Optical whispering gallery mode (WGM) resonators [1,2] combining small size with high $Q$-factor are very attractive for a wide range of applications in photonics. WGMs are commonly imagined as closed-trajectory rays confined within an axisymmetric cavity by the almost total internal reflection from the curved surface of the resonator. WGM resonators are widely used for different types of optical filters, sensors, optoelectronic modulators, etc. [3]. A recently discovered possibility of coherent frequency comb generation and soliton formation in optical microresonators [4,5] opens a way toward very compact spectrometers [6,7], low-noise radio frequency sources [8], and optical frequency synthesizers [9]. Silicon is a promising material for near- and mid-infrared nonlinear WGM-based devices due to its high nonlinear refractive index, $n_2=4.5·{10}^{-18}{\mathrm{m}}^2/\mathrm{W}$ [10], which allows observing nonlinear effects at significantly smaller power levels. Coherent frequency comb generation via soliton mode locking in the mid-infrared band was demonstrated recently in silicon dispersion engineered microring resonators having anomalous group velocity dispersion (GVD) [11]. Although the GVD for the bulk silicon at 1550 nm used in this work is normal, not allowing bright soliton generation, a coherent frequency comb via temporal platicon formation is also possible in this regime [12]. High $Q$-factor resonances of a silicon WGM have been also observed for terahertz frequencies [13], which made a triple resonant scheme for terahertz generation and detection [14] very attractive. It is worth noting that pure silicon is considered as a possible material for test masses of the next generation of gravitational wave detectors [15,16]. WGM resonators are extremely sensitive to both bulk and surface losses, so we regard them as small material loss test samples for the LIGO Voyager project [17,18].

Silicon is a major semiconductor material for modern technology, so samples of high purity and homogeneity are easily available. However, application of silicon microresonators was limited due to the relatively low demonstrated $Q$-factors at some promising spectral ranges. To date, a maximal $Q$-factor up to $Q=2.2\times {10}^7$ was obtained in silicon WGM microresonators manufactured using lithographic techniques [19]. A comprehensive analysis of different optical loss mechanisms in silicon waveguides and microdisks may be found in Refs. [20,21].

In this work we present a novel approach to manufacturing of high-$Q$ silicon microresonators and report experimental observation of the ultra-high $Q$-factor $(Q=1.2\pm 0.1\times {10}^9)$ in 2.5 mm diameter silicon resonators at telecom wavelength (1550 nm). We found that surface treatment chemistry is a critical factor that might explain lower values in earlier experiments. We have also developed an original method that provides efficient excitation of silicon WGM microresonators using a hemispherical silicon coupler. It was demonstrated that this method allows us to obtain up to $\approx 35\%$ coupling efficiency, which is quite sufficient for many practical applications. Note that, while the application of the achieved results for the telecom wavelength may be limited due to the low power threshold defined by two-photon absorption (TPA), a bright perspective in the mid-IR range (2.3–5 μm), which is a domain of interest for many applications in science and technology, may be envisioned, since the TPA limitation is significantly relaxed at this spectral range [22]. It is worth noting, also, that the $Q$-factor of fluoride crystals traditionally used for microresonators degrade in the mid-IR [23], making silicon almost a unique material with the ultra-high achievable $Q$-factor. Using silicon microresonators for the generation of wide frequency combs with extremely low phase noise in the mid-IR [11,24] may provide novel approaches to molecular spectroscopy in the “fingerprint region” [25]. Along with the presented novel hemispherical coupler, microresonators from crystalline silicon may become a basis for a new class of high-power laser sources with narrow linewidth in the mid-IR based on the self-injection locking effect [26–28].

We used high-quality float-zone rectified Si samples exhibiting p-type conductivity with the resistivity $\rho =37\mathrm{k}\mathrm{\Omega }·\mathrm{cm}$. Disk resonators that we manufactured had a thickness of 1.5 mm, diameter of 2.5 mm, and a meridional curvature radii of $1\pm 0.5\mathrm{mm}$. The total $Q$-factor budget $Q$ of a WGM microresonator is determined by the following relation:

Silicon resonators were manufactured by abrasive shaping and the following asymptotic polishing. At the beginning, the technique was identical to the preparation of ${\mathrm{CaF}}_2$ and ${\mathrm{MgF}}_2$ resonators with $Q$-factor above ${10}^9$ [29]. It is reasonable to assume that $Q_{\mathrm{s}.\mathrm{s}}$, analogously to microresonators from other crystals, does not limit $Q$ at this level. $Q_{\mathrm{b}.\mathrm{s}}$ should also be negligible for high-quality crystal. Linear bulk losses in silicon may be evaluated from the resistivity [15]: $\alpha =(0.0454[\mathrm{\Omega }])/\rho$. It could be as low as $\alpha \simeq 1.2\mathrm{ppm}/\mathrm{cm}$ for our material, resulting in

First, samples were manufactured using asymptotic polishing with diamond slurries. The size of the abrasive diamond particles used for the final polishing was less than 30 nm, which is subtle enough to make surface absorption negligible. In our first measurement, the $Q$-factor was determined from Lorentzian resonance FWHM, and found to be $Q\simeq 2·{10}^7$—the value well below the mentioned limits. It was assumed that the surface scattering was a limiting factor, and many efforts were made to reduce it, including diamond turning and heat treatment, with no significant improvement. Results reported in Ref. [31] show strong evidence that the excess absorption resulting from the surface chemistry may be dominant: surface losses of a sample rose from $<1\mathrm{ppm}$ to $\simeq 25\mathrm{ppm}$ after Bindzyl slurry polish and dropped back after polishing with Nalco slurry. As the latter is a silica-based suspension, we changed the surface treatment procedure. Instead of diamond slurry, the final polishing was made by means of OXAPA colloidal silica slurry. This allowed an immediate increase in the measured $Q$ by 2 orders of magnitude. The measurement results are reproducible with no significant degradation of $Q$ during at least 2 days. A possible explanation of the surface loss growth under polishing with diamond slurries is the formation of silicon carbide complexes—the same effect that is responsible for a quick diamond tool degradation in diamond turning of silicon [32].

To test the manufactured microresonators, we developed a new method of excitation of WGMs in silicon microresonators. The high value of silicon refractive index ($n=3.48$ at $\lambda =1550\mathrm{nm}$) does not allow sufficient coupling by means of a fused silica tapered fiber, as is commonly used for WGM resonators. Prism coupling is another efficient and robust way of coupling. The angle of incidence ($\mathrm{\varphi }$) of the input beam to the base of the coupling prism must satisfy phase-matching conditions. The simplest approximation for much larger than the wavelength microresonators is $\mathrm{\varphi }\simeq \arcsin (n_r/n_c)$, where $n_c$ and $n_r$ are prism and resonator material refractive indices, so it is necessary to obtain $n_c>n_r$, because $\mathrm{\varphi }=\frac{\pi }{2}$ for $n_c=n_r$ is not geometrically acceptable. Unfortunately, the choice of applicable transparent materials with $n>3.48$ for 1550 nm is extremely limited. However, a better approximation for the fundamental TE mode and Gaussian incident beam [2,33] gives, for $n_c=n_r$,

The scheme of the experimental setup along with a sketch illustrating our coupling approach is presented in Fig. 1.

 

Fig. 1. Experimental setup. Silicon resonator is 2.5 mm in diameter and silicon hemisphere coupler is 5 mm in diameter.

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A tunable 1550 nm continuous wave laser (Koheras Adjustik) was used for the WGM excitation. After the fiber Faraday isolator and polarization controller, light is emitted to the free space. At the beam splitter, 10% of the laser power is sent to the calibration interferometer with 240 MHz free spectral range (FSR), and the rest is coupled to the resonator through the collimating lens and the hemispherical coupler on a translation stage. The resonator is mounted on an $XYZ$ stage with an additional piezo-electric transducer drive for accurate tuning of the distance between resonator and coupler. The transmitted light is collected on the photodetector. The resonator and the coupler are enclosed in a closed box to prevent resonator contamination. A short-wave infrared camera was used to monitor the alignment.

Because of the relatively large diameters and curvature radii of the resonators, many different families of WGMs can be excited. One may select a desirable mode family by tuning polarization and coupling strength. The transmission spectrum upon the wide slow (0.5 GHz/ms) laser scan is shown in Fig. 2. The maximum observed coupling is $\sim 35\%$—see an example in Fig. 3. The maximal estimated power entering the resonator in this case is $P\simeq 2.5\mathrm{mW}$. Both thermal nonlinearity [1,34] and TPA cause different linewidth shape during the forward and reverse frequency scan. A dip in the transmission [circled in Fig. 3(top)] are clearly seen during the frequency decrease. They are caused by the TPA [35]. Decreasing the pump power, we estimated a low threshold for the $Q$-factor $Q_{\min }>3·{10}^8$ from the linewidth measurement.

 

Fig. 2. Typical transmission spectrum with different mode families. One-third of the resonator’s FSR is covered. Spike above unity is due to an ultra-high-$Q$ mode ringdown.

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Fig. 3. Nonlinear cavity mode profiles during frequency decrease (top) and increase (bottom) for different coupling rates (I, minimum; IV, maximum). Transmission dips at backward scan curves II–IV (circled) indicate TPA.

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Accurate measurement of the $Q$-factor was made using a well-known ringdown method. It allows us to minimize the influence of thermal effects, TPA, and laser frequency noise. We used a fast linear laser frequency scan and a 5 ns rise time photodetector. The laser excites a mode of a resonator, then beating between the outcoupled delayed light from the resonator and the pump laser light reflected from the hemisphere base is observed. By measuring the decay rate of the beatnote, one can obtain the cavity ringdown time. Measurements were made in the undercoupled regime. We obtained $Q$ above ${10}^9$ for many modes in two different resonators. The ringdown trace is presented in Fig. 4. The decay curve obeys an exponential law with more than 99% confidence, yielding $Q\sim 1.2\times {10}^9$. By using Eq. (2) one may obtain the upper limit for the absorption $\alpha \le 120\mathrm{ppm}·{\mathrm{cm}}^{-1}$ in the region of mode propagation. The depth of this layer can be estimated for large azimuthal numbers $m$ as a double depth of the mode center [30]:

 

Fig. 4. Cavity ringdown on the ultra-high-$Q$ mode with $Q$-factor of $Q=\pi c\tau /\lambda =(1.2\pm 0.1)\times {10}^9$, where $\tau$ is the ringdown time. The green line is an exponential fit. The direct ringdown profile is presented in the inset.

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Summing up, we developed the method of manufacturing of high-$Q$ silicon WGM microresonators and observed in them for the first time a $Q$-factor exceeding a billion. It was revealed that use of a polishing technique preventing excessive surface absorption plays a crucial role. The original method of silicon microresonator excitation using a hemispherical silicon coupler was proposed and tested. Coupling efficiency up to 35% was demonstrated. The developed manufacturing method allows the use of silicon WGM-resonator-based devices for terahertz to/from infrared conversion and frequency comb generation.

In addition, the confirmed possibility to obtain low surface absorption in silicon may be applied for the creation of test masses of the future gravitational wave detector (LIGO Voyager) project. Investigation of losses by means of WGM resonators including low temperature measurements will be very useful in the future.