Coherence current, coherence vortex, and the conservation law of coherence

Introducing scalar and vector densities for a mutual coherence function, we present a new conservation law for optical coherence of scalar wave fields in the form of a continuity equation. This coherence conservation law provides new insights into topological phenomena for the complex coherence function. Some properties related to the newly introduced coherence vector density, such as a circulating coherence current associated with a coherence vortex, are investigated both theoretically and experimentally for the first time.