Calculation of SHG in periodically poled crystals by specifying a spatially periodic dependence of the quadratic nonlinearity in a single-domain crystal

A method is developed for calculating SHG in linearly homogeneous periodically poled nonlinear crystals (PPNC) by specifying a spatially inhomogeneous periodic distribution of the quadratic-nonlinearity parameter in the form of a 'small-scale' elliptic sine, whose half-period forms one domain with the characteristic 'microplateau' of the nonlinearity parameter and interdomain walls. It is found that, because the domain length should be equal to the coherence length when the quasi-phase-matching condition is fulfilled, and if the coherence length is calculated in the fixed-field approximation, the dependence of the harmonic amplitude on the longitudinal coordinate has the form of a 'large-scale' elliptic sine with a broad 'macroplateau' corresponding to a certain (in the case of quasi-phase matching, to virtually 100%) transformation; however, the mismatch in a domain is never completely compensated by the reciprocal lattice vector. In this case, the phase trajectories inside one domain have the form of a sequence: an unstable focus, a limit cycle ('macroplateau'), a stable focus. This picture repeats in the next domain. It is shown that the width of the SHG phase-matching curve in a PPNC in the regime of strong energy exchange, taking secondary maxima into account, can be considerably (by several times) larger than the width calculated in the fixed-field approximation.