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

VOGEL et al. Mechanisms of femtosecond laser nanosurgery of cells and tissues 1025laser pulse is approximately proportional to I k ,wherek is thenumber of photons required for multiphoton ionization. Thissimplifying assumption corresponds to the low-intensity approximationof the Keldysh theory and neglects the weakerirradiance dependence of avalanche ionization that usuallydominates plasma formation during the second half of a laserpulse (Fig. 3b). For ϱ max ≤ 5 × 10 20 cm −3 , the proportionalityϱ max ∝ I k has been confirmed by the experimental resultsof Mao et al. [18]. The spatial distribution of the free-electrondensity can thus be expressed as[ ( r2)]ϱ max (r, z) = ϱ max [I(0, 0)] exp −2ka 2 + z2b 2 . (17)FIGURE 7 Irradiance distribution in a confocal laser scanning microscopemeasured by scanning the tip of a scanning near field optical microscopethrough the focal region of a Zeiss axiovert 100/C-Apo ×40 NA = 1.2water-immersion microscope objective. The measurement was performed fora laser wavelength of λ = 488 nm; the isocontour lines refer to 46% of themaximum irradiance (courtesy of Volker Jüngel and Tilo Jankowski, CarlZeiss Jena)relationdl=1 − cos α, (15)(3 − 2cosα − cos 2α)1/2which was derived by Grill and Stelzer for optical setups withvery large solid angles [142]. For NA = 1.3, whichinwatercorresponds to an angle of α = 77.8 ◦ ,wefindl/d = 2.4.A similar value is also obtained from the experimental datain Fig. 7. For λ = 800 nm, the above considerations yield focaldimensions of d = 750 nm and l = 1800 nm.3.2 Irradiance and electron-density distributionswithin the focal volumeThe mathematical form of the diffraction-limitedirradiance distribution in the Fraunhofer diffraction pattern ofa microscope objective (Fig. 6) is too complex for convenientcomputation of the temperature and stress evolution inducedby optical breakdown. We approximate the ellipsoidal regionof high irradiance in the focus by a Gaussian function( r2I(r, z) = I(0, 0) exp[−2a 2 + z2b 2 )], (16)where r and z are the coordinates in radial and axial directions,respectively, and a = d/2 and b = l/2 denote the shortand long axes of the ellipsoid. The boundaries of the ellipsoidcorrespond to the 1/e 2 values of the Gaussian irradiancedistribution.To derive the free-electron distribution ϱ max (r, z) fromthe irradiance distribution I(r, z), we assume that for femtosecondpulses the free-electron density at the end of theFigure 8 shows the irradiance and electron-density distributionsin the focal region according to (16) and (17) forNA = 1.3 and λ = 800 nm, forwhichk = 5. Due to the nonlinearabsorption process underlying optical breakdown, thefree-electron distribution is much narrower than the irradiancedistribution. For λ = 800 nm and breakdown in water, itis narrower by a factor of √ 5 = 2.24, which corresponds toa reduction of the affected volume by a factor of 11.2. Thediameter of the free-electron distribution at the 1/e 2 valuesamounts to 336 nm and the length to 806 nm.It is interesting to note that the influence of the nonlinearityof the absorption process in plasma-mediated surgeryconsiderably reduces the gain in spatial resolution that can beachieved by using a shorter wavelength. For example, whena wavelength of 355 nm is used instead of 800 nm, the widthof the diffraction-limited irradiance distribution decreases bya factor of 2.25 but the plasma diameter decreases by a factorof only 1.42 because the order of the multiphoton processis reduced from 5 to 2 and the irradiance distributionis less strongly narrowed in the process of plasma formation.However, the irradiance range leading to low-densityplasma formation is much broader for the shorter wavelengths(Fig. 5) thus making it easier to “tune” chemical and physicaleffects.When the laser pulse energy is raised above the opticalbreakdown threshold, the spatial distribution of the freeelectrondensity broadens because nonlinear absorption oflaser light occurs upstream of the laser focus and limits theFIGURE 8 Normalized irradiance distribution (a) and electron-density distribution(b) in the focal region for NA = 1.3 andλ = 800 nm that areassumed for the numerical calculations of the temperature and stress evolutioninduced by femtosecond optical breakdown