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1028 Applied Physics B – Lasers and OpticsFIGURE 10 Temperature evolution at the center of the laser focus producedby a series of 800-nm, 100-fs pulses focused into water. (a) 80-MHz repetitionrate, NA = 1.3; (b) 80-MHz repetition rate, NA = 0.6; (c) 1-MHzrepetition rate, NA = 0.6. The volumetric energy density deposited per pulseis always 1 J cm −3 at the focus center. The dashed lines represent thetemperature decay after a single pulse. For comparison, the temperature evolutionduring cw irradiation with the same average power as for the pulsedirradiation is also showndiameter of the initial temperature distribution (1/e 2 values)amounts to 336 nm and the length to 806 nm (Sect. 3.2). Figure10 shows the calculated temperature evolution at the centerof the laser focus when series of 800-nm, 100-fs pulses arefocused into water at different repetition rates (80 MHz and1MHz) and numerical apertures (NA = 1.3 and NA = 0.6).It was assumed that with each pulse an energy density of1Jcm −3 at the center of the initial temperature distribution isdeposited. For other values of the volumetric energy density,the shape of the temperature vs time curve will be the samebut the absolute value of the temperature varies proportionallyto the peak density of absorbed power, A. For comparison,we also calculated the temperature evolution during cw irradiationwith the same average power as for the pulsed irradiation(dotted lines in Fig. 10a–c). For 80-MHz repetition rate,pulsed and continuous energy deposition differ significantlyonly during the first 100 ns.The calculations in Fig. 10a for tightly focused irradiationwith 80-MHz repetition rate reveal that the temperature is only6.8 times larger after a few microseconds than the temperatureincrease caused by a single pulse. This implies that onlya moderate heat accumulation occurs during plasma-mediatedcell surgery. However, when the numerical aperture is reducedfrom NA = 1.3 to NA = 0.6, such as in Fig. 10b, a 45-foldtemperature increase is predicted. Temperature accumulationcan almost entirely be avoided if, at the same NA, the repetitionrate is lowered to 1MHz(Fig. 10c). In this case, thepeak temperature in a long pulse series is only 1.36 timeslarger than after a single pulse. For 1-MHz repetition rate andNA = 1.3, this factor reduces to 1.024.When laser surgery is performed with 80-MHz pulse seriesfocused at NA = 1.3, the boiling temperature of 100 ◦ Cwill, due to the 6.8-fold temperature accumulation, be reachedwhen each individual pulse produces a temperature rise of11.8 ◦ C (starting from 20 ◦ C room temperature). For 800-nm,100-fs pulses this temperature rise requires a free-electrondensity of ϱ c = 2.1 × 10 19 cm −3 , which is reached at an irradianceof 0.51 times the value required for optical breakdown(ϱ cr = 10 21 cm −3 ).The temperature distribution in the vicinity of the laser focusduring application of 80-MHz pulse series is presentedin Fig. 11 for different numerical apertures. The distributionarising from the ellipsoidal focus volume is plotted both inradial and axial directions. For NA = 1.3, the temperaturedistribution remains fairly narrow (FWHM ≈ 600 nm) evenafter a few milliseconds when a dynamic equilibrium betweenenergy deposition and heat diffusion has been established.The rapid decrease of the temperature with increasing distancefrom the laser focus is related to the small size of thefocal volume, which allows for rapid heat diffusion in all directions.By contrast, for NA = 0.6 the steady-state temperaturedistribution is more broadened compared to the singlepulsedistribution, in addition to the stronger increase of thepeak temperature. Both temperature accumulation and broadeningof the temperature distribution can be avoided withrepetition rates ≤ 1MHz.At first sight, the results of our temperature calculationsmight suggest that an irradiance range below the opticalbreakdown threshold exists where predominantly thermal effectsin biological media can be produced. However, oneneeds to consider that about 10 6 free electrons per pulse aregenerated in the focal volume at the irradiance that createsa temperature difference of 11.8 ◦ C per pulse and a peak temperatureof 100 ◦ C after a pulse series of several microseconds(for NA = 1.3). Any thermal denaturation of biomoleculeswill thus always be mixed with free-electron-induced chemicaleffects, and the latter will probably dominate.5.3 Comparison with cw irradiation of linear absorbersSince various researchers have produced cellularmicroeffects using long-pulsed or continuous-wave irradiation[47, 52, 56, 69–71], it is of interest to compare thewidths of the temperature distributions produced by series ofultra-short laser pulses and cw irradiation.Figure 12 shows the evolution of the temperature distributionwhen cw irradiation of 514-nm wavelength is fo-
VOGEL et al. Mechanisms of femtosecond laser nanosurgery of cells and tissues 1029FIGURE 11 Temperature distribution in radial direction (a, c) and axial direction (b, d) produced by series of 800-nm, 100-fs pulses focused into water atnumerical apertures of NA = 1.3(a, b)andNA= 0.6(c, d). The pulse-repetition rate was 80 MHz in both cases, and the volumetric energy density depositedat the focus center was 1 J cm −3 for each pulsecused into a linearly absorbing aqueous medium at NA =1.3. The temperature distribution is slightly broader (FWHM850 nm) than that in Fig. 11a arising from nonlinear absorptionof 80-MHz IR femtosecond pulse trains (FWHM600 nm). The temperature distribution produced by femtosecondpulse trains is narrower because it originates from thefree-electron distribution (18) rather than from the irradiancedistribution (17) that is relevant for linear energy deposition.However, the spatial resolution of femtosecond lasersurgery is not determined by the steady-state temperature distributionbut by the width of the free-electron distributionitself, as we shall see in Sect. 7.1. Therefore, the spatial resolutionof fs laser surgery is considerably better than that ofa cw microbeam.FIGURE 12 Evolution of the temperature distribution in radial directionproduced by continuous energy deposition in a linearly absorbing aqueousmedium at a laser wavelength of 514 nm. The calculations were performedfor the same absorbed average power as in the case of pulsed, nonlinearenergy deposition presented in Fig. 116 Thermoelastic stress generationand stress-induced bubble formation6.1 Calculation of stress distributionand bubble formationThe temperature rise in the focal volume occursduring thermalization of the energy carried by the free electrons,i.e. within a few picoseconds to tens of picoseconds(see Sect. 5.1). This time interval is much shorter than theacoustic transit time from the center of the focus to its periphery.Therefore, no acoustic relaxation is possible during thethermalization time, and the thermoelastic stresses caused bythe temperature rise stay confined in the focal volume, leadingto a maximum pressure rise [2, 155, 156]. Conservation ofmomentum requires that the stress wave emitted from a finitevolume within an extended medium must contain both compressiveand tensile components such that the integral of thestress over time vanishes [155, 157]. The tensile stress wavemay induce fracture of the material even after a temperaturerise too small to produce thermal damage [158]. In water, itwill cause the formation of a cavitation bubble when the tensilestrength of the liquid is exceeded. For cell surgery, thethreshold for bubble formation defines the onset of disruptivemechanisms contributing to dissection.To determine the evolution of the thermoelastic stress distributionin the vicinity of the laser focus, we solved the threedimensionalthermoelastic wave equation. A starting point for
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