Transverse modulational instability ($g^2=\frac{q^2}{2k}(2\gamma I_0-\frac{q^2}{2k})$)

A broad focusing beam can amplify transverse intensity perturbations. Linearizing $i{A}_z+{\nabla }_{\perp }^{2}A/(2k)+\gamma |A{|}^{2}A=0$ about its uniform solution gives the displayed growth rate for transverse wave number $q$. For $\gamma >0$, a finite band grows; noise can develop into filaments. A finite beam and saturating response change the ideal plane-background prediction.