## 4 Multiplexers Based on Cyclically Used Michelson Interferometeres  
A multiplexer based on two interferometers similar to the one of the previous section is shown in Figure 4. It multiplexes the signal from four single mode fibers into respective spatial modes of a few mode fiber. The bottom interferometer includes a phase plate in one of its ports (P1), which introduces a \(\pi\)-phase in the right side of the wavefronts coming from fiber 1, leading to an antisymmetric amplitude along \(X\)-axis. The length of the arms must be adjusted to obtain fully constructive interference in the other port (P2). The light from the single mode fiber 3 (symmetric amplitude from P2) also exits thought P2, leading to a first multiplexing. These signals are coupled into fiber 5, which must support two modes along the horizontal direction. The amplitude unbalance due to the uncompensated reflection hardly matters when the interferometers are used as multiplexers. This unbalance generates a low intensity wave in the exit of port P1 which is unused. Therefore there is a little efficiency reduction in the multiplexed output, but no crosstalk appears. Similarly, a misalignment of the phase plate or a deviation of its phase shift from \(\pi\) due to chromatic dispersion [19] introduces a small symmetrical wave component into the interferometer, but it is sent back to port P1, that is, it does not generate crosstalk on port P2 either. Analogously, the signals from fibers 2 and 4 are combined and coupled to respective modes of the fiber 6. Both fibers 5 and 6 must maintain an horizontal orientation of the second mode. In theory, it can be done with circular core fibers provided that the curves they describe avoid topological coupling between modes. However, EC-FMFs accomplish this task more easily if the long axes of its cores are oriented horizontally (or vertically) at both ends [4]. Remember that these fibers also maintain the beam polarization. The second interferometer (top-right in Figure 4) is similar to the first one, but it multiplexes symmetric and antisymmetric amplitudes along the vertical direction. To do that, the phase plate, the Porro's prism and the compensating prism are rotated 90 degrees; moreover the fiber ends are displaced horizontally (instead of vertically) with respect to the optical axes of the collimating lenses. The beams from fiber 5 are coupled in the antisymmetric modes along \(Y\)-axis of the final few mode  
Figure 4: Two concatenated interferometers to multiplex four modes into a few mode fiber. fiber 7. Although this multiplexer does not cyclically use any interferometer, it is a good introduction to the following system. We can save the last interferometer if the output of fibers 5 and 6 are rotated 90 degrees and then they are introduced again into the first interferometer. In Figure 5 one can see this cyclically used interferometer. Similarly to the MZI with reflective image inverters [13], this interferometer also can multiplex 5 spatial modes instead of 4, albeit in somewhat more complicated scheme as shown in Figure 6. Note that the mode from the fiber SMF3 of Figure 5 never crosses the phase plate, so it generates the fundamental mode of FMF7 output. If we could add a \(\text{LP}_{21a}\) mode (symmetrical along \(X\) and \(Y\) directions) in SMF3, it would behave like the fundamental mode in each multiplexing operation, exiting the interferometer together the other four modes. To that end, we can multiplex \(\text{LP}_{01}\) and \(\text{LP}_{21b}\) modes into an elliptical core fiber by using the interferometer, rotate them \(45^{\circ}\) to generate the \(\text{LP}_{21a}\) mode, and replace the SMF3 with this fiber. The \(\text{LP}_{21b}\) mode can be generated from a \(\text{LP}_{11b}\) one (antisymmetric along vertical direction) entering by port P1 while the \(\text{LP}_{01}\) mode enters by port P2.