We discuss the role of the transverse degrees of freedom in the quadratically nonlinear interaction of two beams (second harmonic generation) in a guiding medium with a parabolic refractive index profile. Discussion of perturbative approach based on solutions of the linear inhomogeneous initial value problem is presented together with numerical results concerning a fully non-linear problem.

We argue that due to the existing self phase modulation of the pumping field the resonance (phase matching) condition cannot be described well in terms of the quasi linear model. The phase matching condition can be satisfied in a fully nonperturbative regime and coincides with the condition of good spatial overlap leading to an efficient coupling between the beams.