# Spectral Broadening in a Continuously Pumped Singly Resonant Second-Harmonic Cavity
## 3 Experimental Results and Discussion
Exploiting changes in the stabilized cavity length, we are able to tune the resonance frequency more than a hundred cold cavity linewidths while keeping the effective frequency detuning negative. At the onset of the internally pumped parametric oscillation, we observe a single pair of widely detuned sidebands, with a frequency shift of approximately 5 THz (tens of thousands of cavity FSRs). As \(|\Delta|\) decreases from the blue side, small frequency comb clusters are generated around the pump and additional sidebands appear at new frequencies. In order to understand the underlying physics, we attempted to model the temporal and spectral dynamics of the OFC generation. The experimentally observed behavior is found not to be well described by the mean-field model [13] that predicts the formation of MI sidebands to occur only close to the pump. However, there is a qualitative agreement with a two-envelope Ikeda map model [13; 15] for the fundamental field \(A_{m}\) and the second-harmonic field \(B_{m}\)  
\[\frac{\partial A_{m}}{\partial z} =\left[-\frac{\alpha_{c1}}{2}-i\frac{k_{1}^{\prime\prime}}{2} \frac{\partial^{2}}{\partial\tau^{2}}\right]A_{m}+i\kappa B_{m}A_{m}^{*}e^{-i \Delta kz}, \tag{1}\] \[\frac{\partial B_{m}}{\partial z} =\left[-\frac{\alpha_{c2}}{2}-\Delta k^{\prime}\frac{\partial}{ \partial\tau}-i\frac{k_{2}^{\prime\prime}}{2}\frac{\partial^{2}}{\partial \tau^{2}}\right]B_{m}+i\kappa A_{m}^{2}e^{i\Delta kz}, \tag{2}\]  
with boundary conditions  
\[A_{m+1}(0)=\sqrt{\theta_{1}}A_{in}+\sqrt{1-\theta_{1}}e^{-i\delta_{1}}A_{m}( L),\qquad B_{m}(0)=0. \tag{3}\]  
Here, m is the roundtrip index, the group-velocity dispersion parameters are the same as in [13], viz., \(k_{1}^{\prime\prime}=0.234\) ps\({}^{2}\)/m and \(k_{2}^{\prime\prime}=0.714\) ps\({}^{2}\)/m, respectively, \(\Delta k^{\prime}=792\) ps/m is the temporal walk-off, \(\Delta k=0\) is the phase mismatch, \(\kappa=11.41\) W\({}^{-1/2}\)m\({}^{-1}\) is the nonlinear coefficient, \(L=15\) mm is the cavity length, \(\theta_{1}=\alpha_{c1}L=\pi/400\) is the coupling coefficient of the fundamental field, \(\alpha_{c2}=\alpha_{c1}\) are the absorption losses, while \(P_{in}=|A_{in}|^{2}\) is the pump power, and \(\delta_{1}=2\pi k_{1}^{\prime}L\Delta\) is the detuning where \(k_{1}^{\prime}\) is the group velocity (i.e., \(\delta_{1}=2\pi\) corresponds to one FSR). The Ikeda map predicts the formation of higher order instability sidebands due to the cavity boundary conditions [32; 36]. Both higher-order modulational (Turing) and parametric (Faraday) instabilities have previously been investigated in the context of cubic Kerr cavity resonators [37; 38]. The presence of large phase shifts can in particular lead to the formation of so-called Arnold tongues for the periodic parameter domains of the instability [39]. However, the observation of such higher-order instabilities has thus far not been reported in quadratically nonlinear resonators. Specifically, the simulations show simultaneously quadratic combs generation close to the pump and SH frequencies together with additional sidebands, as shown in Figure 3, although the model fails to predict the intermediate pair of sidebands around 1045 and 1085 nm (left Figure 3) that are experimentally observed (compare the pink panel of Figure 2). In order to explain the difference in the observed spectra, we suppose that some residual phase mismatch or additional competing processes could act to perturb the dynamics. A hypothesis could be the presence of a significant cubic nonlinearity. By including a strong simultaneous Kerr nonlinearity in our simulations, we can observe a suppression of the narrowband instability that favors the growth of higher-order sidebands. However, we estimate that the intensity of the beam is too low for the cubic nonlinearity to have an appreciable effect and that it would require a more focused beam with higher intensity to make the magnitudes of the cubic and quadratic nonlinearities comparable.