Invited p aper Mechanisms of femtosecond laser nanosurgery of ...

VOGEL et al. Mechanisms of femtosecond laser nanosurgery of cells and tissues 1033FIGURE 17 Stress-wave amplitude for 100-fs pulses and longer pulse durationsas a function of the dimensionless laser pulse energy E/E th .Thepressure amplitudes were measured by means of a hydrophone at 6-mmdistance from the laser focusat 6-mm distance is 0.12 MPa (Fig. 17). The resulting decayconstant is n = 1.13. Previous investigations of complete p(r)curves for spherical shock waves produced by 30-ps pulses of50-µJ energy yielded a very similar constant of n = 1.12 overa large range of propagation distances [96].An estimate of the pressure value at 6-mm distance producedby a 100-fs pulse with threshold energy was obtained byextrapolating the data for E/E th ≥ 1.5 in Fig. 17 to E = E th .It is about 0.008 MPa. The plasma radius at E th was identifiedwith the focal radius of 2.2 µm measured using a knife-edgetechnique [127]. A pressure of p = 0.008 MPa at 6-mm distancecorresponds to a pressure value of 61 MPa at the plasmarim when a decay constant n = 1.13 isassumed,andtoavalueof 56 MPa with a decay constant of n = 1.12, respectively.Our calculations of the thermoelastic stress generation predicta peak pressure of 168 MPa at the bubble-formation threshold(see Sect. 6.3, below). According to Fig. 14, the stress transientthat leaves the heated region in the radial direction hasa peak pressure of ≈ 25% of the maximum compressive amplitudewithin the focal volume. We thus obtain a theoreticalprediction of 42 MPa for the amplitude of the thermoelasticstress wave at the plasma rim. Considering the uncertainties inthe location of the plasma rim and the differences in numericalaperture between experiment and calculation, the agreementbetween experimental results (56–61 MPa) and calculateddata (42 MPa) is very good.Both experiments and calculations reveal that stress confinementin femtosecond optical breakdown results in thegeneration of high pressure values even though the temperaturerise is only relatively small. In a purely thermal process,a temperature rise to, for example, 200 ◦ C starting fromroom temperature would produce a saturation vapor pressureof 1.6MPa. The compressive pressure transient produced bya temperature rise of 180 ◦ C under stress-confinement conditionshas an amplitude of 220 MPa, which is more than twoorders of magnitude larger than the vapor pressure.The situation is different for optical breakdown at longerpulse durations where the stress-confinement condition is notfulfilled. Here, high pressures are always associated with hightemperatures and plasma energy densities. For pulses whichare considerably longer than the thermalization time of thefree-electron energy, a dynamic equilibrium between generationof free electrons and thermalization of their energy is establishedduring the laser pulse [81]. This leads to a high valueof the plasma energy density [80] and a temperature of severalthousand degrees Kelvin [175–177]. The duration of the resultingshock wave is determined by the time it takes for thehigh pressure within the plasma to decrease during the plasmaexpansion [96], which for NA = 0.9 was found to be about25–40 ns [3]. By contrast, the duration of the thermoelasticstress transients is determined by the geometric dimensionsof the breakdown volume, which are in the sub-micrometerrange. This leads to a duration of the stress transients of theorder of or less than 300 ps (Fig. 15). The short duration ofthese stress waves is correlated with a small energy content(see also Sect. 6.4).6.3 Threshold for stress-induced bubble formationA specific feature of the stress transients generatedduring fs optical breakdown is their tensile componentthat is related to the high degree of stress confinement duringenergy deposition. The tensile stress makes it possiblethat a cavitation bubble can be generated by a relatively smalltemperature rise in the liquid. The threshold for bubble formationis defined by the temperature rise leading to a crossingof the kinetic spinodal, as shown in Fig. 13. For λ = 800 nm,NA = 1.3, and a room temperature of 20 ◦ C, the critical temperaturerise and the corresponding critical tensile stress are∆T = 131.5 ◦ C and p =−71.5MPa, respectively. The correspondingcompressive pressure is 168 MPa.The temperature rise of 131.5 ◦ C at the threshold for bubbleformation corresponds to an increase in energy densityof 551 J cm −3 which, according to (18), is produced bya free-electron density of ϱ c = 0.236 × 10 21 cm −3 . This electrondensity is less than the breakdown criterion of ϱ cr =10 21 cm −3 assumed in our numerical calculations and in mostother theoretical studies of plasma formation. The discrepancybetween the threshold values relying on different breakdowncriteria needs to be kept in mind when comparing the resultsof experimental studies, where bubble formation servesas breakdown criterion, with those of numerical simulations.The fact that femtosecond optical breakdown is associatedwith only a relatively small temperature rise explainswhy plasma luminescence is no longer visible for pulse durationsshorter than about 10 ps [122, 128]. For pulse durationslonger than the thermalization time, large amounts of energyare transferred from the free electrons to the heavy particlesduring the laser pulse [81], resulting in a temperature of severalthousand degrees Kelvin, bubble formation, and a brightplasma luminescence [175–177]. By contrast, a peak temperatureof 151.5 ◦ C reached at the threshold for bubble formationwith 100-fs pulses is too low to produce black-body radiationin the visible range of the optical spectrum. Moreover,the recombination radiation of femtosecond-laser-producedplasmas is weak because only one ‘set’ of free electrons is producedthat recombines after the end of the laser pulse. Theenergy density will remain small also at energies above thethreshold for bubble formation because of the shielding of thefocal region by plasma formation upstream of the focus [80,126, 136–139] (see Sect. 3.2). Therefore, bubble formation is