- Email: [email protected]
Generating acoustic waves by laser: theoretical and experimental study of the emission source
Generating acoustic waves by laser: theoretical and experimental study of the emission source J.D.
Aussel*,
A. Le Brun
and J.C.
Baboux+
EDF, Direction Etudes et Recherches, 93206 Saint-Denis, France + INSA, Laboratoire Traitement du Signal et Ultrasons, 69621 Villeurbanne, Received
12 February
France
1988
A laser pulse incident on a material may generate acoustic waves by means of two different phenomena: by thermal expansion at low incident power density (thermoelastic effect); and by vaporizing surface material at high incident power density (ablation effect). A synthesis of various theoretical models representing the acoustic displacements generated by a point laser impact is given, and then validated by interferometer measurements of ultrasonic displacements generated both under ablation conditions and under thermoelastic conditions.
Keywords:
laser generation
of ultrasound;
Introduction Ultrasonic techniques are very widely used in measuring and in non-destructive inspection. These techniques use piezoelectric transducers as emitter/receivers with liquid coupling to the part to be inspected. In cases where conventional methods are difficult to implement e.g. hot products, moving products, samples of complex geometry, or problems of reproducibility due to the coupling liquid, ultrasonic measurements can be made at a distance and without contact by means of laser generation of ultrasound associated with contactless reception such as laser interferometry. In 1963, White’ demonstrated laser generation of acoustic waves in a solid. The first models describing the phenomenon were one-dimensiona12-5, and assumed that the laser impact occupied the entire surface of the material. Since the laser-generated ultrasound was received by means of narrow passband piezoelectric transducers, harmonic directivity patterns were initially used for describing the ultrasonic waves generated by a point laser the use of broadband receivers impact 6-E. Subsequently such as capacitive sensors 9*10, thick piezoelectric discs’ 1 or laser interferometersr2-r5 have made wideband modelling of laser-generated acoustic displacements necessary. We present theoretical models of the acoustic displacements generated by quasi-point laser impact, together
acoustic
source
with experimental displacements
model
interferometric
measurements
Theory The geometry of the problem is shown in Figure I: the laser impulse is incident on the free plane surface z = 0 of a semi-infinite elastic material. The elastic and thermal constants of the material are as follows: 2 and p Lame constants, p density, k thermal diffusivity, K thermal conductivity, C, specific heat capacity, and a coefficient of thermal expansion. The laser pulse arriving at point 0 has an energy Q and varies with time as a d(t) function (where 6 is a Dirac impulse). General formulation
Due to the axial symmetry of the problem, the displacement u( M, t) generated at any point may be expressed@ terms of a scalar potential 4 and a vector potential $ whose only non-zero component extends in a plane perpendicular to MOz and is written $‘”
(1)
The elastic propagation potentials are l6
equations
as a function
* Present address: National Research Council Canada, Industrial Materials Research Institute, 75 Bd de Mortagne, Boucherville, Quebec, Canada
0041-624X/88/050245-1 1 $03.00 @ 1988 Butterworth & Co (Publishers)
Ltd
of these
of the
(3)
Ultrasonics
1988
Vol 26 September
245
Generating
acoustic 06
waves by laser: J.D.
Aussel et al.
Since the laser impact occurs at a point, the density of power absorbed per unit of volume of the material is
(t)
w, = AQ%.6(z)6(1)
(8)
where A is the coefficient of light energy absorption of the material and where s(t) is the Dirac time function. The stress/deformation relationship for an elastic material is
L k,
Oij
c1. P
=
(9)
CijklEkl
where Cijkl is the tensor of the elastic constants of the material and ekl is the tensor of elastic deformation. The elastic deformation tensor may be expressed as a function of temperature
K, Cv, (Y
Ek,
=
TV
T6,,
(10)
where 6,, is the Kronecker symbol. For an isotropic material, with Lam& constants 1 and p, the elastic constants of the material may be expressed by Z
cijkl
=
A6ijdkl
+
PtSiksjl
+
6i16jk)
(11)
The elastic stress due to thermal expansion from Equations (9)-( 11) as follows Figure 1 Geometry of the problem. The laser beam of energy 0 and time dependence 6 (t) is incident on the surface at 0. Due to axial symmetry, cylindrical coordinates, R, 0, I are used
a=* I a* a=* dZ2+;*z+dz2-;I-pZ.()t2=0
*
1
a=* (4)
S
where C, and C, are, respectively, the ultrasound velocities in the material of dilatational (i.e. longitudinal) waves and of equivoluminal (i.e. shear) waves. The initial conditions are (5) The stresses generated under thermoelastic conditions and under ablation conditions are calculated below and are used as boundary conditions for solving Equations (3) and (4). The point of impact 0 is initially assumed to be buried at a depth d beneath the free surface z = -d. Then solutions corresponding to an impact on the free surface are obtained by taking the limit as d -+ 0. Thermoelastic
effect: thermal expansion
stress
With an incident laser pulse of low power the increase in temperature due to the laser sufficiently small to avoid any change of state in material, and elastic stresses are obtained by bulk expansion. The temperature T( M, t) at any point by the thermal conductivity equation5
density, pulse is the solid thermal is given
where wa(M, t) is the density of power absorbed in the material. Assuming that only a small portion of the absorbed energy is converted into an elastic wave, the propagation equation can be solved by using the temperature distribution obtained by solving the conductivity equation independently from the elastic problem’
iaT
V2T-k.Z+K~“=0
246
i
Ultrasonics
(7)
1988
Vol 26 September
is deduced
oij = Cijk,aT6,,
(12)
For the moment of a force dipole equivalent to a point source of volume expansion, Aki and Richards” give Fij = SvCijklEkldV whence, Equation Fii = (31+
(13)
by making ( 11)
the
deformation
explicit
using
TdV
2&
using conductivity
Equation
(14) (7) Rose”
Fii = (32 + 2ll)a,kQH(t)
has obtained (15)
Under thermoelastic conditions, the amount of the force dipole resulting from a laser pulse QS( t) has a step pulse shape H(t). The boundary conditions under thermoelastic effect for a source buried at depth d are on the surface z = -d 0 zz = CJTz= or* = 0
(16)
on the plane z = 0 c zz = grr =(31+
2&Q&S(r)H(t)
(17)
The ablation effect: vaporization stress With a high power density incident laser pulse, the increase in absorbed power due to the laser pulse gives rise to melting and then to vaporization of a small quantity of material at the surface. Readylg has studied the effects of the absorption of laser radiation having a power density WH(t) on an opaque surface, where H(t) is a Heaviside unit step. The laser power density per unit area absorbed by the material is w,(t) = AWH(t) The material
beings to vaporize
(18) after a time t, given by (19)
Generating
where C, is the specific heat capacity of the material, TV is the vaporization temperature, and q is the initial temperature of the material. In order to approach pulse excitation, let us consider a laser pulse having the following power density w(t)=
W[H(t)-H(t-r,)]
where r, is the duration Equation ( 19) the material power densities such that
(20) of the laser will vaporize
1
1’2
pulse. From for absorbed
CT == =
(Awl2
pc + pLC,(
where L is the latent heat of vaporization of the material. In practice, for a laser pulse of finite duration, it is difficult to calculate the amplitude of the normal vaporization stress. Under such conditions, the thermal constants of the material are no longer constant as a function of temperature, and in particular the absorption coefficient A. Further, for an impact of finite size, the vaporization of the gas phase also causes a portion of the material to be ejected while still in the liquid phase, thereby increasing the amplitude of the normal stress due to vaporization. It is also difficult to calculate the variation in stress as a function of time under pulse conditions. At low incident power density, the material drops rapidly to below its vaporization temperature after the end of each laser pulse and as a result the normal stress is pulsed. However, at high power density, vaporization continues for a long time after the end of each laser pulse. The variation in the normal stress as a function of time lies between two limiting cases: the same shape of each laser pulse w(t) and a pulse shape of the Heaviside unit step type. In order to model vaporization conditions, the force equivalent to the vaporization will be assumed to have amplitude f,, to act normally to the surface, and to vary as a pulse 6(t) or as a step H(t). The boundary conditions for an ablation source (normal force) buried at a depth d are then on the surface z = - d c zz = or, = 6,, = 0
(23)
G
(26)
Cl.
(21)
(22)
T, - IJ
waves by laser: J. 0. Aussel et al.
The displacement is a combination of a dilatational, or longitudinal, wave L, an equivoluminal, or shear, wave S and, for points situated in a direction fI relative to the normal Oz at an angle greater than a critical angle 8,, a head wave LS. The head wave comes from the longitudinal wave L at grazing incidence, travelling on the surface which generates a shear wave all long its path in the material in the direction of the critical angle. The critical angle is given by sin8,=-
Under constant vaporization conditions, the vaporization stress is normal to the surface and, for a step power density WH(t) as given in Reference 19 is
acoustic
where C, and C, are S wave and L wave acoustic velocities. Figure 2 shows the relative positions of the wave fronts in the material at a given instant. The arrival times of the three types of waves at a point M situated at a distance R from the point of impact 0 and in a direction 19relative to Oz (Figure 2) are given by
t~s=R[-&sinB+($-&)1’2cost?]
(27)
Analytical calculation of the displacement u(M, t) by inverting the integral solutions have been performed’8,2’,22 only for points situated on the surface (z = 0) and on the axis of symmetry (0 = 0). Displacement at other points can be calculated by numerical integration23s24. Displacement
along the axis of symmetry
Oz
Under thermoelastic conditions, displacement along the axis of symmetry Oz is given for a laser pulse of power
Q(t)W8 (28) where fL and fs are given in the Appendix. Under ablation conditions, displacement along the axis of symmetry for a normal force fad(t) is given by2r
u,(R, 8,t) =
&(%+t$
where gL and gs are given in the Appendix. In practice, the elastic half-space is limited by a plane surface z = R. The displacement measured at the epicentre, i.e. on the axis of symmetry and on the plane z = R, has
on the plane z = 0 (24)
Solving
the propagation
equations
Rose’* has solved these equations for the thermoelastic case and Miklowitz 2o has solved these equations for a normal force f, which case is identical to that which occurs under ablation conditions. The solution is obtained by using Laplace transforms (transient conditions) and Hankel transforms (axial symmetry), and by inverting the resulting transform solutions by the Cagniard-DeHoop method. The displacement u( M, t) is then obtained in the form of double integrals made up as follows
z Figure
u(M,
t) = UL + us + uLs
R
0
(25)
2
Positions of the wavefronts
at a given instant. L,
dilatational wave; S, equivoluminal wave; LS, head wave
Ultrasonics
1988
Vol 26 September
247
Generating acoustic waves by laser: J. D. Aussel et al. a different account Under
equation
when
thermoelastic
conditions’
31+2~ u,‘(R, 8, t) = AQ----a--where fLE and Under
reflections
crk
are
taken
into
uz (nm) 0.1
8
4
fs” are given in the Appendix.
ablation
conditions
uZE(R,O,t)=
(31)
where gLE and gsE are given in the Appendix. The displacements corresponding to pulse conditions 6(t) are thus known. For a laser pulse having finite shape h(t), the displacement is obtained by a convolution of Equations (28)-( 3 1) with h(t). Figure 3 applies to thermoelastic conditions and shows the displacement along the axis of symmetry of a point buried at a depth R = 25 mm; the material is aluminium with an absorption coefficient A = 10%; the laser impulse has a Gaussian shape (duration 20 ns) and its energy Q is 10 mJ. The displacement at the epicentre is about twice the displacement at the buried point in the halfspace. The displacement is negative and corresponds to material being pulled towards the impact surface. Figure 4 is applicable to ablation conditions and shows the displacement along the axis of symmetry of a point buried at a depth R = 25 mm beneath the surface and at the epicentre of a plate of thickness R = 25 mm, for a normal vaporization force of 1 N and a Gaussian pulse shape of duration 20 ns. The Gaussian force corresponds to pulse conditions, i.e. to low rates of vaporization. Figure 4 also shows the displacement generated at high rates of vaporization by a force having a stepped pulse shape H(t).
tL = - R CL
tS=l
Y$
CS
, u,E(nm) 0.1 b
0.05 -
C
0.5-
Surface displacement The largest contribution to surface displacement comes from the surface wave, since the amplitudes of the other waves are much smaller. Under thermoelastic conditions, the surface wave is given by” 32+ 2/1 ak
uZ(8,r, t) = - AQP*-----2+2~
1
Im’*‘*I)
a&
R
R
CL
cs
(32)
K 8nC,‘r2’at
Figure
4
Displacement on the axis of symmetry for aluminium = 1 N, R = 25 mm. (a) Gaussian force, displacement, in the half-spke. (b) Gaussian force, displacement at the epicentre. (c) Step force, displacement at the epicentre under ablation conditions: F
3
l,
0(nm)
O.Ol-
where fk is given in the Appendix. Where v is the Poisson ratio for the material and C, is the velocity of surface waves in the material, a good approximation is given2’ by
o-02-
C = 0.862 + 1.14~ R
0.03I tL = R
by2
t
S
= _ R
CL
Time
CS
Figure 3 Displacement on the axis of symmetry for aluminium under thermoelastic conditions: Q = IO mJ, R = 25 mm. (a) Displacement within a half-space. (b) Displacement at the epicentre of a plate
248
Ultrasonics
1988
Vol 26 September
l+v
‘Cs
(33)
Figure 5 shows the displacement at R = 25 mm from the point of impact on an aluminium half-space, and due to the surface wave for a laser pulse of Gaussian shape, of duration 10 ns, and of incident energy Q = 10 mJ. Under ablation conditions, the surface displacement is given26 by
Generating uz (nm)
acoustic
waves by laser: J. 0. Aussel et al.
uz (nml 3.0-
0.30
a 2.00.20
1.0-
0.10
0.0
0.00
\
-l.O-
-2.o-0.10
-3.0.1 8.0
8.2
8.4
8.6
8.8 9.0 Time (us)
8.2
8.4
8.6
8.8 Time (LB;?
-0.20
uz (nmj 1.0 b -a.30
I
8.0
8.2
I 8.4
I
I
8.6
8.8
1
1
0.0
9.0
Time (Pr) Figure 5 Theoretical displacement. Thermoelastic conditions, surface wave, aluminium. Gaussian laser pulse of 10 ns duration, Q=lOmJandR=25mm
-1.0
-2.0
where gR is given in the Appendix. Figure 6 shows the surface wave for a point normal force of amplitude 1N of Gaussian pulse shape of duration 10 ns, and at R = 25 mm from the point of application of the force on an aluminium half-space, and also for a normal force whose pulse shape is a step H(t).
Experimental To illustrate the accuracy of the above theoretical predictions, a series of experiments are described using interferometric detection. The use of pulsed laser generation with wideband interferometric detection has been described set-up for by several authors 12,14,27. The experimental this work is shown in Figure 7. The generating laser is a Nd:Yag laser operating at a wavelength of 1.06 pm and having a pulse duration of 10 or 20 ns with a maximum energy of 330 mJ. The generating laser pulse is focused on the sample by a set of lenses, and the energy is adjusted by calibrated optical attenuators. The acquisition system is synchronized by a photodetector. A Michelson type laser displacement interferometer is used for reception ” , having a bandwidth of 10 kHz50 MHz. The samples used are made of steel or aluminium. They are plates of different thicknesses, and half cylinders to study the displacements generated in different directions. The surfaces of the samples are polished in order to reflect the reception laser beam as well as possible. Further work is in progress using a Mach-Zehnder
-3.0
8.0
Figure 6 Theoretical displacements. Ablation conditions, surface wave, aluminium. (a) Gaussian normal force of 10 ns duration, amplitude 1 N, R = 25 mm. (b) Stepped normal force, amplitude 1 N.R=25mm
type probe with sensitivity better than 10m4 A HZ-‘/~. This allows the measurement of pulsed displacements to be undertaken on industrial samples with rough surfaces29T30. Power
limit of thermoelastic
and ablation
regimes
Figure 8 shows the displacement measured at the epicentre of a 10 mm thick steel sample. The energy of the laser pulse is constant and the incident power density Won the material is modified by focusing the beam. The duration of the laser pulse is 10 ns. The noise observed at the beginning of the signal is due to triggering the generating laser. In comparison with the theoretical curves in Figures 3 and 4, it can be seen that for power densities in the range 9-141 MW cmm2 the shape of the displacement waveforms seem to reproduce most of the pattern given by the theoretical displacement of the thermoelastic regime. Nevertheless, a peak in L appears
Ultrasonics
1988
Vol 26 September
249
Generating
acoustic
waves by laser: J. D. Aussel et al.
LASER GENERATION
Photodetector
Lens
t
T
Sample
In Figure 9a, at low power densities, the generating source is thermoelastic and the amplitude of the 3 MHZ component increases linearly with the power density, or more precisely with the energy, as predicted by Equation (15). The amplitude at 50” is higher than at O”, as predicted by the directivity patterns of the thermoelastic source. In Figure 9b, for higher power densities, the 3 MHz longitudinal variations at 0 and 50” are no longer linear and have an intersection point at 15 MW cm-‘. The ablation regime begins and is of the same order of magnitude as the thermoelastic regime. In Figure 9c, at higher power densities, the ablation regime is much greater than the thermoelastic regime and the 3 MHz longitudinal amplitude is higher at 0 than at 50”, as predicted by the directivity patterns of the ablation source. The amplitudes of the longitudinal waves do not vary as expected from Equation (22), showing that the stress generating phenomena is more complex than pure vaporization. The power density limit of the two regimes, which depends on the material, can be estimated to be lo-20 MW cm-*. The ablation level can be extracted from Equation (21) which gives a theoretical value of A W = 15 M W cm - ’ for aluminium.
Displacement
Figure 7
Experimental set-up
from N 65 MW cm -L, which is also visible in the first reflection 2L. This peak corresponds to the appearance of vaporization conditions’. The transverse wavefront is not as steep as predicted because the laser impact is not a point. In Figure 8a where the diameter of the impact is 19 mm, the step corresponding to this wavefront practically disappears. It is difficult to observe thermoelastic conditions with a point impact. The generated displacement is proportional to the incident energy Q, and a quasi-point impact would give rise to a power density per unit area which is greater than the vaporization limit. Thus, in order to observe a signal of sufficient amplitude without vaporizing the material, it is necessary to use an impact of non-negligible size relative to ultrasound wavelengths. The signal-to-noise ratio at the beginning of the waveforms is low due to the Q-switching of the generating laser, so that the power limit of the two regimes is difficult to define precisely by observing the L ablation peak outbreak. In order to measure precisely this limit in aluminium, we used a contact piezoelectric transducer as receiver instead of the interferometer, and measured the 3 MHz longitudinal amplitudes generated at 0 and 50” uersus the incident laser power density. These amplitudes are given in Figure 9. The study of laser-generated directivity patterns6*7 has shown that the maximum of the longitudinal mode directivity occurs around 50-60” in the thermoelastic regime, and at 0” in the ablation regime.
250
Ultrasonics
1988
Vol 26 September
of the epicentre
Figure
10 shows
Radial
displacement
the displacement at the epicentre of a 5 mm thick steel sample generated by a laser impact of 20 ns duration, 1 mm diameter, energy 5 mJ, and of incident power density 32 MW cm-‘. The displacement obtained in this way may be considered as being the combination of a displacement due to the thermoelastic effect and of a displacement due to vaporization’. A theoretical curve obtained by combining these two effects can thus be obtained by considering the normal force due to vaporization to be a Gaussian pulse since the vaporization rate is low. At larger incident power densities, the ablation regime becomes dominant. Unlike thermoelastic conditions, the pulse shape of the displacement depends on the incident power density. In comparison with the theoretical Figure 4, Figure 11 shows that at low vaporization rates the normal force is of the pulse type while by increasing the power density it approaches the step type. Vaporization continues after the the end of the laser pulse. By comparing the heights of the longitudinal peaks for several incident power densities with the heights of the theoretical peaks for the displacement generated by forces having a Gaussian pulse shape, the curve of Figure 12 is obtained. This shows the amplitude of the normal force due to vaporization on aluminium and on steel. The normal force has substantially the same amplitude in both materials, giving about half the displacement in steel since its Lame constant p is about twice as large (Equation (29)). at any point
The radial displacement,
i.e. displacement directed towards the impact point for various angles of observation is obtained by interferometric means on semi-cylindrical samples. Radial displacement under thermoelastic conditions. Figure 13a shows the radial displacement generated in
Generating acoustic waves by laser: J. D. Aussel et al.
uz (nm)
a) W = 9 MW/cm*
(nm)
5
b) W =
16 MW/cm*
d) W =
65 MW/cm*
l2 0 -2 -4 -6
uz (nm)
1 /ls-
1P
uz (nm) cl W = 41 MW/cm*
+2 0 -2
1 PS u_ (nm)
\
-5-1
uz (nm) I
-5 -10
1 PS g) W =
h) W = 460 MW/cm*
141 MW/cm*
ow 1 \
L
I
2L
-15
1 PS uz (rim) 22
i) W =
‘zli
1050 MW/cm*
j)W
=
18k
L
1510-
Figure 8
2L
Displacement measured at the epicentre of a 10 mm thick steel sample. Laser pulses of constant energy (330 mJ) and variable
diameter
Ultrasonics
1988
Vol 26 September
251
Generating
acoustic
waves by laser: J. D. Aussel et al. +
t
-0.410
5
I 0 b
1 !Js Thermoelsstic regime
T
1 1 I J
I
-
Time
Figure 10 Intermediate conditions. Epicentre displacement: 5 mm thick steel sample, 1 mm impact diameter, 0 = 5 mJ, power density W, = 32 MW cm-‘, duration 2 TV= 20 ns. (a) Experimental curve. (b) Theoretical curve QT = 2.5 mJ, f, = 1 N Gaussian
Ablation regime
/
0.8 0.6
/
/
,‘b
,i” /I’ d
+-’
Time
___q_-------+
_-+‘--
#v’ 300
Figure 9 Longitudinal waves amplitude at 3 MHz (arbitrary units) versus incident power density in aluminium. (a) Thermoelastic regime, (b) intermediary regime, (c) abalation regime. 0, 6,= 0”; +, 0=50”
aluminium by a laser impact of 3.5 mm diameter, 20 ns duration, and of 16 mJ energy. The types of waves obtained match those predicted: the critical angle in aluminium is about 30”. For directions between 0” and 22.5”, i.e. directions which are less than the critical angle, shear wave fronts and longitudinal wavefronts are observed. For directions between 45” and 77’S”, the appearance of a head wavefront LS is observed which moves closer to the longitudinal wavefront on approaching the surface. The longitudinal peak increases going away from the normal and becomes wider. This widening is due to high frequency diffraction, since the laser impact is not a point when compared with the wavelengths under consideration. This agrees in the thermoelastic case with previously published results3’. Radial
displacement
under
ablation
Figure 13b and c show the radial displacement
252
Ultrasonics
1988 Vol 26 September
conditions.
generated
2
1 i
II
0
I
r
I
I
I
I
1
*
I
Time Figure11 Ablation conditions. Influenceof incident energy power density on the displacement measured at the epicentre: 25 mm thick aluminium, laser pulse duration 20 ns, 1 mm impact diameter
in aluminium at power densities lying in the range lOC-300 MW cme2. Here again, the predicted types of wave are observed: going away from the normal, the amplitude of the longitudinal peak decreases and the waveform approaches the waveform under thermoelastic conditions. It thus appears that ablation conditions are
Generating
acoustic
waves by laser: J. 0. Aussel
et al.
dominant close to the normal to the surface, whereas thermoelastic conditions are more important for points which are at a distance from the normal to the surface.
F, (N) 25
Surfaces
waves
Figure
14 shows the surface displacement of a 20 mm thick steel sample due to a laser impact having a diameter of 2 mm at incident power densities of 30, 200 and 1000 MW cm-‘. Under thermoelastic conditions at 30 MW cm -‘, the pulse shape predicted by Figure 5 is indeed obtained. Under ablation conditions at 200 MW cme2, the displacement waveform corresponds to a normal force of the pulse type, and at 1000 MW cmw2, to a normal force of step-time dependence. The amplitudes and the durations of the surface waves are greater than those predicted because the impact geometry is not a point and the theoretical displacement ought to be integrated as a function of the space distribution of the laser impact.
Figure 12 Maximum normal force under ablation conditions as a function of incident power density: laser pulse of 20 ns duration, 1 mm impact diameter. +, Steel; 0, aluminium
b
a
8 = o”
(nm)
(nm)
e = o”
(nm),
e = o”
(nm)
0 = 22.5’
(nm)
0 = 22.5’
0.5 i
- 0.5 V’
1w 0 = 22.5’
(nm)
3-
1.0 - L
0.5
L 2-
0
- 0.5
0.5 -
I
b
I
1 ccs
e =
(nm)
45O
0
1 PS
(nm)
45O
e =
77.5O
hm)
.
e = 45O
)
1 1.0
-“.5-
1 /ls
e =
-
L
1w
1P (nm) 1.0
J
3 L
Ls
s
e =
(m-n)
77.5O
-t
1P Figure 13 Radial displacement in aluminium R= 25 mm. Laser pulse of 20 ns duration, diameter =a. (a) Thermoelastic regime: W = 8 MW cm-*, a = 3.5 mm. (b) Ablation regime: W = 100 MW cm-*, a = 1 mm. (c) Ablation regime: W = 300 MW cm-*, a = 1 mm
Ultrasonics
1988
Vol 26 September
253
Generating
acoustic
waves by laser: J. D. Aussel et al.
Conclusions The models for the bulk expansion source and for the normal force agree well with the experimental results of displacements generated by a laser under thermoelastic conditions and under ablation conditions. All four types of waves predicted by these models are observed: longitudinal waves, shear waves, head waves, and surface waves. The displacements on the surface and at the epicentre have the same time distributions as those predicted. The displacements at other points show that ablation conditions are dominant close to the normal to the point of impact and that thermoelastic conditions are dominant far from the normal to the point of impact. The waveforms under thermoelastic conditions remain the same for varying energy, whereas under ablation conditions the pulse shape varies as a function of the density of the
incident power. The amplitude and the pulse shape of the vaporizing force could be obtained by deconvolution of the experimental displacements at the epicentre with the theoretical displacement corresponding to a force in the form of a Dirac pulse.
Acknowledgements One of the authors (JDA) wishes to thank the directors of EDF, Etudes et Recherches for the opportunity to prepare his Doctoral Thesis within their facilities, based in part on the work reported here.
References 1
0.3
0.2
6
0.1
0
7
- 0.1
8
9 b
IO
11
12
13 14
15
16 17 18 19 20
21 1 PS Figure
22
14
Surface waves. Displacement normal to the surface: laser pulse of duration 10 ns, 2 mm impact diameter. Observation at 65 mm from point of impact (steel sample). (a) Thermoelastic conditions: W, = 20 MW CIT-~, 0 = 10 mJ. (b) Ablation conditions: W, = 200 MW cm-‘, 0 = 63 mJ. (c) Ablation conditions: W,, = 1000MWcm-2,Q=330mJ
254
Ultrasonics
1988
Vol 26 September
23 24 25
White, R.M. Elastic wave generation by electron bombardment of electromagnetic wave absorption J A$ Phys (1963) 34 2123-2124 White, R.M. Generation of elastic waves by transient surface heating .I Appl Phys (1963) 34( 12) 3559-3567 Ready, J.F. Effects due to absorption of laser radiation J Appl Phys (1965) 36(2) 462-468 Bushnell, J. C. and McCloskey, D.J. Thermoelastic stress production in solids J Appl Phys (1968) 39( 12) 5541-5546 Wet&, G.C. Photothermal generation of thermoelastic waves in composite media IEEE 7fans Ultrason Ferroelectrics Frequency Control (1986) 33(S) 450-461 Hutchins, D.A., Dewhurst, R.J. and Palmer, S.B. Mechanisms of laser-generated ultrasound by directivity pattern measurements Proceedings of Ultrasonics International 81 (1981) 20-25 Hutchins, D.A., Dewhurst, R.J. and Palmer, S.B. Directivity patterns of laser-generated ultrasound in aluminium J Acoust Sac Am (1981) 70(5) 1362-1369 Budenkov, G.A. and Roiko, M.S. Excitation of stress waves in solids by laser beam with finite cross section through thermoelastic effect Defektoskopiya (1984) no. 3 16-24 Dewhurst, R.J., Hut&ills, D.A., Palmer, S.B. and Scruby, C.B. Quantitative measurements of laser-generated acoustic waveforms J Appl Phys (1982) 53(6) 4064-4071 Scruby, C.B., Dewhurst, R.J., Hutchins, D.A. and Palmer, S.B. Quantitative studies of thermally-generated elastic waves in laser-irradiated metals J Appl Phys (1980) 51( 12) 6210-6216 Dewhurst, R.J., Hutchins, D.A., Palmer, S.B. and Scruby, C.B. The performance of thick piezoelectric transducers as wideband ultrasonic detectors Ultrasonics (1983) 21 79-84 Boudarenko, A.N., Drobot, Y.B. and Kruglov, S.V. Optical excitation and detection of nanosecond acoustic pulses in nondestructive testing Defekfoskopiya (1976) no. 6 85-88 Tam, A.C. and Coufal, H. Photoacoustic generation and detection of 10 ns acoustic pulses in solids Appl Phys L&t (1982)42( 1) 33-35 Hutchins, D.A. and Nadeau, F. Non-contact ultrasonic waveforms in metals using laser generation and interferometric detection Proceedings of the 1983 IEEE Ultrasonics Symposium (1983) 1175-1177 Rourkoff, E. and Palmer, C.H. Low-energy optical generation of acoustic pulses in metals and nonmetals Appl Phys Lett (1985) 46(2) 143-145 Ewing, W.N., Jardetzky, W.S. and Press, F. Elastic Wnves in Layered Media McGraw-Hill, New York (1957) 9 Aki, K. and Richards, P.G. Quantitative Seismology. Theory and Methods (Ed Freeman) San Francisco (1980) 1 Rose, L.R.F. Point-source representation for laser-generated ultrasound J Acoustic Sot Am (1984) 75(3) 723-732 Ready, J.F. EfSects of High-Plower ‘La&’ Radiation Academic Press, New York (1971) Ch. 3 67-125 Miklowitz, J. Elastic waves and waveguides Applied Mathematics and Mechanics North-Holland Publishing Co., Amsterdam (1978) 22 231-366 Sinclair, J.E. Epicentre solutions for point multipole sources in an elastic half-space J Phys D (1979) 12 1309-1315 Doyle, P.A. On epicentral waveforms for laser-generated ultrasound J Phys D (1986 j19 1613-1623 Gakenheimer. D.C. Numerical results for Lamb’s point load problem J Appl Mech (1970) 522-524 Willis, J.R. Self-similar problems in elastodynamics Phil Trans Roy Sot London (1973) 274 435-491 Achenbach, J.D. Wave propagation in elastic solids Applied
Generating Mathematics and Mechanics North-Holland Publishing Co., Amsterdam (1973) 16 262-325 Mooney, H.M. Some numerical solutions for Lamb’s problem Bull Seismol Sot Am (1974) 64(2) 473-491 Royer, D. and Die&saint, E. Optical detection of sub-angstrBm transient mechanical displacements Proceedings ofthe 1986 IEEE Ultrasonic Symposium (1986) 527-530 Drain, LE., Speake, J.H. and Moss, B.C. Displacement and vibration measurement by laser interferometry Proceedings ofthe First European Congress on Optics Applied to Metrology SPIE (1977) 136 52-57 Royer, D., Diiulesaint, E. and Martin, Y. Improved version of a polarized beam heterodyne interferometer Proceedings ofthe 1985 IEEE Ultrasonic Symposium (1985) 432-435 Lesne, J.L., L.e Brun, A., Royer, D. and Die&saint, E. High bandwidth laser heterodyne interferometer to measure transient mechanical displacements Proceedings of the International Symposium on the Technologies for Optoelectronics in press Hutchins, D.A. Mechanisms of pulsed photoacoustic generation Can J Phys (1986) 64 1247-1264
26 2-l
28
29
30
31
acoustic
waves by laser: J. D. Aussel
(A7) -4ss~(ss~+cL-z)(ssz+cs-*) gsE= [(2ss*+cs-~)~-4s~(ss*+cL-~)~‘*(ss~+cs-*)~’q~ (A81
with andS,‘=($--$1 Displacement
on the surface
Thermoelastic
regime (2C,_2
fR = cn-4cs-4+
- C,_Z)(
(A9)
1- v)
c,-*cs-z(c,-z-vc,-~)(2c,-*
-cs-+-cR-q
Appendix
(AlO)
Displacement
at the epicentre
Thermoelastic
regime
Ablation regime gR
fL =
- 2s,z(s,z (2q.2 + c,-*)2
+ c,-~)(s,2
-4s,7s,*
+ c,-2)“2
+ cL-2)1’2(sLz
=
glR
+
(All)
g2R
with
+ Cs-2)“2
(A121
(Al) fs=
et al.
S,2(S,2 + Cs-~)“~(2SSZ + cs-2) (2Ss2 + Cs-‘)’
- 4S,2(Sz + C,,-2)1/2(Ssz + Cs-z)l/z
(A13)
H (A21
- s,qs,2 fLE = [(2S,2 + fz-92
+ c,-z)(s,z - 4sLz(s,*
+ c,-z)“z(2s,z
+ c,-2)
+ c,-z)“z(s,2
+ c,-zpq2
and v being the Poisson ratio for the material Iz+ 2/l
QJ=
P (A31 SS2(SS2 + c,-2)“2(ss2 + Cs-2)(2SsZ + cs-2) fs” = [(2&Z + cs-92
- 4S,2(S,2 + cL-2)1’2(ss*
y=tc” R’
’
T=t-
CS
(A14)
R
anda,isarealrootofP+Qx+Rx2+x3=0 where
1
+ cs-*pq2
(A4)
(AW
with
(A16) (A17)
Ablation regime and A1 is such that x (W
2 1-,-2x3 ?
(
P+Qx+Rx2+x3
> =4+&t
A2
----++
1
x + a,
A3
x + a3
for v < 0.263 =4+&+
( flR+ i a,)( aR - i ai) 1
(X+tlR+iUi)(X+aR-iiai)
for v 2 0.263
Ultrasonics 1988 Vol 26 September
255
Aussel*,
A. Le Brun
and J.C.
Baboux+
EDF, Direction Etudes et Recherches, 93206 Saint-Denis, France + INSA, Laboratoire Traitement du Signal et Ultrasons, 69621 Villeurbanne, Received
12 February
France
1988
A laser pulse incident on a material may generate acoustic waves by means of two different phenomena: by thermal expansion at low incident power density (thermoelastic effect); and by vaporizing surface material at high incident power density (ablation effect). A synthesis of various theoretical models representing the acoustic displacements generated by a point laser impact is given, and then validated by interferometer measurements of ultrasonic displacements generated both under ablation conditions and under thermoelastic conditions.
Keywords:
laser generation
of ultrasound;
Introduction Ultrasonic techniques are very widely used in measuring and in non-destructive inspection. These techniques use piezoelectric transducers as emitter/receivers with liquid coupling to the part to be inspected. In cases where conventional methods are difficult to implement e.g. hot products, moving products, samples of complex geometry, or problems of reproducibility due to the coupling liquid, ultrasonic measurements can be made at a distance and without contact by means of laser generation of ultrasound associated with contactless reception such as laser interferometry. In 1963, White’ demonstrated laser generation of acoustic waves in a solid. The first models describing the phenomenon were one-dimensiona12-5, and assumed that the laser impact occupied the entire surface of the material. Since the laser-generated ultrasound was received by means of narrow passband piezoelectric transducers, harmonic directivity patterns were initially used for describing the ultrasonic waves generated by a point laser the use of broadband receivers impact 6-E. Subsequently such as capacitive sensors 9*10, thick piezoelectric discs’ 1 or laser interferometersr2-r5 have made wideband modelling of laser-generated acoustic displacements necessary. We present theoretical models of the acoustic displacements generated by quasi-point laser impact, together
acoustic
source
with experimental displacements
model
interferometric
measurements
Theory The geometry of the problem is shown in Figure I: the laser impulse is incident on the free plane surface z = 0 of a semi-infinite elastic material. The elastic and thermal constants of the material are as follows: 2 and p Lame constants, p density, k thermal diffusivity, K thermal conductivity, C, specific heat capacity, and a coefficient of thermal expansion. The laser pulse arriving at point 0 has an energy Q and varies with time as a d(t) function (where 6 is a Dirac impulse). General formulation
Due to the axial symmetry of the problem, the displacement u( M, t) generated at any point may be expressed@ terms of a scalar potential 4 and a vector potential $ whose only non-zero component extends in a plane perpendicular to MOz and is written $‘”
(1)
The elastic propagation potentials are l6
equations
as a function
* Present address: National Research Council Canada, Industrial Materials Research Institute, 75 Bd de Mortagne, Boucherville, Quebec, Canada
0041-624X/88/050245-1 1 $03.00 @ 1988 Butterworth & Co (Publishers)
Ltd
of these
of the
(3)
Ultrasonics
1988
Vol 26 September
245
Generating
acoustic 06
waves by laser: J.D.
Aussel et al.
Since the laser impact occurs at a point, the density of power absorbed per unit of volume of the material is
(t)
w, = AQ%.6(z)6(1)
(8)
where A is the coefficient of light energy absorption of the material and where s(t) is the Dirac time function. The stress/deformation relationship for an elastic material is
L k,
Oij
c1. P
=
(9)
CijklEkl
where Cijkl is the tensor of the elastic constants of the material and ekl is the tensor of elastic deformation. The elastic deformation tensor may be expressed as a function of temperature
K, Cv, (Y
Ek,
=
TV
T6,,
(10)
where 6,, is the Kronecker symbol. For an isotropic material, with Lam& constants 1 and p, the elastic constants of the material may be expressed by Z
cijkl
=
A6ijdkl
+
PtSiksjl
+
6i16jk)
(11)
The elastic stress due to thermal expansion from Equations (9)-( 11) as follows Figure 1 Geometry of the problem. The laser beam of energy 0 and time dependence 6 (t) is incident on the surface at 0. Due to axial symmetry, cylindrical coordinates, R, 0, I are used
a=* I a* a=* dZ2+;*z+dz2-;I-pZ.()t2=0
*
1
a=* (4)
S
where C, and C, are, respectively, the ultrasound velocities in the material of dilatational (i.e. longitudinal) waves and of equivoluminal (i.e. shear) waves. The initial conditions are (5) The stresses generated under thermoelastic conditions and under ablation conditions are calculated below and are used as boundary conditions for solving Equations (3) and (4). The point of impact 0 is initially assumed to be buried at a depth d beneath the free surface z = -d. Then solutions corresponding to an impact on the free surface are obtained by taking the limit as d -+ 0. Thermoelastic
effect: thermal expansion
stress
With an incident laser pulse of low power the increase in temperature due to the laser sufficiently small to avoid any change of state in material, and elastic stresses are obtained by bulk expansion. The temperature T( M, t) at any point by the thermal conductivity equation5
density, pulse is the solid thermal is given
where wa(M, t) is the density of power absorbed in the material. Assuming that only a small portion of the absorbed energy is converted into an elastic wave, the propagation equation can be solved by using the temperature distribution obtained by solving the conductivity equation independently from the elastic problem’
iaT
V2T-k.Z+K~“=0
246
i
Ultrasonics
(7)
1988
Vol 26 September
is deduced
oij = Cijk,aT6,,
(12)
For the moment of a force dipole equivalent to a point source of volume expansion, Aki and Richards” give Fij = SvCijklEkldV whence, Equation Fii = (31+
(13)
by making ( 11)
the
deformation
explicit
using
TdV
2&
using conductivity
Equation
(14) (7) Rose”
Fii = (32 + 2ll)a,kQH(t)
has obtained (15)
Under thermoelastic conditions, the amount of the force dipole resulting from a laser pulse QS( t) has a step pulse shape H(t). The boundary conditions under thermoelastic effect for a source buried at depth d are on the surface z = -d 0 zz = CJTz= or* = 0
(16)
on the plane z = 0 c zz = grr =(31+
2&Q&S(r)H(t)
(17)
The ablation effect: vaporization stress With a high power density incident laser pulse, the increase in absorbed power due to the laser pulse gives rise to melting and then to vaporization of a small quantity of material at the surface. Readylg has studied the effects of the absorption of laser radiation having a power density WH(t) on an opaque surface, where H(t) is a Heaviside unit step. The laser power density per unit area absorbed by the material is w,(t) = AWH(t) The material
beings to vaporize
(18) after a time t, given by (19)
Generating
where C, is the specific heat capacity of the material, TV is the vaporization temperature, and q is the initial temperature of the material. In order to approach pulse excitation, let us consider a laser pulse having the following power density w(t)=
W[H(t)-H(t-r,)]
where r, is the duration Equation ( 19) the material power densities such that
(20) of the laser will vaporize
1
1’2
pulse. From for absorbed
CT == =
(Awl2
pc + pLC,(
where L is the latent heat of vaporization of the material. In practice, for a laser pulse of finite duration, it is difficult to calculate the amplitude of the normal vaporization stress. Under such conditions, the thermal constants of the material are no longer constant as a function of temperature, and in particular the absorption coefficient A. Further, for an impact of finite size, the vaporization of the gas phase also causes a portion of the material to be ejected while still in the liquid phase, thereby increasing the amplitude of the normal stress due to vaporization. It is also difficult to calculate the variation in stress as a function of time under pulse conditions. At low incident power density, the material drops rapidly to below its vaporization temperature after the end of each laser pulse and as a result the normal stress is pulsed. However, at high power density, vaporization continues for a long time after the end of each laser pulse. The variation in the normal stress as a function of time lies between two limiting cases: the same shape of each laser pulse w(t) and a pulse shape of the Heaviside unit step type. In order to model vaporization conditions, the force equivalent to the vaporization will be assumed to have amplitude f,, to act normally to the surface, and to vary as a pulse 6(t) or as a step H(t). The boundary conditions for an ablation source (normal force) buried at a depth d are then on the surface z = - d c zz = or, = 6,, = 0
(23)
G
(26)
Cl.
(21)
(22)
T, - IJ
waves by laser: J. 0. Aussel et al.
The displacement is a combination of a dilatational, or longitudinal, wave L, an equivoluminal, or shear, wave S and, for points situated in a direction fI relative to the normal Oz at an angle greater than a critical angle 8,, a head wave LS. The head wave comes from the longitudinal wave L at grazing incidence, travelling on the surface which generates a shear wave all long its path in the material in the direction of the critical angle. The critical angle is given by sin8,=-
Under constant vaporization conditions, the vaporization stress is normal to the surface and, for a step power density WH(t) as given in Reference 19 is
acoustic
where C, and C, are S wave and L wave acoustic velocities. Figure 2 shows the relative positions of the wave fronts in the material at a given instant. The arrival times of the three types of waves at a point M situated at a distance R from the point of impact 0 and in a direction 19relative to Oz (Figure 2) are given by
t~s=R[-&sinB+($-&)1’2cost?]
(27)
Analytical calculation of the displacement u(M, t) by inverting the integral solutions have been performed’8,2’,22 only for points situated on the surface (z = 0) and on the axis of symmetry (0 = 0). Displacement at other points can be calculated by numerical integration23s24. Displacement
along the axis of symmetry
Oz
Under thermoelastic conditions, displacement along the axis of symmetry Oz is given for a laser pulse of power
Q(t)W8 (28) where fL and fs are given in the Appendix. Under ablation conditions, displacement along the axis of symmetry for a normal force fad(t) is given by2r
u,(R, 8,t) =
&(%+t$
where gL and gs are given in the Appendix. In practice, the elastic half-space is limited by a plane surface z = R. The displacement measured at the epicentre, i.e. on the axis of symmetry and on the plane z = R, has
on the plane z = 0 (24)
Solving
the propagation
equations
Rose’* has solved these equations for the thermoelastic case and Miklowitz 2o has solved these equations for a normal force f, which case is identical to that which occurs under ablation conditions. The solution is obtained by using Laplace transforms (transient conditions) and Hankel transforms (axial symmetry), and by inverting the resulting transform solutions by the Cagniard-DeHoop method. The displacement u( M, t) is then obtained in the form of double integrals made up as follows
z Figure
u(M,
t) = UL + us + uLs
R
0
(25)
2
Positions of the wavefronts
at a given instant. L,
dilatational wave; S, equivoluminal wave; LS, head wave
Ultrasonics
1988
Vol 26 September
247
Generating acoustic waves by laser: J. D. Aussel et al. a different account Under
equation
when
thermoelastic
conditions’
31+2~ u,‘(R, 8, t) = AQ----a--where fLE and Under
reflections
crk
are
taken
into
uz (nm) 0.1
8
4
fs” are given in the Appendix.
ablation
conditions
uZE(R,O,t)=
(31)
where gLE and gsE are given in the Appendix. The displacements corresponding to pulse conditions 6(t) are thus known. For a laser pulse having finite shape h(t), the displacement is obtained by a convolution of Equations (28)-( 3 1) with h(t). Figure 3 applies to thermoelastic conditions and shows the displacement along the axis of symmetry of a point buried at a depth R = 25 mm; the material is aluminium with an absorption coefficient A = 10%; the laser impulse has a Gaussian shape (duration 20 ns) and its energy Q is 10 mJ. The displacement at the epicentre is about twice the displacement at the buried point in the halfspace. The displacement is negative and corresponds to material being pulled towards the impact surface. Figure 4 is applicable to ablation conditions and shows the displacement along the axis of symmetry of a point buried at a depth R = 25 mm beneath the surface and at the epicentre of a plate of thickness R = 25 mm, for a normal vaporization force of 1 N and a Gaussian pulse shape of duration 20 ns. The Gaussian force corresponds to pulse conditions, i.e. to low rates of vaporization. Figure 4 also shows the displacement generated at high rates of vaporization by a force having a stepped pulse shape H(t).
tL = - R CL
tS=l
Y$
CS
, u,E(nm) 0.1 b
0.05 -
C
0.5-
Surface displacement The largest contribution to surface displacement comes from the surface wave, since the amplitudes of the other waves are much smaller. Under thermoelastic conditions, the surface wave is given by” 32+ 2/1 ak
uZ(8,r, t) = - AQP*-----2+2~
1
Im’*‘*I)
a&
R
R
CL
cs
(32)
K 8nC,‘r2’at
Figure
4
Displacement on the axis of symmetry for aluminium = 1 N, R = 25 mm. (a) Gaussian force, displacement, in the half-spke. (b) Gaussian force, displacement at the epicentre. (c) Step force, displacement at the epicentre under ablation conditions: F
3
l,
0(nm)
O.Ol-
where fk is given in the Appendix. Where v is the Poisson ratio for the material and C, is the velocity of surface waves in the material, a good approximation is given2’ by
o-02-
C = 0.862 + 1.14~ R
0.03I tL = R
by2
t
S
= _ R
CL
Time
CS
Figure 3 Displacement on the axis of symmetry for aluminium under thermoelastic conditions: Q = IO mJ, R = 25 mm. (a) Displacement within a half-space. (b) Displacement at the epicentre of a plate
248
Ultrasonics
1988
Vol 26 September
l+v
‘Cs
(33)
Figure 5 shows the displacement at R = 25 mm from the point of impact on an aluminium half-space, and due to the surface wave for a laser pulse of Gaussian shape, of duration 10 ns, and of incident energy Q = 10 mJ. Under ablation conditions, the surface displacement is given26 by
Generating uz (nm)
acoustic
waves by laser: J. 0. Aussel et al.
uz (nml 3.0-
0.30
a 2.00.20
1.0-
0.10
0.0
0.00
\
-l.O-
-2.o-0.10
-3.0.1 8.0
8.2
8.4
8.6
8.8 9.0 Time (us)
8.2
8.4
8.6
8.8 Time (LB;?
-0.20
uz (nmj 1.0 b -a.30
I
8.0
8.2
I 8.4
I
I
8.6
8.8
1
1
0.0
9.0
Time (Pr) Figure 5 Theoretical displacement. Thermoelastic conditions, surface wave, aluminium. Gaussian laser pulse of 10 ns duration, Q=lOmJandR=25mm
-1.0
-2.0
where gR is given in the Appendix. Figure 6 shows the surface wave for a point normal force of amplitude 1N of Gaussian pulse shape of duration 10 ns, and at R = 25 mm from the point of application of the force on an aluminium half-space, and also for a normal force whose pulse shape is a step H(t).
Experimental To illustrate the accuracy of the above theoretical predictions, a series of experiments are described using interferometric detection. The use of pulsed laser generation with wideband interferometric detection has been described set-up for by several authors 12,14,27. The experimental this work is shown in Figure 7. The generating laser is a Nd:Yag laser operating at a wavelength of 1.06 pm and having a pulse duration of 10 or 20 ns with a maximum energy of 330 mJ. The generating laser pulse is focused on the sample by a set of lenses, and the energy is adjusted by calibrated optical attenuators. The acquisition system is synchronized by a photodetector. A Michelson type laser displacement interferometer is used for reception ” , having a bandwidth of 10 kHz50 MHz. The samples used are made of steel or aluminium. They are plates of different thicknesses, and half cylinders to study the displacements generated in different directions. The surfaces of the samples are polished in order to reflect the reception laser beam as well as possible. Further work is in progress using a Mach-Zehnder
-3.0
8.0
Figure 6 Theoretical displacements. Ablation conditions, surface wave, aluminium. (a) Gaussian normal force of 10 ns duration, amplitude 1 N, R = 25 mm. (b) Stepped normal force, amplitude 1 N.R=25mm
type probe with sensitivity better than 10m4 A HZ-‘/~. This allows the measurement of pulsed displacements to be undertaken on industrial samples with rough surfaces29T30. Power
limit of thermoelastic
and ablation
regimes
Figure 8 shows the displacement measured at the epicentre of a 10 mm thick steel sample. The energy of the laser pulse is constant and the incident power density Won the material is modified by focusing the beam. The duration of the laser pulse is 10 ns. The noise observed at the beginning of the signal is due to triggering the generating laser. In comparison with the theoretical curves in Figures 3 and 4, it can be seen that for power densities in the range 9-141 MW cmm2 the shape of the displacement waveforms seem to reproduce most of the pattern given by the theoretical displacement of the thermoelastic regime. Nevertheless, a peak in L appears
Ultrasonics
1988
Vol 26 September
249
Generating
acoustic
waves by laser: J. D. Aussel et al.
LASER GENERATION
Photodetector
Lens
t
T
Sample
In Figure 9a, at low power densities, the generating source is thermoelastic and the amplitude of the 3 MHZ component increases linearly with the power density, or more precisely with the energy, as predicted by Equation (15). The amplitude at 50” is higher than at O”, as predicted by the directivity patterns of the thermoelastic source. In Figure 9b, for higher power densities, the 3 MHz longitudinal variations at 0 and 50” are no longer linear and have an intersection point at 15 MW cm-‘. The ablation regime begins and is of the same order of magnitude as the thermoelastic regime. In Figure 9c, at higher power densities, the ablation regime is much greater than the thermoelastic regime and the 3 MHz longitudinal amplitude is higher at 0 than at 50”, as predicted by the directivity patterns of the ablation source. The amplitudes of the longitudinal waves do not vary as expected from Equation (22), showing that the stress generating phenomena is more complex than pure vaporization. The power density limit of the two regimes, which depends on the material, can be estimated to be lo-20 MW cm-*. The ablation level can be extracted from Equation (21) which gives a theoretical value of A W = 15 M W cm - ’ for aluminium.
Displacement
Figure 7
Experimental set-up
from N 65 MW cm -L, which is also visible in the first reflection 2L. This peak corresponds to the appearance of vaporization conditions’. The transverse wavefront is not as steep as predicted because the laser impact is not a point. In Figure 8a where the diameter of the impact is 19 mm, the step corresponding to this wavefront practically disappears. It is difficult to observe thermoelastic conditions with a point impact. The generated displacement is proportional to the incident energy Q, and a quasi-point impact would give rise to a power density per unit area which is greater than the vaporization limit. Thus, in order to observe a signal of sufficient amplitude without vaporizing the material, it is necessary to use an impact of non-negligible size relative to ultrasound wavelengths. The signal-to-noise ratio at the beginning of the waveforms is low due to the Q-switching of the generating laser, so that the power limit of the two regimes is difficult to define precisely by observing the L ablation peak outbreak. In order to measure precisely this limit in aluminium, we used a contact piezoelectric transducer as receiver instead of the interferometer, and measured the 3 MHz longitudinal amplitudes generated at 0 and 50” uersus the incident laser power density. These amplitudes are given in Figure 9. The study of laser-generated directivity patterns6*7 has shown that the maximum of the longitudinal mode directivity occurs around 50-60” in the thermoelastic regime, and at 0” in the ablation regime.
250
Ultrasonics
1988
Vol 26 September
of the epicentre
Figure
10 shows
Radial
displacement
the displacement at the epicentre of a 5 mm thick steel sample generated by a laser impact of 20 ns duration, 1 mm diameter, energy 5 mJ, and of incident power density 32 MW cm-‘. The displacement obtained in this way may be considered as being the combination of a displacement due to the thermoelastic effect and of a displacement due to vaporization’. A theoretical curve obtained by combining these two effects can thus be obtained by considering the normal force due to vaporization to be a Gaussian pulse since the vaporization rate is low. At larger incident power densities, the ablation regime becomes dominant. Unlike thermoelastic conditions, the pulse shape of the displacement depends on the incident power density. In comparison with the theoretical Figure 4, Figure 11 shows that at low vaporization rates the normal force is of the pulse type while by increasing the power density it approaches the step type. Vaporization continues after the the end of the laser pulse. By comparing the heights of the longitudinal peaks for several incident power densities with the heights of the theoretical peaks for the displacement generated by forces having a Gaussian pulse shape, the curve of Figure 12 is obtained. This shows the amplitude of the normal force due to vaporization on aluminium and on steel. The normal force has substantially the same amplitude in both materials, giving about half the displacement in steel since its Lame constant p is about twice as large (Equation (29)). at any point
The radial displacement,
i.e. displacement directed towards the impact point for various angles of observation is obtained by interferometric means on semi-cylindrical samples. Radial displacement under thermoelastic conditions. Figure 13a shows the radial displacement generated in
Generating acoustic waves by laser: J. D. Aussel et al.
uz (nm)
a) W = 9 MW/cm*
(nm)
5
b) W =
16 MW/cm*
d) W =
65 MW/cm*
l2 0 -2 -4 -6
uz (nm)
1 /ls-
1P
uz (nm) cl W = 41 MW/cm*
+2 0 -2
1 PS u_ (nm)
\
-5-1
uz (nm) I
-5 -10
1 PS g) W =
h) W = 460 MW/cm*
141 MW/cm*
ow 1 \
L
I
2L
-15
1 PS uz (rim) 22
i) W =
‘zli
1050 MW/cm*
j)W
=
18k
L
1510-
Figure 8
2L
Displacement measured at the epicentre of a 10 mm thick steel sample. Laser pulses of constant energy (330 mJ) and variable
diameter
Ultrasonics
1988
Vol 26 September
251
Generating
acoustic
waves by laser: J. D. Aussel et al. +
t
-0.410
5
I 0 b
1 !Js Thermoelsstic regime
T
1 1 I J
I
-
Time
Figure 10 Intermediate conditions. Epicentre displacement: 5 mm thick steel sample, 1 mm impact diameter, 0 = 5 mJ, power density W, = 32 MW cm-‘, duration 2 TV= 20 ns. (a) Experimental curve. (b) Theoretical curve QT = 2.5 mJ, f, = 1 N Gaussian
Ablation regime
/
0.8 0.6
/
/
,‘b
,i” /I’ d
+-’
Time
___q_-------+
_-+‘--
#v’ 300
Figure 9 Longitudinal waves amplitude at 3 MHz (arbitrary units) versus incident power density in aluminium. (a) Thermoelastic regime, (b) intermediary regime, (c) abalation regime. 0, 6,= 0”; +, 0=50”
aluminium by a laser impact of 3.5 mm diameter, 20 ns duration, and of 16 mJ energy. The types of waves obtained match those predicted: the critical angle in aluminium is about 30”. For directions between 0” and 22.5”, i.e. directions which are less than the critical angle, shear wave fronts and longitudinal wavefronts are observed. For directions between 45” and 77’S”, the appearance of a head wavefront LS is observed which moves closer to the longitudinal wavefront on approaching the surface. The longitudinal peak increases going away from the normal and becomes wider. This widening is due to high frequency diffraction, since the laser impact is not a point when compared with the wavelengths under consideration. This agrees in the thermoelastic case with previously published results3’. Radial
displacement
under
ablation
Figure 13b and c show the radial displacement
252
Ultrasonics
1988 Vol 26 September
conditions.
generated
2
1 i
II
0
I
r
I
I
I
I
1
*
I
Time Figure11 Ablation conditions. Influenceof incident energy power density on the displacement measured at the epicentre: 25 mm thick aluminium, laser pulse duration 20 ns, 1 mm impact diameter
in aluminium at power densities lying in the range lOC-300 MW cme2. Here again, the predicted types of wave are observed: going away from the normal, the amplitude of the longitudinal peak decreases and the waveform approaches the waveform under thermoelastic conditions. It thus appears that ablation conditions are
Generating
acoustic
waves by laser: J. 0. Aussel
et al.
dominant close to the normal to the surface, whereas thermoelastic conditions are more important for points which are at a distance from the normal to the surface.
F, (N) 25
Surfaces
waves
Figure
14 shows the surface displacement of a 20 mm thick steel sample due to a laser impact having a diameter of 2 mm at incident power densities of 30, 200 and 1000 MW cm-‘. Under thermoelastic conditions at 30 MW cm -‘, the pulse shape predicted by Figure 5 is indeed obtained. Under ablation conditions at 200 MW cme2, the displacement waveform corresponds to a normal force of the pulse type, and at 1000 MW cmw2, to a normal force of step-time dependence. The amplitudes and the durations of the surface waves are greater than those predicted because the impact geometry is not a point and the theoretical displacement ought to be integrated as a function of the space distribution of the laser impact.
Figure 12 Maximum normal force under ablation conditions as a function of incident power density: laser pulse of 20 ns duration, 1 mm impact diameter. +, Steel; 0, aluminium
b
a
8 = o”
(nm)
(nm)
e = o”
(nm),
e = o”
(nm)
0 = 22.5’
(nm)
0 = 22.5’
0.5 i
- 0.5 V’
1w 0 = 22.5’
(nm)
3-
1.0 - L
0.5
L 2-
0
- 0.5
0.5 -
I
b
I
1 ccs
e =
(nm)
45O
0
1 PS
(nm)
45O
e =
77.5O
hm)
.
e = 45O
)
1 1.0
-“.5-
1 /ls
e =
-
L
1w
1P (nm) 1.0
J
3 L
Ls
s
e =
(m-n)
77.5O
-t
1P Figure 13 Radial displacement in aluminium R= 25 mm. Laser pulse of 20 ns duration, diameter =a. (a) Thermoelastic regime: W = 8 MW cm-*, a = 3.5 mm. (b) Ablation regime: W = 100 MW cm-*, a = 1 mm. (c) Ablation regime: W = 300 MW cm-*, a = 1 mm
Ultrasonics
1988
Vol 26 September
253
Generating
acoustic
waves by laser: J. D. Aussel et al.
Conclusions The models for the bulk expansion source and for the normal force agree well with the experimental results of displacements generated by a laser under thermoelastic conditions and under ablation conditions. All four types of waves predicted by these models are observed: longitudinal waves, shear waves, head waves, and surface waves. The displacements on the surface and at the epicentre have the same time distributions as those predicted. The displacements at other points show that ablation conditions are dominant close to the normal to the point of impact and that thermoelastic conditions are dominant far from the normal to the point of impact. The waveforms under thermoelastic conditions remain the same for varying energy, whereas under ablation conditions the pulse shape varies as a function of the density of the
incident power. The amplitude and the pulse shape of the vaporizing force could be obtained by deconvolution of the experimental displacements at the epicentre with the theoretical displacement corresponding to a force in the form of a Dirac pulse.
Acknowledgements One of the authors (JDA) wishes to thank the directors of EDF, Etudes et Recherches for the opportunity to prepare his Doctoral Thesis within their facilities, based in part on the work reported here.
References 1
0.3
0.2
6
0.1
0
7
- 0.1
8
9 b
IO
11
12
13 14
15
16 17 18 19 20
21 1 PS Figure
22
14
Surface waves. Displacement normal to the surface: laser pulse of duration 10 ns, 2 mm impact diameter. Observation at 65 mm from point of impact (steel sample). (a) Thermoelastic conditions: W, = 20 MW CIT-~, 0 = 10 mJ. (b) Ablation conditions: W, = 200 MW cm-‘, 0 = 63 mJ. (c) Ablation conditions: W,, = 1000MWcm-2,Q=330mJ
254
Ultrasonics
1988
Vol 26 September
23 24 25
White, R.M. Elastic wave generation by electron bombardment of electromagnetic wave absorption J A$ Phys (1963) 34 2123-2124 White, R.M. Generation of elastic waves by transient surface heating .I Appl Phys (1963) 34( 12) 3559-3567 Ready, J.F. Effects due to absorption of laser radiation J Appl Phys (1965) 36(2) 462-468 Bushnell, J. C. and McCloskey, D.J. Thermoelastic stress production in solids J Appl Phys (1968) 39( 12) 5541-5546 Wet&, G.C. Photothermal generation of thermoelastic waves in composite media IEEE 7fans Ultrason Ferroelectrics Frequency Control (1986) 33(S) 450-461 Hutchins, D.A., Dewhurst, R.J. and Palmer, S.B. Mechanisms of laser-generated ultrasound by directivity pattern measurements Proceedings of Ultrasonics International 81 (1981) 20-25 Hutchins, D.A., Dewhurst, R.J. and Palmer, S.B. Directivity patterns of laser-generated ultrasound in aluminium J Acoust Sac Am (1981) 70(5) 1362-1369 Budenkov, G.A. and Roiko, M.S. Excitation of stress waves in solids by laser beam with finite cross section through thermoelastic effect Defektoskopiya (1984) no. 3 16-24 Dewhurst, R.J., Hut&ills, D.A., Palmer, S.B. and Scruby, C.B. Quantitative measurements of laser-generated acoustic waveforms J Appl Phys (1982) 53(6) 4064-4071 Scruby, C.B., Dewhurst, R.J., Hutchins, D.A. and Palmer, S.B. Quantitative studies of thermally-generated elastic waves in laser-irradiated metals J Appl Phys (1980) 51( 12) 6210-6216 Dewhurst, R.J., Hutchins, D.A., Palmer, S.B. and Scruby, C.B. The performance of thick piezoelectric transducers as wideband ultrasonic detectors Ultrasonics (1983) 21 79-84 Boudarenko, A.N., Drobot, Y.B. and Kruglov, S.V. Optical excitation and detection of nanosecond acoustic pulses in nondestructive testing Defekfoskopiya (1976) no. 6 85-88 Tam, A.C. and Coufal, H. Photoacoustic generation and detection of 10 ns acoustic pulses in solids Appl Phys L&t (1982)42( 1) 33-35 Hutchins, D.A. and Nadeau, F. Non-contact ultrasonic waveforms in metals using laser generation and interferometric detection Proceedings of the 1983 IEEE Ultrasonics Symposium (1983) 1175-1177 Rourkoff, E. and Palmer, C.H. Low-energy optical generation of acoustic pulses in metals and nonmetals Appl Phys Lett (1985) 46(2) 143-145 Ewing, W.N., Jardetzky, W.S. and Press, F. Elastic Wnves in Layered Media McGraw-Hill, New York (1957) 9 Aki, K. and Richards, P.G. Quantitative Seismology. Theory and Methods (Ed Freeman) San Francisco (1980) 1 Rose, L.R.F. Point-source representation for laser-generated ultrasound J Acoustic Sot Am (1984) 75(3) 723-732 Ready, J.F. EfSects of High-Plower ‘La&’ Radiation Academic Press, New York (1971) Ch. 3 67-125 Miklowitz, J. Elastic waves and waveguides Applied Mathematics and Mechanics North-Holland Publishing Co., Amsterdam (1978) 22 231-366 Sinclair, J.E. Epicentre solutions for point multipole sources in an elastic half-space J Phys D (1979) 12 1309-1315 Doyle, P.A. On epicentral waveforms for laser-generated ultrasound J Phys D (1986 j19 1613-1623 Gakenheimer. D.C. Numerical results for Lamb’s point load problem J Appl Mech (1970) 522-524 Willis, J.R. Self-similar problems in elastodynamics Phil Trans Roy Sot London (1973) 274 435-491 Achenbach, J.D. Wave propagation in elastic solids Applied
Generating Mathematics and Mechanics North-Holland Publishing Co., Amsterdam (1973) 16 262-325 Mooney, H.M. Some numerical solutions for Lamb’s problem Bull Seismol Sot Am (1974) 64(2) 473-491 Royer, D. and Die&saint, E. Optical detection of sub-angstrBm transient mechanical displacements Proceedings ofthe 1986 IEEE Ultrasonic Symposium (1986) 527-530 Drain, LE., Speake, J.H. and Moss, B.C. Displacement and vibration measurement by laser interferometry Proceedings ofthe First European Congress on Optics Applied to Metrology SPIE (1977) 136 52-57 Royer, D., Diiulesaint, E. and Martin, Y. Improved version of a polarized beam heterodyne interferometer Proceedings ofthe 1985 IEEE Ultrasonic Symposium (1985) 432-435 Lesne, J.L., L.e Brun, A., Royer, D. and Die&saint, E. High bandwidth laser heterodyne interferometer to measure transient mechanical displacements Proceedings of the International Symposium on the Technologies for Optoelectronics in press Hutchins, D.A. Mechanisms of pulsed photoacoustic generation Can J Phys (1986) 64 1247-1264
26 2-l
28
29
30
31
acoustic
waves by laser: J. D. Aussel
(A7) -4ss~(ss~+cL-z)(ssz+cs-*) gsE= [(2ss*+cs-~)~-4s~(ss*+cL-~)~‘*(ss~+cs-*)~’q~ (A81
with andS,‘=($--$1 Displacement
on the surface
Thermoelastic
regime (2C,_2
fR = cn-4cs-4+
- C,_Z)(
(A9)
1- v)
c,-*cs-z(c,-z-vc,-~)(2c,-*
-cs-+-cR-q
Appendix
(AlO)
Displacement
at the epicentre
Thermoelastic
regime
Ablation regime gR
fL =
- 2s,z(s,z (2q.2 + c,-*)2
+ c,-~)(s,2
-4s,7s,*
+ c,-2)“2
+ cL-2)1’2(sLz
=
glR
+
(All)
g2R
with
+ Cs-2)“2
(A121
(Al) fs=
et al.
S,2(S,2 + Cs-~)“~(2SSZ + cs-2) (2Ss2 + Cs-‘)’
- 4S,2(Sz + C,,-2)1/2(Ssz + Cs-z)l/z
(A13)
H (A21
- s,qs,2 fLE = [(2S,2 + fz-92
+ c,-z)(s,z - 4sLz(s,*
+ c,-z)“z(2s,z
+ c,-2)
+ c,-z)“z(s,2
+ c,-zpq2
and v being the Poisson ratio for the material Iz+ 2/l
QJ=
P (A31 SS2(SS2 + c,-2)“2(ss2 + Cs-2)(2SsZ + cs-2) fs” = [(2&Z + cs-92
- 4S,2(S,2 + cL-2)1’2(ss*
y=tc” R’
’
T=t-
CS
(A14)
R
anda,isarealrootofP+Qx+Rx2+x3=0 where
1
+ cs-*pq2
(A4)
(AW
with
(A16) (A17)
Ablation regime and A1 is such that x (W
2 1-,-2x3 ?
(
P+Qx+Rx2+x3
> =4+&t
A2
----++
1
x + a,
A3
x + a3
for v < 0.263 =4+&+
( flR+ i a,)( aR - i ai) 1
(X+tlR+iUi)(X+aR-iiai)
for v 2 0.263
Ultrasonics 1988 Vol 26 September
255