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There exist a number of textbooks [for example Theuwissen, 1995; Holst, 1998, and references
therein], reports
[for example Lance and Eather, 1993; Eather, 1982], and articles
[for example Janesick et al., 1987, and references therein] on solidstate
imaging with CCD detectors. This section will provide a short summary
of some fundamental concepts required to specify a CCDimaging system
suitable for the needs of auroral and airglow imaging.
Holst [1998] defines the term radiometry, as the
``energy or power transfer from a source to a detector''
while photometry is defined as ``the transfer
from a source to a detector where the units of radiation have been
normalised to the spectral sensitivity of the eye.''
Spectral radiant sterance (radiance)
The basic quantity from which all other radiometric quantities can be
derived is spectral radiant sterance, . Given a
source area, , radiating a
radiant flux, , into a solid
angle, . The spectral radiant
sterance in energy units, ,
then becomes:

(3.1) 
where is the wavelength. Expressing the
spectral radiant sterance in quantum units (
)
the following equation is obtained:

(3.2) 
Here is the frequency, is Planck's
constant and is the speed of light. Please note
spectral radiant sterance (radiance) is not to be confused with
surface brightness which is a photometric unit involving the
characteristics of the human eye [see Holst, 1998, pp.
20,26].
The Rayleigh
In terms of measurement techniques the aurora can be regarded as a
fivedimensional signal with three spatial dimensions, one temporal
and one spectral dimension. The desired physical quantity is usually
the volume emission rate,
, which cannot
be found directly from measurements. However the rate of emission from
a
column along the line of sight is normally just
for any isotropic source with no selfabsorption
[Hunten et al., 1956].
Consider a cylindrical column of crosssectional area 1
extending away from the detector into the source. The volume emission
rate from a volume element of length at distance is
. The contribution to
is given
by:

(3.3) 
Integrating along the line of sight, :

(3.4) 
This quantity is the column emission rate, which
Hunten et al. [1956] proposed as a radiometric unit for the
aurora and airglow. (See also Chamberlain [1995, App.
<]2058#> II#) The unit is named after the fourth
Lord Rayleigh, R. J. Strutt, 18751947, who made the first
measurements of night airglow [Rayleigh, 1930]. (Not to be
confused with his father, J. W. Strutt, 18421919 remembered for
Rayleighscattering etc.) In SIunits
the Rayleigh becomes [Baker and Romick, 1976]:

(3.5) 
The word column denotes the concept of an emissionrate from
a column of unspecified length, as discussed above. It should be noted
that the Rayleigh is an apparent emission rate, not taking absorption
or scattering into account. However, Hunten et al. [1956]
emphasise that ``the Rayleigh can be used as defined without
any commitment as to its physical interpretation, even though it has
been chosen to make interpretation convenient.'' The spectral
radiant sterance (
) in Equation 3.2 can be obtained from
the column emission rate (in Rayleighs) according to
Baker and Romick [1976]:

(3.6) 
Although not a proper SIunit, the Rayleigh is often used in the field
of auroral and airglow measurements. It is also frequently
misunderstood and abused. It is important to remember that the
Rayleigh only is usable when the wavelength is specified. Due to
the plethora of Rayleigh definitions [Baker and Romick, 1976], it is
always wise to state the definition of the unit before using it. In
the following text, the Rayleigh will be used according to the
original definition [Hunten et al., 1956, but in SIunits] as defined
above.
Using the recommended column emission rates in Rayleighs for the
International Brightness Coefficients ( IBC)
as an example, the following spectral radiant sterances are obtained
[Chamberlain, 1995, App. II]: IBCI aurora,
corresponding to 1 kR at 5577 Å is often described as the lowest
column emission rate detectable by the unaided human eye. Usually
this is compared to the luminous incidence of a moonless cloudy night
which is about Lux^{3.1}. By the use of Equations 3.2 and 3.6, the
spectral radiant sterance in energy units () can be
calculated:

(3.7) 
Similarly the brightest IBCIV corresponding to 1 MR at
5577 Å, which is often compared to the luminous incidence of the
fullmoon of about
Lux^{3.1}
becomes:

(3.8) 
Spectral radiant incidence (irradiance)
Spectral radiant incidence (irradiance), , is defined as
radiant power incident per unit area onto a target, in this case
typically the effective aperture of the
optics, , image area, ,
area of CCDdetector, , or the
area of a CCD pixel, .
The transmittance,
, of an optical
system is a function of many parameters, for example wavelength
, viewing angle, temperature, etc., and must be
experimentally determined (See Chapter 4). Here it is enough
to state that the total transmittance is given by the product of the
individual transmittances of the various components of the optical
system:

(3.9) 
Where is exemplified by ,
and which are the transmittance of the atmosphere,
optics and filter, respectively. In the following text
will denote the product of appropriate transmittances according to
Equation 3.9.
Consider an extended source of area, , of
given column emission rate, , imaged by an
optical system, here represented by a single lens with given
focal length, , and
fnumber, . The relation between the
aperturestop, , f and is given by:

(3.10) 
Setting the source at distance from the lens, which is
at the distance from the detector and letting
be the solid angle of the lens aperture
as seen from the source, (assuming small angles), the
photon flux,
, at the lens
aperture then becomes:

(3.11) 
The spectral radiant incidence at the
aperture,
, then
becomes:

(3.12) 
The spectral radiant incidence at the image
plane,
, is
given by (by substituting Equation 3.10 and assuming a circular aperture):

(3.13) 
Noting that, for thin lenses and small angles:

(3.14) 
In this case:

(3.15) 
The lensmagnification formula is:

(3.16) 
where and denote the height of the
image and source respectively. Inserting Equations 3.6 and
3.143.16 into Equation 3.13, the following
equation for
is obtained:

(3.17) 
Note that this equation is only accurate for small angles, an offaxis
image will have reduced incidence compared to an onaxis image. This
is called the ``natural vignetting'',
,
and is usually inserted as a multiplication factor in Equation 3.17, as
required.
However, the actual vignetting is highly dependent on the
characteristics of the optical system. Therefore the vignetting as well as
other aberrations, and the transmittance () must be
experimentally determined. This will be considered further in
Chapter 4 and references therein.
Using Equation 3.17 the number of photons reaching the
image plane,
, can be calculated:

(3.18) 
Here
is the integration time in seconds. The
number of photons hitting the
CCD,
, are:

(3.19) 
Here is the transmittance of the optical window
protecting the CCD, and is the transmittance of the
CCD substrate (in the case of a backside illuminated CCD). Likewise,
the number of photons hitting an individual
pixel,
, is:

(3.20) 
(using Equations 3.183.19 and
Equation 3.20). Again, please remember the caution at the end of
Section 3.1.3
``During the past couple of decades the Charge Coupled
Device (CCD) sensor has gradually replaced the tube type sensors,
such as the vidicon, due to its advantages in size, weight, power
consumption, noise characteristics, linearity, dynamic range,
photometric accuracy, spectral responsitivity, geometric stability,
reliability, and durability'' [Janesick et al., 1987]. As an auroral
imager, the CCD is thus almost an ideal detector for groundbased as
well as for spaceborne instruments. The CCD can be used either as the
primary photon detector, or in an Intensified CCD (ICCD)
configuration. Whether to use a CCD or an ICCD is mainly a matter of
requirements on temporal resolution and signaltonoise
ratio,
. This section will begin with a short review
of a number of parameters describing the performance of a CCD
detector.
Quantum efficiency
The quantum efficiency, ,
is a measure of the number of electrons generated per incident
photon:

(3.21) 
where is the effective quantum yield
(electrons generated, collected, and transferred per interacting
photon per pixel) and is the interacting quantum
efficiency (interacting photons per incident photons per pixel)
[Janesick et al., 1987]. However, for the purpose of this text it is
only needed to consider , which for the purpose of this
text, is the same as the detective quantum efficiency (DQE).
Quantum efficiency for a CCD is wavelength dependent, and is of no
value unless the wavelength is specified. High performance scientific
thinned, backside illuminated antireflection coated CCD devices
might have as high quantum efficiency as 8090% in the visible
region as demonstrated in Figure 3.1.
Figure 3.1:
Quantum efficiency vs. wavelength for the SI003AB CCD (ALIS
ccdcam5) as published in the manufacturers datasheet.
Four curves are presented: with standard antireflection coating
(``StdAR'') UV antireflection coating (``UVAR''), uncoated,
thinned, backside illuminated (``Thinned uncoated'') and
frontside illuminated (``Frontside''). The ALIS imagers have a
standard antireflection coating.

As a comparison, for typical consumer CCDs lies in the range
3070%.
The average number of photoelectrons,
,
obtained from
(Equation 3.20) is:

(3.22) 
Noise
Aurora and airglow are socalled photonlimited signals, where the
quantum nature of light limits the achievable signaltonoise ratio.
Photon arrival follows Poisson statistics, where the variance is equal
to the mean. The sum of the standard deviation of the noise
resulting from the photoelectrons,
, and darkcurrent
noise,
, is denoted shot
noise,
, and is obtained from the
following equation:

(3.23) 
Here
stands for the variance of quantity
and
is the standard deviation.
denotes the mean value.
stands for the Charge Transfer Efficiency (
) where
is the number of transfers. Usually
for reasonably sized CCD devices.
Note that
would be present even for an ideal detector.
Additional noise sources include: reset noise, onchip and offchip
amplifier noise (1/fnoise and white noise). Of particular interest
for lowlight applications is that the 1/fnoise, which is the main
source of the read noise (``noisefloor''), increases in proportion to
the square root of the readout frequency (``pixelclock''), thus
requiring a suitable compromise between noise performance and
framerate. The analoguetodigital converter has a quantisation
noise, also switching transients coupled through the clock signals,
electromagnetic interference, etc., sums up to the total noise.
There is pattern noise due to differences in darkcurrent and photo
response nonuniformities [Holst, 1998]. At high signal levels
the total noise is dominated by pattern noise (pixel to pixel
sensitivity variations within the CCD) for most CCDs, at low signal
levels the read noise (``noise floor'') dominates
[Janesick et al., 1987]. Many of these noise sources can be reduced to
negligible levels by good electronic design practices. In particular,
the darkcurrent (and reset noise) is reduced by cooling the CCD. By
the use of Double Correlated Sampling (DCS) the reset noise
can be almost eliminated. The quantisation noise is eliminated by
using a sufficiently high resolution ADC. It is, however, beyond the
scope of this work to embark onto a detailed analysis of the
noisesources in CCD imagers, instead a simplified noise model from
Holst [1998] is adopted:

(3.24) 
The standard deviation of the noise floor (or read
noise),
, is usually stated in the imager
specification as `read noise', or easily obtained from a zero
exposure. This noise source increases with the squareroot of the
pixel clock frequency, which imposes an
constraint onto
the framerate.
The pattern noise,
, is the sum of
Fixed Pattern Noise (FPN), resulting from pixeltopixel
variations in the darkcurrent, and Photo Response
NonUniformities (PRNU). An approximate worst case value is provided by
Holst [1998]:

(3.25) 
However, for the low signal levels considered here, pattern noise is
neglected. Then the total noise approximation for a bare CCD becomes
(by substituting Equation 3.23 into Equation 3.24):

(3.26) 
It should be remembered that this equation is an approximation.
Signaltonoise ratio
The measured signaltonoise ratio in terms of the digital
output,
, or in terms of
rootmeansquare electrons
is defined as follows:

(3.27) 
Substituting Equation 3.26 the
can now be calculated
with the help of Equations 3.20 and 3.22 and the total noise is given by
Equation 3.26:

(3.28) 
For an ideal photon detector Equation 3.28 becomes:

(3.29) 
The signaltonoise ratio of an ICCD
In the imageintensifier, for each primary photoelectron emitted
by the photocathode (
), the imageintensifier
produces a burst of approximately secondary CCD electrons,
typically by the use of a microchannel plate (MCP) between
the photocathode and phosphor screen. The phosphor screen is then
optically coupled to the CCD by the use of lenses or a fibreoptic
taper^{3.2}.
The number of photons reaching the image plane
(
) is given by Equation 3.18. The average
number of signal electrons generated by the
photocathode,
, for a pixel
area projected onto the photocathode, by the
fibreoptic minification ratio, is given
by [see Holst, 1998, p. 196]:

(3.30) 
After the MCP the number of electrons are amplified with the
average MCP gain,
, i.e. average
number of secondary
photoelectrons,
,
produced per photocathode electron

(3.31) 
The photoelectrons are then converted back to photons by the phosphor
screen with a phosphor efficiency, , and then
reimaged onto the CCD via a fibre optic taper. The transmittance
losses here are denoted . Finally the photons hit the
CCD and are converted to photoelectrons:

(3.32) 
By applying the following approximation:
the
following equation is obtained (using
Equations 3.303.32):

(3.33) 
As with the CCD (see Equation 3.23), the photocathode produces both
photonnoise and darkcurrent noise:

(3.34) 
Another noise source, unique to the ICCD, is the electron
multiplication noise, which is due to the statistical distribution
of the number of secondary photoelectrons^{3.3}.
Taking this uncertainty in
into account, the
combined variance with the photon noise becomes:

(3.35) 
Here, is the microchannel excess noise.
The photocathode darkcurrent noise, is also dependent on
in the same way:

(3.36) 
Taking Equation 3.26 as an approximation for the total CCD noise,
and inserting into it Equations 3.35 and 3.36, for the ICCD noise
sources, leads to the following expression for the total
noise for the ICCD,
:

(3.37) 
By applying Equations 3.33 and 3.37 into Equation 3.27 the following
approximation for the
of an ICCD,
,
emerges:

(3.38) 
As seen, increasing the gain of the image intensifier makes the CCD
noisesources negligible, but does not increase the
. For very high gain, Equation 3.38 is reduced to:

(3.39) 
Threshold of detection and maximum signal
The threshold of detection is usually defined as
while
the Noise Equivalent Exposure,
, is obtained when
. The maximum signal, or Saturation Equivalent
Exposure (SEE) is obtained when the charge well
capacity,
, is reached. This occurs when:

(3.40) 
In most cases the maximum chargewell
capacity,
, is matched to the
maximum ADC output
. For the ICCD case, apart
from the condition above, there is also a saturation level for the
image intensifier to be considered. On the other hand, the
highvoltage for the intensifier can be gated, acting like an
electronic shutter.
The Dynamic Range,
, is defined as the peak
signal divided by the RMS noise and the
DCbiaslevel,
, (if any).
The minimum ADC output,
, is subtracted
in the case of a signed integer output.
is usually expressed in
decibels.

(3.41) 
An approximate theoretical value for
is obtained by dividing the
maximum signal (Equation 3.40) by the total
noise,
, which is found in
Equations 3.26 and 3.37 for the CCD and ICCD case
respectively.)

(3.42) 
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copyright Urban Brändström