Soliton-induced nonlocal resonances observed through high-intensity tunable spectrally compressed second-harmonic peaks

Experimental data of femtosecond thick-crystal second-harmonic generation show that when tuning away from phase matching, a dominating narrow spectral peak appears in the second harmonic that can be tuned over hundreds of nanometers 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.

Figure 1

(a) Experimental SH normalized spectra sweeping the phase-mismatch parameter from negative to positive values (external angle tuned from $-3^{\circ }$ to $+5^{\circ }$ around $\Delta k=0)$. An $L=25$ mm BBO crystal was used pumped with ${\lambda }_1=1.03\mu \mathrm{m}$ 58-fs FWHM 80 GW/cm${}^2$ pulses. (b) The nonlocal theory, predicting $I_2\propto {\left|R\left(\Omega \right)\right|}^2$. (c) Phase-matched sidebands theory, predicting $I_2\propto {\mathrm{sinc}}^2[\Delta k\left(\lambda \right)L/2]$.

Figure 2

The SH resonance wavelengths calculated for type I birefringent $(oo\to e)$ SHG in BBO. (a) Illustration of the phase-matching condition $k_2\left(\omega \right)-2k_1(\omega /2)$ at $+4^{\circ }$ of Fig. 1, reported in the FW reference frame $\tau =t-z{k}_1^{\left(1\right)}$. The SH curve (thin black line) is tunable through $\theta$; this is indicated with thin gray curves for $\theta$ taken $1^{\circ }$ larger and smaller. (b, c) Calculated SH peak wavelengths vs phase mismatch $\Delta k$ for ${\lambda }_1=0.8$ and $1.03\mu \mathrm{m}$, respectively. Solid lines are resonant wavelengths, while dashed lines are the peak values of the Lorentzian shape of the nonresonant response. The self-defocusing regime is indicated where $n_{2,\mathrm{casc}}^I+n_{2,\mathrm{Kerr}}^I<0$.

Figure 3

(a) Experimental evolution of the SH spectrum vs $\Delta k$; the results in Fig. 1 are selected from these data. All spectra are normalized to the same peak value. The baseline of each spectrum indicates the invoked phase mismatch. (b) Quantitative comparison between the theories and the experimentally measured SH peak wavelengths.

Figure 4

Numerical slowly evolving wave approximation [18, 71] simulations using experimental parameters. (a, b) Normalized spectra; in (a) $\left|\mathcal{F}\right[E_1^2\left(\tau \right)\left]\right|$ is also shown (thin lines). (c, d) Normalized temporal intensities (notice the different time axis scalings). An offset is imposed between each curve for clarity of presentation. (e) Propagation dynamics of the $+3^{\circ }$ case. Kerr nonlinearities are from [44].