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Nonlinear self-focusing in laser glass imposes limits on the energy fluence that can be safely transmitted without risking damage. For this reason, it is desirable to strictly limit the peak to average spatial variations of fluence by smoothing schemes such as smoothing by spectral dispersion (SSD). While spatial variations are problematic, the same is not necessarily true of temporal variations since normal group velocity dispersion tends to smooth out temporal peaks caused by spatial self-focusing. Earlier work indicated that increased bandwidth can delay the onset of self focusing. Indeed, a point can be reached at which self phase modulation nonlinearly increases the bandwidth, changing the speckle statistics along with suppressing self focusing. Unfortunately, this study found that a large initial bandwidth (compared with the gain bandwidth) was necessary to achieve this suppression under practical conditions. The full calculation for modulated beams was carried out for one transverse dimension. Two transverse dimensional calculations only treated symmetric beams. The present work reexamines the question of self focusing threshold increases due to high bandwidth by investigating another source of such increase in three dimensional beam breakup -- the bending instability. For simplicity, we consider the behavior of a single space-time speckle. Normal dispersion can lead to splitting of the pulse and delay of self focusing for short enough pulses as noted above. In addition to the self focusing instability, the laser beam is also subject to the so-called bending (sausage like) instability which can spatially disperse the field maxima over time. Because the bending instability breaks an initial axial symmetry, a full three dimensional numerical simulation is required to study it accurately. Such calculations are possible, but costly. We have used a modified 2D nonlinear Schrodinger equation with a high power nonlinearity since this mimics the 3D behavior of the competition between self focusing and bending. This allows a semi-quantitative estimate to be made of the possible significance of the bending instability for suppression of self focusing.
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