Reliable measurements of POFs` optical properties with a low

Reliable measurements of POFs` optical properties with a low
Lukasz Jankowski
Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, D-12205, Berlin, Germany
[email protected]
11th Int. Conf. on Plastic Optical Fibres
18-20 September 2002, Tokyo, Japan, pp. 251-254
Abstract: CCD camera can be fast and robust instrument for POF measurements, especially compared to
scanning techniques. However, most of reasonably priced cameras are constructed to give more qualitative than
quantitative results. We discuss common CCD cameras’ inaccuracies and calibration procedure. (CCD camera,
POF measurement, calibration) © 2002 ICPOF
1. Introduction
Most of the traditional measurement methods of such
POFs’ optical properties as far and near field are
either time-consuming (as scanning techniques) or
require expensive instruments (as goniophotometers
or high-end scientific area scan cameras). On the
other hand, there are many fast and reasonably
priced low-end CCD cameras, but they introduce a
number of inaccuracies that make obtaining
meaningful quantitative results difficult.
However, it is possible to make POFs measurements
with a low-end CCD camera more reliable. In this
paper we discuss most common inaccuracies of such
cameras (paragraph 2), propose calibration
procedure based on individual cells calibration
(paragraph 3) and address common problem of too
small bit depth (paragraph 4). Sample calibration data
obtained for DALSA’s 8 bit CA-D4 camera [1] and a
sample calibrated POF measurement are presented
in paragraphs 5 and 6, respectively.
2. CCD camera inaccuracies
Real CCD cameras introduce a number of
inaccuracies into their measurements. Raw single
CCD cell output n will be mathematically modeled as
nm  d  r m  e ,
Random noise e of each CCD cell is a centered
random variable (i.e. its mean is zero, E[e] = 0). A
common way to estimate the influence of a random
variable is through its sample standard deviation.
Thus, to estimate the random noise we will make a
series of T calibration measurements nt under the
same lighting conditions and compute the matrix of
CCD cells’ sample standard deviations s, where
s2 
As the whole CCD sensor is a matrix of CCD cells,
three matrices of d, r(m) and e parameters have to be
considered for calibration purposes.
Note that response function r(m) and random noise e
in (1) may be wavelength-dependent. As the present
study do not investigate their wavelength dependence,
the calibration and final measurements should be
made with the same light source wavelength.
In case of rapid lighting intensity changes between
neighboring CCD cells a cross-talk effect may occur.
While in most of POF measurements lighting intensity
changes rather slowly with the angle, (1) does not
include cross-talk effects.
1 T
 nt  n
T  1 t 1
andn is an average of all nt measurements. Note that
as n depends on CCD cell excitement m, so may also
s and above computation should be repeated for
different lighting intensities and the maximal s value
should be taken.
As a measure of a single measurement’s uncertainty
we will use 3s level (i.e. 3 or 99.7% certainty level
under normal distribution assumption).
Note that when the measurement is an average of M
snaps, its uncertainty is less and becomes
where m is the CCD cell’s real excitation we like to
measure, d is its constant bias (a result of its dark
current), r(m) represents its response function and e is
its random noise variable. As d represents cell’s bias
at zero lighting intensity, response function r(m) obeys
r 0  0 .
2.1 Random noise
Thus, the undesired effect of too high random noise
may be easily reduced by averaging subsequent
2.2 Dark current
Non-zero dark current results in a non-zero bias
matrix d of any CCD camera. It may be relatively
simple measured, just by taking an average of a
series of T measurements nt made under completely
dark conditions:
1 T
 nt
T t 1
Note that, according to (4), an average of enough
many T snaps should be used to diminish the
influence of the random noise.
2.3 Response function
An ideal CCD sensor would have linear and identical
response functions r(m) of all its CCD cells. However,
for a real CCD sensor cells’ response functions may
depend on the specific cell (non-uniformity) and be
non-linear (intensity-dependence).
Slope coefficient r’(m) of cell’s response function is
called cell’s sensitivity. And normalized matrix of
r ' m 
r ' m 
wherer’(m) is the averaged value of cells’
sensitivities, will be called sensitivity profile of the
CCD sensor (at lighting intensity m).
Note that the cells’ sensitivity (6) may be (and usually
is) non-uniform across the sensor due to the
differences between cells’ response functions r(m),
i.e. the sensitivity may be position-depend. Moreover,
possible non-linearity of the response functions
results in lighting-dependence of the sensitivity, as
the slope of a nonlinear response function changes
with lighting intensity m.
The obvious way to estimate r(m) of a cell is: for every
cell (a) measure few points on r(m); (b) approximate
measured points with a function that obeys (2).
Measurement of points r(m) on the cells’ response
functions may be done in the following way:
Use uniform lighting of CCD sensor (by using
one distant light source or a big integrating
Measure real lighting intensity (e.g. with a
photodiode located next to the sensor). This is
the common value of m (excitement co-ordinate)
for estimating r(m) for all the cells.
Make a series of M snaps, compute an average
measurement and subtract the sensor’s dark
current d, thus obtaining the matrix of
n d .
Those are the r co-ordinates of r(m) points for all
the cells.
Repeat the above measurements with different
values of m.
Special care should be taken when approximating
r(m) with a linear function, as even few percent of
non-linearity may have a considerable effect on
sensor’s sensitivity profile.
2.4 Irregular cells
In a real CCD sensor not all cells are regular: some
may be damaged or dead, response functions r(m) of
others may differ too much from an average or their
random noise may be too high. All those factors make
measurements of that part of CCD cells unreliable;
we will call those cells irregular. Measurements of
irregular cells should be approximated based on
measurement results of neighboring cells.
Irregular cells may be located on basis of previously
computed cells’ calibration parameters, such as
random noise and response function. As an example,
the irregularity criteria for a cell may be as follow:
Cell’s random noise (4) is too big, i.e. it is greater
than mean + 3 of all cells’ random noises.
Mean square approximation error of cell’s
response function is greater than mean + 3 or
r % of all cells’ mean square errors.
Cell’s response function differs too much from an
average response functions (i.e. its coefficients
fall outside the (mean  3) band of all response
functions’ coefficients).
3. Calibration
According to (1), real excitement m of a regular cell
should be computed as
m  r 1 n  d  ,
where n is cell’s raw measurement. In fact, in order to
minimize the effect of cell’s random noise, in place of
single measurement n in (7), an averagen of M
successive measurements should be used:
m  r 1 n  d  .
To decide on the number M of raw measurements to
average, condition (4) should be considered. Note
that as response function r(m) of regular cells should
be a monotonically increasing function, so should also
be r 1 in (7) and (7a).
When real excitement of all regular cells is computed,
excitements of irregular cells may be approximated by
bilinear interpolation of nearest (in the same row and
column) regular cells’ excitements, computed
previously with (7a).
4. Dynamic range
Bit depth of most of low-end CCD cameras is 8 bit, so
their dynamic range is not better than 1:256. For
many POF measurement applications it is not enough.
A simple solution (other than buying a costly high-end
camera) may be as follow: (a) make several
calibrated measurements with different exposure
times; (b) scale down those made at longer exposure
times using the least-square-error method to match
the shortest exposure time measurement; (c) merge
scaled measurements into final measurement. It
should be kept in mind that:
Downscaled measurements’ absolute resolution
increases with the exposure time (as their 3s
levels are also downscaled), so it is better in low
and worse in highly excited areas of the merged
measurement. At most excited sensor areas it
equals the resolution of the shortest exposure
time original measurement (as this one is not
At long exposure times a blooming effect in
highly excited areas may occur.
5. CCD camera example
DALSA’s CA-D4 camera was used to obtain
calibration data and a sample measurement
according to the outlined procedure. The camera’s
technical characterization is:
Bit depth: 8 bit, i.e. 256 gray levels.
Pixel resolution: 1024  1024 cells.
Four series of 32 calibration measurements at 200 ms
exposure time and uniform sensor lighting were
made; the light source used was white, close to CIE
standard illuminant A:
32 measurements under dark conditions with
average raw excitation (dark current) of 8 gray
levels and real lighting intensity 0.018 a.u.
Three series of 32 measurements with average
raw excitations of 30, 120 and 220 gray levels
and measured real lighting intensities of 4.46,
20.10 and 35.70 a.u., respectively.
5.1 Random noise
The camera’s average 3s level was found to be equal
2.4 gray levels and generally evenly distributed
across the sensor (Fig. 1).
Thus, the 99.7 % certainty level (under normal
distribution assumption) of a single measurement is
2.4 gray levels, what results in the dynamic range of
about 5:256. It is considerably less that expected
1:256; random noise takes up 2 bits out of the
camera’s 8 bits.
Fig. 1. Camera’s 3s level at 200 ms exposure time.
5.2 Dark profile
An average of 32 raw measurements made under
dark conditions was computed (Fig. 2).
Note that cell’s average bias d can be as high as 12
gray levels, i.e. almost 5% of the maximum excitation
(255 gray levels). On the left hand side of the sensor
the dark profile is wave-like shaped; this is clearly the
effect of the CCD cells’ row arrangement.
5.3 Sensitivity profile
Four series of measurements result in four points on
each cell’s
approximations resulted in an average mean square
error of 7%, the square function
r m  am2  bm
was used. Note that due to condition (2) there is no
constant term in (8), it is characterized by the dark
current d. By the use of (8) an average mean square
approximation error was reduced to 2%.
Interpolated camera’s sensitivity profile under dark
conditions (i.e. (6) at m=0) shows that cells’ sensitivity
differences are as high as 5-6%.
Fig. 3. Interpolated camera’s sensitivity profile under dark
conditions at 200 ms exposure time.
Note that camera’s sensitivity profile at higher lighting
intensities differs form Fig. 3, as cells’ response
functions (8) are non-linear.
5.4 Irregular cells
According to conditions from paragraph 2.4, camera’s
irregular cells were found. They are represented by
black dots on Fig. 4.
Note vertical dot chains on the left hand side of the
figure. They correspond to the wave-like structure on
the left part of the dark profile (Fig. 2) and are due to
the row arrangement of the CCD cells.
The number of irregular cells was found to be about
3% (30,000) of the total cell number (1,000,000).
Fig. 2. Camera’s dark profile at 200 ms exposure time.
Note that lighting axis still goes up to only 256 a.u., as
it does with raw measurement at single exposure time.
However, the advantages of applied calibration
procedure are clear (see Table 1):
Random noise 3s level is first decreased by a
factor of two (averaging) and further by factors 2,
4, 8 and 16 for all but the most excited sensor
areas (due to the downscaling effect).
Due to less random noise, the absolute
resolution was improved by the factor ranging
from 27 (for low excited areas) to 2 (for most
excited sensor areas).
There is no effect of dark current.
Irregular cells’ measurements were interpolated.
Measurement’s non-linearity was decreased from
7% down to 2%.
Table 1. Summary of measurement improvements
(GL = graylevels).
Fig. 4. Irregular cells. Due to representation
limitations, the number of irregular cells seems to
be higher than in reality (3 % of sensor area).
6. POF measurement example
For demonstration purposes two parallel POFs were
used to illuminate the CCD sensor in a simple setup.
The distances between fibers’ end-faces and sensor
were 15 mm and 5 mm, fiber lengths were 20 cm and
100 cm for fiber No. 1 and 2, respectively.
The calibration procedure described in previous
paragraphs was repeated separately for five exposure
times of the camera (50, 100, 200, 400 and 800 ms).
For each exposure time four snaps were taken to
decrease random noise effect by a factor of two
(according to (4)). Computed calibrated measurement
was scaled to match the 50 ms measurement (Fig. 5).
at 200 ms
(best case)
Average dark
8 GL
0 GL
Irregular cells
Random noise
3s level
2.4 GL
0.09 GL
Dynamic range
256 : 4.8
256 : 0.18
7. Conclusion
Common inaccuracies of a typical low-end CCD
camera were discussed and a reliable calibration
procedure was proposed. As presented procedure is
easy to implement and considerably increases
measurement reliability, it can be used for fast and
reliable POF measurements besides its other
The author would like to thank all his colleagues at
BAM for their support.
[1] DALSTAR CA-D4 Camera Product Info,
[2] NIST/SEMATECH, “Engineering statistics handbook”,
Fig. 5. Sample calibrated measurement.
Fiber No. 2 was lighted perpendicularly to its end face,
so Fig. 5 shows only output of its lower order modes.
As the lighting angle of fiber No. 1 was about 20,
light was transmitted through it only via higher order
modes and a clear ring-like output pattern emerged.
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