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Photothermal IR-Radiometry relies on the
principle of radiation heating and detection of the resulting changes of the
IR emission of the heated sample. Depending on the temporal form of excitation,
there are two different methods: the first method relies on pulsed laser
heating and the second one on modulated laser excitation, with the IR signal
analysed by a Lock-in amplifier. The first method, the pulse technique [1], is
an extremely fast measurement, with the disadvantage, however, of a low
signal-to-noise ratio and the necessity of using high energy deposition rates
on the sample surface, which may easily invoke surface damages. The "modulation
technique", the so-called thermal wave technique [2,3], on the other hand, has
the advantage of a good signal-to-noise ratio because of the use of the
Lock-in amplifier technique, but the measurement time is very long and limits
the application in industry. In many applications in industrial environment,
the materials are low absorbing, resulting only small temperature rises and
only small additional radiation signals, which contributes to privilege the
modulation method.
In most on-line applications in industry, e.g. on a production line,
measurements have to be done on moving samples. Busse et al. have described
thermal wave experiments on a rotating disk in the laboratory [4], whereby the
measured signals have been interpreted only qualitatively. An attempt of
quantitative analysis has been published by Netzelmann [5], presenting
theoretical evaluations of thermal wave propagation on a moving slab with
point-like excitation. The case of a Gaussian heating spot was done
numerically, without any explanation of his model.
Here we present a theoretical description of the signal generation process,
which takes into account the effects of both the velocity of the moving sample
and the Gaussian form of the excitation beam on the sample surface temperature.
In addition, the new model includes the notion of the space-modulation of
heating, which has to be compared with the time-modulation, generally
generated by a chopper, and if the both frequency are equal we have what we
call a "Thermal Resonance Effect", i.e. the amplitude increases with the
frequency at fixed velocity or it increases with the velocity at fixed
amplitude.
Subsequently in Sect. 2 the theory for the case of moving and modulated
excitation of an absorbing semi-infinite solid is presented. In Sect. 3
experimental results measured on a moving fabric in an industrial process are
shown and compared with theoretical calculations. Finally in Sect. 4 possible
improvements of both the theoretical interpretation and the measurement
technique are discussed.
The principle of photothermal IR radiometry
consists of measuring the IR emission of a radiation-heated sample.
Considering only small temperature variations of an opaque sample, one can
assume that the IR emission is proportional to the surface temperature
variation
 In the case of modulated excitation at the modulation frequency
 the induced temperature amplitude
varies at the same frequency
 however
with a phase shift . In order to determine the temperature variations, the
heat diffusion equation
has to be solved with
 the thermal diffusivity,
the mass density, c the specific heat capacity, and
 the absorbed power. The
mathematical solution is based on the theory developed for the Flying Spot
method, respectively for the Photothermal camera [6,7,8] which can be used to
characterize thin films and to localize vertical surface cracks with the help
of moving laser beam excitation and IR detection. A recent state of art of
this technique is given in ref [8].
The geometry of the problem is presented in Figure 1: The excitation source is
given by a modulated laser beam of radius R. The sample is assumed to move
during the measurement at a constant velocity V. An IR detector is focused on
the sample at a distance
from the centre of the laser spot. The coordinate system (X, Y, Z) is related
to the exciting laser beam, and the coordinate system (x, y, z) is a
coordinate system related to the sample. An isotropic and homogeneous sample
with thermal properties independent of the temperature is assumed here, with
negligible heat losses at its surface. To solve equation (1) we first consider
a heating power
described by a delta function, i.e. we look for the Green's function of the
system given for the infinite solid [9, p. 353] by
with t > t' and
 referring to the
excitation source and (x, y, z, t) the measurement position. The final
solution is then the convolution between the Green's function and the actual
heat source distribution [8, 10].
In contrast to the case of a cw laser moving with constant velocity [11, 12],
we now consider a modulated laser spot moving along the x axis.
Assuming that at t =0 the laser is centred at the origin of the sample
reference coordinates, the power distribution at a time t is given by:
where
 is the laser beam power
 ,
the sample's absorptivity,
the Dirac function,
 the
pulsation (i.e. the modulation frequency multiplied by 2
) and a phase shift, which
can be chosen to be zero.
We now solve the problem for a semi-infinite solid located in the half space z
> 0, with negligible heat losses at the surface. The initial temperature is
taken as reference. In this case, the Green's function is determined by the
method of images,
i.e. as a symmetric distribution of the
source with respect to the surface z = 0. To remove the time dependence we use
the co-ordinate system
X, Y, Z related to the laser beam and consider the
condition
with X= x-Vt, Y = y, and Z = z. The solution
for z = 0 is given by
with k the thermal conductivity and where
the dimensionless temperature T*V*
is given by
In equation (6) and (7) the quantity
is the dimensionless Fourier number, V* =
RV/ is a reduced
velocity,
 the reduced
pulsation, and X* = X/R and Y* = Y/R are the dimensionless normalized
coordinates.
The amplitude and phase of expression (7) have been calculated using Simpson's
numerical integration. The two main parameters are the reduced velocity V* and
the reduced pulsation . In order to demonstrate the type of possible
solutions, we here only present the results for a zero offset between the
detection spot and the centre of the excitation beam, i.e. X*=Y*=0.
Figure 2 shows the amplitude and phase of expression (7) as a function of the
reduced pulsation
 for several reduced velocities in a double-logarithmic
scale. For low frequencies, the amplitude does not vary with the frequency,
instead in this frequency range it is the velocity which imposes the value of
the amplitude. At high frequencies, on the other hand, the amplitude mainly
depends on
, i.e. the amplitude decreases with the inverse of the square root
of the frequency. There is also an intermediate range, approximately limited
by V* <
< 10 V*, where the behaviour of the amplitude is more complex. Thus,
it is interesting to see that - for example before reaching the asymptotic
behaviour of the amplitude with the slope of -1/2 in double-logarithmic
representation - the amplitude slightly increases.
The phase converges to zero for low frequencies and to -45° for high
frequencies, which is the phase shift characteristic for one-dimensional heat
propagation in a semi-infinite solid. In this domain, there is no influence of
the velocity on the thermal signal. Thus we can say that for
> 10 V* the
velocity of the sample V* with respect to the laser beam has no effect on the
temperature. This means, for
term V/2R in (9), which has the
dimension of a frequency, is the inverse of the characteristic excitation time
of the laser beam at a given position of the sample and can be called the
space frequency
 Thus if f is much larger than
there is no effect of
the velocity, and in the opposite case, if
is much smaller than it is
the velocity which determines both the amplitude and the phase of the
temperature.
Figure 3 shows the amplitude and the phase of expression (7), calculated as
function of the sample velocity for several values of the reduced pulsation.
For the amplitude, similar to Figure 2, there are three regions to be
distinguished: (i) First for low velocities, where the amplitude is constant
and only varies as function of the frequency. (ii) Secondly for high
velocities, where the amplitude is only velocity-dependent and decreases with
the inverse of the square root of the velocity (compare ref. [7]). In this
case, the amplitude behaves in a similar way as in modulated excitation,
however, at the space frequency (iii) In the intermediate region, a
relative maximum of the amplitude is observed, which can be considered as a
resonance between the time-modulated and the space modulated excitation,
The maximum of the phase sensibility is also achieved for this resonance
condition,whereas the phase converge to zero for high velocities and
to -45° for low velocities.
The experimental set-up, which is shown in
Figure4, has been used for measurements on moving sample of fabrics: Instead
of a laser, a 100 Watt mercury high pressure lamp has herre been used for the
excitation of thermal waves. The light of the lamp has been focused on a
chopper and subsequently on the sample. The spot has got a diameter of about
2.8 mm. The chopper frequency has been varied in a wide range, between 5 Hz
and 20 kHz. A water filter placed between the lamp and the first lens has
mainly been used to prevent infrared radiation of the lamp from falling
directly on the sample and being reflected on the detector. Additionally, the
overheating of the further optical components and of the sample was avoided by
the water filter.
The infrared radiation emitted by the sample has been detected with the help
of a HgCdTe detector using infrared lenses with high transmittance in the far
infrared. The infrared optics has been focused in such a way that heating spot
and detection spot coincided. The device comprising the excitation and
detection of thermal waves was implanted in a drying machine for fabrics,
where the velocity of the fabric could be varied. Details of the complete
measurement process are described in ref. [13].
Figure 5 shows the amplitude measured for different velocities from 0 to 6
m/min at the frequencies of 5.3, 10, 30 and 175 Hz. For a single frequency,
data have been recorded continuously, with the sample velocity changing in
steps after certain time intervals, approximately 100s, during the process.
Theses measurements are compared with the theoretical interpretation using a
thermal diffusivity of 0.085
10-6
m2/s
and an optical absorption coefficient
of 4000 m-1 [13]. The results show a maximum of the photothermal signal,
shifting to higher velocities with increasing modulation frequencies. Good
agreement between the theoretical curves and the measurements has been found.
The maximum, which occurs at low velocities for low frequencies can clearly be
identified, both in the theoretical data and in the measurements. It shifts to
higher velocities with increasing frequency. Furthermore the decreasing signal
amplitude with increasing frequency is in agreement with the theory, with the
curves going down towards zero in the high-frequency limit.
The exact positions of the maxima for the curves at 5.3 and 30 Hz are not
perfectly identical for theory and experiment. This can be explained by a
relatively high uncertainty of the sample velocity,determined by measuring the
time of markers on the fabric moving between two points. Errors of about 5%,
3.1% and 2.7% have been found for velocities of v = 1, 3 and 5 m/min
respectively. Within this range of errors the positions of the maxima in
theory and experiment are nearly in agreement. The difference of signal level
between experiment and theory increases with the modulation frequency, whereby
the measured amplitude is smaller than the calculated one. This is certainly
due to the fact that the detector measures over a larger spot and not at a
single point. Thus with increasing frequency, the lateral expansion of the
thermal wave is getting smaller, contributing to a decreased signal measured
by the IR detector.
The photothermal phases are presented in Figure 6 for the same measurement. As
the electronic signal processing contributes to a phase offset, the absolute
phase values have been corrected so that the phase for V = 0 was -45 deg. The
theoretical curves fit very well the experimental data, taking into account
experimental errors, assumed to be in the range of 5% for the measured phases.
The curve obtained with a modulation frequency of 5.3 Hz shows larger
deviations, although the general behaviour is correct. Here the difference
between theory and experiment may be explained by the fact that the
penetration depths of the thermal wave tend to be in the range of the sample
thickness, respectively in the range of the thickness of the fibers of the
fabrics, which are one order of magnitude smaller than the sample thickness.
Measurements on fabrics moving in an
industrial drying process have successfully been interpreted using a theory
appropriate for a moving modulated heating spot, respectively moving samples.
Rather good agreement between theoretical curves and measurements has been
found both for the photothermal amplitudes and the phases. For a heating spot
of finite diameter it has been found, that the amplitudes do not get their
maximum values at a sample velocity of zero. Instead, they increase with
increasing sample velocity and reach a maximum, when the space frequency
corresponds to the modulation frequency. This is important for the thermal
characterisation of thin layers: for modulation frequency below the space
frequency, it is the velocity which determines the thermal penetration depth.
Thus, for depth resolved thermal waves inspections, the space frequency is the
limiting factor, and this means that with increasing sample velocity the
diameter of the heating spot has to be increased, in order to guarantee the
same depth penetration.
Further improvements of the theory should be possible, taking into account the
effective optical penetration depth of the sample, its finite thickness, and
the detector convolution effects.
Figure 1: Geometry of the theoretical
problem to be solved.
|
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| (a) |
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| (b) |
Figure 2: Photothermal amplitudes (a)
and phases (b) calculated as function
of the reduced pulsation for different normalised velocities.
 |
| (a) |
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| (b) |
Figure 3: Photothermal amplitudes (a)
and phases (b) calculated as function
of the reduced sample velocity for different values of the reduced pulsation
Figure 4: Schematic of a photothermal
measurement system for moving samples
Figure 5: Photothermal amplitudes as
function of the sample velocity V. Solid
lines represent calculated curves, the symbols represent measured data.
Figure 6: Photothermal phases as
function of the sample velocity V
(same measurements data as from Figure 5).
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Faculty of Theoretical and Applied Science, University of Poitiers, France
1992.
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Hermes Sciences Publications, Paris, 2001.
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