*2.3. Stimulated Raman Scattering*

The inelastic scattering of a photon with an optical phonon, which originates from a finite response time of the third-order nonlinear polarization of the material, is called the Raman scattering effect. Spontaneous Raman scattering occurs when a monochromatic light beam propagates in a material like silica. Some of the photons are transferred to new frequencies. The scattered photons may lose (Stokes shift) or gain (anti-Stokes shift) energy.

The left diagram in Figure 2b represents the absorption of a pump photon with energy with the consequent excitation of a molecule from the ground state (*G*) into a higher virtual energy state (*V*). The energy difference between the ground and the excited level is equal to pump photon's energy. In a second step, the molecule falls to an intermediate level (*I*), which is generated by its own periodical oscillations or rotations. This decay is accompanied by a Stokes photon emission. The destruction of the pump and the generation of the Stokes photon happen simultaneously because *V* is a virtual state. The energy difference between the pump and the Stokes photon is equal to the difference between the energy levels *I* and *G*; the remaining energy is the vibrational energy delivered to the molecule.

In contrast to other kinds of nonlinear phenomena where the molecule returns to its ground level after the interaction, an energy transfer between a photon and a molecule takes place here. Raman scattering is a pure gain process and depends on particular material resonances. In crystalline media, these resonances show a very narrow bandwidth. On the other hand, in amorphous silica the molecular vibration modes are overlapped with each other and create a continuum [53]. Contrary to spontaneous emission, SRS can transform a large part of power into a new frequency-shifted wave with the intensity growing exponentially with the propagation distance in the nonlinear (NL) material. Highly efficient scattering can occur as a result of the stimulated version under excitation by an intense laser beam; 10% or more of the energy of the incident laser beam can be converted.

Generally, in WGM resonators the lasing threshold occurs when the cavity round-trip gain equals round-trip loss. The intra-cavity gain coefficient is related to the bulk Raman gain coefficient *gb* (in silica the maximum is 6.5 × <sup>10</sup><sup>14</sup> m/W at 1550 nm) through the equation:

$$\mathcal{g}\_R = \frac{c^2}{\mathcal{C}(\Gamma) 2n^2 V\_{eff}} \mathcal{g}\_b \tag{11}$$

where *Veff* is the effective modal volume and *C*(Γ) is the modal coupling. The threshold pump power can be derived by the gain coefficient, taking into account the power build-up factor in the resonator:

$$P\_{th} = \frac{\pi^2 n^2 V\_{eff}}{\lambda\_p \lambda\_R \mathbb{C}(\Gamma) \mathbb{g}\_R Q^2} \tag{12}$$

Thus, the threshold follows an inverse dependence on the squared quality factor *Q* of the cavity. This explains how an increase in *Q* will cause a two-fold benefit in terms of reducing cavity round-trip losses as well as of increasing the Raman gain, due to the Raman gain dependence on the pump intensity.

Contrary to the parametric effects, Raman scattering is intrinsically phase-matched over the energy levels of the molecule. In other words, it is a pure gain process.

The SRS can be summarized in two parts: (1) the molecular vibrations modulate the refractive index of the medium at the resonant frequency *ω<sup>v</sup>* and the frequency sidebands are induced in the laser field. (2) the Stokes field at frequency *ω<sup>S</sup>* = *ω<sup>L</sup>* − *ω<sup>v</sup>* beats with the laser field to produce the modulation of the total intensity which excites the molecular oscillation at frequency *ω<sup>L</sup>* − *ω<sup>S</sup>* = *ωv*. In this way the two processes reinforce one another. The Raman emission can be thought as a down-conversion of a pump photon and phonon associated with the vibrational mode of the molecule. The anti-Stokes wave is generated together with the Stokes one, through FWM in which two pump photons annihilate themselves to produce Stokes and anti-Stokes photon can occur if 2*ω<sup>L</sup>* = *ω<sup>a</sup>* + *ωs*, providing the total momentum conservation. This leads to the phase-matching condition Δ*k* = 2*k*(*ωL*) − *k*(*ωa*) − *k*(*ωs*) = 0 where *k*(*ω*) is the propagation constant. When the phase mismatch Δ*k* is large, Stokes emission experiences gain whereas the anti-Stokes experiences loss. For a perfect phase-matching, the anti-Stokes wave is strongly coupled to the Stokes one, preventing the growth of the latter.

In 2003 a microcavity-based cascaded Raman laser was demonstrated in WGM silica microspheres with sub-milliwatt pump power [54]. In cascaded Raman oscillation, the Raman signals serve to secondary pump field and generate higher-order Raman waves. As the pump power is increased, the first Stokes line extracts power from the pump until it becomes strong enough to seed the generation of a next Stokes line. This process can continue to generate more Raman peaks. The cascade process can be modeled as coupled harmonic oscillators with the pump and the Raman fields by including higher-order coupling terms(see for instance, [55]) The SRS and cascaded SRS in the infrared region occurs as standing waves because the Raman gain that amplifies the waves is the same for waves traveling in either the forward or the backward direction. In the presence of these phenomena, we have also observed TSFG in the visible, obtaining multicolor emission (red, orange, yellow, and green) by tuning the pump wavelength. Figure 6 shows the spectra measured for each different color and the corresponding microscope picture of the microsphere.

**Figure 6.** Various spectra obtained with different pump wavelengths showing TSFG standing waves generated among cascaded Raman lines with emission at: (**a**) 537.2 nm; (**b**) 578.8 nm, (**c**) 592 nm, and (**d**) 625 nm. Reproduced with modifications from Ref. [35].

In these cases, the pump was high enough to generate several orders of Stokes Raman lines and Raman combs. TSFG and THG can also occur simultaneously. Figure 7 shows the picture of a multicolor emission, one at 519.2 nm and one at 625 nm, corresponding to the THG signal of the pump laser and the TSFG, respectively.

As mentioned before, the red standing wave is the result of TSFG whereas the green traveling wave is TH signal. To fulfill the phase-matching condition and dispersion compensation, we must excite higher-order polar modes (*l*− | *m* |> 1). High polar order modes can be excited by placing the coupling taper far from the equatorial plane where the intensity peaks of these modes are located [1]. These modes also improve mode matching which is another requirement for efficient nonlinear frequency generation. The overlap of the WGM eigenfunctions corresponds to the power of TSFG and THG. In the latter case, the total power is proportional to the overlap of the TH field with the cubic power of the pump field.

**Figure 7.** Optical picture of the microsphere showing the TSFG standing wave in the red and the traveling wave in the green of the THG.

WGMR Raman lasers can be used in sensing applications. WGMR-based lasers have a very narrow linewidth, they are dopant-free and they can attain high detection resolution down to single nanoparticles [56]. However, biochemical sensors need to work at telecom wavelengths where water shows a very high absorption [57,58]. A way to overcome such limitation is to use laser pumps in the visible or near infrared (NIR) [59,60]. Figure 8 shows the SRS line at 807 nm when a microsphere of about 50 μm diameter is pumped at 778 nm.

**Figure 8.** Experimental spectra of cascaded Raman lasing in a microsphere of 50 μm diameter. The laser pump is centered at about 778 nm, the Raman line is centered at about 807 nm.

Stimulated anti-Stokes Raman scattering (SARS) requires phase-matching, to be efficiently generated, and it is thresholdless. The nonlinear polarization of the anti-Stokes wave, *PNL as* , is defined by the relation:

$$P\_{as}^{NL} = \chi^{(3)} E\_P^2 E\_S^\* e^{i(2k\_P - k\_S)z} \tag{13}$$

where *EP* and *ES* are the amplitude of the pump and Stokes waves, respectively; *χ*<sup>3</sup> is the third-order nonlinear susceptibility, and *ki* are the propagation vectors for the pump and Stokes waves [20,21]. Farnesi et al. [25] measured SARS in solid microspherical WGMR in the normal dispersion regime [23,39,46], contrary to well-known theoretical models [44,45], which predicted modulation instabilities and FWM in WGMR only with anomalous dispersion. The cavity boundary conditions introduce an extra degree of freedom, namely the frequency detuning between the pump and the eigenmode of the nonlinear WGMR [23,47]. As stated in [25], there was a negative shift of about 30 MHz of the resonant frequency due to the Kerr effect plus a larger thermo-optic frequency shift. However, the FSR of the resonator showed just a slight change [23]. The linear dispersion also

changes slightly since it is calculated as the variation of the FSR [40]. Similar results have been achieved by Soltani et al. [61]. The group have been able to enhance by a factor of 4 the results of Farnesi et al. [25] using hybrid microspheres. In this particular case, silica microspheres have been functionalized with a layer of gold nanorods in polyethilenglycol (PEG). Farnesi et al. [25] have measured some unusual features, namely strong anti-Stokes components and extraordinarily symmetric spectra. Usually in SRS, the Stokes waves are exponentially enhanced, whereas the anti-Stokes waves are exponentially absorbed [62]. Anti-Stokes Raman components are coupled to the Stokes Raman ones [62,63], independently of the magnitude of dispersion. As a result of the coupling and of an effectively phase-matched hyper-parametric process, the anti-Stokes wave grows along the microsphere directly proportioned to the Stokes wave. When there is perfect phase-matching condition, each eigensolution is an equal combination of Stokes and anti-Stokes components with a power ratio of one. When the phase mismatching conditions deviates from zero gradually, we are in an intermediate case where the anti-Stokes/Stokes power ratio is given by the following equation:

$$\frac{P\_d}{P\_S} = \left| \frac{\gamma qP}{2\gamma qP + \beta\_2 \Omega^2} \right|^2 \tag{14}$$

where *P* is the cavity build-up pump power, Ω/2*π* is the frequency shift between the pump and the first Raman order, *<sup>β</sup>*<sup>2</sup> is the linear dispersion, *<sup>γ</sup>* is the nonlinearity coefficient and *<sup>q</sup>* = (<sup>1</sup> − *<sup>α</sup>*) + *αχ*<sup>3</sup> = 0.82+ *i*0.25 is a complex number that depends on the Raman susceptibility of silica and on the fractional contribution of the electronic susceptibility to the total nonlinear index [26,64].

When there is no phase match, the Stokes and anti-Stokes components are effectively decoupled. This expression is valid for both regimes, normal and anomalous, but the linear dispersion Δ*k* = *β*2Ω<sup>2</sup> must be large. The values obtained from Equation (14) are in close agreement with the experimental ones, given the uncertainties in *β*<sup>2</sup> and *γ*, as it can be seen in Figure 9.

**Figure 9.** *Pa*/*PS* ratio: experimental (solid black squares) and calculated values (solid red circles) The lines are a guide to the eye.

#### *2.4. Stimulated Brillouin Scattering*

SBS is also an inelastic scattering process but it results from the coherent interaction of light photons and acoustic phonons instead of optical phonons. The WGMR acts as a dual photonic-phononic cavity due to the overlap of both waves inside the resonator. SBS is automatically phase-matched because it is a pure gain process, like SRS. The SBS gain coefficient is one of the largest but with a small gain bandwidth [62]. The narrow bandwidth places very stringent conditions on the geometry of resonators since it would require a Brillouin frequency shift that equals the free spectral

range (FSR) of the WGMR. [36,65,66]. We need to excite high order modes [29] with vertical FSR smaller than the fundamental FSR [50] in order to bypass such a stringent condition. SBS can further reduce its threshold power when it is resonantly enhanced, and it can be as low as some micro-watts [65]. The threshold power is directly proportional to the mode volume [67]

$$P\_{th} = \frac{\pi^2 n^2 V\_{eff}}{\lambda\_p \lambda\_s B g\_b Q\_p Q\_s} \frac{1}{1 + \frac{Q\_m \lambda\_m}{2 \pi r}}\tag{15}$$

where the subscripts *p*,*s*, *m* refer to pump, Stokes and mechanical modes, and *B* is the mode overlap. SBS has been demonstrated in silica [67,68] and tellurite [69] microspheres, microbottles [70] and microbubbles [29,71,72]. Hollow and solid spherical WGMR have been used to generate both backward and forward SBS. As with SRS, cascaded SBS can also be generated in WGMR. Even and odd orders are observed in both directions, but showing different lasing efficiency (even (odd) orders are more efficient in forward (backward) direction) [72]. Scattering can occur in forward direction with frequencies in the MHz–GHz range whereas backward are in the GHz range. The SBS frequency in silica glass is about 11 GHz and it scales with the optical one, with a bandwidth in the range of 20–60 MHz at telecom frequencies. In our experiments, the free spectral range (FSR) of our MBR is 141 GHz (diameter about 475 μm) and 105 GHz (diameter about 675 μm). Therefore, we can obtain SBS only by using high order modes, in that case the vertical FSR is much less than the FSR [50]. Frequency combs and SBS can coexist in both forward and backward directions, as Figure 10. SBS efficiency in microbubbles can be so high to allow degenerated FWM from the Brillouin laser line. Figure 10 shows in forward direction, cascaded FWM from the second order Brillouin laser line (1.54522 nm) for a pump wavelength centered at 1.54504 nm

**Figure 10.** Native Kerr comb and second order SBS in forward direction: pump power 72 mW at 1551,344 nm. (Inset: zoom of the spectrum showing the 2nd order SBS laser line and the first pair of FWM lines) Reproduced with modifications from Ref. [73].

## **3. Kerr Switching in Hybrid Resonators**

Inorganic materials still show weak nonlinearity, slow dynamics and the difficulty of discrimination between thermal and Kerr nonlinearity at room temperature limits their performance. Significant advantages can be obtained if organic-inorganic hybrid systems can be used [74,75]. *π*-conjugated polymers are extremely suitable nonlinear optical materials which show structural

flexibility, relative ease of preparation, high *χ*<sup>3</sup> values and high photostability. Hybrid polyfluorene derivatives-silica WGMR have been demonstrated as very good candidates for all-optical switching [12,13] where two beams are present, namely a pump beam that switches a probe [76,77]. The electronic Kerr effect is almost instantaneous (picosecond timescales) and due to the enhancement of the WGMR combined with a strong third-order nonlinearity, the intensities used are well below the damage threshold of the conjugated polymer. The material refractive index *n* and the absorption coefficient *α* depend on the light intensity *I* in the material according to the equations *<sup>n</sup>* = *<sup>n</sup>*<sup>0</sup> + *<sup>n</sup>*<sup>2</sup> *<sup>I</sup>* + *<sup>n</sup>*<sup>4</sup> *<sup>I</sup>*<sup>2</sup> + ... (with the nonlinear refractive index *<sup>n</sup>*<sup>2</sup> ≈ (*χ*3) , *<sup>n</sup>*<sup>4</sup> ≈ (*χ*5), where *χ*<sup>3</sup> and *χ*<sup>5</sup> are the third- and fifth- order nonlinear optical susceptibilities) and *α* = *α*<sup>0</sup> + *βI* (with the nonlinear absorption coefficient *<sup>β</sup>* ≈ (*χ*3)). All-optical switching for a probe signal *Iprobe*, which is resonant with the microsphere, can be realized using a resonant pump beam *Ipump*, which affects the coated cavity resonance position by changing the refractive index of the coating in the corresponding wavelength range [78].

If the *χ*<sup>3</sup> and *χ*<sup>5</sup> are caused by fast electronic Kerr nonlinearity, then, as mentioned before, the nonlinear switching is on a picosecond time scale, which is the most desired situation for the optical switching. However, thermal nonlinearities can restrict the use of the hybrid devices because the spectral response is sensitive to the input power of the probe signal as well [76]. In that case, the light-induced changes in the refractive index can be described phenomenologically by *n*<sup>2</sup> = (*δn*/*δT*)*TL*, where *TL* is the laser-induced change of the temperature of the nonlinear medium, the corresponding switching time being about 10<sup>−</sup>3–10−<sup>5</sup> s. In other words, the thermal switching of a nonlinear medium for the case of a standard Ti-sapphire laser should be the same for the pulsed or CW mode operation regime. Moreover, an intrinsically weak but highly localized probe beam can also participate in this type of the light-induced WGM switching.

Murzina et al. [13] reported on the all-optical switching of WGM in silica microspheres with two types of coatings, an active one based on a Kerr polymeric material (polyfluorene derivative, PF(o)n) [12] and an inert polymer based on an anionic copolymer made of methacrylic acid and methyl methacrylate (Eudragit-<sup>R</sup> L100) [79]. The authors modeled the overlap of the coupled optical field with the polymer layer and verified the role of the probe field experimentally for both polymer coatings.

Figure 11 shows an sketch of the set-up, an optical picture of a microspherical WGMR, with a diameter of about 250 μm and the taper; and the control test with a bare WGMR. A Tunics Plus was used as probe beam, which is a semiconductor external-cavity laser tunable in the spectral range of 1.55–1.6 μm and with 300 kHz linewidth. The pump probe was a Mira 900-f Ti-Sapphire (Coherent). The laser probe was coupled into the WGMR through a homemade tapered fiber whereas the laser pump was coupled into a multimode fiber that illuminated a hemisphere of the WGMR. To make a polymer coverage, the dip coating technique was used. The authors obtained layers of about 100 nm thickness for PF(o) coatings and of about 50 nm thickness for Eudragit coatings. The *Q* values were higher than 10<sup>8</sup> for bare microspheres and higher than 10<sup>6</sup> after polymer coating. To attain adequate solubility in common organic solvents and mesogenic behavior, the polyfluorene derivative, PF(o)n, was functionalized at the C9 position of the fluorine ring with two pendant octyl chains [80]. It shows a maximum absorption at *<sup>λ</sup>abs* = 379 nm, and the measured *<sup>n</sup>*<sup>6</sup> is about 2 × <sup>10</sup>−<sup>10</sup> cm2/W and *<sup>β</sup>* coefficient 7 × <sup>10</sup>−7cm/W [8,14]. As the first step, pump-induced effects on the barre WGMR in were studied. No shift was seen as the averaged pump power was increased up to 30 mW. Thus, we may assume that the Kerr nonlinearity and the thermal nonlinear effects of pure fused silica were negligible for these pump-and-probe levels.

**Figure 11.** (**a**) Experimental pump-and-probe set-up. Left hand side inset: optical image of the WGMR. (**b**) Picture of the microsphere and the coupling taper; (**c**) Frequency center of the WGMR resonance versus pump power. The red line is a guide to the eye. Inset: Zoom of a typical resonance for a bare microsphere, the red line is a Lorentzian fit with a FWHM of about 2.4 MHz. Reproduced with modifications from [12].

Figure 12a shows the WGM spectrum measured for the probe wavelength of 1600 nm and for two different pump powers for the mode-locked regime of the Ti:Sapphire laser. The Ti:Sapphire was tuned at 775 nm to generate the two-photon absorption (TPA) in the PF(o)n coating of the hybrid WGMR. Thermal nonlinearities can be ruled out since no broadening of the resonance, hysteresis or asymmetries could be observed in the transmission spectra.

The results of the pump-and-probe experiments are shown in Figure 12b, where a frequency shift of 2 GHz is obtained in pulsed regime for an average pump power of 35 mW at 775 nm for a probe of 1600 nm. To discriminate the thermal shift from the Kerr shift, we have performed measurements in CW and pulsed regime for the same average pump powers, similarly to a previous work [12]. For the same wavelength and average pump power, the authors obtained a much lower spectral shift of 250 MHz in the CW regime (Figure 12b). In here, we have also tested the influence of the wavelength of the pump beam. Figure 12b also shows a frequency shift of 200 MHz obtained in the mode-locked regime for an average pump power up to 21 mW at 825 nm. The detuning is of the same order of magnitude as the CW regime. We have chosen 825 nm as a pump beam because two-photon absorption (TPA) is not feasible; and because it is also far from the second harmonic of the probe beam. In that case the pump beam acts as a spectrally broad thermal source only.

Figure 12c shows the frequency shift versus pump power for *λprobe* = 1558 nm at two different laser regimes for the PF(o) coated WGMR. At *λpump* = 775 nm a clear quadratic dependency can be observed, whereas at*λpump* = 825 nm the dependency is linear and the detuning is of the same order of magnitude as the CW regime, indicating again that in absence of TPA, the pump acts as a thermal source. It can also be observed that in the case of *λprobe* = 1558 nm the magnitude of the shift is greater than in the case of *λprobe* = 1600 nm, for both regimes, pulsed and CW. Figure 12d shows an almost null red-shift up to 20 mW of pump power for *λprobe* = 1558 nm for a WGMR coated with Eudragit-R L100, an inert polymer.

**Figure 12.** (**a**) Typical WGM spectra measured for a polymer-coated microsphere for two different pump powers: (black) pump laser off and (blue) 32 mW. *λpump* = 775 nm, *λprobe* = 1600 nm. (**b**) Pump power dependence of the detuning of WGM in PF(o)n coated microspheres for mode-locked regime of the Ti-sapphire pump laser: 825 nm (filled squares) and 775 nm (empty circles); and CW at 775 nm (empty downside triangles). (**c**) The probe wavelength is *λprobe* = 1558 nm, switching for CW (filled circles), mode-locked at 775 nm (empty circles) and mode-locked at 825 nm. (**d**) Eudragit coated microspheres for mode-locked regime of the Ti-sapphire pump laser for two different regimes: CW (empty circles) and pulsed (filled squares). The probe wavelength is *λprobe* = 1558 nm. Reproduced with modifications from [13].
