# Mode-Selective Photonic Lanterns for Orbital Angular Momentum Mode Division Multiplexing
## 2 Principle
As shown in Figure 3a, for the mode-group-selective PL, as the Fiber 2 and 3 cores share the same V value, the initial eigenmodes are the in-phase and reverse-phase combination of the LP\({}_{01}\) modes in the Fiber 2 and 3 cores, and their effective refractive indexes are identical at the starting area of the taper. When selectively injecting light into either of the pair of the totally symmetric and identical cores (Fiber 2 or 3 cores), both the degenerate second-order eigenmodes are excited. The two eigenmodes break the degeneracy and bring the difference value of the effective refractive indices in the deciding area of the taper (where the effective index differences of M\({}_{\text{L2}}\) and M\({}_{\text{L3}}\) are larger than \(10^{-4}\), the value normally considered as the floor in order to eliminate MIMO-DSP in PLs [17]), then they return to degeneracy again at the end of the taper. As shown in Figure 3b, each of the local modes shown as the BPM profile is the isometric superposition of the two eigenmodes shown as FEM profiles, and the composition of the output mode in the FMF end depends on the accumulation of the phase differences  
Figure 2: **(a)** The FEM simulation mode indexes evolution along the taper of the standard MSPL. M\({}_{\text{E1}}\), M\({}_{\text{E2}}\), and M\({}_{\text{E3}}\) represent the first- and the two second-order eigenmodes. The M\({}_{\text{EL}}\) represents the leaky mode simulated by FEM. The black dashed line indicates the taper ratio of 0.112. **(b)** The comparison of the FEM/BPM simulated mode profiles at different positions along the taper when selectively injecting light into the Fiber 2 and 3 cores of the standard MSPL. M\({}_{\text{L1}}\), M\({}_{\text{L2}}\), and M\({}_{\text{L3}}\) represent the local modes when individually injecting light into Fiber 1, 2, or 3 cores, respectively. BPM: beam propagation method; FEM: finite element method; MSPL: mode-selective photonic lantern. brought by the two eigenmodes in the deciding area of the taper. The relationship can be described as follows:  
\[\mathrm{M_{L2}}=\mathrm{M_{E2}}+\mathrm{exp}(\mathrm{i}\varphi)\mathrm{M_{E3}} \tag{1}\]  
\[\mathrm{M_{L3}}=\mathrm{M_{E2}}+\mathrm{exp}[\mathrm{i}(\pi-\varphi)]\mathrm{M_{ E3}}, \tag{2}\]  
where \(\mathrm{M_{L2}}\) and \(\mathrm{M_{L3}}\) represent the local modes when individually injecting light into the Fiber 2 or 3 core, \(\mathrm{M_{E2}}\) and \(\mathrm{M_{E3}}\) represent the two second-order eigenmodes, and \(\varphi\) represents the final phase difference between the two parts of the excited eigenmodes generated during the taper. It is already known that one pair of degenerated \(\mathrm{LP_{11}}\) modes which satisfy the \(\pi\)/2 phase difference can be used to generate OAM\({}_{\pm 1}\) modes in optical fibers [7], as follows:  
\[\mathrm{OAM_{-1}}=\mathrm{LP_{11a}}+\mathrm{i}\ \mathrm{LP_{11b}} \tag{3}\]  
\[\mathrm{OAM_{+1}}=\mathrm{LP_{11a}}-\mathrm{i}\ \mathrm{LP_{11b}}. \tag{4}\]  
Therefore, when selectively injecting light into the Fiber 2 or 3 core, if \(\varphi\) equals \(\pi\)/2 (or \(-\pi\)/2), the output mode is one of the OAM\({}_{\pm 1}\) modes, and if \(\varphi\) equals 0 (or \(\pi\)), the output mode is one of the \(\mathrm{LP_{11}}\) modes. The BPM simulation mode profiles of the PLs with lengths of 2 cm, 2.65 cm, and 3.3 cm are shown in Figure 3b-d, as the final taper ratio is determined as 0.112. As described above, when injecting light into either the Fiber 2 or 3 core selectively, the variation of the phase differences leads to the changing of the superposed output mode. For instance, in the simulated PL with a taper length of 2 cm, the excited mode of the Fiber 2 core corresponds to OAM\({}_{-1}\). It becomes \(\mathrm{LP_{11a}}\) in the simulated PL with a taper length of 2.65 cm, and then turns to OAM\({}_{+1}\) when the taper length is 3.3 cm. The output mode profiles of the PLs with different taper lengths are shown in Figure 4, based on the BPM simulation when injecting light into each of the three SMF cores.