Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation

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Abstract
We show that the increase in critical power for elliptic input beams is only 40% of what had been previously estimated based on the aberrationless approximation. We also find a theoretical upper bound for the critical power, above which elliptic beams always collapse. If the power of an elliptic beam is above critical, the beam self-focuses and undergoes partial beam blowup, during which the collapsing part of the beam approaches a circular Townesian profile. As a result, during further propagation additional small mechanisms, which are neglected in the derivation of the nonlinear Schrödinger equation (NLS) from Maxwell’s equations, can have large effects, which are the same as in the case of circular beams. Our simulations show that most predictions for elliptic beams based on the aberrationless approximation are either quantitatively inaccurate or simply wrong. This failure of the aberrationless approximation is related to its inability to capture neither the partial beam collapse nor the subsequent delicate balance between the Kerr nonlinearity and diffraction. We present an alternative two-stage approach and use it to analyze the effect of nonlinear saturation, nonparaxiality, and time dispersion on the propagation of elliptic beams. The results of the two-stage approach are found to be in good agreement with NLS simulations.