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1024 Applied Physics B – Lasers and OpticsFIGURE 5 Maximum free-electron density as a function of irradiance,ϱ max (I/I rate ), for 100-fs pulses at 1064-nm, 532-nm, and 355-nm wavelengths.The normalized threshold (I/I rate = 1) and the corresponding valueof ϱ max are marked by dotted lines‘tuning range’ of the irradiance for each effect. Figure 5 showsthat the tuning range increases for shorter laser wavelengths.3 Irradiance and free-electron distributionswithin the focal volume3.1 Shape of the focal volumeThe temperature and stress distribution in the focalregion depend on the distribution of quasi-free electronsproduced during femtosecond optical breakdown. Therefore,we must first explore the shape of the irradiance and freeelectrondensity distributions within the focal volume beforewe can investigate the resulting temperature and stress effects.Because of the nonlinearity of the breakdown process, thefree-electron distribution is narrower than the irradiance distributionin the focal volume. A description of their relationwill thus also allow us to estimate the possible increase of thespatial precision of the laser effects beyond the level achievablewith techniques that are based on linear absorption.The irradiance distribution in the focal volume of a diffraction-limitedoptical system for a focusing angle of α = 45 ◦ isreproduced in Fig. 6 from the textbook of Born and Wolf [140](α is the half-angle of the light cone such as used in thedefinition of the numerical aperture NA = n 0 sin α). Theisophotes (contour lines for equal irradiance) reveal that thefocal volume in the center of the focal region has an approximatelyellipsoidal shape. A similar structure was obtainedexperimentally when the irradiance distribution ina confocal laser scanning microscope (CLSM) was measuredby scanning the tip of a scanning near field opticalmicroscope (SNOM) through the focal region (Fig. 7), andby a surface-plasmon-based beam-profiling technique [141].For our numerical simulations, the focal volume will thereforebe approximated by an ellipsoid with short axis d and longaxis l.The short axis d of the ellipsoid is identified with the diameterof the central maximum of the Airy pattern in the focalplane that is given byd = 1.22 λNA . (14)The symbol λ refers to the vacuum wavelength of light. Therefractive index of the medium is contained in the value ofthe numerical aperture (NA) of the microscope objective. Theratio l/d of the long and short axes can be obtained from theFIGURE 6 Isophotes (contour lines for equal irradiance) in the focal region of a diffraction-limited microscope objective used to focus a plane wave. Thedashed lines represent the boundary of the geometrical focus. The focusing angle of α = 45 ◦ corresponds to a numerical aperture of NA = 0.94 in water.When the figure is rotated around the u axis, the minima on the v axis generate the Airy dark rings. The figure is taken from Ref. [140], p. 440
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
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