# Wavelength-Dependent Optical Nonlinear Absorption of Au-Ag Nanoparticles
## 3 Results and Discussions
This phenomenon indicates that the sample exhibited SA. It can be seen that the normalized transmittance of the Au-Ag NPs at the focus was the smallest at 600 nm, and the largest at 510 nm. Under the excitation energy of 760 \(\upmu\)J, when the sample cuvette moved gradually to the focus, the normalized transmittance increased first and then decreased, which indicates that the sample exhibited SA at first, and then switched to RSA. Meanwhile, it can also be observed that the amplitude of RSA was the largest at 510 nm and the smallest at 600 nm. The mechanism is explained as follows. SA results from the ground-state bleaching of the plasma absorption band. Under the excitation of a certain energy range, almost all the electrons in the ground state are pumped to the excited state, which greatly weakens the absorption capacity of NPs to external photons. Specifically, the SA was larger near the resonance absorption peak (510 nm) of the sample, while it was relatively small away  
Figure 2: Simulation results for a single Au-Ag NP with R = 24.5–28.5 nm and d = 9–13 nm. **(a)** Absorption spectra of a single Au-Ag NP; inset image: schematic diagram of Au-Ag NPs. **(b)** Plot of peak position versus Au core radius. from the absorption peak. We speculate that this was related to the LSPR. With the further increasing laser energy, free carrier absorption leads to RSA dominating [44]. According to the above results, we can conclude that the Au-Ag NPs had two opposite nonlinear absorptions. The total absorption coefficient can be described as [45]:  
\[\alpha(I)=\frac{\alpha_{0}}{1+(I/I_{S})}+\beta I \tag{1}\]  
where \(\alpha_{0},I,I_{S}\), and \(\beta\) are the linear absorption coefficient, laser intensity, saturation intensity, and nonlinear absorption coefficient, respectively. The laser intensity \(I\) can be defined as:  
\[I=\frac{I_{0}}{1+z^{2}/z_{0}^{2}} \tag{2}\]  
where \(I_{0}\) expresses the laser intensity at the focus. So Equation (1) can be further expressed as:  
\[\alpha(I_{0})=\frac{\alpha_{0}}{1+\frac{I_{0}}{(1+z^{2}/z_{0}^{2})I_{S}}}+\frac {\beta I_{0}}{1+z^{2}/z_{0}^{2}} \tag{3}\]  
The normalized transmittance can be defined as [40]:  
\[T(z)=\sum_{m=0}^{\infty}\frac{\left[\frac{-\beta I_{0}L_{eff}}{1+z^{2}/z_{0}^{ 2}}\right]^{m}}{\left(m+1\right)^{3/2}} \tag{4}\]  
where \(L_{eff}=(1-e^{-\alpha_{0}l})/\alpha_{0}\), \(L_{eff}\) expresses the effective length. Therefore, the theoretical fitting of the experimental data can be carried out by Equation (4). Figure 2 indicates that the theoretical data agreed well with the experimental data. \(I_{S}\) and \(\beta\) could be obtained by theoretical fitting, and the results are listed in Table 1.  
Figure 3: The normalized open-aperture (OA) Z-scan transmittance of the Au-Ag NPs obtained at wavelengths of (**a**) 450 nm, (**b**) 510 nm, (**c**) 550 nm, (**d**) 600 nm. The red and blue dots are experimental data, and the red and blue solid lines are theoretical one. Figure 4 shows the point-fold line charts of wavelength versus saturated absorption intensity, as well as the dependence of wavelength versus nonlinear absorption coefficient, which can intuitively reflect the dependence of saturation intensity and nonlinear absorption coefficient on wavelength. As a comparison, the linear absorption spectra of the Au-Ag NPs are also provided by the black solid line. Figure 4a,b show that the SA intensity decreased significantly when the laser wavelength was around the LSPR absorption peak. Figure 4c shows that \(\beta\) increased significantly when the laser wavelength approached the LSPR absorption peak. We believe that the decrease of SA intensity around the LSPR absorption peak was due to the resonance between the incident photon and a free electron. The increase of \(\beta\) near the LSPR absorption peak was caused by the enhancement of free carrier absorption. In the Z-scan, the laser we used works at a low repetition frequency to avoid heat-induced scattering, but other kinds of scattering could not be avoid.