# Effects of Kerr Nonlinearity in Physical Unlonable Functions
## 3 Results and Discussion  
Let us begin with the effect of Kerr nonlinearity on the speckle generated by a given token. The main findings are summarized in Figure 2, which shows the speckles for a token with defocusing nonlinearity \((n_{2}<0)\), zero nonlinearity \((n_{2}=0)\), and focusing nonlinearity \((n_{2}>0)\). For defocusing nonlinearity, the scattered light is distributed among a larger number of output modes relative to the linear case. By contrast, for focusing nonlinearity, the scattered light tends to concentrate within a small number of output modes, thereby leading to a small number of very bright spots in the pattern (note the different color scales in the density plots). These findings were present for all of the simulations we have performed, and the location as well as the intensities of these bright spots depend on the realization of the random disorder in the token. Moreover, we have found that the concentration of scattered light for focusing nonlinearity becomes weaker for an increasing number of spectral degrees of freedom (e.g., compare Figure 2c to Figure 2f). This was to be expected, because according to Equations (3) and (5), effects of nonlinearity become weaker as we increase \(\mathcal{N}\). Analogous behavior is found as we decrease \(I_{0}\), for fixed \(\mathcal{N}\). Typical PUF-based cryptographic protocols which use optical tokens exploit one way or another the strong dependence of the speckle image on the random internal disorder of the token as well as on the parameters of the input light [1; 2; 3; 5; 9; 10; 12]. In either case, the robustness of a protocol against the cloning of the token is mainly determined by the sensitivity of the speckle image to the cloning. To investigate the role of the nonlinearity in this context, for a fixed reference token, we generated 100 independent random clones, for three different cloning factors \(Q\). For each clone, we calculated the speckle and compared it to the corresponding speckle for the reference token, using the Pearson correlation coefficient [26], which is equal to 1 for perfectly correlated patterns and equal to 0 for totally uncorrelated patterns. Given that the correlation varies from clone to clone, we have analyzed the recorded values in terms of their median, spread and skewness through box and whisker diagrams. A sample of our main findings is depicted in Figure 3, where we show diagrams for a fixed reference token, as a function of the cloning factor \(Q\), for various combinations of parameters. As expected, the correlation decreases with increasing \(Q\), because the quality of the clones decreases. Moreover, we find a large overlap between the red and green boxes for all of the parameters, which suggests that the performance of the linear token (red boxes) is the same as (if not slightly better than) the performance of a token with defocusing nonlinearity (green boxes). By contrast, in the case of focusing nonlinearity (blue boxes), the recorded correlation coefficients tend to be smaller than the corresponding coefficients for a linear token (red boxes) with the same characteristics. In particular, it is worth noting that there is negligible overlap between the blue and the red boxes in all of the cases shown in the figure, and it is only the maximum recorded values which have some overlap with the red boxes. Comparing Figure 3a to Figure 3b, we find that the advantage  
Figure 2: Speckle fields after propagation in a medium with (**a**,**d**) defocusing nonlinearity \(n_{2}=-3\times 10^{-16}\) cm\({}^{2}\)/W, (**b**,**e**) zero nonlinearity, and (**c**,**f**) focusing nonlinearity \(n_{2}=3\times 10^{-16}\) cm\({}^{2}\)/W. Note the different color scales as one goes from negative to positive \(n_{2}\). Parameters: \(L=100\)\(\lambda_{0}\), \(\lambda_{0}=800\) nm, \(I_{0}=2.6\times 10^{13}\) W/cm\({}^{2}\), \(T=100\) fs, \(n_{0}=1.47\), \(\mathcal{N}=3\) (**a**–**c**) and \(\mathcal{N}=5\) (**d**–**f**). In all of the cases, uniform illumination has been considered at the input.