Figure 1B shows the generation principle of an OFC (including degenerate FWM and nondegenerate FWM) and stimulated Raman scattering (SRS). By combining with degenerate and nondegenerate FWM processes, an OFC can be realized as the Kerr frequency comb requires phase matching in a wide wavelength range, which is dominated by the group velocity dispersion (GVD) in the resonator [5], [6], [16]. The total GVD in the silica microsphere resonator should be anomalous in the 1550 nm band, and this can be used to compensate the resonant frequency shift, which is induced by the self-phase and cross-phase modulation excited by the pump light [16]. The total GVD can be expressed by GVD=−(*λ*/*c*)×*d*^{2}*n*_{eff}/*dλ*^{2}, where *c* is the light velocity in vacuum, *λ* is the wavelength in vacuum, and *n*_{eff} is the effective RI [6].

Figure 2A is the scanning electron microscopy (SEM) image of the fabricated functionalized silica microsphere resonator with a diameter of about 248 μm. The rough area on the end surface is coated with iron oxide nanoparticles. In order to verify the existence of the iron oxide nanoparticles on the end surface, we measure both the elemental mapping and the energy dispersive X-ray (EDX) spectrum of the functionalized microsphere resonator. As shown in Figure 2B, the purple spots represent the ferrum (Fe) distribution, which means that the iron oxide nanoparticles only distribute on the end surface far from the WGM locations, and consequently this has no influence on the *Q* factors of the WGMs with low azimuthal mode numbers. The EDX spectrum in Figure 2C also shows that this area consists of oxygen (O), Fe, silicon (Si), and gold (Au). The presence of Au is due to the magnetron sputtering process before SEM analysis. We also measure the basic performance of the functionalized microsphere resonator, and a silica microfiber with a diameter of about 3 μm is selected to couple the pump light into the microresonator [30], [31], [32], [33]. The inset of Figure 2C shows the *Q* factor measurement result, which presents an intrinsic *Q* factor (*Q*_{0}) of 2.1×10^{8}. The ringing phenomenon is derived from the ultrahigh *Q* factor [34], [35].

Figure 2: Characterization of the functionalized microsphere resonators.

(A) Scanning electron microscopic (SEM) image of the functionalized silica microsphere resonator with a diameter of about 248 μm. (B) Ferrum distribution on the end surface of the functionalized microsphere. (C) Measured energy dispersive X-ray spectrum of the coating area. The inset is the measured transmission spectrum. The red line is the theoretical fitting curve. (D) SEM image of the functionalized microsphere resonator with a diameter of about 139 μm.

As shown in Figure 3A, the black line denotes the calculated total GVD curve for the fundamental mode of the silica microsphere resonator with a diameter of 248 μm using the finite element method (COMSOL Multiphysics). We can find that the total GVD in the 1550 nm band is anomalous, and thus we fabricate the functionalized microsphere resonator with a diameter of about 248 μm. We also calculate the total GVD curves of the WGMs with different orders (for details about the GVD calculations, see part S3 in the Supplementary Material). The GVD curves in Figure S3B,C indicate that there are small dispersion differences among different-azimuthal-order WGMs. After that, we measure the parametric oscillation threshold and realize Kerr frequency comb generation using this functionalized microsphere resonator (for details about the experimental setup, see part S4 in the Supplementary Material). During the measurement process, in order to ensure the mechanical stability, the coupling microfiber always gets in touch with the microresonator. To build up the pump power in the microresonator, we finely tune the pump wavelength (*λ*_{p}) and enable it self-locked into a certain WGM. As shown in Figure 3B, *λ*_{p} is located at 1548.45 nm. When the pump power is 0.527 mW, the first equidistant lines symmetrically located at the two sides of the pump wavelength with a spacing of three FSRs are generated through the degenerate FWM process [5].

Figure 3: Kerr frequency comb generation.

(A) Calculated total group velocity dispersion curves of the functionalized silica microsphere resonators with diameters of 248 and 139 μm, respectively. The inset is the fundamental mode field pattern of the microsphere resonator with a diameter of 248 μm. (B–F) Kerr frequency comb formation in the functionalized microsphere resonator with a diameter of 248 μm. (G) Zoom-in spectrum of Figure 3F. (H) Idler power as a function of the pump power. The inset shows the relation between the idler power and the signal power.

Next, we study the dynamic Kerr frequency comb generation by adjusting the pump power. As shown in Figure 3B–F, with the increase of the pump power, cascaded FWM generates more comb lines due to the co-effect of degenerate and nondegenerate FWM [6], [36]. Meanwhile, the comb span increases with the pump power until 1.52 mW. As shown in Figure 3F, the Kerr frequency comb with a span of about 68 nm is achieved. Figure 3G is the zoom-in spectrum of Figure 3F, and we can find that the comb spacing is 2.1 nm, which is equal to the FSR of the microsphere resonator based on the expression: FSR=*λ*^{2}/2*πn*_{eff} *R*, where *R* is the radius of the resonator [37]. After that, the relation between the idler power and the pump power is obtained, which is shown in Figure 3H. We also find that an ultralow parametric oscillation threshold of 0.42 mW is achieved, and the idler power increases almost linearly with the pump power until 1.1 mW. The parametric oscillation threshold of the WGM microresonator can be expressed by [22], [38]:

$${P}_{\text{th}}=1.54\frac{\pi}{2}\mathrm{(}\frac{{Q}_{\text{e}}}{2{Q}_{\text{l}}}\mathrm{)}\frac{{n}^{2}{V}_{\text{eff}}}{{n}_{2}{\lambda}_{\text{p}}{Q}_{\text{l}}^{2}}$$(1)

where *Q*_{e} and *Q*_{l} are the external-coupled and loaded *Q* factors, respectively, *n=*1.45 is the linear RI of silica, *n*_{2}=2.2×10^{−20} m^{2}/W is the nonlinear RI coefficient of silica, *V*_{eff}=16,000 μm^{3} is the fundamental-mode volume, *λ*_{p}=1548.45 nm is the pump wavelength. By substituting the parameters into Eq. (1), the theoretical threshold *P*_{th}=0.23 mW is comparable with the measured threshold. When the pump power further increases, the idler power tends to be saturated because it is converted to other comb lines due to cascaded FWM. We also obtain the relation between the signal power and the idler power as the pump power increases. As shown in the inset of Figure 3H, both the signal and idler sidebands generate simultaneously with nearly equal powers. Therefore, it proves that these two sidebands are generated from the same pump wavelength, which is derived from the parametric interaction in the microresonator [5], [21].

After that, we tuned the comb lines by feeding the control light into the microsphere resonator through its fiber stem. The microsphere resonator is attached on the end of a single mode fiber. As shown in Figure S2, though there is no fiber core in the junction between the fiber stem and the microsphere, the control light can still propagate inside the microsphere along the axial direction of the fiber stem and towards the nanoparticle-coated area. We calculate the electric field distribution of the control light by performing a two-dimensional (2D) simulation using the finite difference time domain (FDTD) method (FDTD solutions). (For details about the simulation result, see part S2 in the Supplementary Material.) From the simulation result shown in the Supplementary Material, the control light directly propagates through the end surface and can be efficiently absorbed by the iron oxide nanoparticles. Figure 4 is the tuning performance of the Kerr frequency comb based on the functionalized microsphere resonator with a diameter of 248 μm when the pump power is fixed at 1.42 mW. Figure 4A–D denotes the corresponding OFCs for different control powers. We find that with the increase of the control power, the whole OFC spectrum has little change apart from the comb line shift. Figure 4E shows the comb line shift vs. the control power. From Figure 4F, which is the zoom-in spectrum of Figure 4E, we can find that the comb line presents a red shift as the control power increases from 0 to 58 mW. During the tuning process, the comb spacing always maintains one FSR. Figure 4G shows the comb line shift and the comb span vs. the control power. The comb line shift increases with the control power, and a tuning range of 0.8 nm is realized when the control power increases to 58 mW, which constitutes 38% of one FSR. The resonance wavelength shift Δ*λ* as a function of the temperature variation Δ*t* can be expressed by [39]:

Figure 4: Controllable Kerr frequency comb.

(A–D) Kerr frequency combs for different control powers. (E) Tuning of the Kerr frequency comb line with the increase of the control power. (F) Zoom-in spectrum of Figure 4E. (G) Comb line shift (red line), comb span (black line), and comb spacing variation (the inset) vs. the control power.

$$\Delta \lambda ={\lambda}_{\text{0}}\mathrm{(}\frac{1}{n}\cdot \frac{\partial n}{\partial t}+\frac{1}{R}\cdot \frac{\partial R}{\partial t}\mathrm{)}\Delta t$$(2)

where *λ*_{0} is the initial resonance wavelength, $\frac{1}{n}\cdot \frac{\partial n}{\partial t}=8.52\times {10}^{-6}{K}^{-}{}^{1},$ and $\frac{1}{R}\cdot \frac{\partial R}{\partial t}=5.5\times {10}^{-7}{K}^{-1}$ [40], [41]. By substituting these parameters into Eq. (2), we calculate the temperature variation as Δ*t=*57 K, which leads to the RI increase of 7×10^{−4} and the radius increase of 0.0039 μm. Such small parameter variations have little influence on the total GVD of the WGMs. As shown in Figure 4G, the comb span has a small variation of about 2 nm, which verifies the fact that the total GVD changes little during the tuning process. According to the FSR expression, it is evidently seen that the comb spacing tends to decrease as *n*_{eff} and *R* increase, which is derived from the increase of the control power and can be seen in the inset of Figure 4G. The maximum comb spacing variation of 141 MHz is calculated corresponding to the control power of 58 mW. Besides, according to our previous work, the response time should be millisecond level [37].

Though the microsphere resonator possesses the advantage for the ultralow-threshold OFC generation, it is difficult to realize the flattened dispersion in a wide wavelength range. Consequently, it is challenging to realize the wide-span Kerr frequency comb in the microsphere resonator. SRS in the silica microresonator is the phenomenon that can generate the Stokes wavelength (*λ*_{stokes}) which is about 13 THz away from the pump wavelength [42]. Therefore, the comb span can be broadened by combining with SRS [38], [43], [44], [45]. Here, we also realize the Raman-Kerr frequency comb with a wider span compared with the Kerr frequency comb. As shown in Figure 2D, we fabricate the functionalized silica microsphere resonator with a diameter of about 139 μm and *Q*_{0} of 3.5×10^{8}. A CO_{2} laser is used to fabricate the microsphere resonator with a relatively small diameter. As shown in Figure 3A, the zero dispersion wavelength of the silica microsphere resonator with a diameter of 139 μm is about 1590 nm. *λ*_{p} is fixed at 1553.7 nm, which is in the normal GVD region, and thus it is difficult to excite degenerate FWM around the pump light [46]. However, in this case, *λ*_{stokes} is located in the anomalous GVD region, which leads to FWM generation near *λ*_{stokes} [38]. As shown in Figure 5A, when the pump power is 0.191 mW, both the sidebands near *λ*_{p} and *λ*_{stokes} are not generated. Figure 5B shows that there are several comb lines near *λ*_{p}, which are derived from nondegenerate FWM between the pump light and stokes comb lines, when the pump power is 0.503 mW. After that, we further increase the pump power to measure the evolution of the Raman-Kerr frequency comb. As shown in Figure 5C, when the pump power is 0.569 mW, more comb lines emerge between *λ*_{p} and *λ*_{stokes}. Furthermore, as shown in Figure 5C–E, the comb span also increases with the pump power. As shown in Figure 5E, when the pump power increases to 0.954 mW, the Raman-Kerr comb with a span of about 164 nm is obtained. Figure 5G is the zoom-in spectrum of Figure 5E, and it shows that the comb spacing is 3.88 nm, which is equal to the FSR of the microsphere resonator. The first-order sideband power and the Raman power as functions of the pump power are also shown in Figure 5F. We can find that the first-order sideband power and the Raman power increase rapidly with the pump power until 0.569 mW. As the pump power further increases, both the first-order sideband power and the Raman power tend to be stable due to the consumption by degenerate and nondegenerate FWM. It is worthy to be noted that the thresholds of the first-order sideband and SRS are roughly equal, which can be verified by the aforementioned theoretical analysis.

Figure 5: Raman-Kerr frequency comb generation.

(A–E) Raman-Kerr frequency comb formation in the functionalized microsphere resonator with a diameter of 139 μm. (F) First-order sideband power and Stokes power as functions of the pump power. (G) Close-up view of Figure 5E.

After that, the tuning of the Raman-Kerr frequency comb is also performed. The pump power is fixed at 0.859 mW. Figure 6A shows the tunable Raman-Kerr comb spectrum when the pump power increases from 0 to 112 mW. As shown in Figure 6B, from the enlarged spectrum near the pump wavelength, we can clearly find that the comb lines present a red shift, consistently, while always maintaining the comb spacing as one FSR. From Figure 6C, we can find that all the comb lines between *λ*_{p} and *λ*_{stokes} are always present with the increase of the control power. Also, from Figure 6D, almost no comb line at the right edge of the Raman-Kerr frequency comb spectrum disappears as the control power increases. Figure 6E shows the comb line shift and the comb span vs. the control power. From the red line, we can find that the comb line shift increases with the control power, and a tuning range of 2.67 nm, which constitutes 69% of one FSR, is achieved when the control power increases to 112 mW. Considering the thermo-optic coefficient and thermal expansion coefficient, this tuning range corresponds to a temperature variation of 190 K, which leads to a RI variation of 0.0023 and a radius variation of 0.0073 μm, and this means that there are little GVD changes. Furthermore, from the black line in Figure 6E, we find that the comb span varies by only about 15.4 nm, which verifies the GVD stability during the tuning process. Besides, as shown in the inset of Figure 6E, the maximum comb spacing variation is calculated as 840 MHz.

Figure 6: Controllable Raman-Kerr frequency comb.

(A) Tuning of the Raman-Kerr frequency comb line with the increase of the control power. (B) (Black line frame in Figure 6A), (C) (red line frame in Figure 6A), and (D) (blue line frame in Figure 6A) Zoom-in spectra of Figure 6A. (E) Comb line shift (red line), comb span (black line), and comb spacing variation (the inset) vs. the control power.

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