# Design and Modelling of a Novel Integrated Photonic Device for Nano-Scale Magnetic Memory Reading
## 2 Materials and Methods
### Mathematical Modelling and Fourier Transformation  
A mathematical model is built in order to simulate the output light intensity vs. light wavelength for the designed devices. The model is given input parameters that are based on IMOS building block performances [21] and FDTD magneto-optical simulations [24] (see Section 2.1). An overview of the model parameters and their brief descriptions are given in Table 1. Additionally, reduced-performance devices with and without noise are simulated with the model to compare the magnetic memory reading capabilities of the devices. These parameters--some standard for all devices and some changing according to the performance levels--are summarized in Tables 2 and 3, respectively. Using the mathematical model, equations which determine the electric field (E-field) components of \(TE\) and \(TM\) modes in branches 1 and 2, are obtained. For simplicity, coefficients addressing the losses of mode propagations, grating couplers and magnetic cladding are combined into the terms \(B_{n}\). For description of other parameters please refer to Table 1.  
\[ \begin{split} E_{TE,1}&=B_{1}\,\cos\left(\alpha \right)\,e^{-i\,\frac{2\pi}{\lambda}n_{TE}L_{1}}\\ E_{TM,1}&=B_{2}\,\sin(\alpha)\,e^{-i\,\frac{2\pi}{ \lambda}n_{TM}(L_{1}-x_{\text{PC}})}\\ E_{TE,2}&=B_{3}\cos(\theta)e^{-i\frac{2\pi}{\lambda}n _{TE}L_{2}}\\ E_{TM,2}&=B_{4}\sin(\theta)e^{-i\left(\frac{2\pi}{ \lambda}n_{TM}(L_{2}-x_{\text{dad}})+\phi\right)}.\end{split}  \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Definitions** \\ \hline \(\lambda\) & Wavelength scanned (nm) \\ \(Loss_{ug}\) & Waveguide propagation loss (assumed to be the same for \(TE\) and \(TM\)) \\ \(n_{\text{TE}}\) & Effective index of \(TE\) mode \\ \(n_{\text{TM}}\) & Effective index of \(TM\) mode \\ \(L_{1}\) & Length of the upper branch (\(\mu\)m) \\ \(L_{2}\) & Length of the lower branch (\(\mu\)m) \\ \(x_{\text{PC}}\) & Distance between polarization converter and left-hand side MMI splitter (\(\mu\)m) \\ \(x_{\text{dad}}\). & Distance between memory bit (cladding) and left-hand side MMI splitter (\(\mu\)m) \\ \(Loss_{clad}\). & Loss due to memory bit (cladding) \\ \(Loss_{C_{TE-TE}}\) & Loss of \(TE\)-selective grating coupler for \(TE\) mode (dB) \\ \(\theta\) & Kerr rotation \\ \(\phi\) & Kerr ellipticity \\ \(\alpha\) & Angle of mode tilt induced by polarization converter (degree) \\ \(Loss_{C_{TM-TM}}\) & Loss of \(TM\)-selective grating coupler for \(TM\) mode (dB) \\ \(Ext_{C_{TM-TE}}\) & Extinction ratio of \(TM\)-selective grating coupler for \(TE\) mode (dB) \\ Noise & Addition of a Gaussian distribution of random noise to the intensity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of model parameters. Figure 1: An unbalanced MZL \(TE\) and \(TM\) mode selective grating couplers are used to couple the light in and out. The polarization converter is taken from [26]. It is important to recall that the interference takes place between the modes whose \(E\)-fields lay in parallel planes and the output light intensity (\(I\)) from devices can be calculated via \(I=\frac{|E|^{2}}{2\,Z_{0}}\), where \(|E|\) is the total E-field amplitude and \(Z_{0}\) is the impedance of the vacuum. The presented equations for \(E\)-field amplitudes reveal that a wavelength sweep of the input light will result in oscillations in intensity. Recall that the information regarding the magnetization direction of the cladding (memory bit type) can be retrieved from the sign of the Kerr rotation and ellipticity (\(\theta\), \(\phi\)). As seen from the equations above, when \(TE\) mode input light is used, information of the memory bit type is visible only in the phase of the \(TM\) mode output light. For an output light intensity vs. wavelength plot that is obtained upon interference of both \(TE\) and \(TM\) modes, two oscillation frequencies, \(\
u_{\text{TE}}\) and \(\
u_{\text{TM}}\) that correspond to these modes are observed. \[