Self-focusing

Remarks

In the paraxial model the beam envelope obeys the two-dimensional cubic focusing nonlinear Schrödinger equation in optics, and the beam power is its conserved $L^2$ norm. For this equation, Weinstein proved that solutions whose $L^2$ norm is smaller than that of the ground state exist for all distances. For larger norms, some solutions blow up at a finite distance. In real media other effects, such as plasma formation or higher-order nonlinearity, stop the collapse.

Example: a Gaussian beam in a medium with $n_2 > 0$ and power well above $P_{\mathrm{cr}}$ narrows as it propagates.

Non-example: in a medium with $n_2 < 0$, the index is lower at the centre of the beam, and the beam spreads faster than by diffraction alone; this is self-defocusing.