Solitons in quadratic nonlinear photonic crystals

We study solitons in one-dimensional quadratic nonlinear photonic crystals with modulation of both the linear and nonlinear susceptibilities. We derive averaged equations that include induced cubic nonlinearities and numerically find previously unknown soliton families. The inclusion of the induced cubic terms enables us to show that solitons still exist even when the effective quadratic nonlinearity vanishes and conventional theory predicts that there can be no soliton. We demonstrate that both bright and dark forms of these solitons are stable under propagation.Comment: 4 pages with 6 figure

Accurate switching intensities and length scales in quasi-phase-matched materials

We consider unseeded Type I second-harmonic generation in quasi-phase-matched (QPM) quadratic nonlinear materials and derive an accurate analytical expression for the evolution of the average intensity. The intensity-dependent nonlinear phase mismatch due to the QPM induced cubic nonlinearity is found. The equivalent formula for the intensity for maximum conversion, the crossing of which changes the nonlinear phase-shift of the fundamental over a period abruptly by $\pi$, corrects earlier estimates by more than a factor of 5. We find the crystal lengths necessary to obtain an optimal flat phase versus intensity response on either side of this separatrix intensity.Comment: 3 pages with 3 figure

The complete modulational instability gain spectrum of nonlinear QPM gratings

We consider plane waves propagating in quadratic nonlinear slab waveguides with nonlinear quasi-phase-matching gratings. We predict analytically and verify numerically the complete gain spectrum for transverse modulational instability, including hitherto undescribed higher order gain bands.Comment: 4 pages, 3 figures expanded with more explanation and mathematical detai

Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression

We study soliton pulse compression in materials with cascaded quadratic nonlinearities, and show that the group-velocity mismatch creates two different temporally nonlocal regimes. They correspond to what is known as the stationary and nonstationary regimes. The theory accurately predicts the transition to the stationary regime, where highly efficient pulse compression is possible.Comment: 3 pages, 2 figures, published verison in Optics Letters. Contains revised equations, including an updated mode

Limits to compression with cascaded quadratic soliton compressors

We study cascaded quadratic soliton compressors and address the physical mechanisms that limit the compression. A nonlocal model is derived, and the nonlocal response is shown to have an additional oscillatory component in the nonstationary regime when the group-velocity mismatch (GVM) is strong. This inhibits efficient compression. Raman-like perturbations from the cascaded nonlinearity, competing cubic nonlinearities, higher-order dispersion, and soliton energy may also limit compression, and through realistic numerical simulations we point out when each factor becomes important. We find that it is theoretically possible to reach the single-cycle regime by compressing high-energy fs pulses for wavelengths $\lambda=1.0-1.3 \mu{\rm m}$ in a $\beta$-barium-borate crystal, and it requires that the system is in the stationary regime, where the phase mismatch is large enough to overcome the detrimental GVM effects. However, the simulations show that reaching single-cycle duration is ultimately inhibited by competing cubic nonlinearities as well as dispersive waves, that only show up when taking higher-order dispersion into account.Comment: 16 pages, 5 figures, submitted to Optics Expres

Plane waves in periodic, quadratically nonlinear slab waveguides: stability and exact Fourier structure

We consider the propagation of broad optical beams through slab waveguides with a purely quadratic nonlinearity and containing linear and nonlinear long-period quasi-phase-matching gratings. An exact Floquet analysis on the periodic, plane-wave solution shows that the periodicity can drastically alter the growth rate of the modulational instability but that it never completely removes the instability. The results are confirmed by direct numerical simulation, as well as through a simpler, approximate theory for the averaged fields that accurately predicts the low-frequency part of the spectrum.Comment: 10 Pages, 13 figures (some in two parts) new version has some typos removed and extra references and explanation adde

On the best compact approximation problem for operators between Lp-spaces

AbstractWe construct (for 1 < p < 2) an operator from lp into Lp which has no nearest compact operator. We also give a sufficient condition for an operator from Lp into Lp (2 < p < ∞) to have a best compact approximant