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. References 1. S.Trokel, Ed., YAG Laser Ophthalmic Microsurgery (Appleton Century Crofts, Norwalk, 1983). 2. M. A. Mainster, D. H. Sliney, C. D. Belcher III, and S. M. Buzney, "Laser Photodisruptors-Damage Mechanisms, In- strument Design and Safety," Ophthalmology 90, 973 (1983). 3. C. A. Puliafito and R. F. Steinert, "Short-Pulsed Laser Microsurgeryof the Eye: BiophysicalConsiderations," IEEE J. Quantum Electron. QE-20, 1442 (1984). 4. H. P. Lrtscher, "Laser-Induced Breakdown for Ophthalmic Microsurgery," in Ref. 1, Chap. 4, p. 39. 5. J. G. Fujimoto, W. Z. Lin, E. P. Ippen, C. A. Puliafito, and R. F. Steinert, "Time-Resolved Studies of Nd:YAG Laser-Induced Breakdown,Plasma Formation, Acoustic WaveGeneration, and Cavitation," Invest. Ophthalmol. Vis. Sci. 26, 1771 (1986). 1 September 1988 / Vol. 27, No. 17 / APPLIEDOPTICS 3667 6. W. Lauterborn and K. J. Ebeling, "High-Speed Holography of Laser-Induced Breakdown in Liquids," Appl. Phys. Lett. 31,663 (1977). 7. C. E. Bell and J. A. Landt, "Laser-Induced High Pressure Shock Waves in Water," Appl. Phys. Lett. 10, 46 (1967). 8. A. Vogel, W. Hentschel, J. Holzfuss, and W. Lauterborn, "Cavi- tation Bubble Dynamics and Acoustic Transient Generation in Ocular Surgery with Pulsed Nd:YAGLasers," Ophthalmol. 93, 1259 (1986). 9. E. S. Sherrard and M. G. Kerr Muir, "Damage to the Corneal Endothelium by Q-Switched Nd:YAG Laser Posterior Capsulotomy," Trans. Ophthalmol. Soc. U.K. 104, 524 (1985). 10. D. Aron-Rosa, J. Aron, J. Griesemann, and R. Thyzel, "Use of the Neodymium:YAGLaser to Open the Posterior Capsule After Lens Implant Surgery," J. Am. Intraoc. Implant Soc. 6, 352 (1980). 17. P. Felix and A. T. Ellis, "Laser-Induced Liquid Breakdown: A Step-by-Step Account," Appl. Phys. Lett. 19, 484 (1971). 18. S. A. Ramsden and P. Savic, "A Radiative Detonation Model for the Development of a Laser-Induced Spark in Air," Nature London 203, 217 (1964). 19. J. W. Daiber and H. M. Thompson, "Laser-Driven Detonation Waves in Gases," Phys. Fluids 10, 1162 (1967). 20. I. Meyer and P. Stritzke, "Expansion of Laser Sparks Produced by a Mode-Locked Nd:Glass Laser," Appl. Phys. 10, 125 (1976). 21. A. J. Alcock, C. De Michelis, K. Hamal, and B. A. Tozer, "Expansion Mechanism in a Laser-Produced Spark," Phys. Rev. Lett. 20, 1095 (1968). 22. Yu. P. Raizer, "Breakdown and Heating of Gases Under the Influence of a Laser Beam," Sov. Phys. Usp. 8, 650 (1966). 23. F. Docchio, P. Regondi, M. R. C. Capon, and J. Mellerio, "Study of the Temporal and Spatial Dynamics of Plasmas Induced in Liquids by Nanosecond Nd:YAG Laser Pulses. 2: Plasma 11. F. Fankhauser, P. Roussel, and J. Steffen, "Clinical Studies on the Efficiency of High Power Laser Radiation upon Some Structure of the Anterior Segment of the Human Eye: First Experiments of the Treatment of Some Pathological Conditions of the Luminescence and Shielding," Appl. Opt. 27, 3669 (1988). 24. F. Docchio, L. Dossi, and C. A. Sacchi, "Q-Switched Laser Irra- Anterior Segment of the Eye by Means of a Q-Switched Laser System," Int. Ophthalmol. 3, 129 (1981). 12. A. Hochstetter, "Lasers in Urology," Lasers Med. Surg. 6, 412 (1986). 13. F. Docchio, C. A. Sacchi, and J. Marshall, "Experimental Inves- Liquids and Membranes," Lasers Life Sci. 1, 87 (1986). 25. M. R. C. Capon and J. Mellerio, "Nd:YAG Lasers: Plasma tigation of Optical Breakdown Thresholds in Ocular Media Under Single Pulse Irradiation with Different Pulse Durations," 26. P. A. Barnes and K. E. Rieckhoff, "Laser-Induced Underwater Lasers Ophthalmol. 1, 83 (1986). 14. S. K. Davi, D. E. Gaasterland, C. E. Cummings, and G. Liesegang, "Pulsed Laser Damage Thresholds in vitro for Intraocular Lenses and Membranes," IEEE J. Quantum Electron. QE-20, 1458 (1984). 15. R. F. Steinert, C. A. Puliafito, and S. Trokel, "Plasma Formation and Shielding by Three Ophthalmic Neodymium-YAGLasers," Am. J. Ophthalmol. 96, 427 (1983). 16. F. Docchio, L. Dossi, and C. A. Sacchi, "Q-Switched Laser Irra- diation of the Eye and Related Phenomena: An Experimental Study. II: Shielding Properties of Laser-Induced Plasmas in Liquids and Membranes," Lasers Life Sci. 1, 105 (1986). diation of the Eye and Related Phenomena: An Experimental Study. I: Optical Breakdown Threshold Determination in Characteristics and Damage Mechanisms," Lasers Ophthalmol. 1, 95 (1986). Sparks," Appl. Phys. Lett. 13, 282 (1968). 27. M. Bass and H. H. Barrett, "Avalanche Breakdown and the Probabilistic Nature of Laser-Induced Damage," IEEE J. Quantum Electron. QE-9, 338 (1972). 28. J. M. Wiesenfeld and E. P. Ippen, "Dynamics of Electron Solvation in Liquid Water," Chem. Phys. Lett. 73, 47 (1980). 29. A. J. Alcock, C. De Michelis, and M. C. Richardson, "Breakdown and Self Focusing Effects in Gases Produced by Means of a Single-Mode Ruby Laser," IEEE J. Quantum Electron. QE-6, 622 (1970). 30. C. De Michelis, "Laser-Induced Gas Breakdown: a Biblio- graphical Review," IEEE J. Quantum Electron. QE-5, 188 (1969). OfOpticscontinuedfrompage3629 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.
© Copyright 2024