# Mode-Selective Photonic Lanterns for Orbital Angular Momentum Mode Division Multiplexing
## 2 Principle  
A taper transition can couple light between one FMF and several SMFs. If the number of SMFs matches the number of spatial modes in the FMF, the transition can have low loss in both directions [13]. Taking a three-mode PL as an example, a standard PL is fabricated by adiabatically tapering three separate SMFs in a low-index glass jacket, as shown in Figure 1a. As unitary coupling between SMFs and one FMF is only possible by optimizing the arrangements of the cores [14], the cross section of the untapered end of the PL is shown in Figure 1b, which can be considered approximately as three isolated fiber cores (Fiber 1, 2, and 3 cores) with a circular cladding and a low-index jacket. The structure is tapered such that the SMF cores nearly vanish, the SMF cladding becomes the new few-mode core, and the low-index capillary becomes the FMF cladding. The tapered end of the PL matches the common two-mode fiber supporting LP\({}_{01}\) and LP\({}_{11}\) modes. The original LP\({}_{01}\) modes in the SMF cores eventually evolve into the LP\({}_{01}\) and LP\({}_{11}\) modes in the FMF core. In this paper, we obtain the eigenmode profiles and effective refractive indexes of the PL at different positions along the taper, simulated by mode analysis using the finite element method (FEM). However, FEM is not able to simulate the many elements of the light transmission in the PL, such as the adiabatic taper criterion and the actual forms of the excited supermodes, i.e., local modes. It is not able to analyze the possible effects brought by the variation of the taper lengths either. Therefore, as a supplement, we use the beam propagation method (BPM) to analyze the evolution of the local modes excited by selectively injecting light into the SMF cores during the taper of the PL, in order to consider the variation of the light transmission conditions. Three-mode MSPLs for LP mode multiplexing can be divided into two types: the standard three-mode MSPL designed with three totally different SMF cores, and the three-mode mode-group-selective PL owning a pair of symmetric SMF cores with a lower initially designed normalized frequency value (V value) compared to the other core. We simulate both these types to analyze the evolution of the eigenmodes and the local modes using FEM and BPM. In this paper, the simulated PL has three cores arranged in an equilateral triangle shape with a core pitch of 42.0 \(\upmu\)m, as the reduced cladding fibers can be used to assist adiabaticity [15; 16]. The core index is 1.4482. For the standard MSPL, the core diameters are 11.0 \(\upmu\)m, 8.65 \(\upmu\)m, and 6.55 \(\upmu\)m, respectively. For the mode-group-selective PL, the core diameters are 11.0 \(\upmu\)m, 6.55 \(\upmu\)m, and 6.55 \(\upmu\)m, as the Fiber 2 core and  
Figure 1: The (**a**) structure and (**b**) cross section of a photonic lantern (PL). Note: (**b**) is not to scale. the Fiber 3 core are identical. The cladding and jacket diameters are 125 and 1116 \(\upmu\)m, and the cladding and jacket indexes are 1.444 and 1.4398, respectively. As shown in Figure 2a, for the standard MSPL, the original eigenmodes of the untapered end are the LP\({}_{01}\) modes in the three separate SMF cores. The second- and third-highest indexes remain separate until the SMF cores nearly vanish and the local modes evolve into the degenerate LP\({}_{11}\) modes. When selectively injecting light into the Fiber 2 or 3 core, as shown in Figure 2b, at the same taper ratio, the local mode shown as the BPM profile is in correspondence with the eigenmode, shown as the FEM profile. On account of the separation of the effective refractive indexes, the second- and third-order local modes individually evolves into the two second-order eigenmodes in the FMF end, i.e., LP\({}_{11a}\) and LP\({}_{11b}\).