Study of the temporal and spatial dynamics of plasmas
induced in liquids by nanosecond Nd:YAG laser pulses.
1: Analysis of the plasma starting times
Franco Docchio, Pietro Regondi, Malcolm R. C. Capon, and John Mellerio
We report on a theoretical and experimental study of the temporal and spatial dynamics of plasmas produced
in liquids by single Nd:YAG laser pulses of nanosecond duration.
This study was motivated by the increasing
attention paid to the phenomenon of optical breakdown and to its related effects on tissues and media in
connection with microsurgicaltechniques developed for ophthalmology and urology. Streak camera recordings of the emissionfrom laser-induced plasmas were taken in distilled and tap water in controlled irradiation
conditions. From streak recordings, plasma starting times as a function of the axial distance from focus, the
overall length of the plasma column,plasma lifetimes, and plasma absorption were derived and analyzed. In
this first paper we analyze the curves of plasma starting time, as a function of the irradiation parameters and
of the properties of the medium. We show that a model obtained by upgrading the theory of the moving
breakdown allows accurate interpretation of the experimental observations.
1L Introduction
In recent years there has been an increasing number
of novel, noninvasive surgical applications of short
pulsed, high power laser pulses in the near infrared.
The Nd:YAG laser, both in the Q-Switching or in the
mode-locking regime, is by far the most popular of the
laser sources for such applications due to its versatility.1"2 Most surgical techniques are based on the phe-
nomenon of dielectric breakdown, induced in transparent or pigmented media and tissues by the intense
electric field associated with the laser pulse. Dielectric breakdown is commonly associated with plasma
formation, 3 ' 4 shock wave propagation, 5 - 8 and particle
ejection.9 In most cases it is the mechanical energy
released from the plasma volume that accounts for
tissue disruption, e.g., thickened lens capsules in the
eye 0 'll and the fragmentation of ureteric and gall
bladder stones.12
Malcom Capon is with Institute of Ophthalmology, Department of
Clinical Ophthalmology, London WC1H 9QS, U.K.; J. Mellerio is
with Polytechnic of Central London, Schoolof Paramedical Studies,
London W1M 8JS, U.K.; the other authors are with UniversitA degli
Studi, Dipartimento di Automazione Industriale, 25060Mompiano,
Brescia, Italy.
Received 29 October 1987.
0003-6935/88/173661-08$02.00/0.
©1988 Optical Society of America.
The increasing importance of the microsurgical use
of dielectric breakdown has rekindled research on the
interaction between high electric fields, oscillating at
optical frequency, and dielectric media, both in the
liquid and solid phase, as well as at interfaces. Of
particular interest are studies of thresholds for breakdown in different media,1 314 of the absorption and
emission properties of the plasma,1 51 6 5 and of the
evolution and propagation of shock waves8"17 in conditions that closely approach those that are found in the
clinical situation.
We report here on a series of experiments on the
spatial and temporal evolution of plasmas induced in
liquids by nanosecond Nd:YAG laser pulses. The experiments were performed using a streak camera with
nanosecond temporal resolution, which also had options for high speed photography. The aim of these
experiments was to study, both spatially and temporally resolved, the (i) evolution of the laser-induced
plasma and (ii) distribution of the recombination luminescence from the plasma volume during the lifetime of the spark, which also gives information on the
dynamics of light absorption by the plasma column.
Streak analysis of laser-induced plasmas has been
performed in the past mainly in gases.'8 -12 In contrast, our study of time- and space-resolved plasma
streaks is, to our knowledge, the first performed in
liquids. Also, in the context of previous experimental
work, two aspects of this study are worthy of consideration. The first is that streak analysis has been performed at a large variety of irradiation conditions.
1 September 1988 / Vol. 27, No. 17 / APPLIEDOPTICS
3661
Second, the use of a relatively short pulse duration
allowed us to make use of a simple theoretical model to
explain the results.
In this first of two papers we present the results of
streak measurements with liquids accepted as models
for ocular media, such as distilled water (a rather im-
purity-free, high breakdown threshold medium), and
tap water (an impurity-rich, low breakdown threshold
medium). We discuss the relation between the starting time of the plasma and its position along the beam
propagation axis by proposing a theoretical model,
which is essentially an upgrading of the moving breakdown model. 2 2
The use of this model allowed precise interpretation
of the experimental results: in particular, it allowed
us to draw a simple relation between the overall length
of the plasma column with respect to the irradiation
conditions and the beam geometry, in good agreement
with the observations.
In the companion paper2 3 we analyze, with the aid of
the moving breakdown model, the time- and spaceresolved light emission characteristics of the plasma.
11. Experimental Apparatus
The experimental apparatus used in the experiments is shown in Fig. 1. The laser system used to
perform the experiments was a LASAG Topaz photodisruptor, an actively Q-switched Nd:YAG laser able
to deliver 12-ns (FWHM) long pulses in a TEMoo mode
configuration. Built into the system were enlarging
optics and the focusing optics which produced a spot of
-13-tm diameter at l/e2 in the liquid (standard focusing).
An additional
lens of 50-mm focal length was
sometimes added to further reduce the focal diameter
to -11-Am (tight focusing). Breakdown was produced
at the center of a quartz cuvette filled with either
doubly distilled water or with tap water.
The spark produced was magnified by a factor of 14
and imaged onto the photocathode of the streak camera. The streak camera/high speed photography system was an HE-700 Imacon image converter system
(Hadland Photonics, Hemel Hempstead, U.K.). The
photocathode was of the Si type, insensitive to any
laser light scattered from the breakdown region. The
streak camera was triggered from the driving pulse of
the laser Pockels cell via a suitable high performance
delay generator (TD3, Hadland Photonics). Streak
rates of 1 or 2 ns/mm at the plane of the Polaroid film
were used. The exposure parameters of the film were
carefully chosen to obtain good linearity at each streak
rate.
High speed photography measurements, preliminarily performed to determine the propagation speed
of the shock wave front, were made using the same
apparatus, as shown with the additional light source
shown in Fig. 1, configured for high speed framing (20
X 106 frames/s). Illumination of the sample was provided by a flashlamp with proper pretriggering with
respect to the laser shot. Also preliminary to making
streak records, the threshold parameters for optical
breakdown in the two media and with the two focusing
3662
APPLIEDOPTICS / Vol. 27, No. 17 / 1 September 1988
~~~~~~~~~~Streak
I
r-L_
-
l
Ob.
P
RamIng
.e~
O
P.C
uartz CellI
Fig. 1. Experimental setup for streak measurements of plasmas
induced in liquids by Nd:YAG laser pulses of 12-ns FWHM dura-
tion. The additional setup used for high speed framing is shown
with dotted lines.
Legends are S.O., schematics of the slit optics;
Obj, expansion objective;P.C., photographic camera.
conditions were determined. Breakdown probability
curves'4 were derived by plotting the number of breakdown events occurring over a large number of shots,
against the energy input. From these curves, the
threshold irradiance values could be estimated: we
defined threshold as that irradiance which produced
breakdown with a probability of 100%. The calibration of the energy meter in the laser instrument was
checked against an EG&G series 581 radiometer. Distilled water and tap water showed markedly different
threshold energies (2.4 and 0.9 mJ, respectively, corresponding to irradiances at the focus of 166 and 61 GW/
cm2 in the standard focusing conditions, and 2.2 and
0.8 mJ, corresponding to 195 and 70.2 GW/cm2 , in the
tight focusing conditions) due to the different impurity content.
The pulse energy was increased from just below
threshold up to the maximum of 15 mJ. At each
energy value a set of streak pictures were recorded and
analyzed.
Ill.
Results
A typical set of streaks obtained for distilled water
for three different values of the input energy of the
Nd:YAG laser, namely, 3, 6.3, and 12 mJ, is shown in
Fig. 2. The plasma produced at 3 mJ, a value corresponding to 1.2 times the threshold irradiance in the
given focusing conditions (focal plane spot size 13 ,im,
standard focusing) occurred consistently at the center
of the focal volume [Fig. 2(a)].
No appreciable wan-
dering of the position of the spark between shots was
seen. At these near threshold conditions, the spark
had a minimum axial length (-60 im). The duration
of the plasma luminescence could be estimated from
the streak to be -10 ns.
A typical spark produced at pulse parameters corresponding to 2.6 times the threshold value [Fig. 2(b)]
was increased in length with respect to the previous
case, originating at the focus and directed back toward
the incident beam. Elongation beyond the focal plane
was seldom observed. The pattern of the streak is
characterized by an abrupt start of the plasma luminescence at every position along the axis of the incoming beam z.
The instant of occurrence of breakdown at higher z
in the plasma streak of Fig. 2(b) is seen to be delayed
300_
200
a)
04
o: _
ri
E
0
0
5
10
00
0
0
E
0
300
F
200
L
100:
0
0
0
b)
_
b)
C)
:0
I,
0
0
I
5
-
I
10
15
x
a'
300 -
15 ns
200
Fig. 3. Streak photographs of plasmas produced in tap water. La-
100
ser parameters are as in Fig. 2: (a)
10 I
II
0
8
I
10
I
185
Fig. 2. Streak photographs of plasmas produced in distilled water
by a 12-ns TEMoo mode Nd:YAG laser for different values of the
parameter A which is defined as the ratio of the peak power, Pma to
the peak power at threshold Pth (see text for the definition of Pth).
The spot size at the focus is 13 ,um. Laser light enters the plasma
(a)
=
1.2, (b)
3=
2.6, (c)
=5.0.
with respect to the instant of breakdown occurrence at
lower z. The figure clearly shows a regular shape of
the locus of the instants of breakdown occurrence vs
the axial coordinate. This curve can be defined as the
plasma starting time curve. The top of the plasma
streak of Fig. 2(b) is almost flat, showing that the
expansion of the spark toward the laser source, after
the end of the pulse, is limited. Due to the flatness of
the top of the spark, the maximum length of the plasma column,
Zmax,
could be easily measured.
By contrast with the stepwise luminescence onset, at
a given z the decrease of the luminescence tails off
more or less smoothly. An interesting feature, which
is also observed in gases,2 3 is that the duration of the
plasma emission is shorter near the focal point and
longer upstream.
= 5.0, (c)
= 10.0.
time curve, a flat streak top, an axial location-dependent luminescence lifetime, are also observed here.
A set of plasma streaks obtained in tap water with
irradiation parameters similar to those of Fig. 2 are
I
20
TIme (11)
region from the top of the picture.
= 1.2, (b)
Moreover, comparing Fig. 2(a) with
Fig. 2(b) it is easy to observe that the luminescence of
the streak at the focus is longer when no plasma exists
in the upstream position.
As the energy is further increased up to five times
the threshold value [Fig. 2(c)], a still increased elonga-
tion of the plasma column toward the incoming beam is
observed. All the other features observed in the case
of the plasma of Fig. 2(b), i.e., a regular plasma starting
shown in Fig. 3. Near threshold, a single tiny spark is
observed as in the case of distilled water [Fig. 3(a)].
However, there is appreciable wandering of the location of the plasma between pulses over a region of 150
gim. With increasing energy [Figs. 3(b) and (c)], the
sparks are characterized by the presence of multiple,
individual plasmas that build up within a main envelope [Fig. 3(b)], or by single plasma columns with a
very irregular starting time curve [Fig. 3(c)]. Despite
these differences, the basic evolution of the streak is
the same as that observed in distilled water. In fact
the overall duration of the streaks is much the same in
both media, the maximum length of the plasma column is also similar [compare Figs. 2(b) and 3(c)], and
the irregular curve of starting times still approaches a
curve similar to that of Fig. 2(b) with superimposed
fluctuations.
The observations of the streaks produced in both
media with different focusing conditions (focal diameter of -11 gm, tight focusing) show that, for similar
ratios of peak irradiance to threshold irradiance, the
streaks maintain a similar profile of starting time, but
with a decreased maximal axial length with respect to
the focal plane. Again, the plasma persisted for the
longest time in the most upstream position.
Figure 4 is a series of pictures at 20
X
106 frames/s of
the breakdown region in distilled water. These show
the speed of propagation of the shock wave in the same
experimental conditions which produced the streaks in
Figs. 2 and 3. From the analysis of the pictures the
speed of propagation of the shock wave within the first
100 ns from plasma initiation may be calculated to be
1 September 1988 / Vol. 27, No. 17 / APPLIEDOPTICS
3663
I
a)
b)
d)
e)
C)
0E
Fig. 4. High speed photographs of the breakdown region in distilled water at a laser peak power that coincides with the threshold
value for breakdown. Laser parameters are as in Fig. 2: (a) t = 0,
(b) t = 50 ns, (c) t = 150 ns, (d) t = 250 ns, (e) t = 350 ns, (f) t = 450 ns.
-4
X
103 m/s, that is, lower by a factor of almost 5 than
that measured in gases,2 4 and similar to that calculated, for the same liquid media, from pump-and-probe
experiments5 and from high speed photography with
laser sources of different duration.6
IV.
Analysis
Recent studies4 3, 24 25 have established the influence
of irradiation conditions and of medium properties (in
particular the impurity content) in the threshold for
optical breakdown in liquids, as has been done in the
past for gases and solids. The observed differences in
the threshold irradiance between impurity-free and
impurity-rich media can be explained by considering
impurities as low-threshold sites of breakdown, where
linear light absorption mechanisms take place to generate the first free electrons in the medium. Also the
different structure of the plasma column observed
in our experiments, with irregularities in the plasma
starting time, wandering of the spark with respect to
the focal plane at threshold, or the existence of multiple, distinct breakdown sites in the plasma column,
can be explained by considering impurities as preferential sites for breakdown initiation, randomly located
within the focal volume.
What emerges from our studies is that, despite the
differences in threshold and structure of the streaks
observed between impurity-free and impurity-rich
media, the basic temporal evolution of the plasma
buildup as a function of the axial coordinate is fairly
similar in both media. Another important feature of
the streaks is that the overall length of the plasma
column is equal, for the two liquids, at equal values of
the ratio of the peak irradiance of the pulse to the
threshold irradiance for breakdown.
To analyze the observed curves of plasma starting
time, and the dependence of the overall plasma length
as a function of the irradiation parameters, we considered the models proposed in the past for the case of
breakdown in gases. In basic terms, three models have
been used to interpret the expansion of the plasma
toward the incoming beam: (i) a radiation-supported
detonation wave, (ii) an ionization wave model, and
(iii) a moving breakdown model. The first model 8"19
interprets the expansion of the laser-induced spark by
means of a mechanism of interaction between the
3664
APPLIEDOPTICS / Vol. 27, No. 17 / 1 September 1988
shock wave which is generated at the point where
breakdown first builds up, expands isotropically, and
gains energy at the expense of the incoming light. The
second model is based on a mechanism of the interaction of light with either free electrons2 6 or ionizing
radiation2 0 produced at the point where the plasma
builds up first, and expanding from that region by
diffusion. The moving breakdown model2 2 needs no
shock wave or charge carrier transport to explain the
plasma evolution toward the direction of the input
beam. It assumes that the breakdown occursindependently at each location on the axis provided that, at
that location, the irradiance of the electric field equals
the threshold value for breakdown. The starting instant of the plasma is then determined by the time
when the irradiance requirements for breakdown to
occur are matched to the threshold of the medium.
Despite the limited favor encountered up to now by
the breakdown model to explain the spatial and temporal dynamics of the plasma in gases, we believe that
such a model could be a candidate for the case of
liquids. The choice was motivated by the following
considerations:
(a) A model based on a light-shock wave interaction
is inadequate to interpret our findings: comparing the
average plasma expansion velocity in distilled water
[from Fig. 2(b) or Fig. 2(c)] with the shock wave veloci-
ty as measured with high speed photography measurements, or with other methods, we obtain that the former is at least 7 times higher than the latter.
(b) Particle velocity is rather limited in liquids.
Velocities of the order of 104 m/s have been measured
in water. 2 6 Also, in water electron-ion recombination
and electron-neutral molecule attachment occur on a
very short time range (fraction of a nanosecond), thus
decreasing the number of electrons that can migrate
outside the plasma region initially formed near the
focal plane. On the other side, transport of ionizing
radiation has been calculated to occur at an average
speed of 3 X 104 M/s (Ref. 20) in a plasma generated in a
gas by a train of mode-locked pulses with an envelope
of -150-ns FWHM. This speed, again, is inadequate
to explain the average, and in particular the initial,
expansion speed of the laser spark.
As to the model of moving breakdown, the intrinsically statistical nature of breakdown and the dynamics
of the processes leading to it proposed for solids,2 7 and
verified to apply in the case of liquids,2 4 justify the
assumption that plasmas build up independently at
each axial location along the beam waist. Breakdown
depends on the ability of the electric field to create free
lucky electrons that initiate the avalanche ionization
process that ultimately leads to an exponential growth
of free carriers. In liquid water, no free electrons are
present: solvation of electrons occurs in the picosecond time domain.2 9 Creation of initial lucky electrons
followslinear absorption by impurities or by electronic
states of solvated electrons, and, although less likely,
multiphoton absorption by water molecules. The
lucky electron approach, where the electric field, oscillating on a plane perpendicular to the beam propaga-
1Z
3.
=
A_
=
ZMax
IUzMax)
1I-'
t4 = 0
a)
I
I
I
z3
z2
Z1
°
1z 2 ))I
U )
I(0)
b)
I (z3 )
t3
AK
IN
LJ'
t2
tl
to
Fig. 5. Diagram illustrating the model of the moving breakdoNvn in
liquids. (a) Geometrical structure of a (half) Gaussian beam with
focus at z = 0. Shaded areas illustrate the spatial distribution co the
power at two distinct locations along the beam axis. (b) In c orrespondence with some axial locations, gives the temporal shape e of a
laser pulse whose peak power is assumed to exceed, at z = 0, the
threshold power required for breakdown in the medium by a f ctor
of 2 ( = 2). The left edge of each shaded area in (b) is the insta nt of
occurrence of breakdown t corresponding to the axial locatic on zi.
At z = Zmax, breakdown occurs only in correspondence with the peak
of the pulse (t = 0). Beyond that point, no breakdown occurs. The
curve of starting time is the locus of all points t(z).
ing beam and (ii) there must be a unique and explicit
dependence of the maximum elongation Zmax on the
beam power and geometric characteristics.
It is important to note that plasma expansion only
occurs in one direction and not in both directions with
respect to the focal point (z = 0). This asymmetry is
expected because absorption of the pulse energy by the
newly formed plasma decreases the amount of power
available for producing breakdown at any point beyond the focal plane.4" 6 This is the so-called shielding
effect: we postpone the discussion of shielding to the
next paper, where we show how this effect can account
for the observed decay pattern of the streaks.
If the laser pulse has a Gaussian amplitude of the
electric field that corresponds to a power (or energy)
distribution of the square Gaussian type, then
P(t) = Pmax- exp[-2(t/2A)
of breakdown, where every layer of thickness dz estab-
lishes its own plasma.
Starting from these assumptions, we now proceed
with the description of the modified moving breakdown model for plasma expansion. The model has
been upgraded with respect to its original formulation,
especially by taking into account the correct temporal
and spatial characteristics of the laser beam. We first
recall that the irradiance is defined as the power divided by the irradiated area and assume that the peak
power of the laser beam, at the geometrical focus
(z = 0), corresponds to the threshold irradiance. In
this case, breakdown is produced only over a small
region around the center of the beam waist. The instant at which breakdown begins coincides with the
peak of the pulse.
Referring to Fig. 5, where now the peak power is a
factor of 2 higher than the threshold value, the irradiance at the focal site first reaches the threshold value
for breakdown at time to, i.e., before the pulse reaches
the peak. At later times during this pulse, as the peak
power of the pulse increases, threshold irradiance requirements for breakdown are met at distances further
away from the focal point:
for example, at location z1 ,
breakdown is reached at an instant t, and so on up to a
value of Zmax corresponding to the point where the
irradiance reaches the threshold value only at the peak
of the pulse (t = 0).
From this it follows that (i) plasma formation occurs
later in time as the value of z increases, i.e., the plasma
initiation site movestoward the direction of the incom-
]}.
(1)
If the beam distribution is also Gaussian, i.e., it is
uniquely defined by the spot size w, the refractive
index n and wavelength X, the irradiance I(z,t) at a
given axial distance z and time t is given by the expression
I(zt) = P(t)/[7rw 2(z)j,
(2)
that is,
I(z,t) = Pmax
jexp[-2(t/2A)
2
tion direction, is supposed to create free initial electrons, may well account for a layer-oriented evolution
2
rw
2
]j.
(3)
(z)
Here, z is the axial distance from the center of the
focal volume and w(z) is the radius of the beam waist at
the location z [being w(z) = wo for z = 0]; t is the time
with zero value at the peak of the pulse; A = FWHM/
2(2 ln2)1/2 where FWHM is the full width at halfmaximum of the pulse power and zo, from the Gaussian
beam law, is given by zo = nrwo/X.
We now define the factor
3
as the ratio of the peak
power of the pulse to the threshold value:
= Pmax/
Pth, where Pth = Ith-7rW2. As defined earlier, the curve
of plasma starting time is the locus t(z) of all the
starting instants t, when breakdown starts, as a function of z and of the operating parameters of the laser.
From Eq. (3), setting the condition I(z,t) = Ith, we
obtain
t(z) = -A12 ln[f3(1 + Z2/zo)J
0
1 12
(4)
The function t(z) is real in the range 0 < z < zo( The corresponding values for t are -A(2 lnof1/2
forz =0 and0forzo(1)1/2. Thequantityto = t(0) =
-A(2 lnof1/2 is the instant at which the plasma is first
produced at the focal site, as a function of parameter .
On the other side, the quantity
1)1/2.
Z... = Z0 (# - 1)1/2
(5)
is the previously defined Zmax, i.e., the maximum length
of the plasma for a given irradiation condition (determined by the value of O3).
The inverse function of t(z) is given by z(t) and has
the form
z(t) = z 0ffl exp[-2(t/2A)
2
] - 11/2.
1 September 1988 / Vol. 27, No. 17 / APPLIEDOPTICS
(6)
3665
*500
16
14
12
01
C
E
E
a, 10
c
0
N
E
to
co
LS
4
2
0
0
50 100 150 200 250 300 350
Axial Distance from Focus (pm)
Fig. 6. Plot of the theoretical curves for plasma starting times,
expressed as the function t(z) - to, i.e., with origin corresponding to
the starting time at z = 0, obtained with the model of moving
breakdown for different values of the ratio a = Pmax/Pth. Laser
parameters are as in Fig. 2. Superimposed on each curve is the
experimental curve of plasma starting time derived from the streak
F
-
photographs of Fig. 2, suitably scaled.
I
To test the validity of the model, in Fig. 6 we plotted
a set of t(z) curves using the temporal, spatial, and
irradiation parameters that correspond to the streaks
of Fig. 2. In the figure the origin for the abscissa
coincides with the time to breakdown to: therefore all
the curves have the same origin. Superimposed to the
theoretical curves of plasma starting time, we plotted
the experimental curves derived from the streaks.
There is excellent agreement of the experimental
curves with the theoretical predictions for the three
I
i
ill
Fig. 7. (a) Dependence of the maximum extension, Zmax, of the
plasma column toward the laser source, on the ratio A for the two
experimental focusing conditions. Solid lines are theoretical curves
of Zmax Vs A [Eq. (5)]. ():
distilled water, standard focusing;
(C): distilled water, tight focusing; (A): tap water, standard focusing; (A): tap water, tight focusing. (b) Comparison between maxi-
mum extent of plasmas produced in distilled water (left) and tap
water (right) at equal a3values of 5.0 (corresponding to the shaded
region in (a).
values of A,with relation to the shape of the curves for
plasma starting times and to the maximal elongation
Zmax.
As shown in Fig. 6, curves of plasma starting
time t(z) are characterized by a nearly parabolic shape,
the main difference being the way they approach the
limit point of abscissa Zmax, where the slope of the curve
grows to infinity. This is consistent with the assumption that no interaction with the plasma can occur
when the pulse irradiance is below threshold. Figure 6
also shows how the limiting parameter Zmax varies
monotonically as a function of A.
Of particular interest is the dependence of Zmax, i.e.,
the maximum elongation of the plasma column, on the
irradiation conditions. Zmax, given by the simple relationship of Eq. (5), does not explicitly depend on the
pulse duration. Therefore, with the geometrical beam
characteristics being equal, the maximum length of the
plasma should not vary with pulse width. An implicit
dependence of Zmax on pulse width comes from the fact
that Pth, which appears in A, depends on the pulse
duration.' 3
3666
APPLIEDOPTICS / Vol. 27, No. 17 / 1 September 1988
Another important aspect related to the characteristics of the curves of plasma starting time t(z) and to Eq.
(5) for the maximum elongation of the plasma column
is the fact that equal curves of plasma starting times
and equal values for Zmax should be expected, when
measured at equal values of the j3parameter, for media
with different threshold breakdown irradiances. In
fact, the expressions for both t(z) and Zmax do not
contain Pmax alone, but only its ratio a to the threshold
value. This aspect is discussed with the aid of Fig.
7(a), where the maximum axial length of the plasma
streaks toward the source laser in distilled and in tap
water, in both standard and tight focusing conditions;
it has been plotted as a function of the parameter A. In
the same figure, solid curves are plots of Zmax as a
function of A for the two cases of wo = 6.5 and 5.5 gim,
respectively. Again, there is good agreement between
experimental points and theoretical curves obtained
with the use of the model of moving breakdown.
Moreover, it is worth noting how, in both focusing
conditions, data for distilled water and tap water are
region at threshold can be made by combining typical
aligned along a single theoretical curve.
breakdown probability
Taking into
account that tap water has a threshold for breakdown
about three times lower than distilled water, this confirms the assumed independence of Zmax on the absolute threshold parameters. The independence of the
physical elongation of the plasma toward the laser
source on the absolute value of the peak irradiance has
further support from Fig. 7(b), where plasma streaks
for distilled and tap water are compared at the value
of 5. Despite the fact that the streak in distilled water
evolves in a continuous way, while the streak in tap
water is a group of two separate plasmas, the overall
elongation is very much the same for the two media.
The above experimental evidence is important in
considering the possible role of nonlinear effects in the
expansion of the breakdown toward the laser source.
Some authors studying breakdown in gases have ex-
plained the multiple spark pattern in the streak by
self-focusing of the laser radiation in the medium.2 9
In our experimental conditions, self-focusing would
not explain the streak elongation in either medium.
Self-focusing depends on the nonlinear dielectric
properties of the bulk of the medium, and to a lesser
extent on those of randomly distributed impurities.
Therefore, the extent to which the focus would move
upstream, if due to self-focusing, should be dependent
on the absolute value of the irradiance of the laser
pulse. This is not found to be so.
Before concluding, we wish to briefly discuss the
limitations of the present formulation of the moving
breakdown model, as well as possible improvements to
overcome them. First, the calculation of the laser
irradiance as the ratio of the power to the surface of
area rw2is based on the limiting assumption that the
power is uniformly distributed throughout the crosssectional area of radius w0. This assumption, leading
to a drastic simplification in the calculations, somehow
contradicts the previous statement that the power has
a Gaussian distribution across any plane perpendicular to the axis of propagation.
Second, in the develop-
ment of the formalism for the model of moving breakdown, the implicit hypothesis was made that the
threshold irradiance Ith is the same for each layer of
thickness dz and area 7rw2(z), regardless of the axial
location z. According to studies performed both in
gases (see, for example, Ref. 30 and references therein)
and in liquids,4 24"13 the threshold irradiance for breakdown depends on the geometry of the irradiated region. Therefore, a more exact analysis should take
into account that Ith is a function of the axial coordinate z.
Finally, the model of moving breakdown in the
present formulation does not give accurate information on the behavior of the plasma column at threshold. In fact, for I = Ith, in Eq. (5) = 1 and Zmax = 0,
which would be unreasonable. However, we note that
the slope of the Zmax Vs curve [Fig. 7(a)] is very steep:
this means that the uncertainty in the length of the
plasma column for near threshold irradiance is very
high. An estimate of the dimension of the plasma
curves, as in Ref. 13, which
reflect the intrinsically statistical nature of the breakdown process, with the model of moving breakdown.
If, for example, one considers a breakdown probability
curve obtained in similar focusing conditions (Fig. 5.10
of Ref. 4), one realizes that, for a value of the irradiance
smaller by 20% than the 100% probability value, the
probability of breakdown occurrence is still very high
(-85%). Given a Gaussian beam distribution with wo
= 6.5 gim, such a value of irradiance would correspond
to an axial distance from the focal plane of 0.5 zo, i.e.,
-80 gim. In other words, there is an 85% probability
that the axial length of the plasma is 80gm long even if
the peak irradiance is set at threshold. The uncertainty in the determination of the axial length of the plasma column decreases proportionally as the /3 value
increases. We are, at present, experimentally investigating the statistical behavior of the laser-induced
sparks to come to an improved mathematical formulation of the model of moving breakdown that accounts
for this aspect.
V.
Conclusions
The model of moving breakdown, in an upgraded
version that takes into account the correct temporal
and spatial distribution of the laser pulse, has proved
to effectively describe the observed curves of breakdown starting time in the streak patterns of Nd:YAG
laser-induced plasma in the nanosecond region. Further work is required to characterize some aspects of
the plasma evolution, in conditions where the possible
role of shock waves or particles with light cannot be
ruled out as it was in this case. It might also be
interesting to check if, in the same irradiation conditions used in these experiments, gases exhibit similar
streak patterns. These experiments could give some
new insight in the still open question of which is the
correct model for plasma evolution in both liquid and
gaseous media.
Financial support was received by F.D. from the
British Council-CNR agreement and by M.R.C.C.
from the Wellcome Foundation. Our thanks are due
to both organizations. We would also like to thank
Sigmacon, Ltd. and especially S. Rondle and J.
Smithers of that company for technical assistance.
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with parental pride. Dr. Morrison is not concerned with
what's new, or what's spectacular, but with the basis of
knowledge.
He presents science as a process, not an event or
a result. "The Ring of Truth" shows that sometimes all it
takes is a passionately involved teacher with an impish look
and a playful air to enthrall an audience with the thinking
behind science. Without the extra touch of personality,
network programmers have not done too badly in presenting
some of the big science-oriented news events. President
Reagan's cancer, the plague of AIDS, the catastrophe of
Chernobyl, the technology of Star Wars have all played on
TV, occasionally exaggerated but on the whole presented
with reasonable concern of scientific reality.
Many science shows are visually more striking than ever
because of TV's harnessing of new technology that takes the
camera into hitherto unseen worlds. "The Infinite Voyage"
used the scanning electron microscope to invade the world of
the very small and an array of radio telescopes to locate
distant stars and help translate patterns of noise into photographs of far-off galaxies. Other programs have used fiber
optics to look into hidden corners inside the human body and
computer graphics to open up ways of depicting and animat3668
APPLIEDOPTICS / Vol. 27, No. 17 / 1 September 1988
ing worlds that hitherto existed only in the imagination.
But in the end, even compelling pictures will not settle in a
viewer's mind and heart without the structural assist of a
good story. It is interesting that TV's medical examiner,
Quincy, played by Jack Klugman, has inspired numerous
high school students to study forensic medicine, or at least to
find out more about this field. Nonfiction can be effective,
too: The removal and opening of the safe of the sunken
Titanic after scientists had pinpointed the wreck drew an
enormously attentive audience. Television accounts of
Robert Oppenheimer's leadership in developing the A-bomb
and his subsequent fall from grace, and of the discovery of
the structure of DNA by James Watson and Francis Crick
were well received because of the narrative techniques used.
Audiences for science programs are still meager compared
with those for shows on sports and religion. Most attempts
at substantially boosting audience size by putting science on
network TV have not met expectations. But there is a
growing constituency, which may influence the mediators of
our culture to consider moving good science programs out of
the intellectual ghetto of public TV and into the commercial
mainstream. Science is important enough to be trivialized.