In this paper we show results of Nd:YAG laser-induced bubbles formed in a one millimeter thick agar gel slab. The nine nanosecond duration pulse with a wave length of 532 nm was tightly focused inside the bulk of the gel sample. We present for the first time a pump-probe laser-flash shadowgraphy system that uses two electronically delayed Nd:YAG lasers to image the the bubble formation and shock wave fronts with nanosecond temporal resolution and up to nine seconds of temporal range. The shock waves generated by the laser are shown to begin at an earlier times within the laser pulse as the pulse energy increases. The shock wave velocity is used to infer a shocked to unshocked material pressure difference of up to 500 MPa. The bubble created settles to a quasi-stable size that has a linear relation to the maximum bubble size. The energy stored in the bubble is shown to increase nonlinearly with applied laser energy, and corresponds in form to the energy transmission in the agar gel. We show that the interaction is highly nonlinear, and most likely is plasma-mediated.
©2008 Optical Society of America
The use of lasers in biological applications has been growing steadly as the technology becomes more stable and afordable. While the laser is not necessarily an efficient method to deliver energy, it is one of the most precise, and can allow for the greatest finesse. The studies that we present here are aimed at the control of laser tissue microprocessing and measuring the secondary effects caused by the use of short pulse lasers; mainly the strong shock waves that are the precursor to the eventual laser induced bubble expansion.
We use agar gel because it presents a simple tissue phantom. The melting point of agar gel is 80–100°C; close the point of cellular damage in animal tissue . It has been well documented that a nanosecond laser focused inside of liquids and gels will cause the formation of a bubble [2, 3]. These studies have also presented an analysis of the growth and collapse of the bubble. The study presented here adds to the literature by giving nanosecond time resolution and seconds of range in a single optical setup, allowing for a precise measure of the bubble growth, collapse, and associated shock wave.
Several authors have done experiments with very high numerical apertures up to NA=1.3 [6, 5]; where their target materials were water, liquids or in vivo; laser pulse durations ranging from fs-ns. Vogel has done several experiments in gels with focusing angles up to 22 ° . In our experiments we wanted to try and avoid the elongated, and conical plasma region that occurs with weak focusing. We used a focusing lens with an NA of 0.5. This lens allowed for tight focusing with an air interface, and a long enough working distance that allowed for the agar to be behind a glass slide; maintaining all interfaces optically smooth, and with the freedom to move the beam focus inside the agar slab. The tight focus of the laser, and working close to the agar’s damage threshold implied a small initial diameter plasma to seed the proceeding bubble formation.
The laser produced plasma precedes the bubble in the gel, which continues to grow long after the laser pulse has finished, which implies that the initial conditions set by the laser completely define (along with the material parameters) the life time of the bubble. Several papers have shown, with picosecond resolution , the beginning of bubble formation in liquids and solids, and other papers were forced to rely on simulations and assumptions for the initial hundreds of nanoseconds [2, 1]; Rau had nanosecond to millisecond time scales, but their probe delay was made with fiber delay for short time scales, and changed to an electronically delayed flash for long times. We wanted a simple system that, without changing any optical components, could image the entire lifetime of the bubble. By using an electronically delayed second laser we achieved over nine orders of magnitude of temporal range; the electronic delay had picosecond resolution making the nanosecond pulse duration the limiting factor for resolution.
The negative aspect of our system was that one exposure could give only one image at a given time delay. Figure 1 shows, for 300 ns delay, the bubble diameter for varying energies; each point was separate exposure on a fresh sample. Other forms of data collection are the use of high speed cameras (limited minimum time delay of 20 µs) , beam deflection methods which trade off spatial resolution for temporal, and thus require multiple exposures to gain spatial information [8, 9], and streak cameras  which are limited in temporal range.
This article presents the experimental setup, results and finally a brief discussion to both interpret and place into current literature our observations.
2. Experimental system
2.1. Laser sources
Our optical system, Fig. 2, consisted of two frequency doubled Continuum Minilite, Q-switched Nd:YAG lasers. One of the two lasers had a half-wave plate polarizer pair to control the power onto the doubling crystal; the laser had been modified to permit the rejected IR beam from the polarizer to escape. We used the green 532 nm beam from the unmodified laser as our pump beam, and the IR 1064 nm beam as the probe. Agar gel has a flat absorption spectrum from 500–1400 nm with an absorption coefficient of less than 4 cm -1. The IR beam was chosen as the probe beam and the green as the pump. The IR probe beam typically had less than 1 µJ of energy per pulse.
The two beams were brought onto the sample co-linear. The green pump beam was roughly collimated before it fell onto the 6mm focal length (NA=0.5) aspherical microprocessing lens (μPROC); the pump beam, which over-filled the lens, was reduced in size by a diaphragm placed before the lens. The IR probe beam was focused by a 35 cm focal length lens (PROBE) to a point before the microprocessing lens such that it was collimated as it propagated through the sample. After the sample we used a 25 mm lens (COL) followed by a 40 cm lens (ETP) to image relay the IR beam onto the CCD placed in the transmitted beam line. The image relay was focused for the IR beam, and the green pump was removed with a red (low pass) coloured filter.
The 35 cm lens (PROBE) in the IR beam line also served a second purpose. The CCD could be placed to have the pump beam retro-reflected from the target (usually the surfaces of the glass slides in the agar target) onto the CCD in another image relay; this setup is commonly refereed to as an equivalent target plane (E.T.P.) system. The E.T.P. was used to set the location of the beam focus in the sample when each new sample was placed. The pump beam energy was monitored with a cross-calibrated pyroelectric energy monitor (Ophir) placed on the green reflection of the IR input beam splitter (the calibration was done with a second energy monitor placed after the microprocessing lens (μPROC); the IR probe beam was filtered out before the energy monitor. The per pulse energy was acquired by the energy monitor display, then imported and stored in a computer. The energies measured were exact and not averages.
The pump laser used had a 1/e 2 pulse duration of 9.2 ns and was focused to a beam waist 1/e 2 radius of 2.3 µm; measured with the ETP system. The probe laser passed the sample with a 1/e 2 radius of 175 µm and a per pulse energy close to 1 µJ.
The timing of the pump to probe delays was controlled with a Stanford Research Instruments DG535 delay generator. The DG535 was triggered by the flash output of the CCD camera which was triggered by either the computer or a hand-held button. The DG535 sent triggers to the flash lamp and Q-switch inputs of both lasers. The pump-probe zero time delay was set with a photodiode placed after the sample holder and read on an oscilloscope; this allowed us to set the zero time delay within 2 ns. The DG535 allowed for delays up to nine seconds with picosecond time resolution. The CCD camera had a shutter time of typically 400 ms, and was set to open about 10 ms before the arrival of the pump pulse.
2.3. Target alignment and preparation
The image relay systems above was used in conjunction with a three axis (x,y,z) computer controlled translation stages; one micron step size. The samples of agar were mixed with a concentration of 1g agar powder per 50 mL of distilled water. Once heated to melting point in a double boiler (to avoid burning) the agar was pored between two microscope slides separated by 1mm spacers. The sample was left to cool before placement onto its gimbled target holder in the microprocessing system. The target holder allowed for adjustment of the surface to beam angle from which we attained a vertical change of less than 50 µm over a horizontal slide of 4 cm. This allowed us to set the beam focus to the center of the 1 mm thick sample and to take hundreds of data points quickly (less than 20 min) with no overlap and or loss of focus location.
The laser was focused in the center of the 1 mm agar sample. One pulse was delivered per site. The time delays were set to observe four distinct regions of interest;
- 0 to 20 ns : Rapidly slowing plasma expansion and coupling to shock wave
- 20 ns to 100 ns with Δt=20 ns: Linear shock wave propagation
- 200 ns to 1 µs with Δt=200 ns: Bubble maximum size
- 1µs to 100 µs with Δt=3 µs: Bubble collapse
- 100 µs to 1 ms with Δt=30 µs: Far field bubble size
Other time delays such as those in Fig. 3 were used to probe a variety of times; Fig. 3 displays the results in a log-log plot of the bubble size vs. time for several typical energies. In the example image shown in the insert of Fig. 3 both the bubble formation and shock wave are visible. The field of view of the CCD camera and the size of the probe beam limit the maximum bubble size that could be measured to be about 450 µm in diameter1 (data in Fig. 3 was made with slightly different imaging lens after the sample (COL) allowing for a slightly larger field of view.)
3.1. Shock wave origin
Rau  has shown evidence that bubble formation is linked to the presence of laser produced plasma in the focal volume of the laser pulse. Deemed plasma-mediated ablation, the topic has been well covered in the area of laser matter interaction and it has recently been applied to medical uses . With plasma mediated ablation, the laser light is not directly absorbed by the agar gel, but it’s energy is absorbed by a plasma that the pulse created through laser-induced ionization. The laser ionizes the material through a combination of multiphoton absorption, and avalanche ionization. Both of these ionization processes imply the presence of a threshold irradiance above which plasma will form. Plasma has a much higher absorption of laser light than the low absorption agar gel, so the sooner in the laser pulse that threshold irradiance is reached, the proportionally more laser energy the ionized volume will absorb. Laser energy is mainly coupled into the low mass electrons in the plasma and from there into the positive ions; this happens on the order of picoseconds. The hot plasma expands at supersonic speeds. The plasma recombines and couples its momentum into a supersonic shock wave within the next 100 ps [13, 1].
If the time for the generation of a shock wave is shorter than the laser pulse producing it, we can look for the time origin of the shock wave and use it to infer when in the laser pulse the plasma was formed. At short times the shock wave front could not be defined in the image. It was not until 10–20 ns after the exposure that a defined shock wave could be extracted. At a delay of 20 ns, the shock wave front expands linearly. This was seen as well by Zysset  and was alluded to as well by Vogel , to list just two examples. Our system could measure the beginning of the shock wave with radii starting close to 20 µm and up to a maximum of 225 µm; the first data point was taken at 20 ns, which corresponded to a radius of 40–60 µm and fell into the constant velocity expansion of the shock wave. From a linear fit of the shock wave front location, the time origin of the shock can be calculated; strictly speaking a shock wave should be losing energy though shock heating, but a fit of the veloctiy to 1/rc gave a value of c=0. In Fig. 4 the location of the shock front for varying times and energies is plotted, and the insert shows the shock velocity vs. per pulse energy.
Assuming that there was a threshold laser irradiance at which a plasma was formed, a fit of the data gave the probe pulse to pump pulse zero delay and shock wave formation threshold energy. For a laser pulse with a gaussian temporal profile:
Each pulse had energy En with peak irradiance I 0,n. According to the above assumption of a plasma threshold, there existed a threshold irradiance Ithreshold that caused the formation of a plasma. Eqn. 1 was solved for tn; the time at which a laser pulse with peak irradiance, In, reached the threshold irradiance for plasma formation; In(tn)=Ithreshold:
In a gaussian pulse shape the total energy and peak irradiance have a linear relation, implying that;
Eqn. 4 was then used as a fit function to the x-intercept found in Fig. 4 to solve for the location of the probe to pump zero time delay, t 0=11ns, and the shock wave formation threshold energy Ethreshold=71 µJ; shown in Fig. 5. Visual observations noted that bubbles were formed for pulse energies above 55 µJ/pulse. A similar phenomenological derivation was performed on laser plasmas by Docchio  in 1988.
3.2. Shockwave velocity
An explosion in water or a solid will produce a shock wave. The shock wave leaves at supersonic speeds from the excited region, and carries with it information about the explosion. The velocity of the shock wave can be used to determine the pressure difference inside and outside the explosion region. Consider the conservation of momentum equation where region 1 is inside the shocked region and region 2 is the surrounding unaffected region. The conservation of momentum and conservation of mass can be stated as;
where ui, ρi, and Pi are for region ‘i’ the velocity of the particles, density, and pressure respectively. If we move the frame of reference to the shock wave moving at speed U, and assuming that the particles in the unaffected region are stationary we can then combine Eqn. 6 and Eqn.5 to get
There is still the unknown particle velocity for the shocked region ui, but for this we can use the Hugoniot equation  to relate the shock velocity, particle velocity and sound velocity C 0 as:
where the constant S, and sound speed C 0, in 10% gel were found by Nagayama  to be 2.0 and 1.52 km/s respectively. Eqn. 9 and Eqn. 8 can be combined to give the pressure differential between regions 1 and 2 as a function of the density and the speed of the shock wave propagation:
3.3. Bubble energy
Of interest as well is the energy that the laser transfers to the medium. Again we use Rau’s  example as a reference; also treated in Vogel . There they use the case explored by Lord Rayleigh for an inertially controlled bubble growth. From there the energy in the bubble can be expressed as a function of the maxium radius and the time of the first collapse of the bubble; Fig. 7 shows several typical bubble evolutions. The energy in the bubble is then;
where EB is the energy in the bubble, Tcol is the time for the first collapse, and Rmax is the maximum bubble radius. It has been shown by Brujan  that a rigid boundary does have an effect on the bubble oscillation frequency. If our sample were water, then Tcol for the larger pulse energies (maximum bubble diameters above 500 µm) could change by up to 10%; in agar gel the modification would be similar to that value. In our case this error was within the measurement error of Tcol. In any event, the effect of the rigid glass boundary on the sample was considered a source of systemic error for the larger pulse energies; the larger error data for bubble closing time was not used because its’ respective Rmax was too big to be imaged onto the CCD. (The transmission and shock wave data were not affected by the boundary because they happen on a time scale too small for any interaction with it.)
In Fig. 8 the ratio of the energy in the bubble to the energy in the laser pulse is shown. The energy in the bubble reaches 0.9% of the laser pulse energy, and the trend in the plot points to a ratio of zero that occurs around 55 µJ/pulse which agrees with the our visual observations. The maximum ratio measured was for energies up to 240 µJ/pulse, limited by the ability to measure the maxium bubble size. Vogel  mentions an energy absorption up to 76%, which corresponds well with the transmission that we measure when the perpulse energywas in excess of 350 µJ/pulse. It should be noted that this is not a complete treatment of the energy in the system as Eqn. 11 uses water coefficients, and does not account for the elastic modulus of the agar. As well the energy in the shock wave is not accounted for. Determination of these factors is part of ongoingwork. It should also be noted that above equation for the bubble energy differs from one used by Vogel , which defines the bubble energy in terms of maximum achieved bubble radius with the vapor and ambient pressure. We used Eqn. 11 because we were able to measure all the parameters it contains (aside from the Hugonoit constants.)
3.4. Far field bubble size
The bubble diameter varied less than 10% from 100 µs to hundreds of seconds; we labeled the quasi-stable diameter the farfield bubble diameter, and measured it at 1ms. This size has a near linear relation to the maximum bubble size, Fig.9. It should also be noted that the bubble does not always collapse to the same center location, and was seen many times to collapse to one side, but its edge did not extend further than the edge of the maximum bubble diameter, and the bubble always retained its proportional size.
We used of a pair of Nd:YAG lasers, where the pump laser is frequency doubled and the probe laser was used in the fundamental 1064 nm. Timing delays were synchronized to a CCD and controlled by a DG535 delay generator. We show the life cycle of a laser generated bubble in agar gel. The formation of the bubble is marked first by the generation of a shock wave, and then the growth, collapse and formation of quasi-stable bubble which lasted for more than a second; the maximum and farfield bubble sizes were shown to be linearly related.
The shock wave has been shown to have been generated before the nine nanosecond laser pulse had finished, and originates at earlier times as the laser pulse energy increases. Using a simplified model to associate the shock wave generation time with an irradiance threshold gave an energy threshold for the formation of a shock wave of 71 µJ/pulse. Moreover, the shock wave velocity has been measured and used to find a pressure difference between the shocked and unshocked regions of 200 MPa at threshold, and up to 500 MPa; where the pressure scales with a near linear dependance on the laser pulse energy.
Using Eqn. 11 the energy in the bubble was inferred and an approximate threshold of bubble formation of 55 µJ/pulse was determined, and corresponded to visual observations. When the energy in the bubble to energy in the laser pulse ratio was considered, an increased absorption of laser power is seen, and corresponds with the measured transmittance of the laser pulse vs incident energy. There is sufficient evidence to suggest, in conjunction with several above mentioned authors, that the formation of bubbles in agar is a plasma mediated interaction, and that the plasma absorption, and shock wave production are an integral part of the interaction. Our system, because of its high spatial and temporal resolution (from nanoseconds to seconds), allowed us to measure with accuracy the plasma induced shock wave and use this data to model the interaction of laser light with the agar gel tissue-phantom. We believe this to be the first time that two electronically synchronized lasers have been used in this type of configuration to measure laser-induced bubbles and shock waves; and allowed us over nine orders of magnitude of temporal range with sub nanosecond resolution.
The authors would like to acknowledge support to the work presented here from CONACYT grants 51839 (CIAM) and 57309, UCMEXUS-CONACYT (Faculty Fellowship S. Camacho-López Sept-Oct 2007), CONACyT-UC MEXUS scholarship for graduate studies 204967 and NSF grant CTS-SGER:0609662. As well support from CICESE postgraduate department for student scholarship and many thanks to Teresa Santiago-Corona and Hogaza Hogaza for both love and support.
|1||1A Gaussian beam can be used to illuminate objects greater then its’ 1/e 2 diameter since there is still sufficient light arriving to the CCD to form an image; the imaging software may need its contrast enhanced.|
References and links
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