Spatial Correlation Singularity of a Vortex Field

Experimental and numerical techniques allowed us to predict and verify the existence of a robust phase singularity in the spatial coherence function when a vortex is present. Though observed in the optical domain, this phenomenon may occur in any partially coherent vortex wave.

Figure 1

A diagram of a partially coherent beam of light transmitted through a vortex mask of transverse diameter $2w$. In the plane $z=z_d$, a conical projection of the source through the center of the mask forms an enclosed circular region. Light inside the circle (point $A$) contains rays from all sectors of the mask, but outside the circle (point $B$) the light contains rays from only adjacent sectors of the mask, as depicted by the dashed lines.

Figure 2

Numerically computed intensity (a)–(c) and cross-correlation (d)–(f) functions for $m=1$ in the far-field plane for various coherence lengths. As the coherence decreases, the dark vortex core fills with diffuse light. However, a ring dislocation persists in the cross-correlation function, and its radial size, $f_r$, increases with decreasing coherence.

Figure 3

A schematic diagram of the experiment showing light from a halogen bulb with a mean wavelength of 800 nm and a bandwidth of 50 nm passing through two apertures separated by a distance, $S=41.5\mathrm{cm}$. The ${\mathrm{Ap}}_2$ plane was imaged with unity magnification by two 50.2 mm focal length lenses (${\mathrm{L}}_1$ and${\mathrm{L}}_2$) onto a phase mask designed to produce an $m=1$ vortex beam. The plane $z=92\mathrm{mm}$ was imaged with a magnification of 3 by two lenses having focal lengths $f=125\mathrm{mm}$ (${\mathrm{L}}_3$) and $f=500\mathrm{mm}$ (${\mathrm{L}}_4$) onto a CCD camera after passing through a wavefront folding interferometer comprising Dove prisms, ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$.

Figure 4

Experimental interferograms of the cross-correlation function in the plane $z=540\mathrm{mm}$ for an (a) $m=1$ vortex beam and (b) $m=0$ nonvortex beam (${\mathrm{L}}_c/w=0.20$ in both cases). Shifted (a) and uniform (b) fringes indicate the respective presences and absences of a ring dislocation of radial size, $R^{\prime }=0.08\mathrm{mm}$.

Figure 5

Experimental (points) and theoretical (line) values of the dislocation radius $R^{\prime }$ in the measurement plane as a function of the relative coherence length ${\mathrm{L}}_c/w$ at the entrance face of the vortex mask.