## 3 Fiber-Optic MAMZI  
In a fiber-optic MAMZI, the attenuation in each arm (\(\tau_{k}\)) can come from the fiber connections, in-line optoelectronic devices, and fabrication defects in the fiber. However, this attenuation is often equalized with external elements, resulting in a common attenuation factor for all arms. Another essential variable is the polarization of the optical fibers, which is omitted in this matrix since commercial polarization controllers can match it for all arms [30]. This adjustment makes \(\tau_{k}\) independent of \(k\), allowing it to be factored, making the phase matrix (Equation (2)) a unitary matrix. To study the controllability of a system consisting of an MBS, we will consider an MBS as  
Figure 1: A multi-arm Mach–Zehnder interferometer. \(x_{k}\) and \(y_{k}\) are inputs and outputs, respectively. MBS: multi-port beam splitter (\(N\) inputs and \(N\) outputs). \(\phi_{k}^{*}\): phase induced in arm \(k\).  
\[MBS=\frac{1}{\sqrt{N}}\left[\begin{array}{cccc}1&1&...&1\\ 1&\alpha_{[2,2]}e^{i(\theta_{[2,2]})}&...&\alpha_{[2,N]}e^{i(\theta_{[2,N]})}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\alpha_{[N,2]}e^{i(\theta_{[N,2]})}&...&\alpha_{[N,N]}e^{i(\theta_{[N,N]})} \end{array}\right], \tag{4}\]  
where \(\alpha_{[k,l]}\) and \(\theta_{k,l}\) are constant parameters representing attenuation and geometric phases induced in the MBS, respectively. Like the Fourier matrix shown in Equation (1), the first row and column contain unit values representing symmetrical splitting ratio and induced phases (\(\alpha_{[1,l]}=\alpha_{[k,1]}=1\) and \(\theta_{1,l}=\theta_{k,1}=0\)). The MBS design and manufacturing determine these conditions.  
Using the MAMZI model from Equation (3), the optical intensity at output \(l\) for an input \(l^{in}=[l_{0},0..0]\) is given by  
\[I_{l}^{out}=\frac{I_{0}}{N^{2}}\Bigg{\{}\sum_{k=1}^{N}\alpha_{[k,l]}^{2}+2\sum _{k=1}^{N-1}\sum_{m=k+1}^{N}\alpha_{[k,l]}\alpha_{[m,l]}cos(\theta_{[k,l]}- \theta_{[m,l]}+\phi_{[k]}^{*}-\phi_{[m]}^{*})\Bigg{\}}. \tag{5}\]  
Equation (5) shows how the MAMZI can be processed as a non-linear system, with the internal phase information \(\phi_{[k,l]}^{*}\) as inputs and the resulting intensity \(I_{l}^{out}\) as outputs, which depends on the optical interference produced by the phases \(\phi_{[k,l]}^{*}\). To evaluate whether the system is controllable, we must check whether the controllability matrix has full rank [31]. The key term in the expression for \(I_{l}^{out}\) is the sum of cosines whose inputs are the angular differences (\(\phi_{k}^{*}-\phi_{m}^{*}\)). Suppose we have a dynamical system with a state vector \(\phi=[\phi_{1}^{*},\phi_{2}^{*},...,\phi_{N}^{*}]\) and with inputs, where each input is \(u=\phi_{k}^{*}-\phi_{m}^{*}\). Then, by incorporating \(u\) into Equation (5) and partially deriving it, we obtain the following:  
\[\frac{dI_{l}}{du_{[k,m]}}=-\frac{2I_{0}}{N^{2}}\Bigg{\{}\sum_{k=1}^{N-1}\sum_ {m=k+1}^{N}\alpha_{[k,l]}\alpha_{[m,l]}sen(\theta_{[k,l]}-\theta_{[m,l]}+u_{[m,k]})\Bigg{\}} \tag{6}\]  
The Jacobian matrix of these partial derivatives will not be zero unless all differences \(u_{[k,m]}\) are such that \(sin(u_{[k,m]})=0\) for all combinations of \((k,m)\). If we consider the space of inputs, as long as there are enough angular variations \(u_{[k,m]}\) that make the terms \(sin(u_{[k,m]})=0\), not all zero simultaneously, the rank of the Jacobian matrix will be complete. Hence, the system will be controllable for the inputs \(u_{[k,m]}\).