## 3 Acquisition of the Fiber's Transmission Matrix  
In the previous section, the experimental principles required for the direct TM acquisition were introduced. It is required to excite every mode of the supported fiber modes sequentially and directly without the need to use a conversion matrix as done in [11] or [12]. Selective mode excitations are especially beneficial for prospective information-technical applications, however the experimental work appears more challenging. The actual procedure of the TM acquisition is based on the work from [12]. To verify the developed technique with the experimental setup from Figure 1, a TM acquisition at an approximately 1 cm long piece of step-index MMF is presented. MMFs at this size have a weakly developed mode scrambling property [11]; thus, an acquired TM corresponds strongly to a diagonal matrix. This characteristic is used to provide a proof-of-principle of the presented system.  
After the complex spatial mode functions from every supported fiber mode have been calculated and translated into a superpixel distribution, a reference phase distribution for the holographic-based decomposition method introduced in Section 2.3 is obtained. In order to measure systematic aberrations, it is necessary to let the object beam, which should be as unmodulated as possible, propagate through the optical setup to measure all the aberrations that act on the object beam. Since the errors, which have to be compensated, are located behind the MMF output, an unmodulated object beam at the fiber output needs to be generated. Due to the weakly developed scrambling at MMFs of approximately 1 cm length, the \(LP_{01}\) mode is excited at the fiber input. The light that propagated through the MMF appears with a sufficiently homogeneous phase distribution at the MMF output and can be obtained holographically at CMOS2. The reconstructed amplitude and phase distributions are shown in Figure 9. Now, the correction can be conducted by calculating \(\phi_{\text{corrected}}=\phi_{\text{measured}}-\phi_{\text{Ref}}\).  
With access to a quantitative phase measurement, the TM acquisition can be executed. The fiber under study supports \(N=\)110 fiber modes considering [28], where \(N\) is calculated by:  
\[N\approx\frac{1}{2}\cdot\left(a\cdot\frac{2\pi}{\lambda_{0}}\cdot\text{NA} \right)^{2}, \tag{9}\]  
with \(a\) being the MMF core radius. The fiber specifications are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Specification** & **Value** \\ \hline \(NA\) & \(0.1\pm 0.015\) \\ \(a\) & \((25\pm 3)\)\(\mu\)m \\ \(\lambda_{0}\) & 532 nm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the used MMF.  
Figure 9: Reconstruction of the reference hologram captured with CMOS2 by exciting an \(LP_{01}\) mode at the fiber input: (**a**) reconstructed phase distribution; (**b**) reconstructed amplitude distribution.  
Hence, the expected TM is of dimensions \(110\times 110\), which means that for each polarization direction a \(55\times 55\) TM can be acquired. Now, every mode contained in the fiber's mode domain has to be excited sequentially at the fiber input. The propagated light fields are superimposed with the reference beam on CMOS2, where holograms are captured. The holograms are reconstructed numerically with the angular spectrum method. Afterwards, the decomposition has to be executed for every shot. The resulting coefficients from the \(i\)-th mode of the mode domain are stored in the \(i\)-th row of the TM. Within the introduced experiment, the TM shown in Figure 10 was measured. Both the input and output vectors with the determined coefficients are arranged following the mode group manner presented in [7]. One result was that every \(\mathrm{LP_{lm}}\) mode belongs to a mode group given by \(2m+l-1\). Modes inside a group mix heavily with each other.