Generation of optical phase singularities by computer-generated holograms.

There has recently been considerable interest in optical fields that exhibit phase singularities, which manifest themselves as isolated dark spots in the modal patterns of certain lasers. Each dark spot has a topological charge that represents the number of 2vr accumulated when the phase gradient is integrated around it. The wave fronts near a singularity have a helical structure, while the field at the singularity must be zero because of the ambiguous phase; hence the dark spot. It has been shown theoretically how, under the influence of nonlinear interactions, frequencydegenerate transverse modes can lock together with a fixed phase difference to produce stable patterns that contain one or more of the singularities.' Such modal patterns have also been observed experimentally.2 -4 Uncontrolled generation of random arrays of singularities has also been reported.5 Here we report a means of generating such singularities in a controlled way, using only the simplest equipment, which amounts to the use of a computergenerated hologram, or zone plate. The hologram that is simulated is that of a modal pattern that contains a set of phase singularities using a reference plane wave. The technique can be extended to produce patterns of great complexity, but in this Letter we mainly concentrate on the circularly symmetric doughnut mode. If one interferes two coherent optical fields El exp(iol) and E2 exp(i0 2 ), the resultant spatial intensity pattern is modulated by a 2ElE 2 cos((k + 02) term, which represents the interference fringes. The desired interference pattern is between a plane wave and the lowest-order hybrid doughnut mode,3 Eor/co exp[-(r/w) 2 ]exp(-ikr 2 /2R ± j0). Here k is the wave number, o) is the spot size, and R is the radius of curvature of the doughnut Gaussian beam. The plus or minus refers to the two possible chiralities of the helical wave. The boundary between a light fringe and a dark fringe occurs when the above cosine term is zero, and when the beams are collinear, this means kr2 ±6 (n ± 1/2)7+ -2R zone plate) by filling in the region where the modulation term is negative. This produces the pattern shown in Fig. 1(a), which may be recognized as a variation on the Fresnel zone plate. Like a normal zone plate, it will tend to form foci at distances such that the waves from all over the transparent areas will arrive in phase, but unlike normal zone plates, symmetry is broken on the axis as waves from diametrically opposite regions will always cancel out. When such a hologram is illuminated with coherent light, a helical wave will be reconstructed. Equation (1) can be generalized for charge p doughnut modes; i.e., kr2 pO = (n + 1/2)irr + 2R (2)