Vortices in random wave fields: Nearest neighbor anticorrelations.

The random wave field (speckle pattern) produced by scattering of coherent optical or other waves from a highly disordered medium consists of a seemingly random array of bright areas due to constructive interference, and dark regions due to destructive interference. The characteristic size of a bright region (speckle spot) corresponds to a coherence area of the field, A„h,a region of nearly constant amplitude and phase. The bright speckle spots have been intensively studied by optical methods for over three decades [1]. The dark areas, on the other hand, have largely been neglected, since they apparently contain nothing of interest. Hidden in these dark regions, however, are phase singularities, isolated points from which contours of constant phase radiate outward in a starlike fashion. Wave-field phase singularities were first described in a fundamental paper by Nye and Berry [2], who showed that such singularities satisfy the wave equation (V +k )%'( xy, z) =0. Writing O'=F(x,y) xexp( —ikz) for a wave that propagates along the z axis yields nondiA'racting solutions F+ =x ~iy =r[cos0 +isine]. Since the phase of the wave is p~ =arg(F+. ) =+ 0, the equiphases of these singularities, which are positive (negative) by convention, form a "star" in which the phase circulates counterclockwise (clockwise). As the wave propagates the phase star rotates, tracing out a spiral in space or time, so that these phase singularities are often referred to as vortices. These vortices are closely related to the spiral solutions of the Ginzburg-Landau equation [3], and have recently been investigated in nonlinear optics and laser physics [4-12]. Writing the general wave function as