Phase-space measurement and coherence synthesis of optical beams

Phase-space optics allows the simultaneous visualization of both space (x) and spatial frequency (k) information. Previous experiments have focused mainly on coherent beams, which are fully described by a two-dimensional complex field (amplitude and phase). In contrast, partially coherent beams inherently have more degrees of freedom and require a four-dimensional description. This description is particularly important for propagation, in which coherence properties determine the intensity evolution. Despite this, most measurements in linear optics, and all those in nonlinear optics, have recorded only the intensity and power spectrum projections (x-space or k-space only). Measuring local coherence remains a challenging problem, especially in the full four-dimensional phase space. In turn, the recording difficulty has limited efforts to generate arbitrary, spatially varying patterns of coherence, despite their usefulness for imaging, illumination and display. We remedy both problems here, using spatial light modulators to create beams with locally varying spatial coherence and to measure their phase-space properties.