# Deep Learning for Computational Mode Decomposition in Optical Fibers
## 3 Results
### Performance of the DNN Using Experimental Data  
It has to be shown that with known FMF parameters it is possible to train a DNN in such a way that a complex-valued MD can be performed based on pure intensity images in an experimental environment. Therefore, random mode combinations can be created assuming the same FMF parameters used in the previous section. The respective light fields are shaped subsequently with the help of a Spatial Light Modulator (SLM) using superpixels and imaged with a camera. In Figure 4, the optical setup used to capture the individual mode field distributions purely intensity-based as well as holographically, is demonstrated. As it has already been introduced in prior work [19], the system is originally intended for selective mode excitation with an MMF. A single-mode fiber (SMF) is used for spatial filtering of a laser beam to ensure the best beam profile for the experiments. The beam is then collimated (CP) and expanded using a beam expander (BE). Since the SLM is only suitable for linear polarization directions, a polarizing beam splitter (PBS) and a half-wave plate (HWP) are used to filter the required polarization direction. Lens L1 is performing an FFT with the light field which is reflected off the surface of the SLM. Since a diffraction grating is added to the phase-mask which is displayed on the SLM, the modulated light field is separated spatially into the respective diffraction orders. The fully modulated light signal is located in the higher diffraction orders, whereas the nonmodulated background is located in the 0 th order. In the Fourier domain, the 1 st diffraction order is filtered spatially with IB1 to let the individual superpixels mix with each other and inverse Fourier transformed with L2 afterwards. The generated light field is then imaged onto a CMOS camera propagating through 2 telescopes (L3-L4 and L4-L5). The captured images are cut out and fed into the network, to predict the chosen mode weights. In addition, it is possible to record the light field distributions for the second experiment series holographically. For this purpose, the laser beam used in the optical setup is additionally split into a reference beam using BS1. For this reason, a beam cleanup is performed for the reference beam using two lenses L4 and L5, as well as a pinhole IB2, in order to generate a beam profile that is as pure as possible. The recorded hologram is then used to determine the individual complex weights according to Equation (4). If purely intensity-based images should be captured, the reference beam has to be blocked. In order to compare the quality of the DNN-based MD with the holography-based MD (Equation (4)), 150 different and random mode field distributions are captured. To ensure a fair comparison between the DNN and the holographic method, the amplitude distributions from the  
Figure 3: Correlation coefficients using Specified Mode Combinations (SMC), respectively, random mode combinations for training the DNN. For better readability, an exemplary selection of the \(\approx\)6000 results are depicted in the figure. holographic measurement were used as input for the DNN. The respective mean relative errors (Equation (10)) are shown in Table 2.  
Using DH, a mean relative error of the phase and amplitude determination of the mode weights of 4.20% and 3.32% could be achieved, respectively. The DNN prediction of the mode weights using the SMC dataset is afflicted with a mean relative error of 9.45% and 6.02% for phase and amplitude, respectively, or 11.03% and 9.56% using random mode combinations for training the DNN. As it is shown in Figure 5a, based on measured, respectively predicted weights, the intensity distributions of the mode fields are computed and correlated with the original intensity distributions (Equation (6)). Using DH, a mean correlation coefficient of 96.97% with \(\sigma=2.26\%\) could be achieved, whereas the mean correlation obtained by using the DNN, which was trained with the SMC dataset, has a value of 95.10% with \(\sigma=1.90\%\).