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Experimental data of femtosecond thick-crystal second-harmonic generation shows that when tuning away from phase matching, a dominating narrow spectral peak appears in the second harmonic that can be tuned over 100's of nm by changing the phase-mismatch parameter. Traditional theory explains this as phase matching between a sideband in the broadband pump to its second-harmonic. However, our experiment is conducted under high input intensities and instead shows excellent quantitative agreement with a nonlocal theory describing cascaded quadratic nonlinearities. This theory explains the detuned peak as a nonlocal resonance that arises due to phase-matching between the pump and a detuned second-harmonic frequency, but where in contrast to the traditional theory the pump is assumed dispersion-free. As a soliton is inherently dispersion-free, the agreement between our experiment and the nonlocal theory indirectly proves that we have observed a soliton-induced nonlocal resonance. The soliton exists in the self-defocusing regime of the cascaded nonlinear interaction and in the normal dispersion regime of the crystal, and needs high input intensities to become excited., Comment: submitted, revised version

We propose an efficient approach to improve few-cycle soliton compression with cascaded quadratic nonlinearities by using an engineered multi-section structure of the nonlinear crystal. By exploiting engineering of the cascaded quadratic nonlinearities, in each section soliton compression with a low effective order is realized, and high-quality few-cycle pulses with large compression factors are feasible. Each subsequent section is designed so that the compressed pulse exiting the previous section experiences an overall effective self-defocusing cubic nonlinearity corresponding to a modest soliton order, which is kept larger than unity to ensure further compression. This is done by increasing the cascaded quadratic nonlinearity in the new section with an engineered reduced residual phase mismatch. The low soliton orders in each section ensure excellent pulse quality and high efficiency. Numerical results show that compressed pulses with less than three-cycle duration can be achieved even when the compression factor is very large, and in contrast to standard soliton compression, these compressed pulses have minimal pedestal and high quality factor.

We interpret the purely spectral forward Maxwell equation with up to 3${^{\rm rd}}$ order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named nonlinear wave equation in frequency domain, includes both quadratic and cubic nonlinearities, delayed Raman effects and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena such as blue/red shifting, non-stationary radiation in accordance with the non-local phase matching condition and dispersive-wave generation are observed and marked, which help improving the experimental knowledge of cascading nonlinear soliton pulse compression.

We study the anisotropic nature of the Kerr nonlinear response in a beta-barium borate (\beta-BaB2O4, BBO) nonlinear crystal. The focus is on determining the relevant $\chi^{(3)}$ cubic tensor components that affect interaction of type I cascaded second-harmonic generation. Various experiments in the literature are analyzed and we correct the data from some of the experiments for contributions from cascading as well as for updated material parameters. We find that the Kerr nonlinear tensor component responsible for self-phase modulation in cascading is considerably larger than what has been used to date. We evaluate the impact of using such a cubic anisotropic response in ultrafast cascading experiments., Comment: Updated version, comments on experiments from the literature welcome