Efficient frequency generation in phoXonic cavities based on hollow whispering gallery mode resonators

We report on nonlinear optical effects on phoxonic cavities based on hollow whispering gallery mode resonators pumped with a continuous wave laser. We observed stimulated scattering effects such as Brillouin and Raman, Kerr effects such as degenerated and non-degenerated four wave mixing, and dispersive wave generation. These effects happened concomitantly. Hollow resonators give rise to a very rich nonlinear scenario due to the coexistence of several family modes.


Experimental set-up and physical principles
The experimental setup is shown in Fig. 1. The laser light from a tunable diode laser (TDL) is amplified with an erbium-doped fiber amplifier (EDFA, IPG) and after passing an attenuator (ATT) and polarization controller (POL) is coupled to the equator of a MBR by means of a tapered fiber, produced in-house too. The laser is tuned into a resonance from high to low frequencies, which results in thermal self-locking 35 of the WGMR mode to the pump laser. A splitter at the end of the taper sends a part of the signal into an optical spectrum analyzer (OSA, ANDO AQ6317B) and part to a photo-detector and oscilloscope (TEKTRONIX). SBS, SRS and FWM were detected on an OSA in forward and backward direction by using a circulator, directly transmitted by the taper fiber. A fiber optical circulator (CIRC) was inserted before the taper in order to extract the feedback signal and observe forward and also in backward direction of lasing. The maximum launched pump power is about 200 mW. We performed the experiments at room temperature and atmospheric pressure.
SBS corresponds to a lattice oscillation that can be described as ω p = ω s + Ω B , a pump photon is scattered into a Stokes photon and an acoustical phonon at the frequency Ω B . Light scattering can occur in both directions, forward with frequencies ranging the MHz-GHz range; and backward with frequencies in the GHz range 9 . The SBS frequency scales with the optical one and it is about 11 GHz in silica glass, with a bandwidth ranging 20 to 60 MHz at telecom frequencies. In our experiments, the free spectral range (FSR) of our MBR is 140 GHz (diameter about 475 μ m) and 100 GHz (diameter about 675 μ m). Therefore, we can obtain SBS only by using mode families with the same azimuthal but different vertical quantum number (their degeneracy is effectively removed because of the strong eccentricity of the bubble), whose FSR is much less than the FSR of the azimuthal number 20 . At elevated pump power, cascaded SBS can happen, generating high-order Stokes lines at ω ns = ω p − nΩ B . The phoxonic MBR supports at least three resonances, two optical -pump and Stokes-and one acoustical, which are highly overlapping.
SRS corresponds to a molecular vibration transition that can be described as ω p = ω s + Ω R , a pump photon is scattered to a Stokes photon and an optical phonon at the frequency Ω R . The SRS frequency scales with the optical one and it is about 10 THz in silica glass, with a bandwidth ranging several THz at telecom frequencies. The Kerr effect is due to an electron cloud oscillation, quasi instantaneous, that can be describe by FWM (hyper-parametrical oscillations). FWM can be degenerated 2ω p = ω s + ω i or not ω p1 + ω p2 = ω s + ω i . FWM is coherent and obeys precise phase matching conditions. In WGMR, the phase matching condition is related to the momentum conservation in the azimuthal mode indices: the frequency spacing matches single or multiple FSR. The presence of anti-Stokes frequencies is a clear indication of a hyper-parametrical oscillation 13,20,36 since SBS and SRS cannot generate anti-Stokes fields. Figure 2 shows an schematic illustration of the energy levels for Brillouin, Raman and Kerr scattering together with an schematic drawing of the multiresonant optical frequency conversion. Figure 3 shows the optical spectrum corresponding to the 475 μ m diameter MBR for a launched power of 80 mW, in backward direction for a pump wavelength of 1552,47 nm and a SBS Stokes line shifted of 0,09 nm (11,2 GHz).

Results
Increasing the launched pump power up to 200 mW, we have obtained cascaded SBS up to the 4 th order in a MBR of diameter about 675 μ m and wall thickness of about 2 μ m. As expected, the Stokes lines are shifted by 11, 22, 33 and 44 GHz for a pump wavelength centered at 1544,614 nm (Fig. 4a). By changing the pump wavelength at 1569,882 nm, we could observe cascaded SBS up to the 3 rd order only, but with a high efficiency in the 2 nd order SBS lasing. In this case, the efficiency is up to 17%.
SRS was also observed contemporaneously to SBS at the same launched pump power of about 200 mW. Figure 5 shows SBS and SRS happening simultaneously at 1550 nm, in forward direction. The Raman Stokes line is separated by 13 THz (110 nm) from the pump, as expected for a wavelength centered at 1550 nm. Degenerated FWM from the stimulated Brillouin laser line is also possible in both forward and backward direction. In forward direction, we observed cascaded FWM from the second order Brillouin laser line (1545,22 nm) for a pump wavelength centered at 1545,04 nm. The FWM lines are separated by a FSR (Fig. 6a). In backward direction, we observed two sets of Stokes and anti-Stokes lines very close to each other. The first set is separated by one FSR from the 2 nd order stimulated Brillouin laser line, the second set is separated by a non-integer multiple of the FSR (Fig. 6b).
We also observed cascaded SBS together with some anti-Stokes components that could correspond to dispersive wave multiplets more than non-degenerated FWM either between the pump and the Raman Stokes or between the Stokes components (Fig. 7).

Discussion
In guiding systems showing translationally invariance, SBS can happen in forward direction (FSBS), were both pump and scattered waves are co-directional and coupled through transverse standing wave phonons; and in backward direction (BSBS), were the waves are counter-propagating and coupled through traveling wave phonons. FSBS processes are usually very weak unless there is a lateral phonon confinement 3,37,38 and even though, they are weaker than BSBS due to superior confinement of longitudinal phonons 3 . However, giant enhancement  of both processes has been theoretically proven when taking into account the contributions of radiation pressure or boundary-induced nonlinearities 3,39 .
WGMR, like nanoscale waveguides 3,38 , microstructured optical fibers 37 and suspended silicon waveguides 40 can efficiently generate FSBS, revealing cascaded Brillouin interactions with characteristics similar to Raman lasing 7,8 . In our case, we have observed cascaded FSBS up to the forth order showing even and odd orders, due to the presence of Rayleigh scattering in the resonator, (see Fig. 4) with high efficiency as expected for thin shells 39 . It is worth to note, that even though we have observed cascaded SBS, we have not observed the correspondent stimulated anti-Stokes Brillouin scattering differently to SRS where stimulated anti-Stokes Raman scattering (SARS) can also occur 36,41 .   We have also observed SRS and FWM occurring simultaneously with SBS. The observed FWM is both degenerated and non-degenerated and it arises from the strong 2 nd order Brillouin laser line in forward and backward direction. Signal and idler are separated by an integer multiple of the FSR, indicating intermodal interactions.
The spatial mode interaction within a microresonator is already a demonstrated mechanism for dispersive wave generation, the optical analog of Cherenkov radiation [42][43][44] . Dispersive wave generation can be, thus, linked to cascaded FWM 42 when high-order dispersion is present in the resonator and due to linearly interacting families of equidistant modes with slightly different FSR 43,44 . In MBR coexist several family modes, giving rise to a very rich nonlinear scenario.

Conclusions
In conclusion, we reported on the simultaneous excitation of Brillouin, Raman and Kerr effects in silica MBR. We showed that phoxonic MBRs can act as enhancement platforms for multiresonant nonlinear phenomena, while keeping the dimensions in the micrometer scale. The coexistence of several family modes in the same cavity favors the fulfillment of the different frequency and phase matching conditions, co-exciting different nonlinear effects. The denser spectra and the capacity of tailoring the dispersion make MBR a very attractive class of WGMR that can be used not only in fundamental studies such as nonlinear optics but also in practical ones such as sensing.

Methods
The MBRs were fabricated from slightly pressurized silica capillaries using a modified fusion splicer, where the electrodes could rotate by 360°, in order to obtain uniform heating of the capillary. The detailed fabrication procedure can be found in ref. 27. Figure 1 shows an optical image of a MBR fabricated with this method, which was created from a capillary with an ID of 200 μ m OD of about 280 μ m (Postnova Z-FSS-200280 capillary). The diameters of the microbubbles used in these experiments range from a minimum diameter of about 475 μ m up to a maximum diameter of about 640 μ m with wall thicknesses are ranging from 3 μ m ± 0.5 μ m and 2 μ m ± 0.5 μ m, respectively. The corresponding quality factors Q are about 3.5 10 7 .