  The ability to measure polarisation, spectrum, temporal dynamics, and spatial
amplitude and phase of optical beams is essential to study fundamental
phenomena in laser dynamics, telecommunications and nonlinear optics. Current
characterisation techniques only apply in limited contexts. Non-interferometric
methods typically lack access to spatial phase, while phase-sensitive
approaches necessitate either an auxiliary reference source or an adequate
self-reference, neither of which is universally available. Regardless of the
reference, deciphering complex wavefronts of multiple co-propagating incoherent
fields remains particularly challenging. Here, we harness the principles of
quantum state tomography to circumvent these limitations. A full description of
an unknown beam is retrieved by measuring its temporally and spectrally
resolved density matrices for both polarisations, using a spatial light
modulator to display projective holograms and a single-mode fibre to guide the
collected signal to a high-speed photodiode and a spectrometer. Despite no
spatial resolution of the detector and the intensity-only character of the
collected signal, the method resolves multiple arbitrary spatial fields within
a single beam, including their phase and amplitude, as well as their spatial
coherence. Leveraging the coherence information unlocks unambiguous
determination of the spectral and temporal evolution of mutually incoherent
fields, even when these spectrally overlap or have an identical time delay. We
demonstrate these hallmark features by characterising the rich spatiotemporal
and spectral output of a vertical-cavity surface-emitting laser diode that has
so far resisted full analysis using existing techniques.
