Fig. 3: Spatial and temporal coherence measurements.

Spatial coherence of the emission from the fundamental Laguerre-Gaussian cavity mode for different excitation pulse energies: a below threshold (\(0.9{P}_{{\rm th}}\)), b above threshold (\(2.4{P}_{{\rm th}}\)), c far above threshold (\(12.4{P}_{{\rm th}}\)). Numbers in the upper right indicate multiplication factors for better visibility. a–c Spatially resolved interference images obtained by overlapping the emission from the mode with a spatially inverted, momentum-shifted copy realized with a Michelson interferometer using a retro-reflector in one arm. d–f Spatial first-order correlation function at zero delay \({g}^{\left(1\right)}(\tau=0,{{{{\boldsymbol{r}}}}}_{{||}})\) extracted by Fourier analysis of a–c. g Effective de Broglie wavelength extracted from panels d-f by a Gaussian fit to the spatial first-order correlations within the extent of the sphere cap-shaped indentation (dashed circle in a). h, i Temporal first-order correlation function \({g}^{\left(1\right)}(\tau,{\bar{{{{\boldsymbol{r}}}}}}_{{||}})\) averaged over \(2{\ \mu {\rm m}}\times 2{\ \mu \rm m}\) at the center of the spatial coherence map in e, f, as a function of interferometer delay. The fits show coherence times of 0.71 ps and 1.06 ps, respectively. Error bars in (g) correspond to 95% confidence intervals of the fit; in (h, i) they correspond to the standard deviation of the averaged \({g}^{\left(1\right)}\) function.