## 3.1 Open aperture Z-scan Measurements

Figures 7 and 8 demonstrate the experimental open-aperture (OA) Z-scan of ITO thin film was performed using fs and an HRR laser at 80 MHz. The nonlinear optical properties of ITO film at different thicknesses of 170 and 280 nm were studied at various laser beam excitation powers ranging from 1.2 to 1.8 W and different excitation wavelengths of 750, 800, and 820 nm. The curves shown in Figs. 7 and 8 all show Reverse Saturable Absorption (RSA). The transmission is symmetric around the focus (Z = 0), indicating an intensity-dependent absorption effect with a minimum transmission at the focus (valley) [47, 48]. The nonlinear absorption coefficient of a sample, subjected to the high-intensity laser (I), is dependent on the intensity delivered to the sample., as indicated by the following equation [49, 50]:

\(\propto \left(\text{I}\right)={\propto }_{0}+{\beta }\text{I}\) [4]

Where \({\propto }_{0}\) is the linear absorption coefficient and ꞵ is the two-photon nonlinear absorption (2PA) coefficient, To determine the NLO absorption coefficient, the OA Z-scan data were fitted by the NLO absorption model given by [51–53]

\({\text{T}}_{\text{O}\text{A}} = 1-\) \(\left.\left( \frac{{\beta } {{\text{I}}_{0 }}^{(\text{n}-1) }{\text{L}}_{\text{e}\text{f}\text{f}}}{{\text{n}}^{\frac{3}{2}}{\left(1+\frac{{\text{Z}}^{2}}{{{\text{Z}}_{0}}^{2}}\right)}^{(\text{n}-1)}}\right.\right)\) [5]

Where I0 is the peak intensity at the focus (Z = 0), and n = 1 for two-photon nonlinear absorption and n = 2 for three-photon nonlinear absorption. Z0 is the Rayleigh length \({\text{Z}}_{0}=\frac{{\pi }{{{\omega }}_{0}}^{2}}{{\lambda }}\). The \({\text{L}}_{\text{e}\text{f}\text{f}} =1-({\text{e}}^{-\text{n}{\propto }_{0}\text{L}}/\text{n}{\propto }_{0}\)), where L is the thickness of the sample. The normalized transmittance (TOA) decreases when the excitation power decreases, as shown in Fig. 7, while in Fig. 8 the TOA decreases with an increase in excitation power.

The observed RSA in the ITO thin films can be explained using the energy levels diagram, as shown in Fig. 9. ITO is made up of indium oxide doped with tin, and indium oxide has a valence band of O2−: 2p and In3+: 3d. The conduction band is made up of 5s and 5p. The conduction band is dominated by indium 5s orbitals, while the valence band is dominated by oxygen 2p electrons [26]. Due to n-type doping of tin impurities, the Fermi energy (Ef) is found a few eV below the conduction band [54]. The majority of carriers in ITO are electrons, which come mostly from the doping of donor Sn and oxygen vacancies. The excited electrons move from the valence band (2p) to the conduction band (5s). RSA is related to excited-state absorption (ESA), free carrier absorption (FCA), two-photon absorption (2PA), and nonlinear scattering [55]. 2PA relies on the excitation wavelength and energy band gap [56]. When ESA takes place, molecules are excited from an already excited state (5s) to a higher excited state (5p). For this to happen, the population of excited states should be high enough to increase the probability of photon absorption from the state [57]. FCA is expected to occur in ITO film because ITO is an n-type semiconductor with free carriers in the 5s that absorb incident photons and excite them to a higher excited state in the conduction band (5p) shown in Fig. 9. When the number of O vacancies and donor levels increase, defect state absorption increases, improving the RSA [58]. From Fig. 9. N0, ND, N1, and N2 denote the population densities corresponding to ground-state (S0), defect state (SD), and first and second excited states. The σ0 is the ground state absorption cross-section; σ1, σ2 are the defect state cross-section and excited-state absorption cross-sections, respectively. The τ3, τ2, and τ1 are the excited-state lifetimes. The total absorption coefficient and pump intensity are used to define the nonlinear propagation and rate equations, which are represented as [59–64]:

\(\frac{\text{d}\text{I}\left(\text{z}\right)}{\text{d}\text{z}}=-\propto \left(\text{I}\right) \text{I}\) [6]

\(\frac{\text{d}{\text{N}}_{0}}{\text{d}\text{t}}=-\frac{{N}_{0 }{\sigma }_{0}I}{hʋ}+ \frac{{N}_{D}}{{\tau }_{1}}\) [7]

\(\frac{\text{d}{\text{N}}_{D}}{\text{d}\text{t}}=\frac{{N}_{0 }{\sigma }_{0}I}{hʋ}-\frac{{N}_{D }{\sigma }_{1}I}{hʋ}+ \frac{{N}_{1 }}{{\tau }_{2}}-\frac{{N}_{D}}{{\tau }_{1}}\) [8]

\(\frac{\text{d}{\text{N}}_{1}}{\text{d}\text{t}}=\frac{{N}_{D }{\sigma }_{1}I}{hʋ}-\frac{{N}_{1 }{\sigma }_{2}I}{hʋ}-\frac{{N}_{1 }}{{\tau }_{2}}+ \frac{{N}_{2 }}{{\tau }_{3}}\) [9]

\(\frac{\text{d}{\text{N}}_{2}}{\text{d}\text{t}}=\frac{{N}_{1 }{\sigma }_{2}I}{hʋ}- \frac{{N}_{2 }}{{\tau }_{3}}\) [10]

The transmitted intensity through the thin film in Eq. 6 can then be expressed as [64]:

\(\frac{\text{d}\text{I}\left(\text{z}\right)}{\text{d}\text{z}}=-[{N}_{0 }{\sigma }_{0}I+{N}_{D }{\sigma }_{1}I+{N}_{1 }{\sigma }_{2}I]\) [11]

According to the Runge-Kutta technique and the Eqs. from 6 to 11, RSA occurs when the σ1 and σ2 are larger than σ0. For simplicity, M1 = σ1/σ0, M2 = σ2/σ1 and M3 = σ2/σ0 are assumed. RSA happens when M1 and M3 are both greater than one [65].

The NLA coefficient as a function of excitation power was estimated from the best fit of the experimental OA data shown in Figs. 7 and 8 using Eq. 5. Figure 10 shows the effect of laser excitation power on the ꞵ of an ITO thin film at thicknesses of 170 and 280 nm and excitation wavelengths of 750, 800, and 820 nm. For each excitation wavelength, as the excitation power increases the ꞵ decreases, due to the many-body effect, and the relaxation time τ3, τ2, and τ1 of excited states gets shorter [65, 66]. When the excitation power increases, the number of free carriers (electrons and holes) also increases. The electrons in the conduction band (5s) and the holes in the valence band (2p) interact with one another. This interaction is called "many-body interactions" [67]. The terms simply describe the Coulomb repulsion between electrons and between holes, and the Coulomb attraction between electrons and holes [68]. As a result, collisions between free carriers in the 5s and 2p increase, scattering of photons and phonons increases, and ꞵ decreases.

As shown in Fig. 8, the normalized transmittance (TOA) decreased as the excitation power increased, but the values of the ꞵ decreased. The RSA represents the contribution from nonlinear absorption as well as nonlinear scattering. Nonlinear scattering increases when excited carrier declines through nonradiative relaxation processes [69]. As shown in Fig. 9, the nonradiative decay of excited carriers via intraband transitions in 5s and 5p results in thermal energy. The thermal energy generated in the ITO film can cause nonlinear scattering and affect the ꞵ. The equation gives the laser light attenuation inside the sample [69, 70].

\(\frac{\text{d}\text{I}\left(\text{z}\right)}{\text{d}\text{z}}=-{\propto }_{0}\text{I}\left(\text{z}\right)-{\beta } {\text{I}}^{2}\left(\text{z}\right)\) [12]

where I(z) is the intensity incident on the sample, and z is the optical path length along with the sample. The scattering coefficient \({\propto }_{s}\) can be used to modify the nonlinear scattering losses in Eq. 12 \(\left[71\right]\).

\(\frac{\text{d}\text{I}\left(\text{z}\right)}{\text{d}\text{z}}=-{\propto }_{0}\text{I}\left(\text{z}\right)-{({\propto }_{s}+{\beta }}_{ }\left){\text{I}}^{2}\right(\text{z})\) [13]

Because of the energy band gap and carrier density, nonlinear scattering occurs significantly more at an ITO thickness of 280 nm than 170 nm at high excitation power. Electrons in the 2p absorb incident photon energy and are excited to a highly excited state at a thickness of 280 nm, whereas, at a thickness of 170 nm, electrons are excited to a low excited level, so the thermal energy is lower than at a thickness of 280 nm, as illustrated in Fig. 9.

The NLO properties of soda–lime glass substrates at different wavelengths and peak intensities were reported in [72]. According to this study, the NLA coefficient of soda-lime glass is 1016 orders of magnitude lower than the NLA coefficient of ITO thin film. Consequently, the contributions of the 1 mm soda–lime glass utilized as a substrate in this study were negligible compared to the NLA coefficient of the ITO thin film.

## 3.4 Nonlinear optical susceptibility of ITO thin films

Using the experimental data of nonlinear refractive index n2 and nonlinear absorption coefficient \(\text{ꞵ}\), the real and imaginary components of the third-order susceptibility \({{\chi }}^{\left(3\right)}\) can be determined. A material's nonlinear susceptibility can be expressed by [76, 77] as:

\({{\chi }}^{\left(3\right)}=\text{R}\text{e}\left[ {{\chi }}^{\left(3\right)}\right]+\text{i} \text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right]\) [14]

Where the real part is associated to the nonlinear refractive index n2 and the imaginary part is associated with the nonlinear absorption coefficient ꞵ. The real nonlinear susceptibility \(\text{R}\text{e}\left[{{\chi }}^{\left(3\right)}\right]\) can be expressed as:

\(\text{R}\text{e}\left[ {{\chi }}^{\left(3\right)}\right]= \frac{{\text{n}}_{0}}{3{\pi }} {\text{n}}_{2}\) [15]

The imaginary nonlinear susceptibility \(\text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right]\) can be expressed as:

\(\text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right]\left(\text{e}\text{s}\text{u}\right)=\left[\frac{{10}^{-7}c{{{\lambda }\text{n}}_{0}}^{2}}{96{{\pi }}^{2}}\right] \text{ꞵ}\) \(\left[16\right]\)

where c is the speed of light, λ is the excitation wavelength of the laser light, and n0 is the linear refractive index of the ITO films. At an excitation wavelength of 800 nm and an excitation power of 1.6 W, the imaginary nonlinear susceptibility \(\text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right]\) was found to be 2.08 × 10− 10 esu and 5.53 × 10− 10 esu for ITO thicknesses of 170 and 280 nm, respectively. The third-order nonlinear optical properties of given materials can also be characterized by a figure of merit (FOM), which depends on the linear absorption 𝛼0 and can be expressed by [78, 79] as.

\(\text{F}\text{O}\text{M}=\left|\frac{ \text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right] }{{{\alpha }}_{0}}\right|\) [17]

Based on the data presented in Table 2, the figure of merit of ITO thin film at a thickness of 280 nm is approximately three times larger than that of 170 nm. The absolute value of the nonlinear susceptibility is expressed as:

\(\left|{{\chi }}^{\left(3\right)}\right|=\sqrt{\left({\text{R}\text{e}\left[ {{\chi }}^{\left(3\right)}\right]}^{2}+{\text{I}\text{m}\left[{{\chi }}^{\left(3\right)}\right]}^{2}\right)}\) [18]

The absolute values of \({{\chi }}^{\left(3\right)}\) are given in Table 2

**Table 2 The nonlinear optical parameters of ITO thin films for different thicknesses at an excitation wavelength of 800 nm and an excitation power of 1.6 W.**