  Spin-orbit coupling in two-dimensional systems is usually characterized by
Rashba and Dresselhaus spin-orbit coupling (SOC) linear in the wave vector.
However, there is a growing class of materials which instead support dominant
SOC cubic in the wave vector (cSOC), while their superconducting properties
remain unexplored. By focusing on Josephson junctions in Zeeman field with
superconductors separated by a normal cSOC region, we reveal a strongly
anharmonic current-phase relation and complex spin structure. An experimental
cSOC tunability enables both tunable anomalous phase shift and supercurrent,
which flows even at the zero-phase difference in the junction. A fingerprint of
cSOC in Josephson junctions is the f-wave spin-triplet superconducting
correlations, important for superconducting spintronics and supporting Majorana
bound states.
