“…In recent years, practical implementations of mode division multiplexing (MDM) on silicon photonics have already been proposed [ 7 , 109 , 110 , 111 ], which promote the demand for direct FMF mode-multiplexing interface on chip. To the best of our knowledge, for higher-order mode interface using GC, thus far only LP 11 mode experimental demonstration has been reported [ 112 , 113 , 114 , 115 , 116 , 117 , 118 , 119 , 120 ]; LP 21 mode coupling using GC is simulated in some works [ 118 , 121 ], but experimental implementations are still desirable. Higher-order FMF modes consist of multiple lobes in its electric field profile, while neighboring lobes have opposite phase.…”