Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains

We investigate the statistical similarity of partially polarized, partially coherent electromagnetic fields in time and frequency domains, and the relationship between statistical similarity and complete coherence. We find that, both in time domain and frequency domain, the complete coherence of two fields is equivalent to the fields being both fully polarized and statistically similar. Unlike in scalar coherence theory, statistical similarity alone is found not to constitute a sufficient condition for complete coherence. We derive the conditions under which spectrally completely coherent fields are also temporally fully coherent, and we point out that temporally completely coherent fields are necessarily fully spectrally coherent at all frequencies. Complete temporal and spectral coherence of electromagnetic fields are found to be related to the recently introduced concept of strict cross-spectral purity, but in contrast to the scalar case, strict cross-spectral purity is not a necessary condition for complete temporal coherence if the fields have different spectral polarization states.