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

1030 Applied Physics B – Lasers and Opticsthe calculation of the thermoelastic stress wave propagation isthe temperature distribution at the end of a single femtosecondlaser pulse that reproduces the free-electron distribution describedby (17). In the following calculations, this temperaturedistribution is characterized by T max , the temperature in ◦ C inthe center of the focal volume. From this temperature distributionthe initial thermoelastic pressure (right after the laserpulse, before the acoustic wave has started to propagate) wascalculated usingp(r) =T∫2 (r)β(T )K(T )T 1dT , (20)where T 1 = 20 ◦ C is the temperature before the laser pulse,and T 2 (r) the temperature of the plasma after the laserpulse, which depends on the location within the focal volume.The temperature dependence of the thermal expansioncoefficient β and the compressibility K was takeninto account, using values for metastable water from [159].The time- and space-dependent pressure distribution p(r, t)due to the relaxation of the initial thermoelastic pressurewas calculated using a k-space (spatial frequency) domainpropagation model [160, 161].Because the heated volume is very small (≈ 0.07 µm 3 )and the region subjected to large tensile stress amplitudesis even smaller (see Fig. 14, below), the presence of inhomogeneousnuclei that could facilitate bubble formation isunlikely. Therefore, we have to consider the tensile strengthof pure water to estimate the bubble-formation threshold infemtosecond optical breakdown. Traditionally, the rupture ofa liquid achieved by tensile stress under isothermal conditionsis called ‘cavitation’ while bubble formation due toheating under isobaric conditions is called ‘boiling’ [162].Such a distinction becomes obscure when targets are bothheated and stretched under conditions of stress confinement.We use the crossing of the ‘kinetic spinodal’ as defined byKiselev [163, 164] as threshold criterion for bubble formation.In the thermodynamic theory of phase transitions, thelocus of states of infinite compressibility (∂p/∂V) T = 0, thespinodal, is considered as a boundary of fluid metastable (superheated)states. Physically, however, the metastable statebecomes short-lived due to statistical fluctuations well beforethe spinodal is reached [159, 165]. The kinetic spinodal is thelocus in the phase diagram where the lifetime of metastablestates becomes shorter than a relaxation time to local equilibrium.If the surface tension is known, the physical boundaryof metastable states in this approach is completely determinedby the equation of state only, i.e. by the equilibrium propertiesof the system [163, 164]. This feature distinguishes the kineticspinodal from the homogeneous nucleation limit derived earlierby Fisher, which depends on the size of the volume underconsideration and the duration of the applied stress [166].Unlike Fisher’s equation, the kinetic spinodal reproduces theshape of the spinodal and is applicable in the entire temperaturerange from room temperature to the critical point. Thenucleation thresholds in cells will probably resemble those inpure water because the biomolecules are too small to serve asboundaries for heterogeneous nucleation.FIGURE 13 (a) Binodal, spinodal, and kinetic spinodal of water as a functionof temperature, calculated with the analytic equation of state of Sauland Wagner [240]. The dotted curve corresponds to the homogeneous nucleationlimit in Fisher’s theory [166], the circles indicate experimental dataof Skripov et al. [159], and the triangles indicate the experimental data ofZheng et al. [167] recalculated in p–T coordinates. Figure reproduced withpermission from Ref. [163]. Copyright 1999 Elsevier Science B.V. (b) Peakcompressive and tensile thermoelastic stresses in the focus center producedby a 800-nm, 100-fs pulse focused into water at NA = 1.3, plotted togetherwith the binodal and the kinetic spinodalFigure 13a presents the saturated liquid/vapor curve (binodal),the spinodal, the kinetic spinodal, and the homogeneousnucleation limit derived by Fisher in a p vs T projection of thethermodynamic phase diagram for water. For comparison, experimentaldata by Skripov et al. on the empirical limit of themetastable region and by Zheng et al. on the tensile strengthof water are also shown [159, 167]. In Fig. 13b, the kineticspinodal is plotted together with the peak compressive andtensile thermoelastic stresses in the focus center that are producedwhen an 800-nm, 100-fs pulse is focused into water atNA = 1.3. The temperature at which the tensile stress curvereaches the kinetic spinodal is defined as the bubble-formationthreshold. For larger laser pulse energies, the kinetic spinodalwill be reached in an increasingly large part of the focalregion.To calculate the dynamics of the cavitation bubble producedafter crossing the kinetic spinodal, first the size ofthe bubble nucleus was determined. It was identified withthe extent of the region in which the negative pressure exceedsthe kinetic spinodal limit p(r, t)