# Effects of Kerr Nonlinearity in Physical Unlonable Functions
## 2 Methods
Let \(E(\mathbf{\rho},z_{2};\lambda_{i})\) denote the two-dimensional speckle field at the output plane, for the \(i\)th component pertaining to wavelengths in the interval \([\lambda_{i}-\delta\lambda_{c}/2,\lambda_{i}+\delta\lambda_{c}/2]\), with \(\lambda_{i}=c/\
u_{i}\). Assuming uniform illumination, the electric field at the \(m\)th output mode is given by  
\[E^{\rm(out)}_{m}(\lambda_{i})\equiv E(\mathbf{\rho}_{m},z_{2};\lambda_{i})=\sqrt{ J(\
u_{i})}\Psi(\mathbf{\rho}_{m},z_{2};\lambda_{i}),\] (4a) where \[\Psi(\mathbf{\rho}_{m},z_{2};\lambda_{i})=\frac{1}{\sqrt{\mathcal{M}}}\sum_{j=1}^ {\mathcal{M}}T^{(\lambda_{i})}_{m_{j}}\] (4b) encapsulates the effect of the internal random disorder of the token on the spatial profile of the electric field at the particular wavelength (see Ref. [6] and references therein). In the discretized model under consideration, the total speckle field at wavelength \[\lambda_{i}\] is given by \[E(\mathbf{\rho},z_{2};\lambda_{i})\equiv\{E^{\rm(out)}_{m}(\lambda_{i}):1\leq m \leq\mathcal{M}\},\] (4c) where \[E^{\rm(out)}_{m}(\lambda_{i})\] is given by Equations ( 4a ) and ( 4b ). What remains is the inclusion of nonlinearity in our model for the randomly disordered token. In general, the propagation of light in a nonlinear inhomogeneous medium is a rather challenging and time-consuming task, because one has to know the details of scatterers (including position, size, and shape). Even if one models successfully a particular token with well-defined disorder, it is not clear at all what the results that will be obtained may imply for other tokens pertaining to different independent realizations of random disorder. Given that our task here is to investigate whether the nonlinearity can make a token more resilient to cloning, we will adopt a generic model, consisting of a thin linear multiple-scattering medium, which is followed by a homogeneous layer with Kerr-type nonlinearity  
Figure 1: Schematic representation of the theoretical model under consideration for the scattering of a Fourier-limited laser pulse from a multiple scattering nonlinear token. (**a**) For a token with spectral correlation bandwidth \(\delta\lambda_{\rm c}\) and a pulse of bandwidth \(\Delta\lambda_{\rm p}\), there are \(\mathcal{N}\simeq\Delta\lambda_{\rm p}/\delta\lambda_{\rm c}\) spectral degrees of freedom, corresponding to wavelengths \(\lambda_{0},\lambda_{\pm 1},\dots\) Different spectral components pertain to independent random monochromatic transmission matrices, and they give rise to uncorrelated monochromatic speckles. (**b**,**c**) The scattering of the \(i\)th spectral component from the nonlinear random token is treated as a sequential process, pertaining to the scattering from a linear multiple-scattering token and the propagation of the resulting fully developed monochromatic speckle in a nonlinear homogeneous medium. (see Figure 1). A similar model has been used successfully in studies related to wavefront shaping techniques in the presence of disorder and nonlinearity [23]. The speckle field \(E(\rho,z_{2};\lambda_{i})\) for the \(i\)th spectral component at the output of the linear random medium is given by Equation (4). The propagation of this monochromatic speckle field in the nonlinear medium of thickness \(L\) will be modeled by the following nonlinear wave equation [24]  
\[\mathbf{i}\frac{\partial\Psi(\mathbf{r};\lambda_{i})}{\partial z}+\frac{1}{2k_{i}n _{0}}\
abla_{\perp}^{2}\Psi(\mathbf{r},\lambda_{i})+k_{i}n_{2}J(\
u_{i})|\Psi(\mathbf{ r},\lambda_{i})|^{2}\Psi(\mathbf{r};\lambda_{i})=0, \tag{5}\]  
where \(\
abla_{\perp}^{2}\equiv\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{ 2}}{\partial y^{2}}\), \(\mathbf{r}\equiv(\rho,z)\), \(k_{i}=2\pi/\lambda_{i}\), \(n_{0}\) is the linear refractive index, and the nonlinear refractive index (Kerr nonlinearity) is denoted by \(n_{2}\), and it is measured in \(\mathrm{cm}^{2}/\mathrm{W}\).