Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law

A generalized bidirectional distribution function (BRDF) that relates the specific intensity of the scattered light from a semi-infinite medium to the specific intensity of the incident light is introduced in the framework of coherence theory. This derivation allows us to obtain from first principles several fundamental properties: First, it is established that the generalized BRDF takes the form of a nonlocal relation between the incident and the scattered specific intensities. This nonlocal structure allows us to account naturally for the lateral shift of a beam. Second, the generalized BRDF is the Fourier transform of the correlation function that describes the memory effect. Third, the Helmholtz principle for specific intensities is derived as a theorem from the reciprocity property of the scattering operator for wave fields. This result allows us to prove Kirchhoff’s law.