Self-guided waves and exact solutions of the nonlinear Helmholtz equation

The paraxial wave theory is known to lead to inaccurate predictions in self-focusing of optical beams. The nonlinear Helmholtz equation describes more accurately wave propagation in dispersive, spatially local, Kerr-type media. We derive rigorous bright and dark solutions to the nonlinear Helmholtz equation in a full three-dimensional form. These expressions are new and unique. The solutions are obtained with a multidimensional extension of the (paraxial) nonlinear Schrödinger equation. We also establish energy conservation laws for both nonlinear wave equations in terms of spatial currents. Our results give insight, for example, into the diffraction and breakup of tightly confined nonlinear fields.