# Wavelength-Dependent Nonlinear Absorption in Palladium Nanoparticles
## 3 Results and Discussion
The results of the Z-scan experiment above can be analyzed via a hypothetical energy level diagram shown in Figure 4. Under the excitation of the laser pulse, the external field causes the collective oscillation of electrons in the conduction band. As the sample approaches the focal point, the light strength increases, which leads to nonlinear absorption. Most of the outermost electrons in the d-band and the quasi-electrons in the sp conduction band near the Fermi-level transitions from the ground state to the excited state, resulting in the reduction of free electrons in the ground state and the plasma bleaching of the ground state. The light intensity is so high that all the electrons available for absorption of photons have been excited. Hence, there are no other electrons to absorb photons. The transmission is the largest at the focal point and SA is formed. With the increase of excitation energy, the electrons in the d-band absorb two photons simultaneously and transition to sp-band, or transition to the excited state. In these processes, there are also two-photon absorption [6, 9]. In the two-photon absorption process, the free carriers in the excited state continue to absorb photons, and the electrons transition from the excited state to the higher second excited state. The transmission increases to a certain extent (this part is SA) before the focal point, and then begins to decline until it reaches the focal point with the maximum light strength and the lowest transmission, forming RSA. The excitation wavelength is from 500 nm to 650 nm, while the SPR wavelength of PdNPs is 286 nm, which makes the nonlinear absorption of PdNPs happen easier at 500 nm and 550 nm, which are closer to SPR under the same excitation energy. At the same excitation wavelength, the excitation energy is larger, and the electrons get the interband transition energy more quickly, which makes it easier to realize the nonlinear absorption of PdNPs. Since 600 nm and 650 nm are far from SPR wavelengths of PdNPs, nonlinear absorption incurs difficulty and results in a small Z-scan signal, as shown in Figure 2c,d. In addition to the qualitative discussion of nonlinearity, it is necessary to analyze quantitatively the nonlinear absorption characteristics of PdNPs. Herein, the total absorption coefficient \(\alpha(I)\) can be expressed using Equation (1) [3]. \[\alpha(I)=\frac{\alpha_{0}}{1+I/I_{s}}+\beta I \tag{1}\]  
Figure 4: Energy diagram for optical transition in Pd NPs. Under the excitation of light pulse, the external field causes the collective oscillation of electrons in the conduction band. Most of the outermost electrons in the d-band and the quasi electrons in the sp-band near the Fermi-level transition from the ground state to the excited state resulting in the reduction of free electrons in the ground state and the plasma bleaching of the ground state. SA is formed. With the increase of excitation energy, the electrons in the d-band absorb two photons simultaneously and transition to sp-band, or transition to the excited state. These processes are two-photon absorption processes. where \(\alpha_{0}\) is the linear absorption coefficient, \(I_{s}\) is the saturation strength, \(\beta\) is the nonlinear absorption coefficient, \(I\) is the laser intensity. The first term on the right side of Equation (1) is SA, and the second term is RSA. The intensity of the Gaussian beam used in the experiment is expressed using Equation (2) [26]. \[I(z)=\frac{I_{0}}{1+z^{2}/z_{0}^{2}} \tag{2}\]  
where \(z\) is the displacement of the sample relative to \(z=0\); \(z_{0}\) is the Rayleigh diffraction length. By combining Equation (1) with Equation (2), the nonlinear absorption coefficient can be expressed using Equation (3). \[\alpha(I)=\frac{\alpha_{0}}{1+\frac{I_{0}}{(1+z^{2}/z_{0}^{2})I_{s}}}+\frac{ \beta I_{0}}{1+z^{2}/z_{0}^{2}} \tag{3}\]  
The relationship between absorption and transmission [25] is expressed using Equation (4). \[T(z)=1-\alpha(I)L_{eff} \tag{4}\]  
where \(L_{eff}\) is the effective length of the sample.