Extending optical filaments with phase-nested laser beams

Extending the length of femtosecond laser filamentation has always been desired for practical applications. Here, we demonstrate that significant extending of a single filament in BK7 glass can be achieved by constructing phase-nested beams. The filamentation and the following energy replenishment are assembled in a single phase-nested beam. The central part of the phase-nested beam is an apertured Gaussian beam, which is focused into one focal spot to produce a short filament. In contrast, the rest of the annular part converges gradually towards the central axis to continuously replenish the energy for supporting the regeneration of filaments. The common-path generating system ensures the stability of generated filaments and easily optimizes the beam parameters to obtain the longest high-quality filament due to its flexibility. In addition, we discuss the significance of continuous replenishment for extending filaments and the potential for generating more extended filaments based on this method.