Fig. 1. (a) Conceptual diagram for cyclic transformation of OAM modes and the high-dimensional Pauli-X gate based on a nonreciprocal MZI. An SPP increases the topological charge of an OAM mode by one to the input lth mode. Optical circulators are used to redirect light reflected off the FP cavity to the third path. This FP cavity transmits the selected OAM mode but reflects other modes. A flipper is used to convert the transmitted (l+1)th mode to the −(l+1)th one. (b) Reflection and transmission spectra of the FP cavity. (c) Phases of the reflected and transmitted fields.

Fig. 2. Schematic of the experimental setup. The light source is a DL pro single-frequency continuous laser. A polarization beam-splitter (PBS) and a QWP form an optical circulator, which is used to redirect light reflected off the FP cavity to the third path [30]. The incident light is horizontally polarized. Light reflected by the FP1 (FP2) cavity passes the QWP twice, transforming into the vertically polarized light, and therefore exits at the reflective end of the PBS1 (PBS2). The thicker purple lines indicate the collimated laser beams.

Fig. 3. Measured transmissions for the FP1. (a) Transmissions of the FP1 cavity for different l modes for a 50 μm and 25 μm incident beam waist, respectively. (b) Transmission spectra of different l modes through FP1. The wavelength interval between the two main peaks in the l=0 mode means the free spectral range (FSR) of the cavity. The transmission spectra for paired modes l=±1,±2, and ±3 are almost identical because the FP1 cavity cannot distinguish the +l and −l modes.

Fig. 4. Experimental results for six-mode cyclic transformation. (a) Measured power of six output OAM modes for six input OAM modes. Each color represents a specific input mode. (b) Measured efficiency E for a different input mode.

Fig. 5. (a) Finesse of the cavity as a function of mirror reflectivity R. (b) Cavity length as a function of the mirror reflectivity R, where we have kept a constant cavity linewidth. The blue dots in (a) and (b) represent the case when the cavity length is set to be 1 mm, which is accessible in experiment, and the corresponding finesse is F=943.