2.1.

Experimental System

A schematic view of the experimental setup is shown in Fig. 1 . The cavitation bubbles were generated inside of a $37\text{-}\mathrm{mm}$ -high water cuvette measuring $55\mathrm{mm}$ at the square-shaped base. At the bottom of the cuvette, a $100\text{-}\mathrm{\mu }\mathrm{m}$ -thick microscope cover glass was mounted to allow unobstructed penetration of the laser radiation into the tank. A Q-switched pulsed Nd:YAG laser (Polaris II, New Wave, Inc.) with a second harmonic generator ( $532\text{-}\mathrm{nm}$ wavelength) was employed. The pulse duration was $5\mathrm{ns}$ , the energy per pulse was $3.75\pm 0.03\mathrm{mJ}$ , and the repetition rate was $0.1\mathrm{Hz}$ . This low repetition rate was chosen to allow the previous microbubble to disappear before the next cavity would form at the same location. The laser beam was focused using a 20x micro-objective lens with a numerical aperture (NA) of 0.40. To provide an optical contact between the objective and the optical window, a small quantity of glycerol was utilized. A single-element transducer (Panametrics-NDT, Inc.) with a $12.7\text{-}\mathrm{mm}$ focal length, an $f$ -number of 2, a $25\text{-}\mathrm{MHz}$ center frequency, and a 40% fractional bandwidth was used both to detect an acoustic emission and to acoustically probe the cavitation bubble. The foci of the objective and the ultrasound transducer were aligned prior to experiment. Using a single laser pulse, the cavitation bubble was formed approximately $3.4\mathrm{mm}$ above the glass surface. One of the lateral sides of the water cuvette was equipped with an additional $100\text{-}\mathrm{\mu }\mathrm{m}$ microscope slide (not shown in Fig. 1) to photograph the microbubbles. All components of the experimental system, including the pulsed laser, an ultrasound pulser-receiver interfaced with the ultrasound transducer, an analog-to-digital (A/D) digitizer (Gage Applied, Inc.), a function generator, and a time-delay device, were synchronized and controlled by a computer.

Fig. 1

Schematic view of the experimental setup. The microbubbles were produced in a water cuvette using a 5-ns laser beam focused by an objective lens through the optical window at the bottom. The ultrasound transducer, positioned at the top, was aligned with the laser beam such that the cavity was located approximately at the focus of the transducer.

In passive mode, acoustic signals generated during laser-induced cavity formation and bubble collapse were detected and recorded. In active mode, the ultrasound transducer sent short ultrasound pulses into the medium and received the ultrasound echo signals reflected from the bubble. We used both passive and active modes together to detect and monitor the behavior of the gas bubble. Once detected by the transducer, the signals were preamplified and recorded using a 12-bit A/D digitizer operating at a $100\mathrm{MS}∕\mathrm{s}$ sampling rate. To probe the cavitation bubble at different times, we used a function generator (model 33250A, Agilent, Inc.) to introduce a user-defined variable delay between the ultrasound and laser pulses.

2.2.

Characteristics of Ultrasound Signals

The recorded ultrasound radiofrequency (RF) signal contained mainly three types of signals: acoustic signatures corresponding to initial cavity formation, ultrasound pulse-echoes from the cavitation bubble, and bubble collapse. Figure 2 presents typical ultrasound RF data that consists of these three parts. The first part, located within the 0 to $20\mathrm{\mu }\mathrm{s}$ range, contains signals related to passive acoustic emission from the laser-induced cavitation bubble during its formation. The first peak near $0.5\mathrm{\mu }\mathrm{s}$ is associated with the laser pulse itself. The energy of light that was not utilized in the formation of the cavity was converted into heat in the coating of the transducer, and an acoustic wave was generated there. The other signals in the first part are related to the laser-induced cavitation bubble. These signals disappeared when the laser beam was blocked mechanically while no other changes were made to the experimental setup, confirming that they originated from the bubble itself. The second large signal at approximately $8\mathrm{\mu }\mathrm{s}$ corresponds to the shock wave that originated at the site of the laser-tissue interaction and propagated toward the transducer. Since the shock wave propagated spherically, the third ultrasound signal, around $12\mathrm{\mu }\mathrm{s}$ , corresponds to the same shock wave that reached the transducer after it was reflected from the optical window at the bottom of the water cuvette. The small-magnitude signals within the 20 to $30\mathrm{\mu }\mathrm{s}$ range are multiple reflections of the same shock wave traveling between the transducer and the optical window of the water cuvette.

Fig. 2

Typical ultrasound RF data record containing both passive and active acoustic signals that correspond to initial cavity formation, ultrasound pulse-echo probing of the bubble, and bubble collapse.

The second part of the record contains the group of signals located within the 50 to $70\mathrm{\mu }\mathrm{s}$ range. These signals correspond to the active ultrasound pulse-echo probing of the cavitation bubble. In this particular case, the ultrasound pulse was sent $50\mathrm{\mu }\mathrm{s}$ after the laser shot, and therefore $50\mathrm{\mu }\mathrm{s}$ after the cavity formation began. The first signal in this group, located at approximately $50\mathrm{\mu }\mathrm{s}$ , is a signal produced by the pulser itself and indicates when the ultrasound pulse was sent. The second and third signals, located at approximately 65 and $70\mathrm{\mu }\mathrm{s}$ , represent the ultrasound pulse reflected from the top surface of the cavity (i.e., the surface closer to the transducer) and the optical window, respectively.

Finally, the third part of the recorded ultrasound signals, located within the 90 to $110\mathrm{\mu }\mathrm{s}$ range, corresponds to the cavity collapse identified by the second passive acoustic emission from the cavity during its collapse. There is also the directly propagated acoustic emission recorded around $97\mathrm{\mu }\mathrm{s}$ , and the reflection of this emission from the optical window at $101\mathrm{\mu }\mathrm{s}$ . The structure of the signals in this group is similar to those in the first group, although the magnitude of the second shock wave is smaller than the first shock wave. Together, the cavity expansion and collapse define the lifespan of the cavity (Fig. 2).

2.3.

Estimation of Location and Size of the Cavitation Bubble

Once the laser beam is focused into the tissue and the threshold fluency of laser-induced optical breakdown is reached, high-density plasma is produced at the site of the laser-tissue interaction. Consequently, a shock wave is generated from the origin of the optical breakdown. The high-density plasma expands very rapidly and becomes a cavitation bubble.¹⁸ The cavitation bubble itself does not affect the propagation of a shock wave because the shock wave travels faster than the edge of the expanding bubble.¹⁸ The time of the shock wave arrival can be used to estimate the location of the laser-tissue interaction based on a time-of-flight principle of distance measurement. This location may be considered as the center of the cavitation bubble. Furthermore, the ultrasound pulses, sent into the medium with a desired time delay, are reflected from the top surface of the bubble (Fig. 2). Therefore, using time-of-flight measurements and accounting for round-trip propagation of the ultrasound pulses, we can assess the position of the propagation surface of the cavitation bubble. Finally, combining these measurements, we can estimate both the location and the size of the cavitation bubble.

Figure 3 demonstrates this approach. Figure 3a describes schematically the properties of a laser-induced cavitation bubble (not drawn to scale) at a certain stage of its evolution. The other graphs in Fig. 3 are the corresponding parts of the RF signal shown in Fig. 2. Figure 3b shows the signal generated by the laser itself and the passive acoustic emission signal located at approximately $8\mathrm{\mu }\mathrm{s}$ . This laser signal can be used as a reference so, the distance $D$ from the origin of the cavity to the ultrasound transducer can be calculated as follows:

Fig. 3

(a) Schematic view of the laser-induced cavitation bubble and definition of parameters describing its size and location. (b) Arrival time of the acoustic (shock wave) emission from the cavitation bubble and (c) the time of flight of the pulse-echo ultrasound can be used to determine the origin and size of the bubble. In these measurements, the laser and the ultrasound pulses are used as reference points, so the estimations are relative to the transducer. However, any stationary object such as an internal boundary (optical window at the bottom) also can be used as a reference point. In this case, the instantaneous radius of the bubble can be determined based on both (d) time delay between shock wave emitted from the cavitation bubble and the reflection of the shock wave from the optical window and (e) time delay between pulse-echo signals from bubble surface and the optical window.

Figure 3c shows the signal from the ultrasound pulser (located near $50\mathrm{\mu }\mathrm{s}$ ) and the pulse-echo ultrasound signal of the active ultrasound probing of the cavitation bubble (located at approximately $65\mathrm{\mu }\mathrm{s}$ ). As in the previous case, this signal from the pulser can be used as a reference. The ultrasound pulsed wave sent by the transducer is reflected from the top surface of the cavitation bubble, i.e., the surface that is closer to the ultrasound transducer. Therefore, the distance $d$ between the transducer and the surface of the cavitation bubble can be estimated as follows:

Since the arrival time of the shock wave emitted from the origin defines the location of the cavitation bubble, and the arrival time of the pulse-echo signal defines the position of its top surface, the bubble size can be estimated by combining these two measurements. From the geometry outlined in Fig. 3a, the radius of the cavitation bubble at the time ${\tau }_d+t_d∕2$ is the difference between the measured distances:

The cavitation bubble is effectively probed at the time determined by the sum of the user-defined time delay $\left({\tau }_d\right)$ and the time required for the ultrasound pulse to reach the bubble $(t_d∕2)$ . If the size of the bubble must be evaluated immediately after it has been formed, then the ultrasound pulse should be initiated prior to the laser pulse, and ${\tau }_d$ would be negative.

The approach presented above is based on differential measurements of the passive acoustic emission from the bubble (relative to the laser pulse) and the active ultrasound echo (relative to the ultrasound pulse). Therefore, these measurements are performed relative to the position of the ultrasound transducer. However, the signals obtained from any stationary reference, such as an internal boundary within the tissue, can also be used to estimate both the location and the size of the laser-induced cavitation bubble. If the distance between the ultrasound transducer and the bottom of the water cuvette $\left(L\right)$ remains the same, then the bubble radius can be defined according to either the transducer or the bottom of the cuvette using the following expressions:

Therefore, in relation to the optical window of the water cuvette, the instantaneous radius of the cavitation bubble is the difference between the position of its upper propagation surface $\left(h\right)$ and the location of its origin $\left(H\right)$ . By using the signals corresponding to the shock wave emitted from the origin of the cavitation bubble and reflected from the bottom of the cuvette [Figs. 2 and 3d], we can calculate the position of the bubble relative to the bottom of the cuvette $\left(H\right)$ as follows:

Finally, the radius of the cavitation bubble can be obtained by combining Eqs. 4, 5, 6:

Hence, there are two possible approaches to measure the bubble location and its radius. The first approach measures both the bubble location and its radius directly from the acoustic signals, while the second approach uses the reflections of the signals from an internal reference surface. Both of these approaches have advantages and limitations. In some cases, the reference surface could be part of a tissue or organ—for example, a boundary of a cornea if a femtosecond laser refractive surgery is considered.⁹ However, if there are no internal boundaries present, the second approach cannot be used. The limitation of the first approach is that the shock wave originated during the plasma formation initially has a propagation speed several times higher than the speed of sound.^{9, 19, 25} The accuracy of the measurements, therefore, will be affected by inaccuracies and variations in the estimation of both the speed of the shock wave and the speed of sound.

To demonstrate the overall capability of the proposed ultrasound method, we used the second approach to measure the location and the radius of the laser-induced cavitation bubbles where the top surface of the optical window of the water cuvette was utilized as the internal boundary. The advantage of such implementation is that the speed of the shock waves does not affect the bubble size estimation if the distance between the transducer and the point of an optical breakdown is greater than the length of the shock-to-acoustic wave conversion. The impact of the variable propagation speed is canceled because both direct and reflected shock waves have exactly the same propagation speed, and the time delay between them [Fig. 3b] is defined only by the speed of the acoustic wave $c$ and the distance between the internal boundary and the origin of the bubble $H$ . Indeed, if $d_s$ is the length of the shock-to-acoustic wave conversion and $c_s$ is an average speed of shock waves within this length, the time required for the shock wave to propagate directly from the point of the optical breakdown to the transducer Fig. 3 is

To monitor bubble dynamics with high temporal resolution and over long periods of time, the ultrasound pulses, synchronized with the laser shots, must be transmitted into the medium at a high pulse repetition rate. The temporal resolution is proportional to the pulse repetition frequency of the ultrasound pulse-echo probe. In our experimental system, we used a maximum pulse repetition frequency of $30\mathrm{kHz}$ , which allowed assessments of the cavitation bubble size approximately every $33\mathrm{\mu }\mathrm{s}$ . Clearly, such temporal probing is too coarse to accurately capture bubble behavior with an average lifespan on the order of $100\mathrm{\mu }\mathrm{s}$ (see Fig. 2). To overcome this technical but not fundamental limitation, the experiments were repeated using a variable time delay ( ${\tau }_d=0$ to $150\mathrm{\mu }\mathrm{s}$ ) between the laser and the ultrasound pulses. Initially, the ultrasound pulses and laser shots were generated simultaneously, so ${\tau }_d$ was equal to zero. Since the speed of sound is much smaller than the speed of light, the ultrasound pulse arrived at the cavity site approximately $8\mathrm{\mu }\mathrm{s}$ after the laser pulse was triggered. The experiment was repeated 50 times, and all ultrasound signals were stored for later off-line processing. The ${\tau }_d$ was then increased by $5\text{-}\mathrm{\mu }\mathrm{s}$ increments and the experiments were repeated 50 times, and the corresponding signals were recorded. Therefore, the dynamic behavior of the cavity was measured every $5\mathrm{\mu }\mathrm{s}$ over a $150\text{-}\mathrm{\mu }\mathrm{s}$ time interval where each measurement included both passive and active ultrasound signatures of the cavitation bubble. However, a new bubble was generated and measured in every experiment, thus requiring statistical analysis of the data. The processes of nanosecond laser-induced optical breakdown, and consequently the formation of the cavitation bubble, are highly irregular.⁵ The modest variations of the laser energy or natural local changes of refraction and absorption coefficients of the medium lead to a wide size distribution of the cavitation bubbles. Therefore, to analyze our data, we divided the entire dataset into smaller subsets based on the cavitation bubbles’ lifespans. It was assumed that the cavitation bubbles within a small range of lifespan were formed under approximately the same conditions, such as laser energy and local optical properties of water.

2.4.

Optical Verification of Ultrasound Measurements

The accuracy of the ultrasound measurements of the laser-induced cavitation bubble size was initially verified using direct optical measurements of the cavitation bubble size at the equilibrium. To avoid any translational motions of the bubbles during cavity formation and oscillation, the experiments were performed in a hydrogel sample made of 10% gelatin. The $10\text{-}\mathrm{mm}$ -thick layer of the liquid gelatin was poured in the water cuvette and stored in a refrigerator for 10 hours to solidify. The sample was then removed from the refrigerator and kept at room temperature for 5 hours to warm up and to reach a uniform temperature distribution. Since the typical bubble sizes induced in the gelatin were smaller than those formed in water, the focal point of the laser beam was moved closer to the bottom optical window such that the typical distance from the origin of the cavity to the glass was about $1\mathrm{mm}$ .

The cavitation bubbles were imaged by an optical microscope—a charge-coupled device (CCD) camera equipped with a 200x objective lens with an NA of 0.1. The working distance of the objective lens was $25\mathrm{mm}$ , so, the optical imaging assembly did not interfere with the ultrasound measurements. Prior to the experiments, the microscope was calibrated using an image analysis micrometer with a $10\text{-}\mathrm{\mu }\mathrm{m}$ step scale (Edmund Optics, Inc.). To assess the accuracy of optical measurement, a $500\pm 2.5\text{-}\mathrm{\mu }\mathrm{m}$ sapphire sphere was embedded into 10% gel and measured 15 times. The measured size of the sphere was $500.5\pm 2.4\mathrm{\mu }\mathrm{m}$ , agreeing well with the expected values.

At the beginning of the experiment, an ultrasound RF signal similar to that shown in Fig. 2 was recorded during the bubble formation. The passive acoustic emission from the cavity was used to calculate the origin of the laser-tissue interaction using Eq. 5. The cavity was then periodically probed using the pulse-echo ultrasound. At the same time, the formed bubble was monitored using the optical microscope. After about 30 minutes, the last ultrasound pulse was sent to probe the cavitation bubble. This particular record was analyzed together with the previously captured passive acoustic emission signal to calculate the bubble size based on Eqs. 6, 7. Simultaneously with the last ultrasound measurement, the bubble inside the gelatin sample was imaged using the CCD camera, and the radius of the cavitation bubble was measured directly from the calibrated optical image. The ultrasound and optical measurements of the microbubble size were compared to initially validate the accuracy of the developed quantitative ultrasound technique. However, due to the limited frame rate of the CCD camera of the optical microscope, an assessment of the temporal accuracy of the ultrasound measurements of the transient bubble size was not possible.

2.5.

Verification of Ultrasound Measurements using Rayleigh Model of Cavitation Bubble Collapse

The temporal accuracy of the developed ultrasound method was verified by comparing the measured behavior of the laser-induced cavitation bubble in water with that estimated from the Rayleigh model of a spherical cavitation bubble collapse in an incompressible liquid.^{26, 27} The equation to describe the radial motion of the cavitation bubble surface is

Integration of Eq. 11 with respect to time gives the collapse time $\left(t_R\right)$ of the cavitation bubble:

To fully describe the dynamic behavior of the laser-induced cavitation bubbles in water, we assumed that the expansion of the cavity and the subsequent collapse of the bubble were symmetric processes, and the Rayleigh model was expanded symmetrically around the point of maximum radius,²⁸ i.e., the lifespan of the cavity was simply twice as long as the cavity collapse time $t_R$ .

Therefore, once the lifespan of the cavitation bubble formed in water was measured (see Fig. 2) and its collapse time was estimated, its temporal behavior could be calculated using the Rayleigh model [Eqs. 11, 12, 13] and compared with the values measured using the ultrasound technique.

3.1.

Optical Verification of Ultrasound Measurements

The ultrasound method was first verified by comparing ultrasound and optical measurements obtained from nine microbubbles generated using a pulse energy of $0.72\mathrm{mJ}$ . Each cavitation bubble was produced in gel and measured both optically and ultrasonically at equilibrium. Generally, due to the nature of the interaction of nanosecond laser pulses with a medium, the dimension of the cavitation bubbles ranged from $85\mathrm{\mu }\mathrm{m}$ to $175\mathrm{\mu }\mathrm{m}$ . Therefore, a comparison between optical and ultrasound measurements was performed for each bubble independently. An example of such analysis is presented in Figs. 4 and 5 .

Fig. 4

Optical image of a laser-induced cavitation bubble. The laser beam was delivered from the bottom and the ultrasound transducer was positioned at the top, as indicated in Fig. 1.

Fig. 5

Temporal characteristics of (a) passive acoustic emission from the cavity, and (b) pulse-echo ultrasound probing of the cavitation microbubble (solid line). Prior to the experiment, the echo signal from the optical window at the bottom of the cuvette was captured [dashed line in Fig. 5b]. These acoustic signals were used to determine the radius of the microbubble at equilibrium.

Figure 4 presents an optical image of the cavitation bubble. Clearly, the bubble is slightly elongated in the vertical direction. Such distortion of the bubble shape from a perfect sphere is not unusual and is determined, among other parameters, by the focusing of the laser beam. The axial radius of the bubble (i.e., the distance between the origin of the bubble and its upper surface) was measured to be $175\mathrm{\mu }\mathrm{m}$ . Given the geometry of the experimental setup, the ultrasound measurements of the laser-induced cavitation bubbles were performed along the ultrasound beam axis, i.e., along the vertical direction in Fig. 4.

Figure 5a shows the measured passive acoustic emission from the laser-induced cavitation bubble formed in gelatin by a single laser pulse. The first signal, located near $8.6\mathrm{\mu }\mathrm{s}$ , represents the shock wave propagating from the origin of the cavitation bubble directly to the ultrasound transducer. The second signal, located at $10.1\mathrm{\mu }\mathrm{s}$ , corresponds to the same shock wave after it is reflected from the optical window at the bottom of the cuvette before it reaches the transducer. The time delay between these two signals was estimated using the normalized cross-correlation procedure. Using the speed of sound of $1500\mathrm{m}∕\mathrm{s}$ , the distance between the origin of the cavitation bubble and the optical window, calculated using Eq. 5, was estimated to be $1.11\mathrm{mm}$ . Figure 5b presents the active ultrasound pulse-echo signals from the optical window before the cavitation bubble was formed and from the cavitation bubble surface after its formation. The first record (dashed line) was captured before the cavitation bubble was produced in the gel. The echo signal, located at approximately $18.5\mathrm{\mu }\mathrm{s}$ , corresponds to the pulse reflected from the optical window at the bottom of the cuvette. The solid line in this figure represents the pulse echo signals from the top surface of the cavitation bubble, located at $16.7\mathrm{\mu }\mathrm{s}$ , as well as from the optical window, located at approximately $18.5\mathrm{\mu }\mathrm{s}$ . Note that the ultrasound pulse-echo signals from the optical window at the bottom of the cuvette, recorded before and after the laser pulse, are slightly different due to the formation of an acoustic inhomogeneity along the propagation path of the ultrasound pulse. Therefore, the distance between the top surface of the bubble and the optical window was measured using the signal from the surface of the cavitation bubble and the signal from the optical window before the bubble formation. Given these temporal estimates and using Eq. 6, the distance from the topsurface of the bubble to the optical window was found to be about $1.29\mathrm{mm}$ . Based on these results, the radius of the cavitation bubble was calculated as $180\mathrm{\mu }\mathrm{m}$ . Therefore, in this particular case, the absolute difference between the ultrasound and optical measurements of the bubble radius was $5\mathrm{\mu }\mathrm{m}$ . That corresponds to a relative difference of 2.9%. For the entire set of experiments, the mean values of absolute and relative differences were $9.5\mathrm{\mu }\mathrm{m}$ and 8.9%, respectively.

3.2.

Verification of Ultrasound Measurements using Rayleigh Model of Cavitation Bubble Collapse

To compare measured bubble evolution with theoretically predicted behavior, another set of experiments was performed and the lifespans of the bubbles in water were measured. In our experimental studies, the overall range of bubble lifespans was $132.4\pm 15.3\mathrm{\mu }\mathrm{s}$ (mean value and standard deviation), corresponding to a maximum cavity radius of $723.5\pm 83.6\mathrm{\mu }\mathrm{m}$ . This large variation in bubble size measurements was related primarily to the repeatability of the nanosecond laser-induced cavitation bubbles and to nonsignificant local changes of refraction and absorption coefficients of the medium. To reduce these possible error sources, the data analysis was performed on small subsets of data identified by a narrow range of cavity lifespans. It was assumed that all cavitation bubbles within a certain range of lifespans $(\pm 2.5\mathrm{\mu }\mathrm{s})$ were formed under approximately the same conditions, such as laser energy and local optical properties of water.

Figure 6 presents the dynamic behavior of the cavitation bubbles in water, obtained from one of the subsets. Only appropriate RF signals corresponding to bubbles with a lifespan ranging between 135 to $140\mathrm{\mu }\mathrm{s}$ were used for these calculations. The number of RF signals and the error bars (plus/minus one standard deviation) were slightly different for each experimental point. The ultrasound measurements were also compared with the Rayleigh model of the cavitation bubble with a lifespan of $137.5\mathrm{\mu }\mathrm{s}$ , corresponding to a maximum bubble radius of $751.4\mathrm{\mu }\mathrm{m}$ . The temporal dependence of the bubble radius was calculated using Eqs. 11, 12, 13, assuming no vapor pressure inside the bubble and $1000\mathrm{kg}∕{\mathrm{m}}^3$ for the water density.

Fig. 6

Measured (time-of-flight method) and modeled (cavity lifespan or collapse time and Rayleigh equation) dynamic behavior of cavities produced by the pulsed nanosecond laser. Only a subset of data corresponding to cavitation bubbles with a lifespan between 135 to $140\mathrm{\mu }\mathrm{s}$ was used. The error bars represent plus/minus one standard deviation of the estimates.

Figure 7 demonstrates the measured dynamics of the laser-induced cavitation bubbles with lifespans ranging between 115 to $120\mathrm{\mu }\mathrm{s}$ , 130 to $135\mathrm{\mu }\mathrm{s}$ , and 140 to $145\mathrm{\mu }\mathrm{s}$ in water. Using Eq. 13, the maximum average sizes of these cavities were calculated as $642\mathrm{\mu }\mathrm{m}$ , $724\mathrm{\mu }\mathrm{m}$ , and $779\mathrm{\mu }\mathrm{m}$ for the bubble lifespans of $117.5\mathrm{\mu }\mathrm{s}$ , $132.5\mathrm{\mu }\mathrm{s}$ , and $142.5\mathrm{\mu }\mathrm{s}$ , respectively. The experimentally measured temporal behaviors of the cavitation bubbles are in good agreement with the theoretical predictions based on the Rayleigh model. Note that since the ratio of the distance $H$ from the bubble origin to the cuvette bottom and the maximum bubble radius $R$ [see Fig. 3a] exceeded 4 for all measured microbubbles, the dynamic behavior of the bubble was not affected by the presence of the cuvette’s glass bottom.

Fig. 7

Comparison of measured (time-of-flight) and modeled (Rayleigh equation) temporal behavior of cavitation bubbles characterized by different lifespans.

The measurements noted above were conducted using the glass bottom as a reference point and $1500\mathrm{m}∕\mathrm{s}$ for the speed of sound. The approach utilizing the transducer as a reference point [Eq. 3] was also applied to measure the evolution of bubbles in water. The same dataset was used for these calculations. Since initially the speed of the shock wave was several times higher than the speed of sound, the bubble radii were underestimated. As expected, this underestimation was constant for all experiments and did not depend on the time delay between the laser and ultrasound pulses. The average difference between the radii obtained with the two methods was $150\pm 32\mathrm{\mu }\mathrm{m}$ (mean value and standard deviation). This difference vanishes if the average speed of the shock wave over the distance from the bubble’s origin to the transducer $\left(12.9\mathrm{mm}\right)$ is assumed to be $1518\mathrm{m}∕\mathrm{s}$ . This assumption is reasonable since the shock wave, initially propagating with a higher speed, becomes an acoustic wave within a few millimeters of propagation^{19, 25}