1. Introduction

Optical frequency combs are light sources that emit equidistant narrow lines in a broad spectrum. Optical frequency comb generation though hyper-parametric oscillations in whispering gallery type micro-resonators has been attracting much attention in the past years, as it is potentially possible to integrate devices on a chip [1–4]. Hyper-parametric oscillation, which is a process that two pump photons convert to a signal photon and an idler photon, requires momentum conservation and energy conservation. In WGM resonators, it occurs when all pump, signal and idler light locate on WGM resonant lines. Unfortunately, WGM resonant lines are not equally spaced because of geometric and material dispersion. Take silica sphere as an example, it has normal geometric dispersion, thus optical frequency comb is allowed to generate at wavelength larger than 1.3 μm where its material dispersion is anomalous, thus the total dispersion can be well balanced.

Many approaches were explored to broaden the wavelength window for generating hyper-parametric oscillation by manipulating material dispersion (using special materials, such as fluoride), geometric dispersion (designing special cavity geometries [2–11]) or introducing a frequency detuning of the pumping light from the mode of the nonlinear resonator in the normal dispersion regime (generally in relatively large cavity size with low dispersion material) [12–15]. In this work, we focus on using micro-bubble resonators (MBRs) to engineer geometric dispersion for hyper-parametric frequency conversions. MBR attracts a lot of attention because of its ultra-high Q, small mode volume and, especially, huge tunable resonance range [16–20]. The thin-wall cavity structure also gives freedom to engineer geometrical dispersion. We found that in a thin-wall MBR, geometric dispersion can be so largely manipulated that hyper-parametric nonlinear process is still possible in silica cavities, whiles for solid spheres, geometric dispersion is too large to be compensated by the material dispersion, hence hyper-parametric oscillation is not allowed.

2. Dispersion and zero dispersion wavelength (ZDW) manipulation

In WGM micro-resonators, momentum conservation for this process is automatically satisfied when pump, signal and idler waves are all on resonance modes of the micro-resonator [21–23]; on the other hand, energy conservation (or frequency matching condition, which is 2υ_p = υ_s + υ_i, with υ being the resonance frequency of the modes, p, s, i represent pump, signal and idler frequency respectively) places strict conditions on cavity dispersion, because the frequency difference between adjacent modes varies as a result of material and geometric dispersion. Dispersion in WGM resonators can be reflected by the variation of free spectral range (FSR), ΔFSR (ΔFSR = (υ_{m + 1}-υ_m) - (υ_m-υ_m-1), where m and υ denote the angular mode number and resonance frequency). Geometric dispersion of a conventional solid core silica WGM resonator is $\Delta FSR\approx -0.41\frac{c}{2\pi nR}{m}^{-5/3}$, where n is the refractive index (RI), R is the cavity radius and m is angular mode number, on the other hand, its material dispersion is$\Delta \text{FSR}=\frac{1}{4{\pi }^2}\frac{c^2}{n^3{R}^2}\text{GVD}$, where GVD is the group velocity dispersion parameter and has positive value at λ> 1.3 μm [3]. However, in MBRs, in addition to the cavity size, bubble thickness also determines the geometrical dispersion. Analogous to abnormal geometric dispersion in a waveguide with tightly confined optical mode [11, 24–27], the geometrical dispersion in MBRs can be manipulated by controlling the degree of light confinement, and anomalous geometric dispersion is possible by choosing the appropriate device parameters. Figures 1(a)-1(b) shows the effective RI and the group RI versus wavelength of a microsphere and a 3 μm wall MBR, the two micro-resonators have the same diameter but different group RI curves.

 

Fig. 1 (a-b) The effective RI (black square) and the group RI (red circle) versus wavelength of a microsphere and a 3 μm wall thickness MBR are plotted. The diameter of the both micro-resonator is 120 μm. (c) Plot of ΔFSR as a function of MBR size at wavelength 1.55 μm, bubble wall thickness varies from 3.5 to 4.5 μm. (d) Plot of ZDW versus MBR wall thickness, MBR size is fixed at 120 μm.

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The resonance wavelengths of MBRs are calculated numerically based on the Mie scattering theory, and the electromagnetic boundary-value problem of the interaction between a monochromatic plane wave and the micro-resonator is solved accurately when the scattering amplitude reaches its maximum value with specified incident wave vector [18, 28]. An iterative approach was used to add the material dispersion contribution into the calculation. After the resonance wavelengths of three adjacent modes are solved, ΔFSR with 10⁻³ MHz precision can be obtained. The calculated relationship between overall dispersion (material and geometric dispersion) of silica MBRs and bubble outer diameter (D) at different bubble thicknesses (T) is plotted in Fig. 1(c). The dispersion curve of a solid silica microsphere is also plotted as a reference. For solid spheres, anomalous dispersion appears when sphere size is larger than 150 μm, smaller spheres have normal dispersion (ΔFSR<0). However, for MBRs, with smaller T, the dispersion curve moves upward to anomalous regime (ΔFSR>0). Therefore the ZDW of MBRs can be controlled by changing the radius and the thickness of a MBR. Figure 1(d) shows the calculated ZDW of a 120 μm diameter MBR with thickness 1−5 μm, it can be seen that ZDW varies from 0.84 to 1.63 μm.

3. Experiment

Both parametric and nonparametric processes exist in WGM micro-resonators. Stimulated Raman scattering (SRS) is a non-parametric oscillation process that converts pump photon to a lower frequency photon, SRS is the dominant nonlinear phenomenon when ΔFSR≠0, because SRS has large gain spectrum bandwidth and does not require phase matching. On the other hand, hyper parametric frequency conversion, such as four wave mixing (FWM), emerges before SRS when dispersion compensation is realized, since its threshold is lower [21–23, 29, 30]. Therefore, in our experiments, the occurrence of FWM and frequency comb (i.e. cascaded FWM in this case) requires anomalous cavity dispersion, and can be a direct experimental proof of the dispersion compensation.

The MBRs are fabricated by the fuse-and-blow technique (See Fig. 2). A fused silica capillary (Polymicro TSP40100) was etched to obtain optimal capillary wall thickness, its outer diameter is 80 μm. Then one end of the capillary is fused, and the other one is connected to a needle cylinder to allow air be purged into the tube. After heating the capillary by the fusion splicer, a MBR is formed. Different size and thickness of MBRs can be obtained with proper gas pressure inside the capillary. By this method, a MBR with 136 μm diameter and 3-4 μm thickness is fabricated.

 

Fig. 2 The fabrication process of a MBR.

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We optically pump the MBR via a 2-3 μm tapered fiber, 3D piezoelectric transducer (PZT) stages are used to control the distance between the tapered fiber and the micro-resonator, and the transmitted signal is sent to an optical spectrum analyzer (OSA) and a photodiode separately. A tunable diode laser (Anritsu Tunics Plus CL) at 1.55 μm is used as the pump source. The Q factor of the WG mode is about 5 × 10⁷ (see Fig. 3(a)). The calculated ZDW of the micro-resonator is below 1.49 μm, thus the MBR works in the anomalous dispersion regime at 1.55 μm.

 

Fig. 3 (a) Transmitted spectrum of a 136 μm diameter and 3-4μm thick wall MBR, the mode Q is 5 × 10⁷. Inset is a photo of the MBR. (b) Transmitted spectrum of the MBR at 3-4 mW optical pump.

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To generate FWM, the pump laser is repeatedly scanned around the first order radial WG mode over 40 pm at the rate 800 Hz. 3-4 mW pump power is coupled into the resonator. As is shown in Fig. 3(b), cascaded FWM, or frequency comb, is observed. The comb spacing is 3.73 nm, which agrees with calculated FSR value (~3.79 nm). The difference between the calculated and observed FSR may be from the thermal effect and measurement error of the cavity size. No SRS signal is observed in this case. This gives direct evidence that the comb is excited by the pump via FWM, and the dispersion compensation in the micro-resonator is achieved.

In comparison, silica microspheres with two diameters (136 μm and 245 μm) are fabricated by heating the tip of a tapered optical fiber with a CO₂ laser, and are pumped by the same method mentioned above. Typical Q factor of the sphere resonator is also above 10⁷. Figure 1(c) and [21] tells that the 136 μm microsphere has normal total dispersion at 1.55 μm, hence, frequency matching condition for FWM is not possible. The 245 μm microsphere on the other hand, has anomalous overall dispersion and thus allows FWM [22]. We pump the spheres in the same way as mentioned before. Figure 4 shows measured transmitted spectra from the two spheres. Frequency comb is clearly observed for the 245 μm microsphere, meanwhile only SRS at 1679 nm is observed for the 136 μm microsphere.

 

Fig. 4 Transmitted spectrum under optical pump. (a) A 136 μm silica sphere, (b) a 245 μm diameter silica sphere. The inset in (a) shows a photo of the 136 μm sphere.

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4. Fine tuning of geometrical dispersion

Fine tuning of anomalous dispersion is necessary to optimize the comb generation, as ΔFSR of several MHz is a typical value for comb generation [21–23]. MBRs provide an additional opportunity for dispersion fine-tuning as they can be filled with liquids. We calculate the geometrical dispersion of a 136 μm MBR with different bubble wall thickness and liquid inside the bubble. The result is shown in Fig. 5. The horizontal axis denotes the RI of the liquid inside the MBR, and the vertical axis denotes the bubble thickness. When the bubble thickness is less than 3.5 μm, anomalous dispersion as large as 10 MHz (equivalent to material dispersion β₂ = 40 ps²/km in the same size micro-resonators) can be obtained. Therefore, in principle, large dispersion tuning is possible.

 

Fig. 5 ΔFSR as a function of RI inside the MBR, and the bubble wall thickness. The diameter of MBR is 136 μm.

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5. Conclusion

In summary, we demonstrated that the geometrical dispersion and the ZDW of WG modes in MBRs can be tuned by changing the radius and the thickness of micro-resonators. As a result, FWM and frequency comb can be generated in new wavelength window that is fundamentally not accessible by using the same size solid microspheres. MBRs are thus a simple, adjustable and ultrahigh Q resonators for generating nonlinear optical processes.