Vacuum-field-induced filamentation in laser-beam propagation

As a laser beam passes through a material medium, it is often observed to break up into a large number of filaments @1#. This filamentation process is initiated by the presence of weak perturbations on the laser wave front @2,3#, which can grow by means of four-wave-mixing processes @4# and become large enough to disturb the overall propagation of the beam. Laser-beam filamentation can be suppressed either by reducing the intensity of the incident laser beam or by using a beam with extremely uniform wave fronts. The maximum intensity that can be transmitted through a given medium is thus determined by the extent to which the wave-front perturbations of the incident beam can be reduced. In this paper we show that vacuum fluctuations of the electromagnetic field @5# constitute a fundamental perturbation to the incident laser field and that filamentation initiated by these quantum fluctuations places a realistic upper limit on the laser intensity that can be transmitted through a given nonlinear optical material without the occurrence of beam breakup. Our theoretical formalism follows closely that of Bespalov and Talanov @6#, which treats the filamentation process classically by considering the gain experienced by a wave-front perturbation on a strong, monochromatic pump beam propagating through a Kerr material. Our treatment differs from theirs in that we consider the optical field to be a quantum-mechanical quantity. The quantum fluctuations of such a field ~i.e., vacuum fluctuations! are necessarily spectrally broadband. Our model thus predicts that quantuminitiated filamentation differs from its classical counterpart in that it is accompanied by a spectral broadening of the transmitted laser field. Our model also differs from its classical counterpart in that it leads to explicit predictions regarding the strength of the fluctuations that initiate the filamentation process. Let us consider the propagation of a laser beam through a Kerr material. We express the positive-frequency part of the total field as