Laser beams that contain phase singularities can be generated with computer-generated holograms, which in the simplest case have the form of spiral Fresnel zone plates.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 fieldsEl exp(iol) and E 2 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

The Z-scan technique is a popular method for measuring degenerate (single frequency) optical nonlinearities using a single laser beam. In order to perform reliable measurements, it is necessary to carefully characterize and control a number of experimental parameters, such as the beam quality, the power and temporal characteristics of the laser, the collection aperture size and position, the sample reflectivity, sample thickness and imperfections in the sample. Failure to control these parameters leads to inaccurate determinations of the nonlinearities. In this paper, we review the theory of Z-scan and examine each of these issues from experimental and theoretical viewpoints. This work will be of interest to anyone who performs Z-scan experiments and to those interested in optical power limiting and nonlinear optical propagation.

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We report the use of a simple interferometric technique which allows direct identification of phase singularities in laser fields . Phase singularities are observed in families of optical patterns formed via cooperative frequency mode locking in a continuous single longitudinal mode Na t ring laser . The interferometric technique complements a previously reported astigmatic imaging method, and is superior in that it can be used to elucidate the structure of the higher order stationary patterns.