m

?

?3

:

(11)

If indeed N

d

˘ 10

16

cm

? 3

, then we should expect a weakening trend of the coupling between signal and aggressor

dipole moment of about 3% per 100ke

?

; and a proportionally weakened coupling if N

d

is smaller. This scaling

of the coupling term z

chan

contributes additional, signi?cant detail (depending on N

d

) at the scale shown in

Figure

3

and inclusion of this e?ect is treated in x

4.3

.

4.2 Aggressor driven modi?cation to drift time and di?usion contribution to correlations

It has been proposed

4

,

5

that the presence of collected conversions at the channel can attenuate the axial term

in the drift ?eld and could help to explain some of the discrepancies seen between an electrostatic treatment

of pixel borders and observed correlations. Guyonnet

4

(x5.3) investigated this further to place upper limits

(< 4%) on the contribution by di?usion - longer collection times - to the total the BF e?ect for focused spots.

In the context of the drift model described here, it seems di?cult to accurately isolate longer collection times

from the e?ect of distorted pixel boundaries that accompany a reduced electric ?eld near the backside window.

Indeed, an initial survey of temperature dependence of the ?at ?eld correlations produced inconclusive results.

In any case, the notion we examine here is that a large aggressor could raise collection times feeding into its

own pixel, e?ectively causing an additional redistribution of charges to neighboring pixels, while the neighboring

pixels do not reciprocate via the same mechanism (they retain their nominal collection times). If signi?cant, this

mechanism could be included by adding another term on the right hand side of Equation

2

.

The same drift calculation used to determine pixel boundaries for cold carriers is used to estimate the e?ect

of the accumulated conversions on subsequent conversions' collection times (and di?usion). Figure

4

shows this

dependence as if the trajectory for cold carriers can be used to compute a drift time and di?usion, for carriers

with temperature matching the substrate's temperature of T = 173 K. Presence of the barriers and the aggressor

are clearly seen as cusps in the ˙(~x) ?eld sampled by the crosscuts shown. When averaged over these linear

traces, positions nominally tied to the central pixel have a modestly increased ˙?

00

that is about 1 part in 400

greater than for positions not tied to the central pixel. We further estimate the net redistributive e?ect when the

central pixel has a larger di?usion ˙ than its neighbors. With ˙

nom

˘ 0:4 (pixels), we sample and average over

all possible Gaussian centroids contained within the pixel to compute the expected contribution to that pixel,

which (for 0:2 < ˙ < 0:6) works out to:

hC

00

(˙)i =

Z

1

0

dx

0

Z

1

0

dy

0

Z

1

0

dx

Z

1

0

dy

1

2ˇ˙

2

exp

?

?

(x? x

0

)

2

+(y? y

0

)

2

2˙

2

?

ˇ 1:0216347 ? 1:7486435˙ + 0:8989589˙

2

:

4

4.1

4.2

diffusion_assist_bf.qdp

σ

along 45

o

diagonal

−40

−30

−20

−10

0

10

20

30

40

4

4.1

4.2

position along segment [μm]

σ

along serial transfer axis

σ

along parallel transfer axis

diffusion

σ

[

μ

m] for fiducial performance

Figure 4. A calculation to estimate the di?usion assisted contribution to the BF e?ect. Notionally, in addition to the

pixel boundaries getting distorted, the presence of the exposure averaged aggressor dipole p? can slow down the carrier

drift toward the channel to increase collection time, boosting the di?usion parameter ˙. For ?ducial sensor performance

(parameters listed in Table

1

but also T = 173 K), the drift calculation was used to sample the launch position dependence

for di?usion. Left: three linear traces are given { one diagonal and two along the address axes { to show ˙ vs. position.

In each case, position zero is the location of an exposure averaged aggressor dipole p? that corresponds to a ?nal, recorded

signal of 100ke

?

. Presence of the electrostatic elements (˘

cs

, ˘

ck

and p?) in speci?c locations print through to produce

the modulations shown. They collectively cause slowdowns and de?ections along the ?eld lines. Right: a graphic

representation of how the linear traces were simulated (sections A-A' , B-B' and C-C' ) where lines of the grid represent

the barriers that form pixel boundaries. An approximate averaging over these linear traces reveal only a modest (0.25%)

increase in ˙ for positions that feed the central pixel containing the aggressor over the neighbors.

The net redistributive e?ect from lag ij = 00 to neighboring pixels would be ?lnq

00

˘ ?˙

@

@˙

lnhC

00

i, or about

? 2:2?10

? 3

(for ˙ ˇ 0:4p and ?˙ ˘ ˙=400). If this \missing" signal were recovered and divided evenly between

the nearest four neighboring pixels on average, the largest in?uence would be seen in lag ij = 10, because the

exposure averaged area increase there (cf. Figure

3

) is small: ˘ 8?10

? 3

, and would impart a ˘ 7%, di?usion

assisted excess over the nominal, pixel area distortion-driven Corr

10

. The term is far less consequential for lag

ij = 01 (˘ 2% excess) and for lag ij = 00 (˘ 1% excess). The ?t to the correlations presented above { Table

1

and Figure

1

{ did not include this as a separate term that would tend to dilute the observed anisotropy between

Corr

10

and Corr

01

in the model. It is unknown at this point whether its inclusion would improve the quality

of the ?t, or if tighter constraints on the Corr

ij

measurements would motivate its inclusion. In any case, the

term's scaling and dependence mimics that of the area distortion model described in Equation

2

and is currently

absorbed in the deterministic, detailed pixel boundary model. A separate treatment of this e?ect may be more

important for devices with smaller backside ?eld strengths, or if these devices were operated with smaller jBSSj.

We are encouraged by what is supported by Figure

2

as a reasonable correspondence between measurement and

calculation shown for Corr

ij

vs. jBSSj { that separate inclusion of this term may be unwarranted.

4.3 Predictions for nonlinearities in Corr

ij

(?)

In the sections above, several terms were described that a?ect the evolution of the dynamic pixel area distortion

model that are currently not well constrained by test stand measurements. These generally in?uence the detailed

relationship between accumulated ?ux and aggressor dipole moment (x

4.1

). In the current case, we have a pattern

of Corr

ij

measured at a single ?at ?eld level ? and variance ?

2

, a drift model parameter list that reasonably

reproduces the correlation pattern when pixel area distortions are mapped via Equation

10

, and a set of pixel

boundary distortions generated for a selection of aggressor levels p. Assuming for the time being that the gain

was accurately determined and that errors on ? and ? are negligible, we investigate how the computed area

distortion patterns can be mapped onto an observable set of parameters. By ?rst assuming a value for N

d

,

expressions for the aggressor p? and z~ are used to determine channel depth in the zero signal limit by solving

iteratively

z

0

=

p?

F

=?

F

1?

?

F

16 ˇ N

d

z

3

0

;

where ?

F

and ?

F

were used in the ?tting procedure (Eq.

10

) used to determine p?

F

. Drift calculations for other

aggressors p?

k

produce the pixel area distortions ?a

ij

(p?

k

). Self-consistent mean & variance pairs (?

0

k

;?

0

k

) are then

calculated, also iteratively, using the equations:

?

0

k

=

p?

k

=z

0

?

1?

?

0

k

16 ˇ N

d

z

3

0

?

=

A

1? B?

0

k

;

?

0

k

=

?

?

0

k

?

2

0

B

B

@

1 ?

?

0

k

?

0

k

?a

00

(p?

k

)

?

1?

?

0

k

16 ˇ N

d

z

3

0

?

?

1?

?

F

16 ˇ N

d

z

3

0

?

1

C

C

A

=

?

?

0

k

?

2

1?

?

0

k

?

0

k

?a

00

(p?

k

)

1? B?

0

k

1? B?

F

!

:

This process allows us to predict the detailed shape of the mean-variance curve as well as the mean-correlation

curves for speci?c lags ij. A similar procedure would be used to allow for (and constrain) a gain error in a

non-degenerate way. This isn't discussed here, but is straightforward to implement, given an additional set of

approximate, (?,?) pairs derived from photon transfer curves.

Figure

5

provides a family of curves that predict the signal dependence of the observable quantities ?

2

? Var

and Corr

ij

for di?erent assumptions of N

d

, again by assuming that an accurate gain was determined to produce

?

F

and ?

F

. Self-consistent values for z

0

for the assumed values of N

d

are also given. It turns out that while the

mean-variance curve is relatively insensitive to the nonlinearities considered in x3.3, the signal level dependence of

the Corr

ij

may the most straightforward indicator for an evolution in the coupling between signal and aggressor.

5. FALSIFIABLE TESTS OF LINEAR EXTRAPOLATIONS OF THE

BRIGHTER-FATTER TEMPLATE: HIGH-CONTRAST LABORATORY TESTS

The situation we described above is that we predict signi?cant departures from linear perturbation models when

we deal with real, high-contrast/high dynamic range data. Di?culty arises from not being able to sample high-

contrast/high dynamic range conditions using ?at ?elds alone: the aggressor scale available tops out near the

square root of the full well. It's surprising, then, that the linear perturbation methods used in astronomical

pixel data pipelines can correct 90% of these dynamical e?ects: that only 10% of the initial BF e?ect remains

uncorrected

13

after the compensation strategy is applied. This is based only on tracking a single \width"

parameter for the PSF's intensity dependence, and does not at all capture the platykurtic distortions to the PSF

pro?le that result from the boundary distortion mechanism. Nonlinear terms due to the variable channel depth

appear to reduce the BF e?ect by about 6% averaged over the exposure for full well (if N

d

= 5 ? 10

15

cm

? 3

);

the direct drift calculations suggest that the linear perturbation underestimates the BF e?ect, anisotropically,

by 10 ? 20% for high-contrast/high dynamic range exposures reaching the same full well in the pixels receiving

the highest ?ux. We expect that sensors using smaller electric ?elds or longer drift distances than these LSST

candidate devices should show correspondingly larger complications.

We consider some lab measurements that could be performed to test the drift model { and the linear pertur-

bation template methodology. Because the template is based only on a single aggressor level p^

F

=p

0

= 0:00572

(for ?

F

= 65ke

?

& ?

F

= 242e

?

), the next-to-leading order terms described in x

4.3

are not carried. We describe

a receiving pixel array with geometric parameters that evolve with time, as the exposure progresses toward full

well. Ratios of images (long vs. short exposure), incremental di?erence images (subtraction of images with

adjacent exposure times), etc., are simulated. Pixel areas at the end of exposure are also recorded { to predict

the e?ect of a ?at ?eld \?ash" at the end of the high-contrast exposure. Di?erences between (high-contrast +

?ash) and (high-contrast only) can be used to measure the pixel area ?eld across the array at the end of high-

contrast exposure. Deviations from these predictions may be interpreted as a superposition of the nonlinearities

0

5×10

4

10

5

1.5×10

5

2×10

5

−0.2

−0.1

0

0.1

Cov

00

relative deviation from linear

flat field level [μ

k

]

area_distortions_to_observables_cov00.qdp

μ

F

(Cov

00

(μ

k

)/Cov

00

(μ

F

))*(μ

F

/μ

k

) − 1

5×10

4

10

5

1.5×10

5

2×10

5

−0.05

0

relative change in observables (finite N

d

):(large N

d

) − 1

flat field level [μ

k

]

area_distortions_to_observables.qdp

μ

F

Cov

00

Corr

ij

N

d

= 1E16 (z

0

= 2.39μm)

N

d

= 5E15 (z

0

= 2.41μm)

(N

d

=5E21: z

0

= 2.37μm)

Figure 5. Aggressor induced, pixel area distortion calculations (?a

ij

(p?

k

)) mapped into observable ?at ?eld statistics to

predict their dependence on ?at ?eld level ?

k

, while allowing for an unknown donor density in the channel (N

d

). Left:

the expected nonlinear term in the variance, ?

2

k

? Cov

00

(?

k

), evaluated for a ?xed channel depth (equivalently, a large

N

d

). Any existing error in the gain determination should result in photon transfer data plotted o? of the locus shown

here. The level called out as ?

F

is the ?ux level at which the correlation pattern was ?t with an adequate pixel area

distortion template drawn using the drift calculation. Similar nonlinear terms in calculations of Corr

ij

were below the

percent level with no clear trend, so these were not shown here. Right: corrections to the Cov

00

and Corr

ij

mappings

that result from a channel depth that evolves with signal level. Two values for donor density in the channel are shown:

N

d

= 1 ? 10

16

and 5 ? 10

15

cm

? 3

. Mapping corrections for Cov

00

tend to be limited to the 2% level, but corrections to

Corr

ij

are independent of lag ij, and would tend to show small positive o?sets in Corr

ij

vs. ?, depending on the sampling

values that are available.

described, namely details of the lag-dependent pixel area evolution with aggressor, combined with details of

the relationship between signal and aggressor, and the evolution of the channel depth. Figures

6

and

7

show

predictions for a focused, Gaussian spot and for interferometric two-slit fringe projections, respectively. NB:

these predictions do not include the detailed, position-dependent drift times for cold carriers (x

4.2

) and only use

the one-to-one position-to-pixel mapping.

6. CONCLUSIONS

By mapping and matching drift calculation results, representing aggressor-victim pixel area distortions, to mea-

sured ?at ?eld correlations, we whittled down and simpli?ed the BF e?ect to just three electrostatic parameters

of the drift ?eld that were not already well constrained by X-ray charge cloud di?usion, for this CCD imaging

sensor. A fourth, self-interaction parameter was also determined, and de?nes the e?ciency by which conversions

collected in the channel can break the symmetry of the drift ?eld to reposition and distort pixel boundaries

encountered for subsequent conversions.

A compact, digital form for the pixel boundary mapping kernel, or Green's function, was used to predict

results for some high-contrast, high dynamic range illuminations of the sensor that could be tested in the

laboratory. We expect measurable deviations from these predictions for at least two reasons: (1) that the pixel

area distortion variation with aggressor amplitude does not increase as the linear, perturbation theory would

predict, and (2) that the self-interaction parameter coupling should evolve with accumulated conversions as an

exposure progresses.

These laboratory measurements, when performed, may provide a basis by which our quantitative understand-

ing of the BF mechanism can be extended to include the high-contrast, high dynamic range domain needed for

precision astronomical corrections.

15

20

25

30

35

0

2×10

4

4×10

4

cts per pixel

pixel address [10μm]

psf_profile_compare.qdp

first half

second half (vs. parallel)

second half (vs. serial)

15

20

25

30

35

0.8

0.85

0.9

0.95

1

area [pix

2

]

pixel parallel address [10μm]

psf_pixel_area_evolution.qdp

end of half exposure

end of full exposure

Figure 6. A high-contrast, high dynamic range simulation of an idealized PSF using linear perturbation of the template.

The illumination used is an isotropic Gaussian with 0.7

00

FWHM centered on the geometric midpoint of four pixels.

Integration continues until full well is reached, which corresponds to AB˘15.2 for LSST's r-band in a 15 second exposure.

Left: a comparison of the accumulated image after the ?rst half of the exposure to the additional accumulation during

the second half. These are linear traces that pass through the PSF centroid. The BF e?ect is seen by comparing linear

traces for the second half of the integration against that for the ?rst half. The total number of counts in the traces for

the second half are typically about 5% lower than for the ?rst half, because counts are also distributed perpendicularly

to the trace. Right: A comparison of the pixel areas resulting from the PSF integration half-way through and after

completion. These can be used to estimate structure in the sky background contribution, and errors in the PSF pro?le if

sky background is subtracted (without using this information). Laboratory data obtained to reproduce these results may

reveal the di?erences predicted in x

4.3

and the limitations of the linear perturbation model used here.

APPENDIX A. COMPARISONS TO SIMILAR EXPRESSIONS IN THE

LITERATURE

Because we aim to generalize and extend what's already in the literature, it should be useful to review here

expressions of similar quantities that have already been published. Note that in the preceding equations, a

ij

indicates pixel areas [e.g., cm

2

], not pixel boundary shift coe?cients perpendicular to boundary axes [e.g.,

pixel/carrier], as in some of the expressions below.

A.0.1 Antilogus et al. (2014)

In their x4.2 treatment of charge responsive pixel boundaries applied to ?at ?eld correlations, the authors don't

distinguish between instantaneous pixel boundary shifts and those shifts implied by statistics of the recorded

image { in other words, the exposure averaged boundary shifts. We ?nd a factor of 2 discrepancy between their

equations and ours if the former interpretation is followed, but perhaps no discrepancy with the latter. Upon

comparing their equations 4.14 and 4.15 against our approximate expressions (Eqs.

4

and

8

respectively):

Cov

?

Q

0

i;j

; Q

0

0 ; 0

?

= 4?V

X

X

a

X

i;j

Cov

ij

ˇ

?

2

Cov

00

d ln a

ij

dq

00

and

Cov

?

Q

0

0 ; 0

; Q

0

0 ; 0

?

=V +4V?

X

X

a

X

0 ; 0

Cov

00

ˇ

?

1?

?

2

d ln a

00

dq

00

ˇ?

?

1+

?

2

d ln a

00

dq

00

?

+O

?

3

?

1

2

d ln a

00

dq

00

?

2

!

:

0

5×10

4

10

5

@ end of step

accum. counts

period1_fringe_integration.qdp

0.99

1

1.01

rel. 1st step

counts increment

0

20

40

60

80

100

0.98

1

1.02

@ end of step

pixel area

parallel pixel address [10μm]

0

5×10

4

10

5

@ end of step

accum. counts

period2_fringe_integration.qdp

0.95

1

1.05

rel. 1st step

counts increment

0

20

40

60

80

100

0.9 0.95 1 1.05 1.1

@ end of step

pixel area

parallel pixel address [10μm]

Figure 7. A high-contrast, high dynamic range simulation of fringe projector illumination to accompany laboratory

measurements. This simulation used linear perturbation of the template and the illumination was chosen to imitate the

available parameter space. In both cases, the peak:valley ratio is set to 3, and the ?nal maximum counts accumulated is

near the full well depth of 100 ke

?

=pixel. Postage stamp images to the left show the image accumulated at readout time.

Upper plots: a fringe period of 28.8 pixels with orientation 82

?

; Lower plots: a fringe period of 7.6 pixels with orientation

140

?

. On the right, for each fringe calculation, the plots show the counts accumulated in 11 steps (top tiers), the 10 ratios

of the incremental counts accumulated divided by counts accumulated in the ?rst step (middle tiers), and the 11 area

curves computed at the end of each step (bottom tiers). The perturbed area ?elds may be validated by subtracting \fringe

only" exposures from \fringe+?at" exposures, which may be the most direct way to probe a dynamic pixel distortion

mechanism. The perturbed area ?eld would be easier to measure, with greater contrast, if the \?at" part of the exposure

could be applied after the \fringe" part, rather than in a simultaneous exposure.

We identify the equivalent instantaneous area distortion coe?cients d ln a

kl

=dq

00

that are a factor of 2 larger

than the exposure averaged area distortion coe?cients, valid for ?at ?eld applications, at least:

1

2

d ln a

ij

dq

00

?

d ln a?

ij

dq

00

˘

V

Eq: 4 : 14

Cov

00

4

X

X

a

X

i;j

1

2

d ln a

00

dq

00

?

d ln a?

00

dq

00

˘

V

Eq: 4 : 15

?

4

X

X

a

X

0 ; 0

+O

?

?

1

2

d ln a

00

dq

00

?

2

!

where it appears that the V in their equations 4.14 and 4.15 may have di?erent de?nitions.

y

A.0.2 Guyonnet et al. (2015)

In their x5.2 parameterization of pixel size variations as a function of ?ux, this paper uses largely the same

notation as in,

3

except that boundary shift coe?cients a

X

i;j

are de?ned di?erently by a factor of 4, such that the

area distortion coe?cients are identi?ed, but the de?nitions for their V in equations 16 and 17 remain to be

aligned:

z

1

2

d ln a

ij

dq

00

?

d ln a?

ij

dq

00

˘

V

Eq: 16

Cov

00

X

X

a

X

i;j

1

2

d ln a

00

dq

00

?

d ln a?

00

dq

00

˘

V

Eq: 17

?

X

X

a

X

0 ; 0

+O

?

?

1

2

d ln a

00

dq

00

?

2

!

:

A.0.3 Gruen et al. (2015)

In their x3.2 (Flat ?eld covariances), the authors cite

3

but yet write down slightly di?erent expressions for the

covariances as a function of lag, and choose a di?erent normalization for the pixel boundary shifts. As before,

we compare their Equations 3.7 and 3.8 to approximate our approximate forms of our Eqs.

4

and

8

, respectively:

Cov(Q

00

;Q

ij

) = 2?

2

X

X = T;B;L;R

a

X

ij

Cov

ij

ˇ

?

2

Cov

00

d ln a

ij

dq

00

and

?Var(Q

00

)=Var? ?=? 4?

2

?

a

R

1 ; 0

+a

T

0 ; 1

?

= +2?

2

X

X = T;B;L;R

a

X

00

Cov

00

? ?ˇ

1

2

?

2

dlna

00

dq

00

+O

?

3

?

1

2

d ln a

00

dq

00

?

2

!

:

Equivalent instantaneous area distortion coe?cients d ln a

kl

=dq

00

are expressed as:

1

2

d ln a

ij

dq

00

?

d ln a?

ij

dq

00

˘

?

Eq: 3 : 7

Cov

00

2

X

X

a

X

i;j

1

2

d ln a

00

dq

00

?

d ln a?

00

dq

00

˘2

X

X

a

X

0 ; 0

+O

?

?

1

2

d ln a

00

dq

00

?

2

!

y

If indeed V

Eq: 4 : 14

= V

Eq: 4 : 15

, then Equations 4.14 and 4.15 can be used together with Eq. 4.4, the sum rule for a

X

i;j

,

to recover Poisson statistics, essentially by rebinning. Coe?cients to the a

X

i;j

terms cancel, leaving ? = V . However, we

believe Equation 4.14 should scale with the recorded variance: V

Eq: 4 : 14

6= ?.

z

Augustin notes that high-quality ?ts to covariances for (i; j) 6= (0; 0) are achieved if the following relation is assumed:

2V

Eq: 16

= ?+Cov

?

Q

0

0 ; 0

; Q

0

0 ; 0

?

where the authors have assumed Var = ? in the ?at image prior to expressing the change in Var in Eq. 3.8.

It should be pointed out that all of the above sets of expressions each internally recover Poisson statistics as

covariances out to large lags ij are summed: The equations of

3

,

4

and

5

have terms that cancel by subtraction,

while (our) equations

3

and

7

are derived from this same principle. Also, the mean{variance relation out to large

signal levels ? may be calculated recursively if ? ln a

ij

(? + ?j?) (Eq.

1

) is computable.

APPENDIX B. ELECTROSTATIC DRIFT MODEL FOR COLD ELECTRONS

Here we brie?y summarize the electrostatic drift ?eld calculation, which is described in greater detail else-

where

8

,

11

,

12

and reproduced here for convenience. Figure

8

shows the assumed pixel geometry and electrostatic

elements in the model, but here only depicts a 2?2 pixel region close to the channel. Collected conversions are

represented by the four \bubble" like structures that hover over positions between pairs of extruded arrows in

two dimensional symmetric arrangement about the potential wells. The potential wells (\bubbles") do not lie

in the plane of the front side clock structure because these devices feature a buried channel. The integrating

and barrier clocks are strip-like equipotentials that extend for long distances along the serial address (i) axis and

provide boundary conditions that justify utilizing the method of images for a small channel depth z

chan

relative

to other relevant dimensions (distance to positions within the drift region, pixel dimension, and the combined

width of adjacent integrating clocks). The accumulated conversions constrained to the potential well will appear,

in the far ?eld approximation, to have an equal and opposite image charge distribution on the opposite side of

the clock plane, acting together as the perturbative, aggressor dipole ?eld denoted p~

ij

.

Similar arguments are used to describe the far-?eld in?uence of the cannel stop ion implants, which, under

depleted operation, act as another dipole moment in the z direction with translational invariance along the

parallel transfer (j) axis. These are denoted ˘~

cs

in Figure

8

. Finally, adjacent integrating and barrier clocks act

as dipole moments con?ned to the i-j plane with translational invariance along the serial transfer (i) axis, shown

as ˘~

ck

.

The predominant component of the drift ?eld is the backdrop ?eld, denoted E~

BD

(z), is a one dimensional

solution of Poisson's equation in the depleted silicon. The in?uence of the periodic and non-periodic contributors

described above can be added in superposition because they explicitly satisfy Gauss' law everywhere except for in

those small volumes that contain ?nite bound charge densities ˆ

b

not described in a the one-dimensional impurity

concentration pro?le N(z), and surface charge densities ˙

f

arising on semiconductor-conductor interfaces with

nonzero normal component of the local electric ?eld. The constant of integration for this backdrop ?eld E~

BD

(z) is

chosen such that a zero backside bias BSS implies a zero electric ?eld strength directly inside the backside surface

of the sensor. In Figure

8

then, the backside window is located a large distance directly above the con?guration

of electrostatic moments shown, i.e. toward where the nine vertical arrows point. Cold carrier pixel boundaries

undergo shifts in response to changes in the positions and magnitudes of the and charge con?guration moments

p~

ij

, ˘~

cs;i

and ˘~

ck;j

.

The equations used for vector integration along the drift ?eld lines are:

E~

tot

(~x) = E~

BD

(z) + ?E~ (~x)

(12)

E~

BD

(z) =

?

1

?

0

?

Si

Z

t

Si

z

dzN

a

(z)? V

BSS

=t

Si

?

z^

(13)

d~l =

?

?

?

EE~~ ((~~xx))

?

?

?

ds

(14)

~x

i +1

= ~x

i

+d~l

(15)

t

coll

(x~

0

) =

Z

~x

0

~x ? k

^=

z

ch

dl

?

e

(E(z);T) jE(z)j

:

(16)

where the cold electron collection time t

coll

(x~

0

) is used to estimate the thermal di?usion at the end of the

axial drift, using the mobility in the small ?eld limit, ˙

2

ˇ 2k

B

T=q

e

? ?

e

(E = 0;T) ? t

coll

(x~

0

). Mobility

Figure 8. The assumed ?eld geometry in the electrostatic model.

parameterizations of Jacoboni

14

were used, although we believe there is mounting evidence in X-ray illumination

data for these sensors to suggest that the velocity saturation e?ect for carriers is not as strong as that model

provides, for the operating conditions in question.

Details of the electrostatic in?uence by periodic barriers, denoted ˘~

cs

and ˘~

ck

in Figure

8

, are contained in

the term ?E~ (~x), and are subdominant for positions far from the channel, ~x ? z^ ˛ z

chan

, but compete with and

can ultimately dominate in?uence of the backdrop ?eld E~

BD

near the channel.

x

The image charge modeling

strategy used, and also the (in?nite) periodic arrangement of the channel stop and clock barrier potentials are

explicitly given in Rasmussen

8

[xx3.2-3.3] and are not reproduced here.

APPENDIX C. SHOELACE FORMULAE UTILIZED

After pixel boundaries are sampled via the drift calculation, they are compiled into lists that comprise polygonal

representations of the pixels. The following formulae were used to compute geometric parameters for each pixel.

While direct mapping (e.g., ~x 2 pixel[i; j] vs. ~x 3 pixel[i; j]) is utilized for certain simulation applications via

e?cient point-in-polygon routines, interpretation of recorded images may be aided with use of ancillary pixel

information according to the recorded signal distribution in the pixels. Polygon representations of pixels in?u-

enced by the recorded signal distributions are straightforward to perform if the signal scale speci?c perturbation

patterns are known. The following shoelace formulae were given in Rasmussen

8

[x3.5] that connect pixel area A

ij

,

pixel astrometric shifts I x

ij

& I y

ij

, and second moments I xx

ij

, I yy

ij

& I xy

ij

, given an ordered set of N vertices

(x; y)

k

where (x; y)

N

? (x; y)

0

. For the pixel boundary calculations represented in this work, we typically worked

with either 15 or 25 points per side (60 ? N ? 100):

x

Any charge con?gurations near the front side potential wells

9

that would motivate carrying higher order terms in a

multipole expansion, are not entertained here. We imagine that such terms would include any ?nite spatial extent in depth

and width of the channel stop implant, and any spatial extent in 3 dimensions of the accumulated signal carriers collected

in the potential well. Such higher order terms necessarily would have a shorter range of in?uence (j?Ej ˘ r

? s

;s ? 4).

At a level where they might be important in the drift calculations, these terms will also in?uence the shapes of charge

clouds residing in adjacent wells, complicating the drift calculation. We plan to neglect such terms until there is su?cient

evidence in the data to suggest their importance.

A

ij

? +

1

2

N

X

? 1

k =0

(x

k +1

y

k

? x

k

y

k +1

);

(17)

I xx

ij

A

ij

? ?

1

12

N

X

? 1

k =0

(y

k +1

? y

k

)(x

2

k

+x

2

k +1

)(x

k

+ x

k +1

);

(18)

I yy

ij

A

ij

? +

1

12

N

X

? 1

k =0

(x

k +1

? x

k

)(y

2

k

+y

2

k +1

)(y

k

+ y

k +1

);

(19)

I xy

ij

A

ij

? +

1

6

N

X

? 1

k =0

(x

k +1

? x

k

)x

k

(y

2

k

+y

2

k +1

+ y

k

y

k +1

)

+

1

24

N

X

? 1

k =0

(x

k +1

? x

k

)

2

(y

2

k

+ 3y

2

k +1

+ 2y

k

y

k +1

);

(20)

I x

ij

A

ij

? ?

1

6

N

X

? 1

k =0

(y

k +1

? y

k

)(x

2

k

+x

2

k +1

+ x

k

x

k +1

);

(21)

I y

ij

A

ij

? +

1

6

N

X

? 1

k =0

(x

k +1

? x

k

)(y

2

k

+y

2

k +1

+ y

k

y

k +1

):

(22)

The sign of Eq.

17

corresponds to a speci?c choice of chirality for the polygonal vertex list. The quanti-

ties above are used to evaluate distortions to pixel area (? ln A

ij

), pixel astrometric shift vectors (e.g., p~

ij

? x^ =

[I x

ij

A

ij

]=A

ij

) and pixel ellipticities (??

1 ;ij

= [I xx

ij

A

ij

? I yy

ij

A

ij

]=[I xx

ij

A

ij

+I yy

ij

A

ij

]; ??

2 ;ij

= 2 I xy

ij

A

ij

=[I xx

ij

A

ij

+

I yy

ij

A

ij

]). It may be possible for existing pixel data pipelines to be retro?tted to take advantage of such book-

keeping information when estimating object parameters, particularly for PSF estimation purposes.

APPENDIX D. A LINEAR PERTURBATION TEMPLATE TO REPRESENT

DYNAMIC PIXEL RESPONSE

The proportional pixel boundary shifts laid out by Antilogus et al.

3

x4.2 (and subsequently Refs.

4

,

5

), uses

constructions that linearly accumulate the in?uence of aggressors in the pixel's vicinity. The coupling coe?cients

are determined using a matrix inversion of constraint equations (containing measured covariances) that utilize

re?ection symmetries, and a sum rule (e.g., Ref.

4

x6.1). Our detailed electrostatic drift calculation may also be

applied, and we can do so while explicitly guaranteeing the continuity equation and one-to-one mapping between

a two dimensional continuous position ?eld and pixel address. In other words, the Greens function doesn't su?er

problems intrinsic to a general arrangement of rectangular pixels that naturally over- and under-claim pixel

\ownership" of the continuous position ?eld.

In the same spirit, we apply the Greens function according to the supposition that all de?ections of pixel

boundaries are perturbations that scale linearly with aggressor amplitude. We refer to application of the linear

perturbation equations collectively as the BF template. Figure

9

illustrates the geometry. The equations used are

as follows, where ?~c

t

k;k +1

are computed distortion vectors of two adjacent corners for the template aggressor p

t

,

nominally separated by a single pixel step along the positive m transfer direction e^

m

: m 2 f0; 1g, e^

m

2 f

?

1

0

?

;

?

0

1

?

g:

?~x

t

l

?e^

m

?

?

?~c

t

k

+

?

l

n? 1

?

?

s e^

m

+ ?~c

t

k +1

? ?~c

t

k

?

?

? e^

m

?~x

t

l

?e^

( m +1)mod2

? ?~c

t

l

? e^

( m +1) mod 2

+ (?d

t

l

? ?d

t

0

);

where ?~x

t

l

, l 2 f0:::n ? 1g are solutions to the electrostatic drift calculation (for template aggressor p

t

) that

form a locus for the pixel boundary of this border, ?d

l

are the boundary distortions perpendicular to the border

axis in the (m + 1) mod 2 direction as shown. As usual, i and j are the lag indices in the m = 0 and m = 1

directions respectively, and s is the pixel spacing. The per-lag template quantities ?~c

t ( i;j )

k

and ?d

t ( ij )

l

are then

compiled by summing over in?uences of aggressors accumulated at the channel according to:

?~c

( q;r )

k

=

X

ij

?

p

q ? i;r ? j

p

t

?

?~c

t ( i;j )

k

?d

( q;r )

l

=

X

ij

?

p

q ? i;r ? j

p

t

?

?d

t ( i;j )

l

~x

( q;r )

l

?e^

m

? ((mmod2==0)?q:r)s+

?

?~c

( q;r )

k

+

?

l

n? 1

?

?

s e^

m

+ ?~c

( q;r )

k +1

? ?~c

( q;r )

k

?

?

? e^

m

~x

( q;r )

l

?e^

( m +1)mod2

? (((m+1) mod 2 == 0)?q : r) s+?~c

( q;r )

l

? e^

( m +1) mod 2

+ (?d

( q;r )

l

? ?d

( q;r )

0

):

In the above, ?~c

( q;r )

k

and ?d

( q;r )

l

are the resulting total perturbations to the pixel corners and boundaries from

multiple aggressors, respectively; and the boundary pairs ~x

( q;r )

l

for this border are combined the boundary pairs

for each of the three other borders to produce a (closed) polygonal description of pixel (q;r). Point-in-pixel

algorithms, as well as the shoelace formulae of Appendix

C

are then readily applied to these distorted pixel

descriptions.

The preceding provides a generalized, 2-dimensional application of drift calculation results as linear pertur-

bations. It is analogous to the simpler, perturbative pixel border shifts described in previous work.

3

{

5

ACKNOWLEDGMENTS

This material is based upon work supported in part by the National Science Foundation through Cooperative

Support Agreement (CSA) Award No. AST-1227061 under Governing Cooperative Agreement 1258333 managed

by the Association of Universities for Research in Astronomy (AURA), and the Department of Energy under

Contract No. DEAC02-76SF00515 with the SLAC National Accelerator Laboratory. Additional LSST funding

comes from private donations, grants to universities, and in-kind support from LSSTC Institutional Members.

REFERENCES

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scope camera design and construction," in [Advances in Optical and Mechanical Technologies for Telescopes

and Instrumentation], Navarro, R. and Burge, J. H., eds., Proc. SPIE 9912 (2016). in press.

[2] Kahn, S., \Final design of the large synoptic survey telescope," in [Ground-Based and Airborne Telescopes

IV], Hall, H. J., Gilmozzi, R., and Marshall, H. K., eds., Proc. SPIE 9906 (2016). in press.

[3] Antilogus, P., Astier, P., Doherty, P., Guyonnet, A., and Regnault, N., \The brighter-fatter e?ect and pixel

correlations in CCD sensors," Journal of Instrumentation 9 (03), C03048 (2014).

[4] Guyonnet, A., Astier, P., Antilogus, P., Regnault, N., and Doherty, P., \Evidence for self-interaction of

charge distribution in charge-coupled devices," A&A 575 , A41 (Mar. 2015).

[5] Gruen, D., Bernstein, G. M., Jarvis, M., Rowe, B., Vikram, V., Plazas, A. A., and Seitz, S., \Character-

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207{231 (June 2015).

[7] Miyazaki, S. Private communication, discussed at Foreground Physical E?ects on LSST Weak Lensing

Science: A Workshop on the Impact of the Last Kiloparsec, University of California at Davis, regarding

measurements made using HSC sensors. (2015).

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sors," Journal of Instrumentation 10 , C05028 (May 2015).

8×10

−4

10

−3

1.2×10

−3

1.4×10

−3

1.6×10

−3

1.8×10

−3

2×10

−3

2.2×10

−3

9.2×10

−4

9.4×10

−4

9.6×10

−4

9.8×10

−4

10

−3

parallel transfer coordinate [cm]

serial transfer coordinate [cm]

template_pixel_contour_map_pd_+2.8189_detail.qdp

ij=10

ij=00

ij=01

ij=11

ij=20

ij=21

Figure 9. An illustration to show the element level information content of the pixel distortion template. This is for the

lower boundary for the pixel with lag ij = 11 (and contains the same information as the upper boundary for the pixel

with lag ij = 10). The (red) ?lled triangles connected by solid lines form the pixel boundary locus ~x

t ( i;j )

l

and divide the

pixel areas labeled by their corresponding lag (ij = 00; 10; 20; 01; 11; 21). The (black) solid line connecting ?lled circles

shows the position of the undistorted pixel boundary (between ij = 10; 11). Open (blue) triangles show positions of the

distorted pixel corners, with (black) solid lines (?~c

t (1 ; 1)

k;k +1

) connecting them to the undistorted corners (?lled circles). With

distorted corner coordinates projected onto the e^

0

(serial, horizontal) coordinate and the separation divided equally (n

samples per side), the e^

1

(parallel, vertical) de?ections ?d

t (1 ; 1)

l

of the border are recorded and stored as a template for

downstream use. With n = 25 as shown here, the template can be stored with modest memory requirements, 32 numbers

per border per lag, and when compiled for an arbitrary recorded charge distribution, 96 (x; y) pairs per pixel that form

a closed polygon. Application of the shoelace formulae (Appendix

C

) provides e?cient distillation of this information to

6 leading geometric terms per pixel. This template was generated for an aggressor amplitude p? = 2:819 p

0

: 492 times the

e?ective exposure averaged, aggressor level ?t described in Table

1

. This corresponds to a readout time aggressor signal

of 119ke

?

, comparable to the full well depth for these sensors.

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astronomy applications," in [Modeling, Systems Engineering, and Project Management for Astronomy VI ],

Proc. SPIE 9150 , 915017 (Aug. 2014).

[13] Lupton, R. Private communication, regarding HSC fork of LSST DM stack (2016).

[14] Jacoboni, C., Canali, C., Ottaviani, G., and Alberigi Quaranta, A., \A review of some charge transport

properties of silicon," Solid State Electronics 20 , 77{89 (Feb. 1977).