A NEW THERMOELASTIC SOURCE MODEL FOR NON-METALS
C. Edwards, T. Stratoudaki and S. B. Palmer
Department of Physics, University of Warwick, Coventry, UK CV4 7AL
ABSTRACT. This paper presents a source model for laser ultrasonic generation in non-metals in the
thermoelastic regime. It is shown that the waveforms can be directly related to the optical absorption
depth of the material. The longitudinal component due to the buried source is shown to be a bipolar
pulse which can be modelled by convolving the temporal form of the laser pulse with the optical
absorption profile and its time delayed inverted reflection from the free surface. The heated disc is
surrounded by cold constraining material so the radial in-plane forces have Heaviside like time
dependence whereas the vertical component only exists during the initial fast rise of the free surface
and therefore can be modelled as a delta function. The predications of the model are compared for a
TEA CO2 laser generation in a plastic, an Excimer laser in glass and a Nd:YAG laser in silicon. This
model also explains difference between constrained and unconstrained surfaces for the first time, in the
constrained case the vertical force is related to the integral of the laser pulse profile and has Heaviside
like time dependence.
INTRODUCTION
A thermolastic laser source is well understood in metals. [1-3]. The source has been
modeled by considering the forces which arise for a heated volume (center of expansion)
in the bulk of the material. This is equivalent to three orthogonal dipoles with Heaviside
time dependence which push against the surrounding unheated constraining material. The
free surface is allowed for by taking into account reflection and mode conversion; as the
source tends to the surface the vertical dipole is almost exactly cancelled out leaving only
the in-plane dipoles (surface center of expansion). While this gives good agreement with
experiment for metals where the light is absorbed in the very thin electromagnetic skin
depth, it is questionable for non-metals where the light is absorbed in the much thicker
optical absorption depth. This paper shows how the vertical dipole does not have
Heaviside time dependence as there is no constraining material in this direction.
LASER HEATING
Ready showed that laser heating can be considered as instantaneous on nanosecond
time scales [4]. The horizontal expansion is constrained by the surrounding unheated
material; cooling is much slower process (especially in a non-metal) hence the lateral
forces have Heaviside time dependence on ultrasonic timescales (10's of jiseconds). In
contrast, there is no vertical constraint. The vertical expansion of the free surface occurs
simultaneously with the laser heating. Inertial vertical forces are only exerted during this
initial rapid motion and therefore must have 8-function time dependence.
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
326
The heated region is imagined to expand free from any constraint, the forces acting
on the surrounding material are the inverse of the forces required to squash the heated
volume back into its original size [5]. The material is free to expand vertically so normal
forces only exist while it is actively expanding. This is represented schematically in Figure
1.
Beer's Law and Optical Absorption Depth
The absorbed laser power (I a b) is given by the negative differential of Beer's Law
Iab(t)-Io(t)(3(l-R)e-pz
(1)
where I0 is the incident laser power, (3 is the optical absorption coefficient and R is the
reflection coefficient. The temperature rise T is related to the time integral of the laser
pulse
(2)
where p and C are the density and specific heat capacity. If the source is constrained by
surrounding material the ultrasonic source strength is related to T, the vertical force
component at a constrained surface is related to the integral of the laser pulse. At an
unconstrained free surface the source strength is related to the differential of T and hence
has the same temporal form as the laser pulse.
The spatial force distribution follows the exponential optical absorption profile, the
overall form of the laser generated ultrasound pulse can be modeled by convolving the
spatial force distribution (for an idealized 8-function laser pulse) with the actual laser pulse
as shown in Figure 2. As v=d/t the rise time of the spatial or optical absorption impulse is
simply l/(3v. The spatial impulse is calculated by summing over z and consists of the direct
longitudinal arrival plus a time delayed inverted, reflection from the surface as shown in
Figure 2.
H(t)
Direct
arrival
Time delayed
inverted
reflection
FIGURE 1: Forces arising from a thermoelastic force at a free surface.
327
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FIGURE 2: The impulse response is simply the convolution of the optical absorption (5(t)) impulse response
with a unit area I(t).
All calculations are carried out on nanosecond times scales as this is the time scale
of the lasers used for laser generated ultrasound. Figure 3 shows the effect of convolving
the optical impulse with progressively longer laser pulses. The maximum amplitude occurs
for the shortest laser pulse, the exponential risetime of the pulse always follows the optical
absorption profile and the laser pulse form affects only the fall time of the pulse. As the
laser pulse lengthens the amplitude reduces and the waveform broadens and appears more
monopolar.
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FIGURE 3: Optical absorption impulse (dashed line) convolved with progressively longer laser pulses.
328
EXPERIMENTAL SETUP
A modified Michelson interferometer, with a bandwidth of 2kHz-80MHz was used
to measure the absolute epicentral displacements [6]. The detection spot was focussed to
less than 50 |im so that the beam diverged in the transparent samples to reduce unwanted
reflections from the far surface. However because of the very large thermal rise associated
with the source region it was necessary to evaporate a thin aluminum layer on the detection
side of the sample in some cases.
EXPERIMENTAL RESULTS
TEA CCh: Plastic
Figure 4a has a large disturbance associated with the huge thermal rise of the
source itself. The sensitivity of an interferometer is related to the collected power from the
reflecting surface. Only a tiny fraction of the light is collected from the generation surface
because of the focussing lens used but as this displacement is much larger than the
epicentral signal, it is of comparable amplitude.. The oscillatory nature of this initial signal
is because it is greater than several fringe shifts whereas the epicentral displacements are
much smaller than the wavelength of the detection laser (532nm Nd:YAG). This
demonstrates the free surface is expanding simultaneously with the laser pulse. This initial
interference can be suppressed by putting a reflective coating on the detection side. The
agreement between the experiment and theory is good, the horizontal dipolar forces
usually associated with the thermoelastic source is not included in the theory. These
horizontal forces are responsible for the negative ramp after the initial longitudinal arrival.
The optical absorption depth was calculated from the measured risetime and longitudinal
velocity of the material. In all cases the time windows showing the expanded and
theoretical waveforms are the same length.
12.0
2500
time GAS)
FIGURE 4: (a) Waveform taken with a pretrigger on a uncoated Perspex sample, (b) waveform from
aluminum coated sample, (c) expansion of L-pulse and (d) theory for 1/p 150 = Jim and FWHM of 100 ns.
329
XeCl Excimer (308nm) : Glass
Figure 5 shows the results obtained using the excimer laser on a glass sample,
again the comparison between experiment and theory is excellent
Nd:YAG: Silicon
The result s for Nd: YAG laser on a silicon sample are shown in Figure 6.
a
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400
500
600
700
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FIGURE 5: (a) Excimer waveform on a glass sample, (b) expansion of the L-pulse and (c) theory for 1/P =
200 |im and FWHM of 40 ns.
0.10
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1000
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FIGURE 6: (a) NdrYAG waveform on a silicon sample, (b) expansion of the L-pulse and (c) theory for 1/P
= 840 |im and FWHM of 10 ns.
330
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FIGURE 7: Calculated waveforms for 1/p =5 Jim and FWHM of 10 ns for free and constrained surfaces.
Silicon actually contracts when exposed to 1.06 Jim radiation [7] as it alters the
population of the conduction and valence bands in the semiconductor. This effect is
responsible for the inversion of the waveforms. The agreement between the experimental
results and the inverted theory is very good.
ENHANCEMENT BY A CONSTRAINED SURFACE
It is well know that applying a transparent constraining layer to a sample surface
enhances the longitudinal wave amplitude by several orders of magnitude. Previous
solutions [8,9] have all assumed Heaviside time dependence for the buried vertical source
component and predicted monopolar pulses for a free surface. The free surface and
constrained surface solutions were derived using Laplace transforms [9]. This approach
predicted that there was only a change in pulse shape but no enhancement of the peak to
peak amplitude for a constrained surface. The present model predicts a large enhancement
as the forces in the constrained surface case do have Heaviside time dependence and the
solution can be derived from the free surface case by integration. Figure 7 shows
calculated waveforms for a hypothetical material where the light is absorbed in 5|im for a
10 ns laser pulse, this is similar to a Nd:YAG on a metal surface where the laser source is
buried to a depth of a few microns by thermal diffusion. The predicted enhancement is 25
times, however the rise time of the free surface pulse is only a few nanoseconds and would
not be properly resolved unless a detector with a bandwidth of around 1 GHz was used
whereas the constrained case is easily resolved with a bandwidth of 100 MHz
SUMMARY AND DISCUSSION
The vertical force in a thermoelastic source at a free surface has delta function time
dependence. Inclusion of this fact in a simple 1-D model enables the longitudinal wave to
be modeled in non-metals. The agreement between experiment and theory is excellent but
could be further improved by using actual temporal profiles of the laser pulse. A short
laser pulse will always produce the greatest amplitude.
This displacement solution is similar to the stress waveforms predicted by Bushnell
and McCloskey [9] using Laplace transform techniques (stress is related to the spatial
differential of displacement). Bushnell and McCloskey assumed the source had Heaviside
331
time dependence (along with many other authors). If they had used delta function time
dependence they would have got the correct solution, indeed their solution is valid for the
constrained case.
ACKNOWLEDGEMENTS
CE wishes to acknowledge Alan Aindow introducing him to the problem of buried
thermoelastic sources while they shared an office in the University of Hull. TS is
supported by the E.U. (Marie-Curie fellowship HPMF-CT-2000-00999).
REFERENCES
1. Scruby C. B. and Drain L. E., Laser-Ultrasonics: Techniques and Applications, Adam
Hilger, Bristol, UK, 1990.
2. Rose L. F. R., J Acoustic Soc. Am. 75 pp.723-733 (1984).
3. Doyle P.A., J. Phys. D : Appl. Phys, 19, pp. 1613-1623 (1986).
4. Ready J F, Effects of high power laser radiation, Acad Press; New York, 1971
5. Aki K. and Richards P. G. Quantitative Seismology, Theory and Methods 1, San
Francisco: Freeman, 1980
6. Bushell A. C., Edwards C. and Palmer S. B., in Review of Progress in QNDE, Vol. 11
eds. D. O. Thompson and D. E. Chimenti, (Plenum Press, 1992), pp. 569-573.
7. Gauster W. B. and Habing D. H., Phys Rev Lett., 18, pp 1058-1061 (1967)
8. Telshow K. L. and Conant R. J., J. Acoust. Soc. Am., .88, pp.1494-1502, (1990).
9. Bushnell J. C. and McCloskey D. J., J. of Appl. Phys., 39, pp. 5541-5546, (1968).
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