# Femtosecond Optical Nonlinearity of Nanodiamond Suspensions
## 3 Results
This can be verified by approximating the experimental data presented in Figure 6 with the well-known equation (see [61]) derived to describe two-photon absorption for the open-aperture z-scan for a Gaussian laser beam and temporally Gaussian pulse:  
\[T_{\alpha,n}=1+\sum_{m=1}^{\infty}\left[-q(z,t=0)\right]^{m}/(m+1)^{3/2}, \tag{2}\]  
where \(q_{0}(z,\ t=0)=\beta I_{0}(t=0)L_{eff}/(1+z^{2}/z_{0}^{2}),L_{eff}=\left(1-e^{ -\alpha L}\right)/\alpha\), \(\alpha=-lnT_{0}/L\), \(I_{0}\)\((t=0)=E_{in}/[\tau w_{0}^{2}\Delta t\frac{\sqrt{\pi}}{2\sqrt{ln2}}]\) (see [58]), \(I_{0}(t)\) is the on-axis irradiance at focus (i.e., z = 0), \(\Delta t\) is the pulse duration (FWHM) of a Gaussian laser pulse, T\({}_{0}\) is the linear transmittance of the studied sample with thickness L. Equation (2) with the first four terms from an infinite series of elements is in good agreement with the experimental z-scan data shown in Figure 6. The values of \(\beta\) found as a result of the approximation of the experimental data, depending on the concentration of nanoparticles C, are shown in Figure 6f and can be expressed as \(\beta(\text{m}/\text{W})=(5.7+22.1\times\text{C}(\text{wt}\%))\times 10^{-15}\). It follows from the approximation, that the defined \(\beta\) does not vanish when the concentration of nanoparticles in suspension is tending to zero confirming the earlier obtained z-scan data for a cell filled with distilled water (see Figure 5a). Based on the above, one can conclude that the two-photon absorption well describes the mechanism of the nonlinear behavior in aqueous suspensions of HP-HT nanodiamonds. The results of the normalized transmittance recorded in the open-aperture z-scan for a fixed concentration of aqueous suspension of detonation nanodiamonds are shown in Figure 7. We have not presented the measurements for the various concentrations as for HP-HT nanodiamonds because those have been already reported [49]. Instead, measurements of the normalized transmittance with variation of the incident laser pulse energy have been performed. In contrast to HP-HT nanodiamonds, suspensions of detonation nanodiamonds exhibit an increase in the nonlinear transmittance with increasing incident light intensity. In the range of incident laser pulse energies, the experimental curves of T\({}_{\alpha,n}(z)\) presented in Figure 7 are symmetrical relative to the focal point z/z\({}_{0}\) = 0. This suggests that the observed phenomenon is not associated with laser-induced irreversible bleaching [62; 63], but originates from the saturable absorption which has been extensively studied in graphene, carbon nanotubes [15; 23; 24; 64; 65] and has also been reported in detonation nanodiamonds with larger particle sizes under femtosecond excitation [50]. Similar to graphene and carbon nanotubes the observed saturable absorption in detonation nanodiamonds is characterized by a short-term decrease of its absorption coefficient during the passing of a light pulse of high intensity and caused by the electronic transition between two energy levels. In accordance with [65; 66], the absorption coefficient at saturable absorption is given by:  
\[\alpha_{SA}(I)=\alpha_{ns}+\alpha_{0}/(1+I/I_{sat}) \tag{3}\]  
where \(\alpha_{ns}\) is the linear absorption coefficient due to the presence of light losses in the medium that are not associated with saturable absorption, \(\alpha_{0}\) is the coefficient characterizing the absorption in a two-level energy system, where the population moves under the action of resonant laser pumping, leading to equalization of the level populations at high intensities (saturation of absorption) [67], \(I_{sat}\) is the saturation intensity, i.e., the intensity at which \(\alpha_{0}\) is halved. It should be noted that the linear absorption coefficient \(\alpha_{SA}(I=0)=\alpha_{ns}+\alpha_{0}\), and at \(I\gg I_{sat}\)\(\alpha_{SA}(I\rightarrow\infty)=\alpha_{ns}\), is true, i.e., even at very high pump intensities, complete bleaching of the medium does not occur.