# Effects of Kerr Nonlinearity in Physical Unlonable Functions
## 2 Methods  
In the diffusive limit [6; 7; 8; 17], the problem can be formulated in terms of \(\mathscr{M}\) input and \(\mathscr{M}\) output transverse spatial modes. For monochromatic laser light at wavelength \(\lambda\), the electric field at the \(m\)th output mode is given by [19; 20; 21; 22]  
\[E_{m}^{\text{(out)}}(\lambda)=\sum_{j=1}^{\mathscr{M}}T_{m,j}^{(\lambda)}E_{j} ^{\text{(in)}}(\lambda), \tag{1}\]  
where \(T_{m,j}^{(\lambda)}\) is the element of the multi-spectral transmission matrix linking the \(m\)th output mode to the \(j\)th input mode for light at wavelength \(\lambda\). In the notation adopted throughout this work, the \(i\)th input and output transverse modes are associated with spatial coordinates \((x_{i},y_{i}):=\rho_{i}\) at the input and output planes, which are located at \(z=z_{1}\) and \(z=z_{2}\), respectively. Hence, \(E_{m}^{\text{(out)}}(\lambda)\) and \(E_{j}^{\text{(in)}}(\lambda)\) are abbreviations to \(E(\rho_{m},z_{2};\lambda)\) and \(E(\rho_{j},z_{1};\lambda)\), respectively. For a given wavelength, the elements of the transmission matrix \(\{T_{m,j}^{(\lambda)}\}\) depend strongly on the realization of the disorder in the token, and they can be treated as independent complex Gaussian random variables, with zero mean and variance given by \(l/L\mathscr{M}\), where \(L\) is the thickness of the token and \(l\ll L\) is the mean free path. For a token with spectral correlation bandwidth \(\delta\lambda_{c}\), the transmission matrices at two different wavelengths \(\lambda_{1}\) and \(\lambda_{2}\), with \(|\lambda_{1}-\lambda_{2}|>\delta\lambda_{c}\), can be treated as independent random matrices, thereby yielding uncorrelated monochromatic speckle patterns at the two wavelengths [19; 20; 21; 22]. Equation (1) refers to the scattering of monochromatic light with wavelength \(\lambda\) from a disordered linear token. To exploit the nonlinearity of a medium, one typically needs ultrashort laser pulses in order to reach high intensities without damaging the medium. The spectrum of an ultrashort laser pulse involves many frequencies, and the power spectral density defines the power in each wavelength of the spectrum. Neglecting the spatial dependence of the field for a moment, for a Gaussian pulse of the form \(E(t)=\sqrt{l_{0}}\exp[-t^{2}/2\tau^{2}]\exp(-\mathrm{i}2\pi\
u_{0}t)\), the electric field at frequency \(\
u=c/\lambda\) is given by  
\[E(\
u)=\sqrt{2\pi l_{0}}\tau\exp\left[-2\pi^{2}\tau^{2}(\
u-\
u_{0})^{2} \right]. \tag{2}\]  
The bandwidth (FWHM) of the pulse \(|E(\
u)|^{2}\) is \(\Delta\
u_{\text{p}}=\sqrt{\ln(2)}/(\pi\tau)\), \(l_{0}\) and \(T=2\sqrt{\ln(2)}\tau\) denote the peak intensity (units of \(\text{W}/\text{cm}^{2}\)) and the FWHM of the pulse in the time domain. The bandwidth can be expressed in terms of the wavelength using the relation \(\Delta\lambda_{\text{p}}=\lambda_{0}^{2}\Delta\
u_{\text{p}}/c\). The spectral density is \(S(\
u)=|E(\
u)|^{2}\), and \(S(\
u)d\
u\) gives the fluence (energy per area) for frequencies between \(\
u-d\
u/2\) and \(\
u+d\
u/2\). Hence, the intensity carried by frequencies within the spectral correlation bandwidth \(\delta\
u_{\text{c}}\) around \(\
u\), can be approximated by  
\[J(\
u):=\frac{S(\
u)\delta\
u_{\text{c}}}{T}=\frac{l_{0}}{\mathscr{N}}\exp \left[-4\pi^{2}\tau^{2}(\
u-\
u_{0})^{2}\right], \tag{3}\]  
where \(\mathscr{N}\equiv\Delta\
u_{\text{p}}/\delta\
u_{\text{c}}\) represents the spectral degrees of freedom, i.e., the number of spectral channels. When a pulse of bandwidth \(\Delta\
u_{\text{p}}\) propagates through a multiple-scattering random token with spectral correlation bandwidth \(\delta\
u_{\text{c}}\), the generated speckle is essentially a superposition of \(\mathscr{N}\) uncorrelated monochromatic speckle patters (see Figure 1a), with the exponent in Equation (3) determining the weight of the different contributions [19; 20; 21; 22].