**2. Theoretical Background**

Both nonlinear refraction and absorption may be (in terms of the lowest order with respect to the field strength) expressed in terms of the cubic nonlinear dielectric susceptibility χ(3). General quantum-mechanical expressions for χ(3) are available in relevant textbooks (compare Reference [6]), but because of the high number of input parameters, their direct usage may be inconvenient in practical applications. Instead, we will make use of simplified expressions that allow for estimating the TPA coefficient β in terms of input parameters that are available from standard (linear) optical thin-film characterization techniques, among them spectrophotometry and ellipsometry.

In this context, the model of Sheik–Bahae et al. [15] provides a simple formula for calculating TPA contributions to nonlinear absorption in semiconducting solids. The material-specific input parameters of the model are the band gap *Eg* as well as the usual linear refractive index *n*. Both these values may be obtained from linear optical spectroscopy. When choosing the remaining, rather material-independent model parameters corresponding to what is recommended in Reference [15], the expression for β may be written as

$$\beta(\nu) \approx \frac{3100\sqrt{21}}{2^5 n^2 E\_\mathcal{S}^3} \frac{\left(\frac{2hc\nu}{E\_\mathcal{S}} - 1\right)^{\frac{3}{2}}}{\left(\frac{2hc\nu}{E\_\mathcal{S}}\right)^5}.\tag{1}$$

Equation (1) yields β directly in cm/GW, provided that both *hc*ν and *Eg* are given in eV (in which *h* represents Planck's constant, *c* is the velocity of light in a vacuum, and ν is the wavenumber, i.e., the reciprocal value of the light wavelength in a vacuum. In order to better understand the parametrization, Figure 1 shows estimated β values corresponding to Equation (1) for the crystalline TiO2 modifications of anatase, rutile, and brookite. The underlying parameters are summarized in Table 1. The crystal refractive indices indicated in Table 1 are polarization-averaged.


**Table 1.** Mass density, band gap energy, and refractive index of selected TiO2 modifications.

Note that, according to this model calculation, the main difference between the different modifications is in the onset energy of the TPA processes. The maximum β value is residually influenced by the choice of the modification, because material modifications with a lower gap tend to have a higher refractive index (compare Figure 2, where this trend is represented graphically for

the sake of clarity), such that in Equation (1), the changes in *n* and *Eg* tend to cancel each other out. For reference purposes, we also include data from two thin-film samples, namely a sample produced by plasma ion-assisted evaporation (PIAD) as well as an ion beam-sputtered (IBS) sample, the latter prepared at Laser Zentrum Hannover LZH (for details, see Reference [17]). IBS is known as a preparation method that yields high-quality optical films with a rather high mass density. Indeed, the IBS data (Figure 2) fall closer to the values reported for the crystalline modifications than the PIAD sample does.

**Figure 1.** Two-photon absorption (TPA) coefficient as calculated from Equation (1) for crystalline TiO2 modifications (solid lines; input data according to Table 1). Symbols: Experimental data reported in References [19,25] for rutile, as obtained from Z-scan techniques.

**Figure 2.** Correlation between optical gap and refractive index for the TiO2 modifications from Table 1.

As previously mentioned, reported values for the nonlinear optical constants of TiO2 are rare. Table 2 presents some published values of nonlinear refractive indices and absorption coefficients.

Note that the experimental β values published in Reference [25] are in rather good agreement with the theory (Figure 1), while those published in Reference [19] exceed the theoretically predicted values for a factor of approximately 5, such that experiment and theory indeed fall into the same order of magnitude anyway.


**Table 2.** Reported values for measured nonlinear TiO2 optical constants. IAD denotes ion-assisted evaporation. C denotes the direction of the rutile optical axis, while E denotes the direction of the electric field vector in the light wave. IBS: Ion beam-sputtered.

As a second approach, we will make use of the beta-distributed oscillator (ß\_do) model [16]. This is a semiempirical model primarily developed for fitting linear optical constants, but again, all of the input parameters can be fitted from linear optical spectra. The dielectric function ε in terms of the ß\_do model is given by

$$\begin{aligned} \varepsilon(\nu) &= \left[ n(\nu) + ik(\nu) \right]^2 = 1 + \frac{I}{\pi} \frac{\sum\_{s=1}^{M} w\_s \left( \frac{1}{r\_s - \nu - i\Gamma} + \frac{1}{r\_s + \nu + i\Gamma} \right)}{\sum\_{s=1}^{M} w\_s}; \\\ w\_s &= \left( \frac{s}{M+1} \right)^{A-1} \left( \frac{M+1-s}{M+1} \right)^{B-1}; s = 1, 2, 3, \dots, M; A, B > 0 \\\ v\_s &= \nu\_a + \frac{V\_b - V\_s}{M+1} s \end{aligned} \tag{2}$$

while the real parameters *J*, Γ, *A*, *B*, ν*a*, and ν*<sup>b</sup>* are free parameters within the ß\_do model [16], and *M* is the number of individual Lorentzian oscillators. Here, *n* and *k* are the linear refractive index and the extinction coefficient, correspondingly. Then, the third-order nonlinear susceptibility χ(3) may be estimated by

$$\begin{aligned} \chi^{(3)} &\approx J\_3 g(\nu) f(\nu)^2 \Big( \sum\_{s=1}^M w\_s \Big)^{-3}; \ \nu < \nu\_a \\\ f(\nu) &= \sum\_{s=1}^M w\_s \Big( \frac{1}{\nu\_s - \nu - i\Gamma} + \frac{1}{\nu\_s + \nu + i\Gamma} \Big) \\\ g(\nu) &= \sum\_{s=1}^M w\_s \Big( \frac{1}{\nu\_s - 2\nu - i\Gamma} + \frac{1}{\nu\_s + 2\nu + i\Gamma} \Big) \end{aligned} \tag{3}$$

which leads us to expressions for the nonlinear refractive index *n*<sup>2</sup> and the TPA coefficient β according to [27]

$$\begin{array}{l} m\_2(\nu) = \frac{3}{4} \frac{\mu\_0 c}{(n^2 + k^2)} \left[ \mathrm{Re} \chi^{(3)} + \frac{k}{n} \mathrm{Im} \chi^{(3)} \right] \\ \beta(\nu) = 3 \frac{\mu\_0 \pi c \nu}{(n^2 + k^2)} \left[ \mathrm{Im} \chi^{(3)} - \frac{k}{n} \mathrm{Re} \chi^{(3)} \right] \end{array} \tag{4}$$

in which μ<sup>0</sup> represents the free space permeability. Note that the expressions written here are also valid in the case where there is still some linear absorption present in the TPA region [27]. In this approach, the parameter *J*<sup>3</sup> is estimated from the generalized millers rule according to References [14,16,28]. The gap values indicated in Table 1 for the PIAD and IBS samples correspond to *hc*ν*a*.
