Point-spread function in depleted and partially depleted CCDs Page: 4 of 8
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longitudinal diffusion. The charges are produced
in a field-free region, and in this approximation
take infinite time to get out. We will deal with
this problem later.
But how serious is our neglect of longitudinal
diffusion in the overdepleted case? An almost-
solution provides insight. If Ez(z) is replaced by
its average value E, in Eq. 6, then a solution which
satisfies the boundary condition at t = 0 isg(zt) = 1 e-(z-Vt)/4Dpt
'z) 47rDptH 0 .C
q v
0.4C
, 0.3C
H
,q 0.2C
0
0. 2C
0.0
0(13)
where v, = p,. The boundary condition
g(zj, t) = 0 is satisfied only for times sufficiently
small that the tail of the distribution do not yet
reach the boundary at z = zj, i.e., roughly for
t < zj/v,. The picture is simple: the 6-function
distribution at the back surface at t = 0 grows
into a Gaussian distribution in three dimensions'
with a = 2Dpt, whose centroid moves with ve-
locity v,. The expanding ball of charge reaches
the potential wells over a fractional transit time
ad/zJ. If this fraction is small compared with
unity (as it turns out to be), then our Gaussian
approximation for the lateral distribution is valid.
5. Resolution dominated by diffusion in a
field-free region
At low bias voltages for the LBNL CCDs and
nearly always for thinned CCDs, the substrate
is not fully depleted. For most optical wave-
lengths and in particular for the blue, light is ab-
sorbed very close to the back surface, and carri-
ers freely diffuse through the undepleted substrate
until they cross the interface, encounter an elec-
tric field, and travel to the CCD potential wells.
As mentioned above, recombination may be ne-
glected for the cases being considered. We con-
sider a point source of charge carriers at the rear
surface of a CCD with a field-free thickness zff.
Carriers are reflected from the rear surface (at
least for the LBNL case), so the problem is equiv-
alent to one with the photoionization source with
twice the intensity at the center of the field-free
substrate with thickness 2zff. It is sufficient to
consider the steady-state solution to the problem,
so that Eq. 5 reduces to Laplace's equation ex-
cept for the 6-function at the origin. We recog-
nize this as equivalent to the potential problem in.0 0.5
u
lf
71)1.0 1.5 2.0 2.5 3.0 3.5 4.0
u = p/z or 4 = x/zffFig. 3.- The radial charge distribution Q(u) and
marginal distribution q(C) (solid curves). The scaled
variables are u = p/zff, where zff is the thickness of
the field-free (undepleted) region and p = x2 + y2,
and C = x/zff. With the extension q(-C) = q(C), the
variance of q(C) is 1. The distributions are normalized
so that f' Q(u)27rudu = 1 and ft q( )d = 1.
which a charge is equidistant between two earthed
planes at z = +zy (Hopkinson 1983). In this case
the gradient at the plane is normal and gives the
electric field, which is in turn proportional to the
charge density on the plate. This is a well-studied
problem. In particular, Jackson (Jackson 1990)
gives two solutions, in his problems 3.17(b) and
3.18(b). For unit total charge on each plane (two
negative unit charges at the origin) the charge dis-
tribution in the potential problem is exactly equal
to the hole distribution at the potential wells in
our diffusion problem. We obtainQ(u)
1 d okkJo (ku)
27r / d cosh k
1~ 00
Z(-1)"(2m -
2n-1(14)
1)Ko((m- 1)ru) ,where J, is the Bessel function regular at the ori-
gin, Kn is the modified Bessel function which is 0
at infinity, u = p/zff, and p = x2 + y2. This
corresponds to Hopkinson's z, = d case (Hopkin-
son 1987).
The integral form converges rapidly for small u,
and only a few terms in the summation given in the
second form are necessary for larger arguments.
For example, 7-place accuracy is obtained with 8
terms for x > 0.5, while only 2 terms are nec-4
I 0.5 x charge fraction
Inside circle of radiu
Half-max at -
= 0.8384
- -
I r Gaussian with same
_ I \Amax as q(4% ( 0.
I -e
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Groom, D. E.; Eberhard, P. H.; Holland, S. E.; Levi, M. E.; Palaio, N. P.; Perlmutter, S. et al. Point-spread function in depleted and partially depleted CCDs, article, October 13, 1999; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc724490/m1/4/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.