Correlation visibility and generalized Siegert relation for random light beams

Phase difference is central to classical coherence theory. With the advancement of various light-field modulation techniques, artificially generated pseudo-thermal light sources or random light beams can exhibit exotic wavefront correlation properties. However, such spatial wavefront correlations cannot be fully characterized using the phase difference alone. For instance, for a pair of conjugate pseudo-thermal beams, the spatial wavefronts exhibit a significant anti-correlation, meaning that the sum of their wavefronts tends to be constant. In this work, we propose the concept of degree of wavefront correlation $p^{(1)}$, ranging symmetrically from $-1$ to $+1$, for numerically calculating the wavefront correlation properties among various pseudo-thermal light sources, and the sign (positive or negative) can be used to determine the tendency-whether it leans toward wavefront-difference or wavefront-sum correlation. Numerical results demonstrate that the classical Siegert relation does not apply to pseudo-thermal light sources that exhibit wavefront-sum correlation properties. To address this, we propose a generalization valid for all Gaussian pseudo-thermal light. Experimentally, we introduce the measurable quantities of correlation visibility $\mathcal{V}_g$ and correlation background $\mu_g$, which form a two-dimensional classification framework $\{\mu_g,\mathcal{V}_g\}$ that enables the experimental characterization of diverse Gaussian pseudo-thermal light using a common-path interferometer and intensity correlation measurement. Furthermore, the correlation visibility $\mathcal{V}_g$ can serve as an observable criterion for a zero-mean, non-circularly symmetric, and jointly Gaussian distribution.