**2. Kerr Effects**

Silica glass is a centrosymmetric material with an odd-order nonlinear polarization which is directly proportional to the nonlinear third-order susceptibility *χ*(3) of the hosting material. The *χ*(3) is a fourth order tensor with 81 elements. The third-order nonlinear polarization is responsible for several nonlinear phenomena, such as THG, FWM, TSFG, the optical Kerr effect and CARS.

#### *2.1. Third-Order Sum-Frequency Generation*

The nonlinear polarization for third-order sum-frequency generation (TSFG) [30] can be written as:

$$\mathbf{P}^{NLS}(\mathbf{r},\omega) = \chi^{(3)}\mathbf{E}\_{\mathfrak{a}}(\mathbf{r},\omega\_{\mathfrak{a}})\mathbf{E}\_{\mathfrak{b}}(\mathbf{r},\omega\_{\mathfrak{b}})\mathbf{E}\_{\mathfrak{c}}(\mathbf{r},\omega\_{\mathfrak{c}}) \tag{1}$$

where *E*(*ωi*) is the electric field amplitude at frequency *ωi*. This equation shows that in the third-order approximation, the radiation at the new frequency *ω* = *ω<sup>a</sup>* + *ω<sup>b</sup>* + *ω<sup>c</sup>* (energy conservation) can be generated by an intense field containing *ωa*, *ωb*, and *ωc*. This general case, or third-order sum-frequency generation (TSFG) has been studied in WGM structures, in particular in liquid droplets, starting from 1989 [31–33]. TSFG is a weak process; even when conditions are optimized, the emission is only 10−<sup>4</sup> times the typical intensity of SRS. A model for TSFG in spherical dielectric microresonators can be based on the work of Chew et al. [34] for emission from a polarization source within a sphere. A particular case of TSFG is the third-harmonic generation (THG) in which the three input frequencies are degenerate and so *ωTGH* = 3*ω*1.

The polarization in Equation (1) can be written as

$$\mathbf{P}^{NLS}(\mathbf{r},\omega) = D \sum\_{jkl} \chi\_{ijkl}^{(3)} E\_{\mathbf{i}}(\mathbf{r},\omega\_{\mathbf{i}}) E\_{\mathbf{b}}(\mathbf{r},\omega\_{\mathbf{b}}) E\_{\mathbf{c}}(\mathbf{r},\omega\_{\mathbf{c}}) \tag{2}$$

where *j*, *k*, *l* are the three orthogonal coordinate directions and *D* is the number of distinct permutation of *ωa*, *ω<sup>b</sup>* and *ωc*. Being silica an isotropic material, only three independent elements can be considered:

$$
\chi\_{i\bar{i}k\bar{l}} = \chi\_{1122}\delta\_{i\bar{j}}\delta\_{k\bar{j}} + \chi\_{1212}\delta\_{i\bar{k}}\delta\_{j\bar{l}} + \chi\_{1212}\delta\_{i\bar{l}}\delta\_{j\bar{k}} \tag{3}
$$

In spherical coordinates (*r*, *θ*, *ψ*), for the transverse electric field (TE) in a sphere, the radial component is zero and *χ*(3) <sup>1111</sup> <sup>=</sup> *<sup>χ</sup>*(3) <sup>1122</sup> <sup>+</sup> *<sup>χ</sup>*(3) <sup>1212</sup> <sup>+</sup> *<sup>χ</sup>*(3) <sup>1221</sup>. In the simplest case of THG, *χ*(3) <sup>1122</sup> <sup>=</sup> *<sup>χ</sup>*(3) <sup>1212</sup> <sup>=</sup> *<sup>χ</sup>*(3) <sup>1221</sup>, and so:

$$
\chi^{(3)}\_{\rm i\,jkl} = \chi^{(3)}\_{\rm 1122} (\delta\_{\rm i\,j}\delta\_{\rm kj} + \delta\_{\rm ik}\delta\_{\rm jl} + \delta\_{\rm il}\delta\_{\rm jk}) \tag{4}
$$

and the *θ* component of the polarization is:

$$P\_{\theta}^{NLS} = 3\chi\_{1122}^3 \left( E\_{a\theta}^3 + E\_{a\theta} E\_{a\phi}^2 \right) \tag{5}$$

The radiations generated by the polarization **P***NLS*(**r** , *ω*), where **r** is the source position, induce additional fields which have to satisfy the boundary conditions at the surface of the sphere. The solution is a combination of spherical Bessel and Hankel functions, *jn* and *h*<sup>1</sup> *<sup>n</sup>*, respectively.

If the three waves generating the TSFG are standing waves, then the output is a standing wave too. The fields (TE) can be written as:

$$\mathbf{E}\_{\mathbf{s}}(\mathbf{r},\omega\_{\mathbf{s}}) = A g\_{n} j\_{n}(k\_{l}r) \left[ \mathbf{Y}\_{nnm}(\theta,\phi) + \mathbf{Y}\_{nnm}^{\*}(\theta,\phi) \right] / 2 \tag{6}$$

where *<sup>A</sup>* is the amplitude factor proportional to <sup>√</sup>*Is*. The field components are labeled *ns* and *ms*, where *s* is *a*, *b* or *c*.

Obtaining the total power at *ω* requires the integration over all *ω*:

$$P\_{n\_3m\_3}^T = \int\_0^\infty P\_{n\_3m\_3}(\omega)d\omega \tag{7}$$

The TSFG power is proportional to the spatial overlap integrals as well as to a frequency overlap integral; the former ones are calculated by integrating the product of the fields of the TSFG mode and of the three generating modes over the sphere volume. In addition to the energy and momentum conservation, in WGMR the pump and the generated frequency must be resonant. When these three conditions are fulfilled, the high quality factor enhances the interaction. The three conditions are quite difficult to be fulfilled simultaneously. The intermodal dispersion of the different spatial WGMs can be used, however, to obtain highly efficient frequency conversion. The inset of Figure 3 shows a picture of the microsphere with the TH signal with the characteristic upper and lower green lobes along the polar direction. As expected, TH signal is codirectional with the pump. Figure 3 shows the emission spectrum of THG at 519.6 nm when pumping with 1556.9 nm.

**Figure 3.** Emission spectrum indicating third-harmonic generation at 519.6 nm when pumping at 1556.9 nm, whereas the inset picture was taken during the spectral measurements. Reproduced with modifications from Ref. [35].

Asano et al. [36] observed THG in silica microbottle resonators. This particular resonator showed interface and surface effects that allowed the simultaneous generation of THG and second harmonic generation (SHG). In this work, the pump powers are over 200 mW, quite high compared to previous works in other types of WGMR. [25]. A way to lower the launched pump power into the microresonator is coating its surface. Dominguez et al. [37] used a similar strategy for second harmonic generation, coating the silica microspheres with a crystal violet monolayer. Chen et al. [38] have published very recently impressive results in terms of efficiency. The authors have coated silica microspheres with a thin layer of 4-[4-diethylamino(styril)]pyridium (DSP) molecules. DSP has a high third-order nonlinear coefficient and the efficiency of THG is 4 order of magnitude higher than the reported in bare silica microspheres. The authors also observed multiemissions due to TSFG.

## *2.2. Four-Wave Mixing*

Third-order four-wave mixing (FWM) is a hyper-parametric oscillation where two pump photons *ωpump* generate a signal *ω<sup>S</sup>* and an idler photon *ωI*. It requires two conditions to be satisfied: the momentum conservation, which is intrinsically satisfied in WGM resonators [39], and the energy conservation, which is not a priori satisfied since the separation between adjacent modes *νFSR* = |*ν<sup>m</sup>* − *νm*+1| can vary due to the material and cavity dispersion. Indeed, only in recent works this process has been observed by coupling a CW laser into microcavities exploiting the Kerr nonlinearity to enable cascaded four-wave mixing. The resonances of the WGMR will also impose that the new generated frequencies will be discrete, creating a frequency comb.

The comb generation can occur in two different ways: as a Type I (or natively mode spaced comb), with sidebands separated by one free spectral range (FSR), and a Type II (or multimode spaced comb), with sidebands separated from the pump by several FSRs. Cascaded FWM preserves the initial spacing to higher-order emerging sidebands thank to the conservation of the energy in the parametric processes [40].

The Kerr comb formation starts by generation of the first symmetrical lines generated in a degenerate FWM process when the parametric gain overcomes the loss of the cavity. The separation of the new lines from the pump depends on the dispersion and the pump power. The threshold of the parametric frequency conversion [24], at which the gain of the excited sidebands is equal to the cavity decay rate, is:

$$P\_{th} = \frac{\kappa^2 n\_0^2 V\_{eff}}{8\eta\omega\upsilon\_0 c n\_2} \tag{8}$$

and the gain of the sidebands, for *Ppump* > *Pth*, can be written as: [41]

$$\mathcal{G} = \sqrt{\kappa^2 (\frac{P\_{abs}}{P\_{th}})^2 - 4 \left(\omega\_0 - \omega\_p + \mu^2 D\_2 - \kappa \frac{P\_{abs}}{P\_{th}}\right)^2} \tag{9}$$

where *Pabs* is the power absorbed by the cavity. At the threshold (*G* = *κ*), from Equation (9) we obtain:

$$
\sqrt{\frac{f^2}{\left|a\_0\right|^2 - 1}} - d\_2 \mu\_{th}^2 + \left|a\_0\right|^2 - \sqrt{\left|a\_0\right|^4 - 1} = 0 \tag{10}
$$

The theory of generation of frequency combs in silica microspheres has been described in a 2010 article by Chembo et al. [42].

The first experimental demonstration of FWM in silica microspheres was done by Kippenberg et al. [43]. The authors showed the generation of a pair of signal-idler photons created by two pump photons separated by one FSR with an emission ratio close to unity. Frequency combs in microspheres were first demonstrated by Agha et al. [44] where the first theoretical model was established. The same group published a broader comb in microspheres [45], but their theoretical model only predicted FWM and modulation instabilities in resonators with anomalous dispersion. However, nonlinear hyper-parametrical oscillations have been also achieved in the regime of normal dispersion [23,25,39,46]. This occurrence is due to the cavity boundary conditions that introduce an additional degree of freedom: the frequency detuning of the pump from the eigenmode of the nonlinear resonator [23,47]. Zhang et al. [10] proposed hybrid silica microspheres for generation of tunable Kerr and Raman-Kerr combs. The authors have coated the polar cap of the microsphere with iron oxide nanoparticles. Since the WGM are excited at the equator of the microspheres, far away from the iron oxide nanoparticles, the high quality factor *Q* was not spoiled. The authors fed the control light into the microsphere through the fiber stem (see Figure 4). This control light was absorbed by the iron oxide nanoparticles and due to a strong photothermal effect, the comb was tuned. The achieved tuning of the Kerr comb was about 0.8 nm whereas the Raman-Kerr comb was tuned about 2.67 nm. The proposed photothermal tuning is an all-optical method and has less disadvantages than the

mechanical methods [48] which present mechanical interferences and need cryogenic temperatures. However, the tuning range achieved by mechanical methods was about 450 GHz at 10 K for a microsphere of 40 μm diameter.

**Figure 4.** (**a**) Generated Kerr combs in a hybrid microsphere for different control light powers, (**b**) zoom-in of the spectra of panel (**a**,**c**) Sketch of experimental set-up showing an iron oxide nanoparticle coated silica microsphere of 248 μm diameter for photothermal tuning of generated optical frequency combs (OFC). The pink ring is the excited whispering gallery mode and the black polar cap represents the coated area with iron oxide nanoparticles. Reproduced with modifications from Ref. [10].

For microbubbles, the first demonstration of cascaded FWM was done by Li et al. [49]. Broader combs in microbubbles were demonstrated by Farnesi et al. [50]. The authors here realized a "Type I" or natively mode spaced comb with sidebands separated by one FSR (see Figure 5a) and a "Type II" or multimode spaced comb with sidebands separated by several FSR (see Figure 5b). Figure 5c shows the FWM pairs in the vicinity of the pump at 1.55264 nm in the backward direction. In this case, the FWM pairs are separated by one FSR. At 14 THz from the pump, centered at 1508 nm, an anti-Stokes comb was observed. In this case, the intensity of the anti-stokes component is high enough to generate its own parametrical oscillation, with a separation smaller than the FSR of the cavity. MBRs are spheroidal WGMR with quite dense spectral characteristics with two nearly equidistant mode families characterized by the same azimuthal but different vertical quantum number. The presence of these two mode families gives the different frequency spacing and the asymmetric spectrum (see Figure 5d [50]).

**Figure 5.** Experimental spectra of: (**a**) a Type I comb, with a frequency offset of 1 FSR, (**b**) a Type II comb, with a frequency offset of 5 FSR, (**c**) FWM in the vicinity of the pump spaced by azimuthal FSR and (**d**) Modulation intensity around the anti-stokes component at 1508 nm , with a frequency offset of 2 vertical FSR (2X0, 12 nm) measured for a microbubble of 475 μm diameter. Reproduced with modifications from Adapted with permission from Ref. [50] c The Optical Society.

Yang et al. [51,52] have experimentally measured frequency combs in the visible (pump wavelength centered at 765 nm) by engineering the dispersion through wall thickness of the microbubbles and degenerate FWM in hollow microbottles.
