## 5 Conclusions  
Although there are several proposals for spatial mode (de)multiplexers for few mode fiber communications, new achromatic and polarization-independent designs are necessary to spatial division multiplexing achieves commercial use in combination with polarization and wavelength division multiplexing. Our demultiplexer, unlike other interferometric ones, is based on a Michelson-like interferometer. It uses a reflective image inverter made by a hollow roof prism mirror or a roof prism in an arm instead of the usual mirror, in such a way that symmetric and antisymmetric modes exit by opposite ports. In order for the system to behave the same for the two polarizations some elements are included. A \(\lambda/4\) retarder must be introduced in an arm, in particular, a Fresnel retarder is desirable as it is achromatic. Besides, any other non-normal incidence in an arm is compensated in the other one with an identical incidence. Such a compensation is not possible in a Mach-Zehnder configuration. A lens in each port is needed to collimating the light from the respective input fibers and focusing it in the output fiber. To minimize the crosstalk between modes, the focal planes of these lenses must match both the mirror and the vertex of the roof prism. The residual crosstalk depends on the quality of the mirrors and the beam splitter.  
If a phase plate is included in the proper port, the system works as a multiplexer that transform the modes of two single mode fibers into two modes (one symmetric and the other antisymmetric) of a few mode fiber. The plate can introduce some losses, but not crosstalk. As the image inverter has  
Figure 6: Generation of a LP\({}_{21a}\) mode multiplexed with a LP\({}_{01}\) mode. The gray box represents the multiplexer of Figure 5. The lines are new recirculating fibers when are out of the box and free space modes inside it. The lower half of the box is port P1 and the upper half is P2, with the input on the left and the output on the right. If we replace the fiber SM3 of Figure 5 with the one containing the multiplexed modes LP\({}_{21a}\) and LP\({}_{01}\), we will obtain five spatial modes in FMF7.  
translation symmetry perpendicularly to the inversion direction, the signal of several pairs of fibers can be multiplexed to the respective output fibers simultaneously with the same interferometer. Besides, these output signals are coupled to few mode elliptical core fibers and recirculated to other inputs. Taking advantage of the fact that the orientation of the modes of these fibers is easily controllable, five spatial modes can be multiplexed in a few mode fiber with a single interferometer. An add-drop mode multiplexer and other configurations based in the same interferometer are suggested.  
In short, we have designed a five spatial mode (de)multiplexer which is achromatic, polarization-insensitive and presents low losses and very low crosstalk. This design can be modified in several ways, so it could be a powerful tool for mode multiplexing.  
All authors have contributed equally to this work. All authors have read and agreed to the published version of the manuscript.  
This research was funded by: Ministerio de Economia, Industria y Competitividad, Gobierno de Espana, grant number AYA2016-78773-C2-2-P; Xunta de Galicia, Conselleria de Educacion, Universidades e FP, Grant GRC number ED431C2018/11; and European Regional Development Fund.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} SDM & Spatial division multiplexing \\ MIMO & Multiple input multiple output \\ DSP & Digital signal processing \\ FMF & Few-mode fiber \\ SMF & Single mode fiber \\ EC & Elliptical core \\ MZI & Mach-Zehnder interferometer \\ TE & Transverse electric \\ TM & Transverse magnetic \\ \end{tabular}