Invited p aper Mechanisms of femtosecond laser nanosurgery of ...
Invited p aper Mechanisms of femtosecond laser nanosurgery of ...
Invited p aper Mechanisms of femtosecond laser nanosurgery of ...
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VOGEL et al. <strong>Mechanisms</strong> <strong>of</strong> <strong>femtosecond</strong> <strong>laser</strong> <strong>nanosurgery</strong> <strong>of</strong> cells and tissues 1025<strong>laser</strong> pulse is approximately proportional to I k ,wherek is thenumber <strong>of</strong> photons required for multiphoton ionization. Thissimplifying assumption corresponds to the low-intensity approximation<strong>of</strong> the Keldysh theory and neglects the weakerirradiance dependence <strong>of</strong> avalanche ionization that usuallydominates plasma formation during the second half <strong>of</strong> a <strong>laser</strong>pulse (Fig. 3b). For ϱ max ≤ 5 × 10 20 cm −3 , the proportionalityϱ max ∝ I k has been confirmed by the experimental results<strong>of</strong> Mao et al. [18]. The spatial distribution <strong>of</strong> 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 <strong>laser</strong> scanning microscopemeasured by scanning the tip <strong>of</strong> a scanning near field optical microscopethrough the focal region <strong>of</strong> a Zeiss axiovert 100/C-Apo ×40 NA = 1.2water-immersion microscope objective. The measurement was performed fora <strong>laser</strong> wavelength <strong>of</strong> λ = 488 nm; the isocontour lines refer to 46% <strong>of</strong> themaximum irradiance (courtesy <strong>of</strong> 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 <strong>of</strong> 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 <strong>of</strong> d = 750 nm and l = 1800 nm.3.2 Irradiance and electron-density distributionswithin the focal volumeThe mathematical form <strong>of</strong> the diffraction-limitedirradiance distribution in the Fraunh<strong>of</strong>er diffraction pattern <strong>of</strong>a microscope objective (Fig. 6) is too complex for convenientcomputation <strong>of</strong> the temperature and stress evolution inducedby optical breakdown. We approximate the ellipsoidal region<strong>of</strong> 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 <strong>of</strong> the ellipsoid. The boundaries <strong>of</strong> the ellipsoidcorrespond to the 1/e 2 values <strong>of</strong> the Gaussian irradiancedistribution.To derive the free-electron distribution ϱ max (r, z) fromthe irradiance distribution I(r, z), we assume that for <strong>femtosecond</strong>pulses the free-electron density at the end <strong>of</strong> 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 <strong>of</strong> √ 5 = 2.24, which corresponds toa reduction <strong>of</strong> the affected volume by a factor <strong>of</strong> 11.2. Thediameter <strong>of</strong> 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 <strong>of</strong> the nonlinearity<strong>of</strong> 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 <strong>of</strong> 355 nm is used instead <strong>of</strong> 800 nm, the width<strong>of</strong> the diffraction-limited irradiance distribution decreases bya factor <strong>of</strong> 2.25 but the plasma diameter decreases by a factor<strong>of</strong> only 1.42 because the order <strong>of</strong> the multiphoton processis reduced from 5 to 2 and the irradiance distributionis less strongly narrowed in the process <strong>of</strong> 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 <strong>laser</strong> pulse energy is raised above the opticalbreakdown threshold, the spatial distribution <strong>of</strong> the freeelectrondensity broadens because nonlinear absorption <strong>of</strong><strong>laser</strong> light occurs upstream <strong>of</strong> the <strong>laser</strong> 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 <strong>of</strong> the temperature and stress evolutioninduced by <strong>femtosecond</strong> optical breakdown