# Femtosecond Optical Nonlinearity of Nanodiamond Suspensions
## 3 Results  
The dependences T\({}_{\rm{oa,n}}\)(z) and t\({}_{\rm{nr}}\)(z) obtained for a cell filled with distilled water (a,b) and an empty cell (c,d) at E\({}_{\rm{in}}\) = 200 nJ are presented in Figure 5a-d. The dependence T\({}_{\rm{oa,n}}\)(z) obtained for a cell with distilled water shows a small dip in the vicinity of z/z\({}_{0}\) = 0 indicating a weak nonlinear absorption. At the same time, the experimental data T\({}_{\rm{oa,n}}\)(z) obtained for an empty cell (Figure 5c) shows no corresponding dip. This means that the observed weak nonlinear absorption for a cell with distilled water occurs because of distilled water and not because of the cell. The nonlinear absorption in distilled water originates from multi-photon absorption [60]. Comparison of the closed/open-aperture data for a cell filled with water (Figure 5b) and an empty cell (Figure 5d) shows that the observed nonlinear refraction leading to self-focusing of the femtosecond laser beam occurs mainly in the quartz walls of the optical cell. The observed cluster of dots at the vicinity of z/z\({}_{0}\) = 0 in Figure 5d arises from the quartz walls-air-quartz walls boundary and not affecting on the amplitude of the normalized transmittance. Closed/open-aperture measurements were also performed for both the studied types of nanodiamonds with the results shown in Figure 5e,f). Adding the nanoparticles to water has no noticeable impact on the nonlinear refraction of the suspensions as one can observe by comparing Figure 5b,d-f) indicating weak nonlinear refraction in both suspensions for given concentrations. Figure 4: Sketch of the z-scan experimental setup. Figure 5: The normalized transmittance of the quartz cell with (**a**) distilled water and (**c**) an empty cell obtained in the open-aperture z-scan. The ratio of the nonlinear transmittance obtained in the closed to the data obtained in the open-aperture z-scan for the quartz cell (**b**) with distilled water, (**c**) with HP-HT nanodiamond suspension, (**f**) with detonation nanodiamond suspension, and (**d**) an empty cell. The measurements were performed at the incident laser pulse energy of 200 nJ. The results of the open-aperture normalized transmittance measurements for aqueous suspensions of HP-HT nanodiamonds are shown in Figure 6. The measurements were performed at the same incident pulse energy of E\({}_{\text{in}}\) = 200 nJ. In order to reveal the dependence of the nonlinear optical properties, i.e., nonlinear absorption on the nanoparticle concentration C, we performed the measurements for the C ranging from 0.1 to 1 wt%. It is shown that the higher the concentration, the stronger the nonlinear absorption, the lower the transmittance at z/z\({}_{0}\). Even at a low concentration of 0.08 wt%, the nonlinear transmittance exhibits a visible dip. It is noteworthy that the dependencies shown in Figure 6 are symmetric relative to z/z\({}_{0}\) = 0 indicating prominent optical limiting properties of femtosecond laser pulses. Figure 6: (**a–e**) Normalized transmittance of suspensions of HP-HT nanodiamonds with different concentrations recorded in the open-aperture z-scan. Dots correspond to the experimental data while solid lines represent the result of the fitting with Equation (2). Figure (**f**) represents the dependence of the calculated two-photon absorption coefficient \(\beta\) as a function of concentration C.  
One can assume that the mechanism of optical limiting in suspensions of HP-HT nanodiamonds is two-photon absorption, in which the absorption coefficient \(\alpha_{TPA}\) of the medium depends on the radiation intensity I according to the equation [61]  
\[\alpha_{TPA}(I)=\alpha+\beta I, \tag{1}\]  
where \(\alpha\) and \(\beta\) are the coefficients of linear and nonlinear absorption, respectively.