# Wavelength-Dependent Nonlinear Absorption in Palladium Nanoparticles
## 3 Results and Discussion
By combining Equation (3) and Equation (4), the normalized transmission can be expressed using Equation (5). Figure 1:0  
Equation (5) is used to fit the experimental data, where \(\alpha_{0}\) = 0.0458 m\({}^{-1}\), \(Leff\) = 0.0013 m, \(z_{0}\) = 15.7 mm and the theoretical fitting results are shown in Figure 2 with solid lines. It can be found that the theoretical fitting curve is in good agreement with the experimental data. Correspondingly, the saturation strength \(I_{s}\) and the nonlinear absorption coefficient \(\beta\) are obtained and shown in Figure 1:1  
In Figure 5a, the linear absorption spectrum of PdNPs and the saturation strength \(I_{s}\) at excitation energies of 630 \(\upmu\)J, 730 \(\upmu\)J and 752 \(\upmu\)J are plotted as double \(Y\)-axis characteristic curves, and the colored dotted lines represent the saturation strength curve varying with the wavelength. The black line is the linear absorption spectrum. In Figure 5b, the characteristic curves of the linear absorption spectrum and the nonlinear absorption coefficient \(\beta\) at the excitation energies of the PdNPs at 630 \(\upmu\)J, 730 \(\upmu\)J and 752 \(\upmu\)J are plotted as a double \(Y\)-axis characteristic curve, and the colored dotted lines represent the nonlinear \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\lambda\) **(nm)** & \(E\) **(\(\upmu\)J)** & \(I_{0}\) **(W/m\({}^{2}\))** & \(I_{s}\) **(W/m\({}^{2}\))** & \(\beta\) **(m/W)** \\ \hline \multirow{4}{*}{500} & 630 & \(2.51\times 10^{13}\) & \(1.32\times 10^{12}\) & \(1.46\times 10^{-11}\) \\  & 730 & \(2.91\times 10^{13}\) & \(4.16\times 10^{11}\) & \(1.1\times 10^{-11}\) \\  & 752 & \(2.99\times 10^{13}\) & \(1.36\times 10^{12}\) & \(2.53\times 10^{-11}\) \\ \hline \multirow{4}{*}{550} & 630 & \(2.51\times 10^{13}\) & \(3.59\times 10^{11}\) & \(5.14\times 10^{-12}\) \\  & 730 & \(2.91\times 10^{13}\) & \(2.43\times 10^{11}\) & \(9.59\times 10^{-12}\) \\  & 752 & \(2.99\times 10^{13}\) & \(1.2\times 10^{12}\) & \(2.8\times 10^{-11}\) \\ \hline \multirow{4}{*}{600} & 630 & \(2.51\times 10^{13}\) & \(1.05\times 10^{12}\) & 0 \\  & 730 & \(2.91\times 10^{13}\) & \(8.32\times 10^{11}\) & 0 \\ \cline{1-1}  & 752 & \(2.99\times 10^{13}\) & \(4.47\times 10^{11}\) & \(4.92\times 10^{-12}\) \\ \hline \multirow{4}{*}{650} & 630 & \(2.51\times 10^{13}\) & \(1.26\times 10^{12}\) & 0 \\  & 730 & \(2.91\times 10^{13}\) & \(8.32\times 10^{11}\) & 0 \\ \cline{1-1}  & 752 & \(2.99\times 10^{13}\) & \(5.99\times 10^{11}\) & \(1.8\times 10^{-12}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nonlinear Optical Parameters of PdNPs. absorption coefficient curve varying with the wavelength. The black line is the linear absorption spectrum. Figure 5a shows that at different energies, with the increase of the wavelength away from SPR, the saturated absorption coefficient first decreases and then increases. From Figure 5b, we can see that when the excitation wavelength is near the SPR peak value, the nonlinear absorption coefficient is larger. At a fixed excitation wavelength, when energy is greater, the nonlinear absorption coefficient is larger. The increase of saturation strength near the SPR wavelength is due to the nonlinear resonant enhancement. When the excitation energy is moderate, the bleaching of the ground state plasmon will lead to SA. However, when the excitation wavelength is different, the number of electrons in the conduction band is different. The wavelength is near SPR; it is easier to realize the ground state plasma bleaching. The greater the excitation energy is, the more electrons will be transferred to the conduction band to realize the ground state plasma bleaching. In the process of plasma ground state bleaching, the absorptivity decreases and the saturation strength increases. At the wavelength far away from SPR, the decrease of the nonlinear absorption coefficient is due to the enhancement of free carrier absorption.