Frequency Comb Generation via Cascaded Second-Order Nonlinearities in Microresonators

Optical frequency combs are revolutionizing modern time and frequency metrology. In the past years, their range of applications has increased substantially, driven by their miniaturization through microresonator-based solutions. The combs in such devices are typically generated using the third-order ${\chi }^{(3)}$ nonlinearity of the resonator material. An alternative approach is making use of second-order ${\chi }^{(2)}$ nonlinearities. While the idea of generating combs this way has been around for almost two decades, so far only few demonstrations are known, based either on bulky bow-tie cavities or on relatively low-$Q$ waveguide resonators. Here, we present the first such comb that is based on a millimeter-sized microresonator made of lithium niobate, that allows for cascaded second-order nonlinearities. This proof-of-concept device comes already with pump powers as low as 2 mW, generating repetition-rate-locked combs around 1064 and 532 nm. From the nonlinear dynamics point of view, the observed combs correspond to Turing roll patterns.

Figure 1

(a) Continuous-wave near-infrared laser light with frequency ${\nu }_0$ is prism coupled into a microresonator (radius $R$). When the phase-matching-criterion for second-harmonic-generation is fulfilled, second-harmonic light at $2{\nu }_0$ is generated. A sequence of cascaded ${\chi }^{(2)}$ processes then leads to the buildup of interlocked near-infrared (red spectrum) and visible (green spectrum) combs. In the comb generation regime, the ordinarily ($o$) polarized pump and extraordinarily ($e$) polarized second-harmonic waves travel with the same group velocity $v_{\mathrm{gr}}$, associated with the repetition rate equaling the free spectral range ${\mathrm{FSR}}_{\mathrm{nl}}$. (b) FSRs of the microresonator vs wavelength: ${\mathrm{FSR}}_p=20.8\mathrm{GHz}$ and ${\mathrm{FSR}}_{\mathrm{sh}}=19.5\mathrm{GHz}$ are the FSRs of the fundamental modes for the pump and second harmonic. (c) Sketch of the experimental setup. OSAs: Optical spectrum analyzers. A detailed description of the experimental procedures is found in [32].

Figure 2

(a) Slow scan of the pump laser frequency $\mathrm{\Omega }$ across the pump mode used for comb generation. Reducing $\mathrm{\Omega }=2\pi {\nu }_0$ reveals comb generation areas marked (a1), (a2), and (a3). At (a1), we observe four sidebands separated by $\simeq 20\times {\mathrm{FSR}}_p$ around the pump (displayed in red) and second-harmonic (displayed in green) frequencies. Bringing $\mathrm{\Omega }$ into the (a2) range reveals dense quasicontinua of modes simultaneously generated around the pump and second-harmonic frequencies. With further reduction of the frequency the system enters a quasistationary state, which is replaced again by the dense quasicontinua in the (a3) range. Subsequently the resonance is lost. (b),(c),(d) A close-up revealing the individual peaks of the sparse spectra being separated by $\simeq 20\times {\mathrm{FSR}}_p$ at both the pump (b) and the second harmonic (c). Quasicontinuous spectra at the pump frequency reveal the separation of neighboring lines by $\simeq {\mathrm{FSR}}_p$ (d), while at the second harmonic the OSA does not allow to resolve features on the FSR scale.

Figure 3

Blue line shows the modulation-instability (MI) growth rate of the upper branch (UB) of the intracavity cw state against the cold-cavity detuning $\delta ={\omega }_p-\mathrm{\Omega }.\mu =\pm 20,\pm 21,\pm 22$ are the mode number offsets with the highest gain within a given interval of $\delta \mathrm{s}$. $\mu \mathrm{s}$ are counted left and right from the mode number $m_p=13112$ corresponding to the cavity resonance ${\omega }_{\mathrm{p}}$ nearest the pump frequency $\mathrm{\Omega }$. $\mu =\pm 20$ matches the experimentally observed spectral lines separated by $\simeq 20$ FSR. Bold black lines and the right vertical axis show the peak power of the intracavity pump cw state vs $\delta$. The bistability interval starts from $\delta /(2\pi )\simeq 3\mathrm{MHz}$ and extends towards larger $\delta \mathrm{s}$. Red line shows MI-growth rate of the lower branch (LB) of the bistability loop. The dashed horizontal line indicates zero of the MI gain. One can see that LB is stable through most of its existence range. Full details of the parameter values used are provided in [32].

Figure 4

Top (bottom) row is for the pump (red) [second-harmonic (green)] field. (a),(d) Numerically calculated spectra for a varying detuning $\delta$ showing three intervals of comb generation. (b),(e) show the near-infrared and green spectra for $\delta /2\pi =0.0699\mathrm{MHz}$ from (a) and (d), respectively. (c),(f) show the spectra for $\delta /2\pi =2.6189\mathrm{MHz}$. Enlarged areas show the individual resonances in the quasicontinua separated by ${\mathrm{FSR}}_{\mathrm{nl}}$ for both the near-infrared and green spectra. To achieve the spectral states shown here the number of resonator round-trips used was 208 000.