## 2 Procedure of Operation
### Holographic Decomposition of Light into the Fiber's Mode Domain  
DH is one possible approach to extract amplitude and phase from an electromagnetic wave using a CMOS digital camera [26]. Applying this technique to the presented experimental setup, the fiber  
Figure 8: For the alignment of the MMF, the reference beam is coupled to the actual output of the MMF and the image from the input fiber facet on CMOS1 is analyzed: (**a**) sub-setup from Figure 1; (**b**) sharp image from an axicon reflected by the input fiber facet and captured with CMOS1; (**c**) sharp image of the input fiber facet and captured with CMOS1; (**d**) after the MMF is aligned properly, concentric interference rings appear on CMOS1 if the back reflected light from the input fiber facet interferes with the reflected light from M5. The sheer diffraction grating is displayed on the SLM. output light field can be measured and analyzed. Therefore, the MMF output signal is superimposed with the reference beam on CMOS2, as shown in Figure 1. The resulting hologram is captured with the camera and numerically reconstructed with the angular spectrum method [27]. In order to receive a quantitative phase measurement, the raw phase data from the measurement has to be relieved from several errors. These unwanted systematic parts of the measured phase data are due to wrapping errors, optical aberrations, phase offsets, MMF surface irregularities or dust on optical components. One approach to determine these error terms is to capture a reference hologram to identify a reference phase distribution, which can be set against the experimental results. In Section 3, it is explained how this is conducted during the TM acquisition. For acquiring a fiber's TM, the holographically measured complex light field has to be decomposed into the fiber's mode domain. Since [PERSON]'s equations are linear differential equations, the light guidance of an arbitrary input field \(\underline{E}_{\text{input}}\) in an MMF can be modeled as a linear sequence approach, where the orthogonal base functions are the complex spatial distributions of the supported fiber modes \(\underline{E}_{n}\)[28]. The proportion of every mode to \(\underline{E}_{\text{input}}\) is determined by their individual weight \(a_{n}\):  
\[\underline{E}_{\text{input}}=\sum_{n=1}^{N}a_{n}\underline{E}_{n}=a_{1} \underline{E}_{1}+a_{2}\underline{E}_{2}+...+a_{N}\underline{E}_{N}\,. \tag{3}\]  
Likewise, the measured field at the fiber output \(\underline{E}_{\text{obtained}}\) consisting of the measured amplitude \(|A|\) and phase \(\phi\) distributions,  
\[\underline{E}_{\text{obtained}}=|A|\cdot e^{i\phi}, \tag{4}\]  
needs to be analyzed considering the supported fiber modes \(\underline{E}_{n}\) from the same mode domain:  
\[\underline{E}_{\text{obtained}}=\sum_{n=1}^{N}c_{n}\underline{E}_{n}=c_{1} \underline{E}_{1}+c_{2}\underline{E}_{2}+...+c_{N}\underline{E}_{N}\,. \tag{5}\]  
The coefficients \(c_{n}\) determine the weight of the individual modes in the measured light field. In order to identify the coefficient of every mode in a measured light field, a linear system of equations is derived by multiplying the complex conjugate base functions \(\underline{E}_{1}^{*}\), \(\underline{E}_{2}^{*}\),..., and integrate over the whole space, named \(R\). The integration is approximated by a two-dimensional sum, thus the camera captures a pixelated image:  
\[