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Characterisation of laser-ultrasound signals from an optical absorption layer within a transparent fluid
ELSEVIER
Characterisation
Ultrasonics 34 (1996) 629-639
of laser-ultrasound signals from an optical absorption layer within a transparent fluid Q. Shan *, A. Kuhn, P.A. Payne, R.J. Dewhurst
The University of Manchester Institute of Science and Technology, DIAS, UMIST, PO Box 88, Manchester, M60 lQD, UK
Received 30 October 1995; Revised 18 January 1996
Abstract
Characteristics of laser-ultrasound signals are presented from photoacoustic interaction with a layered, optically absorbing medium surrounded by a transparent fluid. A thermoelastic model is presented describing the interaction, with signal predictions in the fluid arising from polymer transducer detection. By taking the optical absorption coefficient and finite layer thickness into account, the amplitude and shape of transient elastic waves are calculated for both forward and backward travelling directions. Additionally, wave interaction with the PVDF transducer has been characterised using a discrete-time algorithm for the transducer response. With just three constants to define transducer response characteristics, the response function may be used to predict voltage signals. Good agreement with experimental waveforms is demonstrated, so that the response function may form the basis of system modelling when miniature laser-ultrasound probes are used in various applications. Keywords:
Laser-ultrasound;
Transducers; Photoacoustics;
Optical absorption layers
1. Introduction With current progress in laser technology, fibre-optics and polymer transducers, miniature laser-ultrasound probes may now be constructed as a potential diagnostic tool [ 1,2]. In the field of medicine, it may be feasible to use such a tool as a sensing transducer, prior to some therapeutic use of a laser beam delivered down the same or similar fibre-optic system. For example, it is known that further information is required within blood vessels which are constricted by fatty or calcareous deposits. By using multimode optical fibres for forward-looking laser-ultrasound examination of targets, techniques may, in the future, be combined with therapy such as laser angioplasty to provide greater information at the zone of laser interaction. It is essential therefore to understand the nature of ultrasonic transducer signals which arise from laser absorption in tissue contained within an aqueous or saline environment. Laser-ultrasound signals arising from measurements of laser light absorption in liquids and solids have * Corresponding author. Fax: +44-161-200-4911; e-mail: [email protected]
already been reviewed [ 3,4]. In published literature, such signals are often referred to as either transient elastic waves or stress waves. They can be generated by the use of short duration laser pulses, typically a few nanoseconds long. In pioneering work with tissues, measurement of the optical properties within the tissue lacked a theoretical basis for overall time evolution of the waveform signal shape [ 5-71. More recently [8-lo], there are several accounts of piezoelectric detection of time-resolved stress detection (TRSD) which makes use of an effective optical attenuation coefficient in tissue to help describe the behaviour of laser-induced stress waves. These ultrasonic signals are recognised as a convolution of many factors, such as the optical absorption coefficient, stress relaxation, transmittance through interfaces and the detector response. For studies in layered material, the optical absorption coefficient, layer thickness, and transducer response are crucial to the understanding of time-resolved measurements of ultrasound signals, but there has been no comparison of predicted time evolution with experimental observations. We have therefore developed a one-dimensional model, using optical neutral density filters as phantoms for real biotissues since, for the
0041-624X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PZI SOO41-624X(96)00055-8
Q. Shari et al. 1 Ultrasonics
630
purposes of experimental verification, they possess welldefined optical and thermoelastic properties which facilitate detailed modelling. With short pulse irradiation of an optical absorption layer within a transparent liquid medium, we develop solutions for both the forward travelling and backward travelling waves, expressed in terms of time-resolved pressure changes. These are then combined with a model of piezoelectric transducer performance to provide predictions of time-resolved voltage signals. These predictions are compared with experiments in which the transducer consisted of a polyvinylidene fluoride (PVDF) element. The medium coupling the sample to the transducer was assumed to be non-absorbing to ultrasonic signals.
34 (1996)
629-639
infinity space [4,7,11]. We present below a one dimensional (1-D) model as an approximation to the real experimental conditions. We first examine the problem for a half space boundary condition by assuming the thickness of the sample h -+ 00. The wave equation for pressure can be shown to be of the form [ 61. a%
1
ax2 - 2
azp -/3aH
-=-at2
c,
(1)
at ’
with initial conditions: p(x,O_)=&p(x,O_)=O,
H(x,O_)=O
(2)
and boundary conditions: 2. Modelling of laser-generated acoustic pressures
Pb(O- 9 t) = P&&O+ 9 0,
The sample in which laser-ultrasound was generated consisted of one of a number of different high power, volume absorbing neutral density filters, placed in a transparent liquid. Fig. 1 shows the schematic arrangement in which a PVDF transducer could be positioned at A to monitor forward travelling ultrasonic signals, or at position B to monitor backward travelling signals. Distilled water was used as the coupling medium between the sample layer and transducer position. A model for the prediction of pressure transients which then arise from irradiation of the sample with a short laser pulse is presented below.
- ;
&Ps(O- 7 t) = - f
&.(O+, II
t).
2.1. Solution for halfspace boundary conditions
In Eqs. (l)-(3), p, c, /?, and C, are respectively the acoustic pressure, the sound speed, the isobaric volume expansion coefficient, and the isobaric heat capacity per unit mass. The x spatial coordinate is in the direction shown in Fig. 1, with x = 0 defined as the top of the sample surface. The time coordinate t is defined as t = 0 when the laser pulse first irradiates the sample. Notations with subscripts a and b denote parameters in the sample and water respectively, with p being the density of the medium. H is the heating function, defined as the energy absorbed per time per unit volume. It may be expressed as
Several authors have modelled the acoustic pressure pulses generated by laser pulses incident on a half-
H(x, r)=ae-ax$[u(t)-u(t-A)],
Air = =
II
Fluid B
PVDF Transducer I
Fig. 1. Position of the optical absorbing layer surrounded by a transparent fluid. A PVDF transducer monitors the forward travelling wave at position A, or the backward travelling wave at position B.
(4)
where CIis the optical absorption coefficient, E0 is the radiation energy density at the sample surface, d is the time period during which the energy is uniformly delivered, and u(t) is the unit step function whose value is one when its argument is positive and zero otherwise. Here we assume that the temporal profile of the laser pulse is rectangular, d is the pulse width and &-,/A is the light intensity. When d is small, Eq. (4) gives results that approximate the effect of a Dirac delta function. Later on it will be shown that this approximation is valid for examples shown in this paper. We may note that the photoacoustic response to long duration laser pulses with arbitrary temporal profile can be determined by convolving the delta function response with the specific laser pulse temporal protie. The initial and boundary conditions, Eqs. (2) and (3), assume that the system is at rest before the laser pulse is fired, when t = 0. It is also assumed that the pressure and pressure gradient are continuous across the interface,
Q. Shariet al. / Ultrasonics34 (1996) 629439
and that the pressure is bounded at - co I x I co, which is an implicit condition. A Laplace transformation [ 181 is employed to solve the partial differential equation. The transform pair for pressure p is defined as
631
(12) where
9 (P(X, t); f+ s> = Y(X,s); (5)
(13)
Applying the Laplace transformation on Eqs. (1) and (4) with initial and boundary conditions given by Eqs. (2) and (3), we obtain a nonhomogeneous ordinary differential equation:
By applying the inverse transformation to Eqs. (9) and (lo), the pressure pulses can be expressed in the time domain as:
6p-r(y(X,S);S+t)
=p(x,t).
$y(x,s)-$y(x,s)=
-~em’I(l-eed”).
(6)
P
If d +O, then eeds tends to 1 - ds, so that Eq. (6) can be written as
a2 sy(x,
s2
4 - 2y(x, 4 = -
@Eos
emax,
7
(7)
P
I
where we regard the Laplace parameter s as a constant and x as the independent variable. The solution to Eq. (7) can be derived to be: y = A(s) e-(s/c)x+ B(s) e(s/c)x +
ajlE,c2 eeax CP
(ca)2 ’
%(c~p.cb+P~bpb)e~c~(f+x~ t+t20, (15)
Pb=
(8) where A(s) and B(s) are arbitrary constants with respect to x and are determined by the boundary conditions. The pressure p has to be bounded for all x[ - cc to + co]. Therefore, the first and the third terms of the right-hand side of Eq. (8) can only exist within the absorbing medium of the sample, where x 2 0. Likewise, the second term can only exist in the transparent fluid where x < 0. Eq. (8) can now be written explicitly using parameters related to the corresponding medium. In a semi-infinite half space, with two media, a and b, such that c, and cb represent the sound speed c in the absorbing region and non-absorbing region respectively, we find that in the absorbing medium y, =
/qs)
e-
(s/c‘&
+
afiEoc~ e-Orx s CP
s2- (c,a)2 ’
y, = B(s) e(S/cb)X,x < 0.
i
elsewhere.
0,
In summary, Eqs. (14) and (15) represent the solution of Eq. (1) when the appropriate boundary conditions are taken into account. 2.2. Solution for boundary conditions representing a layer If the sample is a layer within a transparent fluid, as shown in Fig. 1, then this model may be extended to take into account the finite thickness of the absorbing layer, h. The ultrasonic transducers positioned at either A or B are immersed within a fluid and respond to the arrival of pressure transients described by pA and PB respectively.
x 2 0, 2.2.1. Acoustic pulse detected at position B
(9) and in the non-absorbing
Cl?
(14)
s s2 -
t-%0,
medium, we have
One more boundary condition is imposed at the rear optical absorbing interface, where x = h. It may be described as
(10)
From the boundary condition expressed by Eq. (3), we have
-;;Pb(h-,t)=
-$&.(~+,t). LI
(11)
Differences in optical, acoustic and thermal properties between the media at each side of boundary x = h result
Q. Shari et al. / Ultrasonics 34 (1996) 629-639
632
in different solutions compared with those from a half space solution presented in Eqs. (14) and (15). Firstly, if the absorption coefficient is low, the laser energy may not be totally absorbed in the layered medium. This discontinuity in optical absorption coefficient across the interface introduces a discontinuity in Eq. (15). It must be modified to give a pressure wave variation described by:
Pb
=
I
b2[e-c&z
+ h/c,) + ec.a(tz - h/c,)
19
Pa2 =
(t+z>o)*(t+E<; 9 >
0,
Likewise, a later contribution arises from a second reflection at the rear surface of the sample after a further time delay of 2h/c,. Therefore another pressure contribution can be expressed as:
(17)
elsewhere,
where ‘A ’ represents the logical ‘and’ operator. The initial pressure wave no longer has a continuous decay along the x-axis, but is cut off by the interface at x = h. Secondly, when the PVDF transducer is at position B of Fig. 1, the transducer can receive a contribution of pressure arising from pb, the backward travelling transient wave, together with multiple echoes arising from the forward travelling wave p., which suffers partial reflection and possible phase inversion at the boundary x = h. This reflection then propagates towards the transducer, being partially transmitted through the boundary at x = 0. The other component is reflected at this boundary and continues to undergo multiple echoes, with pa gradually fading away with time. Therefore, under layered boundary conditions, Eq. (14) should be modified. Inclusion of a contribution from the first reflection of p. results mathematically in a pressure term of the form:
4 Ce-c,Mh+
h/c.) + e’c,““l -h/d
0,
elsewhere,
(21) where 3h tz=t--+---, co
x
(22)
cb
and b2 is expressed as:
~2=~(,.,p~~bcb)(~“:“~~~~~~.
(23)
In general form, for the ith reflection, the pressure wave can be described by: bi
[e-c,c(ti+h/c,)
+ ef,dti
-h/c,)]
,
1, -c,u(ti
Pai =
-h/c,)
+
e-c,a(t,
+ h/c,)
P.1 =
0,
elsewhere, (24)
0,
where
elsewhere, (18)
where
ti=t-
(2i-1)h+x c,
(25) cb’
and Qi is expressed as: (19)
(20)
where i > 0 is an integer number. Finally, acoustic pressure pulses detected at the position
Q. Shun et al. / Ultrasonics 34 (1996) 629-639
B in Fig. 1 can be expressed as:
(27) i=l
where N > 1 is also an integer number and depends on how many reflections we are interested in. Pressure contributions pa and pai are defined by Eqs. (17) and (24) respectively. 2.2.2. Acoustic pulses detected at transducer position A At transducer position A, only p, and its multiple reflections reach the transducer. To model the first direct arrival of pa, we may first consider the pressure at the rear surface of the absorbing medium and substitute x = h into Eq. (14). The wave travels from the interface x = h towards the transducer position A. Eq. (18) remains valid in describing the first direct arrival to position A, but Eqs. (19) and (20) are modified to take into account the time delay and one transmission across a boundary, so that: +t--,
(28)
633
532 nm was incident on the absorbing layer, taken to be a neutral density glass filter within a non-absorbing water medium. With a laser pulse energy of 5 mJ in a beam diameter of 6.5 mm, solutions to Eqs. (27) and (32) were evaluated and are shown in Fig. 2. Physical parameters used in the calculations are quoted in the figure caption. The upper graph in Fig. 2 represents a solution to Eq. (32) and therefore represents the forward travelling pressure wave reaching a transducer at position A. Similarly, under the same conditions, the lower graph represents a solution to Eq. (27), representing the backward travelling pressure wave reaching a transducer at position B. As expected, a number of pressure discontinuities are displayed which correspond to the effect of the two boundaries of the layered medium. In general, their amplitude gradually decreases with time, but the forward travelling pressure wave (top trace) has the greatest pressure change some time after the initial arrival. It corresponds to the time of arrival at the transducer of elastic wave energy coming from the top of the absorbing layer. This arrives with an acoustic delay time due to the glass thickness, in contrast to any
cb
and
(a) (29)
c,=4r(p.c~;,c,)*
After the direct wave arrival, further arrivals arise from multiple reflections again. For contributions due to the ith reflection, Eq. (24) remains of the same form, but the time parameter is modified to:
CW- l)lh + 14.
ti=t-
G
cb'
(30)
-6 -
’
-6
0
and di is expressed as:
I
0.5
I
1
J
1
1.5
2
6m (b)
4-
a
Hence, the acoustic pressure waves detected at position A in Fig. 1 can be expressed as: PA =
$
Pai-
P d
2-
Pa2
Pa3
. -
-4-
i=l
Eqs. (27) and (32) represent the one-dimensional solutions for acoustic pressure waves generated by a laser within an absorbing layer and subsequently propagating in a non-absorbing fluid to positions B or A as shown in Fig. 1. 2.2.3. Example solutions Any example of predicted pressure waves depends on the properties of laser pulse excitation, as well as the properties of the layered structure. A case was chosen where an 8 ns duration laser pulse at a wavelength of
-6-6
t
1
0
I
0.5
I
1
I
1.5
1
2
Time 6.~4
Fig. 2. Theoretical predictions of ultrasonic pressure waves generated from a 5 mJ laser pulse having a beam diameter of 6.5 mm when incident on a neutral density glass filter contained within distilled water. Physical parameters used in the calculation for water and for the neutral density filter were: p. = 2230 kg m-s, pb = 998.2 kg me3, c,=564Oms-‘, c,=1482.34ms-r, C,=7OOJkgg’K-‘, fi=2.52x lo-‘K-l, a=1250m -I, h = 1.9 mm. Waveforms in (a) represents the forward travelling pressure wave and (b) represents backward travelling wave.
634
Q. Shari et al. / Ultrasonics 34 (1996) 629639
laser energy absorption at the rear surface of the sample, which also forms a pressure wave but without this time delay. Such solutions are only valid if the acoustic pressure wave travels an insignificant distance compared with the optical absorption length during the time of the laser pulse. The time taken for a wave to travel across the absorption length is ~/UC,. In the case described in Fig. 2, this corresponds to 142 ns, which is much longer than the laser pulse duration of 8 ns. Hence the laser pulse is a good approximation to the Dirac delta function, and no convolution due to its finite temporal width is necessary.
3. Model for the PVDF transducer To measure the acoustic pressure pulses shown in Fig. 2, a wideband transducer is required. In potential medical applications, the good acoustic impedance match of polymer films with either water, saline or biological tissue makes transducers based on polyvinylidene fluoride (PVDF) and its copolymers attractive. Their properties have recently been reviewed [ 12,131, and these were the type of transducer element used in our earlier investigations [ 1,2]. When constructed to form a transducer, there are a choice of models used in the literature to predict their performance. Most treat the transducer as an equivalent circuit. The KLM model is one such example [ 141, requiring precise knowledge of the piezoelectric, dielectric and mechanical properties of the transducer. These properties are often difficult to determine so that Lockwood and Foster [lS] have proposed a ‘black box’ approach. They treated the transducer as a black box, disregarding any knowledge of the transducer properties. Another approach is to have a ‘grey box’ model, in which use is made of experimental observations that a polymer transducer has essentially a differential response [ 161, and where the backing material has a classical damping effect. Its voltage output response v(t) to a pressure input p(t) may then be described by a second order ordinary differential equation such that: d2u(r) do(t) dp(t) + C&(t) = Edt2 + 2i% dt dt ’
property and affects the undamped frequency (to be defined later) CO,.It also affects the capacitance of the transducer, which is one of its dielectric properties. In this simple mathematical modelling, the transducer’s transfer function can be simply characterised by only three constant parameters such as c, w,, and E. These three parameters control the transducer’s characteristics through Eq. (33) including contributions from the materials used, its design and its mode of construction. Due to difficulties involved in the independent determination of these parameters for a particular device, Eq. (33) cannot be solved directly for a fabricated transducer. However, the parameters may be determined by the help of experimental measurements derived from an arrangement shown in Fig. 3. Normally, the transducer’s time domain output signal was recorded in a discrete format. Therefore, instead of using a Laplace transformation, a z transformation was used to solve Eq. (33). In the discrete-time domain, Eq. (33) may be expressed by a difference equation such that [ 171. V2u(k) V v(k) + c+(k) --p+ 2bn z
VP(k) = Ez ’
(34)
where k is the current sampling instant, and z is the sampling period. If we define the z transformation pair [ 171 to any time discrete function f(k) such that:
f-(z)= W-(WI,
f(k) = z - ’ CWI
(35)
where z is the discrete transformation variable. By using the property of z transformation of the form: Z[V”f(k)]
= (1 - z-‘)“&‘(z),
(36)
Eq. (34) can be expressed as: V(z) = K()(l - z_‘)P(z) + [K,z_’ + K,z_2]1/(z),
(37)
(33)
where i, CO,, and E are constants of the system. The constant E (in units of V Pa-l SC’) is dependent on the piezoelectric and dielectric properties of the transducer. And c (dimensionless) and w, (in units of rad/s), are determined by the mechanical properties of the transducer film and its backing material. Although we can roughly divide the left- and right-hand side of Eq. (33) into mechanical and electrical behaviour of the transducer, their interrelationship is complicated. For example, the thickness of the transducer film is a mechanical
integrated transducer
\
ND glass filter Experimental tank
I
Fig. 3. Experimental arrangement used to investigate laser-ultrasound generated in an absorbing layer within water serving as a transparent fluid.
Q. Shari et al. / Ultrasonics 34 (1996) 629639
where: K,=
K,=
EZ 1+2~o,r
(38)
+ 0,222’
2(1+ b”4
(39)
1+2&!&z+o,zr~’
and K,=
-
1 1+2~0,z+0,z22’
In Eqs. (37)-(40), K0 is in units of V/Pa, and K1 and K2 are dimensionless. Eq. (37) can be graphically illustrated by a transfer function block diagram as shown in Fig. 4. Such a block diagram can be converted into a simple discrete-time domain algorithm. If we take the pressure pulse p(k) as the transducer input at the kth instant, the transducer output voltage u(k) at the kth instant can be calculated in two steps with the intermediate result denoted as q(k). Firstly, for the second term in Eq. (37), which represents damped oscillation, it follows that q(k) = p(k) + K,q(k - 1) f fbdk
(41)
- 2).
For the first term in Eq. (37), which represents differential operation, it follows that
u(k)=K,Cq(k)-q(k--1)1.
the (42)
Eqs. (41) and (42) form a simple algorithm with only three constants. These three constants were determined by comparing a predicted waveform with a measured voltage output. The procedure used to identify the constants exploited a standard system identification method [ 191. We used Matlab together with its toolbox for system identification. Once the constants were determined for a particular transducer, their values were retained in Eqs. (41) and (42) to predict transducer output voltages for various pressure transient inputs. The experimental arrangement for this characterisation process is described in the following section.
4. The transducer characterisation and discussion Experimental investigations for transducer characterisation were performed using the arrangement shown
vk
Fig. 4. Block diagram used to model the PVDF transducer transfer function. In a z transformation analysis, the transducer can be fully characterised by three constants, K,, K, and K,.
635
in Fig. 3. A frequency doubled, Q-switched Nd : YAG laser delivered pulses of 8 ns duration. They were attenuated by the use of optical filters before being focused with a convex lens into a length of multimode silica fibre, acting as a flexible delivery system. Pulse outputs, from the end of the optical fibre, were measured to be typically in the range of 5-10 mJ, using a Photon Control Calorimeter Model 25V-VIS. They were directed towards a glass neutral density filter placed within a tank of distilled water, the water acting as a constraining and optically transparent medium. Laser pulse energy was partially absorbed by the glass filter, with its beam diameter being approximately 6.5 mm. Under the thermoelastic laserultrasound generation mechanism [ 21, ultrasonic transients were generated within the filter and transmitted through both top or bottom surfaces of the filter into the liquid. They were designated either the backward travelling wave, or the forward travelling wave. The backward travelling wave was detected at position B in Fig. 1 using a concentric transducer at the fibre output. Its design has been described in earlier publications [ 1,2]. Alternatively, another transducer was placed below the target (not shown in Fig. 3) to record the forward travelling wave at position A in Fig. 1. Transducers were based on a 27 nm PVDF polymer film and transducer signal leads were connected to a 50 R input channel of a Tektronix TDS 520 digital storage oscilloscope, in order to maintain a broad-band frequency response of the measurement system. A beam-splitter sent a fraction of the laser beam to a p-i-n photodiode which provided a trigger signal to the oscilloscope. Some properties of glass filters used in the experiments are summarised in Table 1. By using Eq. (32) with filter parameters shown in Table 1, a set of forward travelling pressure wave predictions are presented in Fig. 5. The theory presented in the previous section was able to predict the form of the transient pressure wave arising from filter irradiation. For a transducer in position A of Fig. 1, monitoring forward travelling transients, Fig. 5 shows a set of predictions, for incident laser energies of 6.7 mJ. The time axis is presented from t = 0 with the acoustic delay through the liquid subtracted from the time after laser pulse excitation. The three waves represent transient pressure waveforms which may arise from high (upper trace), middling (middle trace), and low (lower trace)
Table 1 Physical characteristics experiments
of absorbing
glass filters used in the
Filter No
h (mm)
cL(m-l)
p. (kg m-‘)
B (K-l)
1 2 3
2.01 2.08 2.96
5116 1204 252
2470 2430 2420
2.13 x lo-’ 2.16 x 1O-5 2.16 x 1O-5
Q. Shan et al. / Ultrasonics 34 (1996) 629-639
636
________.~. Predictions 20 1
a=5116me’
a=1204
Measurements
-
a=5116 m-’
10 -
2r
m-l
I
a=1204 m-'
a=252 m-' 1
a=252 m-’
0.5 b
5 =
h
0
g-o.5 r" B
-1 -1.5
-4' 0
-2 1
0.5 Time
1.5
2
(ps)
-2.5 0
0.5
1 Time
Fig. 5. Forward travelling pressure waves predicted for three different values of optical absorption coefficients, ~1, associated with the absorbing layer. In each case, the laser input energy remained constant at 6.7 mJ.
optical absorption coefficients. As may be expected, the amplitude of pressure transients decreases in consecutive traces (noting different vertical scales in three waveforms), with the largest amplitudes associated with the sample possessing the highest optical absorption coefficient. In the other cases, step features in the waveform arise from both surfaces of the absorbing media, implying that not all the optical energy was absorbed within the glass filters. Corresponding polymer transducer responses are shown in Fig. 6. These waveforms are quite different in shape, representing the voltage output from transient waveforms shown in Fig. 5. The upper waveforms in Figs. 5 and 6 were used to characterise the forward travelling transducer. The former, representing pressure transients, was taken to be the known transducer system input, and the latter was taken to be the measured system output. The constants of the transducer, K,,, K, and K, were evaluated by using a Matlab system identification tool box running on a Sun workstation. System identification was performed using an output error model structure [ 191. Table 2 summarises the
1.5
2
(ps)
Fig 6. The output of the PVDF transducer in response to the pressure waves pA represented by the waveforms shown in Fig. 5. Measured waveforms are indicated by the solid lines, whilst predicted waveforms based on the three parameter transducer model are indicated by dashed lines.
values of derived transducer constants. Once the constants KO, K, and K, had been evaluated, a predicted curve of transducer response was found to follow the experimental waveforms closely. As shown in Fig. 6, the two waveforms overlay each other satisfactorily. Retaining the same values of Ko, K1 and K2 for the transducer, predictions of transient waveforms derived Table 2 Characteristic constants for transducers backward travelling wave measurements
K,(10-7V/Pa) Ki (dimensionless) K, (dimensionless) Loss function Akaike’s final prediction error criterion
used in either forward or
Forward travelling wave transducer
Backward travelling wave transducer
0.800 + 1.710 + - 0.750 + 8.415 x 8.483 x
0.320 k 1.600 f - 0.700 + 2.387 x 2.425 x
0.014 0.006 0.006 10-a lo-*
0.026 0.036 0.033 1O-g 1O-g
Q. Shan et al. J Ultrasonics 34 (1996) 629-639
from glass filters with different optical absorption coefficients were then compared with experimental measurements. As shown in Fig. 6, there is good agreement between theory and experiment in all cases, supporting the transducer modelling philosophy adopted. Similar studies were also performed for the backward travelling wave derived from the same experimental scheme. With a transducer at position B in Fig. 1, calculated transient waves for incident laser energies of 5.0 mJ are shown in Fig. 7. Corresponding voltage waveforms from the transducer system are shown in Fig. 8. Predicted and measured waveforms again overlap and are in good agreement. Particular constants derived for the transducer from the upper waveform are summarised in Table 2, after adopting a system identification procedure already described above for this different transducer. In the case of low optical absorption coefficient, experimental waveforms possessed significant noise due to the low level of generated acoustic signal. We note that the leading edge of the initial pressure transient (Fig. 7) was sharp, but its amplitude is not as great as the one arising from the first echo within the absorbing layer. The phenomenon is more obvious in Fig. 8. The
“I
631
-
Measurem en1 a=51 16 m-’
F
4
a=1204 m-l
I
a=252
mm1
R
60 t
a B
40
1
a=51 16
8
0.1
S g
0
m-’ -O.l” -0.2
I”
III
”
I
I I
II
1
Y
F
-0.3 '
P
0
z b:
1 Time
0
t
1.5
2
(us)
Fig. 8. Output of the PVDF transducer in response to backward travelling pressure waveforms predicted in Fig. 7. Measured waveforms are indicated by the solid lines, whilst predicted waveforms based on the model are indicated by dashed lines.
1
-4
I 0.5
’ 0
I
I
1
0.5 Time
1.5
2
(ps)
Fig. 7. Backward travelling pressure waves predicted for three different values of optical absorption coefficient a associated with the absorbing layer. In each case the input laser energy was kept constant at 5 mJ.
first positive pulse is always lower in amplitude than the second positive pulse. This feature is a characteristic of backward travelling transients, regardless of the optical absorption coefficient. There are sometimes small pulses with opposite polarity to major features in both Figs. 6 and 8. These secondary features arose from the rear sample surface. For signals arising from the rear sample surfaces, where x = h, it is convenient to define a term, optical thickness D = ah. If the sample was optically thick, when D >>1, all laser pulse energy was essentially absorbed within the sample. In this case, there was no acoustic feature arising from the sample rear surface. Such a situation existed for the upper traces in Figs. 5, 6, 7 and 8. When D <<1, corresponding to an optical thin case, rear sample surface features are pronounced as in the lower traces of Figs. 5, 6, 7 and 8. In the middle traces in these figures, only small pulse feature arose from the rear sample surface, when D = ah = 2.5. We note in passing that, although it is difficult to
Q. Shari et al. / Ultrasonics 34 (1996) 629-639
638
control the properties of PVDF material throughout the manufacturing process of a transducer system, these results have shown that a transducer may be characterised by three constants used in a finite difference equation. These properties may change with time, and repeat experiments using the experimental procedure described above would be capable of measuring this change. With ultrasound generation being independent of ultrasound detection, it would be possible to check the performance of any transducer throughout its lifetime. Once characterised, the transducer’s voltage output sequence, v’(k), can be used to measure qualitatively the real pressure, p’(k), through the inverse use of Eqs. (41) and (42), such that: q’(k) = F
+ q’(k - l), 0
and p’(k) = q’(k) - k,q’(k - 1) - K,q’(k - 2),
(44)
where q’(k) is an intermediate variable. We may take the measured waveform shown on the top trace of Fig. 6 as an example. By using Eqs. (43) and (44), a measured pressure waveform may be deduced, as shown in Fig. 9 as a solid line. It may be compared with the predicted waveform of the top trace of Fig. 5 (expressed as a dashed line in Fig. 9). Reasonable agreement is shown between predicted pressure transient and that derived from measurement by using Eqs. (43) and (44). Two equation pairs, Eqs. (41) and (42) and Eqs. (43) and (44) form a two-directional transformation between pressure measurements and voltage measurements. Eqs. (41) and (42) are useful for the modelling of the transducer’s transfer function. Likewise, Eqs. (43) and (44) are useful in the exploitation of transducer instrumentation. If voltage signals alone are not sufficient to meet the application requirements, these last two equations can be easily implemented by a microprocessor embedded in a portable pressure instrument. The formulas permit lYJ(
I
the evaluation of pressure waveform shape, without the involvement of any FFT operation.
5. Conclusions Laser-generated ultrasonic pressure waves have been modelled in an optical absorbing material with half space boundary conditions. Results have been extended to a single layered structure to show the effect of reflections and phase reversals at the boundary interfaces. The model of the time dependence of the pressure wave has quantitatively related the features of the ultrasonic signals with the properties of an optical absorbing layer. These predicted pressure waveforms are not sufficient to understand signals from a PVDF transducer system. A ‘grey box’ transducer model has also been developed, which was used to predict the response of a PVDF ultrasound transducer to such pressure waves. Using a simple algorithm possessing three constants, transformation of signals from pressure into voltage in the time domain was carried out. It was shown how the forward algorithm can be used to predict the PVDF transducer’s voltage output of the transducer, once the three constants of the transducer were evaluated. An inverse algorithm may be used to determine the shape of unknown pressure waves. Since laser-ultrasound signals in the thermoelastic regime are repeatable, assessment of transducer constants may also provide a simple and convenient means of calibrating new transducers or re-characterising ageing transducers. In future medical applications, tissue in front of the transducer will be complex and consist of more than one layer. For example, it may consist of diseased plaque formed above a layer of healthy arteria wall. Extension of the models above will assist in the understanding of diagnostic signals derived from a photoacoustic source generated within the tissue layer.
Acknowledgement The project was supported by Science and Engineering Research Council contract number GR/K 54733 and the National Heart Research Fund, grant number 004/93. We would also like to acknowledge K.F. Pang for the fabrication of probes used in this study.
References .“_
0
09
1
1.5
2
Time (us)
Fig. 9. Restoration of a pressure wave (solid line) from experimental data with a characterised transducer may be compared with the theoretical prediction (dashed line).
[l] Q.X. Chen, A. Davies, R.J. Dewhurst and P.A. Payne, Electronics Letters 29 (1993) 1632. [Z] Q.X. Chen, R.J. Dewhurst, P.A. Payne and B. Wood, Ultrasonics 32 (1994) 309. [3] A.C. Tam, Rev. Modern Phys. 58 (1986) 381.
Q. Shari et al. / Ultrasonics
[4] M.W. Sigrist, J. Appl. Phys. 60 (1986) R83. [S] F.W. Cross, R.K. Al-Dhahir, P.E. Dyer and A.J. MacRobert, Appl. Phys. Lett. 50 (1987) 1019. [6] F.W. Cross, R.K. Al-Dhahir and P.E. Dyer, J. Appl. Phys. 64 (1988) 2194. [7] P.E. Dyer and R.K. Al-Dhahir, Laser-Tissue Interaction/SPIE 1202 (1990) 46. [S] A.A. Oraevsky, S.L. Jacques and F.K. Tittel, Proc. SPIE 1882 (1993) 86. [ 91 A.A. Oraevsky, IEEE/LEOS Newsletter 8 (1994) 6. [lo] A.A. Oraevsky, S.L. Jacques, R.O. Esenaliev and F.K. Tittel. Proc. SPIE 2323 (1995) 37. [ 1l] S.M. Park, M.I. Khan, H.Z. Cheng and G.L. Diebold, Ultrasonics 29 (1991) 63.
34 (1996)
629-639
639
[12] Y. Xu, Ferroelectric Materials and Their Applications (NorthHolland, Amsterdam, 1991). [ 131 Q.X. Chen, and P.A. Payne, Meas. Sci. Technol. 6 (1995) 249. [14] D.A. Leedom, R. Krimholtz, and G.L. Matthaei, IEEE Trans. Son. Ultrason. SU-18 (1971) 128. [15] G.R. Lockwood and F.S. Foster, IEEE Trans. Son. Ultrason. SU-41 (1994) 225. [ 16) R.J. Dewhurst and W.S.A.R. Al’Rubai, Ultrasonics 27 (1989) 262. [ 171 K. Ogata, Discrete-Time Control Systems (Prentice-Hall, Englewood Cliffs, NJ, 1987). [ 181 F. Oberhettinger and L. Badii, Tables of Laplace Transforms (Springer-Verlag, New York, 1973). [ 193 L. Ljung, System Identification, Theory for The User (Prentice Hall, Englewood Cliffs, NJ, 1987).
Characterisation
Ultrasonics 34 (1996) 629-639
of laser-ultrasound signals from an optical absorption layer within a transparent fluid Q. Shan *, A. Kuhn, P.A. Payne, R.J. Dewhurst
The University of Manchester Institute of Science and Technology, DIAS, UMIST, PO Box 88, Manchester, M60 lQD, UK
Received 30 October 1995; Revised 18 January 1996
Abstract
Characteristics of laser-ultrasound signals are presented from photoacoustic interaction with a layered, optically absorbing medium surrounded by a transparent fluid. A thermoelastic model is presented describing the interaction, with signal predictions in the fluid arising from polymer transducer detection. By taking the optical absorption coefficient and finite layer thickness into account, the amplitude and shape of transient elastic waves are calculated for both forward and backward travelling directions. Additionally, wave interaction with the PVDF transducer has been characterised using a discrete-time algorithm for the transducer response. With just three constants to define transducer response characteristics, the response function may be used to predict voltage signals. Good agreement with experimental waveforms is demonstrated, so that the response function may form the basis of system modelling when miniature laser-ultrasound probes are used in various applications. Keywords:
Laser-ultrasound;
Transducers; Photoacoustics;
Optical absorption layers
1. Introduction With current progress in laser technology, fibre-optics and polymer transducers, miniature laser-ultrasound probes may now be constructed as a potential diagnostic tool [ 1,2]. In the field of medicine, it may be feasible to use such a tool as a sensing transducer, prior to some therapeutic use of a laser beam delivered down the same or similar fibre-optic system. For example, it is known that further information is required within blood vessels which are constricted by fatty or calcareous deposits. By using multimode optical fibres for forward-looking laser-ultrasound examination of targets, techniques may, in the future, be combined with therapy such as laser angioplasty to provide greater information at the zone of laser interaction. It is essential therefore to understand the nature of ultrasonic transducer signals which arise from laser absorption in tissue contained within an aqueous or saline environment. Laser-ultrasound signals arising from measurements of laser light absorption in liquids and solids have * Corresponding author. Fax: +44-161-200-4911; e-mail: [email protected]
already been reviewed [ 3,4]. In published literature, such signals are often referred to as either transient elastic waves or stress waves. They can be generated by the use of short duration laser pulses, typically a few nanoseconds long. In pioneering work with tissues, measurement of the optical properties within the tissue lacked a theoretical basis for overall time evolution of the waveform signal shape [ 5-71. More recently [8-lo], there are several accounts of piezoelectric detection of time-resolved stress detection (TRSD) which makes use of an effective optical attenuation coefficient in tissue to help describe the behaviour of laser-induced stress waves. These ultrasonic signals are recognised as a convolution of many factors, such as the optical absorption coefficient, stress relaxation, transmittance through interfaces and the detector response. For studies in layered material, the optical absorption coefficient, layer thickness, and transducer response are crucial to the understanding of time-resolved measurements of ultrasound signals, but there has been no comparison of predicted time evolution with experimental observations. We have therefore developed a one-dimensional model, using optical neutral density filters as phantoms for real biotissues since, for the
0041-624X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PZI SOO41-624X(96)00055-8
Q. Shari et al. 1 Ultrasonics
630
purposes of experimental verification, they possess welldefined optical and thermoelastic properties which facilitate detailed modelling. With short pulse irradiation of an optical absorption layer within a transparent liquid medium, we develop solutions for both the forward travelling and backward travelling waves, expressed in terms of time-resolved pressure changes. These are then combined with a model of piezoelectric transducer performance to provide predictions of time-resolved voltage signals. These predictions are compared with experiments in which the transducer consisted of a polyvinylidene fluoride (PVDF) element. The medium coupling the sample to the transducer was assumed to be non-absorbing to ultrasonic signals.
34 (1996)
629-639
infinity space [4,7,11]. We present below a one dimensional (1-D) model as an approximation to the real experimental conditions. We first examine the problem for a half space boundary condition by assuming the thickness of the sample h -+ 00. The wave equation for pressure can be shown to be of the form [ 61. a%
1
ax2 - 2
azp -/3aH
-=-at2
c,
(1)
at ’
with initial conditions: p(x,O_)=&p(x,O_)=O,
H(x,O_)=O
(2)
and boundary conditions: 2. Modelling of laser-generated acoustic pressures
Pb(O- 9 t) = P&&O+ 9 0,
The sample in which laser-ultrasound was generated consisted of one of a number of different high power, volume absorbing neutral density filters, placed in a transparent liquid. Fig. 1 shows the schematic arrangement in which a PVDF transducer could be positioned at A to monitor forward travelling ultrasonic signals, or at position B to monitor backward travelling signals. Distilled water was used as the coupling medium between the sample layer and transducer position. A model for the prediction of pressure transients which then arise from irradiation of the sample with a short laser pulse is presented below.
- ;
&Ps(O- 7 t) = - f
&.(O+, II
t).
2.1. Solution for halfspace boundary conditions
In Eqs. (l)-(3), p, c, /?, and C, are respectively the acoustic pressure, the sound speed, the isobaric volume expansion coefficient, and the isobaric heat capacity per unit mass. The x spatial coordinate is in the direction shown in Fig. 1, with x = 0 defined as the top of the sample surface. The time coordinate t is defined as t = 0 when the laser pulse first irradiates the sample. Notations with subscripts a and b denote parameters in the sample and water respectively, with p being the density of the medium. H is the heating function, defined as the energy absorbed per time per unit volume. It may be expressed as
Several authors have modelled the acoustic pressure pulses generated by laser pulses incident on a half-
H(x, r)=ae-ax$[u(t)-u(t-A)],
Air = =
II
Fluid B
PVDF Transducer I
Fig. 1. Position of the optical absorbing layer surrounded by a transparent fluid. A PVDF transducer monitors the forward travelling wave at position A, or the backward travelling wave at position B.
(4)
where CIis the optical absorption coefficient, E0 is the radiation energy density at the sample surface, d is the time period during which the energy is uniformly delivered, and u(t) is the unit step function whose value is one when its argument is positive and zero otherwise. Here we assume that the temporal profile of the laser pulse is rectangular, d is the pulse width and &-,/A is the light intensity. When d is small, Eq. (4) gives results that approximate the effect of a Dirac delta function. Later on it will be shown that this approximation is valid for examples shown in this paper. We may note that the photoacoustic response to long duration laser pulses with arbitrary temporal profile can be determined by convolving the delta function response with the specific laser pulse temporal protie. The initial and boundary conditions, Eqs. (2) and (3), assume that the system is at rest before the laser pulse is fired, when t = 0. It is also assumed that the pressure and pressure gradient are continuous across the interface,
Q. Shariet al. / Ultrasonics34 (1996) 629439
and that the pressure is bounded at - co I x I co, which is an implicit condition. A Laplace transformation [ 181 is employed to solve the partial differential equation. The transform pair for pressure p is defined as
631
(12) where
9 (P(X, t); f+ s> = Y(X,s); (5)
(13)
Applying the Laplace transformation on Eqs. (1) and (4) with initial and boundary conditions given by Eqs. (2) and (3), we obtain a nonhomogeneous ordinary differential equation:
By applying the inverse transformation to Eqs. (9) and (lo), the pressure pulses can be expressed in the time domain as:
6p-r(y(X,S);S+t)
=p(x,t).
$y(x,s)-$y(x,s)=
-~em’I(l-eed”).
(6)
P
If d +O, then eeds tends to 1 - ds, so that Eq. (6) can be written as
a2 sy(x,
s2
4 - 2y(x, 4 = -
@Eos
emax,
7
(7)
P
I
where we regard the Laplace parameter s as a constant and x as the independent variable. The solution to Eq. (7) can be derived to be: y = A(s) e-(s/c)x+ B(s) e(s/c)x +
ajlE,c2 eeax CP
(ca)2 ’
%(c~p.cb+P~bpb)e~c~(f+x~ t+t20, (15)
Pb=
(8) where A(s) and B(s) are arbitrary constants with respect to x and are determined by the boundary conditions. The pressure p has to be bounded for all x[ - cc to + co]. Therefore, the first and the third terms of the right-hand side of Eq. (8) can only exist within the absorbing medium of the sample, where x 2 0. Likewise, the second term can only exist in the transparent fluid where x < 0. Eq. (8) can now be written explicitly using parameters related to the corresponding medium. In a semi-infinite half space, with two media, a and b, such that c, and cb represent the sound speed c in the absorbing region and non-absorbing region respectively, we find that in the absorbing medium y, =
/qs)
e-
(s/c‘&
+
afiEoc~ e-Orx s CP
s2- (c,a)2 ’
y, = B(s) e(S/cb)X,x < 0.
i
elsewhere.
0,
In summary, Eqs. (14) and (15) represent the solution of Eq. (1) when the appropriate boundary conditions are taken into account. 2.2. Solution for boundary conditions representing a layer If the sample is a layer within a transparent fluid, as shown in Fig. 1, then this model may be extended to take into account the finite thickness of the absorbing layer, h. The ultrasonic transducers positioned at either A or B are immersed within a fluid and respond to the arrival of pressure transients described by pA and PB respectively.
x 2 0, 2.2.1. Acoustic pulse detected at position B
(9) and in the non-absorbing
Cl?
(14)
s s2 -
t-%0,
medium, we have
One more boundary condition is imposed at the rear optical absorbing interface, where x = h. It may be described as
(10)
From the boundary condition expressed by Eq. (3), we have
-;;Pb(h-,t)=
-$&.(~+,t). LI
(11)
Differences in optical, acoustic and thermal properties between the media at each side of boundary x = h result
Q. Shari et al. / Ultrasonics 34 (1996) 629-639
632
in different solutions compared with those from a half space solution presented in Eqs. (14) and (15). Firstly, if the absorption coefficient is low, the laser energy may not be totally absorbed in the layered medium. This discontinuity in optical absorption coefficient across the interface introduces a discontinuity in Eq. (15). It must be modified to give a pressure wave variation described by:
Pb
=
I
b2[e-c&z
+ h/c,) + ec.a(tz - h/c,)
19
Pa2 =
(t+z>o)*(t+E<; 9 >
0,
Likewise, a later contribution arises from a second reflection at the rear surface of the sample after a further time delay of 2h/c,. Therefore another pressure contribution can be expressed as:
(17)
elsewhere,
where ‘A ’ represents the logical ‘and’ operator. The initial pressure wave no longer has a continuous decay along the x-axis, but is cut off by the interface at x = h. Secondly, when the PVDF transducer is at position B of Fig. 1, the transducer can receive a contribution of pressure arising from pb, the backward travelling transient wave, together with multiple echoes arising from the forward travelling wave p., which suffers partial reflection and possible phase inversion at the boundary x = h. This reflection then propagates towards the transducer, being partially transmitted through the boundary at x = 0. The other component is reflected at this boundary and continues to undergo multiple echoes, with pa gradually fading away with time. Therefore, under layered boundary conditions, Eq. (14) should be modified. Inclusion of a contribution from the first reflection of p. results mathematically in a pressure term of the form:
4 Ce-c,Mh+
h/c.) + e’c,““l -h/d
0,
elsewhere,
(21) where 3h tz=t--+---, co
x
(22)
cb
and b2 is expressed as:
~2=~(,.,p~~bcb)(~“:“~~~~~~.
(23)
In general form, for the ith reflection, the pressure wave can be described by: bi
[e-c,c(ti+h/c,)
+ ef,dti
-h/c,)]
,
1, -c,u(ti
Pai =
-h/c,)
+
e-c,a(t,
+ h/c,)
P.1 =
0,
elsewhere, (24)
0,
where
elsewhere, (18)
where
ti=t-
(2i-1)h+x c,
(25) cb’
and Qi is expressed as: (19)
(20)
where i > 0 is an integer number. Finally, acoustic pressure pulses detected at the position
Q. Shun et al. / Ultrasonics 34 (1996) 629-639
B in Fig. 1 can be expressed as:
(27) i=l
where N > 1 is also an integer number and depends on how many reflections we are interested in. Pressure contributions pa and pai are defined by Eqs. (17) and (24) respectively. 2.2.2. Acoustic pulses detected at transducer position A At transducer position A, only p, and its multiple reflections reach the transducer. To model the first direct arrival of pa, we may first consider the pressure at the rear surface of the absorbing medium and substitute x = h into Eq. (14). The wave travels from the interface x = h towards the transducer position A. Eq. (18) remains valid in describing the first direct arrival to position A, but Eqs. (19) and (20) are modified to take into account the time delay and one transmission across a boundary, so that: +t--,
(28)
633
532 nm was incident on the absorbing layer, taken to be a neutral density glass filter within a non-absorbing water medium. With a laser pulse energy of 5 mJ in a beam diameter of 6.5 mm, solutions to Eqs. (27) and (32) were evaluated and are shown in Fig. 2. Physical parameters used in the calculations are quoted in the figure caption. The upper graph in Fig. 2 represents a solution to Eq. (32) and therefore represents the forward travelling pressure wave reaching a transducer at position A. Similarly, under the same conditions, the lower graph represents a solution to Eq. (27), representing the backward travelling pressure wave reaching a transducer at position B. As expected, a number of pressure discontinuities are displayed which correspond to the effect of the two boundaries of the layered medium. In general, their amplitude gradually decreases with time, but the forward travelling pressure wave (top trace) has the greatest pressure change some time after the initial arrival. It corresponds to the time of arrival at the transducer of elastic wave energy coming from the top of the absorbing layer. This arrives with an acoustic delay time due to the glass thickness, in contrast to any
cb
and
(a) (29)
c,=4r(p.c~;,c,)*
After the direct wave arrival, further arrivals arise from multiple reflections again. For contributions due to the ith reflection, Eq. (24) remains of the same form, but the time parameter is modified to:
CW- l)lh + 14.
ti=t-
G
cb'
(30)
-6 -
’
-6
0
and di is expressed as:
I
0.5
I
1
J
1
1.5
2
6m (b)
4-
a
Hence, the acoustic pressure waves detected at position A in Fig. 1 can be expressed as: PA =
$
Pai-
P d
2-
Pa2
Pa3
. -
-4-
i=l
Eqs. (27) and (32) represent the one-dimensional solutions for acoustic pressure waves generated by a laser within an absorbing layer and subsequently propagating in a non-absorbing fluid to positions B or A as shown in Fig. 1. 2.2.3. Example solutions Any example of predicted pressure waves depends on the properties of laser pulse excitation, as well as the properties of the layered structure. A case was chosen where an 8 ns duration laser pulse at a wavelength of
-6-6
t
1
0
I
0.5
I
1
I
1.5
1
2
Time 6.~4
Fig. 2. Theoretical predictions of ultrasonic pressure waves generated from a 5 mJ laser pulse having a beam diameter of 6.5 mm when incident on a neutral density glass filter contained within distilled water. Physical parameters used in the calculation for water and for the neutral density filter were: p. = 2230 kg m-s, pb = 998.2 kg me3, c,=564Oms-‘, c,=1482.34ms-r, C,=7OOJkgg’K-‘, fi=2.52x lo-‘K-l, a=1250m -I, h = 1.9 mm. Waveforms in (a) represents the forward travelling pressure wave and (b) represents backward travelling wave.
634
Q. Shari et al. / Ultrasonics 34 (1996) 629639
laser energy absorption at the rear surface of the sample, which also forms a pressure wave but without this time delay. Such solutions are only valid if the acoustic pressure wave travels an insignificant distance compared with the optical absorption length during the time of the laser pulse. The time taken for a wave to travel across the absorption length is ~/UC,. In the case described in Fig. 2, this corresponds to 142 ns, which is much longer than the laser pulse duration of 8 ns. Hence the laser pulse is a good approximation to the Dirac delta function, and no convolution due to its finite temporal width is necessary.
3. Model for the PVDF transducer To measure the acoustic pressure pulses shown in Fig. 2, a wideband transducer is required. In potential medical applications, the good acoustic impedance match of polymer films with either water, saline or biological tissue makes transducers based on polyvinylidene fluoride (PVDF) and its copolymers attractive. Their properties have recently been reviewed [ 12,131, and these were the type of transducer element used in our earlier investigations [ 1,2]. When constructed to form a transducer, there are a choice of models used in the literature to predict their performance. Most treat the transducer as an equivalent circuit. The KLM model is one such example [ 141, requiring precise knowledge of the piezoelectric, dielectric and mechanical properties of the transducer. These properties are often difficult to determine so that Lockwood and Foster [lS] have proposed a ‘black box’ approach. They treated the transducer as a black box, disregarding any knowledge of the transducer properties. Another approach is to have a ‘grey box’ model, in which use is made of experimental observations that a polymer transducer has essentially a differential response [ 161, and where the backing material has a classical damping effect. Its voltage output response v(t) to a pressure input p(t) may then be described by a second order ordinary differential equation such that: d2u(r) do(t) dp(t) + C&(t) = Edt2 + 2i% dt dt ’
property and affects the undamped frequency (to be defined later) CO,.It also affects the capacitance of the transducer, which is one of its dielectric properties. In this simple mathematical modelling, the transducer’s transfer function can be simply characterised by only three constant parameters such as c, w,, and E. These three parameters control the transducer’s characteristics through Eq. (33) including contributions from the materials used, its design and its mode of construction. Due to difficulties involved in the independent determination of these parameters for a particular device, Eq. (33) cannot be solved directly for a fabricated transducer. However, the parameters may be determined by the help of experimental measurements derived from an arrangement shown in Fig. 3. Normally, the transducer’s time domain output signal was recorded in a discrete format. Therefore, instead of using a Laplace transformation, a z transformation was used to solve Eq. (33). In the discrete-time domain, Eq. (33) may be expressed by a difference equation such that [ 171. V2u(k) V v(k) + c+(k) --p+ 2bn z
VP(k) = Ez ’
(34)
where k is the current sampling instant, and z is the sampling period. If we define the z transformation pair [ 171 to any time discrete function f(k) such that:
f-(z)= W-(WI,
f(k) = z - ’ CWI
(35)
where z is the discrete transformation variable. By using the property of z transformation of the form: Z[V”f(k)]
= (1 - z-‘)“&‘(z),
(36)
Eq. (34) can be expressed as: V(z) = K()(l - z_‘)P(z) + [K,z_’ + K,z_2]1/(z),
(37)
(33)
where i, CO,, and E are constants of the system. The constant E (in units of V Pa-l SC’) is dependent on the piezoelectric and dielectric properties of the transducer. And c (dimensionless) and w, (in units of rad/s), are determined by the mechanical properties of the transducer film and its backing material. Although we can roughly divide the left- and right-hand side of Eq. (33) into mechanical and electrical behaviour of the transducer, their interrelationship is complicated. For example, the thickness of the transducer film is a mechanical
integrated transducer
\
ND glass filter Experimental tank
I
Fig. 3. Experimental arrangement used to investigate laser-ultrasound generated in an absorbing layer within water serving as a transparent fluid.
Q. Shari et al. / Ultrasonics 34 (1996) 629639
where: K,=
K,=
EZ 1+2~o,r
(38)
+ 0,222’
2(1+ b”4
(39)
1+2&!&z+o,zr~’
and K,=
-
1 1+2~0,z+0,z22’
In Eqs. (37)-(40), K0 is in units of V/Pa, and K1 and K2 are dimensionless. Eq. (37) can be graphically illustrated by a transfer function block diagram as shown in Fig. 4. Such a block diagram can be converted into a simple discrete-time domain algorithm. If we take the pressure pulse p(k) as the transducer input at the kth instant, the transducer output voltage u(k) at the kth instant can be calculated in two steps with the intermediate result denoted as q(k). Firstly, for the second term in Eq. (37), which represents damped oscillation, it follows that q(k) = p(k) + K,q(k - 1) f fbdk
(41)
- 2).
For the first term in Eq. (37), which represents differential operation, it follows that
u(k)=K,Cq(k)-q(k--1)1.
the (42)
Eqs. (41) and (42) form a simple algorithm with only three constants. These three constants were determined by comparing a predicted waveform with a measured voltage output. The procedure used to identify the constants exploited a standard system identification method [ 191. We used Matlab together with its toolbox for system identification. Once the constants were determined for a particular transducer, their values were retained in Eqs. (41) and (42) to predict transducer output voltages for various pressure transient inputs. The experimental arrangement for this characterisation process is described in the following section.
4. The transducer characterisation and discussion Experimental investigations for transducer characterisation were performed using the arrangement shown
vk
Fig. 4. Block diagram used to model the PVDF transducer transfer function. In a z transformation analysis, the transducer can be fully characterised by three constants, K,, K, and K,.
635
in Fig. 3. A frequency doubled, Q-switched Nd : YAG laser delivered pulses of 8 ns duration. They were attenuated by the use of optical filters before being focused with a convex lens into a length of multimode silica fibre, acting as a flexible delivery system. Pulse outputs, from the end of the optical fibre, were measured to be typically in the range of 5-10 mJ, using a Photon Control Calorimeter Model 25V-VIS. They were directed towards a glass neutral density filter placed within a tank of distilled water, the water acting as a constraining and optically transparent medium. Laser pulse energy was partially absorbed by the glass filter, with its beam diameter being approximately 6.5 mm. Under the thermoelastic laserultrasound generation mechanism [ 21, ultrasonic transients were generated within the filter and transmitted through both top or bottom surfaces of the filter into the liquid. They were designated either the backward travelling wave, or the forward travelling wave. The backward travelling wave was detected at position B in Fig. 1 using a concentric transducer at the fibre output. Its design has been described in earlier publications [ 1,2]. Alternatively, another transducer was placed below the target (not shown in Fig. 3) to record the forward travelling wave at position A in Fig. 1. Transducers were based on a 27 nm PVDF polymer film and transducer signal leads were connected to a 50 R input channel of a Tektronix TDS 520 digital storage oscilloscope, in order to maintain a broad-band frequency response of the measurement system. A beam-splitter sent a fraction of the laser beam to a p-i-n photodiode which provided a trigger signal to the oscilloscope. Some properties of glass filters used in the experiments are summarised in Table 1. By using Eq. (32) with filter parameters shown in Table 1, a set of forward travelling pressure wave predictions are presented in Fig. 5. The theory presented in the previous section was able to predict the form of the transient pressure wave arising from filter irradiation. For a transducer in position A of Fig. 1, monitoring forward travelling transients, Fig. 5 shows a set of predictions, for incident laser energies of 6.7 mJ. The time axis is presented from t = 0 with the acoustic delay through the liquid subtracted from the time after laser pulse excitation. The three waves represent transient pressure waveforms which may arise from high (upper trace), middling (middle trace), and low (lower trace)
Table 1 Physical characteristics experiments
of absorbing
glass filters used in the
Filter No
h (mm)
cL(m-l)
p. (kg m-‘)
B (K-l)
1 2 3
2.01 2.08 2.96
5116 1204 252
2470 2430 2420
2.13 x lo-’ 2.16 x 1O-5 2.16 x 1O-5
Q. Shan et al. / Ultrasonics 34 (1996) 629-639
636
________.~. Predictions 20 1
a=5116me’
a=1204
Measurements
-
a=5116 m-’
10 -
2r
m-l
I
a=1204 m-'
a=252 m-' 1
a=252 m-’
0.5 b
5 =
h
0
g-o.5 r" B
-1 -1.5
-4' 0
-2 1
0.5 Time
1.5
2
(ps)
-2.5 0
0.5
1 Time
Fig. 5. Forward travelling pressure waves predicted for three different values of optical absorption coefficients, ~1, associated with the absorbing layer. In each case, the laser input energy remained constant at 6.7 mJ.
optical absorption coefficients. As may be expected, the amplitude of pressure transients decreases in consecutive traces (noting different vertical scales in three waveforms), with the largest amplitudes associated with the sample possessing the highest optical absorption coefficient. In the other cases, step features in the waveform arise from both surfaces of the absorbing media, implying that not all the optical energy was absorbed within the glass filters. Corresponding polymer transducer responses are shown in Fig. 6. These waveforms are quite different in shape, representing the voltage output from transient waveforms shown in Fig. 5. The upper waveforms in Figs. 5 and 6 were used to characterise the forward travelling transducer. The former, representing pressure transients, was taken to be the known transducer system input, and the latter was taken to be the measured system output. The constants of the transducer, K,,, K, and K, were evaluated by using a Matlab system identification tool box running on a Sun workstation. System identification was performed using an output error model structure [ 191. Table 2 summarises the
1.5
2
(ps)
Fig 6. The output of the PVDF transducer in response to the pressure waves pA represented by the waveforms shown in Fig. 5. Measured waveforms are indicated by the solid lines, whilst predicted waveforms based on the three parameter transducer model are indicated by dashed lines.
values of derived transducer constants. Once the constants KO, K, and K, had been evaluated, a predicted curve of transducer response was found to follow the experimental waveforms closely. As shown in Fig. 6, the two waveforms overlay each other satisfactorily. Retaining the same values of Ko, K1 and K2 for the transducer, predictions of transient waveforms derived Table 2 Characteristic constants for transducers backward travelling wave measurements
K,(10-7V/Pa) Ki (dimensionless) K, (dimensionless) Loss function Akaike’s final prediction error criterion
used in either forward or
Forward travelling wave transducer
Backward travelling wave transducer
0.800 + 1.710 + - 0.750 + 8.415 x 8.483 x
0.320 k 1.600 f - 0.700 + 2.387 x 2.425 x
0.014 0.006 0.006 10-a lo-*
0.026 0.036 0.033 1O-g 1O-g
Q. Shan et al. J Ultrasonics 34 (1996) 629-639
from glass filters with different optical absorption coefficients were then compared with experimental measurements. As shown in Fig. 6, there is good agreement between theory and experiment in all cases, supporting the transducer modelling philosophy adopted. Similar studies were also performed for the backward travelling wave derived from the same experimental scheme. With a transducer at position B in Fig. 1, calculated transient waves for incident laser energies of 5.0 mJ are shown in Fig. 7. Corresponding voltage waveforms from the transducer system are shown in Fig. 8. Predicted and measured waveforms again overlap and are in good agreement. Particular constants derived for the transducer from the upper waveform are summarised in Table 2, after adopting a system identification procedure already described above for this different transducer. In the case of low optical absorption coefficient, experimental waveforms possessed significant noise due to the low level of generated acoustic signal. We note that the leading edge of the initial pressure transient (Fig. 7) was sharp, but its amplitude is not as great as the one arising from the first echo within the absorbing layer. The phenomenon is more obvious in Fig. 8. The
“I
631
-
Measurem en1 a=51 16 m-’
F
4
a=1204 m-l
I
a=252
mm1
R
60 t
a B
40
1
a=51 16
8
0.1
S g
0
m-’ -O.l” -0.2
I”
III
”
I
I I
II
1
Y
F
-0.3 '
P
0
z b:
1 Time
0
t
1.5
2
(us)
Fig. 8. Output of the PVDF transducer in response to backward travelling pressure waveforms predicted in Fig. 7. Measured waveforms are indicated by the solid lines, whilst predicted waveforms based on the model are indicated by dashed lines.
1
-4
I 0.5
’ 0
I
I
1
0.5 Time
1.5
2
(ps)
Fig. 7. Backward travelling pressure waves predicted for three different values of optical absorption coefficient a associated with the absorbing layer. In each case the input laser energy was kept constant at 5 mJ.
first positive pulse is always lower in amplitude than the second positive pulse. This feature is a characteristic of backward travelling transients, regardless of the optical absorption coefficient. There are sometimes small pulses with opposite polarity to major features in both Figs. 6 and 8. These secondary features arose from the rear sample surface. For signals arising from the rear sample surfaces, where x = h, it is convenient to define a term, optical thickness D = ah. If the sample was optically thick, when D >>1, all laser pulse energy was essentially absorbed within the sample. In this case, there was no acoustic feature arising from the sample rear surface. Such a situation existed for the upper traces in Figs. 5, 6, 7 and 8. When D <<1, corresponding to an optical thin case, rear sample surface features are pronounced as in the lower traces of Figs. 5, 6, 7 and 8. In the middle traces in these figures, only small pulse feature arose from the rear sample surface, when D = ah = 2.5. We note in passing that, although it is difficult to
Q. Shari et al. / Ultrasonics 34 (1996) 629-639
638
control the properties of PVDF material throughout the manufacturing process of a transducer system, these results have shown that a transducer may be characterised by three constants used in a finite difference equation. These properties may change with time, and repeat experiments using the experimental procedure described above would be capable of measuring this change. With ultrasound generation being independent of ultrasound detection, it would be possible to check the performance of any transducer throughout its lifetime. Once characterised, the transducer’s voltage output sequence, v’(k), can be used to measure qualitatively the real pressure, p’(k), through the inverse use of Eqs. (41) and (42), such that: q’(k) = F
+ q’(k - l), 0
and p’(k) = q’(k) - k,q’(k - 1) - K,q’(k - 2),
(44)
where q’(k) is an intermediate variable. We may take the measured waveform shown on the top trace of Fig. 6 as an example. By using Eqs. (43) and (44), a measured pressure waveform may be deduced, as shown in Fig. 9 as a solid line. It may be compared with the predicted waveform of the top trace of Fig. 5 (expressed as a dashed line in Fig. 9). Reasonable agreement is shown between predicted pressure transient and that derived from measurement by using Eqs. (43) and (44). Two equation pairs, Eqs. (41) and (42) and Eqs. (43) and (44) form a two-directional transformation between pressure measurements and voltage measurements. Eqs. (41) and (42) are useful for the modelling of the transducer’s transfer function. Likewise, Eqs. (43) and (44) are useful in the exploitation of transducer instrumentation. If voltage signals alone are not sufficient to meet the application requirements, these last two equations can be easily implemented by a microprocessor embedded in a portable pressure instrument. The formulas permit lYJ(
I
the evaluation of pressure waveform shape, without the involvement of any FFT operation.
5. Conclusions Laser-generated ultrasonic pressure waves have been modelled in an optical absorbing material with half space boundary conditions. Results have been extended to a single layered structure to show the effect of reflections and phase reversals at the boundary interfaces. The model of the time dependence of the pressure wave has quantitatively related the features of the ultrasonic signals with the properties of an optical absorbing layer. These predicted pressure waveforms are not sufficient to understand signals from a PVDF transducer system. A ‘grey box’ transducer model has also been developed, which was used to predict the response of a PVDF ultrasound transducer to such pressure waves. Using a simple algorithm possessing three constants, transformation of signals from pressure into voltage in the time domain was carried out. It was shown how the forward algorithm can be used to predict the PVDF transducer’s voltage output of the transducer, once the three constants of the transducer were evaluated. An inverse algorithm may be used to determine the shape of unknown pressure waves. Since laser-ultrasound signals in the thermoelastic regime are repeatable, assessment of transducer constants may also provide a simple and convenient means of calibrating new transducers or re-characterising ageing transducers. In future medical applications, tissue in front of the transducer will be complex and consist of more than one layer. For example, it may consist of diseased plaque formed above a layer of healthy arteria wall. Extension of the models above will assist in the understanding of diagnostic signals derived from a photoacoustic source generated within the tissue layer.
Acknowledgement The project was supported by Science and Engineering Research Council contract number GR/K 54733 and the National Heart Research Fund, grant number 004/93. We would also like to acknowledge K.F. Pang for the fabrication of probes used in this study.
References .“_
0
09
1
1.5
2
Time (us)
Fig. 9. Restoration of a pressure wave (solid line) from experimental data with a characterised transducer may be compared with the theoretical prediction (dashed line).
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Q. Shari et al. / Ultrasonics
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