Barbara Schuppler Integrating Piezoelectric Sensors for Thermoacoustic Computertomography Diplomarbeit Zur Erlangung des akademischen Grades einer Magistra an der Karl-Franzens Universität Graz Naturwissenschaftliche Fakultät vorgelegt bei Ao. Univ. Prof. Dr. Günther Paltauf Institut für Physik Abteilung Magnetometrie und Photonik im April 2007
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First of all, I would like to thank my supervisor, Prof. Dr. Günther Paltauf, for
giving me the opportunity to write my diploma thesis in his group, for supporting
my ideas and supervising my work. Furthermore, thanks go to Prof. Dr. Heinz
Krenn, who warmly integrated me in the group of "Magnetometrie und Photonik",
for the possibility to take part at the seminar Nano and Photonics. Mauterndorf 2006.
Thanks also go to the other students of the group for helping me technically in the
laboratory, for the fruitful discussions and for simply spending a good time together.
Thanks also go to Matthias Skacel, on the one hand, for helping me regarding
electronic and programming issues of the automation, on the other hand, for his
personal encouragement during all the years of studies.
Finally, I would like to thank my family. My brothers always supported my studies.
In the last months, my brother Martin even helped me with improving the quality
of the images included in this thesis. Special thanks go to my grandparents. They
supported me during all the years, financially and personally.
iii
1 Acknowledgments
iv
2 Introduction
Thermo Acoustic Computer Tomography (TACT), also known as photoacoustic or
optoacoustic tomography, is a technology in development for imaging semitranspar-
ent, light scattering media. Even though it is applied mainly to the diagnostics of
biological tissue, it is not yet established in medicine. The basic principle of TACT is
that first short laser pulses (pulse duration ≈ 10nm) are irradiated on the medium,
where the energy of the light gets absorbed in dependence on the optical characteris-
tics of the medium. A rapid thermal expansion of the medium causes an acoustic wave
(thermoelastic effect). This acoustic wave contains information on optical character-
istics and optical structures of the medium. Piezoelectric and optical sensors measure
the pressure signals outside the object. Computer algorithms are used to reconstruct
the distribution of absorbed energy and an image is received.
The choice of the wavelength of the electromagnetic radiation depends on what is
required to be imaged. Using wave lengths in the spectral range of infrared (800 −1200nm), or even microwave (1− 300mm) guaranties the imaging of deep structures,
as required for mammograms, but with a forfeit in contrast. Using light in the visible
spectrum (400 − 700nm) yields a higher contrast, but the penetration depth is only
in the range of some millimeters. For diagnostics of the skin preferably the visible
spectrum is applied [41].
Thermoacoustic computer tomography combines the advantages of optical imaging
and ultrasound imaging. Optical Coherence Tomography (OCT) has a very good res-
olution up to a depth corresponding to the diffusion length of the medium. After this
depth, the propagation of light becomes diffuse. Depending on the type of tissue this
length is about 1mm. Diffuse Optical Tomography (DOT) is the imaging technique
beyond this length. It gives the optical properties of the imaged tissue, but with a
poor resolution.
1
2 Introduction
Medical ultrasonography is a non-invasive imaging technique where ultrasound is
sent into the body where it is reflected from interfaces between tissues and an echo
is returned to a transducer. The used frequencies are between 2 and 13MHz, where
lower frequencies give greater penetration depth but less spatial resolution. The dis-
advantage of this method is that the acoustic contrast between soft tissues is low, but,
as the propagation of ultrasonic waves is less diffuse than of light, the resolution is
higher than when using DOT.
In TACT, incident electromagnetic radiation is transformed via the thermoelas-
tic effect into ultrasonic waves that leave the medium. Because of this, both, high
contrast and high spacial resolution can be obtained. Its advantage over X-ray com-
putertomography is the use of non-ionizing radiation that lies in the range of visible
and infrared light. The warming due to the propagation of the mechanical waves is
very low, as ultrasonic sensors of high sensitivity are used. Also the generation of
thermoelastic pressure is very efficient. Already a temperature rise of 1C leads to a
pressure of 5bar [41]. Its advantage over magnetic resonance is the relatively low cost
[5].
The aim of my diploma thesis was to construct and test a novel TACT set-up
using a piezoelectric line sensor and to develop an control box for the tomographic
experiment. The structure of the written work was chosen to be in chronological order
with the processes that occurred during the project. At the beginning of chapter 3
the focus lies on the propagation of light, then, understanding already absorption
in tissue, the thermoelastic effect will be treated, followed by the propagation of
acoustic waves. In a next step, these acoustic waves arrive at the piezoelectric sensor.
In Chapter 4, the attention will be given to piezoelectricity, its use for sensors and its
practical implementation in the construction of the sensor. Then, the line geometry of
the sensor and its consequences for the reconstruction will be discussed. In Chapter
6, experiments will be presented that allow to determine the characteristics of the
sensor. After describing the set-up of the tomographic experiments, the developed
automation, whose central component is a microcontroller, will be presented. Finally,
in the last chapter, images obtained of phantoms will be shown and discussed.
2
3 Theoretical principles of
Optoacoustics
In this chapter, some historical aspects of optoacoustics and its importance for medical
imaging will be treated. Chronologically following the generation of optoacoustic
waves, beginning with the propagation of light in medium, the processes of absorption
and scattering will be described. Then, processes that transform light into sound,
especially the thermoelastic effect, will be presented. After having understood how
pressure waves are generated, the focus will be on the propagation of acoustic waves
and its consequent influence on their waveform.
3.1 Historic Overview and Future Prospects
In 1880 and 1881 the first reports of experiments on the optoacoustic effect with
solids, liquids and gases were published by Alexander Graham Bell. He illuminated
thin discs with a beam of sunlight and interrupted the beam with a rotating slotted
disc. As the generated acoustic waves were in the range of audible frequencies, he
needed not more than his own ear as measuring instrument [34]. He found out that
by illumination with different wavelengths the measured sound yields a spectrum that
serves to characterize the absorbing components of the material [13].
Incidentally, Bell and his assistants developed the Photophone (Fig. 3.1). A mirror
was fixed on a membrane that served to generate vibrations of the mirror similar to
the vibrations in the voice. A light beam was radiated into this system, and so this
light beam was modulated with the sound. This modulated light beam, then, changed
the electric resistance of a gas cell, that was again transformed into sound. It worked
without wires, but since it only transmitted clearly up to distances of 100m, it did
3
3 Theoretical principles of Optoacoustics
Figure 3.1: Graham Bell´s Photophone [13]
not lead to a successful breakthrough [34].
As in these first experiments an audible acoustic wave was induced by a visible
electromagnetic field, the discoverers called the phenomenon Optoacoustic Effect [34].
In the field of spectroscopy the expression photoacoustic is nowadays more common.
However, the phenomenon was forgotten soon, as no great functional nor scientific
value could be found in the effect. It took fifty years until the optoacoustic effect
experienced a revival for the study of gases [45]. Important progresses were made
since the design and development of the first laser in 1960 [47]. From that time on,
a continuous interest on this effect in various fields of applications was notable and
efforts were made to explain the phenomenon theoretically.
Supporting the theory of Lord Rayleigh (1881), Bell elucidated his observations,
concluding that the principal source for the optoacoustic signal was the mechanical
vibration of the disc that was fixed on the membrane. This mechanical vibration, he
thought, was a result of uneven heating of the beam of light [45]. In the same year,
two more physicists worked on that field. On the one hand, Mercandier shared Bell´s
theory that the reason for the optoacoustic signal was mechanical vibration, on the
other hand, Preece suggested that the optoacoustic effect "is purely an effect of radiant
heat, and it is essentially one due to the changes of volume in vapors or gases produced
by the degradation and absorption of this heat in a confined space"([44]p.517). This
explanation already comes close to modern theories. The first attempt to mathemat-
ically describe the optoacoustic effect was not made until 1973 by Parker [45].
4
3.2 Propagation of Light
Contemporary photoacoustic spectroscopy still works on the same principles as
Bell’s apparatus, but with increased sensitivity by using lasers and microphones. It
can be used for gases, solids and fluids [13]. The actual technical range of appli-
cations of the optoacoustic effect is broad. In environmental engineering, it can be
used to measure the emission of plants and microbes, as well as industrial pollution.
In the field of materials testing, the optoacoustic measuring method is applied rou-
tinely e.g. for the characterization of impurities in semiconductors and to localize
inhomogeneities and cracks in materials [34].
Using radiation-induced ultrasound for biomedical studies was first realized by
Bowen et al. in 1981, utilizing microwaves [35]. Even though in medicine, thermoa-
coustic tomography is yet not utilized, the progress in research shows a high potential.
Sensor methods and reconstruction algorithms for imaging are continuously improving
[5, 10, 42], imaging of small animals and first successful in vivo images were achieved
[52]. Concerning medical applications for humans, thermoacoustic methods have been
tested examining skin vasculatures [29] and breast cancer [32]. Further ideas for the
technical realization are in test stage.
3.2 Propagation of Light
In this chapter, the propagation of light through a medium will be described. The
resulting distribution of light is given by the optical characteristics of the irradiated
medium. These are probabilities for absorption and scattering. The examination of
the absorption is especially important in our case, as this procedure is the reason
for the heating of the media and the resulting generation of a pressure wave. In the
description and explanation of the propagation of light, only the particle nature of
the light will be considered, since wave characteristics like coherence, polarization,
interference and diffraction do not play an important role in the case of optoacoustic
phenomena [34].
5
3 Theoretical principles of Optoacoustics
Figure 3.2: Illustration of the radiation density [34].
3.2.1 Theory of Radiative Transfer
A mathematical ansatz for the description of scattering processes was first developed
by Chandrasekhar [6] for meteorological problems such as the propagation of light in
the atmosphere and in clouds. The aim of the derivation will be to describe processes
like absorption and scattering, as well as phenomena that include both. The theory of
the radiative transfer presented here describes the temporal dependence of the spacial
distribution of light, considering the direction of the beam. This dependence can be
expressed by the radiation density L, which is the power of radiation P per solid angle
dΩ that leaves an area df in direction s. Here the projection of s on the unit vector
~n of the area d~f , which is defined as d~f = ~n · df is considered [34].
The correlations of these vectors are shown in Fig. 3.2. The mathematical definition
of the radiation density is [34]
L :=dP
~s · d~fdΩ. (3.1)
The unit of L is W/m2sr. The irradiance E in a point in space described by ~r can be
defined using the radiation density L:
E(~r) =
∫4π
L(~r, ~s)dΩ. (3.2)
The irradiance has units of an intensity W/m2. Now, how the radiation density tem-
6
3.2 Propagation of Light
porally changes in total within a given volume V for a given beam direction ~s can be
described by
1
clight
∫V
∂L(~r, ~s, t)
∂tdV. (3.3)
This total change of the radiation density in the considered volume V consists of five
components, which form the Radiative Transfer Equation [34]:
The first term describes the part of the radiation density that was lost via the
surface of the volume. The second term describes the part of the radiation density
that was lost due to absorption, where µa is the absorption coefficient. µa describes
the probability for absorption per covered length of path. Its unit is cm−1. For a
purely absorbing medium the Lambert Beer´s Law relates the change of the intensity
I with the absorption coefficient, dependent on the distance z:
dI(z)
dz= −µa · I(z)⇒ I(z) = I0e
−µaz, (3.5)
where I0 is the initial intensity at z = 0. As will be shown in Chapter 3.3, the absorp-
tion coefficient also is important to characterize the transfer from electromagnetic- to
heat- energy.
The third term shows how much radiation density was lost in the considered vol-
ume due to scattering. µs is the scattering coefficient. In analogy to the absorption
coefficient, a relation of µs with the not-scattered intensity Is can be defined as
dIs(z)
dz= −µs · Is(z)⇒ Is(z) = I0e
−µsz. (3.6)
Scattering occurs due to partial reflection, transmission and diffraction, which are
processes that are a result of inhomogeneities of the refractive index n. In biological
tissue, these inhomogeneities are a result of the complex anatomy.
The fourth term in the Radiative Transfer Equation (Eq. 3.4) describes how much
radiation density can be gained in the considered volume due to scattering processes
7
3 Theoretical principles of Optoacoustics
of the surroundings. The function p(~s ′, ~s) gives the probability for a scattering from
a photon that is coming from the direction ~s ′ into the direction ~s in the considered
volume. As this function has a probability character, it has to follow the following
equation[34]: ∫4π
p(cos θ)dΩ = 1, (3.7)
where cos θ = ~s · ~s ′. (3.8)
The Henvey-Greenstein-Function pHG(~s ′, ~s), that was originally derived for the
propagation of light in interstellar nebulae, is also used commonly in tissue-optics:
pHG(~s ′, ~s) = pHG(cos Θ) =1
4π
1− g2
(1 + g2 − 2g cos Θ)3/2, (3.9)
with
g =
∫4π
pHG(cos Θ) cos ΘdΘ, (3.10)
where the parameter g, called anisotropy coefficient, is the average value of the cosine
of the angle of scattering Θ, and can therefore assume values between −1 and 1. For
g = 0, the characteristic of scattering is isotropic in average, which means that an
anisotropy could still exist within the averaged range. If g < 0, a dominant backwards
scattering is the case, if g > 0, the parameter indicates a dominant forward scattering.
In biological tissue, the forward scattering is strongly pronounced.
Considering the size of the particles in relation to the wavelength of the light, two
kinds of scattering can be distinguished. On the one hand, Rayleigh-scattering occurs
when the particle is small in comparison to the wavelength. Then, the scattered light
does not have any preferred direction, but rather comports isotropically (g = 0). On
the other hand, Mie-scattering occurs when the particles are at least the same size
as the wavelength, or bigger. The scattered light propagates dominantly in forward
direction (g → 1).
The fifth and last term ε(~r, ~s, t) of Eq. (3.4) is called source term and describes how
much radiation density the considered volume can gain due to light sources within
the volume. Such could be e.g light that was spontaneously emitted by fluorescence
[34].
To calculate the light transport in inhomogeneous absorbing and scattering media
the use of Monte Carlo Simulations is very common.
8
3.2 Propagation of Light
3.2.2 Diffusion Equation
Approximations are necessary, because the Radiative Transfer Equation is not solvable
precisely by analytical methods. A common ansatz for an approximate solution of
the above equation is to develop L(~r, ~s) and p(~s, ~s ′) with the spherical harmonics
Ylm(Θ, ϕ), so that a system of differential equations of rank 1 (for detailed description
see [33]) is received. Supposing that p(~s, ~s ′) is independent of ϕ, p(~s, ~s ′) can be
developed with Legendre polynomials Pl(cos Θ). The approximations are labeled after
the rank of the term Pl, after which the development is terminated. With the P1-
approximation satisfying results can already be achieved under the condition that
µa << µs(1− g2). This is the case in strongly scattering media, in which the incident
light propagates so diffusely that the irradiance ~E, as well as other functions of Eq.
3.4, lose their dependence on the direction of the incident light beam (~s). This leads
to the diffusion equation, which is a differential equation of rank 2 for the laser fluence
Ψ. Its unit is [Ψ] = J/m2. Ignoring the source term, the stationary diffusion equation
in one dimension along the axis of the laser beam z is [34]
∇ · (Ddiff (~r)∇Ψ(~r)) = −µaΨ(~r), (3.11)
and its solution has a similar form as the Lambert Beer´s law:
Ψ(z) = Ψ0(z)e−µeffz, (3.12)
where µeff is the effective optical energy attenuation coefficient, which is defined as:
µeff =
õa
Ddiff
=√
3µa(µa + µs(1− g)) =√
3µa(µa + µ′s), (3.13)
where Ddiff is the diffusion constant, g is the anisotropy coefficient and µ′s is the
reduced scattering coefficient, defined by the anisotropy factor (µ′s = (1− g)µs). The
laser fluence is a term that is defined for the inside of the medium, as radiant energy
in every point of the space [34]. The definition of the laser fluence not only includes
the light reaching the considered area in the medium from the light source, but also
the scattered light that comes from all directions. Ψ0 is the fluence just underneath
the surface that, due to scattering, can be higher than the incident fluence. This
physical size is not to be confused with the irradiated light, described by the radiant
9
3 Theoretical principles of Optoacoustics
exposure [H(z)] = J/m2. Also for the incident radiant exposure a exponential law can
be defined, in analogy to Lambert Beer´s law:
H(z) = H0e−µaz, (3.14)
whereH0 is the incident radiant exposure at the surface of the absorbing liquid (z = 0).
The energy density is given by the negative gradient of the incident radiant exposure,
W (z) = −dH(z)/dz, or by the product of fluence and absorption coefficient:
W (~r) = µa(~r) ·Ψ(~r) (3.15)
3.3 Generation of Thermoacoustic Waves
3.3.1 Optoacoustic Processes
Many mechanisms exist that lead to the excitation of sound in matter due to inter-
action with laser radiation. In the following, only five of these mechanisms will be
mentioned, where the first two of them are nonlinear effects.
1. Plasma production, caused by a dielectric breakdown produces a shock wave
propagating the medium with supersonic speed. This dielectric breakdown only
happens at laser intensities above 1010W/cm2. It is the most efficient process
for converting electromagnetic energy into acoustic energy. The conversion ef-
ficiency η goes up to 30% for liquids. Unfortunately, this method is not usable
for diagnostic biomedical applications, since the intensities are too high [47].
2. If a threshold, whose value is determined by the thermal properties of the mate-
rial, is exceeded during the generation of acoustic waves, explosive vaporization
sets in. Material ablation occurs, normally accompanied by plasma formation.
This nonlinear effect has a efficiency η of about 1% [47].
3. Electrostriction occurs due to the capability of molecules to be electrically po-
larized by an electromagnetic wave, so that they start moving into and out of
regions with higher light intensity. As a result of these movements, a density
gradient and a following sound wave are generated. In weakly absorbing media,
electrostriction can be an important process [47].
10
3.3 Generation of Thermoacoustic Waves
Figure 3.3: Thermoelastic effect. The illuminated volume absorbs the electromagnetic
energy, which causes a thermal expansion and consequently a pressure field
[10].
4. When light is reflected, absorbed or scattered in a medium a transfer of impulse
of the photons takes place, which results in radiation pressure. In this process,
the radiation pressure itself serves as a sound generating mechanism [47]. This
effect leads to pressures in the range of mbar, but only when the laser intensity
is so high that it would already cause damage in biological tissue. As for ther-
moacoustic tomography laser intensities need to be low, the radiation pressures
are negligible [34].
5. The thermoelastic effect can be described the following way: A short laser pulse
is absorbed in a medium, where a quick heating and thermal expansion take
place. This results in a strain in the body, which relaxes to an acoustic wave.
Its efficiency is rather low (η can reach up to 1 · 10−3%). This effect dominates
the excitation of sound if the laser energies are below the vaporization threshold,
consequently this effect can be utilized for biological applications. Henceforth
the focus will be on the conditions that a system has to fulfill to make the
thermoelastic effect possible and efficient [47].
11
3 Theoretical principles of Optoacoustics
3.3.2 Thermal Confinement and Stress Confinement
The transformation from electromagnetic energy into sound energy is most efficient
when the here presented thermal- and stress- confinement requirements are fulfilled.
The former requires that the laser pulse duration tp is shorter than ttherm, which
means that the time in which the light acts on the medium is shorter than the time for
thermal relaxation. In other words, the time of the deposition of the electromagnetic
energy needs to be so short that the caused enhancement of temperature does not
have enough time to diffuse out of the radiated volume. Expressed mathematically
[49]:
tp < ttherm, (3.16)
ttherm =ρ · cV · δ2
λh, (3.17)
where ρ is the density of the medium, cV is the specific heat capacity, λh is the heat
conduction coefficient and δ is the characteristic length of the radiated volume, also
called penetration depth. In purely absorbing and optically homogeneous media, δ
is identical with the reciprocal of the absorption coefficient µa, whereas in scattering
media, δ equals the reciprocal of the effective attenuation coefficient µeff , which is a
combination of absorption and scattering (see Eq. (3.13)).
The stress confinement demands that the time tp is shorter than the time that the
volume would need to thermally expand. This means that the laser pulses have to be
so short that the radiated medium does not have the possibility to react mechanically
on the energy of the light [41]. This time shall be called acoustic relaxation time tac[49]:
tp < tac, (3.18)
tac =δ
c, (3.19)
where c is the speed of sound. This confinement guarantees that during the radiation
of light, no pressure equalization of the volume with its surrounding occurs. In general,
the heat conduction is not as fast as the acoustic propagation, so that the fulfillment
of the stress confinement is the superordinate condition [41].
To estimate the maximum time that a laser pulse might have, ttherm and tac are
calculated for the case of water. Its thermal and acoustic characteristics come close
12
3.3 Generation of Thermoacoustic Waves
to those of biological tissue, therefore serving as a good estimation [49]:
ttherm =1g/cm3 · 4.187J/K · g · (1 · 10−2cm)2
6 · 10−3WK · cm= 7 · 10−2s, (3.20)
tac =1 · 10−2cm
1.5 · 105cm/s= 6.7 · 10−8s. (3.21)
For tp < tac < ttherm the pulse duration needs to be in the range of some 10ns.
3.3.3 The Correlation of Incident Light and Generated
Pressure
Having explained how light propagates in media and the conditions for the generation
of thermoacoustic waves, the focus of the following will be on the correlation of the
incident light and the generated pressure.
When radiating a light absorbing medium and depositing an energy Q in a volume
V , the medium heats up (∆T ). This increase of temperature is directly proportional
to the deposited energy and inversely proportional to the volume, the density ρ of the
medium and the specific heat at constant pressure cp [49]:
∆T =Q
V · cp · ρ. (3.22)
Using the energy density W (~r) for the energy per volume yields
∆T =W (~r)
cp · ρ. (3.23)
In the general case of gases and liquids, the following relation can be given between
increase of temperature and resulting pressure:
p = −1
κ
(∆V
V
)+β
κ∆T, (3.24)
where β is the cubic expansion coefficient of the medium and κ is its compressibility.
As the stress confinement requires that during the deposition and absorption of the
energy the volume needs to stay constant (∆V = 0) at the time zero
p0 =β
κ∆T (3.25)
13
3 Theoretical principles of Optoacoustics
is valid [49]. Substituting Eq. (3.23) into former equation the relation between the
radiated energy density and the generated pressure becomes apparent:
p0(~r) =β
κ
W (~r)
cp · ρ= Γ ·W (~r). (3.26)
Γ is the dimensionless Grueneisen coefficient, which gives the relation of generated
pressure to the incident radiant exposure. The heating of the medium depends on its
absorption characteristics, therefore, this characteristic (Eq. 3.15) must be included
in the correlation of incident light and generated pressure.
W (~r) = µa(~r) ·Ψ(~r) (3.27)
Substituting former equation into Eq. (3.26) yields [40]
p0(~r) = Γ · µa(~r) ·Ψ(~r). (3.28)
In general, the pressure distribution is given by the distribution of the energy den-
sity. In homogeneous and purely absorbing media, the energy density follows the
distribution of the incident radiant exposure [40].
3.3.4 Thermoacoustic Wave Equation and Solution
So far, the photoacoustic pressure distribution at the time zero (Eq. 3.28) that was
caused by instantaneous heating has been derived. p0 is the source for the following
propagation of a pressure wave, that can be described by the inhomogeneous wave
equation [40]:
∇2ψ(~r, t)− 1
c2∂2ψ(~r, t)
∂t2=
β
ρcpS(~r, t), (3.29)
where ψ is the velocity potential and S(~r, t) is the heat source term that describes the
heat that is generated per unit time and per unit volume and has the unit [S(~r, t)] =
W/m3. As the fulfillment of the thermal confinement, which includes instantaneous
heating, can be assumed, S(~r, t) can be written as the product of the energy density
and a temporal Dirac delta function δ(t):
S(~r, t) = W (~r)δ(t), (3.30)
and substituting Eq. (3.27) for the energy density, the heat source term can be written
as
S(~r, t) = µaΨ(~r)δ(t). (3.31)
14
3.3 Generation of Thermoacoustic Waves
Using former equation, a term for the velocity potential ψ(~r, t) is received, which will
further on serve as the fundamental quantity to derive a solution of the wave equation:
ψ(~r, t) = − t
4πρ
βc2
cp
∫ ∫R=ct
W (~r ′)dΩ, (3.32)
derived from the Green´s function solution of Eq. (3.29). The position vector ~r defines
the point of observation, while ~r ′ defines the point where the source of the pressure
wave is located. When c is the speed of sound in the medium and t is the time of flight
from the pressure source point (~r ′) to the observation point (~r), then R = ct describes
the radius of the sphere around the observation point, over which the integration is
carried out. dΩ is the solid angle element.
The next step in the derivation is to find a correlation between the pressure and the
velocity potential. The pressure p(~r, t) can be derived from the velocity potentialψ
[40]:
p(~r, t) = −ρ∂ψ(~r, t)
∂t. (3.33)
Using Eq. (3.33),(3.32) and (3.26) yields
p(~r, t) =∂
∂t
[t
4π
∫ ∫R=ct
p0(~r′)dΩ
], (3.34)
or
p(~r, t) =βc2
2πcp
∂
∂t
[t
∫ ∫R=ct
W (~r ′)dΩ
]. (3.35)
These equations allow the calculation of the pressure distribution, depending on
the initial energy distribution. The integral can be calculated analytically for geo-
metrically simple sources. Otherwise numerical integrations need to be carried out
[41]. If the laser pulse duration is finite, a convolution of the pressure signal with the
temporal pulse profile is necessary. If the stress confinement is fulfilled (in the range
of 10ns, see Chapter 3.3.2), the pressure signal only suffers negligible changes due to
the laser pulse duration.
15
3 Theoretical principles of Optoacoustics
3.4 Propagation of Acoustic Waves
Henceforth, effects that influence the propagation of acoustic waves, whose generation
was described in the previous section will be considered. Such effects are sound reflex-
ion, refraction, absorption, dispersion, diffraction and in the case of finite amplitudes,
nonlinear acoustic effects.
3.4.1 Reflexion and Refraction
The speed of sound c is dependent on the compressibility κ and the density ρ of the
medium [53]:
c =1√κρ. (3.36)
In analogy to the propagation of light, refraction and reflexion of acoustic waves at
the boundary of two media with the speeds c1 and c2 can be described with
θi = θr, (3.37)
sin θisin θt
=c1c2, (3.38)
where θi is the angle of incidence, θr the angle of reflexion and θt the angle of the
transmitted, refracted acoustic wave.
The intensity Ir of the reflected wave depends on the impedances Z1 and Z2 of the
two media. For the case of normal incidence the intensity is [53]:
IrIi
=
(Z2 − Z1
Z2 + Z1
)2
, (3.39)
where the impedance is given by Z = ρc.
3.4.2 Sound Absorption and Dispersion
The absorption of sound in liquids is determined by the viscosity σ and the thermal
conductivity k. The absorption coefficient is [47] given by
α =8πν2
3c3ρ
[σ +
3
4
(cpcV− 1
)k
cp
], (3.40)
where ν is the frequency, and cp and cV are the specific heats at constant pressure
and at constant volume, respectively. For most liquids, the second term in the square
16
3.4 Propagation of Acoustic Waves
brackets is negligible. The ν2 dependence of the absorption that causes a waveform
distortion is important for all kinds of liquids. An acoustic pulse broadens while
propagating through media as its higher frequency components experience a higher
absorption [47]. In the case of ultrasonic waves, the frequency is in the range of
20kHz to 1GHz and the distances are in the range of several cm. This distortion of
the acoustic waveform can be neglected.
Furthermore, the waveform can be influenced by dispersion when the speed of sound
in a medium is dependent on the frequency of the wave. Sigrist comments that for
ultrasound frequencies the dispersion is negligible. Additionally, d’Arigio points out
that for H2O no dispersion is detectable down to minus 20C [47].
3.4.3 Diffraction of Acoustic Waves
During propagation through media, diffraction distorts the geometrical and temporal
wave profile of an initially plane acoustic wave. Such a plane acoustic wave can be
obtained by radiating a medium of one-dimensional or layered structure with a laser
beam, whose radius is much larger than the optical propagation depth of the medium.
Then, the initial distribution of the energy density exclusively depends of the distance
z of the surface. Integration of Eq. (3.34) yield the solution of a plane wave [41]:
p(z, t) =Γ
2[W (z − ct) +W (z + ct)] . (3.41)
The first term in the square brackets describes a wave in positive z-direction, whereas
the second term describes the wave propagating in negative z-direction. The formula
is only true for acoustically matching materials at the boundary at z = 0, which
guaranties a reflexion free transition.
In the context of diffraction, the wavelength of the acoustic wave λac is the crucial
parameter. It is necessary to distinguish two cases [47]:
• When the penetration depth δ = 1µa
of the laser beam into the liquid is shorter
than the distance z = ctp, the acoustic wavelength is given by λac = 2ctp, where
tp is the laser pulse duration.
• When the penetration depth is longer than z = ctp, the acoustic wavelength is
given by λac = 2µa.
17
3 Theoretical principles of Optoacoustics
Figure 3.4: a) Near field b) Far field [41]
Diffraction limits the generation of plane thermoelastic waves, as it occurs at the
border of the source volume, where the pressure was induced, and respectively at the
border of the laser beam. In z-direction, the distribution of the energy density (W ) is
given by the Lambert Beer´s law. In r-direction, the energy density is nearly constant
and a high gradient at the borders is the source for diffraction. Consequently, the
initially plane wave becomes spherical. In the case of a constant laser intensity across
the beam diameter, the influence of diffraction can be described by the diffraction
parameter [41, 47]
D =
∣∣∣∣zλaca2
∣∣∣∣ , (3.42)
where λac is the acoustic wavelength, z is the distance between sensor and source
and a is the radius of the laser beam. For D < 1 the measurement happens in the
near field, for D > 1 in the far field [41]. In the case of homogeneous absorbers, the
acoustic wavelength is λac = 2/µa, where µa is the absorption coefficient [41].
The boundary between near and far field is located at the propagation distance [40]
zf =D2
4λac. (3.43)
This equation is only valid for acoustic wave lengths that are much smaller than the
diameter of the source.
Fig. 3.4.a shows that if the detector is close to the source, the sphere of integra-
tion with the radius R = ct (see Eq. (3.35)) will reach the depth borders of the
18
3.4 Propagation of Acoustic Waves
source, which are given by the acoustic wave length, but will not reach the lateral
borders. Therefore, the detector measures a planar thermoelastic wave, as no in-
formation originating of the boundaries is received. The detected pressure signal is
directly proportional to the absorbed energy.
Fig. 3.4.b shows the case of a far field measurement. The part of the sphere that
intersects the source is nearly a plane, so that the detector measures the temporal
derivation of the plane wave signal. Therefore, the direct proportionality of pressure
signal and energy distribution is not valid anymore.
Using the near field measurement, a good representation of the depth distribution of
the absorbed energy can be achieved, whereas far field measurements can be utilized
to obtain defined borders of structures with different absorbing characteristics [41].
3.4.4 Nonlinear Acoustic Effects
The nonlinearity parameter B/A, which was introduced by Beyer, gives information
about the nonlinearity of a liquid. Sigrist [47] presents a novel method to determine
B/A. Whereas in previous measurements, one single acoustic pulse was recorded at
several distances of the source, he analyzed numerous pulses of different amplitudes
at the same distance. Therefore, as diffraction effects are dependent on the propa-
gation distance, he could avoid a simultaneous detection of both effects. He could
observe that the wavefronts steepen due to nonlinear effects. This steepening can be
interpreted as an increase of the propagation speed [47]:
c = c0 +p
ρ0c0+B/A · p2ρ0c0
. (3.44)
The first term describes the initial speed of sound for small amplitudes, the second
one the particle velocity and the third term the influence on the speed due to nonlinear
effects. Henceforth, the steepness of the wavefront rises with the pressure, the Beyer
parameter and the distance z.
19
3 Theoretical principles of Optoacoustics
20
4 Piezoelectricity and Piezoelectric
Sensors for TACT
The focus of this chapter lies in the piezoelectric effect. After a general overview,
theoretical descriptions and calculations of special cases of the effect will be presented.
Finally, the attention will be given to the construction and features of the piezoelectric
line sensor that is used in the here presented experiments.
4.1 Principles of Piezoelectricity
The piezoelectric effect describes the interplay of mechanical pressure and electrical
voltage in crystals. The word piezo derives from the Greek word piezein, which means
to press. This effect is based on the phenomenon that a directed deformation of certain
materials causes electric charges on the surface of the material, which is called direct
piezoelectric effect. Then, the barycenter of charge is displaced, and, in consequence,
microscopical dipoles emerge within the unit cell. The summation of these dipoles
over all unit cells of the crystal leads to a macroscopically measurable voltage [1]. In
1880, Jacques and Pierre Curie discovered that in tourmaline crystals, the applied
pressure and the resulting voltage were directly proportional.
In contrast, certain crystals are deformed by a voltage that is impressed on them
(inverse piezoelectric effect). In physics, the piezoelectric effect is at the intersection
of electrostatics and mechanics [9].
Fig. 4.1 shows the most famous piezoelectric crystal, Quartz (SiO2). Every Si-atom
is situated in the center of a tetrahedron of oxygen atoms. The Si-atoms have four
positive elementary charges. The oxygen atoms have two negative charges. Therefore,
the quartz crystal is neutral in total [1].
21
4 Piezoelectricity and Piezoelectric Sensors for TACT
Figure 4.1: a) Quartz crystal [15]. b) Deformation and movement of charges at a
quartz crystal [37].
Having already distinguished between direct and inverse piezoelectric effect, another
differentiation is made concerning the direction in which the force, and the electric field
act. It is called longitudinal piezoelectric effect, if the pressure acts parallel to a polar
axis of the crystal. In contrast, the transversal piezoelectric effect (Fig. 4.1) occurs
when the force acts parallel to a neutral axis. If the force acts in direction of the optical
axis, no piezoelectric effect is measurable [15]. Similarly, the same principles apply to
the inverse piezoelectric effect, which is also classified as longitudinal or transversal,
depending on the direction of the electromagnetic field. Two more characteristics of
the piezoelectric effect shall be mentioned at this point:
• Inverse and transversal piezoelectric effect always occur together. When a crys-
tal is mechanically deformed transversally, an electric field is generated, which,
because of the inverse effect, causes a secondary deformation of the crystal,
which acts against the principal force. This secondary effect is negligible in the
most cases.
• The signs of the piezoelectric effect change when the causation changes its sign.
If e.g. pressure changes to tension, the polarity of the charges change.
The piezoelectric effect can only appear in non-conductive materials. In crystals, a
lack of center of symmetry is the criterion for piezoelectricity [14]. If the crystal
has several polar axes, it is piezoelectric, whereas if it has one single polar axis, the
material is pyroelectric, which means that it can polarize itself spontaneously. By
22
4.1 Principles of Piezoelectricity
changing the temperature of the material, this effect can be enhanced and the surface
charges can be detected [30].
It is common to mathematically describe the piezoelectric effect by the following
coupled equations [14]:
Di = dij · Tj + εTj
ij · Ei and (4.1)
Sj = dij · Ei + sEiij · Tj, (4.2)
where Di is the dielectric displacement, Tj is the mechanical stress, Ei is the electrical
field strength, Sj is the strain, εTj
ij is the permittivity at constant or zero mechanical
stress and sEiij is the elasticity modulus at constant or zero electrical field strength.
The most important material parameter for the piezoelectric effect is the piezoelectric
constant dij that describes the correlation between the electric field strength and the
strain of the material [14]. The index i corresponds to the direction in which the
electric field acts, whereas the indexj gives information about the direction of the
mechanical force [8].
Direct Piezoelectric Effect
Focusing again on Eq. (4.2)and (4.1), the first formula describes the direct piezoelec-
tric effect, where the first term of the formula describes the primary effect, i.e. that
the acting pressure or the mechanical stress causes a dielectric displacement. The
generated electric field E3 again causes a stress on the material that is orientated
in opposite direction to the initial pressure/stress (inverse piezoelectric effect). This
secondary effect is negligible compared to the primary effect. Depending on whether
the effect is lateral or transversal, either d31 or d33 is the constant of main interest.
Inverse Piezoelectric Effect
For the inverse transversal piezoelectric effect, the coupled equations (Eq. (4.2))
reduce to the form
S1 = sE311 T1 + d31E3, (4.3)
where the mechanical force and the electric field strength are normal to each other.
The following formula is used to describe the longitudinal case, where mechanical
23
4 Piezoelectricity and Piezoelectric Sensors for TACT
force and electric field strength act in parallel directions [12]:
S3 = sE333 T3 + d33E3. (4.4)
4.2 The Piezoelectric Line Sensor
At the beginning of this section, the use of PVDF films for sensors and the character-
istics of the used PVDF film will be treated. Then, the construction of the sensor that
was used in the experiments will be illustrated. At the end of this section, attention
will be given to details of the sensor and the calculation of its specifications.
4.2.1 Piezoelectric PVDF Films
Polyvinylidene fluoride (PVDF) films are commonly used as transducers. The field of
applications is wide: infrared sensors, stress gauges, vibration detectors, etc. PVDF
was first created in 1969 . The polymer PVDF has to be prepared to obtain piezo-
electric properties. The film is heated, stretched, and, at the same time, polarized
in an electrical field. The dipoles align with the electrical field. Then, the PVDF
film is cooled down below the Curie-Temperature (80C). At this temperature, the
film stays electrically polarized. If, in the polarization process, the PVDF film is
stretched only in one defined direction under the influence of the electrical field, the
piezoelectric coefficient is higher in this direction. These films are called uniaxial.
Biaxial films have been stretched with the same force in two directions at the same
time. In general, the piezoelectric properties of PVDF films weaken over time due to
aging processes [38].
The thickness of PVDF films is between 6µm and 1mm. The surface coating of
aluminum has a thickness of between 150Å [27] and 500Å [34]. The coating is needed
to contact the piezoelectric material and to pick off the charges. The utilized PVDF
films are polarized in z-direction. Table 4.1 shows important characteristics of the
used PVDF film.
The piezoelectric specifications are affected by temperature influences. After 100
days at room temperature, a depolarization takes place and d33 suffers changes up
to 5%. The same 5% distortion is noticeable after only one day at 60C. Moreover,
24
4.2 The Piezoelectric Line Sensor
thickness of the film dfilm: 25µm [48]
piezoelectric constant d33
for electric fields and mechanical
strain in direction of d: 16pC/N at room temperature [48]
piezoelectric constant d31 = d32
for mechanical strain normal to d: 8pC/N at room temperature [48]
relative dielectric coefficient εr: 11 at 1000Hz [48]
speed of sound c: 2000m/s [27]
density ρ: 1800kg/m3 [27]
elasticity modulus s: 2000MPa [48]
Table 4.1: Important Specifications of PVDF Films.
PVDF is not only piezoelectric, but also pyroelectric. For this reason, it is recom-
mended to take care during measurements that no laser light is radiated directly on
the PVDF surface, since electrostatic induction charges caused by thermal differences
would also show a signal [27].
In the following, some of the advantageous characteristics of PVDF films that make
this medium so suitable for the use as an ultrasound detector will be discussed:
Mechanical Capacitance
PVDF films are very flexible and robust. In lateral direction, these films do not
disrupt under pressures until 180MPa. This value is similar to the capacitance of
bones and hair. The elasticity modulus (see Table 4.1) is ten times higher than of any
biological tissue. This characteristic is important, as laser induced pressure waves can
reach high amplitudes. The mechanical capacitance of PVDF films is sufficiently high
to use PVDF films as pressure sensors for measurements on biological tissues [34].
Acoustic Impedance
Comparing PVDF with other piezoelectric materials, the acoustic impedance is low.
The film used for this work has an impedance of
ZPV DF = c · ρ = 3.6 · 106kg/m2s, (4.5)
25
4 Piezoelectricity and Piezoelectric Sensors for TACT
Figure 4.2: Propagation of acoustic waves through the sensor layers [26].
which is the same order of magnitude as the impedance of water (ZH2O = 1.48 ·106kg/m2s) and biological soft tissue (Ztissue = 1.3 to 1.6 · 106kg/m2s). Due to this
similarity, the acoustic matching conditions that guarantee that a sufficient part of
the pressure amplitude can be transmitted into in the film are provided [34]. If the
the acoustic impedances were of a different order of magnitude, a significant part of
the signal would be reflected at the surface of the sensor and would be lost for the
measurement. In these experiments, the sensor has direct contact to water. The
reflectivity R for the acoustic wave between water and the piezoelectric film is given
by
R =ZPV DF − ZH2O
ZPV DF + ZH2O
=3.6 · 106kg/m2s− 1.48 · 106kg/m2s
3.6 · 106kg/m2s + 1.48 · 106kg/m2s= 0.417. (4.6)
The transmissivity between water and the PVDF film is
T =2 · ZPV DF
ZPV DF + ZH2O
=2 · 3.6 · 106kg/m2s
3.6 · 106kg/m2s + 1.48 · 106kg/m2s= 1.42. (4.7)
Furthermore, it is necessary to consider the transition of the acoustic wave (wave
length λac) from the PVDF film (thickness d = 25µm) to the material that lies behind
the PVDF film in the sensor (Z3). Fig. 4.2 shows the transition of the wave, arriving
at the sensor with an incident angle ϕi. For the case λac < d, the sensor layer is
acoustically transparent. Hereby, the pressure amplitudes before and after the film
are the same. The pressure measured inside the PVDF film is dependent on the
26
4.2 The Piezoelectric Line Sensor
characteristics of the bordering media [26]:
pPV DFpH2O
=p3
pH2O
=2Z ′3
Z ′H2O+ Z ′3
, (4.8)
Z ′i =Zi
cosϕi. (4.9)
This leads to the conclusion that, if the bordering materials have the same acous-
tic impedance, the pressure measured with the PVDF film (pPV DF ) is the same as
the incident, initial pressure. Moreover, it is independent of the incident angle [26].
For this reason, Plexiglas (Polymethylmethacrylat, PMMA), which has an acoustic
impedance of Z3 ≈ 3.2 · 106kg/m2s (value from [18]), was used as the third medium.
4.2.2 The Construction of the Sensor
Preparation of the PVDF Film
After planning the sensor, the first practical step in the construction is the preparation
of the PVDF film. The required geometric form (a rectangle of 4cm × 7mm) of the
film is cut out of the A4-sheet with a sharp razor blade (Fig. 4.3.b). It is essential to
take care that during the cutting of the film the Al-coatings do not come into contact
with each other. Before going on to further steps, it is advantageous to check the film
with a multimeter.
Then, a mask out of adhesive tape is prepared. For this purpose, a rectangle is
drawn onto the tape, which is then applied to a metal bar, where it then is cut with
a razor blade. Thereupon, the mask is removed from the metal bar and fixed on the
film (Fig. 4.3.c), and the two pieces together are placed onto a piece of paper. Now,
one drop of FeCl3 is put on the small rectangle that is to be etched, and the liquid
is evenly distributed using a cotton swab. After several seconds, the FeCl3 is washed
off the PVDF film with distilled water. Finally, the mask is removed from the film
carefully (Fig. 4.3.d).
Realizing the Line
The active surface of the PVDF film is the part of the film where both sides of the film
are in contact to a conducting material. In the case presented here, a line geometry
27
4 Piezoelectricity and Piezoelectric Sensors for TACT
Figure 4.3: Steps in the construction of the sensor.
Figure 4.4: Real picture of the piezoelectric line sensor.
28
4.2 The Piezoelectric Line Sensor
of the sensor is to be obtained. One possibility for realizing a thin line would have
been to remove all of the Aluminum-coating except for the sensor´s contact line. This
could have been achieved either by etching or by ablating with laser pulses. The
disadvantage of these methods is that especially for thin lines under ≈ 300µm width
the electrical contact is easily disrupted.
The chosen method is to first remove the Aluminum on one side of the PVDF
film, as described in Chapter 4.2.2, and then use the quoin of a thin Aluminum
film (200µm) as the electrode. The Aluminum film is pressed between two blocks of
Plexiglas. The surface of the whole block is polished to achieve an even surface (Fig.
4.3.e). Now, the rectangular strip of the piezoelectric material can be glued onto the
Al-line (Fig. 4.3.f). For gluing, a two-component adhesive is used. Is is important
to apply a very thin coating of adhesive and to evenly press the PVDF film onto
the Plexiglas. Moreover, wrinkles of the film should be avoided, and a continuous
end-to-end contact with the electrode is to be obtained. The depth (z-direction) of
the Plexiglas block is chosen to be at least in the range of severalcm. Hereby, pressure
waves that are reflected at the back end of the block would arrive late enough to be
distinguishable from real signals. A detailed explanation of the reasons for the sizes
of certain components of the sensor will be given in Chapter 5.
Achieving Waterproofness
Since in the experiment, the sensor is situated in a water tank, it is necessary to
guide the electric conductors through a waterproof metallic cage which also provides
shielding against electrical noise. One conduction is connected with the Al-film inside
the cage, the second conduction is the cage itself, to which the Al-coated upper side
of the PVDF film is connected. The whole block of Plexiglas, Al-film and PVDF film
is pressed into a cage and sealed with silicone. A tube connected to the cage-box
allows the sensor to be fixated for the experimental set-up and connects the sensor
with a coaxial cable to the amplifier and to the oscilloscope. The cage is equipped
with additional possibilities to screw on further kinds of tubes, so that the sensor is
applicable in various experimental set-ups. Fig. 4.4 shows the constructed sensor.
29
4 Piezoelectricity and Piezoelectric Sensors for TACT
4.2.3 Electric Characteristics of the Sensor and Sensitivity
In this section, the focus of attention will be set on the electric characteristics of the
signal generation, dependent on the features of the used sensor. The aim is to be
able to calculate back from the obtained voltage signal to the original pressure wave
amplitude that arrived at the sensor. By following the pressure wave through the
sensor, the important components that influence the signal can be determined.
Focusing again on Fig. 4.3.a, it is obvious that the PVDF film acts as a plane-
parallel capacitor. Its capacitance is given by [34]
Cfilm = ε0εr ·Afilmdfilm
, (4.10)
where ε0 is the permittivity of free space, εr is the relative dielectric coefficient, Afilmis the active area of the PVDF film (the line), and d the thickness of the dielectric, in