# Femtosecond Optical Nonlinearity of Nanodiamond Suspensions
## 3 Results
The coefficients \(\alpha_{0}\) and \(\alpha_{ns}\) are commonly referred to as saturable and nonsaturable absorptions, respectively. As was established above, the observed multiphoton absorption in pure distilled water is weak and therefore one can neglect its effect on the saturable absorption. In order to find the dependence of nonlinear transmittance on the incident intensity of a suspension of nanoparticles exhibiting saturable absorption the following differential equation should be solved:  
\[dI/dz\prime=-\alpha_{SA}I, \tag{4}\]  
Figure 5:0 (**a–d**) Normalized transmittance of suspensions of detonation nanodiamonds obtained in the open-aperture z-scan for a fixed concentration of 1.25 wt% for various incident laser pulse energies ranging from 106 to 360 nJ. Dots represent the experimental data while solid lines represent the result of the fitting with Equation (5). which is describing the propagation of the light pulse in a nonlinear medium [61] with the absorption coefficient presented in the form (3). For this case, the exact solution of Equation (4) yields to [49]:  
Figure 5:1  
where \(I_{out}\) and \(I_{in}\) are the output and incident radiation intensities. Saturable absorption leads to a slight increase in the intensity at the output of the cell (see Figure 7), therefore, the following expression is valid \(I_{out}=T_{0}\times I_{in}\). For a Gaussian beam during z-scan, the relation \(I_{in}=I_{0}/(1+(z/z_{0})^{2})\) is satisfied. Taking all this into account, one can arrive at the following equation describing the dependence of the normalized transmittance on the sample position:  
Figure 5:2  
where \(A=I_{0}/I_{sat}\) and \(B=\alpha_{0}/\alpha_{ns}\). Equation (6) makes it possible to approximate the experimental dependences shown in Figure 7 with two parameters \(A\) and \(B\). Considering that \(lnT_{0}=-(\alpha_{ns}+\alpha_{0})\times L\), where \(T_{0}\) is the linear transmittance of the suspension, we arrive at the \(I_{sat}=110\) GW/cm\({}^{2}\), \(\alpha_{ns}=2.0\) cm\({}^{-1}\) and \(\alpha_{0}=0.36\) cm\({}^{-1}\). It is seen from Figure 7 that Equation (6) with the defined parameters approximates the experimental data well. It should be noted that the obtained values of \(\alpha_{ns}\) and \(\alpha_{0}\) are defined for the nanoparticle concentration of C = 1.25 wt% and depend linearly on concentration [49], i.e., \(\alpha(\)cm\({}^{-1})=1.6\times\) C (wt%), \(\alpha_{0}\) (cm\({}^{-1})=0.24\times\) C (wt%). This means that for both the HP-HT and detonation nanodiamonds, the nonlinear absorption coefficients, i.e., two-photon absorption \(\beta\) and saturable absorption \(\alpha_{0}\) coefficients, linearly depend on the nanoparticles concentration. The obtained \(\alpha_{ns}\) and \(\alpha_{0}\) are similar to those obtained for the single-digit detonation nanodiamonds [49]. The attained saturation intensity \(I_{sat}\) which is concentration independent [49] is the same order as for the aqueous suspensions of detonation nanodiamonds with an average particle size of 50 nm under the femtosecond pulse excitation [50] but \(11\times 10^{3}\) times higher than that for the single-digit detonation nanodiamonds under the nanosecond pulse excitation [49]. A similar difference in saturation intensities in nanosecond and femtosecond time domains has been observed in a graphene polymer composite [68]. Another important parameter characterizing the saturable absorption properties and the applicability of the materials for the passive mode-locking is the ratio of the saturable to non-saturable losses [65]:  
Figure 5:3  
which should be as high as possible. Since the products \(\alpha_{ns}L\) and \(\alpha_{0}L\) are small, we arrive at \(R=\alpha_{0}/\alpha_{ns}=0.18\) which is lower than for graphene and single-walled carbon nanotubes (\(R\approx 1\)) used for passive mode-locking and Q-switching [65]. It should be added that in our experiments, despite the nonlinear absorption, the nonlinear refraction in the nanodiamond suspension due to heat absorption does not manifest itself.