We introduce a consistent matrix formalism for the characterization of partial polarization and coherence of random, nonstationary electromagnetic beams in time and frequency domains. We derive the temporal and spectral degrees of polarization and the Stokes parameters in terms of the time-domain and frequency-domain polarization matrices. The connections between temporal polarization and spectral coherence on the one hand and spectral polarization and temporal coherence on the other hand are discussed. Additionally, we establish equivalence theorems for fields with different temporal coherence properties to have the same spectral polarization states and for fields with different spectral coherence properties to possess identical temporal polarization. The theory is illustrated by analyzing specific examples of time-domain and frequency-domain electromagnetic Gaussian Schell-model pulsed beams.

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Coherent-mode representation provides physical insight and computational simplification into the analysis of random optical signals. In this work, we present the coherent-mode decomposition for pulsed electromagnetic beam fields. We show that the mode decomposition can be done for any valid space-frequency or space-time coherence matrix representing nonstationary pulsed electric field, and moreover, the spectral and temporal modes are connected via a Fourier transform relation. The analysis also yields the coherent modes of electromagnetic time-domain signals in temporal optics. We present the overall degree of coherence as a measure of the average temporal or spectral and spatial coherence of the beam. Several illustrative examples are discussed analytically and numerically.

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.

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