Quadratic solitons in cubic crystals

Starting from the Maxwell's equations and without resort to the paraxial approximation, we derive equations describing stationary (1+1)-dimensional beams propagating at an arbitrary direction in an optical crystal with cubic symmetry and purely quadratic ( χ (2)) nonlinearity. The equations are derived separately for beams with the transverse-electric (TE) and transverse-magnetic (TM) polarizations. In both cases, eventual equations contain both χ (2) and cubic ( effective- χ (3)) nonlinear terms, the latter ones generated via the cascading mechanism. The final TE equations and soliton solutions to them are quite similar to those in previously known models with mixed χ (2)– χ (3) nonlinearities. On the contrary to this, the TM model is very different from previously known ones. It consists of four first-order equations for transverse and longitudinal components of the electric field at the fundamental and second harmonics. Fundamental-soliton solutions of the TM model are also drastically different from the usual χ (2) solitons, in terms of the parity of their components. In particular, the transverse and longitudinal components of the electric field at the fundamental harmonic in the fundamental TM soliton are described, respectively, by odd and single-humped even functions of the transverse coordinate. Amplitudes of the longitudinal and transverse fields become comparable for very narrow solitons, whose width is commensurate to the carrier wavelength.