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1022 Applied Physics B – Lasers and OpticsTo obtain the evolution of the free-electron density duringa Gaussian laser pulse[ ( ) ]t2I(t) = I 0 exp −4ln2 , (12)τ Lthe total rate equation( )dϱ c dϱc=+dt dtphoto( ) dϱc+dt casc( ) dϱc+dt diff( ) dϱcdt rec(13)was solved numerically for various laser pulse peak intensitiesI using a Runge–Kutta method with adaptive step-sizecontrol. Separate book-keeping was used for the contributionof (5) to evaluate the influence of multiphoton and cascadeionization. The breakdown threshold is defined as the irradianceI rate required to produce a maximum electron densityϱ max during the laser pulse that equals the critical densityϱ cr = 10 21 cm −3 . Besides the time evolution of the electrondensity, we also assessed the dependence of the maximumelectron density on irradiance, by calculating ϱ max asa function of I/I rate .2.3 Evolution of free-electron densityand breakdown thresholdsThe top row of Fig. 3 presents the evolution of thefree-electron density ϱ c during the laser pulse at the opticalbreakdown threshold for 6-ns, 1064-nm pulses, and for100-fs, 800-nm pulses. To facilitate a comparison betweenthe different pulse durations, the time t is normalized withthe respective laser pulse duration τ L . The contribution ofphotoionization to the total free-electron density is plotted asa dotted line. The bottom row of Fig. 3 shows how the maximumfree-electron density achieved during the laser pulsedepends on irradiance.It is obvious that the dynamics of plasma formation isextremely different for nanosecond and femtosecond pulses.With nanosecond pulses, no free electrons are formed forirradiance values below the optical breakdown threshold becausethe irradiance is too low to provide seed electrons bymeans of multiphoton ionization (Fig. 3c). Once the irradianceis high enough to provide a seed electron, the ionizationcascade can start. It proceeds very rapidly owing to the highirradiance (Fig. 3a). The electron density shoots up by nineorders of magnitude within a small fraction of the laser pulseduration until its rise is stopped by recombination, which isproportional to ϱ 2 c . The breakdown threshold is, hence, extremelysharp – either a highly ionized plasma is produced,or no plasma at all. These numerical predictions are supportedby the experimental observation that at the thresholdof nanosecond optical breakdown with IR laser pulsesthe transmission of the focal volume drops abruptly to lessthan 50% of the value without plasma formation [126, 127].The transmission loss for shorter pulse durations is muchless abrupt [80, 126–128].With femtosecond pulses, a much higher irradiance is necessaryfor optical breakdown to be completed during thelaser pulse duration than with nanosecond pulses. This favorsthe generation of free electrons through multiphoton ionizationbecause of its stronger irradiance dependence – ∝ I k asopposed to ∝ I for the cascade ionization rate (see Sect. 2.2).While with nanosecond pulses the total number of free electronsgenerated through avalanche ionization is 10 9 timeslarger than the number generated through multiphoton ionization(Fig. 3a), it is only 12 times larger with 100-fs pulses at800 nm (Fig. 3b). As a consequence of the increasing importanceof multiphoton ionization with shorter pulse durations,there is never a lack of seed electrons for avalanche ionization.An avalanche is initiated at irradiance values considerablylower than the breakdown threshold. The free-electron densityreached at the end of the avalanche depends on irradianceFIGURE 3 Top row: evolution of the free-electrondensity during the laser pulse at the optical breakdownthreshold for 6-ns, 1064-nm pulses and for 100-fs,800-nm pulses. The time t is normalized with respectto the laser pulse duration τ L . The contribution of multiphotonionization to the total free-electron density isplotted as a dotted line. Bottom row: maximum freeelectrondensity ϱ max achieved during the laser pulseas a function of irradiance, for the same laser parameters.The irradiance I is normalized with respect tothe threshold irradiance I rate . The threshold I rate andthe corresponding value of ϱ max are marked by dashedlines
VOGEL et al. Mechanisms of femtosecond laser nanosurgery of cells and tissues 1023in a much smoother way (Fig. 3d) than for ns pulses (Fig. 3c).Therefore, one can generate any desired free-electron densityby selecting an appropriate irradiance value.Figure 4 presents threshold values for irradiance, I rate ,andradiant exposure, F rate = I rate × τ L , required to reach a criticalfree-electron density of ϱ cr = 10 21 cm −3 . The thresholds werecalculated for various wavelengths and pulse durations rangingfrom 10 fs to 10 ns. Two regimes can be distinguished: forτ L < 10 ps, the threshold radiant exposure F rate exhibits onlya weak dependence on pulse duration. This reflects the factthat recombination plays only a minor role during ultra-shortlaser pulses. Therefore, only one set of free electrons is producedthat corresponds to an approximately constant energydensity within the focal volume. This is in accordance with theexperimental threshold criterion of bubble formation that requiresa specific energy density, which varies little with laserparameters. By contrast, for longer pulses more than one setof free electrons is produced and they recombine during thelaser pulse. Here it is the threshold irradiance I rate that remainsapproximately constant, because a minimum irradiance is requiredto provide the seed electrons for the ionization cascadeby multiphoton ionization and to drive the cascade sufficientlyfast to reach the critical free-electron density within the laserpulse duration. As a consequence of the constant threshold irradiance,the radiant exposure threshold and plasma energydensity increase steeply with increasing pulse duration.The predicted form of the F rate (τ L ) dependence qualitativelymatches experimental observations of the pulsedurationdependence of single-shot damage thresholds at surfacesof transparent large-band-gap dielectrics [112, 129] andablation thresholds of corneal tissue [13, 130]. However, studiesin which single-shot thresholds at longer pulse durationsare mixed with multiple-shot thresholds at ultra-short durationsshow a steeper F rate (τ L ) dependence for τ L < 10 ps bothin corneal tissue [131, 132] and dielectrics [116, 123]. Thelower thresholds with multiple exposures are due to accumulativeeffects, the possibility of which is explained by thesmooth ϱ max (I/I rate ) dependence shown in Fig. 3d.The predicted wavelength dependence in the picosecondand nanosecond regimes (increasing threshold with decreasingwavelength) seems to be a little surprising at first sight,because multiphoton processes occur more easily at shorterwavelengths. However, one needs to keep in mind that the cascadeionization rate increases approximately proportionally tothe square of the laser wavelength, as evident from (8).2.4 Low-density plasmas in bulk mediaFIGURE 4 Calculated optical breakdown thresholds (ϱ cr = 10 21 cm −3 )asa function of laser pulse duration for various laser wavelengths. (a) Irradiancethresholds, (b) radiant exposure thesholdsOur numerical calculations for femtosecond breakdownin bulk transparent media indicate that it is possible tocreate low-density plasmas in which the energy density remainsbelow the level that leads to cavity formation in themedium. Experimental evidence for the existence of lowdensityplasmas was recently provided by Mao et al. [18]through measurements of the free-electron density in MgOand SiO 2 . Free electrons are produced in a fairly large irradiancerange below the optical breakdown threshold, witha deterministic relationship between free-electron density andirradiance. Low-density plasmas thus offer the possibility todeliberately produce chemical changes, heating, and thermomechanicaleffects by varying the irradiance. These effects arevery well localized because of the nonlinearity of the plasmaformationprocess, which, for sufficiently small irradiances,allows us to produce a plasma in a volume that is smaller thanthe diffraction-limited focus.For larger irradiances, plasmas in bulk media grow beyondthe region of the beam waist, which is not possible forplasma formation at surfaces [116, 120, 121]. At surfaces, theenergy deposition becomes confined to a thin layer of less than100-nm thickness once the free-electron density reaches thecritical density, because the superficial plasma layer is highlyabsorbing and reflecting [116, 121, 133–135]. By contrast, inbulk media there is no restriction for the region of opticalbreakdown to spread towards the incoming laser beam withincreasing irradiance. At large irradiances, breakdown startsto occur before the femtosecond pulse reaches the beam waist,and both irradiance and beam propagation are influenced bythe plasma generation [21, 136]. These effects shield the focalregion, enlarge the size of the breakdown region, and limitthe free-electron density and energy density reached in the entirebreakdown volume [21, 80, 137–139]. Low-density plasmascan, therefore, easily be produced in bulk media whileat surfaces the self-induced confinement of plasma formationto a thin layer leads to a rapid rise of free-electron densitywith irradiance, and the irradiance range in which low-densityplasmas can be formed is very small [116, 120].The desired chemical or physical effects of low-densityplasmas can be precisely selected if the slope of theϱ max (I/I rate ) curve is small because that offers a large
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