Escape angles in bulk ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ soliton interactions

We develop a theory for nonplanar interaction between two identical type I spatial solitons propagating at opposite, but arbitrary transverse angles in quadratic nonlinear (or so-called ${\chi }^{\mathrm{}\left(2\right)\mathrm{}})$ bulk media. We predict quantitatively the outwards escape angle, below which the solitons turn around and collide, and above which they continue to move-away from each other. For in-plane interaction, the theory allows prediction of the outcome of a collision through the inwards escape angle, i.e., whether the solitons fuse or cross. We find an analytical expression determining the inwards escape angle using Gaussian approximations for the solitons. The theory is verified numerically.